Blejske delavnice iz fizike Bled Workshops in Physics ISSN 1580-4992 Letnik 19, St. 2 vOL. 19, nO. 2 Proceedings to the 21st Workshop What Comes Beyond the Standard Models Bled, June 23- July 1, 2018 Edited by Norma Susana Mankoc Borstnik Holger Bech Nielsen Dragan Lukman dmfa - zaloZniStvo Ljubljana, december 2018 The 21st Workshop What Comes Beyond the Standard Models, 23.- 29. June 2018, Bled was organized by Society of Mathematicians, Physicists and Astronomers of Slovenia and sponsored by Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana Society of Mathematicians, Physicists and Astronomers of Slovenia Beyond Semiconductor (Matjaži Breskvar) Scientific Committee John Ellis, King's College London / CERN Roman Jackiw, MIT Masao Ninomiya, Yukawa Institute for Theoretical Physics, Kyoto University and Mathematical Institute, Osaka-city University Organizing Committee Norma Susana Mankoč Borštnik Holger Bech Nielsen Maxim Yu. Khlopov The Members of the Organizing Committee of the International Workshop "What Comes Beyond the Standard Models", Bled, Slovenia, state that the articles published in the Proceedings to the 21st Workshop "What Comes Beyond the Standard Models", Bled, Slovenia are refereed at the Workshop in intense in-depth discussions. Workshops organized at Bled > What Comes Beyond the Standard Models (June 29-July 9,1998), Vol. 0 (1999) No. 1 (July 22-31,1999) (July 17-31, 2000) (July 16-28, 2001), Vol. 2 (2001) No. 2 (July 14-25, 2002), Vol. 3 (2002) No. 4 (July 18-28, 2003) Vol. 4 (2003) Nos. 2-3 (July 19-31, 2004), Vol. 5 (2004) No. 2 (July 19-29, 2005), Vol. 6 (2005) No. 2 (September 16-26, 2006), Vol. 7 (2006) No. 2 (July 17-27, 2007), Vol. 8 (2007) No. 2 (July 15-25, 2008), Vol. 9 (2008) No. 2 (July 14-24, 2009), Vol. 10 (2009) No. 2 (July 12-22, 2010), Vol. 11 (2010) No. 2 (July 11-21, 2011), Vol. 12 (2011) No. 2 (July 9-19, 2012), Vol. 13 (2012) No. 2 (July 14-21, 2013), Vol. 14 (2013) No. 2 (July 20-28, 2014), Vol. 15 (2014) No. 2 (July 11-19, 2015), Vol. 16 (2015) No. 2 (July 11-19, 2016), Vol. 17 (2016) No. 2 (July 9-17, 2017), Vol. 18 (2017) No. 2 (June 23-July 1, 2018), Vol. 19 (2018) No. 2 > Hadrons as Solitons (July 6-17,1999) > Few-Quark Problems (July 8-15, 2000), Vol. 1 (2000) No. 1 > Selected Few-Body Problems in Hadronic and Atomic Physics (July 7-14, 2001), Vol. 2 (2001) No. 1 > Quarks and Hadrons (July 6-13, 2002), Vol. 3 (2002) No. 3 > Effective Quark-Quark Interaction (July 7-14, 2003), Vol. 4 (2003) No. 1 > Quark Dynamics (July 12-19, 2004), Vol. 5 (2004) No. 1 > Exciting Hadrons (July 11-18, 2005), Vol. 6 (2005) No. 1 > Progress in Quark Models (July 10-17, 2006), Vol. 7 (2006) No. 1 > Hadron Structure and Lattice QCD (July 9-16, 2007), Vol. 8 (2007) No. 1 > Few-Quark States and the Continuum (September 15-22, 2008), Vol. 9 (2008) No. 1 > Problems in Multi-Quark States (June 29-July 6, 2009), Vol. 10 (2009) No. 1 > Dressing Hadrons (July 4-11, 2010), Vol. 11 (2010) No. 1 > Understanding hadronic spectra (July 3-10, 2011), Vol. 12 (2011) No. 1 > Hadronic Resonances (July 1-8, 2012), Vol. 13 (2012) No. 1 > Looking into Hadrons (July 7-14, 2013), Vol. 14 (2013) No. 1 > Quark Masses and Hadron Spectra (July 6-13, 2014), Vol. 15 (2014) No. 1 > Exploring Hadron Resonances (July 5-11, 2015), Vol. 16 (2015) No. 1 > Quarks, Hadrons, Matter (July 3-10, 2016), Vol. 17 (2016) No. 1 > Advances in Hadronic Resonances (July 2-9, 2017), Vol. 18 (2017) No. 1 > Double-charm Baryons and Dimesons (June 17-23, 2018), Vol. 18 (2018) No. 1 > o Statistical Mechanics of Complex Systems (August 27-September 2, 2000) o Studies of Elementary Steps of Radical Reactions in Atmospheric Chemistry (August 25-28, 2001) Contents Preface in English and Slovenian Language............................VII Talk Section........................................................ 1 1 Inflation From Supersymmetry Breaking I. Antoniadis........................................................ 1 2 New Model independent Results From the First Six Full Annual Cycles of DAMA/LIBRA-Phase2 R. Bernabei et al..................................................... 27 3 HS YM and CS Theories in Flat Spacetime L. Bonora........................................................... 58 4 Emergent Photons and Gravitons J.L. Chkareuli, J. Jejelava and Z. Kepuladze.............................. 74 5 A Deeper Probe of New Physics Scenarii at the LHC a. Djouadi.......................................................... 90 6 The Symmetry of 4 x 4 Mass Matrices Predicted by the Spin-charge-family Theory — SU(2) x SU(2) x U(1) — Remains in All Loop Corrections a. Hernandez-Galeana and N.S. Mankoc Borštnik .......................102 7 Extending Starobinsky Inflationary Model in Gravity and Supergravity S.V. Ketov and M.Yu. Khlopov.........................................148 8 Phenomenological Mass Matrices With a Democratic Origin a. Kleppe...........................................................164 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? N.S. Mankoc Borštnik and H.B.F. Nielsen...............................175 10 Do We Find High Energy Physics Inside (Almost) Every Solid or Fluid at Low Temperature? H.B. Nielsen and M. Ninomiya........................................216 11 Electric Dipole Moment and Dark Matter in a CP Violating Minimal Supersymmetric SM T. Shindou..........................................................247 Discussion Section..................................................257 12 On Triple-periodic Electrical Charge Distribution as a Model of Physical Vacuum and Fundamental Particles E.G. Dmitrieff.......................................................261 13 The Correspondence Between Fermion Family Members in Spin-charge-family Theory and Structure Defects in Electrically-charged Tessellations E.G. Dmitrieff.......................................................285 14 Ko - Ko, Do - Do in a Local SU(3) Family Symmetry a. Hernandez-Galeana...............................................299 15 Beyond the Standard Models of Particle Physics and Cosmology M.Yu. Khlopov......................................................314 16 The Ya Matrices, Ya Matrices and Generators of Lorentz Rotations in Clifford Space — Determining in the Spin-charge-family Theory Spins, Charges and Families of Fermions — in (3 + 1 )-dimensional Space D. Lukman and N.S. Mankoc Borštnik..................................327 17 Properties of Fermions With Integer Spin Described in Grassmann Space D. Lukman and N.S. Mankoc Borštnik..................................335 18 Could Experimental Anomalies Reflect Non-perturbative Effects? H.B. Nielsen and C.D. Froggatt........................................359 Virtual Institute of Astroparticle Physics Presentation...................381 19 The Platform of Virtual Institute of Astroparticle Physics in Studies of Physics Beyond the Standard Model M.Yu. Khlopov......................................................383 Poem by Astri Kleppe ...............................................395 20 June Astri Kleppe........................................................397 Preface The series of annual workshops on "What Comes Beyond the Standard Models?" started in 1998 with the idea of Norma and Holger for organizing a real workshop, in which participants would spend most of the time in discussions, confronting different approaches and ideas. Workshops take place in the picturesque town of Bled by the lake of the same name, surrounded by beautiful mountains and offering pleasant walks and mountaineering. Since the 20th workshop we offer every year during or at the end of the workshop a talk to the general audience of Bled. This year the talk with the title "How far do we understand the Universe in this moment?", was given by Norma Susana Mankoc Borštnik in the lecture hall of the Bled School of Management. The lecture hall was kindly offered by the founder of the school Danica Purg. The talk was, due to the schedule constraint at the school, delivered after the workshop already finished. In our very open minded, friendly, cooperative, long, tough and demanding discussions several physicists and even some mathematicians have contributed. Most of topics presented and discussed in our Bled workshops concern the proposals how to explain physics beyond the so far accepted and experimentally confirmed both standard models - in elementary particle physics and cosmology - in order to understand the origin of assumptions of both standard models and be consequently able to make predictions for future experiments. Although most of participants are theoretical physicists, many of them with their own suggestions how to make the next step beyond the accepted models and theories, experts from experimental laboratories were very appreciated, helping a lot to understand what do measurements really tell and which kinds of predictions can best be tested. The (long) presentations (with breaks and continuations over several days), followed by very detailed discussions, have been extremely useful, at least for the organizers. We hope and believe, however, that this is the case also for most of participants, including students. Many a time, namely, talks turned into very pedagogical presentations in order to clarify the assumptions and the detailed steps, analyzing the ideas, statements, proofs of statements and possible predictions, confronting participants' proposals with the proposals in the literature or with proposals of the other participants, so that all possible weak points of the proposals, those from the literature as well as our own, showed up very clearly. The ideas therefore seem to develop in these years considerably faster than they would without our workshops. This year neither the cosmological nor the particle physics experiments offered much new, which would offer new insight into the elementary particles and fields, although a lot of work and effort have been put in, but the news will hopefully come when analyses of the data gathered with energies up to 13 TeV on the LHC will be finished. We also expect that new cosmological experiments will help to resolve the origin of the dark matter. Since the results of the DAMA/LIBRA experiments, presented in this year proceedings, can hardly be explained in some other way than with the signal of the dark matter, it is expected that sooner or latter other laboratories will confirm the DAMA/LIBRA results. Several contributions in this proceedings discuss proposals for the dark matter, suggesting that they might be supersym-metric partners, dark atoms made of dark barions and ordinary barions, the stable neutrons of the second group of four families. Understanding the universe through the cosmological theories and theories of the elementary fermion and boson fields, have, namely, so far never been so dependent on common knowledge and experiments in both fields. The experiments on the LHC and other laboratories around the world might offer the accurately enough mixing matrices for quark and leptons, so that it will become clear whether there is the new family to the observed three as well as several scalar fields, which determine the higgs and the Yukawa couplings, predicted by the spin-charge-family theory. The symmetry in all orders of corrections of the 4 x 4 mass matrices, determined by the scalars of this theory, studied in this proceedings, limits the number of free parameters of mass matrices, and would for accurately enough measured matrix elements of the 3 x 3 sub-matrices of the 4 x 4 mixing matrices predict properties of the fourth family of quarks and leptons. The accurate (n — 1) x (n — 1) submatrix of any n x n matrix, namely, determines the n x n matrix uniquely. The properties of mass matrices of the observed 3 families of quarks are presented, following while taking into account the measured masses of quarks and the standard parametrization of the mixing matrix. It is not surprising that the mass matrices are close to the democratic matrix, since the top quark has much higher mass and is much weaker coupled to the rest of quarks. Also the spin-charge family theory predicts that the 4 x 4 mass matrices of quarks and leptons are close to the democratic ones. This is even less surprising, since the fourth family with the masses close to 1 TeV for leptons and above 1 TeV for quarks is even weaker coupled with the rest three families than it is the third u-quark coupled to the rest of quarks. The new data might answer the question, whether laws of nature are elegant (as predicted by the spin-charge-family theory and also — up to the families — other Kaluza-Klein-like theories and the string theories) or "she is just using gauge groups when needed" (what many models assume, also some presented in this proceedings). Can the higgs scalars be guessed by smaller steps from the standard model case, or they must be recognized in more general theories as it is in the spin-charge-family theory? The evidences obviously tell that fermion fields have half integer spin and the charges in the fundamental representations of the so far observed groups. Shall the study of Grassmann space in confrontation with Clifford space for the description of the internal degrees of freedom for fermions, discussed in this proceedings in the first and second quantization of fields, help to better understand the "elegance of the laws of nature"? While the Clifford space offers the explanation for all the properties of quarks and leptons, carrying the half integer spin and all the charges in the fundamental representations of the groups (which are subgroups of the large enough Lorentz group), with the families included (the properties of which are also explainable by the half integer "family spins and charges"), in the Grassmann space there are the second quantizable fermions, which carry the integer spin and charges, both in adjoint representations of the subgroup of the Lorentz group and no families. And yet there exists the "Dirac-Grassmann" sea. While in the Clifford case one Weyl representation includes in SO(13,1) all the quarks and leptons and anti-quarks and anti-leptons observed so far (as well as the right handed neutrino), in Grassmann space particles and anti-particles are in different representations. Is the working hypotheses that "all the mathematics is a part of nature" acceptable and must be taken seriously? If "nature would make a choice" of the Grassmann instead of the Clifford algebra, all the atoms, molecules and correspondingly all the world would look completely different, but yet possible. Why "she make a choice" of the Clifford algebra? All these is discussed in this proceedings, in order to understand better why the spin-charge-family theory is offering so many answers to the open questions in both standard models. In one of the contributions the higher integer spin Yang-Mills-like gauge fields are studied, allowing infinite number of higher spin states. Such theories might, namely, help to avoid ultraviolet divergences in gauge fields. Also the supersymmetry offers avoiding some of divergences. The analyze is done for the possibility that the LHC would confirm the existence of the supersymmetric partners with masses close or above 1 TeV, as well as of several higgs and of a new family of quarks and leptons. The spin-charge family theory, offering the explanation for all the assumptions of the standard model(s), predicting the fourth family of quarks and leptons around 1 TeV or above, as well as three singlets and two triplets of scalar fields (all with the properties of the higgs with respect to the weak and hyper charges) and four additional families, the lowest of which explains the appearance of the dark matter, does not "see" the supersymmetric particles. Even if the supersymmetry might not be confirmed in the low energy regime, yet the supergravity models, inspired by the string models, can help to better understand the inflation in our universe and the observations at its present stage. The supersymmetry might also help to understand the presence of the (very small) amount of dark energy, of the dark matter, even for primordial formation of black holes. All these is studied in three contributions of this proceedings. It is an interesting observation in this proceedings discussing properties of any material with only the translational symmetry that there must be regions in quasi momentum space where an approximate Weyl equation (relativistic equation for massless particles) determines properties of material, observed, let say, in graphene. The same effect can be observed also in the universe with a strong Hubble expansion. Authors explain this effect with the"homolumo-gap". As we know from several fields of physics, there are many different models, seeming to have very little in common, which explain well the same phenomena. The challenge is to find out what they have in common. There is the contribution in this proceedings treating electrodynamics and linearized gravity in common, causing the spontaneously break of Lorentz invariance by constraints on the electromagnetic and tensor fields. The model is at low energies still in agreement with the observations. Two contributions in the discussion section try to extract properties of quarks and leptons, that is their masses and the forces among them, from the geometrical picture of quarks and leptons carrying the charges of the spin-charge-family theory. How far can such an attempt help to understand our nature? What is the most efficient way to understand our universe? Is it now the time that we should make a new step, as it was the standard model step 50 years ago, with the theory which explains all the assumptions of the standard models? Is the spin-charge-family the right first step beyond the standard models? Will experiments confirm the predictions of this theory? Or should we insist with small steps stimulated by experiments? Is the space-time (3 + 1 )-dimensional? Or d is much larger, infinite? Is the interaction among fermions only gravitational one, manifesting at (3 + 1 ) the gravity and the observed gauge fields, with scalar fields included? Only the theory and the experiment together can answer this question. Since, as every year also this year there has been not enough time to mature the very discerning and innovative discussions, for which we have spent a lot of time, into the written contributions, only two months, authors can not really polish their contributions. Organizers hope that this is well compensated with fresh contents. Questions and answers as well as lectures enabled by M.Yu. Khlopov via Virtual Institute of Astroparticle Physics (viavca.in2p3.fr/site.html) of APC have in ample discussions helped to resolve many dilemmas. The reader can find the records of all the talks delivered by cosmovia since Bled 2009 on viavca.in2p3.fr/site.html in Previous - Conferences. The three talks delivered by: Norma Mankoc Borštnik (Understanding nature with the spin-charge-family theory, making several predictions), Sergey V. Ketov (Starobinsky inflation in gravity and supergravity) and H.B Nielsen(Theory for initial State Conditions), can be accessed directly at http://viavca.in2p3.fr/what_comes_beyond_the_standard_model_2018.html Most of the talks can be found on the workshop homepage http://bsm.fmf.uni-lj.si/. Bled Workshops owe their success to participants who have at Bled in the heart of Slovene Julian Alps enabled friendly and active sharing of information and ideas, yet their success was boosted by vidoeconferences. Let us conclude this preface by thanking cordially and warmly to all the participants, present personally or through the teleconferences at the Bled workshop, for their excellent presentations and in particular for really fruitful discussions and the good and friendly working atmosphere. Norma Mankoc Borštnik, Holger Bech Nielsen, Maxim Y. Khlopov, (the Organizing comittee) Norma Mankoč Borštnik, Holger Bech Nielsen, Dragan Lukman, (the Editors) Ljubljana, December 2018 1 Predgovor (Preface in Slovenian Language) Vsakoletne delavnice z naslovom ,,Kako preseči oba standardna modela, koz-moloskega in elektrosibkega" ("What Comes Beyond the Standard Models?") sta postavila leta 1998 Norma in Holger z namenom, da bi udeleZenci v izčrpnih diskusijah kritično soočali različne ideje in teorije. Delavnice domujejo v Plemljevi hisi na Bledu ob slikovitem jezeru, kjer prijetni sprehodi in pohodi na čudovite gore, ki kipijo nad mestom, ponujajo priloznosti in vzpodbudo za diskusije. Od lanske, 20. delavniče, dalje ponudimo vsako leto med ali ob konču delavniče predavanje za splošno občinstvo na Bledu. Letošnje je imelo naslov "Kako dobro razumemo nase Vesolje v tem trenutku?", ki ga je imela Norma Susana Mankovč Borstnik v predavalniči IEDC (Blejska sola za management). Predavalničo nam je prijazno ponudila ustanoviteljiča te sole, gospa Daniča Purg. Zal je bilo predavanje, zaradi urnika na soli, sele po končani delavniči. K našim zelo odprtim, prijateljskim, dolgim in zahtevnim diskusijam, polnim iskrivega sodelovanja, je prispevalo veliko fizikov in čelo nekaj matematikov. Večšina predlogov teorij in modelov, predstavljenih in diskutiranih na nasših Blejskih delavničah, isče odgovore na vprašanja, ki jih v fizikalni skupnosti sprejeta in s številnimi poskusi potrjena standardni model osnovnih fermionskih in bo-zonskih polj ter kozmoloski standardni model puščata odprta. Ceprav je večina udelezenčev teoretičnih fizikov, mnogi z lastnimi idejami kako narediti naslednji korak onkraj sprejetih modelov in teorij, so še posebej dobrodošli predstavniki eksperimentalnih laboratorijev, ki nam pomagajo v odprtih diskusijah razjasniti resničšno sporočšilo meritev in nam pomagajo razumeti kaksšne napovedi so potrebne, da jih lahko s poskusi dovoljzanesljivo preverijo. Organizatorji moramo priznati, da smo se na blejskih delavničah v (dolgih) predstavitvah (z odmori in nadaljevanji preko več dni), ki so jim sledile zelo podrobne diskusije, naučili veliko, morda več kot večina udelezenčev. Upamo in verjamemo, da so veliko odnesli tudi študentje in večina udelezenčev. Velikokrat so se predavanja spremenila v zelo pedagoške predstavitve, ki so pojasnile predpostavke in podrobne korake, soočšile predstavljene predloge s predlogi v literaturi ali s predlogi ostalih udelezenčev ter jasno pokazale, kje utegnejo tičati šibke točke predlogov. Zdi se, da so se ideje v teh letih razvijale bistveno hitreje, zahvaljujoč prav tem delavničam. To leto eksperimenti v kozmologiji in fiziki osnovih fermionskih in bozonskih polj niso ponudili veliko novih rezultatov, ki bi omogočšili nov vpogled v fiziko osnovnih delčev in polj, čeprav je bilo vanje vlozenega veliko truda. Upamo, da bodo podrobne analize podatkov, zbranih na LHC do energij 13 TeV, prinesle odločujoče rezultate. Pričakujemo tudi, da bodo nove kozmoloske meritve uspele razresiti izvor temne snovi. Ker je dolgoletne rezultate poskusa DAMA/LIBRA, ki so predstavljeni v tem zborniku, le tezško pojasniti drugačše kot s signali temne snovi, je pričšakovati, da bodo tudi ostali poskusi sčasoma potrdili njihove rezultate. Več prispevkov v tem zborniku obravnava različne predloge, ki naj pojasnijo izvor temne snovi: supersimetrični partnerji, temni atomi iz temnih barionov in iz barionov običajne snovi, neutroni iz stabilne pete druzine kvarkov in leptonov, ki pripadajo grupi štirih druzin z druzinskimi kvantnimi števili in so (skoraj) nesklopljeni s spodnjimi štirimi družinami, med katerimi tri že poznamo. Kozmoloska spoznanja in spoznanja v teoriji osnovnih fermionskih in bozonskih poljdelcev se nikoli doslejniso bila tako zelo povezana in soodvisna. Ko bodo z eksperimenti na LHC in v ostalih laboratorijih po svetu uspeli ponuditi dovolj natančne vrednosti za elemente mešalnih matrik za kvarke in leptone, bo znan odgovor na vprasanje ali obstaja poleg opaženih treh tudi četrta družina in več novih skalarnih polj, ki določajo higgsove in Yukawine sklopitve — kar napoveduje teorija spinov-nabojev-druzin. Simetrija popravkov masnih matrik 4 x 4 v vseh redih, ki jo določajo skalarji v tej teoriji in jo obravnava prispevek v zborniku, omeji stevilo prostih parametrov masnih matrik tako, da dovoljnatančno izmerjeni matrični elementi podmatrik 3 x 3 v mesalnih matrikah 4 x 4 omogočijo napoved lastnosti četrte druzine kvarkov in leptonov, saj podmatrika (n — 1) x (n — 1) matrike n x n to enolično določa. Eden od prispevkov v zborniku predstavi studijo, ki določi masne matrike opazenih 3 druzin kvarkov iz standardne parametrizačije mesalne matrike in izmerjenih mass kvarkov. Ne preseneča ugotovitev, da so masne matrike blizu demokratični matriki, saj ima top kvark veliko večjo maso od odtalih kvarkov in je šibko sklo-pljen z ostalimi člani druzin. Tudi teorija spinov-nabojev-druzin pričakovano napove, da so masne matrike 4 x 4 blizu demokratičnim, saj je četrta druzina, ki ima maso nad 1 TeV, sibko sklpopljena s člani ostalih treh druzin. Novi podatki bodo morda dali odgovor tudi na vprasšanje, ali so zakoni narave preprosti (kot napove teorija spinov-nabojev-druzin in tudi ostale teorije Kaluza-Kleinovega tipa, ki pa pojava druzšin ne pojasnijo, pa tudi teorije strun) ali pa narava preprosto "uporabi umeritvene grupe, kadar jih potrebuje" (kar predpostavi veliko modelov, tudi nekateri v tem zborniku). In tudi ali lahko ugibamo pojav higgsovih skalarjev z majhnimi odmiki od standardnega modela fermion-skih in bozonskih polj, ali pa morajo obstoj skalarnih polj pojasniti splosšnejsše teorije (kot je teorija spinov-nabojev-druzin)? Vse meritve doslej potrdijo, da imajo fermioni polsštevilčšne spine ter naboje v fun-damentalnih upodobitvah dosedaj opazšenih grup. Eden od prispevkov obravnava prvo in drugo kvantizačijo fermionskih poljv Grassmannovem prostoru. Fermioni nosijo v Grasmannovem prostoru čelosštevilčšne spine in naboje v adjungirani upodobitvi grup. Bo primerjava lastnosti fermionskih polj, ki "zivijo" v Grassmannovem prostoru, s tistimi, ki zivijo v Cliffordovem prostoru, pripomogla k boljšemu razumevanju "eleganče naravnih zakonov" kot avtorji upajo? Cliffordov prostor ponudi razlago za vse lastnosti kvarkov in leptonov, ki imajo polstevilski spin in vse naboje v fundamentalni upodobitvi grupe (ki so podgrupe v dovolj veliki Lorentzovi grupi), vključno z druzinami (katerih lastnosti lahko prav tako pojasnimo s polstevilskimi "spini in naboji druzin"). V Grassmannovem prostoru pa druga kvantizačija ponudi fermione, ki imajo čelosštevilske spine in naboje, oboje v adjungirani upodobitvi podgrupe Lorentzove grupe in nobenih druzšin. "Dirac-Grassmannovo morje" igra vlogo Diracovega morja. Medtem, ko v Clif-fordovem primeru ena Weylova upodobitev vključuje v SO(13,1) vse kvarke in leptone ter antikvarke in antileptone, ki so jih dosedajopazili (pa tudi desnoročni nevtrino), so v Grassmannovem prostoru delci in antidelci v različnih upodobitvah. Je delovna hipoteza, da je "vsa matematika del narave", sprejemljiva in jo moramo upoštevati? Ce bi "narava izbrala" Grassmannovo namesto Cliffordove algebre, bi atomi, molekule in vse vesolje izgledali drugace. Zakajje ni? Eden od prispevkov pokaze, kako v ravnem prostoru definirati umeritvena polja Yang-Millsovega tipa z višjimi spini, ki ponudijo neskoncno število stanj umer-itvenih poljin omogocijo, da se izognemo divergencam pri visokih energijah. Tudi supersimetrija ponuja moznost obvladovanja nekaterih neskoncnosti. Analiza v prispevku obravnava možnosti, da bi meritve na LHC potrdile obstoj supersimetricnih partnerjev, ki imajo maso nekaj TeV, pa tudi obstoj vecjega števila skalarnih polj in nove druzšine kvarkov in leptonov. Teorija spinov-nabojev-druzšin, ki napove tri singlete and dva tripleta skalarnih polj, pa tudi cšetrto druzšino, ne "vidi" supersimetricnih delcev. Tudi ce poskusi ne potrdijo obstoja supersimetricnih delcev pri nizkih energijah, lahko modeli supergravitacije, ki jih porodi teorija strun, pomagajo razumeti pojav inflacije, ki jo je moralo doziveti v nase vesolje, ker lahko le tako pojasnimo izmerjene lastnosti vesolja. Supersimetrija bi morda lahko pojasnila, zakaj je gostota temne energije tako zelo majhna,pojasnila pa bi tudi prisotnost temne snovi, ter celo tvorbo prvotnih cšrnih lukenj. Te mozšnosti obravnavajo trije prispevki. Zanimivo je, da iz lastnosti poljubne snovi, ki ima samo translacijsko simetrijo, sledi, kot obravanva en prispevek, da v prostoru kvazi gibalne kolicine obstajajo majhna podrocšja, v katerih priblizšna Weylova enacšba (relativisticšna enacšba za brezmasne delce) doloca lastnosti materila, kot je, denimo, grafen. Enak pojav bi lahko opazili v vesolju pri mocšni Hubblovi ekspanziji. Avtorja to pojasnjujeta z vrzeljo"homo-lumo". Na razlicnih podrocjih fizike obstajajo razlicni modeli, ki na videz nimajo veliko skupnega, pa vendar opisujejo iste pojave enako dobro. Prisepvek v zborniku obravnava skupaj elektrodinamiko in linearizirano gravitacijo v modelu elektro-gravitacije,v katerem pogoj na vsako od polj povzrocši spontano zlomitev Lorent-zove invariance. Model je pri nizkih energijah skladen z opažanji. Dva prispevka v sekciji diskusijposkušata iz geometrijske slike v celice porazdeljenih fermionov, ki nosijo razlicne naboje, dolociti silo med fermioni. Do koliksne mere nam lahko tak pristop pomaga razumeti naravo? Kaksna je ucinkovita pot pri razumevanju nasega vesolja? Je dozorel cas, ko lahko napravimo odlocen korak v razumevanju vesolja samo s predlogom teorije, ki pojasni vse privzetke obeh standardnih modelov? Je teorija spinov, nabojev in druzšin pravi predlog? Bodo poskusi potrdili njene napovedi? Ali pa so majhni koraki procš od obeh standardnih modelov bolj varna pot pri nacšrtovanju poskusov? Je prostor-cas štiri razsezen? Ali pa je njegova razseznost mnogo vecja, neskoncna? Je interalcija med fermioni v mnogo razseznem prostoru ena sama, tedajgravitaci-jska, ki se kaze v opazljivem delu vesolja kot vse poznane sile? Samo teorija in eksperiment skupaj lahko odgovorita na ta vprasšanja. Ker je vsako leto le malo casa od delavnice do zaključka redakcije, manj kot dva meseca, avtorji ne morejo izpiliti prispevkov, vendar upamo, da to nadomesti sveZina prispevkov. Cetudi so k uspehu „Blejskih delavnic" najvec prispevali udeleZenci, ki so na Bledu omogocili prijateljsko in aktivno izmenjavo mnenj v osrcju slovenskih Julijcev, so k uspehu prispevale tudi videokonference, ki so povezale delavnice z laboratoriji po svetu. Vprašanja in odgovori ter tudi predavanja, ki jih je v zadnjih letih omogocil M.Yu. Khlopov preko Virtual Institute of Astroparticle Physics (viavca.in2p3.fr/site.html, APC, Pariz), so v izcrpnih diskusijah pomagali razcistiti marsikatero vprasanje. Bralec najde zapise vseh predavanj, objavljenih preko "cosmovia" od leta 2009, na viavca.in2p3.fr/site.html v povezavi Previous - Conferences. Troje letošnjih predavanj, Norma Mankoc Borstnik (Understanding nature with the spin-charge-family theory, making several predictions), Sergey V. Ketov (Starobinsky inflation in gravity and supergravity) in H.B Nielsen (Theory for initial State Conditions), je dostopnih na http://viavca.in2p3.fr/what_comes_beyond_the_standard_model_2018.html Vecino predavanjnajde bralec na spletni strani delavnice na http://bsm.fmf.uni-lj.si/. Najzakljucimo ta predgovor s prisrcno in toplo zahvalo vsem udelezencem, prisotnim na Bledu osebno ali preko videokonferenc, za njihova predavanja in še posebno za zelo plodne diskusije in odlicno vzdusje. Norma Mankoč Borštnik, Holger Bech Nielsen, Maxim Y. Khlopov, (Organizacijski odbor) Norma MankoC Borštnik, Holger Bech Nielsen, Dragan Lukman, (uredniki) Ljubljana, grudna (decembra) 2018 Talk Section All talk contributions are arranged alphabetically with respect to the authors' names. Bled Workshops in Physics Vol. 19, No. 2 A Proceedings to the 21 st Workshop What Comes Beyond ... (p. 1) Bled, Slovenia, June 23-July 1, 2018 1 Inflation From Supersymmetry Breaking I. Antoniadis Laboratoire de Physique Theorique et Hautes Energies - LPTHE Sorbonne Universite, CNRS, 4 Place Jussieu, 75005 Paris, France and Albert Einstein Center, Institute for Theoretical Physics University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland Abstract. I discuss a general class of models where the inflation is driven by supersymmetry breaking with the superpartner of the goldstino (sgoldstino) playing the role of the inflaton. Imposing an R-symmetry allows to satisfy easily the slow-roll conditions, avoiding the so-called n -problem, and leads to two different classes of small field inflation models; they are characterised by an inflationary plateau around the maximum of the scalar potential, where R-symmetry is either restored or spontaneously broken, with the inflaton rolling down to a minimum describing the present phase of our Universe. Inflation can be driven by either an F- or a D-term, while the minimum has a positive tuneable vacuum energy. The models agree with cosmological observations and in the simplest case predict a tensor-to-scalar ratio of primordial perturbations 10-9 < r < 10-4 and an inflation scale 1010 GeV < H* < 1012 GeV. Povzetek. Avtor obravnava razred modelov, v katerih zlomitev supersimetrije povzroči inflacijo, vlogo inflatona pa igra superpartner goldstina (sgoldstino). Avtorjev privzetek, da imajo modeli simetrijo R, omogoci, da je izpolnjen pogoj za 'slow-roll', s cimer se izogne problemu n. Tem pogojem zadostita dve vrsti modelov inflacije z majhnim poljem. Zanje je znacilen inflacijski plato okrog maksimuma skalarnega potenciala, kjer se simetrija R bodisi ohrani ali pa spontano zlomi, inflaton pa se zapelje po potencialu do minimuma, ki opisuje sedanjo fazo nasega vesolja. Inflacijo lahko poganja ali clen F ali clen D, minimum ima pozitivno vakuumsko energijo, ki jo z izbiro parametrov lahko spreminjamo tako, da se ujemajo s kozmoloskimi meritvami. V najpreprostejsem primeru modeli napovedo, da je bilo, ko je bila v zacetku vesolja energijska skala inflacije 1010 GeV < H* < 1012 GeV, razmerje tenzorskih in skalarnih nehomogenosti 10-9 < r < 10-4. Keywords: supersymmetry breaking, R-symmetry, supergravity, cosmology, inflation 1.1 Introduction If String Theory is a fundamental theory of Nature and not just a tool for studying systems with strongly coupled dynamics, it should be able to describe at the same time particle physics and cosmology, which are phenomena that involve very different scales from the microscopic four-dimensional (4d) quantum gravity length of 10-33 cm to large macroscopic distances of the size of the observable Universe ~ 1028 cm spanned a region of about 60 orders of magnitude. In particular, besides the 4d Planck mass, there are three very different scales with very different physics corresponding to the electroweak, dark energy and inflation. These scales might be related via the scale of the underlying fundamental theory, such as string theory, or they might be independent in the sense that their origin could be based on different and independent dynamics. An example of the former constraint and more predictive possibility is provided by TeV strings with a fundamental scale at low energies due for instance to large extra dimensions transverse to a four-dimensional braneworld forming our Universe [1]. In this case, the 4d Planck mass is emergent from the fundamental string scale and inflation should also happen around the same scale [2]. Here, we will adopt a more conservative approach, trying to relate the scales of supersymmetry breaking and inflation, assuming that supersymmetry breaking is realised in a metastable de Sitter vacuum with an infinitesimally small (tuneable) cosmological constant independent of the breaking scale that may be in the TeV region or higher. In a recent work [3], we studied a simple N = 1 supergravity model having this property and motivated by string theory. Besides the gravity multiplet, the minimal field content consists of a chiral multiplet with a shift symmetry promoted to a gauged R-symmetry using a vector multiplet. In the string theory context, the chiral multiplet can be identified with the string dilaton (or an appropriate com-pactification modulus) and the shift symmetry associated to the gauge invariance of a two-index antisymmetric tensor that can be dualized to a (pseudo)scalar. The shift symmetry fixes the form of the superpotential and the gauging allows for the presence of a Fayet-Iliopoulos (FI) term [4], leading to a supergravity action with two independent parameters that can be tuned so that the scalar potential possesses a metastable de Sitter minimum with a tiny vacuum energy (essentially the relative strength between the F- and D-term contributions). A third parameter fixes the Vacuum Expectation Value (VEV) of the string dilaton at the desired (phenomenologically) weak coupling regime. An important consistency constraint of the model is anomaly cancellation which has been studied in [5] and implies the existence of additional charged fields under the gauged R-symmetry. In a subsequent work [6], we analysed a small variation of this model which is manifestly anomaly free without additional charged fields and allows to couple in a straight forward way a visible sector containing the minimal supersymmetric extension of the Standard Model (MSSM) and studied the mediation of super-symmetry breaking and its phenomenological consequences. It turns out that an additional 'hidden sector' field z is needed to be added for the matter soft scalar masses to be non-tachyonic; although this field participates in the supersymmetry breaking and is similar to the so-called Polonyi field, it does not modify the main properties of the metastable de Sitter (dS) vacuum. All soft scalar masses, as well as trilinear A-terms, are generated at the tree level and are universal under the assumption that matter kinetic terms are independent of the 'Polonyi' field, since matter fields are neutral under the shift symmetry and supersymmetry breaking is driven by a combination of the U(1) D-term and the dilaton and z-field F-term. Alternatively, a way to avoid the tachyonic scalar masses without adding the extra field z is to modify the matter kinetic terms by a dilaton dependent factor. A main difference of the second analysis from the first work is that we use a field representation in which the gauged shift symmetry corresponds to an ordinary U(1) and not an R-symmetry. The two representations differ by a Kahler transformation that leaves the classical supergravity action invariant. However, at the quantum level, there is a Green-Schwarz term generated that amounts an extra dilaton dependent contribution to the gauge kinetic terms needed to cancel the anomalies of the R-symmetry. This creates an apparent puzzle with the gaugino masses that vanish in the first representation but not in the latter. The resolution to the puzzle is based on the so called anomaly mediation contributions [7,8] that explain precisely the above apparent discrepancy. It turns out that gaugino masses are generated at the quantum level and are thus suppressed compared to the scalar masses (and A-terms). This model has the necessary ingredients to be obtained as a remnant of moduli stabilisation within the framework of internal magnetic fluxes in type I string theory, turned on along the compact directions for several abelian factors of the gauge group. All geometric moduli can in principle be fixed in a supersymmetric way, while the shift symmetry is associated to the 4d axion and its gauging is a consequence of anomaly cancellation [9,10]. We then made an attempt to connect the scale of inflation with the electroweak and supersymmetry breaking scales within the same effective field theory, that at the same time allows the existence of an infinitesimally small (tuneable) positive cosmological constant describing the present dark energy of the universe. We thus addressed the question whether the same scalar potential can provide inflation with the dilaton playing also the role of the inflaton at an earlier stage of the universe evolution [11]. We showed that this is possible if one modifies the Kahler potential by a correction that plays no role around the minimum, but creates an appropriate plateau around the maximum. In general, the Kahler potential receives perturbative and non-perturbative corrections that vanish in the weak coupling limit. After analysing all such corrections, we find that only those that have the form of (Neveu-Schwarz) NS5-brane instantons can lead to an inflationary period compatible with cosmological observations. The scale of inflation turns out then to be of the order of low energy supersymmetry breaking, in the TeV region. On the other hand, the predicted tensor-to-scalar ratio is too small to be observed. Inflationary models [12] in supergravity1 suffer in general from several problems, such as fine-tuning to satisfy the slow-roll conditions, large field initial conditions that break the validity of the effective field theory, and stabilisation of the (pseudo) scalar companion of the inflaton arising from the fact that bosonic components of superfields are always even. The simplest argument to see the fine tuning of the potential is that a canonically normalised kinetic term of a complex scalar field X corresponds to a quadratic Kahler potential K = XX that brings one unit contribution to the slow-roll parameter n = V "/V, arising from the eK proportionality factor in the expression of the scalar potential V. This problem can be avoided in models with no-scale structure where cancellations arise naturally 1 For reviews on supersymmetric models of inflation, see for example [13]. due to non-canonical kinetic terms leading to potentials with flat directions (at the classical level). However, such models require often trans-Planckian initial conditions that invalidate the effective supergravity description during inflation. A concrete example where all these problems appear is the Starobinsky model of inflation [14], despite its phenomenological success. All three problems above are solved when the inflaton is identified with the scalar component of the goldstino superfield2, in the presence of a gauged R-symmetry [16]. Indeed, the superpotential is in that case linear and the big contribution to n described above cancels exactly. Since inflation arises at a plateau around the maximum of the scalar potential (hill-top) no large field initial conditions are needed, while the pseudo-scalar companion of the inflaton is absorbed into the R-gauge field that becomes massive, leading the inflaton as a single scalar field present in the low-energy spectrum. This model provides therefore a minimal realisation of natural small-field inflation in supergravity, compatible with present observations, as we show below. Moreover, it allows the presence of a realistic minimum describing our present Universe with an infinitesimal positive vacuum energy arising due to a cancellation between an F- and D-term contributions to the scalar potential, without affecting the properties of the inflationary plateau, along the lines of Refs. [3,11,17]. In the above models the D-term has a constant FI contribution but plays no role during inflation and can be neglected, while the pseudoscalar partner of the inflaton is absorbed by the U(1 )R gauge field that becomes massive away from the origin. Recently, a new FI term was proposed [19] that has three important properties: (1) it is manifestly gauge invariant already at the Lagrangian level; (2) it is associated to a U(1) that should not gauge an R-symmetry and (3) supersym-metry is broken by (at least) a D-auxiliary expectation value and the extra bosonic part of the action is reduced in the unitary gauge to a constant FI contribution leading to a positive shift of the scalar potential, in the absence of matter fields. In the presence of matter fields, the FI contribution to the D-term acquires a special field dependence e2K/3 that violates invariance under Kahler transformations. In a recent work [18], we studied the properties of the new FI term and explored its consequences to the class of inflation models we introduced in [16].3 We first showed that matter fields charged under the U(1) gauge symmetry can consistently be added in the presence of the new FI term, as well as a non-trivial gauge kinetic function. We then observed that the new FI term is not invariant under Kahler transformations. On the other hand, a gauged R-symmetry in ordinary Kahler invariant supergravity can always be reduced to an ordinary (non-R) U(1) by a Kahler transformation. By then going to such a frame, we find that the two FI contributions to the U(1) D-term can coexist, leading to a novel contribution to the scalar potential. The resulting D-term scalar potential provides an alternative realisation of inflation from supersymmetry breaking, driven by a D- instead of an F-term. The inflaton is still a superpartner of the goldstino which is now a gaugino within 2 See [15] for earlier work relating supersymmetry and inflation. 3 This new FI term was also studied in [20] to remove an instability from inflation in Polonyi-Starobinsky supergravity. a massive vector multiplet, where again the pseudoscalar partner is absorbed by the gauge field away from the origin. For a particular choice of the inflaton charge, the scalar potential has a maximum at the origin where inflation occurs and a supersymmetric minimum at zero energy, in the limit of negligible F-term contribution (such as in the absence of superpotential). The slow roll conditions are automatically satisfied near the point where the new FI term cancels the charge of the inflaton, leading to higher than quadratic contributions due to its non trivial field dependence. The Kahler potential can be canonical, modulo the Kahler transformation that takes it to the non R-symmetry frame. In the presence of a small superpotential, the inflation is practically unchanged and driven by the D-term, as before. However, the maximum is now slightly shifted away from the origin and the minimum has a small non-vanishing positive vacuum energy, where supersymmetry is broken by both F- and D-auxiliary expectation values of similar magnitude. The model predicts in general small primordial gravitational waves with a tensor-to-scaler ration r well below the observability limit. However, when higher order terms are included in the Kahler potential, one finds that r can increase to large values r ~ 0.015. On general grounds, there are two classes of such models depending on whether the maximum corresponds to a point of unbroken (case 1) or broken (case 2) R-symmetry. The latter corresponds actually to a generalisation of the model we discussed above [11], inspired by string theory [3]. It has the same field content but in a different field basis with a chiral multiplet S K(z,z) + J(z) + J(z), W(z) —> e-l<2j(z)W(z), (1.1) where k is the inverse of the reduced Planck mass, Mpi = k-1 = 2.4 x 1015 TeV. The gauge transformations of chiral multiplet scalars are given by holomorphic Killing vectors, i.e. Sza = eAkA(z), where 0A is the gauge parameter of the gauge group A. The Kahler potential and superpotential need not be invariant under this gauge transformation, but can change by a Kahler transformation SK = eA [ta(z)+ ta(z)] , (1.2) provided that the gauge transformation of the superpotential satisfies SW = eAK2rA(z)W. One then has from SW = WaSza Wak£ =-k2taw, (1.3) where Wa = 9aW and a labels the chiral multiplets. The supergravity theory can then be described by a gauge invariant function G = k2K + log(K6WW). (1.4) The scalar potential is given by V = vf + vd k2K ( VF = eK'K (-3k2ww + vaWgaßvßW) Vd = 2 (Ref)-1AB PaPb, (1.5) where W appears with its Kähler covariant derivative VaW = 3a W(z) + k2(3«K)W(z). (1.6) The moment maps PA are given by Pa = i(kA3«K - Ta). (1.7) In this paper we will be concerned with theories having a gauged R-symmetry, for which rA (z) is given by an imaginary constant rA (z) = i.K-2£,. In this case, k-2£ is a Fayet-Iliopoulos [4] constant parameter. 1.3 Symmetric versus non-symmetric point Here, we present a class of inflation models in supergravity theories containing a single chiral multiplet transforming under a gauged R-symmetry with a corresponding abelian vector multiplet [16]. We assume that the chiral multiplet X (with scalar component X) transforms as: X —> Xe-iqœ. (1.8) where q is its charge, and w is the gauge parameter. The Kahler potential is therefore a function of XX, while the superpotential is constrained to be of the form Xb: K = K(XX), W = K-3fXb, (1.9) where X is a dimensionless field. For b = 0, the gauge symmetry eq. (1.8) becomes a gauged R-symmetry. The gauge kinetic function can have a constant contribution as well as a contribution proportional to ln X f(X)= y + P ln X. (1.10) The latter contribution proportional to p is not gauge invariant and can be used as a Green-Schwarz counter term to cancel possible anomalies. One can show however that the constant p is fixed to be very small by anomaly cancellation conditions and does not change our results [16]. We will therefore omit this term in our analysis below. We are interested in the general properties of supergravity theories of inflation that are of the above form. Before performing our analysis, a distinction should be made concerning the initial point where slow-roll inflation starts. The inflaton field (which will turn out to be p, where X = pe10) can either have its initial value close to the symmetric point where X = 0, or at a generic point X = 0. The minimum of the potential, however, is always at a nonzero point X = 0. This is because at X = 0 the negative contribution to the scalar potential vanishes and no cancellation between F-term and D-term is possible. The supersymmetry breaking scale is therefore related to the cosmological constant as K-2mJ/2 « A. One could in principle assume that the value of the potential at its minimum is of the order of the supersymmetry breaking scale. However, in this case additional corrections are needed to bring down the minimum of the potential to the present value of the cosmological constant, and we therefore do not discuss this possibility. In the first case, inflation starts near X = 0, and the inflaton field will roll towards a minimum of the potential at X = 0. On the other hand, in the second case inflation will start at a generic point X = 0. It is then convenient to work with another chiral superfield S, which is invariant under a shift symmetry S —> S - ica (1.11) by performing a field redefinition X = eS. (1.12) In this case the most general Kahler potential and superpotential are of the form K = K(S + S), W = K-3aebS. (1.13) Note that this field redefinition is not valid at the symmetric point X = 0 for the first case. 1.4 Case 1: Inflation near the symmetric point 1.4.1 Slow roll parameters In this section we derive the conditions that lead to slow-roll inflation scenarios, where the start of inflation is near a local maximum of the potential at X = 0. Since the superpotential has charge 2 under R-symmetry, one has (W) = 0 as long as R-symmetry is preserved. Therefore, (W) can be regarded as the order parameter of R-symmetry breaking. On the other hand, the minimum of the potential requires (W) = 0 and broken R-symmetry. It is therefore attractive to assume that at earlier times R-symmetry was a good symmetry, switching off dangerous corrections to the potential. As similar approach was followed in [22], where a discrete R-symmetry is assumed. Instead, we assume a gauged R-symmetry which is spontaneously broken at the minimum of the potential. While the superpotential is uniquely fixed in eq. (1.9), the Kahler potential is only fixed to be of the form K(XX). We expand the Kahler potential as follows K(X,X) = k-2xx + k-2a(xx )2, W(X) = K-3fXb, f(X) = 1, (1.14) where A and f are constants. The gauge kinetic function is taken to be constant since it was shown that the coefficient |3 in front of the logarithmic term in eq. (1.10) is fixed to be very small by anomaly cancellation conditions [16]. As far as the scalar potential is concerned, the coefficient y can be absorbed in other parameters of the theory. We therefore take y = 1. The scalar potential is given by V = Vf + Vd, (1.15) where Vf = K-4f2(XX )b-èxX (1+AXX ) and — — 2 + (b + XXX (1 + 2AXX )) ~3XX +-1 + 4AXX- (1.16) Vd = k-4 ^[b + XX(1 + 2AXX)]2 . (1.17) The superpotential is not gauge invariant under the U(1 ) gauge symmetry. Instead it transforms as W —> We-iqbw . (1.18) Therefore, the U( 1 ) is a gauged R-symmetry which we will further denote as U(1 )R. From WXkX = —rR k2W, where kX = —iqX is the Killing vector for the field X under the R-symmetry, rR = îk-2£,r with k-2£,r the Fayet-Iliopoulos contribution to the scalar potential, and WX is short-hand for 3W/3X, we find rR = iK-2qb. (1.19) 2 A consequence of the gauged R-symmetry is that the superpotential coupling b enters the D-term contribution of the scalar potential as a constant Fayet-Iliopoulos contribution.4 Note that the scalar potential is only a function of the modulus of X and that the potential contains a Fayet-Iliopoulos contribution for b = 0. Moreover, its phase will be 'eaten' by the U(1) gauge boson upon a field redefinition of the gauge potential similarly to the standard Higgs mechanism. After performing a change of field variables X = peie, X = pe-ie, (p > 0) (1.20) the scalar potential is a function of p, k4v = fV^-1^2 +ap4 (%p2 + (b +1P++A2pA2p4)^ + T (b + p2 + 2Aptl22i) Since we assume that inflation starts near p = 0, we require that the potential eq. (1.21) has a local maximum at this point. It turns out that the potential only allows for a local maximum at p = 0 when b = 1. For b < 1 the potential diverges when p goes to zero. For 1 < b < 1.5 the first derivative of the potential diverges, while for b = 1.5, one has V'(0) = 4f2 + 3q2 > 0, and for b > 1.5, on has V''(0) > 0. We thus take b = 1 and the scalar potential reduces to k4V = f2ep2+Ap4 (-3p2 + (1 7+ +A2pA/4)2) + (1 + P2 + 2Ap4)2 .(1.22) A plot of the potential for A = 1/2, q = 1 and f tuned so that the minimum has zero energy is given in Figure 1.1. Note that in this case the the superpotential is linear W = fX, describing the sgoldstino (up to an additional low-energy constraint) [26]. Indeed, modulo a D-term contribution, the inflaton in this model is the superpartner of the goldstino. In fact, for q = 0 the inflaton reduces to the partner of the goldstino as in Minimal Inflation models [27]. The important difference however is that this is a microscopic realisation of the identification of the inflaton with the sgoldstino, and that the so-called n-problem is avoided (see discussion below). The kinetic terms for the scalars can be written as5 Lkin = —9xX c^Xi^X = -9xx [3^p3^p + p2 (3^9 + qA^) (3^9 + qA^)] . (1.23) It was already anticipated above that the phase 9 plays the role of the longitudinal component of the gauge field A^, which acquires a mass by a Brout-Englert-Higgs mechanism. 4 For other studies of inflation involving Fayet-Iliopoulos terms see for example [24], or [25] for more recent work. Moreover, our motivations have some overlap with [22], where inflation is also assumed to start near an R-symmetric point at X = 0. However, this work uses a discrete R-symmetry which does not lead to Fayet-Iliopoulos terms. 5 The covariant derivative is defined as = — A^kj;, where kX = — iqX is the Killing vector for the U(1) transformation eq. (1.8). K*V 0.2 0.4 0.6 0.8 1.0 Fig.1.1. We now interpret the field p as the inflaton. It is important to emphasise that, in contrast with usual supersymmetric theories of inflation where one necessarily has two scalar degrees of freedom resulting in multifield inflation [28], our class of models contains only one scalar field p as the inflaton. In order to calculate the slow-roll parameters, one needs to work with the canonically normalised field x satisfying dx=^. The slow-roll parameters are given in terms of the canonical field x by e = = 1 fdV/dx\ 2k2 v v 1 d2V/dx2 K2 V (1.24) (1.25) Since we assume inflation to start near p = 0, we expand e = 4 (-4A+L)2 P2 + O(p4), n = 2 ( -4A+2 ) + O(p2), (1.26) where we defined q = fx. Notice that for p C 1 the e parameter is very small, while the n parameter can be made small by carefully tuning the parameter A. Any higher order corrections to the Kahler potential do not contribute to the leading contributions in the expansion near p = 0 for n and e. Such corrections can therefore be used to alter the potential near its minimum, at some point X = 0 without influencing the slow-roll parameters. 2 n A comment on the n -problem in Supergravity A few words are now in order concerning the n-problem [29]. The n problem in N = 1 supergravity is often stated as follows (see for example [30]): If, for instance, a theory with a single chiral multiplet with scalar component 9 is taken, then the Kahler potential can be expanded around a reference location 9 = 0 as K = K(0) + Kv 10-9 (equivalently H* > 107 TeV), based on an model-independent argument. This bound can be lowered at the cost of naturalness between parameters in the potential. 1.5 On the new FI term 1.5.1 Review In [19], the authors propose a new contribution to the supergravity Lagrangian of the form7 Lfi = 2-2 w2w 2 SoS0 T(w2)T(w2) (V (1.38) The chiral compensator field So, with Weyl and chiral weights (Weyl, Chiral) = (1,1), has components S0 = (s0, PLH0, F0) . The vector multiplet has vanishing Weyl and chiral weights, and its components are given by V = (v, Z, H, v^, À, D). In the Wess-Zumino gauge, the first components are put to zero v = Z = H = 0. The multiplet w2 is of weights (1,1), and given by 2 Àplà _ 2 àprà w2 = -ST", w = "S^ • (L39) S0 S0 The components of ÀPLÀ are given by ÀPLÀ = (àPlà ; V2PL ( - • F + iD)À ; 2ÀPlDà + F- • F- - D2) • (1.40) The kinetic terms for the gauge multiplet are given by Lkin = -4 [ÀPlà]f + h.c. • (1.41) The operator T (T) is defined in [34,35], and leads to a chiral (antichiral) multiplet. For example, the chiral multiplet T(w2) has weights (2,2). In global supersymmetry the operator T corresponds to the usual chiral projection operator D2.8 From now on, we will drop the notation of h.c. and implicitly assume its presence for every [ ]F term in the Lagrangian. Finally, the multiplet (Vis a linear multiplet with weights (2, 0) , given by (V)d = (D,DDà,0,DbFab, -DDà, -aCD) • (1.42) 7 A similar, but not identical term was studied in [33]. 8 The operator T indeed has the property that T(Z) = 0 for a chiral multiplet Z. Moreover, for a vector multiplet V we have T(ZC) = ZT(C), and [C]d = 1 [T(C)]f. D The definitions of DDA and the covariant field strength Fab can be found in eq. (17.1) of [23], which reduce for an abelian gauge field to fab = ea1 ebv (23[^Av] + ^Yv]A) D^A = ^ - + lw£bYab - 3iY*A^j A - ^ 1Yabfab + ¿^D) V (1.43) Here, ea^ is the vierbein, with frame indices a, b and coordinate indices v. The fields w^b, b^, and A^ are the gauge fields corresponding to Lorentz transformations, dilatations, and TR symmetry of the conformal algebra respectively, while ■ ^ is the gravitino. The conformal d'Alembertian is given by DC = nab"DaVb. It is important to note that the FI term given by eq. (1.38) does not require the gauging of an R-symmetry, but breaks invariance under Kahler transformations. In fact, a gauged R-symmetry would forbid such a term [19].9 The resulting Lagrangian after integrating out the auxiliary field D contains a term LFI,new = -y (SOSO)2 . (1.44) In the absence of additional matter fields, one can use the Poincare gauge s0 = s0 = 1, resulting in a constant D-term contribution to the scalar potential. This prefactor however is relevant when matter couplings are included in the next section. 1.5.2 Adding (charged) matter fields In this section we couple the term LFI given by eq. (1.38) to additional matter fields charged under the U(1 ). For simplicity, we focus on a single chiral multiplet X. The extension to more chiral multiplets is trivial. The Lagrangian is given by £ = -3 So Soe-3 + [SOW(X)] F - 4 [f(X)ÂPLÀ] F + Lfi, (1.45) with a Kahler potential K(X, X), a superpotential W(X) and a gauge kinetic function f (X). The first three terms in eq. (1.45) give the usual supergravity Lagrangian [23]. We assume that the multiplet X transforms under the U(1), V -> V + A + A, X —> Xe-qA, (1.46) with gauge multiplet parameter A. We assume that the U(1) is not an R-symmetry. In other words, we assume that the superpotential does not transform under the gauge symmetry. For a model with a single chiral multiplet this implies that the superpotential is constant W(X) = F. (1.47) 9 We kept the notation of [19]. Note that in this notation the field strength superfield Wa is given by W2 = APlA, and (V)d corresponds to . D Gauge invariance fixes the Kahler potential to be a function of XeqVX (for nota-tional simplicity, in the following we omit the eqV factors). Indeed, in this case the term LFI can be consistently added to the theory, similar to [19], and the resulting D-term contribution to the scalar potential acquires an extra term proportional to £2 Vd = 1 Re (f(X))-1 (ikxdxK + £,2e3K)2 , (1.48) where the Killing vector is kx = —iqX and f (X) is the gauge kinetic function. The F-term contribution to the scalar potential remains the usual VF = eK(X'X) 3WW + gxXVXWVX"W) . (1.49) For a constant superpotential (1.47) this reduces to Vf = |F|2eK(X'X) (-3 + gXX3XK3XK) . (1.50) From eq. (1.48) it can be seen that if the Kahler potential includes a term proportional to £1 log(XX), the D-term contribution to the scalar potential acquires another constant contribution. For example, if K(X,X)= XX + £1 ln(XX), (1.51) the D-term contribution to the scalar potential becomes Vd = 1 Re (f(X))-1 (qXX + q£ + ^e2K)2 . (1.52) In fact the contribution proportional to £1 is the usual FI term in a non R-symmetric Kahler frame, which can be consistently added to the model including the new FI term proportional to £2. In the absence of the extra term, a Kahler transformation K(X,XH K(X,X) + J(X) + J(X), W(X)-> W(X)e-J(X), (1.53) with J(X) = —£1 ln X allows one to recast the model in the form K(X,X) = XX, W (X) = m3/2X. (1.54) The two models result in the same Lagrangian, at least classically10. However, in the Kahler frame of eqs. (1.54) the superpotential transforms nontrivially under the gauge symmetry. As a consequence, the gauge symmetry becomes an R-symmetry. Note that [18]: 1. The extra term (1.38) violates the Kahler invariance of the theory, and the two models related by a Kahler transformation are no longer equivalent. 2. The model written in the Kahler frame where the gauge symmetry becomes an R-symmetry in eqs. (1.54) can not be consistently coupled to LFI. 10 At the quantum level, a Kahler transformation also introduces a change in the gauge kinetic function f, see for example [36]. 1.6 The scalar potential in a Non R-symmetry frame In this section, we work in the Kahler frame where the superpotential does not transform, and take into account the two types of FI terms which were discussed in the last section. For convenience, we repeat here the Kahler potential in eq. (1.51) and restore the inverse reduced Planck mass k = M—1 = (2.4 x 1018 GeV)-1: K = k-2(XX + £,1 ln XX). (1.55) The superpotential and the gauge kinetic function are set to be constant11: W = k-3F, f(X) = 1. (1.56) After performing a change of the field variable X = pe10 where p > 0 and setting £1 = b, the full scalar potential V = VF + VD is a function of p. The F-term contribution to the scalar potential is given by Vf = 1 F2ep2 p2 (b + P2 P2 (1.57) and the D-term contribution is 2 2 Vd = (b + p2 + £p431 e2 p2) . (1.58) Note that we rescaled the second FI parameter by £ = £2/q. We consider the case with £ = 0 because we are interested in the role of the new FI-term in inflationary models driven by supersymmetry breaking. Moreover, the limit £ —» 0 is ill-defined [19]. The first FI parameter b was introduced as a free parameter. We now proceed to narrowing the value of b by the following physical requirements. We first consider the behaviour of the potential around p = 0, 2 Vd = 2K4 (b2 + 2bp2 + O(p4})+ 2b£p-r (1 + O(p2}) + £2p"r (1 + O(p2))J, (1.59) F2 r 1 Vf = — p2b b2p-2 + (2b - 3) + O(p2) . (1.60) k l Here we are interested in small-field inflation models in which the inflation starts in the neighbourhood of a local maximum at p = 0. In [16], we considered models of this type with £ = 0 (which were called Case 1 models), and found that the choice b = 1 is forced by the requirement that the potential takes a finite value at the local maximum p = 0. Now, we will investigate the effect of the new FI parameter £ on the choice of b under the same requirement. 11 Strictly speaking, the gauge kinetic function gets a field-dependent correction proportional to q2 ln p, in order to cancel the chiral anomalies [11]. However, the correction turns out to be very small and can be neglected below, since the charge q is chosen to be of order of 10-5 or smaller. 2 3 First, in order for V(0) to be finite, we need b > 0. We first consider the case b > 0. We next investigate the condition that the potential at p = 0 has a local maximum. For clarity we discuss below the cases of F = 0 and F = 0 separately. The b = 0 case will be treated at the end of this section. 1.6.1 Case F = 0 In this case VF = 0 and the scalar potential is given by only the D-term contribution V = VD- Let us first discuss the first derivative of the potential: VD = 2 k4 8b 8b, 4bp(1 + O(p2)) + 8f £,p" -1 ( 1 + O(p2)) + ^p-i (! + 0(p2)) (1.61) For VD (0) to be convergent, we need b > 3/4 (note that £ = 0). When b = 3/4, we have VD (0) = 8b2£/3, which does not give an extremum because we chose £ = 0. On the other hand, when b > 3/4, we have VD (0) = 0. To narrow the allowed value of b further, let us turn to the second derivative, VD = 2 k4 4b(1 + 0(p2)) + 8f (4b - 1)*P-2(1 + 0(p2)) +8b ( 8b - i)^2p --2(i+o(p2))" (1.62) When 3/4 < b < 3/2, the second derivative VD(0) diverges. When b > 3/2, the second derivative becomes VD (0) = 2K-4q2b > 0, which gives a minimum. We therefore conclude that to have a local maximum at p = 0, we need to choose b = 3/2, for which we have VD(0)= 3K-V(^ + 1). (1.63) The condition that p = 0 is a local maximum requires £ < — 1. Let us next discuss the global minimum of the potential with b = 3/2 and £ < — 1. The first derivative of the potential without approximation reads 2 2 -,22^ -,22 VD « p(3 + 3£,e3 p + 2£,p2e3 p )(3 + 2p2 + 2£,p2e3p ). (1.64) Since 3 + 3£,e3p2 + 2£,p2e 3p <0 for p > 0 and £ < — 1, the extremum away from p = 0 is located at pv satisfying the condition 3 + 2p2 + 2^p2e3 p2 = 0. (1.65) Substituting this condition into the potential Vd gives Vd (pv) = 0. We conclude that for £ < —1 and b = 3/2 the potential has a maximum at p = 0, and a supersymmetric minimum at pv. We postpone the analysis of inflation near the maximum of the potential in section 1.7, and the discussion of the uplifting of the minimum in order to obtain a small but positive cosmological constant below. In the next subsection we investigate the case F = 0. 2 We finally comment on supersymmetry (SUSY) breaking in the scalar potential. Since the superpotential is zero, the SUSY breaking is measured by the D-term order parameter, namely the Killing potential associated with the gauged U(1), which is defined by D ="WxUx + axW)' (1.66) This enters the scalar potential as VD = D2/2. So, at the local maximum and during inflation D is of order q and supersymmetry is broken. On the other hand, at the global minimum, supersymmetry is preserved and the potential vanishes. 1.6.2 Case F = 0 In this section we take into account the effect of VF; its first derivative reads: b2(2b - 2)p2b-3 + 2b(2b - 3)p2b-1 (1 + O(p2)) VF' = K-4F2 (1.67) For V'(0) to be convergent, we need b > 3/2, for which VD (0) = 0 holds. For b = 3/2, we have VF (0) = (9/4)k-4F2 > 0, that does not give an extremum. For b > 3/2, we have VF (0) = 0. To narrow the allowed values of b further, let us turn to the second derivative, VF'' = K-4F2 b2(2b - 2)(2b - 3)p2b-4 + 2b(2b - 3)(2b - 1)p2b-2( 1 + O(p2)) . (1.68) For 3/2 < b < 2, the second derivative VF'(0) diverges. For b > 2, the second derivative is positive V ''(0) > 0, that gives a minimum (note that VD'(0) >0 as well in this range). We conclude that the potential cannot have a local maximum at p = 0 for any choice of b. Nevertheless, as we will show below, the potential can have a local maximum in the neighbourhood of p = 0 if we choose b = 3/2 and £ < -1. For this choice, the derivatives of the potential have the following properties, V'(0) <0, V''(0)= 3K-4q2(£ + 1)' (1.69) The extremisation condition around p = 0 becomes 9 3K-4q2(£ + 1)p + 9K-4F2 ~ 0. (1.70) So the extremum is at 3F2 P = -4qWT)' (1'71) Note that the extremum is in the neighbourhood of p = 0 as long as we keep the F-contribution to the scalar potential small by taking F2 C q2|£ + 11, which guarantees the approximation ignoring higher order terms in p. We now choose £ < —1 so that p for this extremum is positive. The second derivative at the extremum reads V" ~ 3K-4q2(£ + 1), (1.72) as long as we ignore higher order terms in F2/(q2|£ + 11). By our choice £ < — 1, the extremum is a local maximum, as desired. Let us comment on the global minimum after turning on the F-term contribution. As long as we choose the parameters so that F2/q2 C 1, the change in the global minimum pv is very small, of order O(F2/q2), because the extremisation condition depends only on the ratio F2/q2. So the change in the value of the global minimum is of order O(F2). The plot of this change is given in Fig. 1.2. K*V For F — 0, the local maximum is shifted by a small positive value to pmax — 0. The global minimum now has a positive cosmological constant. In the present case F = 0, the order parameters of SUSY breaking are both the Killing potential D and the F-term contribution FX, which read D« q(f + p2), Fx « Fp1/2ep2/2, (1.73) where the F-term order parameter FX is defined by 1 92>C\ 1 (^ + K2 dKwY (1.74) V 9X 3X J y ' T - 1 pk2K/2 0 M i + Fx--Tf UxaxJ 1 ^ + Therefore, at the local maximum, FX/D is of order O((£ + 1 )-1/2F2/q2) because p there is of order O((£ + 1 )-1 F2/q2). On the other hand, at the global minimum, both D and FX are of order O(F), assuming that p at the minimum is of order O(1), which is true in our models below. This makes tuning of the vacuum energy between the F- and D-contribution in principle possible, along the lines of [16,11]. A comment must be made here on the action in the presence of non-vanishing F and £. As mentioned above, the supersymmetry is broken both by the gauge sector and by the matter sector. The associated goldstino therefore consists of a linear combination of the U(1) gaugino and the fermion in the matter chiral multiplet X. In the unitary gauge the goldstino is set to zero, so the gaugino is not vanishing anymore, and the action does not simplify as in Ref. [19]. This, however, only affects the part of the action with fermions, while the scalar potential does not change. This is why we nevertheless used the scalar potential (1.57) and (1.58). Let us consider now the case b = 0 where only the new FI parameter £ contributes to the potential. In this case, the condition for the local maximum of the scalar potential at p = 0 can be satisfied for — 2 < £ < 0. When F is set to zero, the scalar potential (1.58) has a minimum at pmin = f ln ( — ^r) .In order to have Vmin = 0, we can choose £ = — 2e. However, we find that this choice of parameter £ does not allow slow-roll inflation near the maximum of the scalar potential. Similar to the previous model of section 1.4, it may be possible to achieve both the scalar potential satisfying slow-roll conditions and a small cosmological constant at the minimum by adding correction terms to the Kahler potential and turning on a parameter F. However, here, we will focus on b = 3/2 case where, as we will see shortly, less parameters are required to satisfy the observational constraints. 1.7 Application in Inflation We recall that the the models we described in section 1.4, the inflaton is identified with the sgoldstino, carrying a U(1) charge under a gauged R-symmetry and inflation occurs around the maximum of the scalar potential, where the U(1) symmetry is restored, with the inflaton rolling down towards the electroweak minimum. These models avoid the so-called n-problem in supergravity by taking a linear superpotential, W H 1000 500 00.1 0 0.1 (R90 - Fig. 2.3. Distribution of the percentage variations of R90 with respect to the mean values for all the detectors in DAMA/LIBRA-phase2 (histogram); the superimposed curve is a gaussian fit. See text. For example, the measured rate integrated above 90 keV, R90, as a function of the time has been analysed. Fig. 2.3 shows the distribution of the percentage variations of R90 with respect to the mean values for all the detectors in DAMA/LIBRA-phase2; this has a cumulative gaussian behaviour with u — 1%, well accounted by the statistical spread expected from the used sampling time. Moreover, fitting the time behaviour of R90 including also a term with phase and period as for DM particles, a modulation amplitude AR90 compatible with zero has been found for all the annual cycles (see Ref. [12]). This also excludes the presence of any background modulation in the whole energy spectrum at a level much lower than the effect found in the lowest energy range for the single-hit scintillation events. In fact, otherwise - considering the R90 mean values - a modulation amplitude of order of tens cpd/kg would be present for each annual cycle, that is ~ 100 u far away from the measured values. Similar results are obtained when comparing the single-hit residuals in the (1-6) keV with those in other energy intervals [12]. A further relevant investigation on DAMA/LIBRA-phase2 data has been performed by applying the same hardware and software procedures, used to acquire and to analyse the single-hit residual rate, to the multiple-hit one. Since the probability that a DM particle interacts in more than one detector is negligible, a DM signal can be present just in the single-hit residual rate. Thus, the comparison of single-hit events with multiple-hit events corresponds to compare the cases of DM particles beam-on and beam-off. This procedure also allows an additional test of the background behaviour in the same energy interval where the positive effect is observed. We note that an event is considered multiple-hit when there is a deposition of energy in coincidence in more than one detector of the set-up. The multiplicity can, in principle, range from 2 to 25. A multiple-hit event in a given energy interval, say (1-6) keV, is given by an energy deposition between 1 and 6 keV in one detector and other deposition(s) in other detector(s). The residual rate of events with multiplicity equal or greater than 2 with an energy deposition in the range 1-6 keV is shown in Fig. 2.4; the only procedure applied to multiple-hit events is that used to reject noise events near software energy threshold and is the same used for single-hit events. In particular, in Fig. 2.4 the residual rates of the 1-3 keV e n 0.02 -0.02 - L -.......... i,,,, i,,,, i,,,, i 250 300 350 400 450 500 550 600 650 Time (day) 1-6 keV 650 « 250 300 350 400 450 500 550 60( Time (day) Fig. 2.4. Experimental residual rates of DAMA/LIBRA-phase2 single-hit events (open circles), class of events to which DM events belong, and for multiple-hit events (filled triangles), class of events to which DM events do not belong. They have been obtained by considering for each class of events the data as collected in a single annual cycle and by using in both cases the same identical hardware and the same identical software procedures. The initial time of the figure is taken on August 7th. The experimental points present the errors as vertical bars and the associated time bin width as horizontal bars. Analogous results were obtained for DAMA/Nal (two last annual cycles) and DAMA/LIBRA-phasel [2-5,17,10]. single-hit scintillation events collected during DAMA/LIBRA-phase2 are reported, as collected in a single cycle, together with the residual rates of the multiple-hit events, in the considered energy intervals2. While, as already observed, a clear modulation, satisfying all the peculiarities of the DM annual modulation signature, is present in the single-hit events, the fitted modulation amplitudes for the multiple-hit residual rate are well compatible with zero: (0.0007 ± 0.0006) cpd/kg/keV, and (0.0004 ± 0.0004) cpd/kg/keV, in the energy regions (1-3) keV, and (1-6) keV, respectively. Thus, again evidence of annual modulation with proper features as required by the DM annual modulation signature is present in the single-hit residuals (events class to which the DM particle induced events belong), while it is absent in the multiple-hit residual rate (event class to which only background events belong). Similar results were also obtained for the two last annual cycles of DAMA/NaI [17] and for DAMA/LIBRA-phasel [2-5]. Since the same identical 2 Just for completeness, it is worth noting that the rate of the multiple-hit events is < 0.1 cpd/kg/keV and is dominated by double hit events from residual 40K in the crystals. hardware and the same identical software procedures have been used to analyse the two classes of events, the obtained result offers an additional support for the presence of a DM particle component in the galactic halo. In conclusion, no background process able to mimic the DM annual modulation signature (that is, able to simultaneously satisfy all the peculiarities of the signature and to account for the measured modulation amplitude) has been found or suggested by anyone throughout some decades thus far (see also discussions e.g. in Ref. [1-5,7,8,10]). 2.3.2 The analysis in frequency iU.iU.Lih»., ^.....liiliiiiLiii.ii>MAil,J Frequency (d'1) Frequency (d'1) S 60 o ft. -a (2-6) keV (6-14) keV o0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Frequency (d-1) Fig. 2.5. Power spectra of the time sequence of the measured single-hit events for DAMA/LIBRA-phasel and DAMA/LIBRA-phase2 grouped in 1 day bins. From top to bottom: spectra up to the Nyquist frequency for (2-6) keV and (6-14) keV energy intervals and their zoom around the 1 y-1 peak, for (2-6) keV (solid line) and (6-14) keV (dotted line) energy intervals. The main mode present at the lowest energy interval corresponds to a frequency of 2.74 x 10-3 d-1 (vertical line, purple on-line). It corresponds to a period of ~ 1 year. A similar peak is not present in the (6-14) keV energy interval. The shaded (green on-line) area in the bottom figure - calculated by Monte Carlo procedure - represents the 90% C.L. region where all the peaks are expected to fall for the (2-6) keV energy interval. In the frequency range far from the signal for the (2-6) keV energy region and for the whole (6-14) keV spectrum, the upper limit of the shaded region (90% C.L.) can be calculated to be 10.6 (continuous lines, green on-line). S 60 (2-6) keV 20 02 40 20 To perform the Fourier analysis of the DAMA/LIBRA-phase1 and -phase2 data in a wider region of considered frequency, the single-hit events have been grouped in 1 day bins. Because of the low statistics in each time bin, a procedure described in Ref. [31] has been followed. The whole power spectra up to the Nyquist frequency and the zoomed ones are reported in Fig. 2.5. For the lowest energy interval a clear peak corresponding to a period of 1 year is evident, while in the (6-14) keV energy region the same analysis gives only aliasing peaks. Neither other structure at different frequencies has been observed. £ o Oh 60 3 50 '■â = o Z 40 30 20 10 0 (2-6) keV Range of few years period 0 0.001 0.002 ^Al 0.003 Frequency (d-1) Fig. 2.6. Power spectrum of the annual baseline counting rates for the single-hit events of DAMA/LIBRA-phase1 and DAMA/LIBRA-phase2 in the (2-6) keV energy interval (dotted line, red on-line). Also shown for comparison is the power spectrum reported in Fig. 2.5 (solid line). The calculation has been performed according to Ref. [5]. As can be seen, a principal mode is present at a frequency of 2.74 x 10-3 d-1, that corresponds to a period of ~ 1 year. No statistically-significant peak is present at lower frequencies. This implies that no evidence for a long term modulation is present in the single-hit scintillation event in the low energy range. As regards the significance of the peaks present in the periodogram, we remind that the periodogram ordinate, z, at each frequency follows a simple exponential distribution e-z in the case of the null hypothesis or white noise [32]. Thus, if M independent frequencies are scanned, the probability to obtain values larger than z is: P(> z) = 1 — (1 — e-z)M; in general M depends on the number of sampled frequencies, the number of data points N, and their detailed spacing. It turns out that M is very nearly equal to N when the data points are approximately equally spaced, and when the sampled frequencies cover the frequency range from 0 to the Nyquist frequency [33,34]. The number of data points used to obtain the spectra in Fig. 2.5 is N = 4341 (days measured over the 4748 days of the 13 DAMA/LIBRA-phasel and -phase2 annual cycles) and the full frequencies region up to Nyquist frequency has been scanned. Therefore, assuming M = N, the significance levels P = 0.10, 0.05 and 0.01, correspond to peaks with heights larger than z = 10.6,11.3 and 13.0, respectively, in the spectra of Fig 2.5. In the case below 6 keV, a signal is present; thus, the signal must be included to properly evaluate the C.L.. This has been done by a dedicated Monte Carlo procedure where a large number of similar experiments has been simulated. The 90% C.L. region (shaded, green on-line) where all the peaks are expected to fall for the (2-6) keV energy interval is shown in Fig 2.5; several peaks, satellite of the one year period frequency, are present. The case of the (1-6) keV energy interval can be studied only for DAMA/LIBRA-phase2 and is shown in Ref. [12]; as previously, the only significant peak is that corresponding to one year period. No other peak is statistically significant being below the area obtained by Monte Carlo procedure. In conclusion, apart from the peak corresponding to a 1 year period, no other peak is statistically significant either in the low and in the high energy regions. In addition, for each annual cycle of DAMA/LIBRA-phase1 and -phase2, the annual baseline counting rates have been calculated for the (2-6) keV energy interval. Their power spectrum in the frequency range 0.0002 — 0.0018 d-1 (corresponding to a period range 13.7-1.5 year) is reported in Fig. 2.6; for comparison the power spectrum (solid black line) above 0.0022 d-1 of Fig. 2.5 is shown. The calculation has been performed according to Ref. [5]. No statistically-significant peak is present at frequencies lower than 1 y-1. This implies that no evidence for a long term modulation in the counting rate is present. 2.4 The modulation amplitudes by maximum likelihood approach The annual modulation present at low energy can also be pointed out by depicting the energy dependence of the modulation amplitude, Sm (E), obtained by maximum likelihood method considering fixed period and phase: T =1 yr and t0 = 152.5 day. For such purpose the likelihood function of the single-hit experimental data in the k—th energy bin is defined as: where Nj is the number of events collected in the i-th time interval (hereafter 1 day), by the j-th detector and in the k-th energy bin. Nj follows a Poisson's distribution with expectation value: The bjk are the time-independent background contributions that depend on the energy and on the detector, Mj is the mass of the j-th detector, Ati is the detector running time during the i-th time interval, AE is the chosen energy bin, ejk is the overall efficiency. (2.1) Mdjk = [bjk + Si(Ek)] MjAtiAEejk . (2.2) The signal can be written as: Si(E) = So(E) + Sm(E) • cos w(tt - to) , (2.3) where S0(E) is the constant part of the signal and Sm(E) is the modulation amplitude. The usual procedure is to minimize the function = —2ln(Lk) — const for each energy bin; the free parameters of the fit are the twenty-five (one for each detector) = (bjk + S0) contributions and the Sm parameter. Energy (keV) Fig. 2.7. Modulation amplitudes, Sm, for DAMA/LIBRA-phase2 (exposure 1.13 tonxyr) from the energy threshold of 1 keV up to 20 keV (full triangles, blue data points on-line) -and for DAMA/Nal and DAMA/LIBRA-phase1 (exposure 1.33 tonxyr) [4] (open squares, red data points on-line). The energy bin AE is 0.5 keV. The modulation amplitudes obtained in the two data sets are consistent in the (2-20) keV: the x2 is 32.7 for 36 d.o.f., and the corresponding P-value is 63%. In the (2-6) keV energy region, where the signal is present, the x2/d.o.f. is 10.7/8 (P-value = 22%). The modulation amplitudes obtained considering the DAMA/LIBRA-phase2 data are reported in Fig. 2.7 as full triangles (blue points on-line) from the energy threshold of 1 keV up to 20 keV; superimposed to the picture as open squared (red on-line) data points are the modulation amplitudes of the former DAMA/NaI and DAMA/LIBRA-phase1 [4]. The modulation amplitudes obtained in the two data sets are consistent in the (2-20) keV, since the x2 is 32.7 for 36 d.o.f. corresponding to P-value = 63%. In the (2-6) keV energy region, where the signal is present, the X2/d.o.f. is 10.7/8 (P-value = 22%). As shown in Fig. 2.7 positive signal is present below 6 keV also in the case of DAMA/LIBRA-phase2. Above 6 keV the Sm values are compatible with zero; actually, they have random fluctuations around zero, since the x2 in the (6-20) keV energy interval for the DAMA/LIBRA-phase2 data is equal to 29.8 for 28 d.o.f. (upper tail probability of 37%). Similar considerations have been done for DAMA/NaI and DAMA/LIBRA-phase1 where the x2 in the (6-20) keV energy interval is 35.8 for 28 d.o.f. (upper tail probability of 15%) [4]. The modulation amplitudes for the whole data sets: DAMA/NaI, DAMA/ LIBRA-phase1 and DAMA/LIBRA-phase2 are plotted in Fig. 2.8; the data below 2 keV refer only to DAMA/LIBRA-phase2. It can be inferred that positive signal > 0.05 )/a - , where a are the errors associated to the Sm Fig. 2.10. Histograms of the variable values and (Sm) are the mean values of the modulation amplitudes averaged over the detectors and the annual cycles for each considered energy bin (here AE = 0.25 keV). Each panel refers to a single DAMA/LIBRA detector. The entries of each histogram are 232 (the 16 energy bins in the (2-6) keV energy interval of the seven DAMA/LIBRA-phase1 annual cycles and the 20 energy bins in the (1-6) keV energy interval of the six DAMA/LIBRA-phase2 annual cycles), but 152 for the 16th detector (see Ref. [4]). The superimposed curves are gaussian fits. = 0.5 keV). Thus it is possible to conclude that the effect is well shared between internal and external detectors. To evaluate the hypothesis that the modulation amplitudes obtained for each annual cycle are compatible and normally fluctuating around their mean values a x2 test can be applied. The distribution of these modulation amplitudes are reported in Fig. 2.11, where the x2/d.o.f. are also given; they corresponds to upper tail probability of 5.2%, 97%, 25%, 67% and 72%, respectively. In addition to the x2 test, also the run test has been applied (see e.g. Ref. [35]); it verifies the hypothesis 4 5 7 8 DAMA/LIBRA-phasel <:- DAMA/LIBRA-phase2 (1.04 tonxyr) (1.13 tonxyr) > ■S a 0.05 . 1-2 keV; X2/ d.o.f. = 11.0/5 1 1 t 0 : f ' 1 f i i i i i i i i i i i i 0.05 . 2-3 keV; : 1 1 I X2/ d.o.f. = 4.7/12 J i I 0 i t ' F J 1 F ' t ~.............. 0.U5 3-4 keV; : 1 f , X2/ d.o.f. = 14.8/12 1 { 0 : 1 f " i i II i 4 0.05 1 4-5 keV; : 4 I 1 X2/ d.o.f. = 9.4/12 4 i , 4 f } 0 - t ' " 1 1 1 t } 4 II — "f —"I— 1 0.05 1 5-6 keV; X2/ d.o.f. = 8.8/12 , 4 ^ , I 4 0 1—"j—"f — "f" 1 4 ' 4 II i 0 1 2 3 4 5 6 7 8 9 1011 121314 Annual Cycle Fig. 2.11. Modulation amplitudes of each single annual cycle of DAMA/LIBRA-phasel and DAMA/LIBRA-phase2. The error bars are the la errors. The dashed horizontal lines show the central values obtained by best fit over the whole data set. The x2 test and the run test accept the hypothesis at 95% C.L. that the modulation amplitudes are normally fluctuating around the best fit values. that the positive (above the mean value) and negative (under the mean value) data points are randomly distributed. The lower (upper) tail probabilities obtained by the run test are: 70(70)%, 50(73)%, 85(35)%, 88(30)% and 88(30)%, respectively; this confirms that the data collected in all the annual cycles with DAMA/LIBRA-phasel and phase2 are statistically compatible and can be considered together. 2.5 The phase of the measured modulation effect In order to investigate the phase of the annual modulation effect, it is useful to write the the signal as: Si(E) = So(E) + Sm(E) cos - to) + Zm(E) sin w(ti - to) (2.4) = So(E)+ Ym(E) cos w(tt -t*) releasing the assumption of a fixed phase at t0 = 152.5 day. For DM induced signals: i) Zm ~ 0 (because of the orthogonality between the cosine and the sine functions); ii) Sm ~ Ym; iii) t* ~ t0 = 152.5 day. In fact, these conditions hold for most of the dark halo models with some exceptions (see e.g. Ref. [24,25,27-30]). In Fig. 2.12-left the obtained 2a contours in the plane (Sm, Zm) are shown for the (2-6) keV and (6-14) keV energy intervals considering cumulatively the data of DAMA/Nal, DAMA/LIBRA-phasel and DAMA/LIBRA-phase2. In Fig. 2.12-right instead the obtained 2a contours in the plane (Ym, t*) are depicted. Fig. 2.12 also shows - obviously only for DAMA/LIBRA-phase2 - the 2a contours in the (1-6) keV energy interval. The best fit values in the considered cases (1a errors) for Sm versus Zm and Ym versus t* are reported in Table 2.1. gr o.oi N -0.01 2a contours 6-14 keV 1-6 keV m ? 2-6 keV -0.01 0 0.01 Sm (cpd/kg/keV) 240 220 200 180 ) 160 140 120 100 80 6-14 keV 2a contours ^ 1-6 keV 2-6 keV -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 Ym (cpd/kg/keV) Fig. 2.12. 2a contours in the plane (Sm, Zm) (left) and in the plane (Ym,t*) (right) for: i) DAMA/Nal, DAMA/LIBRA-phase1 and DAMA/LIBRA-phase2 in the (2-6) keV and (6-14) keV energy intervals (light areas, green on-line); ii) only DAMA/LIBRA-phase2 in the (1-6) keV energy interval (dark areas, blue on-line). The contours have been obtained by the maximum likelihood method. A modulation amplitude is present in the lower energy intervals and the phase agrees with that expected for DM induced signals. 0 E (keV) Sm (cpd/kg/keV) Zm (cpd/kg/keV) Ym (cpd/kg/keV) t* (day) DAM, 2-6 6-14 A/NaI+DAMA/L (0.0100 ± 0.0008) (0.0003 ± 0.0005) IBRA-phase1+DA -(0.0003 ± 0.0008) -(0.0009 ± 0.0006) MA/LIBRA-phas (0.0100 ± 0.0008) (0.0010 ± 0.0013) e2: (150.5 ± 5.0) undefined DAMA/LIBRA-phase2: 1-6 (0.0105 ± 0.0011) (0.0009 ± 0.0010) (0.0105 ± 0.0011) (157.5 ± 5.0) Table 2.1. Best fit values (1a errors) for Sm versus Zm and Ym versus t*, considering: i) DAMA/Nal, DAMA/LIBRA-phase1 and DAMA/LIBRA-phase2 in the (2-6) keV and (6-14) keV energy intervals; ii) only DAMA/LIBRA-phase2 in the (1-6) keV energy interval. See also Fig. 2.12. The Zm values, obtained in the hypothesis of Sm set to zero in eq. (2.4), are reported in Fig. 2.13 for DAMA/Nal, DAMA/LIBRA-phase1, and DAMA/LIBRA-phase2; they are expected to be zero. The x2 test of the data supports the hypothesis > 0.05 0.025 S a u g -0.025 -0.05 N 10 12 14 16 18 20 Energy (keV) Fig. 2.13. Energy distribution of Zm for DAMA/NaI, DAMA/LIBRA-phase1, and DAMA/LIBRA-phase2 once setting Sm in eq. (2.4) to zero. The energy bin AE is 0.5 keV. The x2 test applied to the data supports the hypothesis that the Zm values are simply fluctuating around zero, as expected. See text. 0 0 2 4 6 8 that the Zm values are simply fluctuating around zero; in fact, in the (1-20) keV energy region the x2/d.o.f. is equal to 44.5/38 corresponding to a P-value = 22%. Fig. 2.14 shows Ym and t* as a function of the energy for DAMA/NaI, DAMA/LIBRA-phase1, and DAMA/LIBRA-phase2. The Ym are superimposed with the Sm values with 1 keV energy bin. As in the previous analyses, an annual modulation effect is present in the lower energy intervals and the phase agrees with that expected for DM induced signals. No modulation is present above 6 keV and the phase is undetermined. 2.6 Further investigation on possible systematic effects and side reactions in DAMA/LIBRA-phase2 The DAMA/LIBRA-phase2 results - as those of DAMA/LIBRA-phasel and DAMA/NaI - fulfill the requirements of the DM annual modulation signature and investigations on absence of any significant systematics or side reaction effect are already present in the previous sections; however, here the topic is further addressed. Sometimes naive statements are put forwards as the fact that in nature several phenomena may show annual periodicity. However, the point is whether they might mimic the annual modulation signature, i.e. whether they might be not only able to quantitatively account for the observed modulation amplitude but also to contemporaneously satisfy all the requirements of the DM annual modulation signature. This was deeply investigated in the former DAMA/NaI and DAMA/LIBRA-phasel experiments (see e.g. Ref. [16,17,2] and references therein; no one able to mimic the signature has been found or suggested by anyone so far) and will be further addressed in the following for the present DAMA/LIBRA-phase2 data. Firstly, in order to continuously monitor the running conditions, several pieces of information are acquired with the production data and quantitatively analysed; information on technical aspects of DAMA/LIBRA has been given in Ref. [1], where the sources of possible residual radioactivity have also been analysed. % 0.05 Jk 0.025 a 0 "s -0.025 ITs J -0.05 0 2 4 6 8 10 12 14 16 18 20 Energy (keV) 200 « S 150 s- 100 12345678 Energy (keV) Fig. 2.14. Top: Energy distributions of Ym (light data points; red on-line) and of the Sm variable (solid data points; black on-line) for DAMA/NaI, DAMA/LIBRA-phase1, and DAMA/LIBRA-phase2. Here, unlike the data of Fig. 2.8, the energy bin is 1 keV. Bottom: Energy distribution of the phase t* for DAMA/NaI, DAMA/LIBRA-phase1, and DAMA/LIBRA-phase2; here the errors are at 2u. The vertical scale spans over ± a quarter of period around 2 June; other intervals are replica of it. The phase agrees with that expected for DM induced signals at low energy. No modulation is present above 6 keV and thus the phase is undetermined. LIBRA-phase2-2 LIBRA-phase2-3 LIBRA-phase2-4 LIBRA-phase2-5 LIBRA-phase2-6 LIBRA-phase2-7 Temperature (°C) (0.0012 ± 0.0051) -(0.0002 ± 0.0049) -(0.0003 ± 0.0031) (0.0009 ± 0.0050) (0.0018 ± 0.0036) -(0.0006 ± 0.0035) Flux (l/h) -(0.15 ± 0.18) -(0.02 ± 0.22) -(0.02 ± 0.12) -(0.02 ± 0.14) -(0.01 ± 0.10) -(0.01 ± 0.16) Pressure (mbar) (1.1 ± 0.9)10-3 (0.2 ± 1.1 )10-3 (2.4 ± 5.4)10-3 (0.6 ± 6.2)10-3 (1.5 ± 6.3) 10-3 (7.2 ± 8.6) 10-3 Radon (Bq/m3) (0.015 ± 0.034) —(0.002 ± 0.050) —(0.009 ± 0.028) —(0.044 ± 0.050) (0.082 ± 0.086) (0.06 ± 0.11) Hardware rate (Hz) -(0.12 ± 0.16)10-2 (0.00 ± 0.12) 10-2 -(0.14 ± 0.22) 10-2 -(0.05 ± 0.22) 10-2 -(0.06 ± 0.16) 10-2 -(0.08 ± 0.17) 10-2 Table 2.2. Modulation amplitudes (1 a error) obtained - for each annual cycle - by fitting the time behaviours of main running parameters including a possible annual modulation with phase and period as for DM particles. These running parameters, acquired with the production data, are: i) the operating temperature of the detectors; ii) the HP Nitrogen flux in the inner Cu box housing the detectors; iii) the pressure of the HP Nitrogen atmosphere of that inner Cu box; iv) the environmental Radon in the inner part of the barrack from which the detectors are however excluded by other two sealing systems (see text and Ref. [1] for details); v) the hardware rate above single photoelectron threshold. All the measured amplitudes are compatible with zero. In particular, all the time behaviours of the running parameters, acquired with the production data, have been investigated. Table 2.2 shows the modulation amplitudes obtained for each annual cycle when fitting the time behaviours of the values of the main parameters including a cosine modulation with the same phase and period as for DM particles. As can be seen, all the measured amplitudes are well compatible with zero. Let us now enter in some more details. 2.6.1 The temperature The full experiment is placed underground and works in an air-conditioned environment; moreover, the detectors have Cu housing in direct contact with the multi-tons metallic passive shield whose huge heat capacity definitively assures a relevant stability of the detectors' operating temperature [1]. Nevertheless the operating temperature is read out by a probe and stored with the production data, in order to offer the possibility of further investigations. u 1 250 5T Ž 200 -0.1 0 0.1 0 0.2 0.4 (T-)/ r.m.s. of T (oC) Fig. 2.15. left - Distribution of the relative variations of the operating temperature measured during the DAMA/LIBRA-phase2 six annual cycles (histogram); the superimposed curve is a gaussian fit. The standard deviation is 0.2%. Right - Distribution of the root mean square (r.m.s.) detectors' operating temperature variations within periods with the same calibration factors (typically ~ 10 days) during the DAMA/LIBRA-phase2 six annual cycles. The mean value is 0.03 oC. Specific information on the DAMA/LIBRA-phase2 six annual cycles can be derived from Fig. 2.15-left; no evidence for any operating temperature modulation has been observed, as also quantitatively reported in Table 2.2. However, to properly evaluate the real effect of possible variations of the detectors' operating temperature on the light output, we consider the distribution of the root mean square temperature variations within periods with the same calibration factors (typically ~ 10 days); this is given in Fig. 2.15-right cumulatively for the DAMA/LIBRA-phase2 data. The mean value of the root mean square of the variation of the detectors' operating temperature is — 0.03 o C and, considering the known value of the slope of the light output <-0.2%/oC, the relative light output variation is < 10-4, that would correspond to a modulation amplitude < 10-4 cpd/kg/keV (that is <0.5% of the observed modulation amplitude). Moreover, for temperature variations the specific requirements of the DM annual modulation signature (such as e.g. the 4th and the 5th) would fail, while they are instead satisfied by the DAMA/LIBRA-phase2 production data. In conclusion, all the arguments given above quantitatively exclude any role of possible effects on the observed rate modulation directly correlated with temperature. For the sake of completeness, we comment that sizeable temperature variations in principle might also induce variations in the electronic noise, in the Radon release from the rocks and in some environmental background; these specific topics will be further addressed in the following. 2.6.2 The noise Despite the good noise identification near energy threshold and the stringent noise rejection procedure which is used [1,6], the role of a possible noise tail in the data after the noise rejection procedure has been quantitatively investigated. The hardware rate of each detector above a single photoelectron, RHj (j identifies the detector), has been considered. Indeed, this hardware rate is significantly determined by the noise. For the proposed purpose the variable: RH = Ij(RHj — (RHj)), can be built; in the present case (RHj) < 0.2 Hz. The time behaviour of RH during each DAMA/LIBRA-phase2 annual cycle is shown in Fig. 2.16. As can be seen in Fig. 2.17, the cumulative distribution of RH for the DAMA/LIBRA-phase2 annual cycles shows a gaussian behaviour with a = 0.3%, that is well in agreement with that expected on the basis of simple statistical arguments. Moreover, by fitting the time behaviour of RH in the six data taking periods -including a modulation term as that for DM particles - a modulation amplitude compatible with zero is obtained: —(0.061 ±0.067) x 10-2 Hz, corresponding to the upper limit: <0.6 x 10-3 Hz at 90% C.L.. Since the typical noise contribution to the hardware rate of each detector is — 0.10 Hz, the upper limit on the noise relative modulation amplitude is given by: 0-62150HZ3hz - 2.4 x 10-4 (90% C.L.). Therefore, even in the worst hypothetical case of a 10% contamination of the residual noise - after rejection - in the counting rate, the noise contribution to the modulation amplitude in the lowest energy bins would be < 2.4 x 10-5 of the total counting rate. This means that a hypothetical noise modulation could account at maximum for absolute amplitudes less than 10-4 cpd/kg/keV. In conclusion, there is no role of any hypothetical tail of residual noise after rejection. DAMA/LIBRA-phase2_2 DAMA/LIBRA-phase2_3 SB 0.2 £ 0.2 7\' a? a? U 0 U 0 V V 5J si S -0.2 Si -0.2 w" .....,.........,.........,...... w" .....,.........,........., 1 1 6200 6300 6400 6500 6600 6700 6800 time (d) time (d) ^ DAMA/LIBRA-phase2_4 ^ DAMA/LIBRA-phase2_5 SB 0.2 SB 0.2 B B X V 0 X V 0 s? s? M -0.2 M -0.2 w" ........,......... .........,........ w" .....,.........,.........,...... 6900 7000 7100 7200 7300 7400 7500 time (d) time (d) DAMA/LIBRA-phase2_6 DAMA/LIBRA-phase2_7 SB 0.2 SB 0.2 B B S ' ° a V 0 - ......' ■".■■'* '* *'''-'* " ''"'* a V 0 5J -J M -0.2 - M -0.2 t4" ....................................... ir ,.........,...... 7600 7900 Fig. 2.16, detector detector 8000 8200 time (d) (RHj)), where RHj is the hardware rate of each above single photoelectron threshold (that is including the noise), j identifies the and (RHj) is the mean value of RHj in the corresponding annual cycle. 7700 7800 time (d) Time behaviours of RH = Zj (RHj 120 100 u & V S 80 60 40 20 0 EjCRgj - ) (Hz) Fig. 2.17. Distribution of RH during the DAMA/LIBRA-phase2 annual cycles (histogram); the superimposed curve is a gaussian fit. 2.6.3 The calibration factor In long term running conditions the periodical calibrations are performed every ~ 10 days with 241 Am source [1]. Although it is highly unlikely that a variation of the calibration factor (proportionality factor between the area of the recorded pulse and the energy), tdcal, could play any role, a quantitative investigation on that point has been carried out. 1200 1000 C?800[ s = £ 600 400 200 0 - 800 - 700 600 - 1500 = - £ 400 - 300 200 1 ■_____J ^...... 100 ", , , e, -0.05 0 0.05 tdcal .1-0.05 0 e 0.05 0.1 HE Fig. 2.18. Left: Distribution of the percentage variations (etdcal) of each energy scale factor (tdcal) with respect to the value measured in the previous calibration (histogram); the standard deviation is 0.5%. Right: Distribution of the percentage variations (ehe) of the high energy scale factor with respect to the mean values (histogram); the standard deviation is 0.6%. The panels refer to the DAMA/LIBRA-phase2 annual cycles and the superimposed curves are gaussian fits. For this purpose, we define the percentage variation of each energy scale factor (tdcal) with respect to the value measured in the previous calibration: etdcal = tdcatldkcatldkc_alk-1 (here tdcalk is the value of the calibration factor in the k-th calibration). The distribution of etdcal for all the detectors during the DAMA/LIBRA-phase2 annual cycles is given in Fig. 2.18-Left. This distribution shows a gaussian behaviour with ct ~ 0.5%. Since the results of the routine calibrations are properly taken into account in the data analysis, such a result allows us to conclude that the energy calibration factor for each detector is known with an uncertainty C 1 % during the data taking periods. Moreover, the distribution of the percentage variations (eHE) of the high energy scale factor with respect to the mean values for all the detectors and for the DAMA/LIBRA-phase2 annual cycles is reported in Fig. 2.18-right. Also this distribution shows a gaussian behaviour with ct ~ 0.6%. As also discussed in Ref. [2,15,16], the possible variation of the calibration factor for each detector during the data taking would give rise to an additional energy spread (ocal) besides the detector energy resolution (ffres). The total energy spread can be, therefore, written as: a = ^Jo^es + o2al — ffres • [1 + 1 • ()2]; clearly the contribution due to the calibration factor variation is negligible since 2 • (f^^^/E )2 <7.5 x 10-4 20EeV (where the adimensional ratio 20EeV accounts for the energy dependence of this limit value). This order of magnitude is confirmed by a MonteCarlo calculation, which credits - as already reported in Ref. [2,15,16] -a maximum value of the effect of similar variations of tdcal on the modulation amplitude equal to 1 — 2 x 10-4 cpd/kg/keV. Thus, also the unlikely idea that the calibration factor could play a role can be safely ruled out. 2.6.4 The efficiencies The behaviour of the overall efficiencies during the whole data taking periods has been investigated. Their possible time variation depends essentially on the stability of the efficiencies related to the adopted acceptance windows; they are regularly measured by dedicated calibrations [1]. In particular, Fig. 2.19 shows the percentage variations of the efficiency values in the (1-8) keV energy interval for DAMA/LIBRA-phase2. They show a gaussian distribution with a = 0.3%. Moreover, we have verified that the time behaviour of these percentage variations does not show any modulation with period and phase expected for a possible DM signal. In Table 2.3 the modulation amplitudes of the efficiencies in each energy bin between 1 and 10 keV are reported, showing that they are all consistent with zero. In particular, modulation amplitudes -considering the six DAMA/LIBRA-phase2 annual cycles all together - equal to — (0.10 ± 0.32) x 10-3 and (0.00 ± 0.41) x 10-3 are found for the (1-4) keV and (4-6) keV energy bins, respectively; both consistent with zero. Thus, also the unlikely idea of a possible role played by the efficiency is ruled out. Energy (keV) LIBRA-ph2-2 M LIBRA-ph2-3 odulation am] LIBRA-ph2-4 jlitudes (x10 LIBRA-ph2-5 3) LIBRA-ph2-6 LIBRA-ph2-7 1-4 4-6 6-8 8-10 -(0.8 ± 0.7) (0.9 ± 1.0) (0.8 ± 0.8) -(0.3 ± 0.6) (0.7 ± 0.8) (0.9 ± 1.0) -(0.7 ± 0.7) -(0.5 ± 0.5) (0.9 ± 0.8) -(1.3 ± 1.0) (0.6 ± 0.8) -(0.5 ± 0.5) -(1.3 ± 0.8) (0.5 ± 1.0) -(0.1 ± 0.8) -(0.3 ± 0.5) -(0.1 ± 0.8) -(1.0 ± 1.1) -(1.1 ± 0.8) (0.4 ± 0.6) (0.2 ± 0.8) -(0.2 ± 1.0) (0.5 ± 0.8) (0.3 ± 0.6) Table 2.3. Modulation amplitudes obtained by fitting the time behaviour of the efficiencies including a cosine modulation with phase and period as for DM particles for the DAMA/LIBRA-phase2 annual cycles. 2.6.5 The background In order to verify the absence of any significant background modulation, the energy distribution measured during the data taking periods in energy regions not 180: 160: 140 120 0 - 1 100 1 §■ : Oi - fi 80 60: 40: 20 1 0 -0.05 0 0.05 (e-)/ Fig. 2.19. Percentage variations of the overall efficiency values with the respect to their mean values for DAMA/LIBRA-phase2 (histogram); the superimposed curve is a gaussian fit. of interest for DM detection has been investigated. The presence of background (of whatever nature) modulation is already excluded by the results on the measured rate integrated above 90 keV, R90, as a function of the time; the latter one not only does not show any modulation, but allows one to exclude the presence of a background modulation in the whole energy spectrum at a level some orders of magnitude lower than the annual modulation observed in the single-hit events in the (1-6) keV energy region. A further relevant support is given by the result of the analysis of the multiple-hit events which independently proofs that there is no modulation at all in the background event in the same energy region where the single-hit events present an annual modulation satisfying all the requirements of the DM annual modulation signature. These results obviously already account for whatever kind of background including that possibly induced by neutrons, by Radon and by side reactions. ... more on Radon The DAMA/LIBRA detectors are excluded from the air of the underground laboratory by a 3-level sealing system [1]; in fact, this air contains traces of the radioactive Radon gas (222Rn - T^2 = 3.82 days - and of 220Rn - T^2 = 55 s - isotopes, which belong to the 238U and 232Th chains, respectively), whose daughters attach themselves to surfaces by various processes. In particular: i) the walls, the floor and the top of the inner part of the installation are insulated by Supronyl (permeability: 2 x 10-11 cm2/s [36]) and a large flux of HP Nitrogen is released in the closed space of this inner part of the barrack housing the set-up. An Oxygen level alarm informs the operator before entering it, when necessary; ii) the whole passive shield is sealed in a Plexiglas box and maintained continuously in HP Nitrogen atmosphere in slight overpressure with respect to the environment as well as the upper glove box for calibrating the detectors; iii) the detectors are housed in an inner sealed Cu box also maintained continuously in HP Nitrogen atmosphere in slight overpressure with respect to the environment; the Cu box can enter in contact only with the upper glove box - during calibrations - which is also continuously maintained in HP Nitrogen atmosphere in slightly overpressure with respect to the external environment. Notwithstanding the above considerations, the Radon in the installation outside the Plexiglas box, containing the passive shield, is continuously monitored; it is at level of sensitivity of the used Radon-meter as reported in Fig. 2.20. Table 2.2 has already shown that no modulation of Radon is present in the environment of the set-up; moreover, the detectors are further isolated by the other two levels of sealing [1]. £ a o -a « © a o -a « 7700 7800 time (d) time (d) Fig. 2.20. Time behaviours of the Radon in the inner part of the barrack (from which - in addition - the detectors are further isolated by other two levels of sealing [1]) during the DAMA/LIBRA-phase2 annual cycles. The measured values are at the level of sensitivity of the used Radon-meter. In Fig. 2.21 the distributions of the relative variations of the HP Nitrogen flux in the inner Cu box housing the detectors and of the pressure of it are shown as measured during the DAMA/LIBRA-phase2 annual cycles (the typical flux mean value for each annual cycle is of order of ~ 320 l/h and the typical overpressure mean value is of order of 3.1 mbar). (flux-)/ pressure (mbar) Fig. 2.21. Distributions of the HP Nitrogen flux in the inner Cu box housing the detectors and of the pressure of it as measured during the DAMA/LIBRA-phase2 annual cycles (histograms); the superimposed curves are gaussian fits. For clarity the HP Nitrogen flux has been given in terms of relative variations. Possible Radon trace in the HP Nitrogen atmosphere inside the Cu box, housing the detectors, has been searched through the double coincidences of the gamma-rays (609 and 1120 keV) from 214Bi Radon daughter, obtaining an upper limit on the possible Radon concentration in the Cu box HP Nitrogen atmosphere: < 5.8 x 10-2 Bq/m3 (90% C.L.) [2]. Thus, a rate roughly < 2.5 x 10-5 cpd/kg/keV can be expected from this source at low energy. This shows that even an hypothetical, e.g. 10%, modulation of possible Radon in the HP Nitrogen atmosphere of the Cu box, housing the detectors, would correspond to a modulation amplitude < 2.5 x 10-6 cpd/kg/keV (< 0.01 % of the observed modulation amplitude). Moreover, it is worth noting that, while the possible presence of a sizeable quantity of Radon nearby a detector would forbid the investigation of the annual modulation signature (since every Radon variation would induce both the variation in the whole energy distribution and the continuous pollution of the exposed surfaces by the non-volatile daughters), it cannot mimic the DM annual modulation signature in experiments such as the former DAMA/Nal and DAMA/LIBRA-phasel and the present DAMA/LIBRA-phase2 which record the whole energy distribution; in fact, possible presence of Radon variation can easily be identified in this case and some of the six requirements of the DM annual modulation signature would fail. In conclusion, no significant role is possible from the Radon. ... more on side processes As mentioned, possible side reactions have also been carefully investigated and none able to mimic the exploited signature is available; previous results on the topics hold (see e.g. Ref. [5], and references therein). In particular, the case of neutrons, muons and solar neutrinos has been discussed in details in Ref. [7,8], where it has been demonstrated that they cannot give any significant contribution to the DAMA annual modulation result. Table 2.6.5 summarizes the safety upper limits on the contributions to the observed modulation amplitude due to the total neutron flux at LNGS, either from (a, n) reactions, from fissions and from muons and solar-neutrinos interactions in the rocks and in the lead around the experimental set-up; the direct contributions of muons and solar neutrinos are reported there too. Not only the limits are quantitatively marginal, but none of such contributions is able to simultaneously satisfy all the requirements of the exploited signature. Other arguments can be found in Ref. [1-4,7,5,11,8,16,17,15]. Source ®°nk nk tk (neutrons cm 2 s ') Ro,k Ak = Ro,knk (cpd/kg/keV) (cpd/kg/keV) Ak/SSp thermal n (10-2 — 10"' eV) SLOW neutrons epithermal n (eV-keV) 1.08 x 10-6 - 0 -however ^ 0.1 2 x 10-6 - 0 -however ^ 0.1 <8 x 10"6 < 8 x 10"' <3 x 10"3 < 3 x 10"4 < 7 x 10"5 < 0.03 fission, (a, n) —> n (1-10 MeV) —> n from rock FAST (>10 MeV) neutrons —> n from Pb shield (> 10 MeV) v —> n (few MeV) - 0.9 x 10-7 - 0 -however ^ 0.1 - 3 x 10-9 0.0129 end of June - 6 x 10-9 0.0129 end of June - 3 x 10-'° 0.03342* Jan. 4th* <6 x 10"4 < 6 x 10"5 < 5 x 10"4 < 7 x 10"6 < 1.1 x 10"3 < 1.4 x 10"5 < 5 x 10"5 < 1.8 x 10"6 < 5 x 10"3 < 6 x 10"4 < 1.3 x 10"3 < 1.6 x 10"4 direct direct v O^ - 20 m m-2d-1 0.0129 end of June ®°v) - 6 x 10'° v cm-2s-1 0.03342* Jan. 4th* - 10"7 - 10"9 - 10"5 3 x 10"7 - 10"7 3 x 10"5 Table 2.4. Summary of the contributions to the total neutron flux at LNGS; the value, the relative modulation amplitude, r|k, and the phase, tk, of each component is reported. It is also reported the counting rate, R0)k, in DAMA/LIBRA-phase2 for single-hit events, in the (1 — 6) keV energy region induced by neutrons, muons and solar neutrinos, detailed for each component. The modulation amplitudes, Ak, are reported as well, while the last column shows the relative contribution to the annual modulation amplitude observed by DAMA/LIBRA-phase2, Smp - 0.011 cpd/kg/keV. For details see Ref. [8] and references therein. * The annual modulation of solar neutrino is due to the different Sun-Earth distance along the year; so the relative modulation amplitude is twice the eccentricity of the Earth orbit and the phase is given by the perihelion. 2.6.6 Conclusions on possible systematics effects and side reactions No modulation has been found in any possible source of systematics or side reactions; thus, upper limits (90% C.L.) on the possible contributions to the DAMA/LIBRA-phase2 measured modulation amplitude are summarized in Table 2.5. In particular, they cannot account for the measured modulation both because quantitatively not relevant and unable to mimic the observed effect. Source Main comment (see also Ref. [1]) Cautious upper limit (90%C.L.) Radon Sealed Cu Box in HP Nitrogen atmosphere, 3-level of sealing < 2.5 x 10-6 cpd/kg/keV Temperature Air conditioning + huge heat capacity <10-4 cpd/kg/keV Noise Efficient rejection <10-4 cpd/kg/keV Energy scale Routine + intrinsic calibrations <1 - 2 x10-4 cpd/kg/keV Efficiencies Regularly measured <10-4 cpd/kg/keV Background No modulation above 6 keV; no modulation in the (1 - 6) keV multiple-hit events; this limit includes all possible sources of background <10-4 cpd/kg/keV Side reactions From muon flux variation measured by MACRO <3 x 10-5 cpd/kg/keV In addition: no effect can mimic the signature Table 2.5. Summary of the results obtained by investigating possible sources of systematics or of side reactions in the data of the DAMA/LIBRA-phase2 annual cycles. None able to give a modulation amplitude different from zero has been found; thus cautious upper limits (90% C.L.) on the possible contributions to the measured modulation amplitude have been calculated and are shown here. 2.7 Conclusions The data of the new DAMA/LIBRA-phase2 confirm a peculiar annual modulation of the single-hit scintillation events in the (1-6) keV energy region satisfying all the many requirements of the DM annual modulation signature; the cumulative exposure by the former DAMA/NaI, DAMA/LIBRA-phase1 and DAMA/LIBRA-phase2 is 2.46 ton x yr. As required by the exploited DM annual modulation signature: 1) the singlehit events show a clear cosine-like modulation as expected for the DM signal; 2) the measured period is equal to (0.999 ± 0.001 ) yr well compatible with the 1 yr period as expected for the DM signal; 3) the measured phase (145 ± 5) days is compatible with the roughly ~ 152.5 days expected for the DM signal; 4) the modulation is present only in the low energy (1-6) keV interval and not in other higher energy regions, consistently with expectation for the DM signal; 5) the modulation is present only in the single-hit events, while it is absent in the multiple-hit ones as expected for the DM signal; 6) the measured modulation amplitude in NaI(Tl) target of the single-hit scintillation events in the (2-6) keV energy interval, for which data are also available by DAMA/Nal and DAMA/LIBRA-phasel, is: (0.0103 ± 0.0008) cpd/kg/keV (12.9 a C.L.). No systematic or side processes able to mimic the signature, i.e. able to simultaneously satisfy all the many peculiarities of the signature and to account for the whole measured modulation amplitude, has been found or suggested by anyone throughout some decades thus far. In particular, arguments related to any possible role of some natural periodical phenomena have been discussed and quantitatively demonstrated to be unable to mimic the signature (see e.g. Ref. [7,8]). Thus, on the basis of the exploited signature, the model independent DAMA results give evidence at 12.9 a C.L. (over 20 independent annual cycles and in various experimental configurations) for the presence of DM particles in the galactic halo. In order to perform corollary investigation on the nature of the DM particles in given scenarios, model-dependent analyses are necessary3; thus, many theoretical and experimental parameters and models are possible and many hypotheses must also be exploited. In particular, the DAMA model independent evidence is compatible with a wide set of astrophysical, nuclear and particle physics scenarios for high and low mass candidates inducing nuclear recoil and/or electromagnetic radiation, as also shown in a wide literature. Moreover, both the negative results and all the possible positive hints, achieved so-far in the field, can be compatible with the DAMA model independent DM annual modulation results in many scenarios considering also the existing experimental and theoretical uncertainties; the same holds for indirect approaches. For a discussion see e.g. Ref. [5] and references therein. Model dependent analyses, to update the allowed regions in various scenarios and to enlarge the investigations to other ones, will be presented elsewhere. Finally, we stress that to efficiently disentangle among the many possible candidates and scenarios an increase of exposure in the new lowest energy bin is important. The experiment is collecting data and related R&D is under way. References 1. R. Bernabei et al., 2. R. Bernabei et al., 3. R. Bernabei et al., 4. R. Bernabei et al., 5. R. Bernabei et al., 6. R. Bernabei et al., 7. R. Bernabei et al., 8. R. Bernabei et al., , Nucl. Instr. and Meth. A 592, 297 (2008). , Eur. Phys. J. C 56, 333 (2008). , Eur. Phys. J. C 67, 39 (2010). , Eur. Phys. J. C 73, 2648 (2013). , Int. J. of Mod. Phys. A 28,1330022 (2013). , J. of Instr. 7, P03009 (2012). , Eur. Phys. J. C 72, 2064 (2012). , Eur. Phys. J. C 74, 3196 (2014). 3 It is worth noting that it does not exist in direct and indirect DM detection experiments approaches which can offer such information independently on assumed models. 9. DAMA coll., issue dedicated to DAMA, Int. J. of Mod. Phys. A 31 (2016) and references therein. 10. for complete references: http://people.roma2.infn.it/dama/web/publ.html 11. R. Bernabei et al., Eur. Phys. J. C 74, 2827 (2014). 12. R. Bernabei et al., arXiv:1805.10486. 13. 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Bonora International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy, and INFN, Sezione di Trieste Abstract. It is shown that in a flat background one can define higher spin (HS) gauge theories with an infinite number of fields. In particular here HS YM-like in any dimension and HS CS-like theories in any odd dimension are introduced and analyzed. They are invariant under HS gauge transformations which include ordinary U(1) gauge transformations and diffeomorphisms. It is also shown how to recover local Lorentz invariance. The action, equations of motion and conserved currents in the HS YM-like theories are explicitly exhibited. Povzetek. Avtor v prispevku pokaZe, da lahko definira na ravnem ozadju umeritvene teorije visjih spinov z neskončnim stevilom polj. Kot poseben primer uvede in analizira teorije Yang-MIllsovega tipa z višjim spinom v poljubni dimenziji in teorije Cherna-Simmonsa z visjim spinom v poljubni lihi dimenziji. Te teorije so invariantne na umeritvene transfo-macije za visje spine, ki vkljucujejo obicajne transformacije U(1) in difeomorfizme. Pokazze se, kako znova vpeljati lokalno Lorentzovo invarianco. V teorijah Yang-Millsovega tipa z visšjim spinom zapisše akcijo ter enacšbe gibanja in ohranitvene tokove. Keywords: Higher spin theories, Yang-Mills like theories, Chern-Simmons theories, flat spacetime 3.1 Introduction... There are compelling motivations for research to study spin (HS) theories, that is theories with an infinite number of fields with increasing spin. In a theory that unifies all the forces of nature such a feature seems to be inevitable. First (super)string theories have this characteristic. It is well known that the infinite number of fields with increasing spins is related to their good UV behavior. Also the AdS/CFT correspondence indicates that if we wish to resolve the singularities of the theory on the boundary we have to turn to the dual theory, which is a (super)string theory. Other arguments suggest that, when gravity is involved, infinite many local fields of increasing spins are needed in order to avoid possible conflicts with causality [1]. Starting from on these general motivations, in this contribution I will focus on a specific problem, for which for a long time there have been no answers, or only negative ones, in the literature: can one formulate a sensible local massless HS theory in a flat space-time? The standard lore in the literature may be summarized by two objections: first, there are the so-called no-go theorems, which prevent the existence of such theories under rather general conditions; second, the construction of massless HS theories has been so far only successful in AdS spaces. However here I will exhibit examples of HS theories defined in flat spacetime in any dimension, which are massless, gauge invariant and, at least classically, consistent. In [3] and, later on, in [4,7] a method has been proposed to produce HS effective actions by integrating out matter fields coupled to external potentials and quantized according to the worldline quantization. The method consists in computing current correlators, see [5,6], and explicitly determine the effective action. Barring anomalies, we are guaranteed that the result is HS gauge invariant. Unfortunately the method is very cumbersome and the resulting effective action is not guaranteed to be local. In this paper I would like to show that there exists a shortcut. Exploiting the analogy of the HS gauge transformations with the gauge transformations in ordinary non-Abelian gauge theories, one can construct analogous local HS invariants and covariant objects, and in particular actions. In this way one can define (perturbatively) local HS Yang-Mills theories in any dimension and HS Chern-Simons theories in any odd dimension. I will focus in particular on the former. They are characterized by a coupling constant, like the ordinary YM theories. I will show how to define the action, their equations of motion and their conserved quantities. The HS gauge transformations contains in particular the ordinary U(1) gauge transformations and the diffeomorphisms. They do not include the local Lorentz transformations. Since the HS YM-like theories are formulated in a frame-like formalism, local Lorentz transformations are relevant in order to permit their gravitational interpretation. Below I will show how to local Lorentz invariance is hidden in the formalism and how to recover it. 3.2 Higher spin effective action This section is devoted to a concise presentation of the effective action method. The effective action here is defined via the worldline quantization method. This method consists, roughly speaking, in considering the coordinates on which the field depends, as the position of a quantum particle, while the latter is quantized according to the Weyl-Wigner quantization. Let us consider a free fermion theory coupled to external sources. According to the Weyl quantization method for a particle worldline, the full action is expressed as an expectation value of operators So = — m)^ d. (3.1) S = M- Ya (Pa — H a)— m|^> (3.2) We recall that a quantum operator O can be represented with a symbol O (x, u) through the Weyl map O = ddk ddu 0(x,u) eik-(x-X)-iy (u-P) ddx ddy (2n)d (2n)d (3.3) where X is the position operator. The symbol of the product of two operators is the * product (or Moyal product) of the corresponding symbols. In (3.2) Pa is the momentum operator whose symbol is the classical momentum Ua1 . H a is an operator whose symbol is ha (x, u), where ha (x, u) — Y_ — K1".^ (x) u^, ...u. (3.4) n=0 s = n + 1 is the spin and the tensors are assumed to be symmetric in ^ ,..., Any field like ha (x, u), which depends also from the momentum u, will be referred to as master field. One should notice that there are two kind of labels a and They will be interpreted later as flat and curved indices, respectively, but in a flat background they play the same role. Their true nature will illustrated later on. Now one makes the above formalism explicit in (3.2), where we also insert two completenesses J* ddx|x)(x|, and make the identification M(x) = (x|M). Expressing S in terms of symbols one finds S = So + ddu (2n)d ddx ddz eiuz M x + - y • h(x, u) M x - - 2 (3.5) = So + X ddxJ^.^ (x) hS1)"^* (x) s = 1 J The tensor field hS1 ."Sn is linearly coupled to the HS current a n 3ZS1 ...^ M(x + dYaM(x JS,...sn (x) = For instance, for s = 1 and s = 2 one obtains z=0 jai) = MYaM ja2S, — 22 (9(S1 MYa)M - MYa3s,)M) The HS currents are on-shell conserved in the free theory (3.1) p. TaSl •••S* —1 _ a °aj(s) — 0 (3.6) (3.7) (3.8) (3.9) Throughout the paper the position in the phase space are denoted by couples of letters (x, u), (y, v), (z, t), (w, r), the first letter refers to the space-time coordinate and the second the the momentum of the worldline particle. The letters k, p, q will be reserved to the momenta of the (Fourier-transformed) physical amplitudes. 3.2.1 HS gauge symmetries The action (3.2) is trivially invariant under the operation S = (^|O O-1 G O O-1(3.10) where G = —■y • (P — H) — m. So it is invariant under G —> O-1GO, —> O-1|^) (3.11) Writing O = e-iE we easily find the infinitesimal version. 6|^) = ip|^), 6(^| = — i(^|p, (3.12) and Sp = i[p ,G] = iY (G — H) ,p] = y^H (3.13) Let the symbol ofp be e(x,u), then the symbol of [iy •p,p] is ddy(x — -2 |[iy •p p]|x + |) eiyu = —iy-3xe(x,u) (3.14) Similarly Symb([Ha,p]) = [ha(x,u) * e(x,u)j (3.15) where [a * b] = a * b — b * a is the ^-commutator. Therefore, in terms of symbols, 5eha(x,u) = 3ae(x,u) — i[ha(x,u) * e(x,u)] =DaX£(x,u) (3.16) where the covariant derivative defined by Dax = 3a — i[ha(x,u) * ] (3.17) has been introduced. The variation in eq.(3.16) will be referred to hereafter as HSgauge transformation, and the corresponding symmetry HS gauge symmetry. For the transformations of see [4] . It is easy to see that the conservation law in the classical interacting theory DXa Ja(x,u)= 0 (on — shell) (3.18) follows from the above. Using the *-Jacobi identity (which holds also for the Moyal product, because the latter is associative) one can easily get (5£2 6£1 — 6£1 5£2 ) h^(x, u) = i (3a[£1 * £2](x,u)— i[ha(x,u) * [£1 * £2](x,u)]]) = i Dax[£1 *£2 ](x*u) (3.19) i.e. the HS £-transform is of Lie algebra type. 3.2.2 The HS effective action The general formula for the effective action is W[h] = W[0] + £ — n ddxi n=i 1=1 ddut (2n)d X hai (xi,Ui) ...han (Xn,Un) (3.20) where Mn)...,an (xi, ui,..., xn, Un, e) are the n-point functions of the currents Ja, (x1 ,u1),..., Jan (xn,un). W[0] is the constant 0-point contribution, which will be disregarded in the sequel. There are various ways to compute these amplitudes. The most popular is by means of Feynman diagrams. For instance, the 3-point function can be calculated via the Feynman diagram integral (Ja, (xi ,Ui )Ja2 (X2,U2) Ja3 (x3,U3)) ' ddqi ddq2 pi(q1+q2).x1 (2n)d (2n)d — iqi -X2 e-iq2-x3 x Mui -2p - q - ^u2 -^ua -2p - 2q - q2 2 ddp / 1 tr Yai —-Ya2 (2n) p + m Ya2 p - /1 + mYa3 p - q - / 2 + ml , (3.21) to which one must add the cross term. qi, q2 are the momenta of two external outgoing legs. The third one has incoming momentum qi + q2. These amplitudes have cyclic symmetry. The invariance of the effective action under (3.16) is expressed by 0= sew[h] = x ^J n ddxi (dnd (3.22) X Wan)...,an(xi,Ui , . . . ,Xn,Un) D*^1 e(xi ,Ui ) ha2 (x2,U2) . . . han (xu,Un) The generalized equations of motion are obtained by varying W[h] with respect to the master field ha (x, u) . Let us write them in the compact form Fa (x,u) = 0 (3.23) where Fa K u) = Y_ nrr Et ddxi 72nrd waa+l.an Cx, u, xi, ui,..., xn un e) ,u) = > — || d~x n=0 — J i=i ddu1 (2n)d X hai (xi,ui) ...han (xn,un) The EoM's (3.23) are covariant under HS gauge transformation 5eFa(x,u) = i[e(x,u) , Fa(x,u)] (3.24) 1 1 X 3.3 Yang-Mills-like theories 3.3.1 The gauge transformation in the fermion model Let us return to the gauge transformation (3.16) 5eha(x,u) = 3ae(x,u) -i[ha(x,u) , e(x,u)j =D2M*,u) (3.25) and write it down in components. To avoid a proliferation of numerical indices, let us write the expansion of ha (x, u) as ha(x,u) = Aa(x) + xa(x)u^ + ^b^U^Uy + 1C^vAU^UvUA + ... (3.26) As noted above we use two different types of indices. In the expansion (3.4) the indices m,..., are upper (contravariant), as it should be, because in the Weyl quantization procedure the momentum has lower index, since it must satisfy [xa,pv] = iSa. The index a instead is traditionally reserved for a flat index. Of course when the background metric is flat the indices a and ^t are on the same footing, but it is useful to keep them distinct. Let us see why. For the HS gauge parameter we write e(x,u) = e(x) + + ^A^u^ + 3 !^vAu^uvuA + ... (3.27) The transformation (3.25) to the lowest order reads, SAa = 3a£ + ^-3Aa - 9pexa + ... (3.28) Sxa = 3a^v + £• 9xa - 9p^vxa + 9PAaApv - 3AebaAv + ... SbVA = 3aAvA + £^3bavA - 3p£/bapA - 3p£Abapv + Spx^ + 3pxiApV -xa3pAvA +... The next nontrivial order contains terms with three derivatives, and so on. It is natural to compare the previous HS gauge variations with the ordinary gauge, diff,... transformations. To this end let us denote by Aa the standard U(1) gauge field and by e£ = S£ - xa the standard inverse vielbein, and let us restrict the previous general transformation to gauge and diff transformations alone. We have SAa = S (gaAa) = S ((6S - xa) Aa) (3.29) = (-£,-0xa + 0A£axca) Aa + (Sa - xa) + ^Aa) « 3ae + i/Aa - xc^da6 and Sea = S(Sa -xa) = ^ ^ea - aA^ea = •xa - 9ae+9A£,axa (3.30) so that 5jca = S + - a (3.31) where we have retained only the terms at most linear in the fields. From the above we see that the natural identifictions are Aa = A a, xa = X a (3.32) The transformations (3.28) are consistent with the ordinary gauge and diffeo-morphism transformations. Therefore the master field ha can describe in particular the geometry of the gauge theories and the geometry of gravity. The above does not explain the nature of the index a. It is natural to interpret it as a flat index, but this calls for local Lorentz symmetry. This issue will be resumed later on. 3.3.2 Analogy with gauge transformations in gauge theories It should be remarked that in eq.(3.25) and (3.28) the derivative 9a means 9a — 6^3^, not 9a — e^S^ — (e£ — Xa + ...) 9a. In fact the linear correction —xa9a is contained in the term —i[ha(x,u) , e(x,u)], see for instance the second term in the RHS of the first equation (3.28). The obvious remark is that the transformation (3.25) looks similar to the ordinary gauge transformation of a non-Abelian gauge field 6AAa — 3aA +[Aa,A] (3.33) where Aa — A^Ta, A — AaTa, Ta being the Lie algebra generators. In gauge theories it is useful to represent the gauge potential as a connection one form A — Aadxa, so that (3.33) becomes 6aA — dA + [A, A] (3.34) We can do the same for (3.25) 6eh(x,u) — de(x,u) — i[h(x,u) , e(x,u)] = De(x,u) (3.35) where d — 9a dxa, h — hadxa and xa are coordinates in the tangent spacetime, and it is understood that [h(x,u) , e(x,u)] — [ha(x,u) , e(x,u)]dxa We will apply this formalism to the construction of HS CS or YM-like actions. 3.3.3 HS Yang-Mills action In analogy with the ordinary Yang-Mills theory one can introduce the curvature 2-form G — dh — 2[h , h], (3.36) whose components are Gab — 3ahb — 9bha — i[ha * hb] (3.37) Their transformation rule is 5£Gab =-i[Gab , e] (3.38) Next we will consider functionals which are integrated polynomials of G or of its components Gab. In order to exploit the transformation property (3.16) in the construction we need the 'trace property', analogous to the trace of polynomials in ordinary non-Abelian gauge theories. The only object with trace properties we can define in the HS context is ((f * 9» = ddx ddx ddu (2n)d ddu f(x,u) * 9(x,u) f(x,u)9(x,u) = ((9 * f» (3.39) (2n)d From this, plus associativity, it follows that ((f * f2 * ... * fn»» = (-1)ei(£2 + ... + en)^f2 * ... * fn * f, »» (3.40) where et is the Grassmann degree of ft. In particular (([f! , f2 * ... * fn}»» = 0 (3.41) where [ , } is the *-commutator or anti-commutator, as appropriate. This property holds also when the ft are valued in a Lie algebra, provided the symbol (( »» includes also the trace over the Lie algebra generators. Let us return to Gab. From the propery (3.41) it follows that 5e((Gab * Gab»» = —i((Gab * Gab * e - e * Gab * Gab»» = 0 (3.42) Therefore YM(h) = — Jg,((Gab * Gab»» (3.43) is invariant under the HS gauge transformation and it is a well defined functional in any dimension. This construction can be easily generalized to the non-Abelian case, that is when the master field ha is valued in a Lie algebra with generators Ta: ha = h^Ta. See [8] . 3.3.4 HS CS action Using the above properties it is not hard to prove, [7] that CS (h) = n where dt((h * Gt * ... * Gt»» (3.44) o Gt = dht — 2 [ht , ht], ht = th, (3.45) is HS gauge invariant in a space of odd dimension d = 2n — 1. It defines the HS CS action in any odd-dimensional spacetime. з.3.5 Covariant YM-type eom's From(3.43) we get the following eom: 3bGab -i[hb , Gab] =D*Gab = 0 (3.46) which is covariant under the HS gauge transformation 5e (D*Gab) = -i[D*Gab,e] (3.47) In components this equation splits into an infinite set according to the powers of и. Let us expand Gab in the notation of sec.3.3.1. We have Gab = Fab + + leaVu^Uv + 6c^VuvUA + ... (3.48) and express them in terms of the component fields of ha (x, u). For instance, the first eom (O(u0)) is 0 = DAb - 3b3-A + 1 (3CT3-Axb + 3aAa3aXb - 3a3aAbxS - 3^3*a) +^SaA^3ax" - 3bxS + 1 (9AAabba - 3AAbbaa + 3AxSxb - saxEx^)) Ixaf 9a9aAb - 3a3bAa (3.49) 2 +1 (3a3AAaxb + 3AAa3axb - 3a3AAbxaA - 3AAb3axaA) ) +...... The second (»(u1)) □xS - 3a3bx£= 2 (9b(0aAa - 3ffAb + Sax^xa - 3axM (3.50) +3TAb3abbT - 3TAb3bbaT + 3Txbs3axi - 3Txbs3bxa -3T3aAb bbTS + 3T3bAa bbTS - 3T3ax£xbT + 3T3bx£xbT) + ... Ellipses denote terms with a larger number of spacetime derivatives. Let us see a few elementary examples. Consider the case of a pure U(1) gauge field A alone. The equation of motion is 3aFab = DAb - 3b3•A = 0 (3.51) In the 'Feynman gauge' 3 •A = 0 this reduces to DAb = 0. Let us suppose next that only gravity is present. Eq.(3.50) becomes 3aXabs = DxS - 3b3 -xs = 0 (3.52) In the 'Feynman gauge' 3 -xs = 0, (3.52) reduces to DxS = 02. 2 In ordinary gravity (Rsv = 0) we have to impose the DeDonder gauge in order to obtain the same result. Finally, keeping only the spin 3 field the eom becomes 3aBab^v = nbb^ - 3b3abr = 0 (3.53) Again in the 'Feynman gauge' 3abaV = 0 we get IHbb^v = 0. In general we can impose for all the fields the Feynman gauge 3aha (x,u)= 0 (3.54) As is clear from (3.49), for instance, the above eom's are characterized by the fact that at each order, defined by the number of derivatives, there is a finite number of terms. This defines a perturbatively local theory. 3.3.6 Conserved currents The conservation laws of the HS models can be found following the analogy of a current in an ordinary gauge theory or the energy momentum tensor in gravity theories. For instance, if in HS YM we express the invariance of the action under the HS gauge transformation we can write 0 = -4M L(x)D|L-1(x) = L(x)(3| + A|)L-1(x) (3.67) where L(x) is a (finite) LLT. If we choose L = O-1 we get A|(x) -> 0 (3.68) But at this point the LL symmetry is completely fixed. Thus choosing the zero spin connection amounts to fixing the local Lorentz gauge. The connection A^ contains inertial and no gravitational information. It will be referred to as the inertial connection. It is a non-dynamical object (its content is pure gauge). The dynamical degrees of freedom will be contained in the adjoint tensor to be added to A| in order to form a fully dynamical spin connection3. A| is nevertheless a connection and it makes sense to introduce the inertial derivative D| = 3| - 2A| (3.69) which is Lorentz covariant. It is clear that the results ensuing from the effective action method, as well as the HS YM and HS CS theories, are all formulated in a trivial frame setting, eq.(3.62), with a trivial spin connection. In other words the local Lorentz gauge is completely fixed. However from this formalism it is not difficult to recover explicit local Lorentz covariance. 3.4.2 How to recover local Lorentz symmetry Let us restart from the definition of Ja(x, u) Ja(x,u)= Y- 2n+m n!m! 9I ^ M^v, • ••dVn ^(x) n,m=0 gn+m X 9U| • • • 3U|m 3uv, • • • 9uvn 5(u) 9s s = 1 •••uu|s-i ^LHr-1^,..^w9u......9u„ 5(u) (3.70) The splitting of vierbein and spin connection into an inertial and a dynamical part is characteristic of teleparallelism, [9] 70 L. Bonora from which we derive ...,._i (x) = Z ss- — s — 1)| 9(,i ... (x)Ya3^n + 1 ... 3,s-, )^(xp.71) n=0 ( J| Assume now the following LLT ¿Ap =— a = Aab!ab, lab = 4 [Ya,Yb] (3.72) = A and replace in (3.71) the ordinary derivative on p with the inertial covariant derivative 3,^ -> D,^ = ^3, — 2 A, ) i (3.73) and on with Eq.(3.71) becomes 3,i-> D,i| = + A, (3.74) J'aSi,...,._, (x) (3.75) s 1 ' -1)T = z ss-(1(s— 1)| d(,1 . . . d,n p (x)yad,n + 1 . . . n=0 ( )! Now, given 6aA, = — 3,A + T [A,, A] (3.76) and (3.12), it is easy to prove that 6a(D,^) = — 2A(D,P), S(D|» = 2 (D,P)A (3.77) The same holds for multiple covariant derivatives Sa(D,, ... D,^P) = 2a(D,, ... D,^P), etc. It follows that SAJ 'as(i1...,s-1 (x) ■1 _(—1)n sS— n=0 = Aab(x) J'b,1...,s-1 (x) (3.78) = — Z ^1(7—1)! D(,, ... d,nPi(x)lYa,A]D,1+1 ... d,s-, )P(x) Therefore the interaction term CO p Sint = X ddx J' & (x)ha^ (3.79) s = 1 J is invariant under (3.12) and (3.76) provided öAha^ ...^n (x) = A% (x) hb^ ...^n (x) (3.80) On the other hand, writing S ' — So — ddx^ (iYa( 3a - 2 Aa) - m)^ (3.81) instead of S0, also SO turns out to be invariant under LLT. So, provided we define LLT via (3.12) and (3.76), S' = SO + Sint is invariant. Replacing simple spacetime derivatives 3a with the inertial ones Da everywhere is not enough. As pointed out above instead of 3a = we should write 3a = eaa(x)3a, where eaa(x) is a purely inertial frame. Moreover, whenever it appears, we should rewrite Aa(x) = eaa(x)Aa(x). With this new recipes all inconsistencies disappear. For instance MDaJb) = AaC(DcJb)+ AbC(DaJc) Therefore 5A(/nabDaJb) = 0. Likewise 5AGab = AaCGcb + AbCGac (3.82) which implies the local Lorentz invariance of GabG ab Summary. The HS effective action approach fixes completely the local Lorentz gauge. This is due the fact that in its formalism (and in the general in the HS YM and CS formalism) the choice e£ = 5 a and Aa = 0 for the inertial frame and connection, is implicit. However the same formalism offers the possibility to recover the LL invariance by means of a simple recipe: 1. replace any spacetime derivative, even in the * product, with the inertial covariant derivative, 2. interpret any flat index a attached to any object Oa as ea(x)Oa. Anticipating future developments we add that in the process of quantization e£(x) and Aa(x) will be treated as classical backgrounds. 3.5 Conclusions The main message of this paper is that it is possible to construct field teory models of Yang-Mills type with infinite many HS fields in flat spacetime in any dimension. It is also possible to construct similar models of Chern-Simons type in any odd dimensional flat spacetime. We have seen that of such models we can define the actions, invariant under HS gauge transformations, which encompass the ordinary gauge transformations and the diffeomorphisms. It was also shown that although the local Lorentz gauge is fixed in this formalism, local Lorentz invariance can be easily implemented. We can derive sensible eom's. A more detailed account and further developments are contained in related papers[7,8] : for instance one can introduce matter master scalar and fermion fields, and realize the analog of Higgs mechanism; one can also introduce ghosts, and carry out the BRST quantization and develop the practical machinery for perturbative calculations via Feynman diagrams. All these results may be at first surprising, because, as noted in the introduction, there exist no-go (Weiberg-Witten) theorems forbidding the existence of interacting massless HS theories in flat spacetime (for a review see [10]). A full discussion of this problem will be given in [8] . Here let us simply notice that such theorems are based on a set of hypotheses, which are very plausible in ordinary field theories, but can be circumvented in theories like the ones introduced here. For instance two basic requirements are the minimal coupling of the matter fields to gravity and the polynomial structure of the energy-momentum tensor. It turns out that none of these requirements is realized in HS YM-like theories: gravity is non-minimally coupled to HS fields and the energy-momentum tensor is not a polynomial of the fields, but a series. Acknowledgments I would like to thank my collaborators Maro Cvitan, Predrag Dominis Prester, Stefano Giaccari and Tamara T. Stemberga. I would like to thank Norma Susana Mankoc Borstnik and Holger Nielsen for inviting me to give this talk at the Workshop. References 1. X. O. Camanho, J. D. Edelstein, J. Maldacena and A. Zhiboedov, Causality Constraints on Corrections to the Graviton Three-Point Coupling, JHEP 1602, 020 (2016) [arXiv:1407.5597 [hep-th]]. 2. M. A. Vasiliev, Consistent equation for interacting gauge fields of all spins in (3+1)-dimensions, Phys. Lett. B 243, 378 (1990); Properties of equations of motion of interacting gauge fields of all spins in (3+1)-dimensions, Class. Quant. Grav. 8 1387 (1991) ; Algebraic aspects of the higher spin problem, Phys. Lett. B 257,111 (1991); More on equations of motion for interacting massless fields of all spins in (3+1)-dimensions, Phys. Lett. B 285, 225 (1992). 3. X. Bekaert, E. Joung and J. Mourad, "Effective action in a higher-spin background," JHEP 1102 (2011) 048 4. L. Bonora, M. Cvitan, P. Dominis Prester, S. Giaccari, M. Paulisic and T. Stemberga, Worldline quantization of field theory, effective actions and L«, structure,, JHEP 1804, 095 (2018) 5. L. Bonora, M. Cvitan, P. Dominis Prester, S. Giaccari, B. Lima de Souza and T. Stemberga, One-loop effective actions and higher spins, JHEP 1612 (2016) 084. 6. L. Bonora, M. Cvitan, P. Dominis Prester, S. Giaccari, and T. Stemberga, One-loop effective actions and higher spins. II JHEP 1801 (2018) 080. 7. L. Bonora, M. Cvitan, P. Dominis Prester, S. Giaccari and T. Stemberga, HS in flat spacetime. The effective action method, [ArXiv:1811.04847] 8. L. Bonora, M. Cvitan, P. Dominis Prester, S. Giaccari and T. Stemberga, HS in flat spacetime. YM-like theories, to appear. 9. R. Aldrovandi and J. G. Pereira Teleparallel gravity. An introduction, Springer, Dordrecht Heidelberg New York London 2013, and references therein. 10. X. Bekaert, N. Boulanger and P. Sundell, How higher-spin gravity surpasses the spin two barrier: no-go theorems versus yes-go examples, Rev. Mod. Phys. 84 (2012) 987 doi:10.1103/RevModPhys.84.987 [arXiv:1007.0435 [hep-th]]. Bled Workshops in Physics Vol. 19, No. 2 A Proceedings to the 21 st Workshop What Comes Beyond ... (p. 74) Bled, Slovenia, June 23-July 1, 2018 4 Emergent Photons and Gravitons J.L. Chkareuli, J. Jejelava and Z. Kepuladze Center for Elementary Particle Physics, Ilia State University, 0162 Tbilisi, Georgia E. Andronikashvili Institute of Physics, 0177 Tbilisi, Georgia Abstract. Now, it is already not a big surprise that due to the spontaneous Lorentz invariance violation (SLIV) there may emerge massless vector and tensor Goldstone modes identified particularly with photon and graviton. Point is, however, that this mechanism is usually considered separately for photon and graviton, though in reality they appear in fact together. In this connection, we recently develop the common emergent electrograv-ity model which would like to present here. This model incorporates the ordinary QED and tensor field gravity mimicking linearized general relativity. The SLIV is induced by length-fixing constraints put on the vector and tensor fields, A^ = ±MA and = ±MH (Ma and MH are the proposed symmetry breaking scales) which possess the much higher symmetry than the model Lagrangian itself. As a result, the twelve Goldstone modes are produced in total and they are collected into the vector and tensor field multiplets. While photon is always the true vector Goldstone boson, graviton contain pseudo-Goldstone modes as well. In terms of the appearing zero modes, theory becomes essentially nonlinear and contains many Lorentz and CPT violating interaction. However, as argued, they do not contribute in processes which might lead to the physical Lorentz violation. Nonetheless, how the emergent electrogravity theory could be observationally differed from conventional QED and GR theories is also briefly discussed. Povzetek. Avtorji so razvili model za elektrogravitacijo, ki vsebuje običajno kvantno elek-trodinamiko in tenzorsko polje gravitacije. Slednje predstavlja linearizirano splosno teorijo relativnosti. Spontano krsitev Lorentzove invariance sprožijo s predpisom za vektorska in tenzorska polja: A^ = ±MA in = ±Mh (Ma in MH sta predlagani skali zlomitve simetrije). Predpis prinese mnogo visjo simetrijo kot jo ima Lagrangeva gostota modela. Dvanajst Goldstonovih delcev tvori multiplete vektorskih in tenzorskih polj. Foton je vedno pravi vektorski Goldstonov bozon, graviton pa vsebuje tudi psevdo Goldstonove nacine. Model postane tako nelinearen in vsebuje vrsto interakcij, ki zlomijo Lorentzovo in CPT simetrijo, ki pa ne vodijo do fizikalne zlomitve Lorentzove simetrije. Avtorji komentirajo, v cem se elektrogravitacija razlikuje od elektrodinamike in gravitacije. Keywords: Spontaneous symmetry violation, Lorentz invariance violation, emergent field theory. 4.1 Introduction While Lorentz symmetry looks physically as an absolutely exact spacetime symmetry, the spontaneous Lorentz invariance violation (SLIV) suggests a beautiful scenario where massless vectors and/or tensor fields emerge as the corresponding zero modes which may be identified with photons, gravitons and other gauge fields [1-3]. Though they appear through condensation of the pure gauge degrees of freedom in the starting theory their masslessness are provided by their Nambu-Goldstone nature [4-12] rather than a conventional gauge invariance. 4.1.1 Emergent vector fields theory In order to violate Lorentz invariance one necessarily needs field(s) being sensitive to the spacetime transformations, as vector or tensor fields are. They can evolve vacuum expectation value which fixes direction of the violation in the spacetime and create the corresponding condensate. Therefore, if there is an interaction with this condensate one could expect Lorentz violation to be manifested physically. If we want to arrange spontaneous Lorentz violation by the vector field, we could start, as usual, with the potential terms in the Lagrangian L = - - V; V = A (A2 - n^Ml)2 (4.1) = ; A2 = A^A2 ; n2 = where n2 is an unit constant vector specifying character of Lorentz violation. If n2 is time-like vector, we have time-like violation breaking SO(1,3) to SO(3). If n2 is space-like vector, we have space-like violation breaking SO(1,3) to SO(1,2). We started with gauge invariant kinetic term, but since potential violates gauge invariance anyway, we could have started with general kinetic terms Lk = a (3aAp)2 + b (3aAa)2 (4.2) but problem arising here is a propagating ghost mode, which we get ride off with the gauge invariant form of kinetic terms. Such a system of vector field with potential, generally appears not stable, its energy is not bound from below unless phase space is restricted with condition Aj2 - L(a) = — -ff — 26(n«aa)2 — ^ 9va2 (4.8) 6 is Lagrange multiplier setting orthogonality condition for the vector Goldstone field, thus treating it as gauge fixing one. In general, we have here pletora Lorentz 2 ~ and CPT violating couplings like -f2Vn29va2 in the higher orders, especially if charged currents are introduced as well, but it appears in all physical processes (photon-photon, matter-photon, matter-matter interactions) at least in the tree and one loop level, there is no sign of physical Lorentz invariance violation. Looks like Lorentz invariance is realized in nonlinear fashion and Lorentz breaking condition (4.3) is treated like a nonlinear gauge choice for vector field [16,17]. Consideration of the spontaneous Lorentz violation scenarios for non-Abelian vector fields meet same challenges, though consequently lead to the same conclusions as in the Abelian vector field case, despite the fact that there are some significant differences as well. The length fixing constraint adapted for non-Abelian vector fields in fact violates not only Lorentz symmetry, but an accidental symmetry SO(N, 3N) of the constraint itself (here N defines unitary symmetry group of vector fields) which is much higher than symmetry of the theory Lagrangian. This gives extra massless modes which together with the true Lorentzian Goldstone complete the whole gauge multiplet of the non-Abelian theory taken [18]. 4.1.2 Emergent tensor field gravity Actually, for the tensor field gravity we can use the similar nonlinear constraint for a symmetric two-index tensor field H2^ = n2MH , H2v = n2 = = ±1 (4.9) (where n2V is now a properly oriented unit Lorentz tensor, which supposedly specifies vacuum expectation values, while MH is the proposed scale for Lorentz U(H)= Ah (H2 -n2MH)2 (4.10) violation in the gravity sector) which fixes its length in the same manner as it appears for the vector field (4.3). Again, the nonlinear constraint (4.9) may in principle appear from the standard potential terms added to the tensor field Lagrangian liml _ ^ — n mH) in the nonlinear a-model type limit when the coupling constant AH goes to infinity. Just in this limit the tensor field theory appears stable, but doing so, we are effectively excluding corresponding Higgs mode from the theory and it does not lead to physical Lorentz violation [19]. This constraint (4.9), like the non-Abelian vector field, has higher symmetry then the kinetic term, particularly SO(7,3). So, spontaneous symmetry violation breaks not only Lorentz symmetry, but also this SO(7,3) and therefore produces also PGM-s, but in contrast to vector field, when we had only two channels of Lorentz symmetry violation to SO(3) or SO (1,2) and three true Goldstone modes always, for tensor field we have more possibilities. If we write down constraint in more details Hjiv = H2o + H2=j + (V2Ht=j )2 — (V2Hot)2 = n2 MH = ±MH (4.11) we see that if only one component of the tensor field should acquire vacuum expectation value (assuming minimal vacuum configuration) we have following alternatives: (a) noo = 0 , SO(1,3) —» SO(3) (b) nt=j = 0 , SO(1,3)-> SO(1,2) (4.12) (c) nt=j = 0 , SO(1,3) —» SO(1,1) for n2 = 1 and (d) not = 0 , SO(1,3)-> SO(2) (4.13) for n2 = —1. For a, b cases we have three true Goldstone modes and for c, d we have five, since only one generator of Lorentz transformations remains unbroken. While in b, c, d cases physical graviton consists, at least partially, from true Goldstone modes, in case a only true goldstones are Hot components, thus physical graviton will be constructed from PGM-s. One should notice that pseudo-Goldstone nature of some components of tensor multiplet poses no threats and generally in contrast to the scalar pseudo-Goldstone modes they do not acquire mass duo to the quantum effects, if diffeomorphism (diff) invariance is present. So, we are putting (4.9) on the tensor field mimicking linearized general relativity 1 L = L(H)+ ls - ^h^tS^ (4.14) where L(H) = 23ah^3ah^ - l3AHtT3AHtT - 3AHAv3^H^ + 3vHtr32H^ (4.15) Here Htr stands for the trace of Hiv (Htr = niVHiV) and L(H) is invariant under the diff transformations SH^ = + , Sx^ = e(x), (4.16) while LS and Ti^ are the Lagrangian and corresponding energy momentum tensor of whatever is gravitating, (vector fields, matter). In case, vector field is considered L(A) = -TiV(A) = -F^ + 4niVFa|3FaP (4.17) where L(H) is fully diff invariant, but that is not the case for other parts of Lagrangian and diff invariance is satisfied only proximately, but they become more and more invariant when the tensor field gravity Lagrangian (4.14) is properly extended to GR with higher terms in H-fields included1. Once tensor field acquires vacuum expectation value, we can expand it into Goldstone mode n2h2 Hív = hiv +n^vMH + O(1/MH), n • h = 0 (4.18) Here hiv corresponds to the pure emergent modes satisfying the orthogonality condition and h2 = hivhiv, n • h = nivhiv. Lets specify once again that hiv consists of Goldstone and PGM-s. Only case, when physical graviton will consists of only Goldstone mode is when Lorentz invariance is fully broken, we have six emergent goldstone modes and other pseudo Goldstone components is gauged away by fixing remaining gauge freedom (more about supplementary conditions below). Such a scenario can not be achieved by minimal vacuum configuration. Nevertheless, whether tensor field will be defined only by Goldstone modes or by a mixture with PGM-s, hole tensor multiplet always stays strictly massless. A particular case of interest is that of the traceless VEV tensor niv = 0 (4.19) in terms of which the emergent gravity Lagrangian acquires an especially simple form (see below). It is clear that the VEV in this case can be developed on several Hiv components simultaneously, which in general may lead to total Lorentz violation with all six Goldstone modes generated. For simplicity, we will use sometimes this form of vacuum configuration in what follows, while our arguments can be applied to any type of VEV tensor niv. Alongside to basic emergent orthogonality condition in (4.18) one must also specify other supplementary conditions for the tensor field hiv (appearing eventually as possible gauge fixing terms in the emergent tensor field gravity). We have 1 Such an extension means that in all terms included in the GR action, particularly in the QED Lagrangian term , (—g)1/2g^vgAPF^AFvp, one expands the metric tensors g^v = n^v + H,v/Mp, g^v = — H^v/Mp + H^HV/Mp + • • • taking into account the higher terms in H-fields. remaining three degrees of gauge freedom. Usually, spin 1 states in tensor field is gauged away by the conventional Hilbert-Lorentz condition + q3vhtr = 0 (4.20) (q is an arbitrary constant, giving for q = -1/2 the standard harmonic gauge condition), because spin-1 component always has negative contribution in energy and therefore it is desirable action. However, as we have already imposed the emergent constraint (4.18), we can not use the full Hilbert-Lorentz condition (4.20) eliminating four more degrees of freedom in Otherwise, we would have an "over-gauged" theory with a non-propagating graviton. In fact, the simplest set of conditions which conform with the emergent condition n • h = 0 in (4.18) turns out to be 3p(a^hvp - 3vh^p) = 0 (4.21) This set excludes only three degrees of freedom2 in and, besides, it automatically satisfies the Hilbert-Lorentz spin condition as well. Putting parameterization (4.18) into the total Lagrangian given in (4.14), one comes to the truly emergent tensor field gravity Lagrangian containing an infinite series in powers of the modes. For the traceless VEV tensor without loss of generality, we get the especially simple form L = l3Ah^3AVv - löAHtr3Ahtr - 3aHAv3^V + + Mh sas^V - 9Ahtr +ls - (^mh^v + h^v The bilinear field term + 8MH h2n„v\ T, 2mh I MP -|XV |AnvA n|An n2 T^ + o(i/mH) 3|h23vh2 (4.22) Mh MP "n|vTs |v (4.23) in the third line in the Lagrangian (4.22) merits special notice. This term arises from the interaction term with tensor field. It could significantly affect the dispersion relation for the all the fields included in , thus leading to an unacceptably large Lorentz violation if the SLIV scale MH were comparable with the Planck mass MP. However, this term can be gauged away [19] by an appropriate redefinition of the fields involved by going to the new coordinates x1 —} x1 + (4.24) In fact, with a simple choice of the parameter function £,^(x) being linear in 4- coordinate 2 The solution for a gauge function ^(x) satisfying the condition (4.21) can generally be chosen as + 3^0, where 0(x) is an arbitrary scalar function, so that only three degrees of freedom in are actually eliminated. 2 n ™ = Mn"Vxv , (4.25) the term (4.23) is cancelled by an analogous term stemming from the kinetic term in LS. On the other hand, since the diff invariance is an approximate symmetry of the Lagrangian L we started with (4.14), this cancellation will only be accurate up to the linear order corresponding to the tensor field theory. Indeed, a proper extension of this theory to GR1 with its exact diff invariance will ultimately restore the usual dispersion relation for the vector field and other matter fields involved. We will consider all that in significant detail in the next section. So, with the Lagrangian (4.22) and the supplementary conditions (4.18) and (4.21) lumped together, one eventually comes to a working model for the emergent tensor field gravity [19]. Generally, from ten components of the symmetric two-index tensor four components are excluded by the supplementary conditions (4.18) and (4.21). For a plane gravitational wave propagating in, say, the z direction another four components are also eliminated, due to the fact that the above supplementary conditions still leave freedom in the choice of a coordinate system, x^ —» x^ + £,^(t — z/c)* much as it takes place in standard GR. Depending on the form of the VEV tensor caused by SLIV, the two remaining transverse modes of the physical graviton may consist solely of Lorentzian Goldstone modes or of pseudo-Goldstone modes, or include both of them. This theory, similar to the nonlinear QED [14], while suggesting an emergent description for graviton, does not lead to physical Lorentz violation [19]. 4.1.3 Length Fixing Constraints and Nonlinear Gauge We have overviewed above the SLIV scenarios for vector and tensor fields and could see that, though the well motivated length fixing constraint for a given field causes spontaneous Lorentz violation, somewhat counterintuitively, in physical processes, Lorentz symmetry appears intact. Therefore we rightfully suspect that the Lorentz breaking constraint condition acts effectively as a gauge fixing condition. To prove or disprove whether this suspicion is reasonable one either should check the SLIV effects in the corresponding physical processes in all orders, that looks unrealistic, or has to find some generic argument, particularly find a solution for gauge function or, at least, prove that such a solution exists. In case of vector field Aa and Lorentz breaking condition A a = nj| MA , the corresponding equation for gauge function S is (Aa + 3aS)2 = n2Ml (4.26) This equation is nonlinear and its exact solution for arbitrary Aa is not yet found. However, to our fortune, it is well known that this equation taken for time-like violation case (n^ = 1) is in fact the Hamilton-Jacobi equation for the relativistic particle, which moves in the external electromagnetic field. An action for such a system is given by S = M/dXOdX« - Aadxa (MVuOU« - Aaua) dT (4.27) where t is evolution parameter and ua = ^jy. In this case, even though we do not have exact solution for that, we know that an action S describes a physical system and therefore it has a solution for an arbitrary electromagnetic field Aa. Analogously, for the space-like np (np = -1) our basic equation (4.26) might be considered as the Hamilton-Jacobi equation for a hypothetical tachyon moving in the external electromagnetic field S = M^-dxadxa - Aadxa = (MV-u«u» - Aaua) dx (4.28) So, though this action can only correspond to a hypothetical particle, which is not discovered so far, theoretically it might exist at least as a free particle state. At this point we are unable to solve (4.26) exactly nor for time-like, neither for space-like cases, but we can check that ultra-relativistic particle and tachyon (in the limit of very large momenta, when particle velocity vp —» c from below and tachyon velocity vt —» c from above) have somewhat similar equations of motions ¿P, = -J^FU ¿pi = -Foi + Fu (4.29) dt ' - with the electromagnetic field flipped for tachyon (pi stands for the corresponding three-momenta). No dependent, one believes or not in an existence of charged tachyon one might at least can take this similarity as a hint that in space-like case, similar to a time-like violation, we are dealing with effectively nonlinear gauge fixing condition. For the tensor field, diff gauge invariance also could only fully be approved, when corresponding gauge function £, is found, which satisfies the following equation + 3a£,ß + 3ߣ,a)2 = ±MH (4.30) While we do not have a heuristic argument like that we had above for the vector field time-like SLIV case, we can provide some arguments very similar to its space-like violation case leading again to the mainly intuitive suggestion. So, to conclude, though the above discussion looks highly suggestive towards the vector and tensor field constraints, (4.3) and (4.9), to consider them as the be nonlinear gauge choices, they are not yet, sure, the rigorous proofes. Therefore, presently the only way to check whether these constraints are just gauge choices or not is actually related to seeking of the SLIV efects by a direct analysis of the corresponding physical processes. 4.2 Electrogravity model Usually, an emergent gauge field framework is considered either regarding emergent photons or regarding emergent gravitons, but in nature they do not exist in separate framework, they are different parts of one picture and therefore the most natural thing is to discuss them as such. For the first time, we consider it regarding them both in the so-called electrogravity theory where together with the Nambu QED model [14] with its gauge invariant Lagrangian we propose the linearized Einstein-Hilbert kinetic term for the tensor field preserving a diff invariance (more details can be found in our recent paper [20]). We show that such a combined SLIV pattern, conditioned by the constraints (4.3) and (4.9), induces the massless Goldstone modes which appear shared among photon and graviton. One needs in common nine zero modes both for photon (three modes) and graviton (six modes) to provide all necessary (physical and auxiliary) degrees of freedom. They actually appear in our electrogravity theory due to spontaneous breaking of high symmetries of our constraints. While for a vector field case the symmetry of the constraint coincides with Lorentz symmetry SO(1,3), the tensor field constraint itself possesses much higher global symmetry SO(7,3), whose spontaneous violation provides a sufficient number of zero modes collected in a graviton. As we understand already these modes are largely pseudo-Goldstone modes since SO (7,3) is symmetry of the constraint (4.9) rather than the electrogravity Lagrangian whose symmetry is only given by Lorentz invariance. 4.2.1 Constraints and zero mode spectrum Before going any further, let us make some necessary comments. Note first of all that, apart from dynamics that will be described by the total Lagrangian, the vector and tensor field constraints (4.3,4.9) are also proposed to be satisfied. In principle, these constraints, like in previous cases, could be formally obtained from the conventional potential introduced in the total Lagrangian. The most general potential, where the vector and tensor field couplings possess the Lorentz and SO (7,3) symmetry, respectively, must be solely a function of A2 = A2A2 and = H2VH2V. Indeed, it cannot include any contracted and intersecting terms like as Htr, H2VA2AV and others which would immediately reduce the above symmetries to the common Lorentz one. So, one may only write U(A,H) = Aa(A2 - n2MA)2 + AH(H2v -n2MH)2 + AAHA2H^ (4.31) where Aa,h,ah stand for the coupling constants of the vector and tensor fields, while values of n2 = ±1 and n2 = ±1 determine their possible vacuum configurations. As a consequence, an absolute minimum of the potential (4.31) might appear for the couplings satisfying the conditions AA,H > 0 , AAAH > AAH/4 (4.32) However, as in the pure vector field case discussed in section 1, this theory is generally unstable with the Hamiltonian being unbounded from below unless the phase space is constrained just by the above nonlinear conditions (4.3, 4.9). They in turn follow from the potential (4.31) when going to the nonlinear a-model type limit Aa,h —> oo. In this limit, the massive Higgs mode disappears from the theory, the Hamiltonian becomes positive, and one comes to the pure emergent electrogravity theory considered here. We note again that the Goldstone modes appearing in the theory are caused by breaking of global symmetries related to the constraints (4.3,4.9) rather than directly to Lorentz violation. Meanwhile, for the vector field case symmetry of the constraint (4.3) coincides in fact with Lorentz symmetry whose breaking causes the Goldstone modes depending on the vacuum orientation vector n^, as can be clearly seen from an appropriate exponential parametrization for the starting vector field (4.5). However, in the tensor field case, due to the higher symmetry SO (7,3) of the constraint (4.9), there are much more tensor zero modes than would appear from SLIV itself. In fact, they complete the whole tensor multiplet in the parametrization (4.18). However, as was discussed in the previous section, only a part of them are true Goldstone modes, others are pseudo-Goldstone ones. In the minimal VEV configuration case, when these VEVs are developed only on the single A^ and components, one has several possibilities determined by the vacuum orientations n^ and . There appear the twelve zero modes in total, three from Lorentz violation itself and nine from a violation of the SO (7,3) symmetry that is more than enough to have the necessary three photon modes (two physical and one auxiliary ones) and six graviton modes (two physical and four auxiliary ones). We could list below all possible cases corresponding n — n values, the timelike-spacelike SLIV, when n0 = 0 and nt=j = 0, the spacelike-timelike (nonzero nt and n00), spacelike-spacelike diagonal (nonzero nt and ni=j) and spacelike-spacelike nondiagonal (nonzero nt and nt=j) cases, but for brevity, instead we only list the most interesting cases corresponding to minimal and maximal Lorantz symmetry breaking. (1) When both n^ = 0 and n^ = 0, wether ^ is time or space component we have minimally broken Lorentz invariance and only three broken generators and therefore three Goldstone modes and all of them is collected into the photon, while components of haß needed for physical graviton and its auxiliary components can be only provided by the pseudo-Goldstone modes following from the symmetry breaking SO(7,3) —» SO(6,3) related to the tensor-field constraint (4.9). (2) For the case, when nt = 0 and nßY = 0 (one of the nondiagonal space components of the unit tensor is nonzero), when i = ß = y Lorentz symmetry appears fully broken so that the photon a^ has three Goldstone components , while the graviton is collected by the rest of true Goldstone and PGM-s. (3) Only case when both physical photon and graviton h- consists of true Goldstone modes is when n0 = 0 and nt=j = 0, but some gauge degrees of freedom for a graviton are given by the PGM states stemming from the symmetry breaking of the tensor-field constraint (4.9). In any case, while photon may only contain true Goldstone modes, some PGM-s appear necessary to be collected in graviton together with some true Goldstone modes to form full tensor multiplet. 84 J.L. Chkareuli, J. Jejelava and Z. Kepuladze 4.2.2 The Model In the previous section and Generally in emergent tensor field gravity theories we considered the vector field A^ as an unconstrained material field which the emergent gravitons interacted with, but now in electrogravity model we propose that the vector field also develops the VEV through the SLIV constraint (4.3), thus generating the massless vector Goldstone modes associated with a photon. We also include the complex scalar field 9 (taken to be massless, for simplicity) as an actual matter in the theory £(9) = D.9 (D.9)*, D. = 3. + ieA. . So, the proposed total starting electrogravity Lagrangian is £tot = L(A) + L(H) + L(9) + Lint(H, A, 9) (4.33) (4.34) where L(A) and L(H) are U(1) gauge invariant and diff invariant vector and tensor field Lagrangians, while the gravity interaction part 1 Mp Lint(H,A,9) = -^H.v[T .v(A) + T ^(9)] (4.35) contains the tensor field couplings with canonical energy-momentum tensors of vector and scalar fields. In the symmetry broken phase one goes to the pure Goldstone vector and tensor modes, a^ and respectively, Which is thoroughly discussed in the previous sections (4.8), (4.22). At the same time, the scalar field Lagrangian £(9) in (4.34) is going now to £(9) = n2 3. + iea. + ieMAn^ - ie a n. 9 2M - A (4.36) while tensor field interacting terms (4.35) in £int (H, A, 9) convert to Lint = -M ( h.v + MhU^V 2MHh "^v T.v ( a. a2n. ) + T.v (9) 2M A (4.37) where the vector field energy-momentum tensor is now solely a function of the Goldstone a. modes. 2 2 4.2.3 Emergent electrogravity interactions To proceed further, one should eliminate, first of all, the large terms of the false Lorentz violation being proportional to the SLIV scales MA and MH in the interaction Lagrangians (4.36) and (4.37). Arranging the phase transformation for the scalar field in the following way 9 —» 9 exp[-ieMAn.x. (4.38) one can simply cancel that large term in the scalar field Lagrangian (4.36), thus coming to n 2 D^ — ie——a2n, I 9 2M A (4.39) where the covariant derivative D, is read from now on as D, = 3,+iea,. Another unphysical set of terms (4.23) appear from the gravity interaction Lagrangian Lint (4.37) where the large SLIV entity MHn,v couples to the energy-momentum tensor. They also can be eliminated by going to the new coordinates (4.24), as was mentioned in the previous section. For infinitesimal translations £,(x) the tensor field transforms according to (4.16), while scalar and vector fields transform as = ^S^, 5a, = £,a3aa, + 3^av , (4.40) respectively. One can see, therefore, that the scalar field transformation has only the translation part, while the vector one has an extra term related to its nontrivial Lorentz structure. For the constant unit vector n, this transformation looks as 5n, = 3,£vnv, (4.41) having no the translation part. Using all that and also expecting that the phase parameter £A is in fact linear in coordinate x, (that allows to drop out its highderivative terms), we can easily calculate all scalar and vector field variations, such as 5 (D,9) = £a3a(D,9) + 3,£ada9, 5f,v = + 3,£AfAv + 3v£Af,A (4.42) and others. This finally leads to the total variations of the above Lagrangians. Whereas the pure tensor field Lagrangian L(H) (4.15) is invariant under diff transformations, 5L(H) = 0, the interaction Lagrangian Lint in (4.34) is only approximately invariant being compensated (in the lowest order in the transformation parameter £,) by kinetic terms of all the fields involved. However, this Lagrangian becomes increasingly invariant once our theory is extending to GR1. In contrast, the vector and scalar field Lagrangians acquire some nontrivial additions 5L(A) = £a3aL(A) 1 — 2 (3,£a + 3A£,) f,VfV + MX (fV0,Va2 + 1fpv3pv KaA)^ a,avn2 Dv9 + "2M^nAjA 5L(9) = £a3Al(9) + (3,£v + 3v £,) where j, stands for the conventional vector field source current (4.43) J, = ie[9*D,9 — 9 (D,9)1 (4.44) while Dv9 is the SLIV extended covariant derivative for the scalar field Dv9 = Dv9 — ie a nv9 (4.45) 2Ma 2 2 The first terms in the variations (4.43) are unessential since they simply show that these Lagrangians transform, as usual, like as scalar densities under diff transformations. Combining these variations with Lint (4.37) in the total Lagrangian (4.34) one finds after simple, though long, calculations that the largest Lorentz violating terms in it mH £,A + 3a£^ mp 2 -f^fV - f^a2 + 2D> (D^)* v MA A (4.46) will immediately cancel if the transformation parameter is chosen exactly as is given in (4.25) in the previous section. So, with this choice we finally have for the modified interaction Lagrangian 1 MP Lint(h, a, 9) = -—h^vT^v(a, 9) + 1 MP MA £1 + 1 MP MH £2 + Mh MP Ma £3 "(4.47) where £1 = n2h fv 3 a2 nJ + n ^V 4fAp3Apa2 + nAJA £2 = 2n2h2n^v [—ff + 2DV9 (D^)*] £3 = n n^A 2fpv9pv (a^aA) — (a^aA)nVJv (4.48) Thereby, apart from a conventional gravity interaction part given by the first term in (4.47), there are Lorentz violating couplings in £1,2,3 being properly suppressed by corresponding mass scales. Note that the coupling presented in £3 between the vector and scalar fields is solely induced by the tensor field SLIV. Remarkably, this coupling may be in principle of the order of a normal gravity coupling or even stronger, if MH > MA. However, appropriately simplifying this coupling (and using also a full derivative identity) one comes to £3 ~ n2 (n^aA) np [3Vfvp — Jp] (4.49) that after applying of the vector field equation of motion turns it into zero. We consider it in more detail in the next section where we calculate some tree level processes. 4.3 The lowest order SLIV processes The emergent vector field Lagrangian (4.8) and emergent gravity Lagrangian in (4.22) taken separately present in fact highly nonlinear theory which contains lots of Lorentz and CPT violating couplings. Nevertheless, as it was shown in [19,16,17] in the lowest order calculations, they all are cancelled and do not manifest themselves in physical processes. As we talked about earlier, this may mean that the length-fixing constraints (4.3,4.9) put on the vector and tensor fields appear as the gauge fixing conditions rather than a source of an actual Lorentz violation. In the context of electrogravity model, which contains both photon and graviton as the emergent gauge fields, this means that only source of new physics can be (4.47). Even if suspicion that length fixing constraints are nonlinear gauge choices is true, for Lorentz invariance to be realized anyway, U(1) and diff gauge transformations should commute in the symmetry broken phase and then we could claim that L1 and L2 in (4.47) will have no physical effects, but there is also (4.48), which is proportional to diff transformation parameter and strictly speaking it is not zero Lagrangian. So, in this picture to be logically sound and consistent we should check all interactions in the (4.47) anyway. For that one properly derive all necessary Feynman rules and then calculate the basic lowest order processes, such as photon-graviton scattering and their conversion, photon scattering on the matter scalar field and other, that has been throughly carried out in our paper mentioned above [20] where can be found all necessary details. These calculations explicitly demonstrate that all the SLIV effects in these processes are strictly cancelled. This appears due to an interrelation between the longitudinal graviton and photon exchange diagrams and the corresponding contact interaction diagrams. So, physical Lorentz invariance in all processes is left intact. Apart, many other tree level Lorentz violating processes related to gravitons and vector fields (interacting with each other and the matter scalar field in the theory) may also appear in higher orders in the basic SLIV parameters 1/MH and 1/MA, by iteration of couplings presented in our basic Lagrangians (4.22, (4.47)) or from a further expansions of the effective vector and tensor field Higgs modes (4.7, 4.18) inserted into the starting total Lagrangian (4.34). Again, their amplitudes appear to cancel each other, thus eliminating physical Lorentz violation in the theory. Most likely, the same conclusion could be expected for SLIV loop contributions as well. Actually, as in the massless QED case considered earlier [16], the corresponding one-loop matrix elements in our emergent electrogravity theory could either vanish by themselves or amount to the differences between pairs of similar integrals whose integration variables are shifted relative to each other by some constants (being in general arbitrary functions of the external four-momenta of the particles involved) which, in the framework of dimensional regularization, could lead to their total cancellation. So, after all, it should not come as too much of a surprise that emergent electrogravity theory considered here is likely to eventually possess physical Lorentz invariance provided that the underlying gauge and diff invariance in the theory remains unbroken. 4.4 Conclusion We have combined emergent photon and graviton into one framework of electro-gravity. While photon emerges as true vector Goldstone mode from SLIV, graviton at least partially consists of PGM-s as well, because alongside of Lorentz symmetry much bigger global symmetry of (4.9) SO(7,3) is broken as well. Configuration of true Goldstone and PGM-s inside graviton solely depends on VEV-s of vector and tensor fields. So, in total 12 massless Goldstone modes are born to complete photon and graviton multiplets with an orthogonality conditions n^a^ = 0, = 0 in place. Emergent electrogravity theory is nonlinear and in principal contains many Lorentz and CPT violating interactions, when expressed in terms of Goldstone modes. Nonetheless, all non-invariant effects disappear in all possible lowest order physical processes, which means that Lorentz invariance is intact and hence Lorentz invariance breaking conditions (4.3,4.9) act as a gauge fixing for photon and graviton, instead of being actual source of physical Lorentz violation in the theory. If this cancellation occurs in all orders (i.e. (4.3, 4.9) are truly nonlinear gauge fixing conditions), then emergent electrogravity is physically indistinguishable from conventional gauge theories and spontaneous Lorentz violation caused by the vector and tensor field constraints (4.3,4.9) appear hidden in gauge degrees of freedom, and only results in a noncovariant gauge choice in an otherwise gauge invariant emergent electrogravity theory. From this standpoint, the only way for physical Lorentz violation to take place would be if the above gauge invariance were slightly broken by near Planck scale physics, presumably by quantum gravity or some other high dimensional theory. This is in fact a place where the emergent vector and tensor field theories may drastically differ from conventional QED, Yang-Mills and GR theories where gauge symmetry breaking could hardly induce physical Lorentz violation. In contrast, in emergent electrogravity such breaking could readily lead to many violation effects including deformed dispersion relations for all matter fields involved. Another basic distinction of emergent theories with non-exact gauge invariance is a possible origin of a mass for graviton and other gauge fields (namely, for the non-Abelian ones, see [18]), if they, in contrast to photon, are partially composed from pseudo-Goldstone modes rather than from pure Goldstone ones. Indeed, these PGM-s are no longer protected by gauge invariance and may properly acquire tiny masses, which still do not contradict experiment. This may lead to a massive gravity theory where the graviton mass emerges dynamically, thus avoiding the notorious discontinuity problem [21]. So, while emergent theories with an exact local invariance are physically indistinguishable from conventional gauge theories, there are some principal distinctions when this local symmetry is slightly broken which could eventually allow us to differentiate between the two types of theory in an observational way. Acknowledgements We would like to thank Colin Froggatt, Archil Kobakhidze and Holger Nielsen for useful discussions and comments. Z.K. wants to thank participants of the 21st Workshop "What Comes Beyond the Standard Models?" (23-30 June, Bled, Slovenia) for interesting and useful discussions, as well as the organizers for such a productive and working environment. This work is partially supported by Georgian National Science Foundation (grant No. YS-2016-81). References 1. J.D. Bjorken, Ann. Phys. (N.Y.) 24 (1963) 174. 2. P.R. Phillips, Phys. Rev. 146 (1966) 966. 3. T. Eguchi, Phys.Rev. D 14 (1976) 2755; 4. Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345; J. 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Chkareuli, J.G. Jejelava, G. Tatishvili, Phys. Lett. B 696 (2011) 126; J.L. Chkareuli, C.D. Froggatt and H.B. Nielsen, Nucl. Phys. B 848 (2011) 498. 20. J.L. Chkareuli, J. Jejelava, Z. Kepuladze, Eur. Phys. J. C78 (2018) 156; e-Print: arXiv:1709.02736 [hep-th]. 21. H. van Dam and M. J. G. Veltman, Nucl. Phys. B 22 (1970) 397. Bled Workshops in Physics Vol. 19, No. 2 A Proceedings to the 21 st Workshop What Comes Beyond ... (p. 90) Bled, Slovenia, June 23-July 1, 2018 5 A Deeper Probe of New Physics Scenarii at the A. Djouadi ** Laboratoire de Physique Theorique, Universite Paris-Sud and CNRS, F-91405 Orsay, France Abstract. The implications of the discovery of a Higgs boson at the LHC with a mass of 125 GeV are summarised in the context of the Standard Model of particle physics and in new physics scenarios beyond it, taking the example of the minimal supersymmetric Standard Model extension, the MSSM. The perspectives for Higgs and new physics searches at the next LHC upgrades as well as at future hadron and lepton colliders are then briefly summarized. Povzetek. Avtor povzame implikacijo odkritja higgsovega bozona z maso 125 GeV na pospesevalniku LHC na standardni model osnovnih delcev ter na nekatere modele, ki poskusajo narediti nov korak v fiziki osnovnih delcev. Kot primer omeni minimalno su-persimetricno razsiritev standardnega modela znano kot MSSM. Pregleda obete za iskanje znakov nove fizike v naslednji nadgradnji LHC in na bodočih leptonskih in hadronskih pospesevalnikih. Keywords: Higgs boson, new physics scenarios, supersymmetry, MSSM 5.1 Introduction The ATLAS and CMS historical discovery of a particle with a mass of 125 GeV [1] and properties that are compatible with those of a scalar Higgs boson [2,3] has far reaching consequences not only for the Standard Model (SM) but also for new physics models beyond it. In the SM, electroweak symmetry breaking is achieved spontaneously via the Brout-Englert-Higgs mechanism [2], wherein the neutral component of an isodoublet scalar field acquires a non-zero vacuum expectation value v. This gives rise to nonzero masses for the fermions and the electroweak gauge bosons while preserving the SU(2) x U(1) gauge symmetry. One of the four degrees of freedom of the original isodoublet field, corresponds to a physical particle [3]: a scalar boson with JPC = 0++ quantum numbers under parity and charge conjugation. The couplings of the Higgs boson to the fermions and gauge bosons are related to the masses of these particles and are thus decided by the * Presented at 20th Bled Workshop in 2017. Contribution received too late for inclusion in 2017 Proceedings. ** E-mail: abdelhak.djouadi@th.u-psud.fr LHC * symmetry breaking mechanism. In contrast, the Higgs mass itself MH, although expected to be in the vicinity of the weak scale v « 250 GeV, is undetermined. Let us summarise the known information on this parameter before the start of the LHC. A direct information was the lower limit MH > 114 GeV at 95% confidence level (CL) established at LEP2 [4]. Furthermore, a global fit of the electroweak precision data to which the Higgs boson contributes, yields the value MH = 92— GeV, corresponding to a 95% CL upper limit of MH < 160 GeV [4]. From the theoretical side, the presence of this new weakly coupled degree of freedom is a crucial ingredient for a unitary electroweak theory. Indeed, the SM without the Higgs particle leads to scattering amplitudes of the W/Z bosons that grow with the square of the center of mass energy and perturbative unitarity would be lost at energies above the TeV scale. In fact, even in the presence of a Higgs boson, the W/Z bosons could interact very strongly with each other and, imposing the unitarity requirement leads to the important mass bound MH < 700 GeV [5], implying that the particle is kinematically accessible at the LHC. Another theoretical constraint emerges from the fact that the Higgs self-coupling, A 0, up to the TeV scale implies that Mh > 70 GeV. If the SM is to be extended to the Planck scale Mp ~ 1018 GeV, the requirements on A from finiteness and positivity constrain the Higgs mass to lie in the range 130 GeV < MH < 180 GeV [6]. This narrow margin is close to the one obtained from the direct and indirect experimental constraints. The discovery of the Higgs particle with a mass of 125 GeV, a value that makes the SM perturbative, unitary and extrapolable to the highest possible scales, is therefore a consecration of the model and crowns its past success in describing all experimental data available. In particular, the average mass value measured by the ATLAS and CMS teams, MH = 125J ± 0^24 GeV [7], is remarkably close to the best-fit of the precision data which should be considered as a great achievement and a triumph for the SM. In addition, a recent analysis that includes the state-of-the-art quantum corrections [8] gives for the condition of absolute stability of the electroweak vacuum, A(MP) > 0, the bound MH > 129 GeV for the present value of the top quark mass and the strong coupling constant, mt"p = 173^2 ± 0^9 GeV and as(MZ) = 0J184 ± 0^0007 [4]. Allowing for a 2ct variation of mtxp, one obtains MH > 125^6 GeV that is close to the measured MH value [7]. In fact, for an unambiguous and well-defined determination of the top mass, one should rather use the total cross section for top pair production at hadron colliders which can unambiguously be defined theoretically; this mass has a larger error, Amt ~ 3 GeV, which allows more easily absolute stability of the SM vacuum up to MP [9]. Nevertheless, the SM is far from being perfect in many respects. It does not explain the proliferation of fermions and the large hierarchy in their mass spectra and does not say much about the small neutrino masses. The SM does not unify in a satisfactory way the electromagnetic, weak and strong forces, as one has three different symmetry groups with three coupling constants which shortly fail to meet at a common value during their evolution with the energy scale; it also ignores the fourth force, gravitation. Furthermore, it does not contain a particle that could account for the cosmological dark matter and fails to explain the baryon asymmetry in the Universe. However, the main problem that calls for beyond the SM is related to the special status of the Higgs boson which, contrary to fermions and gauge bosons has a mass that cannot be protected against quantum corrections. Indeed, these are quadratic in the new physics scale which serves as a cut-off and hence, tend to drive MH to very large values, ultimately to MP, while we need MH = 0(100 GeV). Thus, the SM cannot be extrapolated beyond 0(1 TeV) where some new physics should emerge. This is the reason why we expect something new to manifest itself at the LHC. There are three avenues for the many new physics scenarios beyond the SM. There are first theories with extra space-time dimensions that emerge at the TeV scale (the cut-off is then not so high) and, second, composite models inspired from strong interactions also at the TeV scale (and thus the Higgs is not a fundamental spin-zero particle). Some versions of these scenarios do not incorporate any Higgs particle in their spectrum and are thus ruled out by the Higgs discovery. However, the option that emerges in the most natural way is Supersymmetry (SUSY) [10] as it solves most of the SM problems discussed above. In particular, SUSY protects MH as the quadratically divergent radiative corrections from standard particles are exactly compensated by the contributions of their supersymmetric partners. These new particles should not be much heavier than 1 TeV not to spoil this compensation [11] and, thus, they should be produced at the LHC. The Higgs discovery is very important for SUSY and, in particular, for its simplest low energy manifestation, the minimal supersymmetric SM (MSSM) that indeed predicts a light Higgs state. In the MSSM, two Higgs doublet fields Hu and Hd are required, leading to an extended Higgs consisting of five Higgs bosons, two CP-even h and H, a CP-odd A and two charged H± states [12]. Nevertheless, only two parameters are needed to describe the Higgs sector at tree-level: one Higgs mass, which is generally taken to be that of the pseudoscalar boson MA, and the ratio of vacuum expectation values of the two Higgs fields, tan |3 = vd/vu, expected to lie in the range 1 < tan | < 60. The masses of the CP-even h, H and the charged H± states, as well as the mixing angle a in the CP-even sector are uniquely defined in terms of these two inputs at tree-level, but this nice property is spoiled at higher orders [13]. For MA ^ MZ, one is in the so-called decoupling regime in which the h state is light and has almost exactly the SM-Higgs couplings, while the other CP-even H and the charged H± bosons become heavy, MH « MH± « MA, and decouple from the massive gauge bosons. In this regime, the MSSM Higgs sector thus looks almost exactly as the one of the SM with its unique Higgs boson. Nevertheless, contrary to the SM Higgs boson, the lightest MSSM CP-even h mass is bounded from above and, depending on the SUSY parameters that enter the important quantum corrections, is restricted to M™* < 130 GeV [13] if one assumes a SUSY breaking scale that is not too high, MS < O (1 TeV), in order to avoid too much fine-tuning in the model. Hence, the requirement that the MSSM h boson coincides with the one observed at the LHC, i.e. with Mh « 125 GeV and almost SM-like couplings as the LHC data seem to indicate, would place very strong constraints on the MSSM parameters, in particular the SUSY-breaking scale MS. This comes in addition to the LHC limits obtained from the search of the heavier Higgs states and the superparticles. In this talk, the implications of the discovery of the Higgs boson at the LHC and the measurement of its properties will be summarised and the prospects for the searches of new physics, in particular in the SUSY context, in the future will be discussed. 5.2 Implications: Standard Model and beyond In many respects, the Higgs particle was born under a very lucky star as the mass value of« 125 GeV allows to produce it at the LHC in many redundant channels and to detect it in a variety of decay modes. This allows detailed studies of the Higgs properties. 5.2.1 Higgs production and decay We start by summarizing the production and decay at the LHC of a light SM-like Higgs particle, which should correspond to the lightest MSSM h boson in the decoupling regime. First, for MH « 125 GeV, the Higgs mainly decays [14] into bb pairs but the decays into WW* and ZZ* final states, before allowing the gauge bosons to decay leptonically W—and Z—)U (£ — e, are also significant. The H—>t+t- channel (as well as the gg and cc decays that are not detectable at the LHC) is also of significance, while the clean loop induced H —» yy mode can be easily detected albeit its small rates. The very rare H —» Zy and even H —» yr channels should be accessible at the LHC but only with a much larger data sample. On the other hand, many Higgs production processes have significant cross sections [15-17]. While the by far dominant gluon fusion mechanism gg —» H (ggF) has extremely large rates (« 20 pb at y/s — 7-8 TeV), the subleading channels, i.e. the vector boson fusion (VBF) qq —» Hqq and the Higgs-strahlung (HV) qq —> HV with V — W, Z mechanisms, have cross sections which should allow for Higgs studies of the already at y/s > 7 TeV with the « 25 fb-1 data collected by each experiment. The associated process pp —»ttH (ttH) would require higher energy and luminosity. This pattern already allows the ATLAS and CMS experiments to observe the Higgs boson in several channels and to measure some of its couplings in a reasonably accurate way. The channels that have been searched are H —»ZZ* —» 4£±, H —} WWm 2£2v, H —» yy where the Higgs is mainly produced in ggF with subleading contributions from Hjj in the VBF process, H —» tt where the Higgs is produced in association with one (in ggF) and two (in VBF) jets, and finally H —} bb with the Higgs produced in the HV process. One can ignore for the moment the low sensitivity H —»m^ and H —» Zy channels. A convenient way to scrutinize the couplings of the produced H boson is to look at their deviation from the SM expectation. One then considers for a given search channel the signal strength modifier ^ which for the H —»XX decay mode measures the deviation compared to the SM expectation of the Higgs production cross section times decay branching fraction |j.xx. ATLAS and CMS have provided the signal strengths for the various final states with a luminosity of « 5 fb-1 for the 2011 run at Vs = 7 TeV and « 20 fb-1 for the 2012 run at Vs = 8 TeV. The constraints given by the two collaborations, when combined, lead to a global signal strength ^atlas = 1.18 ± 0.15 and ^cms = 1.00 ± 0.14 [7]. The global value being very close to unity implies that the observed Higgs is SM-like. Hence, already with the rather limited statistics at hand, the accuracy of the ATLAS and CMS measurements is reaching the 15% level. This is at the same time impressive and worrisome. Indeed, the main Higgs production channel is the top and bottom quark loop mediated gluon fusion mechanism and, at y/s = 7 or 8 TeV, the three other mechanisms contribute at a total level below 15%. The majority of the signal events observed at LHC, in particular in the search channels H —» yy, H —» ZZ* -> 4£, H —> WWm 2£2v and to some extent H —» tt, thus come from the ggF mechanism which is known to be affected by large theoretical uncertainties. Indeed, although cr(gg —» H) is known up next-to-next-to-leading order (NNLO) in perturbative QCD (and at least at NLO for the electroweak interaction) [15,16], there is a significant residual scale dependence which points to the possibility that still higher order contributions cannot be totally excluded. In addition, as the process is of O(a^) at LO and is initiated by gluons, there are sizable uncertainties due to the gluon parton distribution function (PDF) and the value of the coupling as. A third source of theoretical uncertainties, the use of an effective field theory (EFT) approach to calculate the radiative corrections beyond NLO should also be considered [15]. In addition, large uncertainties arise when u(gg —»H) is broken into the jet categories H+0j, H + 1j and H+2j [18]. In total, the combined theoretical uncertainty is estimated to be Ath « ±15% [16] and would increase to Ath « ±20% if the EFT uncertainty is also included. The a priori cleaner VBF process will be contaminated by the gg —»H+2j mode making the total error in the H+ jj "VBF" sample also rather large [18]. Hence, the theoretical uncertainty is already at the level of the accuracy of the cross section measured by the ATLAS and CMS collaborations. Another drawback of the analyses is that they involve strong theoretical assumptions on the total Higgs width since some contributing decay channels not accessible at the LHC are assumed to be SM-like and possible invisible Higgs decays in scenarios beyond the SM do not to occur. In Ref. [17], following earlier work [19] it has been suggested to consider the ratio DXX = crp (pp —) H —> XX)/cp(pp —» H —» VV) for a specific production process P and for a given decay channel H XX when the reference channel H VV is used. In these ratios, the cross sections and hence, their significant theoretical uncertainties will cancel out, leaving out only the ratio of partial decay widths which are better known. The total decay width which includes contributions from channels not under control such as possible invisible Higgs decays, do not appear in the ratios DXX. Some common experimental systematical uncertainties such as the one from the luminosity measurement and the small uncertainties in the Higgs decay branching ratios also cancel out. We are thus left with only with the statistical and some (non common) systematical errors [17]. The ratios DXX involve, up to kinematical factors and known radiative corrections, only the ratios |cX|2/ |cV|2 of the Higgs reduced couplings to the particles X and V compared to the SM expectation, cX = gHXX/gHXX. For the time being, three independent ratios can be considered: Dyy, DTT and Dbb. In order to determine these ratios, the theoretical uncertainties have to be treated as a bias (and not as if they were associated with a statistical distribution) and the fit has to be performed for the two ^ extremal values: Md|exp ± ^Md/Mdlth with « ±20% [20]. A large number of analyses of the Higgs couplings from the LHC data have been performed and in most cases, it is assumed that the couplings of the Higgs boson to the massive W, Z gauge bosons are equal to gHzz = gHWW = cV and the couplings to all fermions are also the same gHff = cf. However, as for instance advocated in Ref. [21] to characterize the Higgs particle at the LHC, at least three independent H couplings should be considered, namely ct, cb and cV. While the couplings to W, Z, b, t particles are derived by considering the decays of the Higgs boson to these particles, the Htt coupling is derived indirectly from o(gg —»H) and BR(H —»yy), two processes that are generated by triangular loops involving the top quarks in the SM. One can assume, in a first approximation, that cC = ct and cT = cb and possible invisible Higgs decays are absent. In Ref. [21], a three-dimensional fit of the H couplings was performed in the space [ct, cb, cV], when the theory uncertainty is taken as a bias and not as a nuisance. The best-fit value for the couplings, with the a/s = 7+8 TeV ATLAS and CMS data turns out to be ct = 0.89, cb = 1.01 and cV = 1.02, ie very close to the SM values. 5.2.2 Implications of the Higgs couplings measurement The precise measurements of Higgs couplings allow to draw several important conclusions. i) A fourth generation fermions is excluded. Indeed, in addition to the direct LHC searches that exclude heavier quarks mb /, mt / < 600 GeV [23], strong constraints can be also obtained from the loop induced Higgs-gluon and Higgs-photon vertices in which any heavy particle coupling to the Higgs proportionally to its mass will contribute. For instance the additional 4th generation t' and b' contributions increase d(gg —» H) by a factor of « 9 at LO but large O(GFm2,) electroweak corrections should be considered. It has been shown [23] that with a fourth family, the Higgs signal would have not been observable and the obtained Higgs results unambiguously rule out this possibility. ii) The invisible Higgs decay width should be small. Invisible decays would affect the properties of the observed Higgs boson and could be constrained if the total decay width is determined. But for a 125 GeV Higgs, rH = 4 MeV, is too small to be resolved experimentally. Nevertheless, in pp —» VV —» 4f, a large fraction of the Higgs cross section lies in the high-mass tail [24] allowing to to put loose constrains r^/r^m ~ 5-10 [25]. The invisible Higgs decay width rf^v can be better constrained indirectly by a fit of the Higgs couplings and in particular with the signal strength in the H -> ZZ process: |j.zz « r(H -> ZZ)/r*f with r*f = r+rsm; one obtains r^/rf^ <50% at 95% CL with the assumption Cf = cy = 1 [ 20]. A more model independent approach would be to perform direct searches for missing transverse energy. These have been conducted in pp —» HV with V-> jj, H and in VBF, qq -> qqET leading to BRinv < 50% at 95%CL for SM-like Higgs couplings [7]. A more promising search for invisible decays is the monojet channel gg —> Hj which has large rates [26]. While the most recent monojet ATLAS and CMS searches are only sensitive to BRinv ~ 1, more restrictive results can be obtained in the future. The Higgs invisible rate and the dark matter detection rate in direct astro-physical searches are correlated in Higgs portal models and it turns out that LHC constraints are competitive [27] with those derived from direct dark matter search experiments [28]. iii) The spin-parity quantum numbers are those of a standard Higgs. One also needs to establish that the observed Higgs state is indeed a CP even scalar and hence with JPC = 0++ quantum numbers. For the spin, the observation of the H —» yy decay rules out the spin-1 case [29]. The Higgs parity can be probed by studying kinematical distributions in the H —» ZZb —» 4£ decay channel and in the VH and VBF production modes [30] and with the 25 fb-1 data collected so far, ATLAS and CMS found that the observed Higgs is more compatible with a 0+ state and the 0- possibility is excluded at the 98%CL [7]. Other useful diagnostics of the Higgs CP nature that also rely on the tensorial structure of the HVV coupling can be made in the VBF process [31]. Nevertheless, there is a caveat in the analyses relying on the HVV couplings: a CP-odd state has no tree-level VV couplings [32]. In fact, a better way to measure the Higgs parity is to study the signal strength in the H VV channels and in Ref. [20] it was demonstrated that the observed Higgs has indeed a large CP component, > 50% at the 95%CL. In fact, the less unambiguous way to probe the Higgs CP nature would be to look at final states in which the particle decays hadronically, e.g. pp —» HZ —» bb 11 [32]. These processes are nevertheless extremely challenging even at the upgraded LHC. 5.2.3 Implications for Supersymmetry We turn now to the implications of the LHC Higgs results for the MSSM Higgs sector and first make a remark on the Higgs masses and couplings, which at tree-level depend only on MA and tan |3, when the important radiative corrections are included. In this case many parameters such as the masses of the third generation squarks mj., m^. and their trilinear couplings At, Ab enter Mh and M H through quantum corrections. These are introduced by a general 2 x 2 matrix AM2j but the leading one is controlled by the top Yukawa coupling and is proportional to m4, logMS with MS = , mj2 the SUSY-breaking scale and the stop mixing parameter Xt [13]. The maximal value Mmax is then obtained for a decoupling regime MA ~ O(TeV), large tan large MS that implies heavy stops and maximal mixing Xt ^\/6MS [33]. If the parameters are optimized as above, the maximal Mh value reaches the level of 130 GeV. It was pointed out in Refs. [34,35,21] that when the measured value Mh = 125 GeV is taken into account, the MSSM Higgs sector with only the largely dominant correction discussed above, can be again described with only the two parameters tan |3 and MA; in other words, the loop corrections are fixed by the value of Mh. This observation leads to a rather simple but accurate parametrisation of the MSSM Higgs sector, called hMSSM. The reduced couplings of the CP-even h state (as is the case for the heavier H) depend in principle only on the angles | and a (and hence tan | and MA), cV = sin(| —a), c0 = cos a/sin c£ =— sin a/cos while the couplings of A and H± (as well as H in the decoupling regime) to gauge boson are zero and those to fermions depend only on for tan | > 1, they are enhanced ( 104 GeV, are required to obtain Mh = 125 GeV and even higher in most cases ii) Implications from the production rates of the observed state. Besides the corrections to the Higgs masses and couplings discussed above, there are also direct corrections to the Higgs couplings and the most ones are those affecting the hbb vertex [40] and the stop loop contributions to the gg —>h production and h—>yy decay rates [41]. A fit of the ct, cb and cV couplings shows that the latter are small [20]. In turn, ignoring the direct corrections and using the input Mh « 125 GeV, one can make a fit in the plane [tan MA]. The best-fit point is tan | = 1 and Ma = 550 GeV which implies a large SUSY scale, Ms = 0(100) TeV. In all, cases one also has Ma > 200-350 GeV. iii) Implications from heavy Higgs boson searches. At high tan | values, the strong enhancement of the b, t couplings makes that the ® = H/A states decay domi-nantly into t+t- and bb pairs and are mainly produced in gg —»® fusion with the b-loop included and associated production with b-quarks, gg/qq —»bb + ® [42]. The most powerful LHC search channel is thus pp —» gg + bb —» ® —» t+t-. For the charged Higgs, the dominant mode is H± —» tv with the H± light enough to be produced in top decays t —»H+b —»Tvb. In the low tan | regime, tan | < 3, the phenomenology of the A, H, H± states is richer [34]. For the production, only gg —» ® process with the dominant t and sub-dominant b contributions provides large rates. The H/A/H± decay pattern is in turn rather involved. Above the tt (tb) threshold H/A —»tt and H+ —»tb are by far dominant. Below threshold, the ^WW, ZZ decays are significant. For 2Mh < MH < 2mt (MA > Mh + MZ), H —} hh (A —} hZ) is the dominant H(A) decay mode. But the A —» tt channel is still important with rates > 5%. In the case of H±, the channel H+ —> Wh is important for MH± < 250 GeV, similarly to the A —>hZ case. In Ref. [34] an analysis of these channels has been performed using current information given by ATLAS and CMS in the context of the SM, MSSM [43] or other scenarios. The outcome is impressive. The ATLAS and CMS H/A —» t+t-constraint is extremely restrictive and MA < 250 GeV, it excludes almost the entire intermediate and high tan |3 regimes. The constraint is less effective for a heavier A but even for MA « 400 GeV the high tan | > 10 region is excluded and one is even sensitive to MA « 800 GeV for tan | > 50. For H±, almost the entire MH± < 160 GeV region is excluded by the process t —» H+b with the decay H+ —» tv. The other channels, in particular H —» VV and H/A —»tt, are very constraining as they cover the entire low tan | area that was previously excluded by the LEP2 bound up to Ma « 500 GeV. Even A —» hZ and H —» hh would be visible at the current LHC in small portions of the parameter space. 5.3 Perspectives for Higgs and New Physics The last few years were extremely rich and exciting for particle physics. With the historical discovery of a Higgs boson by the LHC collaborations ATLAS and CMS, crowned by a Nobel prize in fall 2013, and the first probe of its basic properties, they witnessed a giant step in the unraveling of the mechanism that breaks the electroweak symmetry and generates the fundamental particle masses. They promoted the SM as the appropriate theory, up to at least the Fermi energy scale, to describe three of Nature's interactions, the electromagnetic, weak and strong forces. However, it is clear that these few years have also led to some frustration as no signal of physics beyond the SM has emerged from the LHC data. The hope of observing some signs of the new physics models that were put forward to address the hierarchy problem, that is deeply rooted in the Higgs mechanism, with Supersymmetric theories being the most attractive ones, did not materialize. The Higgs discovery and the non-observation of new particles has nevertheless far reaching consequences for supersymmetric theories and, in particular, for their simplest low energy formulation, the MSSM. The mass of approximately 125 GeV of the observed Higgs boson implies that the scale of SUSY-breaking is rather high, at least O(TeV). This is backed up by the limits on the masses of strongly interacting SUSY particles set by the ATLAS and CMS searches, which in most cases exceed the TeV range. This implies that if SUSY is indeed behind the stabilization of the Higgs mass against very high scales that enter via quantum corrections, it is either fine-tuned at the permille level at least or its low energy manifestation is more complicated than expected. The production and decay rates of the observed Higgs particles, as well as its spin and parity quantum numbers, as measured by ATLAS and CMS with the « 25 fb-1 data collected at a/s = 7+8 TeV, indicate that its couplings to fermions and gauge bosons are almost SM-like. In the context of the MSSM, this implies that we are close to the decoupling regime and this particle is the lightest h boson, while the other H/A/H± states must be heavier than approximately the Fermi scale. This last feature is also backed up by LHC direct searches of these heavier Higgs states. This drives up to the question that is now very often asked: what to do next? The answer is, for me, obvious: we are only in the beginning of a new era. Indeed, it was expected since a long time that the probing of the electroweak symmetry breaking mechanism will be at least a two chapters story. The first one is the search and the observation of a Higgs-like particle that will confirm the scenario of the SM and most of its extensions, that is, a spontaneous symmetry breaking by a scalar field that develops a non-zero vev. This long chapter has just been closed by the ATLAS and CMS collaborations with the spectacular observation of a Higgs boson. This observation opens a second and equally important chapter: the precise determination of the Higgs profile and the unraveling of the electroweak symmetry breaking mechanism itself. A more accurate measurement of the Higgs couplings to fermions and gauge bosons will be mandatory to establish the exact nature of the mechanism and, eventually, to pin down effects of new physics if additional ingredients beyond those of the SM are involved. This is particularly true in weakly interacting theories such as SUSY in which the quantum effects are expected to be small. These measurements could be performed at the upgraded LHC with an energy close to a/s = 14 TeV, in particular if a very high luminosity, a few ab-1, is achieved [43,44]. At this upgrade, besides improving the measurements performed so far, rare but important channels such as associated Higgs production with top quarks, pp—>ttH, and Higgs decays into m+m- and Zy states could be probed. Above all, a determination of the self-Higgs coupling could be made by searching for double Higgs production e.g. in the gluon fusion channel gg —> HH [45]; this would be a first step towards the reconstruction of the scalar potential that is responsible of electroweak symmetry breaking. This measurement would be difficult at the LHC even with high-luminosity but a proton collider with a/s = 30 to 100 TeV could do the job [44]. In a less near future, a high-energy lepton collider, which is nowadays discussed in various options (ILC, TLEP, CLIC, |M-collider) would lead to a more accurate probing of the Higgs properties [46], promoting the scalar sector to the very high-precision level of the gauge and fermion sectors achieved by the LEP and SLC colliders in the 1990s [4]. At electron-positron colliders, the process e+e- —} HZ, just looking at the recoiling Z boson allows to measure the Higgs mass, the CP parity and the absolute HZZ coupling, allowing to derive the total decay width r^. One can then measure precisely, already at a/s « 250 GeV where a(e+e- —» HZ) is maximal, the absolute Higgs couplings to gauge bosons and light fermions from the decay branching ratios. The important couplings to top quarks and the Higgs self-couplings can measured at the 10% level in the higher- order processes e+e- —» ttH and e+e- —> HHZ at energies of at least 500 GeV with a high-luminosity. Besides the high precision study of the already observed Higgs, one should also continue to search for the heavy states that are predicted by SUSY, not only the superparticles but also the heavier Higgs bosons. The energy upgrade to «14 TeV (and eventually beyond) and the planed order of magnitude (or more) increase in luminosity will allow to probe much higher mass scales than presently. In fact, more generally, one should continue to search for any sign of new physics or new particles, new gauge bosons and fermions, as predicted in most of the SM extensions. In conclusion, it is not yet time to give up on SUSY and more generally on New Physics but, rather, to work harder to be fully prepared for the more precise and larger data set that will be delivered by the upgraded LHC. It will be soon enough to "philosophize" then as the physics landscape will become more clear. Acknowledgements: I thank the organisers for their invitation to give a talk at the conference. 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Mankoc Borštnik1 1 Department of Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia 2 Departamento de Física, ESFM - Instituto Politecnico Nacional. U. P. "Adolfo Lopez Mateos". C. P. 07738, Ciudad de Mexico, Mexico Abstract. The spin-charge-family theory [1-7,9-12,15-17,19-24] predicts the existence of the fourth family to the observed three. The 4 x 4 mass matrices — determined by the nonzero vacuum expectation values and the dynamical parts of the two scalar triplets, the gauge fields of the two groups of SU(2) determining family quantum numbers, as well as of the three scalar singlets with the family members quantum numbers (Ta = (Q, Q', Y')), — manifest the symmetry SU(2) x SU(2) x U(1). All scalars carry the weak and the hyper charge of the standard model higgs field (± 1, ^ 1, respectively). It is demonstrated, using the massless spinor basis, that the symmetry of the 4 x 4 mass matrices remains SU(2) x SU(2) x U(1) in all loop corrections, and it is discussed under which conditions this symmetry is kept under all corrections, that is with the corrections induced by the repetition of the nonzero vacuum expectation values included. Povzetek. Teorija spinov-nabojev-druzin [1-7,9-12,15-17,19-24] napove cetrto družino k doslej opazenim trem. Masne matrike 4 x 4 — določajo jih dva skalarna tripleta, ki sta umeritveni polji dveh grup SU(2) (tripleti dolocajo družinska kvantna stevila), ter trije skalarni singleti s kvantnimi stevili družinskih canov Ta = (Q, Q', Y') vsak s svojimi nenicelnimi vakuumskimi pricakovanimi vrednostmi ter kot dinamicna polja — imajo simetrijo SU(2) x SU(2) x U(1). Vsi skalarji — oba tripleta in vsi trije singleti — imajo enake sibke in hipernaboje kot higgsova polja v standardnem modelu (± 1, ^ 1). Avtorja pokažeta, da ostane simetrija masnih matrik 4 x 4 enaka SU(2) x SU(2) x U(1) v vseh redih popravkov, ki jih dolocajo dinamicna polja. Obravnavata pa tudi vkljucitev ponovitve nenicelnih vakuumskih pricakovanih vrednosti v vseh redih in spremembo simetrije, ki jo te ponovitve povzrocijo. Keywords:Unifying theories, Beyond the standard model, Origin of families, Origin of mass matrices of leptons and quarks, Properties of scalar fields, The fourth * This is the part of the talk presented by N.S. Mankoc Borstnik at the 21st Workshop "What Comes Beyond the Standard Models", Bled, 23 of June to 1 of July, 2018. family, Origin and properties of gauge bosons, Flavour symmetry, Kaluza-Klein-like theories PACS:12.15.Ff 12.60.-i 12.90.+b 11.10.Kk 11.30.Hv 12.15.-y 12.10.-g 11.30.-j14.80.-j 6.1 Introduction The spin-charge-family theory [1-12,15-17,19-24] predicts before the electroweak break four - rather than the observed three — coupled massless families of quarks and leptons. The 4 x 4 mass matrices of all the family members demonstrate in this theory the same symmetry [1,5,4,21,22], determined by the scalar fields originating in d > (3 + 1): the two triplets — the gauge fields of the two SU(2) family groups with the generators N L, T, operating among families — and the three singlets — the gauge fields of the three charges (xa = (Q, Q', Y ))) — distinguishing among family members. All these scalar fields carry the weak and the hyper charge as does the scalar higgs of the standard model: (± 2 and t 2, respectively) [1,4,24]. The loop corrections alone, as well as corrections including the repetition of the nonzero vacuum expectation values in all orders, make each matrix element of mass matrices dependent on the quantum numbers of each of the family members. Since there is no direct observations of the fourth family quarks with masses below 1 TeV, while the fourth family quarks with masses above 1 TeV would contribute according to the standard model (the standard model Yukawa couplings of the quarks with the scalar higgs is proportional to , where m^ is the fourth family member (a = u, d) mass and v the vacuum expectation value of the scalar higgs) to either the quark-gluon fusion production of the scalar field (the higgs) or to the scalar field decay too much in comparison with the observations, the high energy physicists do not expect the existence of the fourth family members at all [25,26]. One of the authors (N.S.M.B) discusses in Refs. ([1], Sect. 4.2.) that the standard model estimation with one higgs scalar might not be the right way to evaluate whether the fourth family, coupled to the observed three, does exist or not. The ut-quarks and dt-quarks of an ith family, namely, if they couple with the opposite sign to the scalar fields carrying the family (A, i) quantum numbers and have the same masses, do not contribute to either the quark-gluon fusion production of the scalar fields with the family quantum numbers or to the decay of these scalars into two photons. The strong influence of the scalar fields carrying the family members quantum numbers to the masses of the lower (observed) three families manifests in the huge differences in the masses of the family members, let say ut and dt, i = (1,2,3), and families (i). For the fourth family quarks, which are more and more decoupled from the observed three families the higher are their masses [22,21], the influence of the scalar fields carrying the family members quantum numbers on their masses is in the spin-charge-family theory expected to be much weaker. Correspondingly the u4 and d4 masses become closer to each other the higher are their masses and the weaker are their couplings (the mixing matrix elements) to the lower three families. For u4-quarks and d4-quarks with the similar masses the observations might consequently not be in contradiction with the spin-charge-family theory prediction that there exists the fourth family coupled to the observed three ([28], which is in preparation). But three singlet and two treplet scalar fields offer also other explanations. We demonstrate in the main Sect. 6.2 that the symmetry SU(2) x SU(2) x U(1), which the mass matrices demonstrate on the tree level, after the gauge scalar fields of the two SU (2) family groups triplets gain nonzero vacuum expectation values, keeps the same in all loop corrections. We discuss also the symmetry of mass matrices if all the scalar fields, contributing to mass matrices, have nonzero vacuum expectation values. We use the massless basis. In Sect. 6.4 we present shortly the spin-charge-family theory and its achievements so far. All the mathematical support appears in appendices. Let be in this introduction stressed what supports the spin-charge-family theory to be the right next step beyond the standard model. This theory can not only explain — while starting from a very simple action in d > (13 + 1), Eqs. (6.35) in App. 6.4, with massless fermions (with the spin of the two kinds, Ya and ya, one kind taking care of the spin and of all the charges of the family members (Eq. (6.4)), the second kind taking care of families (Eqs. (6.34, 6.50))) coupled only to the gravity (through the vielbeins and the two kinds of the spin connections fields dabafac and d>abafac, the gauge fields of Sab and Sab (Eqs. (6.35)), respectively — all the assumptions of the standard model, but also answers several open questions beyond the standard model. It offers the explanation for [4-6,1,7,9-12,15-17,19-24]: a. The appearance of all the charges of the left and right handed family members and for their families and their properties. b. The appearance of all the corresponding vector and scalar gauge fields and their properties (explaining the appearance of higgs and the Yukawa couplings). c. The appearance and properties of the dark matter. d. The appearance of the matter/antimatter asymmetry in the universe. This theory predicts for the low energy regime: i. The existence of the fourth family to the observed three. ii. The existence of twice two triplets and three singlets of scalars, all with the properties of the higgs with respect to the weak and hyper charges, what explains the origin of the Yukawa couplings. iii. There are several other predictions, not directly connected with the topic of this paper. The fact that the fourth family quarks have not yet been observed — directly or indirectly — pushes the fourth family quarks masses to values higher than 1 TeV. Since the experimental accuracy of the 3 x 3 submatrix of the 4 x 4 mixing matrices is not yet high enough [32], it is not yet possible to calculate the mixing matrix elements among the fourth family and the observed three 1. Correspondingly it is not possible yet to estimate masses of the fourth family members by 1 The 3 x 3 submatrix, if accurate, determines the 4 x 4 unitary matrix uniquely. fitting the experimental data to the free parameters of mass matrices, the number of which is limited by the symmetry SU(2) x SU(2) x U(1), predicted by the spin-charge-family [22,21]. If we assume the masses of the fourth family members, the matrix elements can be estimated from the measured 3 x 3 submatrix elements of the 4 x 4 matrix [22,21] 2. The more effort and work is put into the spin-charge-family theory, the more explanations of the observed phenomena and the more predictions for the future observations follow out of it. Offering the explanation for so many observed phenomena — keeping in mind that all the explanations for the observed phenomena originate in a simple starting action — qualifies the spin-charge-family theory as the candidate for the next step beyond the standard model. The reader is kindly asked to learn more about the spin-charge-family theory in Refs. [2-4,1,5,6] and the references therein. We shall point out sections in these references, which might be of particular help, when needed. 6.2 The symmetry of the family members mass matrices The mass term Y.s=7 8 ^Ysp0s Eq. (6.3), of the starting action, Eq. (6.35), manifests in the spin-charge-family theory [4,1,5,6] the SU(2) x SU(2) xU(1) symmetry. The infinitesimal generators of the two family groups namely commute among themselves, {N L, T1}- = 0, Eq. (6.8), and with all the infinitesimal generators of the family members groups, {fAi, Ta}- = 0, (xa = (Q* Q Y')), Eq. (6.9). After the scalar gauge fields, carrying the space index (7* 8), of the generators NL and f1 of the two SU(2) groups gain nonzero vacuum expectation values, spinors (quarks and leptons), which interact with these scalar gauge fields, become massive. There are the scalar gauge fields, carrying the space index (7* 8), of the group U(1) with the infinitesimal generators Ta (=(Q* Q * Y')), which are responsible for the differences in mass matrices among the family members (ui* di* ei* i( 1* 2* 3*4), i determines four families). Their couplings to the family members depends strongly on the quantum numbers (Q* Q Y'). It is shown in this main section that the mass matrix elements of any family member keep the SU(2) x SU(2) xU(1) symmetry of the tree level in all corrections (the loops one and the repetition of the nonzero vacuum expectation values), provided that the scalar gauge fields of the U(1) group have no nonzero vacuum expectation values. In the case that the scalar gauge fields of the U(1) group have nonzero vacuum expectation values, the symmetry is changed, unless some of the scalar fields with the family quantum numbers have nonzero vacuum expectation values. We comment on all these cases in what follows. Let us first present the symmetry of the mass term in the starting action, Eq. (6.35). 2 While the fitting procedure is not influenced considerably by the accuracy of the measured masses of the lower three families, the accuracy of the measured values of the mixing matrices do influence, as expected, the fitting results very much. We point out that the symmetry SU(2)x SU(2) belongs to the two SO(4) groups — to sO(4)so(3 1) and to SO(4)so(4). The infinitesimal operators of the first and the second SO (4) groups are, Eqs. (6.40, 6.41), 1 21 N + (= NL):= 1 (S23 + iS01,S31 + iS02,S12 + iS03), f1 : = 1 (S58 - S67, S57 + S68, S56 - S78), (6.1) respectively. U(1) contains the subgroup of the subgroup SO(6) as well as the subgroup of SO (4) (SO(6) and SO(4) are together with SO(3,1) the subgroups of the group SO(13,1)) with the infinitesimal operators equal to, Eq. (6.42), t4 = -1 (S910 + S11 12 + S1314), t1 = 1 (S58 - S67, S57 + S68, S56 - S78), f2 = 1 (S58 + S67, S57 - S68, S56 + S78). (6.2) There are additional subgroups SU(2) x SU(2), which belong to sO(4)gO(3 1) and so(4)so(4), Eqs. (6.40, 6.41), the scalar gauge fields of which do not influence the masses of the four families to which the three observed families belong according to the predictions of the spin-charge-family theory3. All the degrees of freedom and properties of spinors (of quarks and leptons) and of gauge fields, demonstrated below, follow from the simple starting action, Eq. (6.35), after breaking the starting symmetry. Let us rewrite formally the fermion part of the starting action, Eq. (6.35), in the way that it manifests, Eq. (6.3), the kinetic and the interaction term in d = (3 + 1) (the first line, m = (0,1,2,3)), the mass term (the second line, s = (7,8)) and the rest (the third line, t = (5,6,9,10, ••• ,14)). Lf = ii>Ym(pm - X 9AiTAiAmi)^ + A,i {Y_ i|>YsP0s + s=7,8 { X ^YtP0t , (6.3) t=5,6,9,...,14 where p0s = Ps - 2Ss s"ds's"s - 2Sabd>abs, P0t = Pt - 2St't"t - 2Sabdabt 4, with m € (0,1,2,3), s G (7,8), (s',s") G (5,6,7,8), (a,b) (appearing in SSab) 3 The gauge scalar fields of these additional subgroups SU(2) x SU(2) influence the masses of the upper four families, the stable one of which contribute to the dark matter. 4 If there are no fermions present, then either dabc or dabc are expressible by vielbeins faa [[2,5], and the references therein]. We assume that there are spinor fields which determine spin connection fields - dabc and idabc. In general one would have [6]: P0a = faaP0a + je {Pa, Efaa}-, P0a = Pa - 2 Ss s"ds 's"a - 2 Sab id aba. Since the term 2E{Pa, Efaa}- does not influece the symmetry of mass matrices, we do not treat it in this paper. run within either (0,1,2,3) or (5,6,7,8), t runs e (5, •••,14), (t ',t") run either € (5,6,7,8) or e (9,10, • • •, 14)5. The spinor function ^ represents all family members, presented on Table 6.3, of all the 21 = 8 families, presented on Table 6.4. In this paper we pay attention on the lower four families. The first line of Eq. (6.3) determines in d = (3+1) the kinematics and dynamics of spinor (fermion) fields, coupled to the vector gauge fields. The generators TAi of the charge groups are expressible in terms of Sab through the complex coefficients cAlab (the coefficients cAlab of TAi can be found in Eqs. (6.38, 6.2)6, TAi = £ CAiab Sab , (6.4) a,b fulfilling the commutation relations |xAi, TBj}- = isABfAijkxAk • (6.5) They represent the colour (x3i), the weak (x1i) and the hyper (Y) charges 7. The corresponding vector gauge fields Am1 are expressible with the spin connection fields cUstm, Eq. (6.44)8 Am = £ cAist wstm • (6.6) s,t The second line of Eq. (6.3) determines masses of each family member (ui, di,vi, ei). The scalar gauge fields of the charges — those of the family members, determined by Sab and those of the families, determined by Sab — carry space index (7,8). Correspondingly the operators y0ys, appearing in the mass term, transform the left handed members of any family into the right handed members of the same family, what can easily be seen in Table 6.3. Operators Sab transform one family member of a particular family into the same family member of another family. Each scalar gauge fields (they are the gauge fields with space index s > 5) are as well expressible with the spin connections and vielbeins, Eq. (6.45) [2]. The groups SO(3,1), SU(3), SU(2)i, SU(2)ii and U(1 )„ (all embedded into SO(13 + 1)) determine spin and charges of spinors, the groups sU(2)^0(3 1), 5 We use units h = 1 = c 6 Before the electroweak break there are the conserved (weak) charges T1 (Eq. (6.38)), T3(Eq. (6^2) and Y := t4 + t23 (Eqs. (6.38, 6.2) and the non conserved charge Y' := —t4 tan2 £2 + t23 , where £2 is the angle of the break of SU(2) II from SU(2) I x SU(2) II x U(1 )II to SU(2)I x U(1 )I. After the electroweak break the conserved charges are T3 and Q := Y + T , the non conserved charge is Q' := — Y tan2 £1 + T , where -&1 is the electroweak angle. 7 There are as well the SU(2)ii (T2i, Eq. (6.38)) and U(1 )ii (t4, Eq. (6.2)) charges, the vector gauge fields of these last two groups gain masses when interacting with the condensate, Table 6.5 ([1,4,5] and the references therein). The condensate leaves massless, besides the colour and gravity gauge fields in d = (3 + 1), the weak and the hyper charge vector gauge fields. 8 Both fields, Ai^1 and Am\ are expressible with only the vielbeins, if there are no spinor fields present [2]. Eqs (6.1), SU(2)so(4)/ Eqs. (6.1), (embedded into SO(13 + 1 )) determine family quantum numbers 9. The generators of these latter groups are expressible by S ab fAi = £ cAiab Sab , (6.7) a,b fulfilling again the commutation relations {-fAi, fBj}_ = i5ABfAijkTfAk , (6.8) while {TAi,-fBj}_ = 0. (6.9) The scalar gauge fields of the groups SU(2)i (= su(2)so(3 1 ) with generators NL, Eq. (6.40)), su(2)i (= Su(2)so(4), with generators f1, Eq. (6.41)) and U(1) (with generators (Q, Q', Y'), Eq. (6.43)) are presented in Eq. (6.45) 10. The application of the generators f1, Eq. (6.41), NL, Eq. (6.40), which distinguish among families and are the same for all the family members, is presented in Eqs. (6.49, 6.51, 6.13). The application of the family members generators (Q, Q ', Y') on the family members of any family is presented on Table 6.1. The contribution of the scalar gauge fields to masses of different family members strongly depends on the quantum numbers Q, Q ' and Y' as one can read from Table 6.1. In loop corrections the contribution of the scalar gauge fields of (Q, Q ', Y') is proportional to the even power of these quantum numbers, while the nonzero vacuum expectation values of these scalar fields contribute in odd powers. r qi,r y ti r t23 y ' q ' l y t'3 y ' q ' ur dr vr e r 2 1 3 0 — 1 2 1 3 0 — 1 if if 1 2 1 i i i 2 2 (1-3 tan2 «2 ) — j (1 + 3 tan2 «2 ) 22 (1 + tan2 «2 ) 22 (-1 + tan2 «2 ) —2 tan2 « 1 33 tan2 « 1 0 tan2 « 1 ui dl vl e l 1 11 1 1 2 1 i 1 1 2 — i tan2 «2 — 6l tan2 «2 1 tan2 «2 ' tan2 «2 2 (1 — 3 tan2 «1 ) — j (1 + 33 tan2 « f ) 1 (1 + tan2 «f ) — 2 ( 1 — tan2 « i ) Table 6.1. The quantum numbers Q, Y, t4 , Y', Q', t23 , t13 , Eq. (6.43), of the family members ulr, ylr of one family (any one) [6] are presented. The left and right handed members of any family have the same Q and t4, the right handed members have t13 = 0, and t23 = 2r for (uR, vR) and — 1 for (dR, eR), while the left handed members have t23 = 0 and t13 = 2 for (uL, vL) and — 1 for (dL, eL). vR couples only to Aj as seen from the table. 9 SU(3) do not contribute to the families at low energies. We studied such possibilities in a toy model, Ref. [18]. 10 All the scalar gauge fields, presented in Eq. (6.45), are expressible with the vielbeins and spin connections with the space index a > 5, Ref. [2]. There are in the spin-charge-family theory 2( + 7-1 = 8 families 11, which split in two groups of four families, due to the break of the symmetry from SO (7,1) into SO(3,1) xSO(4). Each of these two groups manifests SU(2)gO(3 1) xSU(2)gO(4) symmetry [6]. These decoupled twice four families are presented in Table 6.4. The lowest of the upper four families, forming neutral clusters with respect to the electromagnetic and colour charges, is the candidate to form the dark matter [20]. We discuss in this paper symmetry properties of the lower four families, presented in Table 6.4 in the first four lines. We present in Table 6.2 the representation and the family quantum numbers of the left and right handed members of the lower four families. Since any of the family members (uL R, R, R, eL R) behave equivalently with respect to all the operators concerning the family groups su(2)so(1 3) x su(2)so(4), the last five columns are the same for all the family members. We rewrite the interaction, which is in the spin-charge-family theory responsible for the appearance of masses of fermions, presented in Eq. (6.3) in the second 78 78 line, in a slightly different way, expressing y7 = ((+) + (-)) and correspondingly 78 78 Y8 =-i((+)-(-)). Lmass = 1 &LY0 (±) (- Z T" A± - Z ^ A±'+ ^ , + , - A A i Ta = (Q,Q',Y'), fAi = (1L, f1), 78 1 Y0 (±)= Y01 (y7 ± iY8), A a \ „a ,..st ,,,st ,,,st -,- • ...st ± = C st d ± , d ± = d 7 T id 8 , st AA = Z CAab CDab± , CDab± = d>ab7 T idab8 . (6.10) ab In Eq. (6.10) the term ps is left out since at low energies its contribution is negligible, A determines operators, which distinguish among family members — (Q, Q', Y')12, their eigenvalues on basic states are presented on Table 6.1 — (/A, i) represent the family operators, determined in Eqs. (6.40, 6.41, 6.42). The detailed explanation can be found in Refs. [4,5,1]. 78 78 Operators TAi are Hermitian ((xAi)t = TAi), while (y0 (±))t = y0 (t). If the scalar fields AAi are real it follows that (AAi)t = AAi. 11 In the break from SO(13,1 ) to SO(7,1 ) x SO(6) only eight families remain massless, those for which the symmetry SO(7,1 ) remains. In Ref. [18] such kinds of breaks are discussed for a toy model. 12 (Q, QY') are expressible in terms of (t13,t23,t4) as explained in Eq. (6.43). The corre- sponding superposition of œss ± fields can be found by taking into account Eqs. (6.38, 6.2). While the family operators T11 and NL commute with y° (±), {Sab,Scd}_ =0 for all (a, b, c, d), the family members operators (t13, t23) do not, since S78 does 78 78 . 78 . / /. not (S78y0 (t) = —Y° (t) S78). However [^Y0 (t) (Q,Q',Y')A^Q'Q ,Y VR]t = VR (Q,Q', Y') A±q,q ',y ')f y° (±) ^L = VR (QR, Qlk ,YRk) A±Q'Q ''Y ^R 6k)l, where (QR, Q >k, YRk) denote the eigenvalues of the corresponding operators on the spinor state ^R. This means that we evaluate in both cases quantum numbers of the right handed partners. 78 But, let us evaluate —j < u[ + uR|0a|u[ + uR > ^, with Oa = Y.+,_ Y° (±) (t4A478 +t23A2738 +t13A1738 ). One obtains -1= {6 (A- +A+)+A-3 +A+3}. Equivalent (±) (±) (±) V2 evaluations for |d[ + dR > would give -= {1 (A- + A+) — A-3 - A+3}, while for neutrinos we would obtain —= {— = (A- + A+) + A-3 + A+3} and for e1 we would obtain —= {— = (A- + A+) — A-3 — A+3}. Let us point out that the fields include also coupling constants, which change when the symmetry is broken. This means that we must carefully evaluate expectation values of all the operators on each state of broken symmetries. We have here much easier work: To see how does the starting symmetry of the mass matrices behave under all possible corrections up to we only have to compare how do matrix elements, which are equal on the tree level, change in any order of corrections. In Table 6.2 four families of spinors, belonging to the group with the nonzero values of N L and T1, are presented. These are the lower four families, presented also in Table 6.4 together with the upper four families 13. There are indeed the four families of and presented in this table. All the 213+1-1 members of the first family are represented in Table 6.3. The three singlet scalar fields (AQ, aQ , A^') of Eq. (6.10) contribute on the 78 Q 78 Q' tree level the "diagonal" values to the mass term— y° (t) QAQ +y° (t) Q 'AQ 78 ' +Y° (t) Y'A ^ — transforming a right handed member of one family into the left handed member of the same family, or a left handed member of one family into the right handed member of the same family. These terms are different for different family members but the same for all the families. Since Q = (t13 + t23 + t4) = (S56 + t4), Y' = (—t4 tan2-S2 + t23) and Q' = (—(t4 + t23) tan2 + t13) — is the standard model angle and is the corresponding angle when the second SU(2) symmetry breaks — we could use 78 Q 78 Q ' 78 instead of the operators (y° (t) QAQ +y° (t) Q 'AQ +y° (t) Y'A ^ ) as well 78 78 78 the operators (y° (±) t4 A±, y° (±) t23 A±3, y° (±) t13 A±3), if the fact that the coupling constants of all the fields, also of dabs and dabs, change with the break of symmetry is taken into account. 13 The upper four families have the nonzero values of N R and T2. The stable members of the upper four families offer the explanation for the existence the dark matter [20]. Let us denote by — aa the nonzero vacuum expectation values of the three singlets for a family member a = (u1, v1, d1, e1), divided by the energy scale (let say TeV), when (if) these scalars have nonzero vacuum expectation values and we use the basis 11 + >: aa = -{« +^ I 78 , 1 ^y0 (±) [Q < aQ > +Q' < aQ > +Y' < A± >]|j + j > 2}51j + h.c., ' (6.11) Each family member has a different value for aa. All the scalar gauge fields Aq8 , Aq8 , Ay78 have the weak and the hypercharge as higgs scalars: (± 1, ^ 1, (±) (±) (±) respectively). Tf13 Tf23 IN L N l Tf4 ur ur ur ur 03 12 56 78 (+i) W | W (+8) | | ^ ^ ^ 03 12 56 78 [+i](+) I [+](+) I I... 03 12 56 78 (+i) [+] I (+) [+] I I- 03 12 56 78 [+i]( + ) I (+) [+] I I- 03 12 56 78 [-i] [+] I [+] [-] I I • • • 03 12 56 78 (-i) (+) I [+] [-] II-- 03 12 56 78 [-i] [+] I (+)(-) II-- 03 12 56 78 (-i)(+) I (+) (-) II-" -1 0 -1 0 -1 2 0 2 0 2 -1 0 1 0 -1 2 0 2 0 2 1 0 -1 0 -1 2 0 2 0 2 1 0 1 0 -1 2 0 2 0 2 Table 6.2. Four families of the right handed uR1 with the weak and the hyper charge (t13 = 0, Y = 2) and of the left handed uR1 quarks with (t13 = 2, Y = 6), both with spin 1 and colour (t33, t38) = [(1/2,1/(2^3), (-1/2,1/(2^3), (0, —1/( V3)] charges are presented. They represent two of the family members from Table 6.3 — uR1 and uR1 — appearing on 1st and 7th line of Table 6.3. Spins and charges commute with NR, t11 and t4, and are correspondingly the same for all the families. Transitions among families for any family member are caused by (NL AN l± and f11 A1 @), what manifests the symmetry SUNl (2) x SUti (2). There are corrections in all orders, which make all the matrix elements of the mass matrix for any of the family members a dependent on the three singlets (t4A4, t23A23, t13A13), Eq. (6.11). i > r(3,1) S12 x13 T23 T33 T38 T4 Y Q (Anti)octet, r (/,1) = ( — 1) 1 , r (6) = (1 ) — 1 of (anti) quarks and (anti) leptons 1 uR1 03 12 56/8 910 1112 1314 (+i) [+] 1 [+] ( + ) || (+) [ —] [ —] 1 1 2 0 1 2 1 2 1 2 s3 1 2 3 2 3 2 uR1 03 12 56/8 910 1112 1314 [ —i] ( —) 1 [+] ( + ) ii (+) [ —] [ —] 1 1 2 0 1 2 1 2 1 2 s3 1 2 3 2 3 3 dR1 03 12 56/8 910 1112 1314 (+i) [+] i ( — ) [ —] ii (+) [ —] [ —] 1 1 2 0 1 2 1 2 1 2 s3 1 1 3 1 3 4 ¿R1 03 12 56/8 910 1112 1314 [ —i] ( —) i ( — ) [ —] ii (+) [ —] [ —] 1 1 2 0 1 2 1 2 1 2 s3 1 1 3 1 3 5 dL1 03 12 56/8 910 1112 1314 [ —i] [+] i ( —) ( + ) ii (+) [ —] [ —] -1 1 2 1 2 0 1 2 1 2 s3 1 1 6 1 3 6 ¿L1 03 12 56/8 910 1112 1314 (+i) ( —) i ( — ) (+) ii ( + ) [ —] [ —] -1 1 2 1 2 0 1 2 1 2 s3 1 1 6 1 3 7 uL1 03 12 56/8 910 1112 1314 [ —i] [+] i [+] [ —] ii ( + ) [ —] [ —] -1 1 2 1 2 0 1 2 1 2 s3 1 1 6 2 3 8 uL1 03 12 56/8 910 1112 1314 (+i) ( — ) i [+] [ —] ii (+) [ —] [ —] -1 1 — 2 1 2 0 1 2 1 2 s3 1 6 1 6 2 3 9 uR2 03 12 56/8 910 1112 1314 (+i) [+] i [+] ( + ) 11 [ —] ( + ) [ —] 1 1 0 1 — 1 1 1 2 2 10 uR2 03 12 56/8 910 1112 1314 [ —i] ( —) i [+] ( + ) 11 [ —] ( + ) [ —] 1 1 2 0 1 2 1 2 2 y3 2 s3 1 2 3 2 3 11 dc2 dR 03 12 56/8 910 1112 1314 (+i) [+] i ( — ) [ —] 11 [ —] ( + ) [ —] 1 1 2 0 1 2 1 2 1 2 s3 1 1 3 1 3 12 dR2 03 12 56/8 910 1112 1314 [ —i] ( —) i ( — ) [ —] 11 [ —] ( + ) [ —] 1 1 2 0 1 2 1 2 1 2 s3 1 1 3 1 3 13 dL2 03 12 56/8 910 1112 1314 [ —i] [+] i ( —) ( + ) 11 [ —] ( + ) [ —] -1 1 2 1 2 0 1 2 1 2 s3 1 1 6 1 3 14 dL2 03 12 56/8 910 1112 1314 (+i) ( —) i ( — ) (+) 11 [ —] ( + ) [ —] -1 1 2 1 2 0 1 2 1 2 s3 1 1 6 1 3 15 uL2 03 12 56/8 910 1112 1314 [ —i] [+] i [+] [ —] i i [ —] ( + ) [ —] -1 1 2 1 2 0 1 2 1 2 s3 1 1 6 2 3 16 uL2 03 12 56/8 910 1112 1314 (+i) ( — ) i [+] [ —] 11 [ —] ( + ) [ —] -1 1 2 1 2 0 1 2 1 2 s3 1 1 6 2 3 17 uc3 uR 03 12 56/8 910 1112 1314 (+i) [+] i [+] ( + ) 11 [ —] [ —] ( + ) 1 1 2 0 1 2 0 1 s3 1 2 3 2 3 18 uc3 uR 03 12 56/8 910 1112 1314 [ —i] ( —) i [+] ( + ) 11 [ —] [ —] ( + ) 1 1 2 0 1 2 0 1 s3 1 2 3 2 3 19 d c3 dR 03 12 56/8 910 1112 1314 (+i) [+] i ( — ) [ —] 11 [ —] [ —] ( + ) 1 1 2 0 1 2 0 1 s3 1 1 3 1 3 20 d c3 dR 03 12 56/8 910 1112 1314 [ —i] ( —) i ( — ) [ —] 11 [ —] [ —] ( + ) 1 1 2 0 1 2 0 1 s3 1 1 3 1 3 21 dL3 03 12 56/8 910 1112 1314 [ —i] [+] i ( —) ( + ) 11 [ —] [ —] ( + ) -1 1 2 1 2 0 0 1 s3 1 1 6 1 3 22 dL3 03 12 56/8 910 1112 1314 (+i) ( —) i ( — ) (+) 11 [ —] [ —] ( + ) -1 1 2 1 2 0 0 1 S3 1 1 6 1 3 23 uL3 03 12 56/8 910 1112 1314 [ —i] [+] i [+] [ —] i i [ —] [ —] ( + ) -1 1 2 1 2 0 0 1 S3 1 1 6 2 3 24 uL3 03 12 56/8 910 1112 1314 (+i) ( — ) i [+] [ —] 11 [ —] [ —] ( + ) -1 1 2 1 2 0 0 1 S3 1 6 1 6 2 3 25 VR 03 12 56/8 910 1112 1314 (+i) [+] i [+] ( + ) ii (+) ( + ) ( + ) 1 1 2 0 1 2 0 0 1 2 0 0 26 VR 03 12 56/8 910 1112 1314 [ —i] ( —) i [+] ( + ) ii (+) ( + ) ( + ) 1 1 — 2 0 1 2 0 0 1 — 2 0 0 27 eR 03 12 56/8 910 1112 1314 (+i) [+] i ( — ) [ —] ii (+) ( + ) ( + ) 1 1 2 0 1 — 2 0 0 1 — 2 — 1 — 1 28 e R 03 12 56/8 910 1112 1314 [ —i] ( —) i ( — ) [ —] ii (+) ( + ) ( + ) 1 1 — 2 0 1 — 2 0 0 1 — 2 — 1 — 1 29 e L 03 12 56/8 910 1112 1314 [ —i] [+] i ( —) ( + ) ii (+) ( + ) ( + ) -1 1 2 1 2 0 0 0 1 2 — 1 — 1 30 e L 03 12 56/8 910 1112 1314 (+i) ( —) i ( — ) (+) ii ( + ) ( + ) ( + ) -1 1 2 1 2 0 0 0 1 2 — 1 — 1 31 VL 03 12 56/8 910 1112 1314 [ —i] [+] i [+] [ —] ii ( + ) ( + ) ( + ) -1 1 2 1 2 0 0 0 1 2 — 1 0 32 VL 03 12 56/8 910 1112 1314 (+i) ( — ) i [+] [ —] ii (+) ( + ) ( + ) -1 1 2 1 2 0 0 0 1 2 — 1 0 33 ¿L1 03 12 56/8 910 1112 1314 [ —i] [+] i [+] ( + ) i i [ —] ( + ) ( + ) -1 1 2 0 1 2 1 2 1 2 s3 1 6 1 1 3 34 ¿L1 03 12 56/8 910 1112 1314 (+i) ( — ) i [+] ( + ) 11 [ —] ( + ) ( + ) -1 1 2 0 1 2 1 2 1 2 s3 1 6 1 3 1 3 35 uL1 03 12 56/8 9 10 1112 13 14 [ —i] [+] i ( — ) [ —] i i [ —] ( + ) ( + ) -1 1 2 0 1 2 1 2 1 2 s3 1 6 2 3 2 3 36 uL"1 03 12 56/8 910 1112 1314 (+i) ( — ) i ( —) [ —] 11 [ —] ( + ) ( + ) -1 1 — 2 0 1 — 2 1 — 2 1 2 a/3 1 — 6 2 — 3 2 — 3 37 d c 1 dR 03 12 56/8 910 1112 1314 (+i) [+] i [+] [ —] i i [ —] ( + ) ( + ) 1 1 2 1 2 0 1 — 2 1 2 -/3 1 — 6 1 — 6 1 3 38 ¿R1 03 12 56/8 910 1112 1314 [ —i] ( — ) i [+] [ —] i i [ —] ( + ) ( + ) 1 1 — 2 1 2 0 1 — 2 1 2 -/3 1 — 6 1 — 6 1 3 39 uR1 03 12 56/8 910 1112 1314 (+i) [+] i ( — ) ( + ) 11 [ —] ( + ) ( + ) 1 1 2 1 2 0 1 2 1 2 s3 1 6 1 6 2 3 Continued on next page ia > (Anti)octet, F (7,1) = 1 )1 , r (6) = (1) - of (anti)quarks ant (anti)leptons 03 12 56 78 10 1112 13 14 [-i] ( -) i ( -) ( + ii -] ( + ) ( + ) 03 12 56 78 10 1112 13 14 [-i] [+] i [+] ( + ii ( + ) [-] ( + ) 03 12 56 78 10 1112 13 14 (+i) ( -) i [+] ( + ii + ) [-] ( + ) 03 12 56 7 8 10 1112 13 14 [-i] [+] i ( ) [ - ii ( + ) [-] ( + ) 03 12 56 78 10 1112 13 14 (+i) ( -) i ( ) [ - ii + ) [-] ( + ) 03 12 56 78 10 1112 13 14 (+i) [+] i [+] [ - ii ( + ) [-] ( + ) 03 12 56 78 10 1112 13 14 [-i] ( -) i [+] [ - ii ( + ) [-] ( + ) 03 12 56 78 10 1112 13 14 (+i) [+] i ( ) ( + ii + ) [-] ( + ) 03 12 56 78 10 1112 13 14 [-i] ( -) i ( ) ( + ii + ) [-] ( + ) 03 12 56 78 10 1112 13 14 [-i] [+] i [+] ( + ii ( + ) ( + ) [-] 03 12 56 78 10 1112 13 14 (+i) ( -) i [+] ( + ii + ) ( + ) [-] 03 12 56 78 10 1112 13 14 [-i] [+] i ( ) [ - ii ( + ) ( + ) [-] 03 12 56 78 10 1112 13 14 (+i) ( -) i ( ) [ - ii + ) ( + ) [-] 03 12 56 78 10 1112 13 14 (+i) [+] i [+] [ - ii ( + ) ( + ) [-] 03 12 56 78 10 1112 13 14 [-i] ( -) i [+] [ - ii ( + ) ( + ) [-] 03 12 56 78 10 1112 13 14 (+i) [+] i ( ) ( + ii + ) ( + ) [-] 03 12 56 78 10 1112 13 14 [-i] ( -) i ( ) ( + ii + ) ( + ) [-] 03 12 56 78 10 1112 13 14 [-i] [+] i [+] ( + ii [ ] [-] [-] 03 12 56 78 10 1112 13 14 (+i) ( -) i [+] ( + ii -] [-] [-] 03 12 56 78 10 1112 13 14 [-i] [+] i ( ) [ - ii [ ] [-] [-] 03 12 56 78 10 1112 13 14 (+i) ( -) i ( ) [ - ii -] [-] [-] 03 12 56 78 10 1112 13 14 (+i) [+] i ( ) ( + ii -] [-] [-] 03 12 56 78 10 1112 13 14 [-i] ( -) i ( ) ( + ii -] [-] [-] 03 12 56 78 10 1112 13 14 (+i) [+] i [+] [ - ii [ -] [-] [-] 03 12 56 78 10 1112 13 14 [-i] ( -) i [+] [ - ii [ -] [-] [-] t37t7 s12 t13 t23 t33 t38 1 - 2 1 -2 0 1 -2 1 2 1 2 0 1 2 1 2 1 2 s3 1 2 0 1 2 1 2 1 2 s3 1 2 0 1 2 1 2 1 2 s3 1 2 0 1 2 1 2 1 2 s3 1 2 1 2 0 1 2 1 2 s3 1 2 1 2 0 1 2 1 2 s3 1 2 1 -2 0 1 2 1 2 s3 1 -2 1 -2 0 1 2 1 2 s3 1 2 0 1 2 0 1 s3 1 2 0 1 2 0 1 s3 1 2 0 1 2 0 1 s3 1 2 0 1 2 0 1 s3 1 2 1 2 0 0 1 s3 1 -2 1 2 0 0 1 s3 1 2 1 -2 0 0 1 s3 1 -2 1 -2 0 0 1 s3 1 2 0 1 ? 0 0 1 2 0 1 ? 0 0 1 0 1 ? 0 0 1 -2 0 1 -2 0 0 1 2 1 -2 0 0 0 1 -2 1 -2 0 0 0 1 2 1 ? 0 0 0 1 2 1 2 0 0 0 t4 Y Q 1 1 2 1 6 1 3 1 3 1 6 1 3 1 3 1 6 2 3 2 3 1 6 2 3 2 3 1 6 1 6 1 3 1 6 1 6 1 3 1 1 2 1 1 2 1 6 1 3 1 3 1 6 1 3 1 3 1 6 2 3 2 3 1 6 2 3 2 3 1 6 1 6 1 3 1 1 1 3 1 1 2 1 1 2 1 ? 1 1 1 ? 1 1 1 ? 0 0 1 2 0 0 1 2 1 2 0 1 2 1 2 0 1 ? 1 ? 1 1 2 1 2 1 Table 6.3. Thelefthanded(r 03,1 group, manifesting the subgroup SO (7, 1 1, Eq. (6.53)) multiplet of spinors — the members of the fundamental representation of the SO ( 13, 1 ) of the colour charged quarks and anti-quarks and the colourless leptons and anti-leptons — is presented in the massless basis using the technique presented in App. 6.5. It contains the left handed (F (3,1) = — 1 ,App.6.5)weak(SU (2 ) j )charged(T13 = ± 2 Eq. (6.38)), and SU (2 ) n chargeless (t t23 _ , 0, Eq. (6.38)) quarks and leptons and the right handed ( F ( 3,1 7 1, weak (SU (2 ) j) chargeless and up and down (± 2, respectively). Quarks distinguish from leptons only 33 _ r < 1 —1— ) ( — 1 —1— ) (o--L )i Eq (6 2)) SU (2 ) jj charged (t in the SU(3) x U(1) part: Quarks are triplets of three colours (c 1 = (t33,t38) carrying the "fermion charge" (t4 = 6 ,Eq. (6.2)). The colourless leptons carry the "fermion charge" (t4 = —2). The same multipletcontains also the left handed weak (SU (2 ) j) chargeless and SU(2)jj charged anti-quarks and anti-leptons and the right handed weak (SU (2 ) j) charged and SU(2)jj chargeless anti-quarks and anti-leptons. Anti-quarks distinguish from anti-leptons again only in the SU(3)xU(1 ) part: Anti-quarks are anti-triplets, carrying the "fermion charge" (t4 = —6). The anti-colourless anti-leptons carry the "fermion charge" (t4 = 2). Y = (t23 + t4 ) is the hyper charge, the electromagnetic charge is Q = ( t13 + Y). The states of opposite charges (anti-particle states) are reachable from the particle states besides by Sab also by the application of the discrete symmetry operator c^/* P, presented in Refs. [43,44]. The vacuum state, on which the nilpotents and projectors operate, is not shown. The reader can find this Weylrepresentation also in Refs. [5,15,16,4] and in the references therein. Taking into account Table 6.3 and Eqs. (6.49, 6.58) one easily finds what do 78 operators y0 (±) do on the left handed and the right handed members of any 40 R c 2 L 41 42 L c 2 L 43 c 2 L 44 45 R c 2 R 46 47 R c 2 R 48 49 L c 3 L 50 c 3 L 51 52 L c 3 R 53 54 R c 3 R 55 56 R 57 L 58 L 59 L 60 L 61 k 62 k 63 R 64 R 2 3 family i = (1,2,3,4). ,0 78 > = > , Y0 (-) ^Ur,Vr - - i H-uL,VL 78 Y0 (+) K.,Vl > = Kr,Vr 78 Y0 ( + ) ^TdR,eR > = Wll,el >, 78 >, Y dl,e, > = ^Tdr,er > . (6.12) We need to know also what do operators ('f1± = t11 ± if12, t13) and (N± = ] ± iNNL, INL) do when operating on any member (uL,R, vL,R, dL,R, eL,R) of a particular family i = (1,2,3,4). Taking into account, Eqs. (6.47, 6.48, 6.58, 6.60, 6.51, 6.40, 6.41), 03 12 56 78 N± =-(Ti)(±), NL = 1 (S12 + iS03), r1± = (t) (±)(t) 1 13 = 1 (S56 - S78), ab ab -k) (k) = -inaa ab ab ab (k) [k] = i (k), ab 1 ab [k], T ab ab (k) (k)= 0, naa (k) = 2 (Ya + Vyb ab ab (k) [-k]= 0, ab [k]= 1 (1 + kYaYb)< (6.13) one finds N + |V > = >, N + |V >= 0, N- |^2 > = l^1 >, N- l^1 >= 0, N + |^3 > = >, N- > = |V >, T1+ |V > = |^3 >, N + |^4 >= 0, N- |V >= 0, >= 0, f1 + |^3 > = >, T1-|V >= 0, =1^1, .4 r1+ U ,2 |^4 > = |V >, > = |V >, 1 ?1- 1,1.2 |^2 >= 0, >= 0, f1+ U ,4 ]NL |V > = -2 |V >, ]NL |^2 |^2 > 2 ]N3|^3 > = -2|^3 >, N3|^4 >=+1|^4 >, t13|V > = -1^1 >, t13|V >=-2|^2 >. f13|V > = + 2|^3 >, f13|^4 >=+ 2|^4 > (6.14) independent of the family member a = (u, d, v, e). The dependence of the mass matrix on the family quantum numbers can easily be understood through Table 6.2, where we notice that the operator INL transforms the first family into the second (or the second family into the first) and the third family to the fourth (or the fourth family into the third), while the operator f1 § transforms the first family into the third (or the third family into the first) and the second family into the fourth (or the fourth family into the second). The application of these two operators, INL and f1is presented in Eq. (6.14) and demonstrated in the diagram 'V V2 V V4 J f1 ±. (6.15) The operators N L and f 1 3 are diagonal, with the eigenvalues presented in Eq. (6.14): INL has the eigenvalue — 2 on 1 > and |V3 > and + \ on |V2 > and |V4 >, while f 13 has the eigenvalue — 1 on 1 > and |V2 > and +1 on |V3 > and |V4 >. If we count 1 as a part of these diagonal fields, then the eigenvalues of both operators on families differ only in the sign. The sign and the values of Q, Q' and Y' depend on the family members properties and are the same for all the families. _nl± _N 3 _ iS Let the scalars (A 78 a, /A 7l , A 78 , A '738 ) be scalar gauge fields of the opera- (±) (±) (±) (±) tors (IN±, INl, T^, f13), respectively. Here A 78 = A7 ^i A8 for all the scalar gauge fields, while A^ = 1 (A™l1 Ti A^^ respectively, and Afj = 1 (A 1718 A 1728 ), (±) (±) (±) (±) (±) (±) respectively. All these fields can be expressed by clabc, as presented in Eq. (6.45), provided that the coupling constants are the same for all the spin connection fields of both kinds, that is if no spontaneous symmetry breaking happens up to the weak scale. We shall from now on use the notation A^1 instead of A^j for all the operators (zt) with the space index (7,8). In what follows we prove that the symmetry of the mass matrix of any family member a remains the same in all orders of loop corrections, while the symmetry in all orders of corrections (which includes besides the loop corrections also the repetition of nonzero vacuum expectation values of the scalar fields) remains unchanged only under certain conditions. In general case the break of symmetry can still be evaluated for small absolute values of aa, Eq. (6.11). We shall work in the massless basis. Let us introduce the notation O for the operator, which in Eq. (6.10) determines the mass matrices of quarks and leptons. The operator O is equal to, Eq. (6.10), O = ^ y0 (±) (- ^ Ta A± - ^ TAi AAi), + ^ a A i Ta A± = (QAQ,Q' aQ', Y' A±'), TAi AAi = (T1 i Ai\NL ANLi), {Ta,Tp}_ = 0, {TAi,TBj}- = iSAB fijkTAk , {Ta,TB}- = 0. (6.16) Each of the fields in Eq. (6.16) consists in general of the nonzero vacuum expectation value and the dynamical part: AAi = (< A^- > +A±i(x), < A^Li > +ANLi(x), < A± > +A± (x)), where a common notation for all three singlets is used, since their eigenvalues depend only on the family members (a = (u, d, v, e)) quantum numbers and are the same for all the families. We further find that 78 78 78 {y0 (±), T4}- = 0, {y0 (±), T1}- = 0, {y0 (±),NL}- = 0, 78 78 78 78 {y0 (±), T13}- = —2 y0 (±) S78 , {y0 (±), T23}- = +2y0 (±) S78. (6.17) To calculate the mass matrices of family members a = (u, d, v, e) the operator O must be taken into account in all orders. Since for our proof the dependence of the operator O on the time and space does not play any role (it is the same for all the operators), we introduce the dimensionless operator O, in which all the degrees of freedom, except the internal ones determined by the family and family members quantum numbers, are integrated away 14. Then the change of the massless state of the ith family of the family member a of the left or right handedness (L,R), r >, changes in all orders of corrections as follows 00 ( —1)n n2n+1 UI^R > = i X ( (2n + 1), KR > . (6.18) n=0 ( + )! In Eq. (6.18) |^nLiR1 > represents the internal degrees of freedom of the ith i (1,2,3,4), family state for a particular family member a in the massless basis. The mass matrix element in all orders of corrections between the left handed ath family member of the ith family < I and the right handed ath family member of the jth family >, both in the massless basis, is then equal to U I^Ri >• Only an odd number of operators O2n+1 contribute to the mass matrix elements, transforming > into I^j*5 > OT opposite. The product of an even number of operators O2n does not change the handedness and consequently 14 O is measured in TeV units (as all the scalar and vector gauge fields). If the time evolution is concerned then O = O • (t — t0)/TeV is in units h = 1 = c dimensionless quantity. We assume that also the integration over space coordinates is in < 1 > already taken into account, only the integration over the family and family members is left to be evaluated. contributes nothing. Correspondingly without the nonzero vacuum expectation values of scalar fields all the matrix elements would remain zero, since only nonzero vacuum expectation values may appear in an odd orders, while the contribution of the loop corrections always contribute to the mass matrix elements an even contribution (see Fig. (6.1)). Our purpose is to show how do the matrix elements behave in all orders of corrections CO (_i)n 4 4 < = i x 72_rn7 i L o i ><^ak,i L o i ^ak2 > n=0 ( '' ki=1 k2 = 1 4 <^akni y o> x . (6.19) yL k =1 Let be repeated again that all the matrix elements <^£kl |Ol^2 > or < ^kiiL k2 = 1 only evaluate the internal degrees of freedom, that is the family and family members ones, while all the rest are assumed to be already evaluated. Since the mass matrix is in this notation the dimensionless object, also all the scalar fields are already divided by the energy unit (let say 1 TeV). We correspondingly introduce the dimensionless scalars (AQ, AQ , A^'), A^, AN L. The only operators Ta, distinguishing among family members, are (t4, t13, t23 ), included in Q = (t13 + Y), Y = (t23 + t4), Q' = (t13 _ Ytan2 and in Y' = (t23 _ t4 tan2 $2). All the operators contributing to the mass matrices of 78 each family member a have a factor y° (±), which transforms the right handed family member to the corresponding left handed family member and opposite. When taking into account O2n+1 in all orders, the operators Ta A", Ta = (Q, Q', Y'), contribute to all the matrix elements, the diagonal and the off diagonal ones. To simplify the discussions let us introduce a bit more detailed notation O = x O1 = Oa + Oi3 + ON3 + O+ ONl@ i 78 , Oa y0 (±) (Q aQ,q ' aQ ,y ' A±), +,- 78 i3 A i3 O13 =—Y_ Y0 (±) T13 A ._ 78 ONl3 =— ^ Y0 (it) N3 AN O 7S = — X y0 (±) T7S A± O= — X y0 (±) N± ANL± . (6.20) We shall use the notation for the expectation values among the states < ^{J =< i|, I^R >= Ij > for the zero vacuum expectation values and the dynamical parts as follows: 78 i. < i|Oa|j > =< i| y0 (±) Ta(< A± > +A±(x))|j >. -13 , 78 _ ii. = +A±3(x))|j >. -NL3 78 _ _ iii. < i|O Ij >=< i| _ L+,- y0 (±) NL(< ANl3 > +ANL3(x))|j >. iv. < i|Oi±|j > =< i| _ L+,- y0 (±) ti@(< A> +A±±(x))|j >. v. < i|ONL±|j >=< i| _ L+,_ Y0 (±) N±(< ANl± > +ANL±(x))|j >. „a 78 . vi. < i|Odia|i >=< i| L+,- Y0 (±) {Ta(< A± > +A± (x))_ t13(< A±3 > _M3 f — Nl3 .A nl3f (< A± >,< A±3 >,< ANl3 >,< A>,< AAN'L± >) represent nonzero +A±3(x)) — N3(< ANl3 > +ANL3(x))}|i >. Nl ~ T, ~ N 3 ~ 1W ~ Nl± vacuum expectation values and (A± ( x), AA± (x), AAnl (x), AA± (x), AA± (x)) the corresponding dynamical fields. In the case i. < A± > represent the sum of the vacuum expectation values of (QaAQ±), Q/aAQ'(±), Y/aAj±)) of a particular family member a, where (Qa, Q/a, Y/a) are the corresponding quantum numbers of a family member a. A" (x) represent the corresponding dynamical fields. In the case vi. we correspondingly have for the four diagonal terms on the tree level, that is for n = 0 in Eq. (6.19) (after taking into account Eq. (6.14): < 1 | OOiaI 1 > = aa — (ai + a2), < 2 | O^ I 1 > I2 >= aa — (ai — a2), < 3 | O^ I3 >= aa + (a1 — a2) and < 4|Oaia|4 >= aa + (a1 + a2), where (a1, a2, aa) represent the nonzero vacuum expectation values of 2 ^ (< AA 1+ > + < AA 1— >), 2 ^ (< AAN+3 > + < AaN-)3 >), 2 ^72 (< Aa+) > + < A(-j >), all in dimensionless units. We are now prepared to show under which conditions the mass matrix elements for any of the family members keep the symmetry SU(2) x SU(2) x U(1) at each step of corrections, what means that the values of the matrix elements obtained in each correction respect the symmetry of mass matrices on the tree level. We use the massless basis |^L,r >, making for the basis the choice (I^L > + >). , The diagrams for the tree level, one loop and three loop contributions of the operator O, determining the masses of quarks and leptons, Eqs. (6.16, 6.20), are presented in Fig. (6.1). 0 -M- ! rR3 1 + <5/ O i>R é L Fig. 6.1. The tree level contributions, one loop contributions (not all possibilities are drawn, the tree level contributions occurs namely also to the left or to the right of the loop, while to O three singlets and two triplets, presented in Eq. (6.16), contribute) and two loop contributions are drawn (again not all the possibilities are shown up). Each (i, j, k, l, m... ) determines a family quantum number (running within the four families — (1,2,3,4)), a denotes one of the family members (a = (u, v, d, e)) quantum numbers, all in the massless basis . Dynamical fields start and end with dots •, while x with the vertical slashed line represents the interaction of the fermion fields with the nonzero vacuum expectation values of the scalar fields. 6.2.1 Mass matrices on the tree level Let us first present the mass matrix on the tree level for an ath family member, that is for n = 0 in Eq. (6.19). Taking into account Eq. (6.14) one obtains for the diagonal matrix elements on the tree level (for n = 0 in Eq. (6.19)) [aa - (a 1 + a2), aa - (a 1 - a2), aa + (a 1 -a2), aa + (a 1 + a2)], respectively. The corresponding diagrams are presented in Fig. (6.2). a i — a2 + a a ^L1 n1 -a1 + a2 + aa ►-X-►- ^f ^a2 i? ai — a2 + a a -X- a1 + a2 + aa rL4 a4 R Fig. 6.2. The tree level contributions to the diagonal matrix elements of the operator Ô Jia, Eq. (6.20). The eigenvalues of the operators N L and T13 on a family state i can be read in Eq. (6.14). Taking into account Eq. (6.14) one finds for the off diagonal elements on the tree level: < V |..|V > =< V|..|^4 > =< >f= < > =< ANlb >, < V |..|^3 > =< ^2|..|^4 > =< V|..|V >f= < ^4|..|^2 > =< A1b >. The corresponding diagrams for < ^1|..|^2 >, < ^2|..|^1 >, < ^2|..|^3 > and < ^3|..|^2 > are presented in Fig. (6.3). The vacuum expectation values of this matrix elements on the tree level are presented in the mass matrix of Eq.(6.22). (5 -x— ô î+ -x- ^f^t ) ^rW) (5 —X- < (^R3) ô1--X- Fig. 6.3. The tree level contributions to the off diagonal matrix elements of the operators ^ ^ n 0 ~ ± iiil O and O NLH, Eq. (6.20) are presented. The application of the operators N L and T □ on a family state i can be read in Eq. (6.14). The contributions to the off diagonal matrix elements < 1..|^4 >, < ^2|..|^3 >, < ^3|..|^2 > and < ^4|..|^1 > are nonzero only, if one makes three steps (not two, due to the left right jumps in each step), that is indeed in the third order of correction. For < 1..|^4 > we have (in the basis ^ O^l > + >) and with the ~NtS 1 ~NL0 ~Nl0 notation < A LB >= (< A > + < A f-1a >) after we take intoaccount that l(+) 78 Y0 (±) transform the right handed family members into the left handed ones and opposite): < V | t 1 S < A1 ± > |Vk >< Vk| N ± < A N l@ > |V4 > < V4I (ai + + aa)|V4 >. There are all together six such terms, presented in Fig. (6.4), since the diagonal term appears also at the beginning as (-a 1 — a2 + aa) and in the middle as (a 1 — a2 + aa), and since the operators Y.+,- T1 ± < A1 S > and Y. + - 11S < ANls > appear in the opposite order as well. We simplify the notation from |Vk > to |k >. Summing all these six terms for each of four matrix elements (< 11..|4 >, < 2|..|3 >, < 3|..|2 >, < 4|..|1 >) one gets (taking into account Eqs. (6.19, 6.14)): < 11..|4 > = aa < A1 b > < ANlb > , < 2|..|3 > = aa < A 1b > < ANlb > , < 3|..|2 > = aa < A 1 b > < ANlb > , < 4|..|1 > = aa < A 1b >< Anlb >. (6.21) Each matrix element is in Eq. (6.21) divided by 3!, since it is the contribution in the third order! One notices that < 4|..|1 >t=< 1|..|4 > and < 3|..|2 >t=< 2|..|3 >. These matrix elements are included into the mass matrix, Eq. (6.22). To show up the symmetry of the mass matrix on the lowest level we put all the matrix elements in Eq. (6.22). "M(o) = -a 1 -a2 +aa _» I in t ffl ^ iixi-^a „a I —«I i+a2+a a a«<ÀAT œ> ,a« > a aa a 1 —««2 +a a 1 +«2+aa (6.22) Mass matrix is dimensionless. One notices that the diagonal terms have on the tree level the symmetry < ^ 11 > + < >= 2 aa = < > + < >, and that in the off diagonal elements with "three steps needed" the contribution of the fields, which depend on particular family member a = (u, d, v, e), enters. We also notice that < >f=< l..^1 >. We see that < 1|..|3 >=< 2|..|4 > =< 3|..|1 >t=< 4|..|2 >t,that < 11..|2 >=< 3|..|4 >=< 2|..|1 >t=< 4|..|3 >t and that < 4|..|1 >t=< 11..|4 > and < 3|..|2 >t=< 2|..|3 >, what is already written below Eq. (6.21), < i|..|j > denotes < >. In the case that a =< A 1b >=< A 1b >= e and < A'lb >=< A'lb >= d, which would mean that all the matrix elements are real, the mass matrix simplifies to Mo) /—a i — a 2 + a0 d e b d -a i + a 2 + aa b e ai eb be a2 + aa d d a i + a2 + aaJ (6.23) with b = aaed. 0diag X 0 —x- ONL-—X- n1 n1 n3 O 1- -x- Odiag X (5nl- -x- n4 O1- ONL- Od"iag -X--X-—x-► VL n4 K Odiag ONL- 01-X--X-—-X— vl ^f n2 K ONL- Od"iag 01- —x——x——x—► vl n2 K 0Nl- 01- Od"iag —X——x——x— ^L < n4 ^t Fig. 6.4. The tree level contribution to the matrix element < > is presented. One comes from < V | to |^4 > in three steps: < V | Y+,- < Ä^S > Yk >< Ü+- N± < Änl± > |^4 > < (ai + a2 + aa)|V >. There are all together six such terms, since the diagonal term appears also at the beginning as (— a1 — a2 + aa) and in the middle as (a1 — a2 + aa), and since the operators Y + - < Ä1^ > and - N± < Änl± > appear in the opposite order as well. 6.2.2 Mass matrices beyond the tree level We discuss in this subsection the matrix elements of the mass matrix in all orders of corrections, Eq. (6.19), the tree level, n = 0, of which is presented in Eq. (6.22). The tree level mass matrix manifests the SU(2) x SU(2) x U(1) symmetry as seen in Eq. (6.22), with (< 1 |x|1 > + < 4|x|4 >) — (< 2|x|2 > + < 3|x|3 >) = 0 and < 1 |x|3 >=< 2|x|4 >=< 3|x|1 >t =< 4|x| 1 > and with (< 1|xxx|4 >, < 2|xxx|3 >, < 3|xxx|2 >, < 4|xxx|1 >) related so that all are equal if < A> and < A> are real. Let us repeat that the generators of the two groups which operate among families commute: {t1l, N5l}_ = 0, and that these generators commute also with generators which distinguish among family members: {t1t, xa}_ = 0, {xa, N5L}- = 0, where Ta represents (Q,Q', Y') (or t4,t23,t13). To study the symmetry SU(2) x SU(2) x U(1) of the mass matrix, Eq. (6.22), in all orders of loop corrections, of repetition of nonzero vacuum expectation values and of both together — loop corrections and nonzero vacuum expectation values — we just have to calculate at each order of corrections the difference between each pair of the matrix elements which are equal on the three level, as well as the Hermitian conjugated difference of such a pair. Since the dependence of all the scalar fields on ordinary coordinates are in all cases the same, we only have to evaluate the application of the operators to the internal space of basic state, that is on the space of family and family members degrees of freedom. Correspondingly we pay attention only on this internal part — on the interaction of scalar fields with the space index (7,8) with any family member of any of four families separately with respect to their internal space. The dependence of the mass matrix elements on the family member quantum numbers appears through the nonzero vacuum expectation value aa, Eq. (6.22), as well as through the dynamical part of OOa, Eq. (6.20). We demonstrate in this subsection how does the repetition of the nonzero vacuum expectation values of the scalar fields and loop corrections in all orders influence matrix elements, presented on the tree level in Eq. (6.22). In the case that aa = 0 (that is for < AQ >= 0, < AQ' >= 0 and < AY' >= 0) the symmetry in all corrections, that is in all loop corrections and all the repetition of nonzero vacuum expectation values of the scalar fields, and of both — the loop corrections and the repetitions of nonzero vacuum expectation values nonzero of all the scalar fields except aa — keep the symmetry of the tree level, presented in Eq. (6.22). We prove in this subsection that in the case that < AQ >= 0, < AQ >= 0 and < Ay >= 0, that is for aa = 0, the symmetry of mass matrices remains unchanged in all orders of corrections: the loop ones of dynamical fields — AQ, AQ , AY , ANl , A1 — in the repetition of nonzero vacuum expectation values of the scalar fields carrying the family quantum numbers — < ANl > and < A1 > — and of all together. The symmetry of mass matrices remains in all orders of corrections the one of the tree level also if aa = 0 while a 1 = 0 and a2 = 0. The symmetry changes if the nonzero vacuum expectation values of all the scalar fields are nonzero. In the case, however, that aa = 0, the matrix elements, which are in the lowest order proportional to aa in Eq. (6.22), remain zero in all orders of corrections, while the nonzero matrix elements become dependent on family members quantum numbers due to the participations in loop corrections in all orders of the dynamical fields Aq, Aq ' and AY'. We study in what follows first the symmetry of mass matrices in all orders of corrections in the case that aa = 0, and then the symmetry of the mass matrices, again in all orders of corrections, when aa = 0. We also comment that the symmetry of the tree level remain the same in all orders of corrections, if aa = 0, while ai = 0 = a2. Mass matrices beyond the tree level, if aa = 0 We study corrections to which the scalar fields which distinguish among families, contribute — with their nonzero vacuum expectation values < ANl > and < A1 > and their dynamical parts ANl and A1 — while we assume aa = 0 (aa denotes the vacuum expectation values to which the tree singlet fields, distinguishing among family members, contribute, that is (< Aq >, < Aq >, < Ay >), taking into account the loop corrections of the corresponding dynamical parts (AQ, AQ , AY ) in all orders. We show that in such a case — that is in the case that aa = 0 while all the other scalar fields determining mass matrices have nonzero vacuum expectation values (ai = 0, a2 = 0, < A1l@ >= 0, < A1 S >= 0) — the matrix elements, evaluated in all orders of corrections, keep the symmetry of the tree level. We also show, that in this case the off diagonal matrix elements, represented in Eq. (6.22) as (aa < A 1b >< A1lb >, aa < A 1b >< A1lffl >, aa < A 1 ffl >< A 11 lb >, aa < A 1 >< A 11 >), remain zero in all orders of corrections. Let us look how the corrections in all orders manifest for each matrix element separately. i. We start with diagonal terms: < .....|V >, i = (1,2,3,4). On the tree level the symmetry is: {< v i < 6d«ia > i v > + < v4i < 6d«a > i v >} - {< v2I < 6d«ia > i v > +{< v3i < 6daia > IV3 >} = 0. i.a. It is easy to see that the tree level symmetry, {< V i < 66aia > iV > + < v4i < 6 d«a > iv4 >} - {< v2i < 6 daia > i v2 > + < v3i < 0 aia > i v3 >} = 0, remains in all orders of corrections, if only the nonzero vacuum expectation values _ - _ - 78 - _ - of < A13 >= a 1 and < ANl3 >= a2 contribute in operators y0 (±) t1 3 < A1 3 > 78 1 and y0 (±) NNL < A 11 l3 >. At, let say, (2k + 1)st order of corrections we namely have {(-(a 1 +a2))(2k+1)+(a 1 +a 2)(2k+1)} - {(-(a 1 - a 2))(2k+1)+(a 1 -a 2)(2k+1)} = 0. i.b. The contributions of the dynamical terms, either (AQ, AQ , AY ) or (A13, A11l3) do not break the three level symmetry. Each of them namely always appears in an even power, Fig. (6.1), changing the order of corrections by a factor of two or 2n ( iAa i 2(n-k-1), i A13i 2k, i A11 l3| 2l), where (n - k - 1, k, 1) are nonnegative integers, while TAa represents (Qa, Q/a, Y/a). The contribution to | Aa | 2m,m = (n - k - 1), origins in the product of | AQ| 2(m-P-r) • | AQ'| 2p • | AY'| 2r. Again (m -p - r, p, r) are nonnegative integers. i.c. There are also other contributions, either those with only nonzero vacuum expectation values or with dynamical fields in addition to nonzero vacuum expectation values of scalars, in which 61 S and 6Nl@ together with all kinds of diagonal terms contribute. Let us repeat again what do the operators 01S and ONl@, Eq. (6.20), do when they apply on V1. The operators O1 b transforms V1 into V3 and V2 into V4. Correspondingly the states V1 and V4 take under the application of O1 b the role of V2 and V3, while V2 and V3 take the role of V1 and V4, all carrying the correspondingly changed eigenvalues of t1 3. The operator 011 lffl transforms V1 into V2 and V3 into V4. Correspondingly the states V1 and V2 take under the application of 011l® the role of V3 and V4, while V3 and V4 take the role of V1 and V2, carrying the correspondingly changed eigenvalues of nl3. Either the dynamical fields or the nonzero vacuum expectation values of these scalar fields, O1 S and 01 L±l, must in diagonal terms appear in the second power or in nx the second power. We easily see that also in such cases the tree level symmetry remains in all orders. i.c.1. To better understand the contributions in all orders to the diagonal terms, discussing here, let us calculate the contribution of the third order corrections either from the loop or from the nonzero vacuum expectation values to the diagonal matrix elements < > under the assumption that aa = 0. Let us evaluate the contributions of the operators < 013 >, ONl3, 01@ and ONl@ in the third order. We see that t1b transforms V3 into V1 and V4 into V2, while t1b transforms V2 into V4 and V1 into V3- We see that N^ transforms V2 into V1 and V4 into V3, while N® transforms V1 into V2 and V3 into V4. It then follows that {< V |xxx|V > + < ^4|xxx|^4 >} — {< ^2|xxx|^2 > + < ^3|xxx|^3 >} = 0, where xxx represent all possible acceptable combination of < 01@ >, < ONl@ > and the diagonal terms < 013 > and < 0Nl3 >. One namely obtains that the contribution of {< V |xxx|V > + < ^4|xxx|^4 >} = {| < A1B > |2 [—2(a1 + a2) + (a1 — a2)] + | < ANlb > |2[—2(a1 + a2) — (a1 — a2)] + (—(a1 + a2)3) + | < A1B > |2[+2(a1 + a2) —(a1 — a2)] +1 < ANl- > |2[+2(a1 + a2) + (a1 — a2)J + (a1 + a2)3} = 0, and for {< V2|xxx|V2 > + < V3|xxx|V3 >} one obtains = {| < A1B > |2[—2(a1 — a2) + (a1 + a2)] + | < ANlb > |2[—2(a1 — a2) — (a1 + a2)] + (—(a1 — a2)3) + | < A1B > |2[+2(a1 — a2) — (a1 + a2)] + | < ANl- > |2[+2(a1 — a2) + (a1 + a2)] + (a1 — a2)3} = 0. Also the dynamical fields keep the tree level symmetry of mass matrices. To prove one only must replace in the above calculation | < A1B > |2 by |A1B|2 and | < ANlb > |2 by |ANlb|2. To the diagonal terms the three singlets contribute in absolute squared values (|Aq|2, |Aq |2, |Ay |2, each on a power, which depend on the order of corrections. This makes all the diagonal matrix elements, < V1|.....|V1 >, < V2|.....|V2 >, < V3|.....|V3 > and < V4|.....|V4 >, dependent on the family member quantum numbers. Such behaviour of matrix elements remains unchanged in all orders of corrections, either due to loops of dynamical fields or due to repetitions of nonzero vacuum expectation values. The reason is in the fact that the operators < 01@ > and < 0Nl@ > exchange the role of the states in the way that the odd power of diagonal contributions to the diagonal matrix elements always keep the symmetry {< V1 |U|V1 > + < V4|U|V4 >} — {< V2|U|V2 > + < V3|U|V3 >} = 0. These proves the statement that corrections in all orders keep the symmetry of the tree level diagonal terms in the case that aa = 0. ii. Let us look at matrix element < V1|.....|V3 > and < V2|.....|V4 > in Eq. (6.22), where we have on the tree level < 1 |x|3 >=< 2|x|4 > and < 3|x|1 >=< 4|x|2 >=< 1 |x|3 >t. We again simplify the notation < V1|.....> into < i|...|j >. The two matrix elements — < 1 |x|3 >, < 2|x|4 > — are on the tree level denoted by < A1B >, while < 3|x|1 > and < 4|x|2 > are denoted by < A1B >. We have to prove that corrections, either of the loops kind or of the repetitions of the nonzero vacuum expectation values or of both kinds in any order keeps the symmetry of the tree level. ii.a. Let us start with the corrections in which besides < A1 b > in the first power only < A13 >= a1 and < ANl3 >= a2 contribute, the last two together appear in an even power so that all three together contribute in an odd power. The contribution of (< 1|x|1 >)2k+1 = (—(a1 + a2))2k+1 in the (2k + 1)th order is up to a sign equal to (< 4|x|4 >)2k+1 = (a1 + a2)2k+1, where k is a nonnegative integer, while the contribution of (< 2|x|2 >)2k+1 = (—(a1 — a2))2k+1 is up to a sign equal to (< 3|x|3 >)2k+1 = (a1 — a2)2k+1. In each of the matrix elements, either < 11.....|3 > or < 2|.....|4 >, both factors together, (—(a1 + a2))m (a1 — a2)n in the case < 11.....|3 > and (—(a1 — a2))m (a1 + a2)n in the case < 2|.....|4 >, with (m + n) an even nonnegative integer (since together with < A1 b > must be of an odd integer corrections to take care of the left/right nature of matrix elements) one must make the sum over all the terms contributing to corrections of the order (m + n + 1). It is not difficult to see that the contribution to < 11.....|3 > is in any order of corrections equal to the contributions to the same order of corrections to <2|.....|4>. ii.a.1. To illustrate the same contribution in each order of corrections to < 11.....|3 > and to < 2|.....|4 > let us calculate, let say, the third order corrections. The contribution of the third order to < 1 |xxx|3 > is — J < A1b > {(ai + a2)2 + (a1 — a2)2 — (a1 — a2)(a1 + a2)} and the contribution of the third order to < 2|xxx|4 > is — J- < A1B > {(a1 —a2)2 + (a1 +a2)2 — (a1 +a2)(a1 — a2)}, that is the contributions in the third order of < 1 |xxx|3 > and < 2|xxx|4 > are the same. ii.b. One can repeat the calculations with < A1B > and the dynamical fields A1 b and A1ffl, with or without the diagonal nonzero vacuum expectation values. In all cases all the contributions keep the symmetry on the tree level due to the above discussed properties of the diagonal terms. All the dynamical terms must namely appear in absolute values squared in order to contribute to the mass matrices, as shown in Fig. 6.1. To the diagonal terms the three singlets contribute in absolute squared values (|AQ|2, |AQ |2, |AY |2), each on some power, depending on the order of corrections. This makes the matrix element < 11.....|3 > and < 2|.....|4 >, < 3|.....|1 > and < 4|.....|2 >, dependent on the family members quantum numbers. In all cases all the contributions keep the symmetry on the tree level. ii.c. The Hermitian conjugate values < 1|.....|3 >*=< 2|.....|4 > have the transformed value of < A1B >, that means that the value is < A1ffl >, provided that the diagonal matrix elements of the mass matrix are real, keeping the symmetry of the matrix elements < 11.....|3 > =< 2|.....|4 > in all orders of corrections. These proves the statement that corrections in all orders keep the symmetry of the tree level of the off-diagonal terms < 11.....|3 > and < 2|.....|4 > and of their Hermitian conjugated matrix elements in the case that aa = 0. iii. Let us look at matrix element < 11.....|2 > and < 3|.....|4 > in Eq. (6.22), where we have on the tree level < 1 |x|2 >=< 3|x|4 >. These two matrix elements are on the tree level denoted by < ANlb >. We have to prove that corrections, either the loop corrections or the repetitions of the nonzero vacuum expectation values or both kinds of corrections, in any order, keep the SU(2) xSU(2) x U(1) symmetry of the tree level. The proof for the symmetry of these matrix elements is carried out in equivalent way to the proof under ii. . iii.a. Let us start with the corrections in which besides < AN lb > in the first power also only < A13 >= a 1 and < ANl3 >= a2 contribute. The sum of powers of the last two a must be even, so that a correction would be of an odd power due to the left/right transitions. Again the contributions of both diagonal terms, < 1|x|1 > and < 4|x|4 >, in any power — (< 1|x|1 >)2k+1 = (—(a 1 + a2))2k+1 and (< 4|x|4 >)2k+1 = (a 1 + a2)2k+1, where k is a nonnegative integer — differ only up to a sign when they appear in an odd power and are equal when they appear in an even power. These is true also for the contributions of < 2|x|2 > and < 3|x|3 > since (< 2|x|2 > )2k+1 = (—(a 1 — a2))2k+1 is up to a sign equal to (< 3|x|3 >)2k+1 = (a 1 — a2)2k+1. If they appear with an even power, they are equal. In each of the (m + n + 1 )th order corrections to the matrix elements, either < 11.....|2 > or < 3|.....|4 >, where (—(a 1 + a2))m (—(a 1 — a2))n contribute to < 11.....|2 > and (a 1 — a2)m (a 1 + a2)n contribute to < 3|.....|4 >, the two contributions are again equal, since both m and n are even nonnegative integers. iii.a.1. Let us, as an example, calculate the fifth order corrections to the tree level contributions of < 1 |x|2 > =< ANlb >. The contribution of the fifth order < 1 |xxxxx|2 > to < 1|x|2> is 5r < ANlb > {(—(a 1 —a2))4 + (—(a 1 +a2))4 +3(—(a 1 + a 2))(—(a 1 — a2))3+6(—(a 1 + 52 ))2(—(a 1 — a2))2 + 3(—(a 1 + a2))3(—(a 1—a2))},and the contribution of the fifth order < 3|xxxxx|4 > to < 3|x|4 > is 517 < ANlb > {(a 1 + a2)4+(a 1 — a 2)4 + 3(a 1 — a 2)(a 1 + a 2)3+6(a 1 — a2 )2(a 1 + a 2)2 + 3(a 1 — a 2)3 (a 1 + a2)}, which is equal to the contribution of the fifth order in the case of < 1 |xxxxx|2 >. iii.b. One can repeat the calculations with dynamical fields (ANlb, A 11l®) in all orders and with < A1 b > and with the diagonal nonzero vacuum expectation values and with the diagonal dynamical terms, paying attention that the dynamical fields contribute to masses of any of the family members only if they appear in pairs. To the diagonal terms the three singlets (AQ, AQ , AY ) contribute in the absolute squared values (|AQ |2, |AQ |2, |AY |2), each on a power, which depends on the order of corrections. In all cases all the contributions keep the symmetry on the tree level. iii.c. The proof is valid also for < 2|.....|1 >=(<1|.....|2 >) + and < 4|.....|3 >= (< 3|.....|4 >)t in any order of corrections. Namely, if diagonal mass matrix elements are real then in the matrix elements < 2|.....|1 > only < A11 lb > of the matrix element < 11.....|2 > must be replaced by < ANl® >. These proves the statement that corrections in all orders keep the symmetry of the tree level off-diagonal terms < 11.....|2 > and < 3|.....|4 > in the case that aa = 0. iv. It remains to check the matrix elements < 1 |.....|4 >, < 2|.....|3 >, < 3|.....|2 > and < 4|.....|1 > in all orders of corrections. The matrix elements on the third power, (< 1|xxx|4 >, < 2|xxx|3 >, < 3|xxx|2 >, < 4|xxx|1 >), appearing in Eqs. (6.21, 6.22), are for aa = 0 all equal to zero. It is not difficult to prove that these four matrix elements remain zero in all order of loop corrections. The reason is the same as in the above three cases, i., ii., iii.. The proof that the symmetry SU(2) x SU(2) x U(1) of the tree level remains unchanged in all orders of corrections, provided that aa = 0, is completed. There are in all these cases the dynamical singlets contributing in the absolute squared values (|AQ|2, |AQ |2, |AY |2 — each on a power, which depend on the order of corrections — which make that all the matrix elements of a mass matrix, except the (< 1|.....|4 >, < 2|.....|3 >, < 3|.....|2 >, < 4|.....|1 >) which remain zero in all orders of corrections, depend on a particular family member. Mass matrices beyond the tree level if aa = 0 We demonstrated that for aa = 0 the symmetry of the tree level remains in all orders of corrections, the loops corrections and the repetitions of nonzero vacuum expectation values of all the scalar fields contributing to mass terms, the same as on the tree level, that is SU(2) x SU(2) x U(1). Let us denote all corrections to the diagonal terms in all orders, in which the nonzero vacuum expectation values in all orders as well as their dynamical fields in all orders contribute when aa = 0 as: -(a + aa ):=<^a1|....|^a1 >,-(a -a,2):=<'a2|....|'a2 >, (a 1 - a*) :=< ^a3|....|^a3 >, (a 1 + a*) :=< ^a4|....|^4 >. We study for aa = 0 how does the symmetry of the diagonal and the off diagonal matrix elements of the family members mass matrices change with respect to the symmetry on the tree level, presented in Eq. (6.22), in particular for small values of |aa| in comparison with the contributions of all the rest of nonzero vacuum expectation values or of dynamical fields. We discuss diagonal and off diagonal matrix elements separately. The symmetry of all depends on aa. i. Let us start with diagonal terms: < .....>. On the tree level the symmetry is for aa = 0: {< V | < Oaia > > + < ^4| < oda >' >} -{< < oaia > ' > +{< v| < Oaia > ' >}=0. i.a. Let us evaluate the matrix elements < 'R1 ^-'a1 >• Let us denote for a while, just to simplify the derivations, n1 = aa - (a 1 + a2), n2 = aa - (a 1 - a2) n3 = aa + (a 1 - a2) n4 = aa + (a 1 + a2). One finds < 'a11—|'a1 >=[aa - (JJ 1 + JJ 2)] ^[(aa)3 - 3(aa)2(a1 + J2)+ 3(aa)(J1 + a.2)2] +1 [(aa)5 - 5(aa)4(a1 + J2)+ 10(aa)3(J1 + a.2)2 - 10(aa)2(Jj1 + JJ2)3 +5(aa)(a 1 + J2)4]- ••• . (6.24) Assuming that |aa| << (|(a 1|, |(a2|) it follows 3 f~ ~ >2 5 3T 1 + a 2)2 + 5! < 'a11—|'a1 >=-(a + J2)+ aa{1 - - (a +12)2 + - (a 1 + a.2)4 - ^(I 1 + a.2)6 + •••}. (6.25) Correspondingly we obtain for < 'a41—f r4 > in the limit that |aa| << (JJ 11, |a2 3 f~ ~ >2 5 3T 1 + a 2)2 + 5 7 <'a4|....|'a4 >=+(J + JJ 2) + aa{1 - - (J +a.2)2 + = (a 1 +J2)4 7! (a 1 + Ž2)6 + •••}. (6.26) For < ^a2|....|^i?2 > one obtains in the limit that |aa| << (|(a1|, |a2|) 3 5 < ^a2|...W2 >= -(a -12) + aa{i - -(a - a2)2 + -(a - a2)4 - ^(a - a2)6 + ••• (6.27) And for < > one obtains in the limit that |aa| << (|(a1|, |(a2|) the expression 3 5 < ^3|....|^f >= -(I1 -12)+ aa{1 - -(a -12)2 + -(a -12)4 7 7\ Finally we obtain -^(a -12)6 + •••}. (6.28) (<^|...K > + <^a4|...|^a4 > (<^a2|...|^a2 > + <^a3|...|^a3 >) = 1 4 aaa1 I2{1 - jj[(I1)2 + (a2)2]} + ••• . (6.29) The term with (aa)2 drops away. For small |aa| the term (aa)3 might be negligible. It is obvious that for aa = 0 the diagonal matrix elements do not keep the tree level symmetry of mass matrices (which is (< ^f11... ^r1 > + < ... |^R4 > ) - (< ^2|... > + < ^3|... >) = 0). But one sees as well thaR the contributions of higher terms to asymmetry are getting smaller and smaller and for |aa| << (|a11, |a2|) and for (|a11, |a2|) < 1, the first term is dominant and the non symmetry can be evaluated. ii. Let us look at the matrix element < 1|.....|3 > and < 2|.....|4 > in all orders of corrections in the case that aa = 0 (on the tree level, Eq. (6.22), < 1 |x|3 > =< 2|x|4 >=< 3|x|1 >*=< 4|x|2 >t) and let in this case < A1B > represent the matrix elements i< 11.....|3 > and < 2|.....|4 > in both cases in all orders of corrections. We namely showed that in this case the matrix element < 11.....|3 > is ~ 10 equal to < 2|.....|4 >= < A >. We now allow aa = 0. 10 Taking into account that in the case that aa is zero < A > includes all the corrections in all orders and that also a2 includes the corrections in all orders, we find (<^f1|...|^a3 > - <^a2|...|^a4 >) = < A10 > (1 + 3 a%{1 - j (I2)2 + •••} . (6.30) It is obvious that for aa = 0 also the non diagonal matrix elements do not keep the tree level symmetry of mass matrices (< ^f11... |^r3 > - < ... |^R4 >) = 0, which is not zero any longer). But one sees as well that the contributions of higher terms to asymmetry are getting smaller and smaller and for |aa| << |a2|, for | ia2| < 1, the first term in corrections is dominant. One can correspondingly evaluate the amount of non symmetry. iii. Let us look also at the matrix element < 11.....|2 > and < 3|.....|4 >, first in all orders of corrections in the case that aL = 0 (on the tree level, Eq. (6.22), < 1 |x|2 > =< 3|x|4 >=< 2|x|1 >*=< 4|x|3 >*) and let in this case < ANl0 > represent the matrix elements < 11.....|2 > and < 3|.....|4 > in all orders of corrections. We namely showed that in the case that aL = 0 the matrix element < 11.....|2 > is ~ 1 0 equal to < 3|.....|4 >= < A >. We now allow aL = 0. N1 0 Taking into account that for aL = 0 the matrix element < A l > includes corrections in all orders and that also a2 includes in this case corrections in all orders, one finds (<^|...|^a2 > — <^a3|...|^a4 >) = < ANl0 > (1 + 8aaai{1 — 5(ai )2 + • • •) . (6.31) It is obvious that for aL = 0 also these non diagonal matrix elements do not keep the tree level symmetry of mass matrices (< 1... |^r3 > — < ^f2| ... |^r4 >= 0 is no longer the case). But one sees as well that the contributions of higher terms to asymmetry are getting smaller and smaller and for |aa| << |a 11 and for |a 11 < 1, the first term in corrections is dominant and the non symmetry, the difference < ^a11... |^r3 > — < ^l2| ... |^r4 > can be evaluated. iv. It remains to check the matrix elements < 1|.....|4 >, < 2|.....|3 >, < 3|.....|2 > and <4|.....|1 >. The matrix elements which are nonzero only in the third order of corrections, (< 1|x|4 >= 0 =< 2|x|3 >= 0 =< 3|x|2 >=< 4|x|1 >, the first nonzero terms are < 1 |xxx|4 >, < 2|xxx|3 >, < 3|xxx|2 >, < 4|xxx|1 >), appearing in Eqs. (6.21, 6.22), which are for aL = 0 all equal to zero in all orders of corrections. We again take into account that for aL = 0 the matrix element < A1 @ > and < ANl± > include the corrections in all orders and that also a 1 and a2 include the corrections in all orders. We find when aa = 0 <^fi|...|^a4 > <^l2|...|^L3 > -10 ~N l0 ~ 10 ~N lffl < A >< A > < A >< A > <^a4|...|^a1 > _ <^a3|...|^a2> -¡•lffl ~N lffl ~iffl ~N l0 < A >< aa > < aa >< A > 3 —aa{1 — ^[(ai)2 + (a2)2] + •••} . (6.32) One sees that these off diagonal matrix elements keep the relations from Eq. (6.22) at least in the lowest corrections. We demonstrated that the matrix elements of the mass matrix of Eq. (6.22) do not keep the symmetry of the tree level in all orders of corrections if aL = 0, but the changes can in the case that (|aa|, |a 11, |a2|) are small in comparison with unity be estimated. Mass matrices beyond the tree level if a" = 0, while a1 = 0 = a2 One can easily see that the mass matrix of Eq. (6.22) keeps the symmetry in all orders of corrections also if aa = 0 and a 1 = 0 = a2. One obtains in this case for the diagonal terms < "j^U"r1 >, for each of four families (i = (1,2,3,4)) the expression <"aiU >= aa— 3{(aa)3 + aa(| < AA1 0 > |2 + | < A*l0 > |2 + |Aa|2 + |AAT3|2 + | AA1 0|2 + |A* l3| 2 + | AA* l01 2)} + 1 {(aa)5 + (aa)3(| < AA10 > |2 + | < AA*l0 > |2 + |Aa|2 + A3|2 + A0|2 + 5! |AA* l 312 + |AA n l0|2) + aa(| < AA 10 > |4 +1 < AA*l0 > |4 + |Aa|4 + |AA 13|4 + |AA 10|4 + |AAN l 314 + |AA* l0|4 + • • • + | < AA10 > |2| < AA* l 0 > |2 + •••)+ ••• } — 7{(aa)7 + (aa)5(| < AA 10 > |2 + • • •) + • • •} + • • • . (6.33) Let us denote the above expression for the diagonal terms < "j^U|"r1 >, which takes into account corrections in all orders while assuming a 1 = 0 = a2, with aa. (The definition of the scalar fields is presented in Eq. (6.20)). Let us add that the choice that the third components of the scalar fields A1 and A111l have no vacuum expectation values — < A 13 >= a 1 = 0, < A111 l3 >= a2 = 0 — does not seem a meaningful choice. Namely, if all the components of the two triplets, A1 and A111l, influencing the family quantum numbers of the four families, would have no vacuum expectation values, all the families would have the same mass, determined by aa and the contributions in all orders of corrections of the dynamical scalar fields, A 1, A111l and aa =< Aa > and the dynamical part of Aa. Let be added, however, that the choice < A 1S >= 0, < A^l@ >= 0 and aa = 0, while a 1 = 0 = a2, makes all the matrix elements of the mass matrix, Eq. (6.22), different from zero. 6.3 Conclusions In the spin-charge-family theory to the 4 x 4 mass matrix of any family member (that is of quarks and leptons — the observed three families namely form in the spin-charge-family theory the 3 x 3 submatrices of these predicted 4 x 4 mass matrices) the two scalar triplets (A1, Al1 l) and the three scalar singlets (AQ, AQ , A J'), s = (7,8), contribute, all with the weak and the hyper charge of the standard model higgs (±2, T1, respectively). The first two triplets influence the family quantum numbers, while the last three singlets influence the family members quantum numbers. The only dependence of the mass matrix on the family member (a = (u, d, v, e)) 78 Q 78 Q' quantum numbers is due to the operators y0 (±) QAQ, y0 (±) Q 'AQ and 78 ' 78 Y0 (±) Y'A± . The operator y0 (±), appearing at the contribution of the two triplet scalar fields as well as at the three singlet scalar fields, transforms the right handed members into the left handed ones, or opposite, while the family operators transform a family member of one family into the same family member of another family. We demonstrate in this paper that the matrix elements of mass matrices 4 x 4, predicted by the spin-charge-family theory for each family member a = (u,d,v,e), keep the symmetry SU(2)SO(4)i+3 x SU(2)SO(4),weak, x U(1) in all orders of corrections under the assumption that either the vacuum expectation values of three singlets < Aa >= aa are equal to zero, Subsect. 6.2.2, aa = 0, while all the other scalar fields — A1, ANl — can have for all the components nonzero vacuum expectation values, or that aa does not need to be zero, aa = 0, but then the two third components of the two scalar triplets, < A13 >= a1, < ANl3 >= a2, Subsect. 6.2.2, must be zero, a1 = 0, a2 = 0. For the case that the two triplets and the three singlets have for all components nonzero vacuum expectation values we represent the symmetries of the mass matrices in dependence of the order of corrections, Subsect. 6.2.2. In the first case, when aa = 0, to any order of corrections all the components of the two triplet scalar fields contribute, either with the nonzero vacuum expectation values or as dynamical fields or as both in all orders of corrections, while the three singlet scalar fields contribute only as dynamical fields. In this case the corrections keep the symmetry of the three level in all orders of corrections. The contributions of the dynamical fields of the three singlets in all orders of loop corrections — together with the contributions of the two triplets which interact with spinors through the family quantum numbers either with the nonzero vacuum expectation values or as dynamical fields — make all the matrix elements dependent on the particular family member quantum numbers. Correspondingly all the mass matrices bring different masses to any of the family members and correspondingly also different mixing matrices to quarks and leptons. However, the choice aa = 0 keeps the four off diagonal terms, which are proportional to aa in Eq.(6.22), equal to zero in all orders of correction. In the second case, when a1 = 0, a2 = 0, in any order of corrections the three singlet scalar fields contribute either with nonzero vacuum expectation values or as dynamical fields, while the two triplets scalar fields contribute with the nonzero vacuum expectation values and the dynamical fields, except the two of the triplet components — A13 and ANl3 — which contribute only as dynamical fields. The symmetry of the tree level is kept in all order of corrections, this choice makes, however, all the diagonal terms to remain equal in all orders of corrections. When all the singlets and the triplets have for all the components nonzero vacuum expectation values (aa = 0, a1 = 0, a2 = 0, < ANl@ = 0 > < AT@ >= 0) the symmetry of the tree level changes, but we are still able to determine the symmetry of mass in all orders of corrections, that is of the loop ones and the repetition of the nonzero vacuum expectation values, expressing the matrix elements of mass matrices with a few parameters only, due to the fact that the symmetry of the mass matrices limit the number of free parameters. In the case that |aa| is small (in comparison with |a11 and |a2|), the higher order corrections drop away very quickly. When fitting the free parameters of mass matrices to the observed masses of quarks and leptons and their 3 x 3 submatrices of the predicted 4 x 4 mixing matrices, we are able to predict the masses of the fourth family members as well as the matrix elements of the fourth components to the observed free families, provided that the mixing 3 x 3 submatrices of the predicted 4 x 4 mass matrices of quarks and leptons are measured accurately enough — since the (accurate) 3 x 3 submatrix of a 4 x 4 matrix determines 4 x 4 matrix uniquely [21,22]. This means that although we are so far only in principle able to calculate directly the mass matrix elements of the 4 x 4 mass matrices, predicted by the spin-charge-family, yet the symmetry of mass matrices, discussed in this paper, enables us — due to the limited number of free parameters — to predict properties of the four family of quarks and lepton to the observed three families, that is the masses of the fourth families and the corresponding mixing matrices [21,22]. We only have to wait for accurate enough data for the 3 x 3 mixing (sub)matrices of quarks and leptons. Let us add that the right handed neutrino, which is a regular member of the four families, Table 6.3, has the nonzero value of the operator Y'AY only. 6.4 Appendix: Short presentation of the spin-charge-family theory This section follows similar sections in Refs. [1,4-7]. The spin-charge-family theory [1-7,9-12,15-17,19-24] assumes: a. A simple action (Eq. (6.35)) in an even dimensional space (d = 2n, d > 5), d is chosen to be (13 + 1). This choice makes that the action manifests in d = (3 + 1) in the low energy regime all the observed degrees of freedom, explaining all the assumptions of the standard model, as well as other observed phenomena. There are two kinds of the Clifford algebra objects, Ya's and Ya's in this theory with the properties. {Ya,Yb}+ = 2nab {Ya,Yb}+ = 2nab {Ya,Yb}+ = 0. (6.34) Fermions interact with the vielbeins faa and the two kinds of the spin-connection fields — daba and daba — the gauge fields of Sab = 4(Ya Yb - Yb Ya) and Sab = 4(Ya Yb - Yb Ya), respectively. The action A = ddx E 1 $ YaPca^)+ h.c. + ddx E (aR + aR), (6.35) in which poa = faa poa + 2e (P«, Ef"a}-, P0a = Pa — J Sab ^aba — 2 Sab Daba, and R = 2 {fa[afpb] (^aba,P — ¿caa ¿"bp)} + h.C., R = 1 {fa[afpb] (¿0aba,p — CDcaa CD%P)} + h.C. 15, introduces two kinds of the Clifford algebra objects, Ya and Ya, {Ya, Yb}+ = 2nab = {Ya, Yb}+. faa are vielbeins inverted to eaa, Latin letters (a, b,..) denote flat indices, Greek letters (a, |3,..) are Einstein indices, (m, n,..) and v,..) denote the corresponding indices in (0,1,2,3), while (s, t,..) and (ct, t, ..) denote the corresponding indices in d > 5: eaafpa = ¿a , eaafab = , (6.36) E = det(eaa). b. The spin-charge-family theory assumes in addition that the manifold M(13+1' breaks first into M(7+1) x M(6) (which manifests as SO(7,1) xSU(3) xU(1)), affecting both internal degrees of freedom — the one represented by Ya and the one represented by Ya. Since the left handed (with respect to M(7+1') spinors couple differently to scalar (with respect to M(7+1') fields than the right handed ones, the break can leave massless and mass protected 2((7+1 )/2-1' families [36]. The rest of families get heavy masses 16. c. There is additional breaking of symmetry: The manifold M(7+1' breaks further into M(3+1)x M(4). d. There is a scalar condensate (Table 6.5) of two right handed neutrinos with the family quantum numbers of the upper four families, bringing masses of the scale 1 |v2R >2) 0 0 0 1 —1 0 0 0 1 —1 0 0 0 1 ru,viii ^ i_viii . i (|V1R >1 |e2R >2) ruviii . „VIII . I (|e1R >1 |e2R >2) 0 0 0 0 —1 —1 —1 0 0 0 —1 —1 —2 —2 0 1 —1 0 0 0 1 0 1 —1 0 0 0 1 Table 6.5. This table is taken from [5]. The condensate of the two right handed neutrinos vR, with the VUIth family quantum numbers, coupled to spin zero and belonging to a triplet with respect to the generators T2i, is presented together with its two partners. The right handed neutrino has Q = 0 = Y. The triplet carries T4 = —1, T23 = 1, T4 = —1, NR = 1, nL = 0, Y = 0, Q = 0, T31 = 0. The family quantum numbers are presented in Table 6.4. The stable of the upper four families is the candidate for the dark matter, the fourth of the lower four families is predicted to be measured at the LHC. 6.5 Appendix: Short presentation of spinor technique [1,4,11,13,14] This appendix is a short review (taken from [4]) of the technique [11,42,13,14], initiated and developed in Ref. [11] by one of the authors (N.S.M.B.), while proposing the spin-charge-family theory [2,4,5,7,9,1,15,16,10-12,17,19-24]. All the internal degrees of freedom of spinors, with family quantum numbers included, are de-scribable with two kinds of the Clifford algebra objects, besides with Ya's, used in this theory to describe spins and all the charges of fermions, also with Ya's, used in this theory to describe families of spinors: {Ya,Yb}+ = 2nab , {Ya,Yb}+ = 2nab , {Ya,Yb}+ = 0. (6.46) We assume the "Hermiticity" property for Ya's (and Ya's) = naaYa (and Y= naaYa), in order that Ya (and Ya) are compatible with (6.34) and formally unitary, i.e. ya ^ ya = I (and ya ^Ya = I). One correspondingly finds that (Sab)^ = naanbbSab (and (Sab)t = naanbbSab). Spinor states are represented as products of nilpotents and projectors, formed as odd and even objects of ya's, respectively, chosen to be the eigenstates of a Cartan subalgebra of the Lorentz groups defined by ya's ab 1 n aa ab 1 i (k): = 2(ya + n^Yb), [k]:= 2 (1 + kyayb), (6.47) where k2 = naanbb. We further have [4] ab 1 naa ab ab 1 i ab Ya (k): = 2(yaya + ^yV) = naa [—k], ya m:= 2(ya + kyayayb) =(—k), ab 1 «aa ab ab 1 i ab Ya (k): = —i-(ya + Vyb)ya = —inaa [k], Ya [k]:= i-(1 + ryaYb)ya = —i (k), 2 ik 2 k (6.48) where we assume that all the operators apply on the vacuum state |Vo}- We define ab ^ ab ab ^ ab a vacuum state |V0 > so that one finds < (k) (k) >= 1 , < [k] [k] >= 1. ab ab ab We recognize that ya transform (k) into [—k], never to [k], while ya transform ab ab ab (k) into [k], never to [—k] ab ab ab ab ab ab ab ab ya (k)= naa [—k], yb (k)= —ik [—k], ya [k]=(—k), yb [k]= —iknaa (—k), ab ab ab ab ab ab ab ab yia (k) = —inaa [k], -yb (k) = —k [k], y^ [k] = i (k), y1b [k] = —knaa (k) (6.49) The Clifford algebra objects Sab and Sab close the algebra of the Lorentz group Sab := (i/4)(yayb — ybya), Sab : = (i/4)(YaYb — YbYa), (6.50) {Sab,Scd}_ = 0, {Sab,Scd}_ = i(nadSbc +nbc Sad —nacSbd —nbdSac), {S ab,S cd}_ = i(nadSSbc + nbcSad — nacSbd — nbdSac). ab ab One can easily check that the nilpotent (k) and the projector [k] are "eigenstates" of Sab and S ab ab 1 ab ab 1 ab Sab (k)= 2k (k), Sab [k]= 2k [k], ab 1 ab ab 1 ab Sab (k)= 2 k (k), Sab [k]= — 2 k [k], (6.51) where the vacuum state |V0} is meant to stay on the right hand sides of projectors ab and nilpotents. This means that multiplication of nilpotents (k) and projectors ab [k] by Sab get the same objects back multiplied by the constant 1 k, while Sab ab ab multiply (k) by | and [k] by (—|) (rather than by |). This also means that when ab ab (k) and [k] act from the left hand side on a vacuum state |"^0) the obtained states are the eigenvectors of Sab. The technique can be used to construct a spinor basis for any dimension d and any signature in an easy and transparent way. Equipped with nilpotents and projectors of Eq. (6.47), the technique offers an elegant way to see all the quantum numbers of states with respect to the two Lorentz groups, as well as transformation properties of the states under the application of any Clifford algebra object. Recognizing from Eq.(6.50) that the two Clifford algebra objects (Sab,Scd) with all indexes different commute (and equivalently for (Sab, Scd)), we select the Cartan subalgebra of the algebra of the two groups, which form equivalent representations with respect to one another S03,S12,S56, ••• ,Sd_1d, if d = 2n > 4, S 03,S 12,S56, ••• ,Sd-1 d, if d = 2n > 4. (6.52) The choice of the Cartan subalgebra in d < 4 is straightforward. It is useful to define one of the Casimirs of the Lorentz group — the handedness r ({FJ Sab}_ = 0) (as well as f) in any d = 2n r(d) :=(i)d/2 n (VnaaYa), if d = 2n, a f(d) :=(i)(d_1)/2 ^ (Vn^Y^, if d = 2n. (6.53) a We understand the product of Ya's in the ascending order with respect to the index a: y0y1 • • • Yd. It follows from the Hermiticity properties of Ya for any choice of the signature naa that r^ = r, r2 = I.( Equivalent relations are valid for f.) We also find that for d even the handedness anticommutes with the Clifford algebra objects Ya ({Ya, H+ = 0) (while for d odd it commutes with Ya ({Ya, r}_ = 0)). Taking into account the above equations it is easy to find a Weyl spinor irreducible representation for d-dimensional space, with d even or odd 17. For d even we simply make a starting state as a product of d/2, let us say, only nilpotents ab (k), one for each Sab of the Cartan subalgebra elements (Eqs.(6.52, 6.50)), applying it on an (unimportant) vacuum state. Then the generators Sab, which do not belong to the Cartan subalgebra, being applied on the starting state from the left 17 For d odd the basic states are products of (d — 1 )/2 nilpotents and a factor (1 ± F). hand side, generate all the members of one Weyl spinor. 0d 12 35 d-1 d-2 (kod)(kl2)(k35) ••• (kd-1 d-2) I^0 > 0d 12 35 d-1 d-2 [-kod][-kl2](k35) ••• (kd-1 d-2) I^0 > Od 12 35 d-1 d-2 [-k0d](k12)[-k35] ••• (kd-1 d-2) I^0 > 0d 12 35 d-1 d-2 [-k0d](k12)(k35) ••• [-kd-1 d-2] I^0 > od 12 35 d-1 d-2 (k0d)[-k12][-k35] ••• (kd-1 d-2) I^0 > . (6.54) All the states have the same handedness r, since {r, Sab}- = 0. States, belonging to one multiplet with respect to the group SO(q, d - q), that is to one irreducible representation of spinors (one Weyl spinor), can have any phase. We could make a choice of the simplest one, taking all phases equal to one. (In order to have the usual transformation properties for spinors under the rotation of spin and under Cn Pn,some of the states must be multiplied by (-1).) The above representation demonstrates that for d even all the states of one irreducible Weyl representation of a definite handedness follow from a starting ab state, which is, for example, a product of nilpotents (kab), by transforming all ab mn ab mn possible pairs of (kab)(kmn) into [-kab][-kmn]. There are Sam, San, Sbm,Sbn, which do this. The procedure gives 2(d/2-1' states. A Clifford algebra object Ya being applied from the left hand side, transforms a Weyl spinor of one handedness into a Weyl spinor of the opposite handedness. We shall speak about left handedness when r = -1 and about right handedness when r = 1. While Sab, which do not belong to the Cartan subalgebra (Eq. (6.52)), generate all the states of one representation, Sab, which do not belong to the Cartan subalgebra (Eq. (6.52)), generate the states of 2d/2-1 equivalent representations. Making a choice of the Cartan subalgebra set (Eq. (6.52)) of the algebra S ab and S ab. (S03 s 12 S56 S78 S910 Sn '2 S^ M ) (S 03 S12 S 56 S78 S9 10 S" 12 S13 14 ) a left handed (r t13,1' = -1) eigenstate of all the members of the Cartan subalgebra, representing a weak chargeless uR-quark with spin up, hyper charge (2/3) and colour (1/2,1/(2%/3)), for example, can be written as 03 12 56 78 9 1011 1213 14 (+i)(+) I (+)( + ) II (+) [-] [-] l^0> = 1 (Y0 - y3)(y1 + iY2)I(Y5 + iY6)(Y7 + iY8)II (Y9 + iY10)(1 -iYnY12)(1 -iY13Y14)I^0>. (6.55) This state is an eigenstate of all Sab and Sab which are members of the Cartan subalgebra (Eq. (6.52)). The operators Sab, which do not belong to the Cartan subalgebra (Eq. (6.52)), generate families from the starting uR quark, transforming the uR quark from Eq. (6.55) to the uR of another family, keeping all of the properties with respect to Sab unchanged. In particular, S01 applied on a right handed uR-quark from Eq. (6.55) generates a state which is again a right handed uR-quark, weak charge-less, with spin up, hyper charge (2/3) and the colour charge (1/2,1/(2%/3)) ; 01 03 12 56 78 91011121314 03 12 56 78 91011121314 (+i)(+) I (+)(+) II (+) [-] [-] = -2 [+i][ + ] I (+)(+) II (+) [-] [-1(6.56) One can find both states in Table 6.4, the first uR as uR8 in the eighth line of this table, the second one as uR7 in the seventh line of this table. Below some useful relations follow. From Eq.(6.49) one has ab cd i ab cd au cd i Sac (k)(k) = --naancc [—k][—k], Sac (k)(k)= -naancc [k][k], ab cd ab cd ab cd ab cd ab cd ab cd Sac [k][k] = - (-k)(-k), 2 [k][k]=-- (k)(k), ab cd ab cd 2 ab cd ab cd Sac (k)[k] =-2^aa [-k](-k), (k)[k]= -2naa [k](k), ab cd ab cd ab cd ab cd Sac [k](k) = 2ncc (-k) [-k], [k](k)= 2ncc (k) [k] . (6.57) We conclude from the above equation that Sab generate the equivalent representations with respect to Sab and opposite. We recognize in Eq. (6.58) the demonstration of the nilpotent and the projector ab ab character of the Clifford algebra objects (k) and [k], respectively. ab ab ab ab (k)(k) =0, ab ab (k)(-k)= naa ab [k], ab ab (-k)(k)= = naa [ ab -k], ab ab (-k)(-k) ab ab [k] [k] II [k a ]b ab ab [k][-k] = 0, ab ab [-k][k] = 0, ab ab [-k][-k]= ab ab (k) [k] =0, ab ab [k](k) = ab (akb) , ab ab (-k)[k]= ab (-k), ab ab (-k)[-k]= ab ab (k) [-k] ab = ( k) , ab ab [k](-k): = 0, ab ab [-k] (k) = 0, ab ab [-k](-k) = (6.58) Defining ab ab (±i) = 2(Ya T Yb), one recognizes that ab ab (±1)= ^(Ya ± iYb), [±i]= ^(1 ± YaYb), [±1]= 2 (1 ± iYaYb)- ab ab (k)(k) = 0, ab ab ab ab ab ab ab ab (-k) (k)= -inaa [k], (k) [k]= i (k), (k) [-k]= 0. (6.59) c c c Below some more useful relations [15] are presented: 03 12 03 12 + ± 03 12 03 12 56 78 56 78 T1± = (T)(±)(T) , T2^ = (T)(T)(T) , 56 78 56 78 T1± = (T)(±)(T), T2t = (T)(T)(T) . (6.60) In Table 6.4 [4] the eight families of the first member in Table 6.3 (member number 1) of the eight-plet of quarks and the 25th member in Table 6.3 of the eight-plet of leptons are presented as an example. The eight families of the right handed u1r quark are presented in the left column of Table 6.4 [4]. In the right column of the same table the equivalent eight-plet of the right handed neutrinos v1R are presented. All the other members of any of the eight families of quarks or leptons follow from any member of a particular family by the application of the operators N± L and t(21 Eq. (6.60), on this particular member. The eight-plets separate into two group of four families: One group contains doublets with respect to N R and T2, these families are singlets with respect to NL and T1. Another group of families contains doublets with respect to N L and T1, these families are singlets with respect to N R and T2. The scalar fields which are the gauge scalars of N R and T2 couple only to the four families which are doublets with respect to these two groups. The scalar fields which are the gauge scalars of N L and T1 couple only to the four families which are doublets with respect to these last two groups. After the electroweak phase transition, caused by the scalar fields with the space index (7,8), the two groups of four families become massive. The lowest of the two groups of four families contains the observed three, while the fourth remains to be measured. The lowest of the upper four families is the candidate for the dark matter [1]. References 1. N.S. Mankoc Borstnik, "Spin-charge-family theory is offering next step in understanding elementary particles and fields and correspondingly universe", Proceedings to the Conference on Cosmology, Gravitational Waves and Particles, IARD conferences, Ljubljana, 6-9 June 2016, The 10th Biennial Conference on Classical and Quantum Relativistic Dynamics of articles and Fields, J. Phys.: Conf. Ser. 845 012017 [arXiv:1607.01618v2]. 2. D. Lukman, N.S. Mankoc Borstnik, "Vector and scalar gauge fields with respect to d = (3 + 1) in Kaluza-Klein theories and in the spin-charge-family theory", Eur. Phys. J. C, 77 (2017) 231[arXiv:1604.00675] 3. N.S. Mankoc Borstnik, H.B.F. Nielsen, "The spin-charge-family theory offers understanding of the triangle anomalies cancellation in the standard model", Forschritte Der Physik -Progress of Physics (2017) 1700046, [arXiv:1607.01618] 4. N.S. 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Mankoc Borstnik, "The spin-charge-family theory explains why the scalar Higgs carries the weak charge ±-j and the hyper charge + ^ ", Proceedings to the 17th Workshop "What Comes Beyond the Standard Models", Bled, July 20 - 28, 2014, p.163-182, [arxiv:1409.7791, arxiv:1212.4055]. 25. A. Ali in discussions and in private communication at the Singapore Conference on New Physics at the Large Hadron Collider, 29 February - 4 March 2016. 26. M. Neubert, in duscussions at the Singapore Conference on New Physics at the Large Hadron Collider, 29 February - 4 March 2016. 27. A. Lenz, "Constraints on a fourth generation of fermions from higgs boson searches", Advances in High Enery Physics 2013, ID 910275. 28. N.S. Mankoc Borstnik, H.B.F. Nielsen, "Do the present experiments exclude the existence of the fourth family members?", Proceedings to the 19th Workshop "What comes beyond the standard models", Bled, 11-19 of July, 2016, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. 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Lee, World Scientific, Singapore 1983, T. Appelquist, A. Chodos, P.G.O. Freund (Eds.), Modern Kaluza-Klein Theories, Reading, USA: Addison Wesley, 1987. 48. D. Lukman, N. S. Mankoc Borstnik, H. B. Nielsen, New J. Phys. 13 (2011) 10302 [arXiv:1001.4679v4]. Bled Workshops in Physics Vol. 19, No. 2 A Proceedings to the 21 st Workshop What Comes Beyond ... (p. 148) Bled, Slovenia, June 23-July 1, 2018 7 Extending Starobinsky Inflationary Model in Gravity and Supergravity S.V. Ketov1'2'3 * andM.Yu. Khlopov4 ** 1 Physics Department, Tokyo Metropolitan University, Minami-ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan 2 Kavli Institute for the Physics and Mathematics of the Universe (IPMU), The University of Tokyo, Chiba 277-8568, Japan 3 Research School of High Energy Physics, Tomsk Polytechnic University, Lenin ave. 30, Tomsk 634050, Russia 4 Institute of Physics, Southern Federal University Stachki 194, Rostov on Don 344090, Russia Abstract. We review some recent trends in the inflationary model building, the super-symmetry (SUSY) breaking, the gravitino Dark Matter (DM) and the Primordial Black Holes (PBHs) production in supergravity. The Starobinsky inflation can be embedded into supergravity when the inflaton belongs to the massive vector multiplet associated with a (spontaneously broken) U(1 ) gauge symmetry. The SUSY and R-symmetry can be also spontaneously broken after inflation by the (standard) Polonyi mechanism. Polonyi particles and gravitinos are super heavy and can be copiously produced during inflation via the Schwinger mechanism sourced by the Universe expansion. The overproduction and instability problems can be avoided, and the positive cosmological constant (dark energy) can also be introduced. The observed abundance of the Cold Dark Matter (CDM) composed of gravitinos can be achieved in our supergravity model too, thus providing the unifying framework for inflation, supersymmetry breaking, dark energy and dark matter genesis. Our supergravity approach may also lead to a formation of primordial non-linear structures like stellar-mass-type black holes, and may include the SUSY GUTs inspired by heterotic string compactifications, unifying particle physics with quantum gravity. Povzetek. Avtorja obravnavata nekaj novejsih modelov inflacije, zlomitve supersimetrije, temne snovi, ki jo sestavljajo gravitini in nastajanja prvotnih crnih lukenj v supergravitaciji. Inflacija Starobinskega se pojavi v supergravitaciji, ce je inflaton del masivnega vektorskga multipleta, ki spontano zlomi umeritveno simetrijo U(1). Supersimetrijo in simetrijo R lahko po inflaciji spontano zlomi tudi mehanizem Polonyija. Izredno masivni delci Polony-ija in gravitini, lahko nastanejov dovolj velikih kolocinah med inflacijo z mehanizmom Schwingerja. S tem se avtorja izogneta problemu prevelike produkcije tezkih delcev in nestabilnosti ter pojasnita tudi pozitivno kozmolosko konstanto (temno energijo). Njun model s supergravitacijo razlozi opazeno pogostost hladne temne snovi (CDM), ce jo sestavljajo gravitini in ponudi razlago za nastanek in potek inflacije, zlomitev supersimetrije, temno energijo in temno snov. Njun model lahko pojasni tudi nastanek prvotnih nelin-eranih struktur, kot so crne luknje, ki imajo maso enake masi običajnih zvezd, in morda * E-mail: ketov@tmu.ac.jp ** E-mail: khlopov@apc.in2p3.fr vključuje supersimetricne teorije velikega poenotenja (GUT), izvirajoče iz kompaktifikacije heterotskih strun, kar bi poenotilo fiziko delcev in kvantno gravitacijo. Keywords: inflation, modified gravity, supergravity, cold dark matter, dark energy, supersymmetry breaking, primordial black holes 7.1 Introduction The Cosmic Microwave Background (CMB) data collected by the Planck collaboration [1-3] favours the slow-roll single-field inflationary scenarios, with an approximately flat scalar potential. The celebrated Starobinsky model [4] does provide such scenario, and relates its inflaton (called scalaron in this context) to the particular extension of Einstein-Hilbert gravity with the extra higher derivative term given by the scalar curvature squared, R2. However, a theoretical explanation of fundamental origin of the Starobinsky model is still missing. The viable inflationary dynamics is driven by the R2 term dominating over the (Einstein-Hilbert) R term. This is related to a missing UV completion of the non-renormalizable (R+R2) gravity. The interesting and ambitious project for string phenomenology would be to provide a derivation of the Starobinsky model from the first principles. A first step towards this is an embedding of the Starobinsky model into four-dimensional N = 1 supergravity. In the supergravity framework, the inflaton (scalaron) can mix with other scalars, and this mixing may ruin any initially successful inflationary mechanism. The inflationary model building based on supergravity in the literature usually assumes that inflaton belongs to a chiral (scalar) supermultiplet [5-7]. However, there is the alternative to this assumption: inflaton can also belong to a massive N = 1 vector multiplet. The vector multiplet-based approach avoids stabilization problems related to the inflaton (scalar) superpartner, as the way-out of the standard n-problem. The scalar potential of a vector multiplet is given by the D-term instead of the F-term. The minimal supergravity models, with inflaton belonging to a massive vector multiplet, were proposed in Refs. [8,9]. Then any desired values of the CMB observables (the scalar perturbations tilt ns and the tensor-to-scalar perturbations ratio r) can be recast from the single-field (inflaton) scalar potential proportional to the derivative squared of arbitrary real function J. However, in these models, the vacuum energy is vanishing after inflation, thus restoring supersymmetry, and only a Minkowski vacuum is allowed. The way-out of this problem was proposed in [10,11] by adding a Polonyi (chiral) superfield with a linear superpotential [12], leading to a spontaneous SUSY breaking and allowing a de-Sitter vacuum after inflation. A successful model of inflation in supergravity should also be consistent with the Cold Dark Matter (CDM) constraints and the Big Bang Nucleosynthesis (BBN). For example, many supergravity scenarios are plagued by the so-called gravitino problem. Gravitinos can decay, injecting hadrons and photons during the BBN epoch, which may jeopardize the good Standard Model prediction of nuclei ratios [13-16]. In very much the same way, the Polonyi (overproduction) problem and its relation to the BBN results were extensively discussed in the literature [17-22]. In addressing these issues, the mass spectrum and the soft SUSY parameters are important. The leading (WIMP-like) dark matter production mechanisms and decay channels are selected from the mass pattern, and have either thermal or non-thermal origin. In this paper, we review a class of the minimalistic Polonyi-Starobinsky (PS) N = 1 supergravity models for inflation, with the inflaton belonging to a (massive) vector multiplet. These models can avoid the overproduction and BBN problems, while accounting for the right amount of CDM composed of gravitinos. In our analysis, we assume that the Polonyi field, inducing a spontaneous SUSY breaking at a high energy scale, and the gravitino, as the Dark Matter (DM) particle, are both super-heavy. The main mechanism producing DM is given by the Schwinger-type production sourced by inflationary expansion. After inflation, Polonyi particles rapidly decay into gravitinos. We find that gravitinos produced directly from Schwinger's production and from Polonyi particles decays, can account for the correct abundance of Cold Dark Matter. Another aspect is an inclusion of the (mini) Primordial Black Holes (PBHs) that may have been copiously produced in the early Universe, and later may have evaporated into gravitinos and other Standard Model particles [23-27]. A large amount of mini PBHs cannot be produced in our model when the other scalar and pseudo-scalar partners of inflaton are not participating in the inflationary dynamics. The Starobinsky inflaton entails a scalar potential shape that cannot lead to a large number of PBHs, because it does not allow for amplifying instabilities and has no exit out of inflation with a first order phase transition. It is still possible that dynamics of other scalar fields changes this picture. In this case, the extra moduli can exit from inflation via ending in false minima. The tunneling process from a false minimum to the true one sources the production of bubbles related to the first order phase transition. As regards the (solar mass type) PBHs, their production in the early Universe is possible in our supergravity approach after a certain deformation of the Starobinsky scalar potential. We envisage a unification of the inflaton in a vector multiplet and the Supersymmetric Grand Unified Theories (SUSY GUTs), whose gauge group has at least one abelian factor, such as the flipped SU(5)xU(1) model arising from the compactified heterotic superstrings or the intersecting D-branes. 7.2 Starobinsky model of (R + R2) gravity Starobinsky model of inflation is defined by the action [4] where we have introduced the reduced Planck mass MP[ = 1/V8nGN ~ 2.4 x 1018 GeV, and the scalaron (inflaton) mass m as the only parameter. We use the spacetime signature (—, +, +, +,). The (R + R2) gravity model (7.1) can be considered as the simplest extension of the standard Einstein-Hilbert action in the (7.1) 7 Extending Starobinsky Inflationary Model in Gravity and Supergravity 151 context of (modified) F(R) gravity theories with an action m2 C ■ • LPl Sf = "2" d4^v/-gF(R) , (7.2) in terms of the function F(R) of the scalar curvature R. The F(R) gravity action (7.2) is classically equivalent to S[g,v,x] = M2 [F'(x)(R - x)+ F(x)] (7.3) with the real scalar field x, provided that F " = 0 that we always assume. Here the primes denote the derivatives with respect to the argument. The equivalence is easy to verify because the x-field equation implies x = R. In turn, the factor F' in front of the R in (7.3) can be (generically) eliminated by a Weyl transformation of metric that transforms the action (7.3) into the action of the scalar field x minimally coupled to Einstein gravity and having the scalar potential v = (M21 xFM—M . (7.4) (7.5) 2 ) F'(x)2 Differentiating this scalar potential yields dV ( M2l ^ F''(x)[2F(x)- xF'(x)] dx \2 (F '(x))3 The kinetic term of x becomes canonically normalized after the field redefinition x(0 and F''(R) >0, (7.8) and they are obviously satisfied for Starobinsky model (7.1) for R > 0. Differentiating the scalar potential V in Eq. (7.4) with respect to 9 yields 2 dV = dV dx = Mpl d9 dx d9 2 xF'' + F' - F' 2xF' - FF„ :/2 c/3 where we have dx dx dF ' dF ' /dF ' V2 F ' d9 dF' d9 d9 / dx V3MPl F'' £ , (7.9) d9 (7.10) This implies ^ = Mpi dp 11 a/6F '2 (7.11) Combining Eqs. (7.4) and (7.11) yields R and F in terms of the scalar potential V, R = F = VédV 4V Mpi dip M21 V6 dV 2V MP1 dp M21 exp 3|/Mpi 2 exp I 2^ 3i/Mpi (7.12) (7.13) These equations define the function F(R) in the parametric form, in terms of a scalar potential V (<), i.e. the inverse transformation to (7.4). This is known [28] as the classical equivalence (duality) between the F(R) gravity theories (7.2) and the scalar-tensor (quintessence) theories of gravity (7.7). In the case of Starobinsky model (7.1), one gets the famous potential 41 V (p) = 7 M21m2 exp | -\l 3i/Mpi (7.14) This scalar potential is bounded from below (non-negative and stable), and it has the absolute minimum at < = 0 corresponding to a Minkowski vacuum. The scalar potential (7.14) also has a plateau of positive height (related to inflationary energy density), that gives rise to slow roll of inflaton in the inflationary era. The Starobinsky model (7.1) is the particular case of the so-called a-attractor inflationary models [29], and is also a member of the close family of viable inflationary models of F(R) gravity, originating from higher dimensions [30]. A duration of inflation is measured in the slow roll approximation by the e-foldings number 1 CV* A/ (7.15) Ne 1 MH V . V7 dp where p* is the inflaton value at the reference scale (horizon crossing), and pend is the inflaton value at the end of inflation when one of the slow roll parameters Mp2l V' and nv(i) = Mpl ( V ' V (7.16) is no longer small (close to 1). The amplitude of scalar perturbations at horizon crossing is given by [31] A = V3 12npM6l(V*')2 3m2 8n2Mpl sinh4 (v^M; p* (7.17) The Starobinsky model (7.1) is the excellent model of cosmological inflation, in very good agreement with the planck data [1-3]. The planck satellite mission measurements of the Cosmic Microwave Background (CMB) radiation [1-3] give the scalar perturbations tilt as ns « 1 + 2nv — 6ev « 0.968 ± 0.006 and restrict 2 2 the tensor-to-scalar ratio as r « 16eV < 0.08. The Starobinsky inflation yields r « 12/N « 0.004 and ns « 1 — 2/Ne, where Ne is the e-foldings number between 50 and 60, with the best fit at Ne « 55 [32,33]. The Starobinsky model (7.1) is geometrical (based on gravity only), while its (mass) parameter m is fixed by the observed CMB amplitude (COBE, WMAP) as m « 3 • 1013 GeV or m « 1.3 • 10-5 . Mpi A numerical analysis of (7.15) with the potential (7.14) yields [31] 3^/Mpl ln ( 3Ne 5.5 , 3^end/Mpl ln 2 jj (4 + 3V3) (7.18) 0.5 , (7.19) where Ne « 55 has been used. 7.3 Starobinsky inflation in supergravity Let us introduce a set of two chiral superfields H) and a real vector superfield V coupled to the supergravity sector, with the following Lagrangian:1 £ = d292£ j3(DD - 8R)e-3(K+2J) + 1waWa + W(®) j + h.c. , (7.20) where R is the chiral scalar curvature superfield, E is the chiral density superfield, (Da, Da) are the superspace covariant spinor derivatives, K = K(®,®) is the Kahler potential, W(®) is the superpotential, Wa = — 4 (DD — 8R)DaV is the abelian (chiral) superfield strength, and J = J(He2gVH) is a real function with the coupling constant g. The Lagrangian (7.20) is invariant under the supersymmetric U(1) gauge transformations H —> H' = e-igZH , H->H' = eigZH , (7.21) V^V' = V + 2(Z — Z) , (7.22) the gauge parameter of which, Z, is itself a chiral superfield. The chiral superfield H can be gauged away via the gauge fixing of these transformations by imposing the gauge condition H = 1. Then the Lagrangian (7.20) gets simplified to £ = d292£ j 3(DD - 8R)e-1 (K+2J) + 1waWa + wj + h.c. (7.23) After eliminating the auxiliary fields and moving from the initial (Jordan) frame to the Einstein frame, the bosonic part of the Lagrangian (7.23) reads [10]2 e-1 L = - 1r-Kaa3mA3mA-lfmnFmn-2J"3mC3mC-1J"BmBm-V, (7.24) 1 We use the standard notation [34] for supergravity in superspace. 2 The primes and capital latin subscripts denote the derivatives with respect to the corre- sponding fields. 154 S.V. Ketov and M.Yu. Khlopov with the scalar potential V = g2J'2 + eK+2J j ka\ (wA + kaw)(wA + kAAW) - (3 - WW j (7.25) in terms of the physical fields (A, C, Bm), the auxiliary fields (F, X, D) and the vector field strength Fmn = DmBn — Bm. As is clear from Eq. (7.24), the absence of ghosts requires J "(C) >0, where the primes denote the differentiations with respect to the given argument. We restrict ourselves to the Kahler potential and the superpotential of the Polonyi model [12]: K = O® , W = + (3) , (7.26) with the parameters | and (. Our model includes the single-field (C) inflationary model, whose D-type scalar potential is given by V (C) = g2 (J' )2 (7.27) in terms of arbitrary function J(C), with the real inflaton field C belonging to a massive vector supermultiplet. The Minkowski vacuum conditions (after inflation) can be easily satisfied when J' = 0, which implies [12] (A) = V3 — 1 and ( = 2 — V3 . (7.28) This solution describes a stable Minkowski vacuum with spontaneous SUSY breaking at arbitrary scale (F) = The related gravitino mass is given by m3/2 = |e2-73+J . (7.29) There is also a complex (Polonyi) scalar of mass ma = 2|ie2-V3 > 2m3/2 (7.30) and a massless fermion in the physical spectrum. The inequality in Eq. (7.30) is saturated in the original Polonyi model [12] but it is not the case in our model when (J) < 0. As regards the early Universe phenomenology, our model has the following theoretically appealing features: • there is no need to "stabilize" the single-field inflationary trajectory against scalar superpartners of inflaton, because our inflaton is the only real scalar in a massive vector multiplet, • any values of CMB observables ns and r are possible by choosing the J-function, • a spontaneous SUSY breaking after inflation occurs at arbitrary scale • there are only a few parameters relevant for inflation and SUSY breaking: the coupling constant g defining the inflaton mass, g ~ minf., the coupling constant I defining the scale of SUSY breaking, | ~ m3/2, and the parameter ( in the constant term of the superpotential. Actually, the inflaton mass is constrained by CMB observations as minf. ~ 0(10-6), while |3 is fixed by the vacuum solution, so that we have only one free parameter ^ defining the scale of SUSY breaking in our model (before studying reheating and phenomenology). The D-type scalar potential associated with the Starobinsky inflationary model of (R + R2) gravity arises when [9] 3 J(C) = 2 (C - ln C) (7.31) that implies J'(C) = - (1 - C-1) and J"(C) = - (C-2) >0. (7.32) According to (7.24), a canonical inflaton field ^ (with the canonical kinetic term) is related to the field C by the field redefinition C = exp (yi/3^ . (7.33) Therefore, we arrive at the (Starobinsky) scalar potential VStar.W = (l - e-^*)2 with m2f. = 9g2/2 . (7.34) The full action (7.20) of this PS supergravity in curved superspace can be transformed into a supergravity extension of the (R + R2) gravity action by using the (inverse) duality procedure described in Ref. [9]. However, the dual super-gravity model is described by a complicated higher-derivative field theory that is inconvenient for studying particle production. Another nice feature of our model is that it can be rewritten as a supersym-metric (abelian and non-minimal) gauge theory coupled to supergravity in the presence of a Higgs superfield H, resulting in the super-Higgs effect with simultaneous spontaneous breaking of the gauge symmetry and SUSY. Indeed, the U(1) gauge symmetry of the original Lagrangian (7.20) allows us to choose a different (Wess-Zumino) supersymmetric gauge by "gauging away" the chiral and anti-chiral parts of the general superfield V via the appropriate choice of the su-perfield parameters Z and Z. Then the bosonic part of the Lagrangian in terms of the superfield components in the Einstein frame, after elimination of the auxiliary fields and Weyl rescaling, reads [11] e-1 L = - 1r - kaa* amAamA - 4FmnFmn - 2Jhfv3mh3mh - 1 jv? BmBm + iBm(Jvh3mh - Jvh9mh)- V , (7.35) where h, h are the Higgs field and its conjugate. The standard U(1) Higgs mechanism arises with the canonical function J = 2 he2Vh, where we have chosen g = 1 for simplicity. As regards the Higgs sector, it leads to e-1 LHiggs = -3mh3mh + iBm(h3mh - h3mh) - hhBmBm - V . (7.36) After changing the variables h and h as h = — (p + v)eiZ, h = — (p + v)e-iZ , (7.37) where p is the (real) Higgs boson, v = (h) = (h) is the Higgs VEV, and Z is the Goldstone boson, the unitary gauge fixing of h —» h' = e-iZh and Bm —» Bm = Bm + 3mZ, leads to the standard result i 1 1 -> e-1LHiggs = - = 3mp3mp - = (p + v)2BmBm - V . (7.38) The Minkowski vacuum after inflation can be easily lifted to a de Sitter vacuum (Dark Energy) in our model by the simple modification of the Polonyi sector and its parameters as [11] (A) =l-5-1)+6+O(62), 15 =(2--7)+jt7)6+O(62), (739) where 6 is a very small deformation parameter, 0 < 5 C 1. It leads to a positive cosmological constant Vo = 6 = m2/26 (7.40) and the superpotential VEV (W) = ^((A) + p) = ^a + b - =6) , (7.41) where a = ( %/3 -1 ) and b = (2 - %/3) provide the SUSY breaking vacuum solution to the Polonyi parameters in the absence of a cosmological constant. The full scalar potential (7.25) is a sum of the D- and F-type terms, while there is a mix of the inflaton - and Polonyi-dependent terms in the F-type contribution. This mixing leads to instability of the (Starobinsky) inflationary trajectory that is supposed to be driven by the D-term only. This issue was resolved in Ref. [35] where a modification of the original PS supergravity action (7.20) was proposed via adding the generalized Fayet-Iliopoulos term and modifying the J-function (7.31). 7.4 Super heavy gravitino dark matter The complete set of equations of motion in our supergravity model (Sec. 3) is very complicated. In this section, we consider only the leading order with respect to the inverse Planck mass. In addition, we neglect the coupling of Polonyi and gravitino particles to the inflaton, and introduce the effective action of the Polonyi field in the Friedmann-Lemaitre-Robertson-Walker (FLRW) background (in comoving coordinates) as I[A] = dt d3Xy (a2 - (VA)2 - MAa2 - ZRA2 ) , (7.42) where the non-minimal coupling constant of the Polonyi field to gravity is equal to Z = 1, A is the Polonyi field, MA stands for its mass, R is the Ricci scalar, and a is the FLRW scale factor. The mode decomposition of the Polonyi field reads A(x) = d3k(2n)-3/2a-1(n) bkhk(n)eikx + bkhk(n)e-ikx , (7.43) where the conformal time coordinate n is introduced, b,b are the (standard) creation/annihilation operators, and the coefficient functions h, h+ are normalized as follows: hkhk* _ hkhk = i . (7.44) Because of Eqs. (7.42) and (7.43), the equation of motion of the modes is a " hk(n) + ^k(n)hk(n) = 0, where w2 = 5 — + k2 + MAa2 , (7.45) and h" = d2h/dn2. Equation (7.45) can be conveniently rescaled by using some reference scales a(n*) = a* and H(n*) = H* as follows: hfc'(n) + (k2 + b2 a2)hk(n)= 0, (7.46) in terms of the rescaled quantities n = na*H* , a = a/a* , k = k/(H*a*) . The leading order of the gravitino action coincides with the massive RaritaSchwinger action, = d4xei|> ffRff{^}, (7.47) where the gravitino kinetic operator has been introduced as = m3/2YffV^v + iYffVpDv^p , (7.48) and the supercovariant derivative is = -rp^p + + 1^^abYab, (7.49) in the Y-notation y^1 = y^1 . Since the supergravity torsion is of the second order with respect to the inverse Planck mass, we ignore it in the leading order approximation. The F(pV can be represented by the standard symmetric Christoffel symbols that are actually cancelled from the Rarita-Schwinger action (7.47). The Rarita-Schwinger action leads to the gravitino equation of motion, (iD _ m3/2)^ _ (iZV + m2/2Y^) Y ■ ^ = 0 . (7.50) In the flat FLRW background, Eq. (7.50) reduces to iY—n3— = _ (m3/2 + i^Y0) Y—3—^ , (7.51) 158 S.V. Ketov and M.Yu. Khlopov where = 2aa-1e,[aeb] , e, = a(n)6, , m.3/2 = m.3/2 (n) . (7.52) A solution to Eq. (7.51) is ^(x)=J d3p(2n)-3(2p0)-1 ^e^b,(n,A)akA(n) + e-ikxbC(n, A)akA(n)} . A (7.53) We find that the equations of motion for the 3/2-helicity gravitino modes have the same form as that of Eq. (7.45), namely, b,(n,A) + C(k,a)b, (n,A) + w2(k,a)b,(n,A) = 0, (7.54) where we have introduced the notation a ' C(k,a)b,(n,A) = -2iYVikiYvn3nb, - 2Yv(m3/2 + i-Y^V^b, , (7.55) w2 (k, a)/2 = k2 + m3/2 + y0m3/2 - ^. (7.56) Following a procedure similar to the standard one in the case of Dirac and KleinGordon equations, we can reformulate the mode equations of motion in our case as Pv Pvb,(n,A) = 0, (7.57) where we have introduced the projector operator Pv = iYvn - Yviki - ^3/2 + i^y0) Yv = 0. (7.58) The dynamics of the gravitino and Polonyi fields during inflation necessary lead to their quantum production. The number density of produced particles can be calculated by using a Bogoliubov transformation, hk1 (n) = akhk° (n) + Pkhk110 (n). (7.59) This transformation is performed from the vacuum solution selected by the boundary conditions at n = nin, corresponding to the initial time of inflation, to the final time n = nf, when the particles creations process from inflation stops. In the inflationary epoch, the dynamical regime is a '/a2 C MPl and MPl ba/k C 1. This implies that we can consider the extremes as nin = -œ and nf = +00, performing a WKB semiclassical approximation. By assuming these boundary conditions, the energy density of the Polonyi particles produced during inflation reads 1 pA(n) = MAnA(n) = MAH,f an) ) pa , (7.60) where pa = i dkk21 p k |2 . (7.61) 0 The inflaton mass sets the characteristic energy scale for the Hubble constant, calculated at fixed cosmological time t = tf: H2(tf) ~ m2, p(tf) ~ m2M2!. We propose the following formula for Polonyi particles (energy-density and Polonyi mass) produced during inflation [36]: t,2> 8n/MAA ATrehA nA(tf) (QAh2/^h2) = f^if J MpPf , (7-62) where MA is the Polonyi mass, nRh2 ~ 4.31 x 10-5 is the radiation energy density at today's temperature To, HAh2 is the energy density of the produced Polonyi fields, all in the units of the critical energy density. There is about 8th-orders-of-magnitude suppression of the energy density. The normalized power spectrum PA cannot provide such suppression with our values for MA and Hinf. However, it comes from the dilution factor (a.)-3 = (af/a0-3 in Eq. (7.60). To get the gravitino and Polonyi masses, we have to add a few cosmologi-cal assumptions about the relevant parameters of the reheating process and, in particular, about the reheating temperature Treh. The cosmological parameters can be fixed by specifying the e-foldings number N e in the range between 50 and 60. For a more precise estimate of the CDM abundance, we choose Ne = 55, as in Sec. 2. This implies ns = 0.964, r = 0.004, minf = 3.2 • 1013 GeV and Hinf = nM^Pg/2 = 1.4 • 1014 GeV. In our scenario, well below the inflaton mass scale the low-energy effective field theory is given by the Standard Model (SM) that has the effective number of d.o.f. as g* = 106.75. It is reasonable to assume that all the SM particles originated from perturbative inflaton decay via the (Starobinsky) universal reheating mechanism, whose reheating temperature is known [37,38]: ( 90 \ 1/4 Treh = V^totMp = 3 • 109 GeV. (7.63) On the other hand, the reheating temperature for heavy gravitino is given by [39] Treh = 1.5 • 108 GeV (*0f y^f2. (7.64) Combining Eqs. (7.63) and (7.64) we get the gravitino and Polonyi masses as follows: m3/2 = (7.7 ± 0.8) • 1012 GeV and MA = 2e-(I>m3/2 > 2m3/2 . (7.65) 7.5 Primordial Black Holes in supergravity PBHs may be formed in the early Universe by collapse of primordial density perturbations resulting from inflation, when these perturbations re-enter the horizon and are large enough, i.e. when gravity forces are larger than pressure, in general. Apart from being considered as another (non-particle) source for DM, some PBHs (of stellar mass type) are also considered as the candidates for the gravitational wave effects caused by the binary black hole mergers observed by LIGO/Virgo collaboration [40,41]. The PBH mass Mpbh is related to the perturbations scale k by Carr's formula [42] *„ 4nH-3 ( y \( g* \-1 ( k/(2n) V2 Mpbh = Yp — k M0 {^{f^ (JT^j , (7.66) whose coefficient y = 3-3/2 k 0.2, the (normalized) energy density is almost equal to the (normalized) entropy density g* k 3.36, and M0 stands for the Solar mass, M0 k 2 x 1033 g. The PBHs abundance f = nPBH/nc is proportional to the amplitude of the scalar perturbations Pz, while for the LIGO events one finds k/(2n) ~ 10-9 Hz, Pz ~ 10-2 and f ~ 10-2, as the regards the orders of their magnitudes [40,41]. The value of 10-9 Hz corresponds to 106 Mpc-1. In a single-field inflation, relevant perturbations are controlled by inflaton 2 2 scalar potential, so that large fluctuations PR k K- (2^) are produced when the slow roll parameter £ = r/16 goes to zero, i.e. when the potential has a near-inflection point where V' k V" k 0 . (7.67) Since we want a copious PBH production along with observationally consistent CMB observables, we should "decouple" these events, and demand the existence of another ("short") plateau in the scalar potential after the inflationary plateau towards the end of inflation. This is not the case for the Starobinsky inflation with the scalar potential (7.14), however, it can be easily achieved in a more general framework. Our supergravity framework in Sect. 3 is an example of such framework, because it leads to a single-field inflation governed by arbitrary function J, so that the associated inflaton scalar potential is given by V = g- (J')2. As an example, let us consider the inflaton scalar potential r^ = (1 + £ - e-a* - £,e-13*2) 2 , (7.68) Vo V / which is a deformation of the Starobinsky potential (7.14) with a = \J2/3 and the new real parameters |3 > 0 and £ > 0. The Starobinsky potential (7.14) is recovered when £ = 0. The scalar potential (7.68) falls into our supergravity framework, has Minkowski minimum at ^ = 0 and the inflationary plateau for large positive But, in addition, it also has an inflection point in the "waterfall" region between the inflationary plateau and the Minkowski vacuum. Indeed, the conditions (7.67) result in two equations, ae-a* + 2£p^e-p*- = 0 (7.69) and a2e-a* - 2£pe-p*- + 4£pVe-**2 = 0 , (7.70) respectively. They imply a quadratic equation on a^ + 1 - 2|3^2 = 0, (7.71) whose solution is given by a + Va2 + 4|3 =-4p-> 0 . (7.72) Then the remaining condition above is solved by 4 = 2M+ ' ("3) Of course, there are many other possibilities to choose the scalar potential having the form of a real function squared. We just showed that it is possible to combine a viable (Starobinsky-like) inflation with a viable (stellar mass type) PBHs production in the context of supergravity. 7.6 Conclusion Our results lead to the intriguing unifying picture of CDM, dark energy (positive cosmological constant) and cosmological inflation, in which their parameter spaces are linked to each other. This scenario also suggests the interesting phenomenology in the ultra high energy cosmic rays: super heavy Polonyi particles may decay into the SM particles, as the secondaries, in top-bottom decays. Cosmological high energy neutrinos from the primary and secondary decay channels can be tested by IceCube and ANTARES experiments. Another interesting outcome is that some (stellar mass type) PBHs remnants produced from the supergravity fields can compose part of the CDM halo coexisting with gravitinos. In this scenario, gravitational wave signals from the PBHs mergers can be envisaged, with intriguing implications for LIGO/VIRGO experiments. In short, gravitational wave experiments may provide us with precious indirect information about the scalar sector of the inflationary supergravity. Finally, the intriguing possibility exists for a unification of the inflaton in the vector multiplet, and the SUSY GUTs such as the flipped SU(5) x U(1) model arising from (Calabi-Yau) compactified heterotic superstrings or the intersecting D-branes. Acknowledgements The work by SVK on gravity and supergravity is supported by the Competitiveness Enhancement Program of Tomsk Polytechnic University in Russia. This work is also supported by a Grant-in-Aid of the Japanese Society for Promotion of Science (JSPS) under No. 26400252, and the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. 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Carr, "The Primordial black hole mass spectrum", Astrophys. J. 201,1(1975). Bled Workshops in Physics Vol. 19, No. 2 Proceedings to the 21 st Workshop What Comes Beyond ... (p. 164) Bled, Slovenia, June 23-July 1, 2018 8 Phenomenological Mass Matrices With a Democratic Origin A. Kleppe * SACT, Oslo Abstract. Taking into account the available data on the mass sector, and without any preconceptions about a specific matrix texture, we obtain quark mass matrices with a kind of democratic underpinning. Our starting point is a factorization of the "standard" parametrization of the Cabibbo-Kobayashi-Maskawa mixing matrix, from which we derive this specific type of quark mass matrices. Povzetek. Avtorica uporabi razpolozljive podatke o masah delcev in obicajno parametriza-cijo mesalne matrike Cabibba, Kobayashija in Maskawe ter poisce, ne da bi vnaprej privzela kakrsnokoli zahtevo za simetrijo, masne matrike za kvarke. Izkaze se, da so zelo zblizu demokraticnim matrikam. Keywords: Mass matrices, flavour symmetry, democratic texture 8.1 Mass states and flavour states In this project, we take a rather phenomenological approach to the quark mass sector, by assuming that the quark mass matrices can be derived from a simple factorization of the Cabbibo-Kobayashi-Maskawa (CKM) mixing matrix [1], /Vud Vus Vub \ V = I Vud Vus Vub ) \Vud Vus Vub/ which appears in the charged current Lagrangian Lcc = — LY^L W, + h.c. (8.1) where 4 and 4' are fermion fields with charges Q and Q — 1, correspondingly. Lcc is usually interpreted as an interaction between left-handed physical particles with charge Q and superpositions of left-handed physical particles of charge Q — 1 , e.g. between a (left-handed) up-sector quark and a superposition * E-mail: kleppe@nbi.dk of (left-handed) down-sector quarks. But it can just as well be interpreted as interactions between flavour states f, f', Lcc = - 272 fLY^ fLW + h.c. (8.2) where f = U1"' = U', and UU= V The reason we emphasize this is that f, f' appear in the mass Lagrangian Lmass = fMf + f M'f' = ^D^ + ^ 'D >', (8.3) where f, f' are quark flavour states with charge 2/3 and -1/3, respectively, and ^' are the corresponding mass states. The mass matrices in the weak basis are denoted by M = M(2/3) and M' = M'(-1/3), which in the mass bases correspond to the diagonal matrices D = diag(mu, mc, mt) and D' = diag(md,ms,mb). It is the form of the mass matrices M and M' in the weak basis that we are looking for, in the hope that they can shed light on the mechanism behind the hierarchical fermion mass spectra. In the context of weak interactions it is thus crucial to distinguish between mass states and flavour states, the flavour states being the eigenstates of the weak interactions, and the mass eigenstates correspond to the "physical particles" that take part in strong and electromagnetic interactions. The picture is that the flavour states all live in the same weak basis in flavour space, while the mass states of different charge sectors live in their separate mass bases. We go from the weak basis to the mass bases of the charge 2/3- and charge -1/3-sector, respectively, by rotating the mass matrices M(2/3) and M'(-1/3) by the unitary matrices U and U', which are factors of the CKM-matrix, V = UU'". M —> UMU1" = D = diag(mu,mc,mt) (8.4) M' -> U'M'U= D' = diag(md,ms,mb) We can always assume that the mass matrices are Hermitian [3], and diagonalized by hermitian unitary matrices. Since V = UU= 1, the up-sector mass basis is different from the down-sector mass basis, and the CKM matrix bridges the two mass bases. It can be argued that flavour states merely exist in our fantasy, since they are not directly measurable. This line of thought is however defied by the neutrinos. Whereas in the quark sector there is a distinction between flavour states, where mass states are perceived as "physical" and the weakly interacting flavour states are defined as mixings of these physical particles, in the lepton sector the situation is quite different. This is due to the fact that as far as we know, neutrino mass states never appear on the scene - in the sense that they never take part in interactions, but merely propagate in free space. The neutrinos vc,v^,vT are flavour states, but we nontheless perceive them as "physical", because they are the only neutrinos that ever appear in interactions, i.e. they are the only neutrinos that we "see". A neutrino is defined by the charged lepton with which it interacts: what we call the electron-neutrino vc is the superposition of neutrino mass states which appears together with the electron, and likewise for ^ and t; in that sense the conservation of the lepton number is a tautology. The only mixing matrix that occurs in the lepton sector is the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix U which exclusively operates on neutrino states, 'Ve\ /V! V„ | = U(PMNS) I V2 \V3, where (vi , v2, v3) are mass eigenstates, and ( ve, V^, vt) are the weakly interacting flavour states. In the lepton sector, the charged currents are thus interpreted as (e, t) interacting with the neutrino flavour states (ve, v^, vT) - and the charged leptons are consequently defined as being both flavour states and mass states. 8.2 Factorizing the weak mixing matrix The usual procedure in establishing an ansatz for the quark mass matrices is to hypothesize a mass matrix of a specific form. Here we instead look for a "natural" factorization of the Cabbibo-Kobayashi-Maskawa mixing matrix, hoping to find the "correct" rotation matrices U and U' that diagonalize the mass matrices M and M'. The CKM matrix can of course be parametrized and factorized in many different ways, and different factorizations correspond to different rotation matrices U and U', and correspondingly to different mass matrices M and M'. We choose what we perceive as the most obvious and "symmetric" factorization of the CKM mixing matrix, following the well-known standard parametrization [2] with three Euler angles a, ß, 29, V = CßC20 -Cßsas2eel5 - -iS\ SßC2e S2ee sßCa sßsa S2eelS + CßC a saC2e Cßsa CaC2e = UU 't (8.5) -CßCas2eelS + sßsa -sßCas2eelS This corresponds to the diagonalizing rotation matrices for the up- and down- sectors '10 0 U = W I 0 cos a sin a v0 — sin a cos ay ~iY \ /cos 9 0 sin 9N 0 1 0 eiY/ V — sin 9 0 cos 9> Wt Cee-iY 0 see-iY WI —saseeiY Ca saCeeiY | Wt (8.6) i—CaseeiY —sa CaCee iY , and U ' = W = W ^cos ß — sin ß 0^ sin ß cos ß 0| 0 0 1 -iY eiY , cos 9 0 — sin 9 0 1 0 | Wt sin 9 0 cos 9 'CßCee-iY sßCee-iY s0e iY —sß Cß 0 -Cßsee sßsee iY -iY (8.7) Wt Cee iY respectively, where W = W(p) is a unitary matrix which is chosen is such a way that the same phase y appears in the mass matrices of both charge sectors, i.e. a matrix of the form (0 cos p ± sin p^ 1 0 0 0 t sin p cos p cos p 0 ± sin p 0 1 0 yT sin p 0 cos p cos p ± sin p 0 0 0 1 yT sin p cos p 0y Here the value of the parameter p is unknown, whereas a, 9 and y correspond to the parameters in the standard parametrization, with y = 5/2, 6 = 1.2 ± 0.08 rad, and 29 = 0.201 ± 0.011°, while a = 2.38 ± 0.06° and | = 13.04 ± 0.05°. In our factorization scheme, a and | are the rotation angles operating in the up-sector and the down-sector, respectively. With the rotation matrices U(a, 9, y, p) and U'(|,9,y, p), we obtain the mass matrices for the up- and down-sectors, respectively, M = U^diag(mu,mc,mt)U and M' = U'^diag(md,ms,mb)U' For the up-sector this gives 'M„ MT2 MT3\ M = 1 M2I M22 M23 | ,M3! M32 M33/ xc^ + Ys^ = Wt(p) | Zse eiY Zse e-iY (X — Y)CeseN Y — 2Zcot2a —ZCe eiY | W(p) JX — Y)Cese —ZCe e-iY Xs^ + YCJe (8.8) e e where X = mu, Y = mc sin2 a + mt cos2 a, Z = (mt - mc) sin a cos a = \J (mt - Y)(Y - mc), and mu,mc, mt are the masses of the up-, charm- and top-quark; and W(p) is a unitary one-parameter matrix. Analogously for the down-sector mass matrix, /M' 1 M'2 M' M' = I M21 M22 M23 \M2 1 M32 M33, / X's2 + Y'c2 Z'ce eiY (X' - Y')cese\ = Wf(p) I Z'ce e-iY Y' + 2Z'cot2| -Z'se e-iY I W(p) (8.9) V(X' - Y')cese -Z'se eiY X'c2 + Y's2 / where X' = mb, Y' = md cos2 | + ms sin2 Z' = (ms - md) sin | cos | = \J(ms - Y')(Y' - md), and md, ms, mb are the masses of the down-, strange- and bottom-quark, respectively. The two mass matrices thus display similar textures. With Y = mc sin2 a + mt cos2 a, Z = (mt - mc) sin a cos a, Y' = md cos2 | + ms sin2 and Z' = (ms - md) sin | cos we can moreover write m-u = X, mc = Y - Z cot a, mt = Y + Z tan a, (8 10) md = Y' - Z' tan ms = Y' + Z' cot mb = X', (. ) 8.3 The matrix W There are of course many ways to chose a one-parameter unitary matrix, but we choose a matrix W(p) which conveniently gives mass matrices with the same phase y for both charge sectors, W(p) = sin p 0^ 01 sin p cos p 0 cos p 0 (8.11) This gives the up-sector mass matrix Xc2 + Ys2 M = Wf I Zse e iY JX - Y)cese -Zce e-iY Zse e-iY (X - Y)ceseN Y - 2Z cot 2a -Zce eiY xs2 + yc2 w = (X cos2 |i + Ysin2 |i (Y - X) sin |icos |i -Zsin |i e iY = I (Y - X) sin | cos |X sin2 | + Y cos2 | -Z cos |i e-iY Z sin | e iY Z cos | e iY F (8.12) where | = p - 0, X = mu, Y = mc sin2 a + mt cos2 a, Z = y/(mt - Y)(Y - mc) and F = Y - 2Zcot2a = mcc;a + mts;a. Now, depending on the value of ^ = p — 0, we get different matrix textures, e.g. for p — 0 = 0 or n, we get the simple form 'X M(0,n) = | 0 ,0 0 Y -Zeiy and for p — 9 = n/2, equally simple M(n/2) = Y0 0X -Zeiy 0 0 -Ze-iy -Ze-iyNl 0 F (8.13) (8.14) Applying the same procedure to the down-sector, we get the down-sector mass matrix M' = W(p) X's0 + Y'c0 Z'ce eiY (X' — Y')c0s0\ Z'c0 e-iY Y' + 2Z'cot2ß —Z's0 e-iY | W(p) = V(X' — Y')cese —Z'se eiY X'c0 + Y's0 J 'X' sin2 h' + Y' cos2 h' (X' — Y') sin h' cos h' Z' cos h' eiY = | (X' — Y') sin h' cos h' X' cos2 h' + Y' sin2 h' —Z' sin h' eiY Z' cos h ' e-iY —Z' sin h' e-iY F' (8.15) where ^' = p+0, X' = mb, Y' = md cos2 |3+ms sin2 Z' = — Y')(Y' — md) and F' = Y' + 2Z' cot 2| = mds| + msCg. Again, different ^'-values correspond to different matrices, e.g. for ^' = p + 0 = 0 or n, we get M '(0,n) = Y' 0 Z 'eiyNl 0 X' 0 Z'e-iY 0 F' (8.16) and for h' = P + 9 = n/2, we get X' 0 0 M'(n/2) = | 0 Y' —Z'eiY 0 —Z 'e-iY F' (8.17) t 8.4 Texture Zero Mass Matrices The matrices (8.13) and (8.14), as well as (8.16) and (8.17), make us wonder if our scheme is compatible with quark mass matrices of texture zero. The study of texture zero matrices is driven by the need to reduce the number of free parameters, since the fermion mass matrices are 3x3 complex matrices, which without any constraints contain 36 real free parameters. It is however always possible to perform a unitary transformation that renders an arbitrary mass matrix Hermitian [3], so there is no loss of generality in assuming that the mass matrices are Hermitian, reducing the number of free parameters to 18. This is still a very large number, which in the end of the 1970-ies prompted Fritzsch [6], [7] to introduce "texture zero matrices", i.e. mass matrices where a certain number of the entries are zero. Since then, a huge amount of articles have appeared, with analyses of the very large number of (different types of) texture zero matrices and their phenomenology. In the course of this work, a number of of texture zero matrices have been ruled out. A handful of matrices have however been singled out as viable [8], which among the texture 4 zero matrices are: A B 0N B* DC 0 C* 0, ABC B* ,C* D 0 00 A0B 0 0 C ,B* C* D/ 0 C 01 C* A B 0 B* D, 0 0 CN 0 A B C* B* D D C BN C* B* 00 0A while A 0 0 0 C B 1 and ,0 B* D, A0B 0 C 0 B* 0 D are among the matrices that are ruled out. In our scheme this precisely corresponds to the matrices (8.13), (8.14), (8.16) and (8.17), which gives a constraint on the angle P, 1 (8.18) p = ^Nn±9 where N G Z, ruling out the matrices M (2 Nn - 9) and M' (1 Nn + 9). This implies that our mass matrices M and M' are not of texture zero. Instead, they display a kind of democratic texture [4], a feature that has merely been outlined in our earlier project [5]. 8.5 Democratic mass matrices In the Standard Model, fermions get their masses from the Yukawa couplings by the Higgs mechanism. We know that the fermion masses within one charge sector are very different, but there is no apparent reason why there should be a different Yukawa coupling for each fermion of a given charge. Taking the difference between the weak basis and the mass bases into account, the democratic philosophy proclaims that in the weak basis, the fermions of a given charge should have identical Yukawa couplings, just like they have identical couplings to the gauge bosons of the strong, weak and electromagnetic interactions. The democratic hypothesis thus implies that in the weak basis the quark mass matrices for both charge sectors have an initial, "democratic" form Mo = k I 1 1 1 I = kN (8.19) \1 1 V where k has dimension mass; and the mass spectrum (0,0,3k) reflects the phenomenology of the fermion mass spectra with one very big and two much smaller mass values - in the mass basis. In the weak basis the matrix M0 = kN is however totally flavour symmetric, in the sense that the flavour states ft of a given charge are indistinguishible and the initial mass Lagrangian reads 3 Lmass = kfNf = ^ kftfj i=1,j=1 which is a totally flavour symmetric situation, with a discrete flavour symmetry under the cyclic permutation group Z3 operating on the mass matrix. That the Yukawa couplings are identical for all the flavours, while the mass eigenvalues are so completely different is a reminder of the difference between flavour states and mass states. The democratic symmetry is unchanged if we add a diagonal matrix diag(X,X,X) to kN, since the new democratic mass matrix M0 = kN + diag(X,X,X) still corresponds to a completely flavour symmetric mass Lagrangian, 3 3 3 Lmass = fMof = k ^ fifj + X ^ ff = (k + X) ^ fifj (8.20) i,j = 1 i=1 i=1 Moreover, since the up-sector mass matrix and the down sector mass matrix in this assumed democratic initial stage are structurally identical, the mixing matrix is equal to unity, so there is no CP-violation. In order to obtain the final mass spectra with the three hierarchical non-zero values, the initial democratic symmetry must be broken in such a way that we get a mixing matrix and masses that all agree with data. In the democratic scenario an ansatz thus consists of a specific choice for the flavour symmetry breaking scheme. In our approach, it however comes out of the formalism, without any presupposition of a democratic texture or a specific breaking scheme. 8.5.1 Reparametrizing the mass matrices By reformulating the matrix elements Mu, M22, M' 1, and M 22 in the quark mass matrices (8.12) and (8.15), using the relations Xc2 + Ys2 = (Y - X)s2 + X, Xs2 + Yc2 = (Y - X)c2 + X, and X's ¡2, + Y'c ¡2, = (Y' - X')c ¡2, + X ', and X'c* + Y's* = (Y' - X')sj2, + X', the mass matrices can be rewritten in a way that reveals a kind of "democratic substructure", Xc2 + Ys2 (Y - X)s^c „ -Zs „ (Y - X)s2C 2 Xs2 + Yc 2 -Zc 2 e-iY e -iy\ M = Zs 2 eiY -Zc , , eiY M- (8.21) 'sin |i = B cos | GeiY> n i r 111 rrr 'sin |i X cos | + X Ge-iY / X + Ay F 172 A. Kleppe and = B' where and X's,, + Y'c,, (X' - Y')s,,c,, Z'cM M' = I (X' - Y')s,,c,/ X'c2 + Y's2 -Z's Z 'c, cos , - sin , G /e-iY 2 / 2 i i i i i i iii eiY eiY I _ (8.22) -Z's,, e-iY F' cos 2 / -sin ,' I + GVY . X/+A' G _ G' = X = mu, ( = p (mt - me)saca B _ Y - X = mcs?, + mtc mu (mcs2 + mtc2 - mu)' X' = mb, (' _ p + e, (ms - md)spcp A _ (mc - mu)(mt - mu) (mcs2 + mtc2 - mu) , B' = Y' - X' _ mss2 + mdc2 A' _ (md - mb)(ms - mb) mb, (mdc2 + mss2 - mb)' (mdc2 + mss2p - mb)' a = arctan ^ = 2.38 ± 0.06°, ( = arctan (= 13.°4 ± °.°5°. The matrices of the two charge sectors thus display great similarities. That A = 0 and A' = 0 moreover means that mc = mu, mt = mu, md = mb and ms = mb, and with the additional condition mc = mt and md = mb, we almost have the prerequisite for CP-violation - which basically says that CP-violation occurs once there is a third family (and a complex phase). 2 X 2 cc 8.6 Discussion We interpret the structure displayed by (8.21) and (8.22) as the result of an in initial democratic matrix, where the flavour symmetry undergoes a stepwise breaking, each step corresponding to one term. If we consider the up-sector, the first term comes from /1 1 1\ /sin ( \ /1 1 1\ /sin ( \ Mo _ k lilija Mi _ B I cos ( I I 1 1 1 I I cos ( I , \1 1 1) \ GeiV \1 1 y \ Ge-iY/ (8.23) where k and B both have the dimension mass. This first symmetry breaking step really corresponds to shifting the flavours in such a way that fi —» s,f 1, f2 —» c,f2, f3 —> Ge-iYf3. The mass spectrum still consists of two massless and one massive state, but the flavour symmetry is partially broken, with the mass Lagrangian Lmass _ fMi f _ XiXi + XXiX2 + XX2Xi + X2X2 _ (Xi + XX2)(xi + X2), wherex1 _ B(s,fi + c,f2), x2 _ BGe-iYf3. The original total flavour symmetry is thus broken down to the partial flavour symmetry f1 ^ f2, but there is still only one non-vanishing eigenvalue. In the next step, by shifting the origin from diag(0,0,0) to diag(X, X, X), we obtain a mass spectrum with one very heavy, massive state, and two lighter states with mass X, i.e. /sin |i \ /1 1 1\ /sin |i \ /X \ M^ M2 = B I cos | I I 1 1 1 I I cos | I + I X I V Ge1^/ \1 1 1/ \ \ XJ (8.24) where X has dimension mass. In the last step, the remaining degeneracy in the mass spectrum (X, X, X + B(G2 + 1)) is subsequently broken, by adding the term diag(0,0, A), where A has dimension mass. We argue that this last breaking is necessitated by the principle of minimal energy, in analogy with the Jahn-Teller effect. M3 = B ( sln " cos * Ge0 ( 1 1 1 K sln ^ cos " Ge-iO + ( X X X ) + ( 0 0 A ) (8.25) We identify our scheme as a democratic scenario, where the flavour symmetry is broken in the specific way described above. 8.7 Numerical values In order to get a notion of the sizes of the parameters B, G, X, A, we calculate their values for quark masses at different Using quark masses at MZ, [9], [10], [11] mu(MZ) = 1.24MeV, mc(MZ) = 624MeV, mt(MZ) = 171550MeV md(MZ) = 2.69MeV, ms(MZ) = 53.8MeV, mb(MZ) = 2850MeV ( ) we get the numerical values for the parameters: up-sector d-sector B = 171254MeV « mt cos2 a B ' = -2844.71MeV « 2md - mb G = 0.0414 G ' = -0.0039 X = 1.24MeV X ' = 2850MeV A = 623.83MeV « mc cos a A' = —2798.76MeV « ms - md - mb and as before, we use the angles a = 2.38° and |3 = 13.04°. We would also like to establish some numerical value, or at least a range, for the parameter p. Our initial assumption was that the matrices (8.6), (8.7) which diagonalize the up-sector and down-sector mass matrices, are given by the factorization of the Cabibbi-Kobayashi-Maskawa matrix (8.5). The parameters of the CKM matrix are well-known, so the only remaining "steering-parameter" is p. The angles | and |' in the mass matrices of the up- and d-sector depend on p, whose value is unknown. We have the constraint p = jNn ± e (8.27) which excludes some values of p, but it remains unknown what value(s) p actually takes. 8.8 Conclusion By factorizing the "standard parametrization" of the CKM weak mixing matrix in a very natural and straightforward way, we obtain mass matrices with a type of democratic texture that can be derived from a democratic matrix, followed by a well-defined scheme for breaking the primary flavour symmetry. This democratic texture unexpectedly emerges from our factorization of the weak mixing matrix, there is no presupposition about what form our resulting mass matrices would have, and no assumptions other than our factorization scheme and the choice of the unitary matrix W(p). References 1. M. Kobayashi, T. Maskawa; Maskawa (1973), "CP-Violation in the Renormalizable Theory of Weak Interaction", Progress of Theoretical Physics 49 (2): 652-657. 2. L.L. Chau and W.-Y. Keung (1984), "Comments on the Parametrization of the Kobayashi-Maskawa Matrix", Phys. Rev. Letters 53 (19): 1802. 3. D. Emmanuel-Costa and C. Simoes, Phys. Rev. D 79, 073006 (2009). 4. H. Harari, H. Haut and J. Weyers, Phys. Lett. B 78, 459 (1978). 5. A. Kleppe, "A democratic suggestion", hep-ph/1608.08988. 6. H. Fritzsch, Phys. Lett. B 70, 436 (1977), Phys. Lett. B 73, 317 (1978). 7. H. Fritzsch, "Texture Zero Mass Matrices and Flavor Mixing of Quarks and Leptons", hep-ph/1503.07927v1. 8. S. Sharma, P. Fakay, G. Ahuja, M. Gupta, Phys. Rev. D 91, 053004 (2015). 9. Matthias Jamin, private communication. 10. M. Jamin, J. Antonio Oller and A. Pich, "Light quark masses from scalar sum rules", arXiv:hep-ph/0110194v2. 11. FLAG Working Group, "Review of lattice results concerning low energy particle physics" (2014), hep-lat/1310.8555v2. 12. http://pdg.lbl.gov/2017/tables/rpp2017-sum-quarks.pdf Bled Workshops in Physics Vol. 19, No. 2 JLV Proceedings to the 21 st Workshop What Comes Beyond ... (p. 175) Bled, Slovenia, June 23-July 1, 2018 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? * N.S. Mankoc Borštnik1 and H.B.F. Nielsen2 1 Department of Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia 2 Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, Copenhagen 0, Denmark Abstract. This is a discussion on fermion fields, the internal degrees of freedom of which are described by either the Grassmann or the Clifford anticommuting "coordinates". We prove that both fields can be second quantized so that their creation and annihilation operators fulfill the requirements of the commutation relations for fermion fields. However, while the internal spins determined by the generators of the Lorentz group of the Clifford objects Sab and Sab (in the spin-charge-family theory Sab determine the spin degrees of freedom and S ab the family degrees of freedom) are half integer, the internal spin determined by Sab (expressible with Sab + Sab) is integer. Nature "made" obviously the choice of the Clifford algebra, at least in the so far observed part of our universe. We discuss here the quantization — first and second — of the fields, the internal degrees of freedom of which are functions of the Grassmann coordinates 0a and their conjugate momenta, as well as of the fields, the internal degrees of freedom of which are functions of the Clifford ya. Inspiration comes from the spin-charge-family theory ([1,2,9,3], and the references therein), in which the action for fermions in d-dimensional space isequal to J ddx E J (ipyapoa^) + h.c., with poa = faaP0a + Je {Pa, Ef*a}-, P0a = Pa - JSab^aba - 1 Sab^aba. We write the basic states as products of those either Grassmann or Clifford objects, which allow second quantization for fermion fields, and look for the action and solutions for free fields also in the Grassmann case in order to understand why the Clifford algebra "wins in the competition" for the physical (observable) degrees of freedom. Povzetek. Avtorja obravnavata razliko med fermionskimi polji, katerih interne prostostne stopnje opisemo bodisi z Grassmannovimi bodisi s Cliffordovimi antikomutirajocimi "koordinatami". DokaZeta, da lahko v obeh primerih poiscemo kreacijske in anihilacijske operatorje, ki zadoscajo komutacijskim relacijam za fermionska polja v drugi kvantizaciji. Obe vrsti opisa fermionskih polj se vseeno bistveno razlikujeta: notranji spini, določeni z generatorji Lorenztove grupe Cliffordovih objektov Sab in Sab (v teoriji spinov-nabojev-druzin dolocajo Sab spinsko kvantno stevilo ter s tem spine in naboje kvarkov in leptonov, Sab pa dolocajo družinska kvantna stevila), imajo polstevilcen spin, medtem ko je notanji spin, ki ga dolocajo Sab (izrazljivi z Sab + Sab), celostevilcen. Narava je ocitno "izbrala" Cliffordovo algebro (vsaj v opazljivem delu vesolja). Avtorja obravnavata prvo in drugo kvantizacijo polj, katerih notranje prostostne stopnje opiseta s funkcijami Grassmannovih * This article is the expanded part of the talk presented by N.S. Mankoc Borstnik at the 21st Workshop "What Comes Beyond the Standard Models", Bled, 23 of June to 1 of July, 2018. koordinat 0 a in ustreznih konjugiranih momentov, pa tudi polja, katerih notranje prostostne stopnje so opisane s funkcijami Cliffordovih koordinat ya. Uporabo za opis fermionov v Grassmannovem prostoru je navdihnila teorija spinov-nabojev-druzin ([1,2,9,3], in reference v njih), v kateri akcijo v d-razseZnem prostoru opise eden od avtorjev (N.S.M.B.) z J ddx E 1 (iyap0ap + h.c., s kovariantnim odvodom poa = faap0a + je (pa, Efaa}-, P 0a = pa — 1 Sab^aba — j S ab & aba. Bazna stanja iščeta kot produkt bodisi Grassmannovih bodisi Cliffordovih "koordinat", ki dopusčajo drugo kvantizacijo, ponudita akcijo za prosta polja tudi v primeru Grassmannovih koordinat, da bi bolje razumela, zakaj je v tekmi za fizikalne prostostne stopnje "zmagala" Cliffordova algebra. Keywords: Second quantization of fermion fields, Spinor representations, Kaluza-Klein theories, Discrete symmetries, Higher dimensional spaces, Beyond the standard model PACS:11.30.Er,11.10.Kk,12.60.-i, 04.50.-h 9.1 Introduction This paper is to look for the answers to the questions: Why our universe "uses" the Clifford rather than the Grassmann coordinates, although both lead in the second quantization procedure to the anti-commutation relations required for fermion degrees of freedom? Is the answer that the Clifford degrees of freedom offer the appearance of families, the half integer spin and the charges as observed so far for fermions, while the Grassmann coordinates offer the groups of (isolated) integer spin states with the charges in the adjoint representations and no families? Can the choice of the Clifford degrees of freedom explain why the simple starting action of the spin-charge-family theory of one of us (N.S.M.B.) [9,3,5,8,4,6,7] is doing so far extremely well in manifesting the observed properties of the fermion and boson fields in the observed low energy regime? The questions are too demanding that this paper could offer the answers. We are trying only to make first steps towards understanding them. Our working hypothesis is that "nature knows all the mathematics", accordingly therefore also both — the Grassmann and the Clifford "coordinates". In a trial to understand why Grassmann space "was not the choice of nature" to describe the internal degrees of freedom of fermions, we see that ya's and ya's of the spin-charge-family theory enable to describe not only the spin and charges of fermions, but also the existence of families of fermions (in the first and second quantized theory of fields). This work is a part of the project of both authors, which includes the fermion-ization procedure of boson fields or the bosonization procedure of fermion fields, discussed in Refs. [11,12,14] for any dimension d (by the authors of this contribution, while one of them, H.B.F.N. [13], has succeeded with another author to do the fermionization for d = (1 + 1)), and which would hopefully help to better understand the content and dynamics of our universe. In the spin-charge-family theory [9,3,5,8,4,6,7] — which offers explanations for all the assumptions of the standard model, with the appearance of families, the scalar higgs and the Yukawa couplings included, offering also the explanation for the matter-antimatter asymmetry in our universe and for the appearance of the dark matter — a very simple starting action for massless fermions and bosons in d = (1 + 13) is assumed, in which massless fermions interact with only gravity, the vielbeins faa (the gauge fields of moments pa) and the two kinds of the spin connections (^aba and &aba, the gauge fields of the two kinds of the Clifford algebra objects Ya and Ya, respectively). A = [ ddx E 1 (if YaPoaf) + h.c. + ddxE (aR + aR), (9.1) with poa = faaPoa + 2e {Pa,Efaa}-, P0a = Pa - 2Sab ^aba - JSab^aba and R = 1 {fafb] (^abcc,p - CUCaa + h.C., R = \ {fa[af2b] (Waba,p - ^caa ^Cbp)} + h.c.. The two kinds of the Clifford algebra objects, Ya and Ya, {Ya,Yb}+ = 2nab = {Ya,Yb}+, {Ya,Yb}+ = 0. (9.2) anticommute (Ya and Yb are connected with the left and the right multiplication of the Clifford objects, there is no third kind of the Clifford operators). One kind of the objects, the generators Sab = 4(Ya Yb - Yb Ya), determines spins and charges of spinors of any family, another kind, Sab = 4 (Ya Yb -Yb Ya), determines the family quantum numbers. Here 1 fa[afPb] = faafPb - fabfPa. There are correspondingly two kinds of infinitesimal generators of the Lorentz transformations in the internal degrees of freedom — Sab for SO(13,1) and Sab for SO(13,1) — arranging states into representations. The scalar curvatures R and R determine dynamics of the gauge fields — the spin connections and the vielbeins, which manifest in d = (3 + 1) all the known vector gauge fields as well as the scalar fields [5] which explain the appearance of higgs and the Yukawa couplings, provided that the symmetry breaks from the starting one SO(13,1) to SO(3,1) x SU(3) x U(1). The infinitesimal generators of the Lorentz transformations for the gauge fields — the two kinds of the Clifford operators and the Grassmann operators — operate as follows, Eq. (9.25) {Sab, Ye}- = —i (nae Yb - nbe Ya), {Sab, Ye}- = -i(nae Yb -nbe Ya), {Sab, ee}_ = -i(nae eb -nbe ea), {Mab, Ad...£...9}_ =-i (nae Ad...b...g - nbe Ad...a...g), (9.3) faa are inverted vielbeins to eaa with the properties eaafab = 5%, eaafpa = 5^, E = det(eaa). Latin indices a, b,.., m, n,.., s, t,.. denote a tangent space (a flat index), while Greek indices a, .., |i, v, ..ff, T,.. denote an Einstein index (a curved index). Letters from the beginning of both the alphabets indicate a general index (a, b, c,.. and a, Y,.. ), from the middle of both the alphabets the observed dimensions 0,1,2,3 (m, n,.. and v,..), indexes from the bottom of the alphabets indicate the compactified dimensions (s, t,.. and ff, T,..). We assume the signature r|ab = diag{1, —1, —1, • • • , -1}. where Mab are defined by a sum of Lab plus either Sab or S ab, in the Grassmann case Mab is Lab + Sab, which appear to be Mab= Lab + Sab + Sab, as presented later in Eq. (9.26). We discuss in what follows the first and the second quantization of the fields, the internal degrees of freedom of which are determined by the Grassmann coordinates 0a, as well as of the fields, the internal degrees of freedom of which are determined by the Clifford coordinates Ya (or Ya) in order to understand why "nature has made a choice" of fermions of spins and charges (describable in the spin-charge-family theory by subgroups of the Lorentz group expressible with the generators Sab) in the fundamental representations of the groups (which interact in the spin-charge-family theory through the boson gauge fields — the vielbeins and the spin connections of two kinds), rather than of fermions with the integer spins and charges. We choose correspondingly either ea's or Ya's (or Ya's, either Ya's or Ya's [6,7,9]) to describe the internal degrees of freedom of fields. In all these cases we treat free massless fields; masses of the fields in d = (3 + 1) are in the spin-charge-family theory due to their interactions with the gravitational fields in d > 4, described by the scalar vielbeins or spin connection fields [[1,2,9,3,5,8,4,6,7], and the references therein]. 9.2 Observations helping to understand why Clifford algebra manifests in the observable d = (3 + 1) We present in this section properties of fields with the integer spin in d-dimensional space, expressed in terms of the Grassmann algebra objects, and the spinor fields with the half integer spin, expressed in terms of the Clifford algebra objects. Since the Clifford algebra objects are expressible with the Grassmann algebra objects (Eqs. (9.17,9.18)), the norms of both are determined by the integral in Grassmann space, Eqs. (9.28, 9.31)2. a. Fields with the integer spin in Grassmann space A point in d-dimensional Grassmann space of real anticommuting coordinates ea, (a = 0,1,2,3,5,..., d), is determined by a vector {ea} = (e0,e1,e2,e3,e5,...,ed). A linear vector space over the coordinate Grassmann space has correspondingly the dimension 2d, due to the fact that (eai )2 = 0 for any at e (0,1,2,3,5,..., d). Correspondingly are fields in Grassmann space expressed in terms of the Grassmann algebra objects d B = ^ aaia2...ak eai ea2 ...eak |^og >, at < ai+i , (9.4) k=0 2 These observations might help also when fermionizing boson fields or bosonizing fermion fields. where |^og > is the vacuum state, here assumed to be |^og >= |1 >, so that -gfrl^0g >= 0 for any ea. The Kalb-Ramond boson fields aa, a2...ak are antisymmetric with respect to the permutation of indexes, since the Grassmann coordinates anticommute {ea,eb}+ = o. (9.5) The left derivative -dp on vectors of the space of monomials B(e) is defined as follows -A- B(e) = dBM, sea ( ) aea , de?3eb}+ B = 0,forall B. (9.6) Defining p0a = ig|p it correspondingly follows {p0a,peb}+ = 0, {p0a,eb}+ = inab , (9.7) The metric tensor nab (= diag(1, —1, —1,..., —1)) lowers the indexes of a vector {ea}: ea = nab eb, the same metric tensor lowers the indexes of the ordinary vector xa of commuting coordinates. Defining 3 (ea)f = ^naa = — ipeanaa, (9.8) it follows aea (—)f = naa ea, (pea )f = — inaaea. (9.9) oea rnnliw nmnprfipc fla anrl ^r\rri}cr"\r\nrlinrrK7 _ aeQ' Making a choice for the complex properties of ea, and correspondingly of g|p, as follows {ear = (e0 ,e], —e2,e3, —e5,e6,..., —ed-1,ed), {^r = (A -d- —A -d- —A -d- —-d-) (910) xae/ (ae0, se1, se2, ae3, ae5, ae6aed-1, aed), v' ' it follows for the two Clifford algebra objects Ya = (ea + ), and Ya = i(ea — o|p), Eqs. (9.17, 9.18), that Ya is real if ea is real, and imaginary if ea is imaginary, while Ya is imaginary when ea is real and real if ea is imaginary, just as it is required in Eq. (9.23). We define here the commuting object yG, which will be useful to find the action for Grassmann fermions, Eq. (9.37), and the appropriate discrete symmetry operators for this purpose — (Cg, Tg, Vg) in ((d — 1) + 1 )-dimensional space-time 3 In Ref. [2] the definition of was differently chosen. Correspondingly also the scalar product needed a (slightly) different weight function in Eq. (9.28). and (Cn, Tn, Pn) in (3 + 1) space-time — while following the definitions of the discrete symmetry operators in the Clifford algebra case [21] yG = (1 - 2eanaa^) 00a inaa YaYa , {yg,yG}- = 0. (9.11) Index a is not the Lorentz index in the usual sense. yG are commuting operators — {yG,yG}- = 0 for all (a,b) — as expected. They are real and Hermitian. yG1 = YG , (yG)* = YG . (9.12) Correspondingly it follows: yG^yg = I, ygyg = I.I represents the unit operator. By introducing [2] the generators of the infinitesimal Lorentz transformations in Grassmann space as Sab = (eapeb - ebp0a), (9.13) one finds {Sab, Scd}- = i{sadnbc + Sbcnad - Sacnbd - Sbdnac}, Sabt = naanbbSab. (9.14) The basic states in Grassmann space can be arranged into representations with respect to the Cartan subalgebra of the Lorentz algebra as presented in Ref. [2,15]. The state in d-dimensional space, for example, with all the eigenvalues of the Cartan subalgebra of the Lorentz group of Eq. (9.84) equal to either i or 1 is: (e0 - e3)(e1 + ie2)(e5 + ie6) ••• (ed-1 + ied)|^o9 >,with |^o9 >= |1 >. All the states of the representation, which start with this state, follow by the application of those Sab, which do not belong to the Cartan subalgebra of the Lorentz algebra. S01, for example, transforms (e0 - e3)(e1 + ie2)(e5 + ie6) • • • (ed-1 + ied)|^og > into (e0e3 + ie1ie2)(e5 + ie6) • •• (ed-1 + ied)|^og >, while S01 - iS02 transforms this state into (e0 + e3)(e1 - ie2)(e5 + ie6) ••• (ed-1 + ied)|^og >. b. Fermion fields with the half integer spin and the Clifford objects Let us present as well the properties of the fermion fields with the half integer spin, expressed by the Clifford algebra objects d F = ^ aflla2...ak Yai Ya2 ...Yak |^oc >, at < ai+1 , (9.15) k=0 where |^oc > is the vacuum state. The Kalb-Ramond fields aai a2...ak are again in general boson fields, which are antisymmetric with respect to the permutation of indexes, since the Clifford objects have the anticommutation relations, Eq. (9.2), {Ya,Yb}+ = 2nab . (9.16) A linear vector space over the Clifford coordinate space has again the dimension 2d, due to the fact that (yai )2 = naiai for any at G (0,1,2,3,5,..., d). One can see that Ya are expressible in terms of the Grassmann coordinates and their conjugate momenta as Ya = (9a - ip0a). (9.17) We also find Ya Ya = i (6a + ip0a), (9.18) with the anticommutation relation of Eq. (9.16) for either Ya and Ya {Ya,Yb}+ = 2nab , {Ya,Yb}+ = 0. (9.19) Taking into account Eqs. (9.8, 9.17, 9.18) one finds (Ya)f = Yanaa, (Y a)f = Y anaa, YaYa = naa , Ya(Ya)f = I, YaYa = naa , ya(Ya)f = I, (9.20) where I represents the unit operator. Making a choice for the 0a properties as presented in Eq. (9.10), it follows for the Clifford objects {yT = (y0,y\ -Y2,Y3, -Y5,Y6,..., -Yd-1,Yd), {Y T = (-Y0, -Y1 ,Y2, -Y 3,Y5, -Y6,...,Yd-1, -Yd), (9.21) All three choices for the linear vector space — spanned over either the coordinate Grassmann space, or over the vector space of Ya, as well as over the vector space of Ya — have the dimension 2d. We can express Grassmann coordinates 0a and momenta p0a in terms of Ya and Y a as well4 ea = 1 (Ya - iYa), 7\ 1 ^ = 1 (Ya + iYa). (9.22) 3ea 2 It then follows ea11 >= nab|1 >. Correspondingly we can use either Ya or Ya instead of ea to span the vector space. In this case we change the vacuum from the one with the property afal^o9 >= 0 to |^oc > with the property [2,7,9] <^oclYa|^oc > = 0, Ya|^oc >= iYa|^oc >, YaYb|^oc >=-iYbYa|^oc >, YaYb| 4oc > | a=b =-YaYb| 4oc >, YaYb| ^oc > | a=b = n^fooc > . (9.23) 4 In Ref. [28] the author suggested in Eq. (47) a choice of superposition of ya and Ya, which resembles the choice of one of the authors (N.S.M.B.) in Ref. [2] and both authors in Ref. [16,17] and in present article. 182 N.S. Mankoc Borstnik and H.B.F. Nielsen This is in agreement with the requirement Ya F(y)|^oc >:= ( ao Ya + aa, Ya Yai + aa,a2 Ya Yai Ya2 + ••• + aa,...ad Ya Yai ••• Yad ) I^OC >, Ya F(y) I^oc >:= ( i aoYa - i aa, Yai Ya + i aa, a2 Yai Ya2 Ya + • • • + i (-1)d aa, -ad Yai ••• Yad Ya )I^oc > . (9.24) We find the infinitesimal generators of the Lorentz transformations in Clifford space Sab = ^(YaYb - YbYa), Sabt = naanbbsab, Sab = 4(YaYb - YbYa), Sabt = naanbbSab, (9.25) with the commutation relations for either Sab or Sab of Eq. (9.14), if Sab is replaced by either Sab or Sab, respectively, while Sab = Sab + Sab , {Sab,S cd}_ = 0. (9.26) The basic states in Clifford space can be arranged in representations, in which any state is the eigenstate of the Cartan subalgebra operators of Eq. (9.84). The state, for example, in d-dimensional space with the eigenvalues of either S03, S12, S56,..., Sd-1 d or S03, S12, S56,..., Sd-1 d equal to 2 (i, 1,1,..., 1 ) is (y0-"3)(y1 + i"2)(Y5 + iY6) • • • (Yd-1 + iYd), where the states are expressed in terms of Ya. The states of one representation follow from the starting state by the application of Sab, which do not belong to the Cartan subalgebra operators, while S ab, which operate on family quantum numbers, cause jumps from the starting family to the new one. 9.2.1 Norms of vectors in Grassmann and Clifford space Let us look for the norm of vectors in Grassmann space d B = aa,a2...ak 9a' 9a2 . . . 9ak I^og > k=0 and in Clifford space F = Y_ aa, a2...ak Yal Y^ . . . Y^oc >, d aa k=0 where |^og > and |^oc > are the vacuum states in the Grassmann and Clifford case, respectively. In what follows we refer to Ref. [2]. a. Norms of the Grassmann vectors Let us define the integral over the Grassmann space [2] of two functions of the Grassmann coordinates < B|C >, < B|9 >=< 9|B >*, by requiring {d9a,eb}+ = 0, dde e0e1•••ed = 1, dea = o, deaea = 1, dde = ded...de0, ^ = + ek), (9.27) with d flc aeQ ec = nac. We shall use the weight function w = njL0(-d^ + ek) to dd 1xK,...bk Cb,...bk k=0 J (9.28) define the scalar product < B|C > r d < b|c > = dd-1xddea w< B|e > < e|c >= £ where, according to Eq. (9.8), follows: d < B|e >=< ^og| £ a*a, ...appeap napap • • • peai na] ai. p = 0 The vacuum state is chosen to be |^og >= |1 >, as taken in Eq. (9.4). The norm < B|B > is correspondingly always nonnegative. b. Norms of the Clifford vectors Let us look for the norm of vectors, expressed with the Clifford objects F = Yt aQ] a2...ak Ya] Ya2 ... Yak|^oc >, where |^o9 > and |^oc > are the two vacuum states when the Grassmann and the Clifford objects are concerned, respectively. By taking into account Eq. (9.20) it follows that (Yai Ya2 ... Yak= yaknakak ... Ya2-qa2a2Ya]na] ai , (9.29) since Ya Ya = naa. We can use Eqs. (9.27, 9.28) to evaluate the scalar product of two Clifford algebra objects < y|F >=< (ea-ipea)|F > and equivalently for < (ea-ipea)|G >. These expressions follow from Eqs. (9.17, 9.18, 9.20)). We must then choose for the vacuum state the one from the Grassmann case — |^oc >= |^og >= |1 >. It follows < F|G > = dd-1xddea w < F|y > < y|G >= y_ k=0 ' dd 1 x aa] ...ak bb]...bk . (9.30) {Similarly we obtain, if we express F = Y k=0 aa] a2...ak Ya] Ya2 . ..Yak |^oc > -k=0 bb] b2...bk and G = Yd=0 bb]b2...bk Yb] Yb2 .. .Ybk|^oc > and take |^oc >= |^o9 >= |1 >, the scalar product < F|G > = dd-1xddea w < F|y > < Y|G >= Y_ k=0 ' d xaa,...ak ab,...bk .} (9.31) Correspondingly we can write dd0a w(aaia2...ak Yai y"2 ...Yak )t(aaia2...ak Ya' y"2 ...Yak ) = ao1a2...al aai a2...ak . (9.32) The norm of each scalar term in the sum of F is nonnegative. c. We have learned that in both spaces — Grassmann and Clifford — norms of basic states can be defined so that the states, which are eigenvectors of the Cartan subalgebra, are orthogonal and normalized using the same integral. Studying the second quantization procedure in Subsect. 9.2.3 we learn that not all 2d states can be represented as creation and annihilation operators, either in the Grassmann or in the Clifford case, since they must — in both cases — fulfill the requirements for the second quantized operators, either for states with integer spins in Grassmann space or for states with half integer spin in Clifford space. 9.2.2 Actions in Grassmann and Clifford space Let us construct an action for free massless particles in which the internal degrees of freedom will be described: i. by states in Grassmann space, ii. by states in Clifford space. In the first case the internal degrees of freedom manifest the integer spin, in the second case the internal degrees of freedom manifest the half integer spin. While the action in Clifford space is well known since long [22], the action in Grassmann space must be found. We shall represent it here. In both cases we look for actions for free massless states in ((d — 1) + 1) space 5. States in Grassmann space as well as states in Clifford space will be organized to be — within each of the two spaces — orthogonal and normalized with respect to Eq. (9.27). We choose the states in each of two spaces to be the eigenstates of the Cartan subalgebra — with respect to Sab in Grassmann space and with respect to Sab and Sab in Clifford space, Eq. (9.84). In both spaces the requirement that states are obtained by the application of creation operators on the vacuum states — bf obeying the commutation relations of Eq. (9.48) on the vacuum state |^og >= |1 > in Grassmann space, and 6f obeying the commutation relation of Eq. (9.60) on the vacuum states |^oc >, Eq. (9.67), in Clifford space — reduces the number of states, in Clifford space more than in Grassmann space. But while in Clifford space all physically applicable states are reachable by either Sab (defining family members quantum numbers) 5 In (3 + 1) space the mass is due to the interaction of particles with the scalar fields, with which the particles interact in ((d — 1) + 1) space. or by Sab (defining family quantum numbers), the states in Grassmann space, belonging to different representations with respect to the Lorentz generators, seem not to be connected. a. Action in Clifford space In Clifford space the action for a free massless object must be Lorentz invariant A = ddx 1 (^y° YaPa^)+ h.c., (9.33) pa = i ad^, leading to the equations of motion YaPal^a > = 0, (9.34) which fulfill also the Klein-Gordon equation YaPaYbPblC > = papaic >= 0, (9.35) for each of the basic states >. Correspondingly y° appears in the action since we pay attention that Sabt y0 = y0 Sab , SV = y°s-1 , S = e-2-ab(s-+L-). (9.36) We choose the basic states to be the eigenstates of all the members of the Cartan subalgebra, Eq. (9.84). Correspondingly all the states, belonging to different values of the Cartan subalgebra — they differ at least in one value of either the set of Sab or the set of Sab, Eq. (9.84) — are orthogonal with respect to the scalar product defined as the integral over the Grassmann coordinates, Eq. (9.27), for a chosen vacuum state. Correspondingly the states generated by the creation operators, Eq. (9.65), on the vacuum state, Eq. (9.67), are orthogonal as well (both last equations will appear later). b. Action in Grassmann space We define here the action in Grassmann space, for which we require — similarly as in the Clifford case — that the action for a free massless object 9 , 1 A = 2 { ddx dde w (^(1 - 29°—°) - (9apa + naaeatPaM, (9.37) is Lorentz invariant. We use the integral also over 9a coordinates, with the weight function w from Eq. (9.27). Requiring the Lorentz invariance we add after the operator yG (Yg = (1 - 29agfa )), which takes care of the Lorentz invariance. Namely Sabt (1 - 2e° de° ) = (1 - 2e° de° ) Sab, St (1 - 29° ¿0) = (1 - 29° ¿0) S-1 , S = e-tœ"b(L +S ), (9.38) while ea, and pa transform as Lorentz vectors. The equation of motion follow from the action, Eq. (9.37), 1 7\ 7\ j[(i -2e0^) ea + ((i -2e0^) ea)t] pai^f > = 0, (9.39) as well as the Klein-Gordon equation {(i - 2e0w)eapalt ebpbl^f > = papal^e >= 0, (9.40) for each of the basic states >. c. We learned: In both spaces — in Clifford and in Grassmann space — there exists the action, which leads to the equations of motion and to the corresponding Klein-Gordon equation for free massless particles. In both cases we use the operator, which does not change the Clifford or Grassmann character of states. We shall see that, if one identifies the creation operators in both spaces with the products of odd numbers of either ea (in the Grassmann case) or ya (in the Clifford case) and the annihilation operators with their Hermitian conjugate operators, the creation and annihilation operators fulfill the anticommutation relations, required for fermions. The internal parts of states are then defined by the application of the creation operators on the vacuum state. But while the Clifford algebra defines spinors with the half integer eigenvalues of the Cartan subalgebra operators of the Lorentz algebra, the Grassmann algebra defines states with the integer eigenvalues of the Cartan subalgebra. 9.2.3 Second quantization of Grassmann vectors and Clifford vectors States in Grassmann space as well as states in Clifford space are organized to be — within each of the two spaces — orthogonal and normalized with respect to Eq. (9.27). All the states in each of spaces are chosen to be eigenstates of the Cartan subalgebra — with respect to Sab in Grassmann space, and with respect to Sab and Sab in Clifford space, Eq. (9.84). In both spaces the requirement that states are obtained by the application of creation operators on vacuum states — bf obeying the commutation relations of Eqs. (9.42, 9.48) on the vacuum state |^og >= |1 > for Grassmann space, and ba obeying the commutation relation of Eq. (9.60) on the vacuum states |^oc >, Eq. (9.67), for Clifford space — reduces the number of states arranged into the representations of the Lorentz group. The reduction of degrees of freedom depends on whether d = 2(2n+1) or d = 4n, n is a positive integer. The second quantization procedure with creation operators expressed by the product of Grassmann or Clifford objects requires that the product has an odd number of objects. We shall pay attention in this paper almost only to spaces with d = 2(2n +1)6. 6 The main reason that we treat here mostly d = 2(2n + 1) spaces is that one Weyl representation, expressed by the product of the Clifford algebra objects, manifests in d = (1 + 3) all the observed properties of quarks and leptons, if d > 2(2n + 1), n = 3. We define in Grassmann space creation operators by an odd number of factors of superposition of 9 a's and annihilation operators by Hermitian conjugation of the corresponding creation operators. In Clifford space we define creation operators by an odd number of factors of superposition of Ya's and the annihilation operators by Hermitian conjugate creation operators. Each basic state is a product of factors chosen to be eigenstates of the Cartan subalgebra of the Lorentz algebra. But while in Clifford space all physically applicable states are reachable either by Sab or by S ab, the states, belonging to different groups with respect to the Lorentz generators, in Grassmann space two different representations of the Lorentz group are not connected by the Lorentz operators. Let us construct creation and annihilation operators for the cases that we use a. Grassmann vector space, b. Clifford vector space. We shall see that from 2d states in either of these two spaces there are reduced number of states generated by the creation operators, which fulfill the requirements for the creation and their Hermitian conjugate annihilation operators. a. Quantization in Grassmann space There are 2d states in Grassmann space, orthogonal to each other with respect to Eq. (9.27). To any coordinate there exists the conjugate momentum. We pay attention in what follows mostly to spaces with d = 2(2n +1), although also spaces with d = 4n will be treated. In d = 2(2n +1) spaces there are drr states, Eq. (9.51), 2 ' 2 ' divided into two separated groups of states, all states of one group reachable from a starting state by Sab. These states are Grassmann odd products of eigenstates of the Cartan subalgebra. We use these products to define the creation operators and their Hermitian conjugate operators as the annihilation operators, fulfilling requirements of Eq. (9.41, 9.42). Let us see how it goes. If bft is a creation operator, which creates a state in the Grassmann space when operating on a vacuum state |^og > and bf = (bft)t is the corresponding annihilation operator, then for a set of creation operators bft and the corresponding annihilation operators bf it must be bfi^og > = 0, bf^og > = 0. (9.41) We first pay attention on only the internal degrees of freedom — the spin. Choosing ba = gja it follows ba = 9a, Afl 9 a 99a ' |bea,bbt} + i^og > = ^abl^og > , {b0a,bb}+i^og > = 0, {69at,bbt}+^og > = 0, bfa0 l ^og > = 9a| ^og >, l ^og > = 0. (9.42) The vacuum state |^og > is in this case |1 >. The identity I (I1" = I) can not be taken as a creation operator, since its annihilation partner does not fulfill Eq. (9.41). We can use the products of superposition of ea's as creation and products of superposition of 's as annihilation operators provided that they fulfill the requirements for the creation and annihilation operators, Eq. (9.48), with the vacuum state |^og >= |1 >. In general they would not. Only an odd number of ea in any product would have the required anticommutation properties. It is convenient to take products of superposition of vectors ea and eb to construct creation operators so that each factor is the eigenstate of one of the Cartan subalgebra member of the Lorentz algebra (9.84). We can start with the creation operators as products of d states 6ajb_ — —72(eai ± eebl). Then the corresponding annihilation operators have f factors of b^b = —2 ( aeat ± e* a eb ), e = i, if naiai — nbibi and e — —1, if naiai — nbibi. 1 In d — 2(2n + 1), n is a positive integer, we can start with the state 1 1 > — (—)d (e0-e3)(e7 + ie2)(e5 + ie6)••• (ed-1 + ied)|1 > . (9.43) 12 The rest of states, belonging to the same Lorentz representation, follows from the starting state by the application of the operators Scf, which do not belong to the Cartan subalgebra operators. Let us add that in d — 4n we should start with the state 1 1 > |4n — (—)d-1 (e0-e3)(e1 + ie2)(e5 + ie6)••• (ed-3 + ied-2)ed-1 ed| 1 > . (9.44) 2 Again the rest of states, belonging to the same Lorentz representation, follow from the starting state by the application of the operators Scf, which do not belong to the Cartan subalgebra operators. i. Taking into account Eqs. (9.8, 9.9, 9.43) one can propose the following starting creation operator and the corresponding annihilation operator 6t01t — (—=)d (e0-e3)(e1 + ie2)(e5 + ie6)••• (ed-1 + ied), v2 b?1 — (—)d( 9 -)...(J___) bt \[i) ( 3ed-1 L aed) ( ae0 ae3), for d — 2(2n + 1), 601t — (--=)d-1 (e0 - e3)(e1 + ie2)(e5 + ie6) ••• (ed-3 + ied-2 )ed-1ed, v2 b?1 — (—) d-1-?__( 9 - i 9 )...(J___^) bt ) aed aed-1 ( aed-3 L aed-2) ( ae0 ae3), for d — 4n. (9.45) The rest of the creation operators belonging to this group in either d — 2(2n +1) or in d — 4n follows by the application of all the operators Sef, which do not belong to the Cartan subalgebra operators. The corresponding annihilation operators follow by the Hermitian conjugation of a particular creation operator. One finds, for example for d = 2(2n + 1), 6®1f = (-=)d-1 (e0e3 + ie1e2)(e5 + ie6) ••• (ed-1 + ied), 1 ra0Q3 , ,-Q1Q2wa5 , rad-1 VT 1 9 3, ,3 3 .3 3 be1 = (_)d-1 (___- i——) b (v2) ( sed-1 L sed) (se3 se0 L se2 se1). (9.46) For d = 4n one finds equivalently 2 0 3 1 2 5 6 d-3 d-2 d-1 d b®11" = (—)d-2(e0e3 + ie1e2)(e5 + ie6)---(ed-3 + ied-2) ed-1 ec 1 3 3,3 .3,, 3 3 .3 3 be1 = (—) d-2—__r_(_r__i_r_)•••(—__— - i—__—) b (^/2) sed sed-1 ( sed-3 L sed-2) (se3 se0 L se2 se1). (9.47) It was taken into account in the above two equations that S01 transforms ()2 (e0 — e3)(e1 + ie2) into (e0e3 + ie1 e2) and that any Sac (a = c), which does not belong to Cartan subalgebra, Eq.(9.82), transforms (^)2(ea + ieb)(ec + ied) (a = c and a = d, b = c and b = d, naa = nbb) into ^(eaeb + eced). The states are normalized and the simplest phases are chosen. One finds that Sab(ea ± eeb) = ^i nr(ea ± eeb), e = 1 for naa = 1 and e = i for naa = —1, while either Sab or Scd, applied on (eaeb ± eeced), gives zero. Although all the states, generated by creation operators, which include one (I ± eeaeb) or several (I ± eeai ebl) • • • (I ± eeakeak), are orthogonal with respect to the scalar product, Eq.(9.28), their Hermitian conjugate values include I*, which, when applying on the vacuum state |^og >= |1 >, does not give zero. Correspondingly such creation operators do not have appropriate annihilation partners, which would fulfill Eqs. (9.41, 9.42). However, creation operators which are products of several e's, let say n with n = 2,4... d — 1 — always of an even number of e's, since Sab is a Grassmann even operator, eai • • • ean (factors eaeb can be "eigenstates" of the Cartan subalgebra operators provided that Sab belong to the Cartan subalgebra: Sabeaeb|1 >= 0) — can appear in the expression for a creation operator, provided that the rest of expression has an odd number of factors (d — n (with "eigenvalues" either (+1 or —1) or (+i or —i), as can be seen in the states of Eqs. (9.45, 9.46, 9.47)). Then such creation and annihilation operators fulfill the relations, we skip the index 1 in bf1 and in be1t {6?,j}+^og > = 6ij |^og >, {bf,bf}+|^og > = 0|^og >, {eiei,6®i}+|^og > = 0|^og >, b?t|*og > = l^j > b,el*og > = 0|^og >. (9.48) It is not difficult to see that states included into a representation, which started with bet as presented in Eq. (9.45) for d = (2n + 1)2 and 4n spaces, have the properties, required by Eq. (9.48): i.a. In any d-dimensional space the product -gear • • • 9^ak, with all different at (also if all or some of them are equal, since (g|a )2 = 0), if applied on the vacuum |1 >, is equal to zero. Correspondingly the second equation and the last equation of Eq. (9.48) are fulfilled. i.b. In any d space the product of different 0as — 0ai 0a2 • • • 0ak with all different 0a's (at = aj) for all at and aj — applied on the vacuum |1 > is different from zero. Since all the 0's, appearing in Eqs. (9.45,9.46,9.47) are different, forming normalized states, the fourth equation of Eq. (9.48) is fulfilled. i.c. The third equation of Eq. (9.48) is fulfilled provided that there is an odd number of 0 s in the expression for a creation operator. Then, when in the anticommutation relation different 0a's appear (like in the case of d = 6 {000305, 010206}+), Such a contribution gives zero. When two or several equal 0's appear in the anticommutation relation, the contribution is zero (since (0a)2 = 0). i.d. Also for the first equation in Eq. (9.48) it is not difficult to show that it is fulfilled only for a particular creation operator and its Hermitian conjugate: Let us show this for d = 1 +3 and the creation operator (0°—03) 0102 and its Hermitian conjugate (annihilation) operator: {^ -g|r (- -g|r), ^(0° - 03) 01 02}+. Applying (-g|o - -g|3) on (0° - 03) gives two, while -g|r -g|r applied on 0102 gives one. ii. There is additional group of creation and annihilation operators which follows from the starting state |^e2 > |2(2n+1) = (-L) 2 (0° + 03)(01 + i02)(05 + i06) • • • (0d-3 + i0d-2)(0d-1 + i0d) , 2 for d = 2(2n + 1), |^|2 > |4n = (—)d-1 (0° + 03)(01 + i02)(05 + i06) • • • (0d-3 + i0d-2) 0d-1 0d , 2 for d = 4n. (9.49) These two states can not be obtained from the previous group of states, presented in Eqs. (9.43,9.44) by the application of Sef, since each Sef changes an even number of factors, never an odd one. Correspondingly both starting states form a new group of states, the first in d = 2(2n + 1), the second in d = 4n. All the rest states of this new group of states in either d = 2(2n + 1) or in d = 4n follow from the starting one by the application of Sef. The corresponding creation and annihilation operators are b®2f = (—)2 (e0 + e3)(e1 + ie2)(e5 + ie6)••• (ed-1 + ied), v2 £02 , 1 -,-d 9 • 9 , d d boi =(—) 2 (-^^r —) •••(+ ^e3), for d = 2(2n + 1), 602t = (—) 2-1 (e0 + e3)(e1 + ie2 )(e5 + ie6) ••• (ed-3 + ied-2)ed-1 ed, 01 A/2 £02 1 4-1 d d d . 9 d 9 bo2 = (—) 2 aed aed-1 ( aed-3 - i aed-2) •••( deo + ae3), 01 \[i) 9ed 9ed-1 ( 9ed-3 9ed-2r"( 9e0 + 9e3- for d = 4n. (9.50) As in the first case all the rest of creation operators can be obtained from the starting one, in each of the two kinds of spaces, by the application of Sac, and the annihilation operators by the Hermitian conjugation of the creation operators. Also all these creation and annihilation operators fulfill the requirements for the creation and annihilation operators, presented in Eq. (9.48). One can choose as the starting creation operator of the second group of operators by changing sign instead of in the factor (e0 — e3) in the starting creation operator of the first group in any of the rest of factors in the product. In each case the same group will follow. Let us count the number of states with the odd Grassmann character in d = 2(2n + 1). There are in (d = 2) two creation ((e0 ^ e1, for nab = diag(1, —1)) and correspondingly two annihilation operators (-g0o T "dor), each belonging to its own group with respect to the Lorentz transformation operators, both fulfill Eq. (9.48). It is not difficult to see that the number of all creation operators of an odd Grassmann character in d = 2(2n + 1)-dimensional space is equal to d! d, d, • 2 ! 2 ! We namely ask: In how many ways can one put on j places d different ea's. And the answer is — the central binomial coefficient for xd 1 d — with all x different. This is just ¿drr. But we have counted all the states with an odd 2 ! 2 ! Grassmann character, while we know that these states belong to two different groups of representations with respect to the Lorentz group. Correspondingly one concludes:There are two groups of states in d = 2(2n + 1) with an odd Grassmann character, each of these two groups has 1 dd? (9.51) 2 2 ? 2 ? members. In d = 2 we have two groups with one state, which have an odd Grassmann character, in d = 6 we have two groups of 10 states, in d = 10 we have two groups of 126 states with an odd Grassmann characters. And so on. Correspondingly we have in d = 2(2n + 1)-dimensional spaces two groups of creation operators with i ¿ddr members each, creating states with an odd 2 '2 ' Grassmann character and the same number of annihilation operators. Creation and annihilation operators fulfill anticommutation relations presented in Eq. (9.48). The rest of creation operators [and the corresponding annihilation operators] have rather opposite Grassmann character than the ones studied so far — like e0e1 [afr afo] in d = (1 + 1) (e0 T e3)(e1 ± ie2) [(dfr T i¿fr)(afo T ¿fr], e0e3e1e2 [sW afr sW aa°] in d = (3 + 1). All the states |^f >, generated by the creation operators, Eq. (9.48), on the vacuum state |^og > (= |1 >) are the eigenstates of the Cartan subalgebra operators and are orthogonal and normalized with respect to the norm of Eq. (9.27) > = Sij. (9.52) If we now extend the creation and annihilation operators to the ordinary coordinate space, the relations among creation and annihilation operators at one time read {b10(x),bet(x')}+|^g > = Si S(X - X')|^og >, {bf (X), bf (X')}+|^og > = 0|^og >, {610t(x),eet(x')}+|^og > = 0|^og >, bf (X)|^og > = 0 |^og > |^og > = |1 > . (9.53) Again the index 1 or 2 in (Bf1, bft1) or in (bf2, bft2) is kept. b. Quantization in Clifford space In Grassmann space the requirement that products of eigenstates of the Cartan subalgebra operators represent the creation and annihilation operators, obeying the relations of Eq. (9.48), reduces the number of states from 2d (allowed in the first quantization procedure) to two isolated groups of i ¿ddr (There is no operator 2 2 '2 ' that determines the family quantum number and would connect both isolated groups of states.) Let us study what happens, when, let say, Ya's are used to create the basis and correspondingly also to create the creation and annihilation operators. Let us point out that Ya is expressible with ea and its derivative (Ya = (ea + afr~)), Eq. (9.17), and that we again require that creation (annihilation) operators create (annihilate) states, which are eigenstates of the Cartan subalgebra, Eq. (9.84). We could as well make a choice of Ya = i(ea - dp) instead of Ya's to create the basic states 7. We shall follow here to some extent Ref. [19]. 7 In the case that we would choose ya's instead of ya's, Eq.(9.17), the role of ya and ya should be then correspondingly exchanged in Eq. (9.92). 9 Why Nature Made a Choice of Clifford and not Grassmann Coordinates? 193 Making a choice of the Cartan subalgebra eigenstates of Sab, Eq. (9.84), ab 1 naa ab 1 i (k):= 2 (Ya + , [k]:= -(1 + -yV) , (9.54) ab where k2 = naanbb, recognizing that the Hermitian conjugate values of (k) and ab [k] are ab ^ ab ab ^ ab (k) = naa (-k), [k] =[k], (9.55) while the corresponding eigenvalues of Sab, Eq. (9.56), and Sab, Eq. (9.101), are ab 1 ab ab 1 ab Sab (k) = -k (k), Sab [k]= -k [k] ab k ab ab k ab Sab (k) = ^ (k), Sab [k]= -^ [k], (9.56) we find in d = 2(2n + 1) that from the starting state with products of odd number of only nilpotents 03 12 35 d-3 d-2 d-1 d M > l2(2n+1) = (+i) ( + ) (+) ■ ■ ■ ( + ) ( + ) l^oc > , (9.57) having correspondingly an odd Clifford character8, all the other states of the same Lorentz representation, there are 2d-1 members, follow by the application of Scd (which do not belong to the Cartan subalgebra) on the starting state 9, Eq. (9.84): SCd l^1 > l2(2n+1) = l^i > l2(2n+1). The operators Scd, which do not belong to the Cartan subalgebra of Eq. (9.84), generate states with different eigenstates of the Cartan subalgebra (S03, S12, S56, • • • , Sd-1 d), we call the eigenvalues of their eigenstates the "family" quantum numbers. There are 2d -1 families. From the starting new member with a different "family" quantum number the whole Lorentz representation with this "family" quantum number follows by the application of Sef: Sef Scd> l2(2n+1) = l^t > l2(2n+1). All the states of one Lorentz representation of any particular "family" quantum number have an odd Clifford character, since neither Scd nor Scd, both with an even Clifford character, can change this character. We are interested only in states with an odd Clifford character, in order that the corresponding creation operators defining these states when being applied on an appropriate vacuum state, and their annihilation operators, will fulfill anticommutation relations required for spinors with half integer spin. We shall discuss the number of states with an odd Clifford character after defining the creation and annihilation operators. 8 We call the starting state in d = 2(2n + 1) > l2(2n+1), and the starting state in d = 4n > l4n. 9 The smallest number of all the generators Sac, which do not belong to the Cartan subal- gebra, needed to create from the starting state all the other members, is 2 d -1 - 1. This is true for both even dimensional spaces - 2(2n + 1) and 4n. For d = 4n the starting state must be the product of one projector and 4n — 1 nilpotents applied on an appropriate vacuum state, since we again require that the corresponding creation and annihilation operators fulfill the anticommutation relations. Let us start with the state 03 12 35 d-3 d-2 d-1 d > |4n = (+i) ( + ) (+) • • • ( + ) [+] l^oc > , (9.58) All the other states belonging to the same Lorentz representation follow again by the application of Scd on this state > |4n, while a new family starts by the application of Scd|^1 > |4n and from this state all the other members with the same "family" quantum number can be generated by SefScd on > |4n: SefScd > |4n = > Un, All these states in either d = 2(2n + 1) space or d = 4n space are orthogonal with respect to Eq. (9.27). However, let us point out that (Ya)^ = Yanaa. Correspondingly it follows, ab ^ ab ab ^ ab Eq. (9.55), that (k) = naa (—k),and [k] =[k]. Since any projector is Hermitian conjugate to itself, while to any nilpotent ab (k) the Hermitian conjugated one has an opposite k, it is obvious that Hermitian conjugated product to a product of nilpotents and projectors can not be accepted as a new state 10. The vacuum state |^oc > ought to be chosen so that < ^oc|^oc >= 1, while 03 12 56 78 all the states belonging to the physically acceptable states, like [+i][+][—][—] d-3 d-2 d-1 d • • • (+) (+) |^oc > in d = 2(2n + 1), must not give zero for either d = 2(2n + 1) or for d = 4n. We also want that the states, obtained by the application of ether Scd or Scd or both, are orthogonal. To make a choice of the vacuum it is needed to know the relations of Eq. (9.88). It must be ab t ab < ^oc| • • • (k) • • • | • • • (k') • • • |^oc > = §kk' , ab^ ab < ^oc| • • • [k] • • • | • • • [k'] • • • |^oc > = §kk' , ab^ ab <^oc| ••• [k] ••• | ••• (k') ••• |^oc > = o. (9.59) Our experiences in the case, when states with the integer values of the Cartan subalgebra operators were expressed by Grassmann coordinates, teach us that the requirements, that creation and annihilation operators must fulfill, influence the choice of the number of states, as well as of the vacuum state. 03 12 35 d-3 d—2d-1 d 10 We could as well start with the state > |2(2n+1) = (—i)(—)(—) ••• (—) (—) i^oc > 03 12 35 d-3 d-2d-1 d for d = 2(2n + 1) and with > |4n =(—i)(—)(—) ••• (—) [—] |^oc > in the case of d = 4n. Then creation and annihilation operators will exchange their roles and also the vacuum state will be correspondingly changed. Let us first repeat therefore the requirements which the creation and annihilation operators must fulfill (br^n+iioc > = ^ Woe >, {br^+iioc >=oiioc >, {b«Tt,6Prt}+|ioc > = oiloc >, bnioc >=oiioc >, bfYtiloc > = i ir > , (9.60) paying attention at this stage only at the internal degrees of freedom of the states, that is on their spins. Here (a, (,...) represent the family quantum number determined by Sac and (i,j,...) the quantum number of one representation, determined by Sac and index y is to point out that these creation operators represent Clifford rather than Grassmann objects. In what follows we shall skip the index y, since either states or creation and annihilation operators carry two indexes, while in Grassmann case there is no family quantum number. From Eqs. (9.57, 9.58) is not difficult to extract the creation operator which, when applied on the vacuum state for either d = 2(2n + 1) or d = 4n, generates the starting state . i. One Weyl representation We define the creation b]t — and the corresponding annihilation operator b = (b] t)t — which when applied on the vacuum state i poc > create a vector of one of the two equations (9.57, 9.58), as follows 03 ] 2 56 d- ] d b ] t : = (+i)(+)(+) ••• (+) , d- ] d 56 ] 2 03 b ] : = (—) ••• (—)(—)(-i), for d = 2(2n + 1), o3 12 56 d-3 d-2 d-1 d b ] t : = (+i)(+)(+) ••• (+) [+] , t d- ] ,dd-2 d-3 56 1 2 03 b] : = [+] (-) ••• (-)H(-i), for d = 4n. (9.61) We shall call the | ^oc >, when operating on the vacuum state, the starting vector of the starting "family". Now we can make a choice of the vacuum state for this particular "family" taking into account Eq. (9.88) 03 12 56 d-1 d | 4oc > = [-i][-][-] ••• [-] I 0>, for d = 2(2n + 1), 03 12 56 d-3 d-2 d-1 d I^oc > = [-i][-][-] ••• [-] [+] |0>, for d = 4n, (9.62) n is a positive integer, so that the requirements of Eq. (9.60) are fulfilled. We see: The creation and annihilation operators of Eq. (9.61) (both are nilpotents, (6 1 *)2 = 0 and (6] )2 = 0), 6] * (generating the vector > when operating on the vacuum state) gives 6 ] *|"^oc >= 0, while the annihilation operator annihilates the vacuum state 6 ] >= 0, giving {6 ] , 6 ] *}+|^oc >= |^0c >, since we choose the appropriate normalization, Eq. (9.54). All the other creation and annihilation operators, belonging to the same Lorentz representation with the same family quantum number, follow from the starting ones by the application of particular Sac, which do not belong to the Cartan subalgebra (9.82). We call 6] * the one obtained from 6 ] * by the application of one of the four generators (S0 1, S02, S3 1, S32). This creation operator is for d = 2(2n + 1) equal to 03 1 2 35 d- 1 d 03 1 2 56 d- 1 d 6 1 * =[-i] [-](+) • • • (+) , while it is for d = 4n equal to 62* =[-i] H (+) • • • [+] • All the other family members follow from the starting one by the application of different Sef, or by the product of several Sgh. We accordingly have 61 * « Sab..Sef6 1 *, 61 = 61 *|^oc > will be normalized. We recognize that [19]: i.a. (61 *)2 = 0 and (61 )2 = 0, for all i. ab cd To see this one must recognize that Sac (or Sbc, Sad, Sbd) transforms (+) (+) to ab cd [-] [-], that is an even number of nilpotents (+) in the starting state is transformed into projectors [-] in the case of d = 2(2n + 1). For d = 4n, Sac (or Sbc, Sad, Sbd) ab cd ab cd transforms (+)[+] into [-](-). Therefore for either d = 2(2n + 1) or d = 4n at least one of factors, defining a particular creation operator, will be a nilpotent. For d = 2(2n + 1) there is an odd number of nilpotents, at least one, leading from the dg d-1 d starting factor (+) in the creator. For d = 4n a nilpotent factor can also be (-) d-1d d-1 d (since [+] can be transformed by Se d-1, for example into (-) ). A square of at least one nilpotent factor (we started with an odd number of nilpotents, and oddness can not be changed by Sab), is enough to guarantee that the square of the corresponding (61*)2 is zero. Since 61 = (61*)*, the proof is valid also for annihilation operators. i.b. 6|*|^oc >= 0 and 6||^oc >= 0, for all i. To see this in the case d = 2(2n +1) one must recognize that 61* distinguishes from 61 * in (an even number of) those nilpotents (+), which have been transformed ab 1 ab ab into [-]. When [-] from 6t* meets [-] from |^oc >, the product gives [-] back, d—1 d and correspondingly a nonzero contribution. For d = 4n also the factor [+] can d-1 d be transformed. It is transformed into (-) which, when applied to a vacuum d-1 d d-1 d d-1 d state, gives again a nonzero contribution ( (-) [+] = (-) , Eq. (9.88)). In the case of b1 we recognize that in b1 * at least one factor is nilpotent; that of the same type as in the starting bj — (+) — or in the case of d = 4n it can be d-1 d also (-) . Performing the Hermitian conjugation (bt* )*, (+) transforms into (-), d-1 d d-1 d d-1 dd-1 d while (-) transforms into (+) in c^. Since (-)[-] gives zero and (+) [+] also gives zero, b? |^oc >= 0. i.c. {b1 *, b1 *}+ = 0, for each pair (i,j). There are several possibilities to be discussed. A trivial one is, if both b1* and 61 * have a nilpotent factor (or more than one) for the same pair of indexes, say kl kl kl (+). Then the product of such two (+) (+) gives zero. It also happens, that bt* 03 kl mn has a nilpotent at the place (kl) ([-] • • • (+) • • • [-] • • •) while b * has a nilpotent 03 kl mn = Sijl^oc > . To prove this we must recognize that b1 = b1 Sef ..Sab and i^* = Sab..Sefb1. Since any b1 |^oc >= 0, we only have to treat the term b1 b1 *. We find b1 b1 * to one. When Sef • • • SabSlm • • • Spr are proportional to several 03 kl np products of Scd, these generators change b1* into (+) • • • [-] • • • [-] • • •, making 1 1 kl kl the product b? b * equal to zero, due to factors of the type (-)[-]. In the case of d—1 d d—1 d d = 4n also a factor [+] (-) might occur, which also gives zero. We saw and proved that for the definition of the creation and annihilation operators, Eq. (9.61), for states in Eqs. (9.57, 9.58) and further for all the rest of creation and annihilation operators, Eq. (9.63), and for the choice of the vacuum states, Eq. (9.62), all the requirements of Eq. (9.60) are fulfilled, provided that creation and correspondingly also the annihilation operators have an odd Clifford character, that is that the number of nilpotents in the product is odd. For an even number of factors of the nilpotent type in the starting state and accordingly in the starting b? *, an annihilation operator b? would appear with all factors of the type [-], which on the vacuum state (Eq.(9.62)) would not give zero. ii. Families of Weyl representations Let bf* be a creation operator, fulfilling Eq. (9.60), which creates one of the (2d/2—1) Weyl basic states of an a-th "family", when operating on a vacuum state |^oc > and let ba = (bf*)* be the corresponding annihilation operator. We shall now proceed to define bf* and bf from a chosen starting state (9.57,9.58), which b? * creates on the vacuum state |^oc >. When treating more than one Weyl representation, that is, more than one "family", we must take into account that: i. The vacuum state chosen to fulfill requirements for second quantization of the starting family might not and it will not be the correct one when all the families are taken into account. ii. The products of S ab, which do not belong to the Cartan subalgebra set of the generators S ab, when being applied on the starting family , generate the starting member ^f of each of the remaining families. There is correspondingly the same number of "families" as the number of vectors of one Weyl representation, namely 2d/2—1. Then the whole Weyl representation of a particular family ^ f follows again with the application of Sef, which do not belong to the Cartan subalgebra of Sab on this starting a family state. Any vector > follows from the starting vector, Eqs. (9.57, 9.58), by the application of either Sef, which change the family quantum number, or Sgh, which change the member of a particular family (as it can be seen from Eqs. (9.90,9.102)) or with the corresponding product of Sef and Sef > . (9.64) Correspondingly we define bf* (up to a constant) to be bf* < S ab • •• S efSmn •• • Spr b1* < Smn ••• Sprb1*Sab ••• Sef. (9.65) This last expression follows due to the property of the Clifford object ya and correspondingly of Sab, presented in Eqs. (9.92, 9.93). For bf = (6°'t)t we accordingly have = (bat)t = 6«^oc > will be normalized. We ought to generalize the vacuum state from Eq. (9.62) so that ba |^oc >= 0 and bf |^oc >= 0 for all the members i of any family a. Since any Seg changes ef gh ef gh ab ab ab ab ab (+) (+) into [+] [+] and [+] t =[+], while (+) t (+)=[-], the vacuum state |^oc > from Eq. (9.62) must be replaced by |^oc > = 03 12 56 d—1 d 03 12 56 d—1 d 03 12 56 d—1 d [-i][-][-] ••• [-] +[+i][+][-] ••• [-] +[+i][-][+] ••• [-] + -|0>, for d = 2(2n + 1), |^oc >= 03 12 35 d—3 d—2d-1 d 03 12 56 d—3 d—2 d-1 d [-i][-][-] ••• [-] [+] +[+i][+][-] ••• [-] [+] + -"|0>, for d = 4n, (9.67) n is a positive integer. There are 2d—1 summands, since we step by step replace all ab ef 03 12 35 d—1 d 03 12 35 possible pairs of [-] • • • [-] in the starting part [-i] [-] [-] • • • [-] (or [-i] [-] [-] d—3 d—2d—1 d ab ef • • • [-] [+] ) into [+] • • • [+] and include new terms into the vacuum state so that the last 2n + 1 summands have for d = 2(2n + 1) case, n is a positive integer, only one factor [-] and all the rest [+], each [-] at different position. For d = 4n d—1 d 03 12 35 d—3 d—2d—1 d also the factor [+] in the starting term [-i] [-] [-] • • • [-] [+] changes to d—1 d _ [-] . The vacuum state has then the normalization factor 1/\/2d/2—1. There is therefore 2 d—1 2 d—1 (9.68) number of creation operators, defining the orthonormalized states when applying on the vacuum state of Eqs. (9.67) and the same number of annihilation operators, which are defined by the creation operators on the vacuum state of Eqs. (9.67). S ab connect members of different families, Sab generates all the members of one family. We recognize that: ii.a. The above creation and annihilation operators are nilpotent — (b^)2 = 0 = (ba)2 — since the "starting" creation operator b1t and annihilation operator are both made of the product of an odd number of nilpotents, while products of either Sab or S ab can change an even number of nilpotents into projectors. Any is correspondingly a factor of an odd number of nilpotents (at least one) (and an even number of projectors) and its square is zero. The same is true for ii.b. All the creation operators operating on the vacuum state of Eq. (9.67) give a non zero vector — b^ |^oc >= 0 — while all the annihilation operators annihilate this vacuum state — ba >= 0 for any a and any i. It is not difficult to see that 6a |^oc >= 0, for any a and any i. First we recognize that whatever the set of factors Smn • • • Spr appear on the right hand side of the annihilation operator 6] in Eq. (9.66), it leaves at least one factor [-] ab ab unchanged. Since b] is the product of only nilpotents (-) and since (-)[-] = 0, this part of the proof is complete. Let us prove now that bTat |^oc >= 0 for any a and any i. According to Eq. (9.65) the operation Smn on the left hand side of b]t, with (m, n,..), which does not belong to the Cartan subalgebra set of indices, transforms the term 03 12 lm nk d-1 d 03 12 lm nk d-1 d [-i] [-] • • • [-] • • • [-] • • • [-] (or the term [-i] [-] • • • [-] • • • [-]...... [+] ) into 03 12 lm nk d-1 d 03 12 lm nk the term [-i][-] • • • (+) • • • (+) • • • [-] (or into the term [-i][-] • • • (+) • • • (+) d-1 d lm lm nk nk ...... [+] ) and 61 on such a term gives zero, since (+) (+)= 0 and (+) (+)= 0. Let us first assume that Smn is the only term on the right hand side of 61t and that none of the operators from the left hand side of bD11t in Eq. (9.65) has the indices m, n. It is only one term among all the summands in the vacuum state (Eq. (9.67)), which gives non zero contribution in this particular case, namely the 03 12 lm nk d-1 d 03 12 lm nk d-1 d term [-i] [-] • • • [+] • • • [+] • • • [-] (or the term [-i] [-] • • • [+] • • • [+]...... [+] ). lm nk lm nk lm lm Smn transforms the part • • • [+] • • • [+] • • • into • • • (-) • • • (-) • • • and since (+) (-) lm gives nu [+], while for the rest of factors it was already proven that such a factor on 61t forms a b1t giving non zero contribution on the vacuum, Eq. (9.62), the proof is complete. It is also proved that what ever other Sab but Smn operate on the left hand side of 61t the contribution of this particular part of the vacuum state is nonzero. If the operators on the left hand side have the indexes m or n or both, the contribution on this term of the vacuum will still be nonzero, since then such a Smp will transform lm 1 lm lm lm the factor (+) in 6/ into [-] and [-](-) is nonzero, Eq. (9.88). It was proven that operating on the vacuum |^oc > of Eq. (9.67) gives a nonzero contribution. The vacuum state has namely a term which guarantees a non zero contribution for any possible set of Smn • • • Spr operating from the right hand side of 61t (that is for each family) (what we achieved just by the transformation cd gh cd gh of all possible pairs of [-], [-] in the vacuum into [+], [+]). (When we speak about 03 d [-] also [-i] is understood.) It is not difficult to see that for each "family" of 2d-1 families it is only one term among all the summands in the vacuum state |^oc > of Eq. (9.67), which gives a nonzero contribution, since whenever [+] appears on a ab ab wrong position, that is on the position, so that the product of (+) from b11 and [+] from the vacuum summand "meet", the contribution is zero. ii.c. Any two creation operators anticommute: {6°^, 6?t}+ = 0. According to Eq. (9.65) we can rewrite {6tat, 6j?t}+, up to a factor, as {Smn • • • Spr61 tSab • • • Sef, Sm'n' • • • Sp 'r'61 tSa'b' • • • Se'f'}+. Whatever the product Sab • • • SefSm'n' • • • Sp'r' (or Sa b' • • • Se f'Smn • • • Spr) is, it always transforms an even number of (+) in t^t into [-]. Since an odd number of nilpotents (+) (at least one) remains unchanged in this right + after the application of all the Sab in the product in front of it, or d— 1 d d-1 d [+] transforms into (-) , and since the left t^ + is a product of only nilpotents (+) in d = 2(2n + 1), or an odd number of nilpotents and [+] for d = 4n, while d-1 dd-1 d [+] (-) = 0, the anticommutator for any two creation operators is zero. ii.d. Any two annihilation operators anticommute: {6a, 62}+ = 0. According to Eq. (9.66) we can rewrite {6a, 62}+, up to a factor, as {Sab • • • Sef61 Smn • • • Spr, Sa'b' • • • Se'f'61 Sm'n' • • • Sp'r'}+. Whatever the product Smn • • • SprSa'b' • • • Se'f' (or Sm'n' • • • Sp'r'Sab • • • Sef) is, it always transforms an even number of (-) in 61 into [+]. Since an odd number of nilpotents (-) (at least one) remains unchanged d-1 d in this 61 after the application of all the Sab in the product in front of it or [+] d-1 d is transformed into (-) , and since 61 on the left hand side is a product of only nilpotents (-) for d = 2(2n + 1) (or an odd number of nilpotents and [+] ab ab ab ab for d = 4n), while (-)(-)= 0 and [+][-]= 0, the anticommutator of any two annihilation operators is zero. ii.e. For any creation and any annihilation operator it follows: {6f, 62}+ |^oc >= 5a25tj |^oc >. Let us prove this. According to Eqs. (9.65, 9.66) we may rewrite {6f,6f+} + up to a factor as {Sab • • • Sef61 Smn • • • Spr, Sm 'n' • • • Sp 'r' 61+Sa'b' • • • Se'f'}+. We distinguish between two cases. It can be that both Smn • • • SprSm n • • • Sp r and Sa b • • • Se f Sab • • • Sef are numbers. This happens when a = ( and i = j. Then we follow i.b.. We normalize the states so that < >= 1. The second case is that at least one of products Smn • • • SprSm n • • • Sp r and ab ab ab ab Sa b • • • Se f Sab • • • Sef is not a number. Then the factors like (-) [-] or [+] (-) or ab ab (+) [+] make the anticommutator equal to zero. And the proof is completed. Let us extend the creation and annihilation operators to the ordinary coordinate space {6«(x),6f + (x')}+ |^oc > = §¡2si S(x- x')|^oc >, {6f (X),6f(x')}+ |^oc > = 0 l^oc >, {6ta+(x),6f+(x')}+ l^oc > = 0 l^oc >, 6f(X)|^oc > = 0 l^oc >, 6f+(x)|$oc > = lC(x) >, (9.69) with the vacuum state |^oc > defined in Eq. (9.67). c. Discrete symmetries in Grassmann space and in Clifford space in d and in d = (3 + 1) space Let Yp[¥p] be the creation operator creating a fermion in the state ¥p (which is a function of x) and let ¥p (x) be the second quantized field creating a fermion at position x either in the Grassmann or in the Clifford case. Then ¥p(x) ¥p(x)d(d-1)x, (9.70) describes on a vacuum state a single particle in the state ¥ {¥p[¥p] — [ ¥p(x) ¥p(x)d(d-1)x}|vac > so that the anti-particle state becomes {C¥p[¥pos] — ¥p(x)(C¥pos(x))d(d-1)x}|vac > . We distinguish in d-dimensional space two kinds of dicsrete operators C, P and T operators with respect to the internal space which we use. In the Clifford case we have [21] Ch — n Ya K, yaea TH — Y0 n Ya KIxc , ya£K -p(d-1) — Y0 T ' H — Y Ix , Ixxa —-xa , Ix0xa — (-x0,x), Ixx —-x, Ix3xa — (x0,-x1,-x2,-x3,x5,x6,...,xd). (9.71) The product Ya is meant in the ascending order in Ya. In the Grassmann case we correspondingly define Cg — n yGk, ygeaTa TG — yG n YG KIxC, yg £Mya PGd-1) — YG Ix , (9.72) yG is defined in Eq. (9.11) as yG — (1 - 2eana^^^-), (9.73) 3ea while Ixxa — —xa, Ixcxa — (-x0,x), Ixx — -x, Ix3xa — (x0,-x1,-x2,-x3,x5,x6,...,xd). Let be noticed, that since yG (— -inaa Yaa) is always real as there is YaiYa, while Ya is either real or imaginary, we use in Eq. (9.72) Ya to make a choice of appropriate yG. In what follows we shall use the notation as in Eq. (9.72). Let us define in the Clifford case and in the Grassmann case the operator "emptying" [7,9] (arxiv:1312.1541) the Dirac sea, so that operation of "emptyingN" after the charge conjugation CH in the Clifford case and "emptying G" after the charge conjugation CG in the Grassmann case (both transform the state put on the top of either the Clifford or the Grassmann Dirac sea into the corresponding negative energy state) creates the anti-particle state to the starting particle state, both put on the top of the Dirac sea and both solving the Weyl equation, either in the Clifford case, Eq. (9.34), or in the Grassmann case, Eq. (9.39), for free massless fermions "emptyingN" = ya K in Clifford space, "emptyingG" = yG K in Grassmann space, (9.74) although we must keep in mind that indeed the anti-particle state is a hole in the Dirac sea from the Fock space point of view. The operator "emptying" is bringing the single particle operator Ch in the Clifford case and CG in the Grassmann case into the operator on the Fock space in each of the two cases. Then the anti-particle state creation operator — [Yp] — to the corresponding particle state creation operator — can be obtained also as follows YjjYp] |vac > = Ch |vac >= Y^X) (Ch V*)) d(d-1)x |vac > , Ch = "emptyingN" • Ch (9.75) in both cases. The operators CH and CG Ch = "emptyingN" • Ch , Cg = "emptyingNG" • Cg , (9.76) operating on Yp (X) transforms the positive energy spinor state (which solves the corresponding Weyl equation for a massless free fermion) put on the top of the Dirac sea into the positive energy anti-fermion state, which again solves the corresponding Weyl equation for a massless free anti-fermion put on the top of the Dirac sea. Let us point out that either the operator "emptyingN " or the operator "emptyingNG" transforms the single particle operator either CH or CG into the operator operating in the Fock space. We use the Grassmann even, Hermitian and real operators yG, Eq. (9.11), to define discrete symmetry in Grassmann space, first in ((d + 1) — 1) space and then in (3 + 1) space, as we did in [21] in the Clifford case. In the Grassmann case we do this in analogy with the operators in the Clifford case [21] CNG = Yg KIx6x8...xd , tng=Yg n Kix°1 Ax° ix5x7 ...xd ' G v d P(dG-1) = tGÜ YsGIX , > (d 1) _„,0 s=5 CNG = YG >Ix6x8...xd yG enys d > (d 1 ) _ 0 cngpNg 1 = Yg n yG Ixs Ix6x8...xd ) yG e3ys,s=5 CngTngPNG1' = n YG IxK. (9.77) rsGe3ya Let us try to understand the Grassmann fermions in the case d = 5 + 1, before the break, as well as after the break of d = 5 + 1 into d = 3 + 1, when the fifth and the sixth dimension determine the charge in d = 3 + 1. There are two decuplets in this case [15], both of an odd Grassmann character, which can be second quantized. The two triplets in the first decuplet— ^3) and (^4, ^5, ^6) — both solving the Eq. (9.39) for massless free fermions in Grassmann space with the space function e-ipa*a. The Grassmann even opoerator operator CngPiNg-1 ' transforms with pa = (|p0|, 0,0, |p3|, 0,0) into the antiparticle state ^6, with the positive energy |p01 and with — |p31, for example. Correspondingly transforms CngpNu-1 ' the particle state ^3 with the positive energy and into the antiparticle state ^4 with the positive energy, and the particle ^3 into the positive energy antiparticle state ^4. All belong to the same representation. Applying the Grassmann even operators on one of the states of one the decuplets — Cg (= yGyG, Eq. (9.72)), Cng^G ' (= yGyGyGyG ^ I*s K, Eq. (9.72)) — one remains within the same decuplet. To get the positive energy antiparticle states the operator emptingN in (d — 1) + 1 and emptingNG in d = (3 + 1) are needed, Eqs. (9.74, 9.76). The reader can find more discussions in Refs. [15,21]. d. What do we learn in the second quantization procedure in Grassmann and in Clifford space We proved that basic states in both spaces can be written by creation operators operating on an appropriate vacuum state. The creation and annihilation operators fulfill in both spaces anticommutation relations as required for fermions, Eqs (9.48, 9.60). In both spaces the creation operators are chosen to create states that are eigenstates of the corresponding Cartan subalgebra of the Lorentz algebra, the generators of which are Sab, Eq. (9.13), for the Grassmann case and (Sab, Sab), first generating spins and the second families, Eq. (9.25), for the Clifford case. I decuplet S°3 S12 S56 1 (e° - e3)(e' + ie2)(e5 + ie6) i 1 1 2 (e°e3 + ie1 e2)(e5 + ie6) 0 0 1 3 (e° + e3)(e1 - ie2)(e5 + ie6) -i -1 1 4 (e° - e3)(e1 - ie2)(e5 - ie6) i -1 5 (e°e3 - ie1 e2)(e5 - ie6) 0 0 6 (e° + e3)(e1 + ie2)(e5 - ie6) -i 1 7 (e° - e3)(e1e2 + e5e6) i 0 0 8 (e° + e3)(e'e2 - e5e6) -i 0 0 9 (e°e3 + ie5 e6)(e1 + ie2) 0 1 0 10 (e°e3 - ie5e6)(e1 - ie2) 0 -1 0 II decuplet S°3 S12 S56 1 (e° + e3)(e1 + ie2)(e5 + ie6) -i 1 1 2 (e°e3 - ie1 e2)(e5 + ie6) 0 0 1 3 (e° - e3)(e1 - ie2)(e5 + ie6) i -1 1 4 (e° + e3)(e1 - ie2)(e5 - ie6) -i -1 5 (e°e3 + ie1 e2)(e5 - ie6) 0 0 6 (e° - e3)(e1 + ie2)(e5 - ie6) i 1 7 (e° + e3)(e1e2 + e5e6) -i 0 0 8 (e° - e3)(e'e2 - e5e6) i 0 0 9 (e°e3 - ie5e6)(e1 + ie2) 0 1 0 10 (e°e3 + ie5e6)(e1 - ie2) 0 -1 0 Table 9.1. The creation operators of the decuplet and the antidecouplet of the orthogonal group SO(5,1 ) in Grassmann space are presented. Applying on the vacuum state >= |1 > the creation operators form eigenstates of the Cartan subalgebra, Eq. (9.84), (S03, S12, S56). The states within each decouplet are reachable from any member by Sab. The product of the discrete operators Cng (= n^y* Yg I*6*8...*d) P^V (= Yg 0^=5 YgI*s) transforms, for example, ^ï into ^6, ^2 into ^5 and ^3 into ^4. Solutions of the Weyl equation, Eq. (9.39), with the negative energies belong to the "Grassmann sea", with the positive energy to the particles and antiparticles. While in the Grassmann case the vacuum state is simple, >= |1 >, in the Clifford case the vacuum state is a sum of products of 22-1 projectors, Eq. (9.67). In 2(2n+1 )-dimensional spaces there are in the Clifford case 22-1 states in one representation reachable from (any) starting state by S ab, while S ab transform each of these states changing its family quantum number. There are correspondingly 2 2-1 x 2 2-1 states reachable with either Sab or Sab. Each state is obtained by the corresponding creation operator on the vacuum state and is annihilated by its Hermitian conjugate operator. In 2(2n + 1 )-dimensional spaces there are in the Grassmann case two decoupled groups with 1 ^jrr states in each representation. Each of states can be 2 ■ 2 ■ obtained by the corresponding creation operator and is annihilated by its Her-mitian conjugated operator. While all of 22-1 x 22-1 states in Clifford space are reachable by even Clifford objects, either Sab or Sab, in Grassmann space the two groups of representations can not be reached by an even number of Grassmann objects. 9.3 Conclusions We have learned in the present study that one can use either Grassmann or Clifford space to express the internal degrees of freedom of fermions in any even dimensional space, either for d = 2(2n + 1) or d = 4n. In both spaces the creation operators and their Hermitian conjugated annihilation operators fulfill the anticommutation relation requirements, needed for fermions, provided that they are expressed as odd products of either Grassmann (9a, (9a)t = dpHaa, Eq. (9.8)) or Clifford objects (either Ya = (0a + dr~), Eq. (9.17) and correspondingly Yat = Yanaa, or Ya = i(ea — dp), Eq. (9.18), and correspondingly Y= Yanaa). But while in the Clifford case states appear in the fundamental representations of the Lorentz group, carrying half integer spins, the states in the Grassmann case are in adjoint representations of the Lorentz group. The Clifford case, offering two kinds of the Clifford objects (y[ and Ya), enables to describe besides the spin degrees of freedom of fermion fields also their family degrees of freedom. The Grassmann case offers only one kind of objects. Assuming that "nature has both choices" for describing the internal degrees of freedom of fermion fields, the question arises why Grassmann choice is not chosen, or better, why the Clifford choice is chosen. In the case that spin degrees in d > 5 manifest as charges in d = (3 + 1), fermions in the Grassmann case manifest charges in the adjoint representations. On the other hand in the Clifford case — this is used in the spin-charge-family theory, which takes the Lorentz group SO(13,1) — the spin and charges appear in the fundamental representations of the corresponding groups, offering also the family degrees of freedom. We present in this paper the action describing free massless particles with the internal degrees of freedom describable in Grassmann space, Eqs. (9.37, 9.38). The action leads to the equation of motion analogous to the Weyl equation in Clifford space, fulfilling the Klein-Gordon equation. Since the Clifford objects Ya and Ya are expressible with the Grassmann coordinates 0a and their conjugate moments gf^, either basic states in Grassmann space, Eq. (9.4), or basic states in Clifford space, Eq. (9.15), can be normalized with the same integral, Eq. (9.27, 9.28, 9.30). To understand better the difference in the description of the fermion internal degrees of freedom with either Clifford or Grassmann space, let us replace in the starting action of the spin-charge-family theory, Eq. (9.1), using the Clifford algebra to describe fermion degrees of freedom, the covariant momentum p0a = faa POa, P0a = Pa - 2Sabdaba - 1Sabdaba, with poa = Pa - ^ SabHaba, where Sab _ Sab + Sab, Eq. (9.26), and Haba are the spin connection gauge fields of Sab (which are the generators of the Lorentz transformations in Grassmann space!), while faa p0a replaces the ordinary momentum when massless objects start to interact with the gravitational field through the vielbeins and the spin connections. Let us add that varying the action with respect to either daba or d>aba when no fermions are present, one learns that both spin connections are uniquely determined by the vielbeins ([9,3,5] and references therein) and correspondingly in this particular case ^aba — aba . Let us use instead of pa in the action for free massless fields using Grassmann space to describe the internal degrees of freedom, Eq. (9.37), the above covariant momentum p0a — faa (pa — 1 SabHaba). One finds in this case that the representations of the Lorentz group in d — 2(2n + 1) — 13 + 1 and their subgroups SO(7,1), SU(3) and U(1) are all in the adjoint representations of the groups. The spin-charge-family theory (using Clifford objects) offers the explanation for all the assumptions of the standard model of elementary fields, fermions and bosons, vector and scalar gauge fields, with the appearance of families included, explaining also the phenomena like the existence of the dark matter [10], of the matter-antimatter asymmetry [4], offering correspondingly the next step beyond both standard models — cosmological one and the one of the elementary fields. We do notice, however, that the Grassmann degrees of freedom do not offer the appearance of families at all. We also notice that the second quantization procedure allows in d — 2(2n +1)-dimensional space for each member of a Weyl representation in Clifford space (for each of 2d-1 "family member") 2d-1 "families", all together therefore 2d-1 x 2d-1 basic states which can be second quantized, according to this paper. From 2d Clifford objects, only those of an odd Clifford character contribute to the second quantization — half of them as creation and half of them as annihilation operators, 2d -1 projectors from the rest of objects form the vacuum state. We notice that in case of Grassmann space and d — 2(2n + 1) only twice two isolated groups of 2 ¿dd, states of an odd Grassmann character can be second 2 d i d i quantized. To come to the low energy regime the symmetry must break, first from SO(13,1) to SO(7,1) x SU(3) x U(1) and then further to SO(3,1) xSU(3) x U(1), in both spaces, in Grassmann and in Clifford. In Clifford case there are two kinds of generators and correspondingly two kinds of symmetries. We learned in Refs. [2325] that when breaking symmetries only some of families stay massless and correspondingly observable in d = (3 + 1). This study is indeed to learn more about possibilities that "nature has". One of the authors (N.S.M.B.) wants to learn: a. Why is the simple starting action of the spin-charge-family theory doing so well in manifesting the observed properties of the fermion and boson fields? b. Under which condition can more general action lead to the starting action of Eq. (9.1)? c. What would more general action, if leading to the same low energy physics, mean for the history of our Universe? d. Could the fermionization procedure of boson fields or the bosonization procedure of fermion fields, discussed in Ref. [12] for any even dimension d (by the authors of this contribution, while one of them (H.B.F.N. [13]) has succeeded with another author to do the fermionization for d = (1 + 1)) tell more about the "decisions" of the universe in the history? Although we have not yet learned enough to be able to answer these questions, yet we have learned at least that the description of the fermion internal degrees of freedom in Grassmann space would not offer families, and would not be in agreement with the spin and charges and other observations so far. We also learned that if there are no fermion present only one kind of dynamical fields manifests, since either Daba or D aba are uniquely expressed by vielbeins ([9] Eq. (C9) and references therein), which could mean that the appearance of the two kinds of the spin connection fields might be due to the break of symmetries. 9.4 Appenix: Lorentz algebra and representations in Grassmann and Clifford space The Lorentz transformations of vector components 0a, Ya, or Ya, which all could be used to describe internal degrees of freedom of fields with the anticommutation relations of fermions, and of vector components xa, which are real (ordinary) commuting coordinates: 9'a = Aab eb, Y'a = Aab Yb, Y'a = Aab Yb and xa = Aab xb, leave forms Qa,a2...al 9ai 9a2 . . . 9ai, aa,a2...ai Yai Ya2 . . . Y^, aa,a2...ai Y^ Ya2 ...Yai and haia2...ai xai xa2 ...xai, i = (1,...,d), invariant. While 6a1a2...ai (= naibina2b2 . ..naib 6blb2".bi) is a symmetric tensor field, aai a2...ai (= naiblna2b2 .. .naibi ablb2...bi) are antisymmetric tensor Kalb-Ramond fields. The requirements: x a x bnab = xc xdncd, 9 'a9 'b£ab = 9c9d£cd, y'aY/b£ab = YcYd£cd and Y/aY/b£ab = YcYd£cd lead to A% Acdnac = nbd. Herenab (in our case nab = diag(1, -1, -1,..., -1)) is the metric tensor lowering the indexes of vectors ({xa} = nabxb, {9a} = nab 9b, {Ya} = nab Yb and {Ya} = nab Yb) and £ab is the antisymmetric tensor. An infinitesimal Lorentz transformation for the case with detA = 1, A00 > 0 can be written as Aab = 5£ + wab, where wab + wb a = 0. According to Eqs. (9.17,9.18, 9.25) one finds, Eq. (9.3), {Ya,Scd}- = 0 = {Ya,Scd}- , {Ya, Scd}- = {Ya,Scd}- = i (nacYd - nadYc), {Y a, Scd}- = {y a,Scd}- = i (nacY d - nadYc). (9.78) Comments: In cases with either the basis 9a or with the basis of Ya or Ya the scalar products — the norms < B|B > and < F|F > (where < 9|B >, Eq. (9.4), and < y|F >, Eq. (9.15), are vectors in Grassmann and Clifford space, respectively) — are non negative and equal to Y.d=0 J" dd-1x6b1 ...bkhb1 ...bk. 9.4.1 Lorentz properties of basic vectors What follows is taken from Ref. [2] and Ref. [9], Appendix B. Let us first repeat some properties of the anticommuting Grassmann coordinates. An infinitesimal Lorentz transformation of the proper ortochronous Lorentz group is then 50c = -^WabSab0c = ^ca0a , §YC = -2wabSabYC = WCaYa , §YC = -2wabSab^c = WCaYa , 5xc = -2^abLabxc = WCaXa , (9.79) where dab are parameters of a transformation and Ya and Ya are expressed by 0a and g^ in Eqs. (9.17, 9.18). Let us write the operator of finite Lorentz transformations as follows S = e-2-ab(Sab+Lab). (9.80) We see that the Grassmann 0a and the ordinary xa coordinates and the Clifford objects Ya and Ya transform as vectors Eq. (9.80) 0'c = e-2^ab(Sab+Lab) 0c e j ^ab(Sab+Lab) = 0C - 2wab{Sab, 0C}- + • • • = 0C + WCa0a + • • • = ACa0a , x'c = Acaxa , Y'c = ACaYa , Y'c = AcaYa . (9.81) Correspondingly one finds that compositions like Yapa and Yapa, here pa are Pa (= iala), transform as scalars (remaining invariants), while Sab dabc and Sab d)abc transform as vectors. Also objects like 1 R = and R = 1 f"[afPb] (dab«,R - dcaa dCb3) R = 1fa[afpb] (ddaba,P - ddcaadd%p) from Eq. (9.1) transform with respect to the Lorentz transformations as scalars. Making a choice of the Cartan subalgebra set of the algebra Sab, Sab and Sab, Eqs. (9.13, 9.17, 9.18), S°3 s12 s56 Sd-1 d S°3 s12 s56 ^ ^ ^ sd-1 d S °3, S12, S56, ••• ,S d-1d, (9.82) one can arrange the basic vectors so that they are eigenstates of the Cartan subalgebra, belonging to representations of Sab, or of Sab and Sab, with ab from Eq (9.82). 9.5 Appendix: Technique to generate spinor representations in terms of Clifford algebra objects We shall briefly repeat the main points of the technique for generating spinor representations from Clifford algebra objects, following Ref. [16]. We advise the reader to look for details and proofs in this reference. We assume the objects y-, Eq. (9.17), which fulfill the Clifford algebra, Eq (9.16). {Ya,Yb}+ = I 2nab, for a, b G {0,1,2,3,5, ••• , d}, (9.83) for any d, even or odd. I is the unit element in the Clifford algebra, while {Ya, Yb}± = YaYb ± YbYa. We accept the "Hermiticity" property for Ya's, Eq. (9.20), Ya^ = naaYa, leading to y^Y- = I. Assuming the relation of Eq. (9.17) this last relations follow. The Clifford algebra objects Sab close the Lie algebra of the Lorentz group {Sab, Scd}_ = i(nadSbc + nbcSad - nacSbd - nbdSac). One finds from Eq.(9.20) that (Sab)t = naanbbSab and that {Sab,Sac}+ = 1 naanbc. Recognizing that two Clifford algebra objects Sab, Scd with all indexes different commute, we select (out of many possibilities) the Cartan sub algebra set of the algebra of the Lorentz group as follows S0d,S12,S35, ••• ,Sd-2d-1, if d = 2n, S12,S35, ••• ,Sd-1d, if d = 2n + 1. (9.84) To make the technique simple, we introduce the graphic representation [16] as follows ab 1 naa (10: = 1 (Ya + VYb), ab 1 i [k]: = - (1 + kYaYb), (9.85) where k2 = naanbb. One can easily check by taking into account the Clifford algebra relation (Eq. (9.83)) and the definition of Sab (Eq. (9.25)) that if one multiplies ab ab from the left hand side by Sab the Clifford algebra objects (k) and [k], it follows that ab 1 ab Sab (k)= 2k (k), ab 1 ab Sab [k]= ^k [k] . (9.86) ab ab This means that (k) and [k] acting from the left hand side on anything (on a vacuum state |^0), for example) are eigenvectors of Sab. We further find ab ab ab ab Ya (k) = naa [-k], Yb (k) = -ik [-k], ab ab ab ab Ya [k] =(-k), Yb [k] = -iknaa (-k) . (9.87) ab cd ab cd abcd ab cd ab cd It follows that Sac (k)(k)= — |naancc [—k] [—k], Sac [k][k]= 2 (—k)(—k), Sac (k)[k] = ab cd ab cd ab cd —2naa [—k](—k), Sac [k](k)= 2ncc (—k)[—k]. It is useful to deduce the following relations ab ab (k) (k) = 0, ab ab (k)(—k)= naa ab [k], ab ab (—k)(k)= :^aa [ ab —k], ab ab (—k)(—k)= 0, ab ab [k] [k] ab = [k], ab ab [k][—k]= 0, ab ab [—k][k]= 0, ab ab [—k][—k]= ab [—k], ab ab (k) [k] = 0, ab ab [k](k)= ab (akb) , ab ab (—k)[k]= ab (—k), ab ab (—k)[—k]= 0, ab ab (k) [—k] ab = (k), ab ab [k]( —k): = 0, ab ab [—k](k)= 0, ab ab [—k](—k)=( ab —k) (9.88) We recognize in the first equation of the first row and the first equation of the second row the demonstration of the nilpotent and the projector character of the ab ab Clifford algebra objects (k) and [k], respectively. Whenever the Clifford algebra objects apply from the left hand side, they always ab ab ab ab ab ab transform (k) to [—k], never to [k], and similarly [k] to (—k), never to (k). We define in Eq. (9.62) a vacuum state |^oc > so that one finds ab ^ ab < (k) (k) >= 1 , ab ^ ab < [k] [k] >= 1 (9.89) Taking the above equations into account it is easy to find a Weyl spinor irreducible representation for d-dimensional space, with d even or odd. (We advise the reader to see Ref. [16].) For d even, we simply set the starting state as a product of d/2, let us say, only ab nilpotents (k) for d = 2(2n+1), Eq. (9.57), or nilpotents and one projector, Eq. (9.58), for d = 4n, one for each Sab of the Cartan subalgebra elements (Eq. (9.84)), applying it on the vacuum state, Eq. (9.62). Then the generators Sab, which do not belong to the Cartan subalgebra, applied to the starting state from the left hand side, generate all the members of one Weyl spinor. od 12 35 d—1 d— -2 (kod)(ki2)(k35) • ••(kd— 1d —2) l^oc > od 12 35 d—1 d— -2 [—kod][—ki2](k35) • ••(kd— 1d —2) l^oc > od 12 35 d—1 d— 2 [—kod](ki2)[—k35] • ••(kd— 1d —2) l^oc > od 12 35 d—1 d— 2 (kod)[—kl2][—k35] ••• [—kd-1 d—2] l^oc >, for d = 2(2n + 1 ), n = positive integer. (9.90) d-1 d-2 [kd-1 d-2] |^oc > , d-1 d-2 [kd-1 d-2] |^oc > , d-1 d-2 [kd-1 d-2] |^oc > , od 12 35 d-1 d-2 (kod)[ k.12][ k.35] ••• [kd-1 d-2] |^oc >, ford — 4n , n — positive integer . (9.91) 9.5.1 Technique to generate "families" of spinor representations in terms of Clifford algebra objects When all 2d states are considered as a Hilbert space, we found in this paper that for d even there are 2d/2-1 "families members" and 2d/2-1 "families" of spinors, which can be second quantized. (The reader is advised to se also Ref. [2,26,16,17,27,9].) We shall pay attention on only even d. One Weyl representation form a left ideal with respect to the multiplication with the Clifford algebra objects. We proved in Ref. [9], and the references therein that there is the application of the Clifford algebra object from the right hand side, which generates "families" of spinors. Right multiplication with the Clifford algebra objects namely transforms the state with the quantum numbers of one "family member" belonging to one "family" into the state of the same "family member" (into the same state with respect to the generators Sab when the multiplication from the left hand side is performed) of another "family". We defined in Ref.[17] the Clifford algebra objects Ya's as operations which operate formally from the left hand side (as Ya's do) on any Clifford algebra object A as follows YaA — i(-)(A)AYa , (9.92) with (-)(A) — -1, if A is an odd Clifford algebra object and (-)(A) — 1, if A is an even Clifford algebra object. Then it follows, in accordance with Eqs. (9.17, 9.18, 9.19), that Ya obey the same Clifford algebra relation as Ya. (Y~aY~b + YbY~a)A — -ii((-)(A))2A(YaYb + YbYa) — I • 2nabA (9.93) and that Ya and Ya anticommute (Y~aYb + YbY~a)A — i(-)(A)(-YbAYa + YbAya) — 0. (9.94) We may write Od 12 35 (kod)(k12)(k35) • •• Od 12 35 [-kod][-k12](k35) • • • Od 12 35 [-kod](k12)[-k35] • • • {Ya, Yb}+ — 0, while {Y~a, Yb}+ — I • 2nab (9.95) One accordingly finds ab ab ab ab ab ab ,7b rib. _ ,• ab „b Y~a (k): = -i (k) Ya = -inaa [k], Yb (k): = -i (k) Yb = -k [k], ab ab ab Yb [k]: = i [k] Yb = -knaa (k) If we define ab ab ab Y~a [k]: = i [k] Ya = i (k), it follows Sab = 4 [Ya,Yb] = 1 (YaYb - YbYa), SabA = A1 (YbYa - YaYb), 7 ab (9.96) (9.97) (9.98) manifesting accordingly that Sab fulfil the Lorentz algebra relation as Sab do. Taking into account Eq. (9.92), we further find {Sab,Sab}_ = 0, {Sab ,Yc}- = 0, {Sab,Yc}- = 0. (9.99) One also finds {Sab,r}_ = 0, {Ya,r}_ = 0, for d even, r(d) :=(i)d/2 ^ (VnaaYa), if d = 2n, (9.100) where handedness r ({r, Sab}_ = 0) is a Casimir of the Lorentz group, which means that in d even transformation of one "family" into another with either Sab or Ya leaves handedness r unchanged. We advise the reader also to read [2] where the two kinds of Clifford algebra objects follow as two different superpositions of a Grassmann coordinate and its conjugate momentum. We present for Sab some useful relations ab Sab (k) = ab cd ac [k][k] = k ab k (k), ab k ab Sab [k] = -k [k], ab cd 2 (k)(k) = 2naan ab cd cc [k] [k], ab cd ab cd ab cd ab cd -2 (k) (k), Sac (k)[k] = -2naa [k](k), Sa [k](k) = 2nc ab cd (k) [k] . (9.101) We transform the state of one "family" to the state of another "family" by the application of Sac (formally from the left hand side) on a state of the first "family" for a chosen a, c. To transform all the states of one "family" into states of another "family", we apply Sac to each state of the starting "family". It is, of course, sufficient to apply Sac to only one state of a "family" and then use generators of the Lorentz group (Sab) to generate all the states of one Dirac spinor d-dimensional space. ab ab One must notice that nilpotents (k) and projectors [k] are eigenvectors not only of the Cartan subalgebra Sab but also of Sab. Accordingly only Sac, which c do not carry the Cartan subalgebra indices, cause the transition from one "family" to another "family". The starting state of Eq. (9.90) can change, for example, to 0d 12 35 d-1 d-2 [k0d][k12](k35) ••• (kd-1 d-2), (9.102) if S01 was chosen to transform the Weyl spinor of Eq. (9.90) to the Weyl spinor of another "family". References 1. N. Mankoc Borstnik, "Spin connection as a superpartner of a vielbein", Phys. Lett. B 292 (1992) 25-29. 2. N. Mankoc Borstnik, "Spinor and vector representations in four dimensional Grassmann space", J. of Math. Phys. 34 (1993), 3731-3745. 3. N.S. Mankoc Borstnik, "Spin-charge-family theory is offering next step in understanding elementary particles and fields and correspondingly universe", Proceedings to the Conference on Cosmology, Gravitational Waves and Particles, IARD conferences, Ljubljana, 6-9 June 2016, The 10th Biennial Conference on Classical and Quantum Rela-tivistic Dynamics of Particles and Fields, J. Phys.: Conf. Ser. 845 012017 [arXiv:1409.4981, arXiv:1607.01618v2]. 4. N.S. Mankoc Borstnik, "Matter-antimatter asymmetry in the spin-charge-family theory", Phys. Rev. D 91 065004 (2015) [arxiv:1409.7791]. 5. N.S. Mankoc Borstnik, D. Lukman, "Vector and scalar gauge fields with respect to d = (3 + 1) in Kaluza-Klein theories and in the spin-charge-family theory", Eur. Phys. J. C 77 (2017) 231. 6. N.S. Mankoc Borstnik, "The spin-charge-family theory explains why the scalar Higgs carries the weak charge ± 1 and the hyper charge ^ 1", Proceedings to the 17th Workshop "What comes beyon2d the standard models", B2led, 20-28 of July, 2014, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Zaloznistvo, Ljubljana December 2014, p.163-82 [ arXiv:1502.06786v1] [http://arxiv.org/abs/1409.4981]. 7. N.S. Mankoc Borstnik N S, "The spin-charge-family theory is explaining the origin of families, of the Higgs and the Yukawa couplings", J. of Modern Phys. 4 (2013) 823 [arxiv:1312.1542]. 8. N.S. Mankoc Borstnik, H.B.F. Nielsen, "The spin-charge-family theory offers understanding of the triangle anomalies cancellation in the standard model", Fortschrite der Physik, Progress of Physics (2017) 1700046. 9. N.S. Mankoc Borstnik, "The explanation for the origin of the Higgs scalar and for the Yukawa couplings by the spin-charge-family theory", J.ofMod. Physics 6 (2015) 2244-2274, http://dx.org./10.4236/jmp.2015.615230 [http://arxiv.org/abs/1409.4981]. 10. G. Bregar and N.S. Mankoc Borstnik, "Does dark matter consist of baryons of new stable family quarks?", Phys. Rev. D 80, 083534 (2009) 1-16. 11. N.S. Mankoc Borstnik, H.B.F. Nielsen, "Fermionization in an Arbitrary Number of Dimensions", Proceedings to the 18th Workshop "What comes beyond the standard models", Bled, 11-19 of July, 2015, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Zaloznistvo, Ljubljana December 2015, p. 111-128 [http://arxiv.org/abs/1602.03175]. 12. N. S. Mankoc Borstnik, H.B. Nielsen, "Fermionization, Number of Families", Proceedings to the 20th Workshop "What comes beyond the standard models", Bled, 9-17 of July, 2017, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Zaloznistvo, Ljubljana, December 2017, p.232-257. 13. H. Aratyn, H.B. Nielsen, "Constraints On Bosonization In Higher Dimensions", NBI- HE-83-36, Conference: C83-10-10.2 (Ahrenshoop Sympos.1983:0260), p.0260 Proceedings. 14. H.B. Nielsen, M. Ninomya, "Dirac sea for bosons, I,II", Progress of the theoretical Physics, 113, 606 [hepth/0410218]. 15. D. Lukman, N.S. Mankoc Borstnik, "Representations in Grassmann space", to appear in arxiv. 16. N.S. Mankoc Borstnik, H.B.F. Nielsen, J. of Math. Phys. 43, 5782 (2002) [hep-th/0111257]. 17. N.S. Mankoc Borstnik, H.B.F. Nielsen, J. of Math. Phys. 44 4817 (2003) [hep-th/0303224]. 18. N.S. Mankoc Borstnik and H.B. Nielsen, Phys. Rev. D 62,044010 (2000) [hep-th/9911032]. 19. N.S. Mankoc Borstnik and H.B. Nielsen, "Second quantization of spinors and Clifford algebra objects", Proceedings to the 8th Workshop "What Comes Beyond the Standard Models", Bled, July 19 - 29, 2005, Ed. by Norma Mankoc Borstnik, Holger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFA ZaloZnistvo, Ljubljana December 2005, p.63-71, hep-ph/0512061. 20. "Why nature made a choice of Clifford and not Grassmann coordinates", Proceedings to the 20th Workshop "What comes beyond the standard models", Bled, 9-17 of July, 2017, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA ZaloZnistvo, Ljubljana, December 2017, p. 89-120 [arXiv:1802.05554v1v2]. 21. N.S. Mankoc Borstnik and H.B.F. Nielsen, "Discrete symmetries in the Kaluza-Klein theories", JHEP 04:165, 2014 [arXiv:1212.2362]. 22. P.A.M. Dirac Proc. Roy. Soc. (London), A 117 (1928) 610. 23. D. Lukman, N.S. Mankoc Borstnik and H.B. Nielsen, "An effective two dimensionality cases bring a new hope to the Kaluza-Klein-like theories", New J. Phys. 13:103027, 2011. 24. D. Lukman and N.S. Mankoc Borstnik, "Spinor states on a curved infinite disc with nonzero spin-connection fields", J. Phys. A: Math. Theor. 45:465401, 2012 [arxiv:1205.1714, arxiv:1312.541, hep-ph/0412208 p.64-84]. 25. D. Lukman, N.S. Mankoc Borstnik and H.B. Nielsen, "Families of spinors in d = (1 + 5) with a zweibein and two kinds of spin connection fields on an almost S2", Proceedings to the 15th Workshop "What comes beyond the standard models", Bled, 9-19 of July, 2012, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Zaloznistvo, Ljubljana December 2012,157-166, arxiv.1302.4305. 26. A.Borstnik Bracic, N. Mankoc Borstnik,"The approach Unifying Spins and Charges and Its Predictions", Proceedings to the Euroconference on Symmetries Beyond the Standard Model", Portoroz, July 12-17, 2003, Ed. by Norma Mankoc Borstnik, Holger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFA Zaloznistvo, Ljubljana December 2003, p. 31-57, hep-ph/0401043, hep-ph/0401055. 27. A. Borstnik Bracic, N. S. Mankoc Borstnik, "On the origin of families of fermions and their mass matrices", hep-ph/0512062, Phys Rev. D 74 073013-28 (2006). 28. M. Pavsic, "Quantized fields a la Clifford and unification" [arXiv:1707.05695]. Bled Workshops in Physics Vol. 19, No. 2 A Proceedings to the 2 1 st Workshop What Comes Beyond ... (p. 216) Bled, Slovenia, June 23-July 1, 2018 10 Do We Find High Energy Physics Inside (Almost) Every Solid or Fluid at Low Temperature? H.B. Nielsena and M. Ninomiyab a Niels Bohr Institute, University of Copenhagen, 17 Blegdamsvej, DK 2100 Copenhagen Denmark b Advanced Mathematical Institute Osaka-city University, Sugimoto 3-3-138, Sumiyoshi-ku Osaka, 1558-8585, Japan and Yukawa Institute for Theoretical Physics, Kyoto University Kyoto. 606-8502, Japan Abstract. It is an old idea of ours (H. B. Nielsen "Dual Models",section 6 "Catastrophe Theory Program", Scottish University Summer School, 1976) that a most general material with only translation symmetry, but otherwise no symmetries should generically (in general) have some small regions in quasi momentum space, where you "see" an approximate Weyl equation behavior. The Weyl equation is the relativistic equation for a (left handed) neutrino. This remark means that one could imagine, that there were behind the Standard Model of High energy physics, a very general crystal model with very little symmetry. Even for the Yang Mills or electrodynamics types fields a similar philosophy is possible. There are though some problems with this solid-state type of model beyond the Standard model, for which we thought have some remedy by means of homolumo gap effects. By making use of relativistic quantum field theory on the lattice we predicted theoretically very high magneto-conduction due to Adler-Bell-Jackiw chiral anomaly effect - so called Nielsen-Ninomiya effect (or mechanism) in gapless parity violating material. Nowadays this kind of material such as chiral or Weyl semimetal and the effect are detected by experiments. Povzetek. Avtorja obravnavata idejo HBN (H. B. Nielsen "Dual Models", razdelek 6 "Catastrophe Theory Program", Scottish University Summer School, 1976), da obstajajo v najbolj splosnem modelu za snov, ki ima le translacijsko simetrijo, majhna območja v prostoru kvazi gibalne kolicine, v katerih pribliZno velja Weylova enacba. Ker velja Weylova enacba za relativisticno gibanje (levorocnih) nevtrinov, predlagata, da razsirjeni standardni model gledamo kot zelo splosen model za kristal z zelo malo simetrijami. Podoben pristop uporabita za primer elektromagnetnega polja in vsa Yang-Millsova polja. Tezave, ki se pri tem pojavijo, omilita s "homo-lumo" vrzelmi. Uporaba relativisticne kvantne teorije polja na resetki napove visoko magnetno prevodnost, ki jo sprozi kiralna anomalija Adler-Bell-Jackiwa, ter s tem pojav Nielsen-Ninomiye: visoko magnetno prevodnost v snoveh, ki krsijo parnost, med obema pasovoma pa ni vrzeli. Te lastnosti materialov merijo v Weylovih (kiralnih) polkovinah. Keywords: Weyl equation, homo-lumo gap Introduction The authors, in particular H. B. N. have through many years the dream, that it is not important what the (most) fundamental laws of Nature might be, because almost certainly the same effective laws would come out anyway: This philosophy is called "Random Dynamics". Inside a piece of matter - crystal, glass, ... - one should then at very low temperature according to this dream find the Standard Model. Recently one is about to find Cases of Relativity-behaving Quasi-particles: A material, e.g. graphene, with such simulations of relativistic particles as we talk about. Materials with relativistic particles simulated as quasiparticles may be very applicable to say high conductivity purposes,... Some of our publications: • H. B. Nielsen and M. Ninomiya, "No Go Theorem for Regularizing Chiral Fermions," Phys. Lett. 105B, 219 (1981). • H. B. Nielsen and M. Ninomiya, "Absence of Neutrinos on a Lattice, 1. Proof by homotopy theory" Nucl. Phys. B 185, 20 (1981). • H. B. Nielsen and M. Ninomiya, "Absence of Neutrinos on a Lattice. 2. Intuitive Topological Proof," Nucl. Phys. B 193,173 (1981). • As for the initiation of Random Dynamics, See "Fundamentals of Quark Models". Proceedings: 17th Scottish Universities Summer School in Physics, St. Andrews, Aug 1976, I.M. Barbour, A.T. Davies (Glasgow U.);1977 - 588 pages; Edinburgh: SUSSP Publ. (1977);Conference: C76-08-01; Contributions: Dual Strings, Holger Bech Nielsen (Bohr Inst.). Aug 1974, 71 pp.;NBI-HE-74-15 In the last section the idea of "Random Dynamics " is introduced based on finding Weyl equation in "whatever". The present paper consists as part I and part II. The part I: Relativity Theory found in solid state. and The part II "What comes beyond Topological Insulator - Nielsen-Ninomiya Effect (or Mechanism) due to ABJ Anomaly -" Part I: Relativity-Theory found in Solid State Physics I-1 Introduction I-2 Automatic: a pet-thought: Natural laws come by themselves! ("Random Dynamics") I-3 General: A very general world with (only) momentum conservation. I-4 Graphene: Example Graphene. I-5 Heusler: Half-metals, Heusler compounds. I-6 Wang: Thoughts about making materials having models of relativistic particles inside. I-7 Doubling: Nielsen - Ninomiya theorem about doubling of such relativistic particles unavoidably on the lattice. great future; hope of seeing high energy physics in low temperature materials not out, but not quite finished. material simulates relativistic quantum field theory. I-8 Further: Further Developments of our "Random Dynamics" I-9 Conclusion for part I The part II: What comes beyond Topological Insulator -Nielsen-Ninomiya Effect (or Mechanism) due to ABJ Anomaly II-1 : Introduction II-2 : 1+1 dimensional Example II-3 : 3+1 dimensional case Weyl (or chiral) Fermion Adler-Bell-Jackiw Anomaly II-4 : Parity non-invariant, Zero-gap material II-5 : Transfer from Left- to Right- comes by Adler-Bell-Jackiw anomaly II-6 : Further arguments II-7 : Conclusions Appendix A : Necessary properties of quantum field theory in this paper Appendix B : Adler-Bell-Jackiw anomaly in continuum spacetime I-2 Automatic Our Old Work in 1976: Dreams Laws of Nature Automatic "Dual Strings. Fundamentals of Quark Models." by H. B. Nielsen, in Scottish University Summer School in Physics, St. Andrews, 1976 (There H.B.N. still mainly is talked on String theory, but at the end a general (fermion) Hamiltonian is studied.) Assumed was translational invariance, at least with respect to a lattice say, and thus a (quasi) momentum conservation, but with respect to the "internal degrees of freedom" there is a very general theory, though assuming there being essentially a finite (discrete). system of states(representing possibly spin and band degrees of freedom.). (Trivial) Generic Considerations on Fermion Dispersion relations (1976). We ignore all conservation laws except for • Energy conservation and Hamiltonian development. • Momentum Conservation. • Particle (number) conservation. • Free approximation (first). • Smoothness, (so that e.g H(p) is differentiable and continuous as function of P.) • Generic: i.e. no fine-tuned values of parameters, and consider a single particle equation: i-^(p,t)= H(PMP), (10.1) where for each value of the momentum p the H(p) is a Hermitian matrix. Relativity and Dimensionality of Space time being 3+1 come out Automatically! A priori - with no fine-tuning (=generically) - the Fermi surface would put itself at separate eigenvalues; but if for some reason ( e.g. "homlumo-gap effect") the Fermi-level were just where n = 2 levels meet, then in a small neighborhood the shape of the dispersion relations would be given by taking H(p) to be n x n = 2 x 2. We then Taylor expand H(p) « H(p°) + ^ aaV^ + ... (10.2) a,a where cra are the Pauli-matrices and the unit matrix = 1. The "vierbein" Va is a set of expansion coefficients for H(p) as function of the components pa (strictly speaking ^ =1,2,3; here). Hermitian matrix, Provided Fermi-level at Degeneracy n = 2 leads to Weyl Equation in 3+1 Dimensions. In the old days we argued that in a general physics universe the Hubble expansion would finally lead to the Fermi-level approaching an n = 2 degenerate levels energy; but now H. B. N.'s Zagreb group - I.Andric, L. Jonke, D. Jurman, and HBN - have studied in general, what is called "Homolumo-gap Effect" meaning the by Jahn and Teller[1] first proposed effect, that the electrons filling the Fermi-sea would back react such as to increase the homolumo gap between the lowest unoccupied (LUMO) and the highest occupied (HOMO) state. This effect goes in the direction to make metals not occur, and make every materials become an insulator, but the gapless semiconductor may be too hard for the homolumo-gap effect to dispense with. Note that this hope for getting automaticly a Weyl-equation like theory had, when using just Hermitean Hamiltonian marices and looking at the n = 2 degeneracy possibility, the consequence that there came only three spatial dimensions functioning the relativistic way, because there were only 3 Pauli matrices. Somehow arguing that the dimensions for which there are no Pauli matrices will lead to essentially zero velocity for the fermion/quasi-electron in these directions and that such dimensions will not be observed, we have come to 3+1 dimensions as an additional prediction from the very general starting theory! With time-reversal symmetry imposed dimension prediction gets modified. Symmetry Square Pauli M. Dimension Field TP (TP)2 = 1 Ox, Oz 2+1 Real - (TP)2 =-1 Ox, Oy, Oz 3+1 Complex TP 5 of them 5+1 Quaternions Table 10.1. The symmetry assumed in line 1 and 3 is the combination of time reversal T and parity P to TP, which leaves the momentum p invariant but is an antilinear operator effectively conjugating the complex numbers in the matrix. If then Fermi-level falls at n = 2 degenerate levels in addition to the Kramers-Kronig doubling in the 3rd case, one gets by Taylor expanding the 2 x 2 resolved into Pauli-matrices, and a generalized Weyl equation results corresponding to the in fourth column denote space + time dimensions. Actually the effective theory is naturally written in terms of the in column 5 mentioned division-algebra(= field). Fundamentally in many Dimensions, but in Most dimensions the Fermion Run with Zero Velocity, we Ignore them. In the for fundamental physics ideal situation of no extra T or TP symmetry the Hamiltonian matrix H(p) is just a generic(~ random) Hermitian matrix (with complex matrix elements), and it predicts at the two levels degenerate point -hoped to be favored at the Fermi-surface by either Hubble expansion or homolumo-gap-effect - that the Fermion only moves with appreciable velocity in as many spatial dimensions as there are Pauli-matrices. We hope that the dimensions in which the velocity gets zero, can/shall be ignored. If the zero-velocity dimensions are ignored, then we have remarkable agreement: The number of dimensions in which the generic double degeneracy neighborhood has the fermions move just corresponds to experimental number of dimensions 3+1 and to having relativity and rotational invariance! If TP (or T) is good symmetry and (TS)2 = Ithen H(p) must have real matrix elements. This is the case in which we in a crystal - with PT symmetry say - completely ignore the usual spin as being decoupled so as to be totally ignored. In this case we get the effective dimensionality, if we ignore the zero-velocity directions: 2 + 1 This means that the relativistic effective fermion should appear "generically" (automatically) even in only 2 spatial dimensions. With Genuine Spin= 1 Electrons and Unbroken Time reversal, the "Quaternion Case" If T or TP good symmetries, and spin 1 included, then T2 = (TP)2 = —1 we have generally doubling of all levels according to Kramers-Kronig rule.[2] So double degeneracy is already there generally and nothing special. In this case we shall therefore instead consider that we can get 4 times degenerate levels sporadically. If we go to such a 4-times degenerate point in momentum space, we could elegantly go to a quaternion 2 x 2 matrices (quaternions are writable as 2 x 2 complex matrices, so that 2 x 2 quaternion matrices can be equivalent to 4 x 4 complex matrices with some restriction. Dimension of non-zero velocity directions: 5 + 1 I-3 Graphene Graphene denotes the layer of carbon like the ones in graphite taken as seperate, i.e. it is 2(space)dimensional material. The quasi electrons running in the graphene layers actuall do show dispersion relations behaving how we above argued for the case with time reversal but ignoring the spin leading to the effective space time dimension 2+1. On the following picture 10.1 one sees the lattice structure of graphene: The next figure 10.2 is supposed to generally illustrate a metal, an insulator and a material with a Dirac-like quasi particle (on the figure 10.1). Even just making a two-layer of graphene complicates the situation and the work by Gammelgaard on the next figure 10.3 illustrates a gap appearing: Fig. 10.1. (2+1)-dimensional Example is Graphene. m jt k Thirt rs ro^i, ijfdiit IÎ--C insuLiicr-s. tun w panUfy-l Jkd mhcL ufrt^e ncuL Fig. 10.2. Fig. 10.3. Putting Double Layer Produces Gap. The left dispersion law is for a double layer of graphene; the right for single layer. (Gammelgaard). The next figures 10.4 illustrate calculation of the dispersion relations for quasi-electrons in graphene by the model described just below. Since we have a 2 space dimension material the energy can be the orbital direction up in the perspective while the two spatial momentum components form the basis plane of the three-dimensional perspective figure: Fig. 10.4.. The Dirac points are of course the points where two branches of the dispersion relation meet with a cone shape. (Fig. 10.5): Fig. 10.5. Dispersion Relation of Graphene The electronic properties of graphene can be described using a simple tight binding model. The electrons in the covalent bonds form deep fully filled valence bands, and thus their effects on the conductivity can be safely disregarded. The unhybridized p orbital is only slightly perturbed by the neighboring atoms. Therefore, the wave function of an electron in the system can be written as a Linear Combination of Atomic Orbitals (LCAO). Using these orbitals as the basis set to represent the wave function, the Hamiltonian that governs the dynamics of the electron is given by: H = ^ £i|^i ><^i| + ^ ^ ti(|^i ><^j| + ><^i|) (10.3) i I {}l whereei represents the onsite energy at the atom, > the i'th atomic orbital, {< i, j >}i the set of couples of lth-nearest neighbors, and tt the hopping parameter between them. In Graphene the Fermi- surface just Lies at the Double degenerate Point So in graphene by symmetry one really get a simulation of a 2+1 dimensional massless Weyl/Dirac fermion, also w.r.t. the placing of the fermi surface. If we think of just the generic case of a very general theory there will typically be no reason why the fermi surface should be just at the Weyl point (with the double degeneracy). We have, however, speculated on two mechanisms, which might make the fermi-surface be driven towards the degeneracy point: • If the world in question has a strong Hubble expansion, then filled states above the degeneracy point would be gradually emptied and holes below the degeneracy point would be also gradually be expanded away/attenuated. • "Homolumo-gap-effect" - meaning that the fermions act back onto the various degrees of freedom that can be adjusted in the lattice in which the fermions run. This back action will be so as to in the ground state arrange to lower the energies of filled fermi states. Thereby arise the so called Homolumo-gap, or rather it gets expanded by this back action "homolumo-gap-effect". In the case that we have degeneracy point that is somehow topologically stabilized, as one might say of the Weyl points discussed here, it may not be possible for the homolumo-gap-effect to really produce a gap. In stead we expect that it will only bring the fermi surface to coincide with the degeneracy point; that would namely lower the filled states as much as possible with the "topological ensurance" of the degeneracy point. I-4 Heusler Heusler Compound Mn2CoAl is a Spin Gapless Semicondutor: Siham Oardi, G.H. Fecher, C. Felser and J. Kubler (arXiv:1210.0148v1 [cond-mat.mtrl-sci], 29 Sep. 2012.) investigated the Heusler compound Mn2CoAl. They gave the article the name Realization of spin gapless semiconductors: the Heusler compound Mn2CoAl. In halfmetallic ferromagnets you have so to speak metal as far as the electrons with one direction of the spin is concerned, but insulator w.r.t. to the elctrons with the opposite spin direction. Now it may further happen that we instead of the metallic we get a gapless semiconductor, namely if we have a degeneracy point as we discussed above. Once there is effectively only one spin of the electron one escapes the time reversal symmetry. Thus in such halfmettals there is a better chance to find Weyl points. The following figure 10.6 illustrates dispersion relation along a piecewise straight curve in momentum space for the two different spin directions along the magnetization axis for the compound Mn2 CoAl. The dispersion relation for the two different spin orientations are printed respectively red and blue: Momentum k Fig. 10.6. Band structure of Mn2CoAl, Majority spin red. In the following figure 10.7 are then as function of temperature given some carrier properties of this material Mn2CoAl: On the following page from Lakhan Baisly et al. as figure 7 in their article we see the density of electron levels (DOS) for the two spin orientations seperately. In the in red shown DOS there can be seen crudely a gap, so for this spin orientation we have the insulator. For the other spin orientation - shown with the positive ordinate pointing upwards there is also a dip at the fermilevel, but now the DOS is going non-zero immediately by going away from the fermilevel. So for this spin we rather have the gapless semiconductor behavior. The strong dependence of the conductivity as function of the magnetic field is just what one expects due to the Adler-Bell-Jackiw-anomaly-effect described more in part II of the present article below. These figures are from: Siham Ouardi et al. "Realization of Spin Gapless Semiconductors: The Heusler Compound Mn2CoAl" DOI: 10.1103/PhysRevLett.110.100401. Zero Gap Material with Quadratic Energy Dispersion (this is by fine tuning) HgTe is one of the few materials wherin this quadratic dispersion law zero gap has been found, since 1950's. Pbi_xSnxTe, Pbi_xSnxSeandBixSbi_x are zero-gap materials (with quadratic disp.). But really one - Wang, Dou, and Zhang - expects that all narrow gap semiconductors by some doping or pressure could be tuned to have zero gap (with Fig. 10.7. Majority spin and Minority spin. Calculated with spin orbit coupling. quadratic dispersion law). Then they call for finding a non-toxic material of this kind. I-5 Wang Physical Chemistry; Chemical Physics Controllable electronic and magnetic properties in a two-dimensional ger-manene heterostructure Run-wu Zhang, Wei-xiao Ji, Chang-wen Zhang,* Sheng-shi Li,b Ping Li, Pei-ji Wang, Feng Lia and Miao-juan Rena Author affiliations Abstract The control of spin without a magnetic field is one of the challenges in developing spintronic devices. Here, based on first-principles calculations, we predict a new kind of ferromagnetic half-metal (HM) with a Curie temperature of 244 K in a two-dimensional (2D) germanene Van der Waals heterostructure (HTS). Its electronic band structures and magnetic properties can be tuned with respect to external strain and electric field. More interestingly, a transition from HM to bipolar-magnetic-semiconductor (BMS) to spin-gapless-semiconductor (SGS) in a HTS can be realized by adjusting the interlayer spacing. These findings provide a promising platform for 2D germanene materials, which hold great potential for application in nanoelectronic and spintronic devices. Fig. 10.8. Fig. 10.9. Hall conductivity as function of magnetic field. ju F 1 r i L i i I Fig. 10.10. Magnetoresistence as function of a magnetic field. I-6 Doubling Nielsen-Ninomiya's No-go theorem The authors are very proud of, that we have shown a theorem saying: When one makes the mentioned "relativistic fermions of Weyl-type" (=chi-rale fermion) on a lattice (so e.g. in a crystal) then you always get equally many right-spinning and left-spinning Weyl-type particle(species). This theorem is a great challenge for those wanting to make a lattice model (with calculational purposes) for a theory with massless (or almost massless) quarks, let alone the Standard Model. By having 3 K +3 K' Dirac-points of Compensating Handedness Our Doubling Theorem Realized in Graphene. I-7 ABJ Anomaly In the article H. B. Nielsen and M. Ninomiya, "Adler-Bell-Jackiw Anomaly And Weyl Fermions In Crystal," Phys. Lett. 130B, 389 (1983). doi:10.1016/0370-2693(83)91529-0 we have put forward how to understand intuitively the Adler-Bell-Jackiw anomaly and how it should be possible to see it in crystals. Indeed now it has -presumably- been found in Na3Sb in its three dimensional form; at least the characteristic property that this anomaly can lead to a negative magnetoresistance seems justified for this material as should be seen from the following figure 10.12: It is clearly seen for the low temperatures that there is a dramatic peak in the resistance when the magnetic field is small, whereas the resistance becomes appreciably smaller when the magnetic field is switched on. The lower of the two figures shows the resistance in the direction of the magnetic field. It is indeed important that this increased conductivity goes in the direction of the magnetic field and thus there is a dependence of the magnetoresistance as a function also of the angle between the magnetic field and the direction of the electric field. This subject will be explained in more detail in part II. Fig. 10.11. Our Doubling Theorem Realized in Graphene. Fig. 10.12. I-8 Further Further Developments of Our "Random Dynamics" Further speculations, calculations, supporting the idea of getting the Standard Model out as a - say low energy limit - of/from almost whatever the (most) "fundamental" physical laws (say complicated) might be: • A low energy boson system - with only momentum conservation... like the general fermion system considered - gives (in free approximation) free Maxwell equations. • Remarkably: All species of particles in the Standard Model except the Higgs boson are eihter Yang-Mills particles or chiral fermions; so they would all be massless except for effects due to the Higgs field! This is just what one gets by asking for the low energy limit in the general theory! I-9 Conclusion • Hope that the type of relativistic chiral fermions, one finds in high energy physics Standard Model in fact comes by itself - and even points to the right dimensionality 3+1, which just is the right one-; but there are a couple of "small" problems (different species of particles have in first go different "maximal" velocities) • Now adays the phenomenon is about being found in real materials, graphene etc. One can make relativity models chemically It should be especially stressed that the negative magneto-resistance due to the Adler Bell Jackiw anomaly has been seen in N03S6 II. What comes beyond Topological Insulator ? -"Nielsen-Ninomiya Effect" due to Adler-Bell Jackiw chiral Anomaly- II-1 Introduction In part I we mainly argued about "Gapless Semiconductor" "Topological Insulator" and this subject has been very rapidly developing presently. We now, in this part II, argue chiefly a new application of relativistic quantum field theory. Specifically, We investigate in condensed matter (in nano-scale= 10-9m) how the Relativistic Quantum field theory Effect can appear and can be detected in material science. Theoretically this effect was predicted already 35 years ago in 1983 by the present authors • (H. B. N and M. N.) in a High Energy Theoretical Physics journal, Physics Letters B Vol. 130, issue 6 p 389 (1983), entitled "The Adler-Bell-Jackiw anomaly and Weyl Fermions in a Crystal". • Prior to the above paper one of the authors (M. N) was invited to give talks in the International Workshop on "Lattice Field Theory" in Saclay, Paris and Subsequently held XXI International Conference on High Energy Physics, Paris July 26-31, 1982 (so called "Rochester Conference series"), where he talked about Weyl fermions on lattices and the ABJ-anomaly. In solid material there offen appears crystal lattice structure. Thus we are forced to use lattice field theory which has been well developped in high energy physics. In this formulation the crucial facts for us are the following: Suppose At each lattice site we put one Weyl fermion e.g. ¥L (Left-handed one). Our Nielsen-Ninomiya Theorem states that there should appear equally many right handed and left handed Weyl fermions - looking in momentum space at different momentum values -. In the simplest construction resulting from just "naively" replacing derivatives by differences on the lattice our theorem is implemented by there appearing 2d species (d: space dimension). Therefore in 3 space dimensions it turs out that there should be 8 species of Weyl (or chiral) fermions. Furthermore 4 of them are left-handed ¥L and rest 4 species are right-handed chiral fermions. That is to say on the lattice there should be pairwise (left-handed and right-handed) chiral fermions. Therefore we are not able to construct chiral theory with for instance only one handed fermion on the lattice. Thus it leads to the very important consequence in high energy physics. In reality the Standard Model or, unified model of, weak and electromagnetic interactions called "Glashow-Salam-Weinberg model", or "Standard Model" of Weak and Electromagnetic Interaction cannot be constructed on the lattice! The reason is that in the Standard Model all the fermions are left-handed chiral fermions, while no right-handed fermion at all. The experimental results performed so far are all well in agreement with the standard model predictions. If one takes serious the proposal of a new law of nature by one of us and various collaborators, "Multiple Point Principle", one can even claim an indication for, that the Standard Model contrary to the expectation of many of our colleagues, should be valid up to an energy scale of the order of 1018 GeV (rather close to the Planck scale): One of the authors (H. B. N.) made together with C. D. Froggatt a theoretical calculation of mH with recourse to the just mentioned "multiple point principle (MPP)". The value is in very good agreement with experimental value at LHC (Large Hadron Collider in CERN, Geneva) mH ~ 125GeV. See e.g. H. B. Nielsen and M. Ninomiya "Degenerate vacua from unification of second law of thermodynamics with other laws; The derivation of Multiple point principle" Int. J. Mod. Phys. A23 (2008) 919 DOI: 10.1142/S02177510839682, in which an argument for among other things is given MPP from a model with the action taken to be complex rather than real as it is normal. If the Standard Model shall as from this suggestion from Multiple Point Principle etc. be valid only with tiny corrections if any almost up to the Planck scale, it would be even more mysterious that we could not put it on a lattice because of its chiral particles. Really we could -it looks -hardly regularize it with any sensible cut off! Quite a mystery. [3] 2) ABJ anomaly on a lattice Condensed matter researchers except for high energy physicists (including some nuclear theorists), may not have heard of the Adler-Bell-Jackiw or chiral anomaly. Therefore we briefly explained ABJ anomaly in continuum space in Appendix A. Here we turn to our nano-scale material case. In the material there is a lattice structure Fig. 10.13. 2 Fig. 10.13. Lattice structure. In this 3 dimensional lattice on each sites we put one Weyl or Chiral electron e.g. eL (Left handed electron),then according to the Nielsen-Ninomiya Theorem, there should appear somehow so many of them, that there are equally many right haned and left handed ones. In fact we get in the simplest case 4 eL as well as 4 eR. To understand band structure, we go to the momentum space. Note that due to the lattice translational invariance the momentum is conserved modulo multiple of the unit length of reciprocal lattice. The Brillouin zone in the momentum space is topologically equivalent to the hypertorus S1 x S1 x S1. In such a topological structure of crystal lattice, the Adler-Bell-Jackiw anomaly explained for continuum spacetime in appendix B, is easily understood also , as was presented in PLB 130 n06, (1983) by the present authors. II-2 1 + 1 dimensional example For simplicity, as an example the 1 space 1 time dimensional case is considered. Right chiral (Weyl) fermion obeys lattice Weyl eg. i d ^R(na) = 2a iyR((n+1)a)-¥L((n-1)a)] where n = 0, ±1, ±2, ••• denote sites and a is a lattice space. This can be easily solved and the dispersion relation is given by w = (a) sin pa. Thus near p = 0 there is a RH (RH = right handed) species with the dispersion law w « p and further there is a LH (LH =left handed) species near n with the dispersion law w «-(p- a). These situations are illustrated in the following Fig. 10.14. Note that due to topology of momentum space, there is a periodicity modulo 2n. (e.g. points p = - a and n are identified) II-3 3 + 1 dimensional case This 1 + 1 dimension example clearly tells us, that in lattice theory there appear equal number of RH and LH chiral (or Weyl) fermion species (really in 1+1 dimension one should rather talk about right mover and left mover, because there is no genuine handedness in 1+1 dimensions) . It is not completely straightforward to generalize to 3 + 1 dimensions, but with use of the appriate mathematics of homotopy (group) theory one make the analogous theorem in 3+1 or in even higher dimensions to the theorem in 1+1 that in a period real function has pass zero in positive and in negative direction equally many times per period. Fig. 10.14. II-3 (a) Weyl (or chiral) Fermion In generic chiral (Weyl) fermion theory which obeys iY (-X) = HY(-X) = wY(-X) We assume that the generic Hamiltonian satisfy the following four conditions: (1) Locality of interaction in the sense that H(—-if) —» 0 as |"X-if )| —> large fast enough that the Fourie transform of H(—) has continuous first derivative. (2) Translational invariance in the lattice (3) Hermiticiti of H (reality of S) (4) Furthermore an assumption is that the charge (=lepton number in our case) is bilinear in the fermion field. Under these conditions in the generic H case we gave a rigorous proof in terms of the Homotopy theory in topology in 1981 (see, II-1). II-3 (b) Adler-Bell-Jackiw anomaly on a lattice Let us go into the Adler-Bell-Jackiw (ABJ) anomaly on the lattice in the continuum spacetime. We reviewed this anomaly in continuum spacetime in Appendix B. Here we argue for the lattice version of the ABJ anomaly. Firstly we as as an example let us explain the 1 + 1 dimensional lattice Weyl (chiral) fermion. In the lattice RH chiral or Weyl electron system, we put on an external uniform electric field E in x-direction denoted by A1 = E in temporal gauge (A0 = 0). Then the Weyl eq. reads idtYr(x) = (-idX - A 1)YR(x). The dispersion law is given by c(p) = p. In the classical eq. of the electron in the presence of the electric field is p = eE so that the RH electron in quantum theory is given by cc = p = eE. Therefore the creation rate of the RH electrons per unit time and unit length is determined by a change of the Fermi surface that separates the filled and unifilled states as shown in Fig. 10.15. Fig. 10.15. We denotes the quantization length L, then the density of states per unit momentum is given by . Therefore the rate of change of the RH electron number Nr is given by Nr = ^ • wfS where wfs denotes the rate of energy take up of the RH electron fermi surface per fermion, i.e. eE. Therefore we obtain RH electron creation is given by NR = E per unit length (namely for L=1). This is the ABJ anomaly. Thus the chiral charge QR defined as the total number of RH particles (over the fermisea minus the number of holes) is not conserved: QR = NR = E In the same manner the annihilation rate of LH electrons with w = -p is derived as Nl =- 2n E This means that creation rate of the LH anti-electron is given as N L = 2en E By adding both, the anomaly of the Dirac electrons is Nr + Nl = nE, and thus Q5 = nE To proceed to the 3 + 1 dimension case, we should calculate the energy levels in the presence of an external uniform magnetic field, e.g. in the z-direction so that A2 = Hx, and A^ = 0 otherwise. Thus we consider the equation for the two component RH electron field ¥R [idt -(? - e?)?] Yr(X) = 0 This eq. can be solved by introducing an auxiliary field ® as ^R = Thus t ; 3 i£ + - eA)-0j he eq. for ® is given by i§ -(- - eA)-0 • i§ + (- - eA)-0 ® = 0 This eq. reduces to the harmonic oscillation tpe eq, H3X7)2 + (eH)2(x' + PH) + (p3)2 + eHaa] to = œ2to with aa = ±1 The energy eigenvalues œ are given by the Landau levels as follows w(n,CT3 ,pa) = ± [2eH(n + \ ) + (pa)2 + (eHo^)]1 with n = 0,1,2, ••• , except for the n = 0 and a3 = — 1 mode. Here œ(n = 0, a = — 1,p3) = ±p3. The eigenfunction is of the form ®na3 (x) = Nnas (x) X exp(—ip2X2 — ip3X3) XeXp( — leH(x' + fH)2) X Hm(x' + PH )x(a3) where Nno-3 is normalization constant and x(a3) denotes the eigenfunctions of Pauli spin 03 : x(1) = ^0) and x(-1) = ^^ 01 Thus the solution of the eq. for Two-component RH electron ¥R becomes the relations ^Rn+1'ff3=-1) = n^,^-1 ¥n,a3 = 1 for n = 0,1,2, • • • The zero mode n = 0 is ¥Rn=°,a3 = -1) = 0 with w = -p3. Therefore the ground state energy of ¥R is given by w(n = 0, ct3 = -1, p3) = p3 The energy eigenvalue for the other modes are d(n = 0, CT3,P3) = ± [2eH(n + 2) + (P3)2 + eH^]2 These dispersion laws are depicted in the Fig. 10.16. Fig. 10.16. In the next step an external uniform electric field E is turned on along the same direction parallel to H. For the zero mode (n = 0, ct3 = -1) the dispersion law is the same as that for 1 + 1 dimensions. Thus the creation rate of the particles is calculated in a similar manner. We should note that the electric field E is switched on adiabatically,and there is no particle creation in the n = 0 modes. The density of the state in momentum space in the magnetic field direction is for quantization length L L 4^2, and thus the creation rate (=the ABJ anomaly) is expressed as NR = LLn1 dfs (n = 0,^3 = -1,P3) 2 = 4^EH For the LH electrons annihilation rate of LH anti electron is e2 IN L = -—EH 4n R and the creation rate of the LH anti particle is given by 2 N L = HH = Q L In the case of the Dirac electron 2 N R + N L = HH = Q 5 II-3 (c) Generic Case We again look at a generic case of which Hamiltonian is given by NxN local Her-mitian matrix. The N discrete energy eigenvalues are determined by the following eigenvalue eq. N £ Hki(-pM1' ("p) = WiW) (i = 1, • • • , N) i=i Here we assume that the ith level Vi(-P) and (i + 1 )th level are degenerate. The eigenvalue ) are assumed to be degenerate with the (i + 1) level at several different points in momentum space, which are denoted as (^d(Pd), "Pd) in the dispersion space P), "P). The ith and (i + 1 )th levels are described by d submatrix H(2) ("P): it has the ith and (i + 1 )th entries of NxN matrix H. We then expand H(2) ("P) in powers of ("P —-Pd) around are of the degenerate point (^d("PPd),Pd). In the expansion of H(2) ("P) is given H(2) ("P) = H(2) ("Pd) + ("P " "Pd)^- d + O(("P " "Pd)2). The derivative term is expressed by the Pauli matrices 1_+ cra), (a = 1,2,3) andL = 2 x 2 unit matrix, as 9H(2) " " d = M7 d)-+ vkw d )ffa 3"P k Here V are the constants depending on "Pd. Thus near "P = "Pd, H(2) ("P) takes the form H(2) ("P) = Wd1-+ ("P " "Pd)"1+ ("P " "Pd)kVk^a The eigenvalue eq. of the ith and (1 + i)th energy eigenvalues near "P = "P d H(2)("P )u = cuu. This is rewritten by using a new set of variables P = " " "d, P0 = ^ " ^d " P"" as PV"' u = P0u If we introduce K0 = p0 and k = ±p"V Were ± correspond to the sign of det V. For simplicity we may take as an example Vka = v5ka (k, a = 1,2,3). The above eigenvalue eq. becomes Ik "^u = ±k0 u Where the dispersion law (k0)2 = v2k2. Thus, it is = v2p2 In this way RH and LH Weyl eq. describes the 2 energy levels near degeneracy point in (^("P), "P) space correspond to a species of Weyl fermions contained in the theory. Our theorem tells that RH and LH degeneracy points appear necessarily as a pair because of the Brillouin zero structure (topology). The theorem was proved by only topological arguments together with locality, as was shown our papers in 1981. The doubling of the Weyl fermions are illustrated in Fig. II-4 (page 18). II-4 Parity non-invariant zero-gap material We assume that we have found a parity non invariant material (i.e. a crystal should be of non-centrosymmetric symmetry; e.g. BiTeI form a non-centrosymmetric crystal. Best might be a triclinic pedial class with no point symmetry at all.) with zero-gap, which can be simulated by a Weyl, fermion theory with a dispersion law = v2p2. The effect analogous to the ABJ anomaly gives rise to a peculiar behavior of the conductivity of the electric current in the presence of the magnetic field. It is enough to consider one conduction band The valence band (negative energy state) is assumed to be completely filled. In the absence of external field, the single electron distribution function in the thermodynamical equilibrium is of the form f0("P) = [1 + exp[(^(p) — u)/kT]]-1 In the presence of E and H = 0 there occurs a small deviation from thermodynamical equilibrium so that f = f0 + 6f, and the E field accelerates the electrons in the same direction and then ( s ) = V9V drift 9Pz At the same time the accelerated electrons get scattered back into some states in the same cone. We assume that f fills back into f0 exponentially with a relaxation time t0 so that 6f « e-T Then f =4(f—f0) 9t/ coll T0 Therefore the steady state condition is (§{)drift = — (df u) (Boltzmann eq.). The sol. of this is in the lowest order in E f(— )= f0 M + eET0 Then the longitudinal current density is given by 1 L3 To = TT L(-e)vzf(lP)(#deg. Pts) Where vz = dpr and (#deg. pt) denotes the number of deg. pts (= degeneracy points). In the low temperature approximation f°(w) = 0(u — w) so that 1 u2 J° = ^e2E(U_)T°(#deg. pt) the relaxation time is given in terms of transition probability of electron from the state with—P into one with—P', W(—p —> —P') by To = L3 L ^ w * ^') p' We assume that the interaction between the electron and the ionized impurities is given by the screened Coulomb potential (pot.) of the from . ,. _i— 4ne2 \ e Yo v * )= t w With the screening length y° and k the dielectric constant. Computing t° in the first order perturbation we obtain the current as / v \2 /,,4x r ß T-1 T = 2(^ J0 3nm V4neV W ) ln(1 + ß)- (#deg. pt) 1 + p_ With p = ^er^ (#deg. pt) and ni the density of impurity. Next compute the magneto-conductivity when H parallel to E is so strong that only the lowest states n = 0, ct3 = — 1 with dispersion law w = vpz or w = —vpz near the RH and LH degeneracy point are filled the ABJ anomaly effect will cause the movement in the momentum space of electrons from the lowest Landau level (n = 0, ct3 = — 1) at the one deg. pt. (=degeneracy point) in the LH cone to the corresponding one (n = 0, ct3 = — 1) in the RH cone (at the RH deg.pt.). Thus these moved electrons will give raise to a deviation from the thermodynamical equilibrium, that can be expressed by the different chemical potentials for the electrons at the RH degeneracy pt., ur and at the LH one uL. If one had calculated the relaxation time in the approximation where only one degeneracy point at a time was relevant -such as we did above in the H = 0 case-we would have found 1 = 0. This comes out of such a calculation due to the energy conservation factor 5(w — w') = V5(pz — pZ) contained in W(PZ — PZ) which makes (23) give T = 0. However we cannot neglect scattering processes involving two degeneracy point. II-5 Transfer from LH to RH cones by Adler-Bell-Jackiw Anomaly The mechanism for the electric current with both E can H switched on peculiarly different from the one with a negligibly weak H. In the presence of strong H the lattice anomaly of the ABJ anomaly takes place: transfer of the particles from the LH degeneracy pt. to the RH one acts as a drift term, i.e. N |drift in the Boltzmann equation. On the other hand for negligible H each degeneracy points act independently. By the ABJ anomaly the Fermi energy level |J.R in the RH cone goes up compared to that of the H = 0 case ^ and in the LH cone is lowered. See Fig. 10.16 (a) (1 + 1 dim. case) and Fig. 10.17 (3 + 1 dim. case) Fig. 10.17. In order that the system is in the steady state the excess electrons by the ABJ anomaly in the RH cone must be scattered back to the another state. But they can not be scattered back into the state in the same come! because, as was explained above t = oo. Therefore they must transfer into the states in another cone; that is from the RH cone into the LH cone. We may call this the intercone scattering and we denote the corresponding relaxation time by ti. If the intercone transition probability W(pz —» pZ) from RH cone into the LH cone is calculated, then the collision term is given by NR|coll = L HpZ [f(pz)- fo(Pz)] L Lp^ W(pz-> pZ) - -Pr(Nr nR) Here NR and NR denote the total electron numbers in the RH cone above the degeneracy energy in the H — 0 and H = 0 cases respectively. Thus T- = L lP3 W (pz - pz) 1 The generation of a current associated with the ABJ anomaly can be shown by the following energy conservation argument. ABJ anomaly indicates that electrons are transferred from the LH cone into the RH cone by the rate of (2 EH Per unit Time,per unit volume.: Notice that the dispersion law is continuous and the RH and LH cones are connected smoothly as shown Fig. 10.17. Since the Fermi level energies are |R > |L the transfer costs the energy 2 (2njTEH(|R — |L). This energy must be taken from the E field by the presence of a current JA determined by the energy balance as 2 EJa = (2nF eH(M-R — Ml) At the zero temperature, in the RH cone f0(^) = 0(|mr — and thus NR = AL PyPz f0(^) eH Hr PyPz > (2n)2 R + - h)dNR Inserting this into Boltzmann eq. N R Idrift — — N R |col We obtain uR — uL — evExi 2 Therefore JA — ev (27t)2 EHti (#deg.pt.) Here the subscript A stands for the anomalous current the one associated with the analogue of the ABJ anomaly. In the definition of ti we may approximate W(— —» pZ) = W(— — p ') r n —2 So that W(pz — pZ) = (^)2ni (— — — ')2 + Yr 2n5(w — w') with ^ = —> T2HJ Y EkH (#deg.pt.). According to — = —P ——Pd, p0 — ^ — — , we have —P ——P' — T^d ——P d — P' — — / _\ _\ where p and p are oscillating around P d and P d: since they are order of 1 — — / -| (eH) 2. We may ignore the oscillatory part ( P — P ) and -2 term in the denominator of W(pz — pz) when compared to the distance of the RH and LH deg. pts d — ^pd .In this approximation we obtain Ja — ()2 (f d — f d)4(#deg.pt.) We then obtain the ratio of the conductivity that is defined by f — ctE as 2a _ 3 i v CT0 16 I ^ , ,4 r -| (ïï) [!nd + P) — T+ë] (—d — — d)4 By these results, for the intercom relation time ti the electrons must travel a "long distance" in momentum space. Thus ti is expected to be a large value compared to tq for H = 0. Therefore ^ given above is large. v II-6 Further arguments So far we have presented our own theoretical predictions in 1983 although we believed sooner or later our predicted "Nielsen-Ninomiya" mechanism (or effect) will be proved by experiment. Indeed after almost 35 year later Princeton University group led by Prof. N. Phuan Ong and R. Cava, found chiral anomaly in crystalline material. This surprising news in science community appeared in an article by Catherine Zandonella, • office of the Dean of Research, in Science, September 3, 2015 entitled Research at Princeton: Long-sought chiral anomaly detected in crystalline material (science). At the almost same time, scientist's article entitled. • "Evidence for the chiral anomaly in the Dirac semimetal Na3Bi" By J. Xiong Satya K. Kushwaha, Tian Liang, J. W. Kritzan, M. Hirsehberger, Wulin Wang, R. J. Cava, X. P. Oug, Science Express, 03 , September 2015. and • "Signature of the chiral anomaly in a Dirac semimetal - a current plume steered by J. Xiong, S. K. Kushwaraha, T. Liang, J. W. Krizan, Wudi Wang, R. J. Cava and N. P. Ong Since then the works on this subject is really under rapidly developing mainly in Experiments, also theories: e. g. Dirac cones,and Weyl semimetals. We believe in the rather near future we shall see some machines using "Nielsen-Ninomiya Mechanism (or Effect). See e.g also [4]. II-7 Conclusions In the present article we present the viewpoint at two exceptional high energy theoretical physicists new eras of condensed matter. In the first part I we mainly considered "Topological Insulator" from random dynamics point of view. The essential point is that in generic Fermion dispersion relations i.e. in (almost) all solids or fluids at low temperature we can derive the recently found properties of Topological insulators such as graphene etc. In the 2nd point II, we present what comes beyond topological insulator. We believe that the Adler-Bell-Jackiw anomaly effect in the chiral non invariant gapless material, causes that • magnetic conductance is enhanced very much (ideally permanent current) • Chiral electron (chiral fermion in general) in lattice of the gapless material runs with a fixed speed. (This fixed speed is what in the relativity theory analogue is the speed of light.) This is so, because we by analogy can apply the relativistic quantum field theory. To make any apparatus using the above theory will be widely opened to not only condensed matter, but chemistry, beyond artificial division such as, physics chemistry engineering etc. Acknowledgements One of the authors (H. B. N.) acknowledges the Niels Bohr Institute, Copenhagen University for allowance to work as emeritus and for economical support to visit the conference in New York for which this work is the proceeding. M. N. acknowledges Advanced Mathematical Institute Osaka City University, and Yukawa Institute for Theoretical Physics, Kyoto University as emeritus. M. N. also acknowledges the present research is supported by the JSPS Grant in Aid for Scientific Research No. ISKO 5063. Appendix A We consider electron in quantum field theory (Relativistic quantum mechanics.) We present only necessary properties in Appendix A The electron in the relativistic quantum field theory it is usually described as ( Y h Dirac field Y = I ^ ) where YL and YR are 2 component fields. Now the electron has intrinsic spin —. Thus electron has the angular momentum then —, whose value are half integers, and the spin components is S take values ± 2. For massless fermions the right YR and the left YL componets in the (free) Dirac equation gets seperated, and we actually even find that the spin direction is the same as that of electron movement for the right components YR and the opposite for the left components YL. Let us start with the Dirac field such as an electron in the quantum field theory. The electron has intrinsic spin 2 of fermion obeying the free Dirac eq. (iY^ — me) Yd = 0 (II — 1) thereafter we ignore electron mass unless described. Our notation is that of the textbook of Bjorken-Drell " Relativistic Quantum Fields". For our purpose we list up relevant notations below • The 3 + 1 dimensional flat space metric (tensor): /10 0 0 \ 0 —1 0 0 g 0 0 -10 \0 0 0 -1J The y matrices are Y0 = 0- 0 Y V 0 -_ were L= 10 01 and Y1 = (J^i o ) 1 = 1,2,3 Here ct1 denotes 2 x 2 Pauli matrices andL = ^ 11 Furthermore Y5 = Y5 = (note (y5)2 = 1) The 4 component Dirac field is denoted as YD(p,s) and when there is no interactions obeys the free Dirac eq. as 244 H.B. Nielsen and M. Ninomiya (iy^ - m)¥ = 0 where 3^ = , in momentum representation Here s denote intrinsic spin |s| = 2 ¥d("p, —) obeys p¥d(p,s) = 0 with 4 component Dirac field we may describe ¥D(P>S)=((P>S) Where ¥L and ¥R are 2 component spinor respectively the eigenvalue solution of free Dirac eq. (II — 1) is the form of ¥(p,s) = 2 + m2 E / \ E / E > 0 ~E „ n " o . hoe E < 0 Fig. 10.18. Dirac's "hole theory". We adopt the Dirac's "hole theory". In this theory often used in condensed matter as dispersion relation, the negative states are all filled, while the hole in the Dirac sea is antiparticle, i.e. positron e+. In solid state physics where one has say a crystal lattice, which from the quantum field theory is discretized, so therefore we are interested in discretizing the quantum field theory here. The Dirac fermion wave function ¥D (—P, s) has 4-components: 2 degree of freedom as that energy can have plus or minus. Furthermore the electron has intrinsic spin of which value is |s| = 2 .In the massless case spin/(vector) direction can be either the direction of the electron motion or the opposite. We then define for describing "chirality". It is usually distinguished by this quantity. That is to say = +1 or — 1. Customary +1 is named Left moving- and —1 case is Right moving-Weyl or chiral fermion denoted ¥L and respectively. (The Lorentz or Poincare group of spacetime in 3 + 1 dim Hermann Weyl investigated in detail and the basis is 2 component spinor called Weyl spinors ¥L and In terms of these 4 component Dirac field ¥ such handed components ( ¥l 0 can be constructed (¥D = ( ^ )) Appendix B We are now ready to discuss about Adler-Bell-Jacklin anomaly. In quantum field theory there are various symmetries. One of the most interesting symmetries is chiral (or axial) symmetry. That is the interaction of Dirac field ¥D with electromagnetic field A^ is given by S = J d4xY(x) [iy^ (3^ + ieA^(x))]yd (x) (*) in the case of massless electron., where YD = Y^y0. It has chiral symmetry which may be obvious, if we rewrite (*) in terms of YL and YR as the Dirac eq. can be written as 0 i(3o + ui(3i + ieAi)) )f yl N = 0 i(3o - ui(3i + ieAi)) 0 J {Yr J ' In this way the equations of YL and YR are separately given by the following Weyl equations i(3o - ai(3i + Ai))YL = 0 and i(3o + ai(3i + Ai))YR = 0. In these forms it is evident that the theories are invariant under the following infinitesimal Weyl transformations YL -> (1 - ia^ - P^ )yL yR^ (1 - ia^ + p^)yR Where ai and pi (i = 1,2,3) are infinitesimal transformation parameters, restricted to leave the normalization of the Weyl fields invariant. This Weyl or Chiral transformation is broken due to quantum effect in quantum field theory. There were several suggestive articles, but explicit manifestation is presented by • S. Adler, Phys. Rev. 177 (1969) 2426 and • J. S. Bell and R. Jackiw Nuovo Cimento 60A (1969) 4. Furthermore the method of path integral formulation this ABJ anomaly is due to nor-invariance of the path integral measure • K. Fujikawa Phys. Rev. Lett. 42 1195 (1979) Phenomenologically this ABJ anomaly is really important. It has been observed by experiments. n0 meson decays into 2 photons. When we approximate n0 as being massless, this decay process is expressed as the following diagram, triangle diagram of Feynman diagram If chiral symmetry is not broken, this diagram turns out to give zero. Thus this decay is not allowed. However, experimentally this decay process certainly exists. This is the evidence that Adler-Bell-Jackiw anomaly does exist. In high energy, physics the ABJ anomaly is expressed as the non-conservation chiral current Jjj such that _eaßy6p F. aß Fyö daJ5 ____ 0 > a = 16n2 Here the chiral current J^ is defined as Ja = lime j1? (x + f )YaY5exp -ieJX+1 dzA(z) We might perform the calculation to show that the above triangle diagram is non-zero due to the ABJ anomaly. But we have instead in subsection 10 alluded to a derivation of the ABJ-anomaly by using how particles are pumped up or down from or to the fermi-sea (in high energy physics the Dirac sea). 1(x + f References 1. Jahn, H. A.; Teller, E. (1937). "Stability of polyatomic molecules in degenerate electronic states. I. Orbital degeneracy". Proc. R. Soc. A. 161 (A905): 220-235. Bib-code:1937RSPSA.161..220J. doi:10.1098/rspa.1937.0142. 2. Kramers, H. A., Proc. Amsterdam Acad. 33,959 (1930) E. Wigner, Über die Operation der Zeitumkehr in der Quantenmechanik, Nachr. Akad. Ges. Wiss. Gottingen 31, 546-559 (1932), http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002509032. 3. "Weyl particles, weak interactions & origin of geometry", Nuclear Physics B -Proceedings Supplements Volume 29, Issues 2-3, December 1992, Pages 200-246. https://doi.org/10.1016/0920-5632(92)90021-J. 4. Qiang Li, Dmitri E.Kharzeev, "Chiral magnetic effect in condensed matter systems", Nuclear Physics A 956 December 2016, Pages 107-111. Bled Workshops in Physics Vol. 19, No. 2 A Proceedings to the 21 st Workshop What Comes Beyond ... (p. 247) Bled, Slovenia, June 23-July 1, 2018 11 Electric Dipole Moment and Dark Matter in a CP Violating Minimal Supersymmetric SM T. Shindou * Division of Liberal-Arts, Kogakuin University Nakanomachi 2665-1, Hachioji, 192-0015, Tokyo, Japan Abstract. We consider a dark matter scenario in the minimal supersymmetric standard model with CP violation where the Bino-like neutralino is a dark matter and its annihilation cross section is enhanced enough to reproduce the observed relic abundance of the dark matter through heavy Higgs bosons exchange. In this benchmark scenario, we examine the electric dipole moments of the electron, the mercury, and the neutron. We also consider the spin-independent cross section for the dark matter scattering with nuclei. We show that the electric dipole moments will be very powerful tool to explore the parameter space in this model, even when most of the new particles are very heavy. Povzetek. Avtor obravnava model za temno snov v okviru minimalnega supersimetricnega standardnega modela s kršitvijo CP, v katerem temno snov tvori vrsta nevtralina z dovolj velikim sipalnim presekom za anihilacijo z izmenjavo tezkih Higsovih bozonov, da da njegova gostota ustreze izmerjeni pogostosti temne snovi. V tem modelu oceni električne dipolne momente elektrona, jedra zivega srebra in nevtrona. Obravnava od spina neodvisne sipalne preseke za sipanje te temne snovi na jedrih. Ugotovi, da je elektricni dipolni moment elektrona koristno orodje za raziskavo prostora parametrov tega modela tudi v primeru, ce je vecina delcev v tem supersimetricnem modelu zelo masivnih. Keywords: dark matter, neutralino, EDM, MSSSM 11.1 Introduction Though there is no evidence of supersymmetry (SUSY) at the LHC experiments, SUSY is still an attractive candidate of physics beyond the Standard Model (SM). There are several motivations to consider the minimal SUSY Standard Model (MSSM) than it in the SM. For example, (i) the gauge coupling unification is improved in the MSSM, (ii) quadratic divergence in the scalar sector is cancelled, (iii) spin-0 scalar fields are naturally introduced, (iv) MSSM provides a well-defined ultraviolet picture of type-II two Higgs doublet model, (v) If R-parity is unbroken, the lightest SUSY particle (LSP) can be a dark matter (DM) candidate, and so on. Among such attractive motivations, we focus on the point (v). In the SM, there are several unsolved problems and one of the most serious problems is absence of * E-mail: shindou@cc.kogakuin.ac.jp the DM candidate. In the MSSM, all the SM particles are R-parity even and all the SUSY partner particles are R-parity odd, so that the lightest R-parity odd particle cannot decay. Therefore, unbroken R-parity guarantees the stability of the LSP which can be a DM. Several different candidates can be considered in the MSSM such as the neutralino, the gravitino, the axino, the saxion and the sneutrino. In this talk, we briefly review the analysis studied in Ref. [1] where a neutralino DM scenario is considered. In the neutralino DM scenario, the relic abundance of the LSP tends to be much more than the observed value. In order to realise the observed relic abundance of the DM, a mechanism to enhance the annihilation cross section of LSP is necessary. For example, following scenarios are sometimes considered: (i) neutralinos annihilate significantly through SU(2) gauge interaction, or (ii) annihilation cross section of Bino-like neutralino is enhanced with a particular mass spectrum of other associated particles. In the former class, one possible case is the Higgsino-like neutralino DM scenario with the mass of about 1 TeV. In this scenario, phenomenology such as the direct detection of DM, contribution to the EDMs, and collider signals have been studied in Ref. [2]. There is another possibility that a neutralino DM whose main component is Bino annihilates through heavy Higgs boson resonance [3-7]. We, here, focus on the second case. In this scenario, masses of the heavy Higgs boson are about twice of the mass of the neutralino DM. This Bino-like neutralino also contains small Higgsino component so that the neutralino can directly be searched through Higgs bosons exchange by the spin-independent scattering off nucleus [8]. We consider the MSSM with CP violating phases. In this case, the CP violating phases can significantly affect the electric dipole moments (EDM). Therefore the EDMs are powerful tools to explore the CP violating phases in the model. In this talk, we examine the electron EDM, the nucleon EDM, and the mercury EDM. CP phases can also contribute to the DM-nucleon spin-independent scattering cross section. Since the pseudo scalar exchange process is strongly suppressed in the non-relativistic limit, the spin-independent cross section is suppressed with a significant size of CP phase. 11.2 The benchmark of our analysis The superpotential and the soft SUSY breaking terms in the MSSM are given by[9] W = eab[(ye)ijHaLtbEj + (yd)ijHaQtbDj + (yu)ijH2a QbUj (11.1) - ^HaHb], and Lsoft =- MBB - MW«Wa - M23GAgA - m2H, H|aHa + m2H2 H?aH2a - q?La(M2 )tJq?L - ^a(M?)ij??L -iJ-iR(MUu- diR(M?)ijd?R - eiR(Mj2)ijgiR - £ab [(Te)ijHa£tbLejR + (TdlijHfqtbLdjR +(Tu)ijHaqbL-UjR + m2HaHb + h.c.] , (11.2) respectively. In the following, we ignore the Yukawa couplings except for the third generation quarks and leptons. Then yt, yb, and yT denote the Yukawa couplings of top, bottom, and tau, respectively. We also neglecting the flavor mixing in the soft SUSY breaking terms, we take flavor diagonal soft scalar masses as M?. = (M? )ii, M? = (M?)ii, M2. = (M2)ii, M2. = (M2 )ii, and M^. = (M?)ii. For the trilinear couplings, A parameters defined by (Tu)33 = ATyt, (Td)33 = ATyb, and (Te )33 = ATyT are used. Since we consider the CP violating case, the each parameter in the above superpotential and the soft SUSY breaking Lagrangian can be a complex number. The mass of the SM-like Higgs boson in the MSSM is calculated by the input parameters in the superpotential and the SUSY breaking Lagrangian. In order to reproduce the observed mass value mh = 125 GeV, we take tan |3 := (H2)/(H } = 30 and we fix the stop mass parameters as M?3 = 7 TeV, Mt := Mu3 = 7 TeV and At = 10 TeV. The other SUSY particles are irrelevant to the mass of the SM-like Higgs boson as well as the DM relic density. Therefore we can take their masses much heavier than stop. In such a case, they are decoupled from low energy observables. Here we take masses of the other sfermions as 100 TeV and M2 = M3 = 10 TeV. In our analysis, we focus on the Bino-like DM with the Higgs funnel scenario so that the heavy Higgs boson mass is close to twice the mass of the DM. In the scenario, the Bino-like neutralino rapidly annihilate through the heavy Higgs bosons resonance and the appropriate cosmic abundance for DM is reproduced. In addition, the masses of heavier neutral Higgs bosons, mH and mA, are close to the charged Higgs boson mass mH± in the MSSM. Thus we fix mH± to be twice of Bino mass parameter Mi. Note that the X-X-Higgs boson coupling depends on non-vanishing Higgsino component in the neutralino. We choose to reproduce the correct amount of DM relic density as nDMh2 = 0.1198 ±0.0015 [10]. As a consequence of these fact, both the Bino mass |M11 and the Higgsino mass should be of the order of TeV. We consider M1 as a free parameter and solve from the measured dark matter energy density. In the following, we summarise our benchmark parameter set: |M?| = |M3| = 10 TeV, M?,,2 = Mai,2 = Maii2,3 = M^,2,3 = Meli2,3 = 100 TeV, Mq3 = Mt = 7 TeV, At = 10 TeV, mH± = 2M1, tan | = 30. (11 3) (11 4) (11 5) (11 6) (11 7) (11 8) The other A-terms are zero. With this parameter set, CP phases in the five parameters, M1, M2, M3,At), may be relevant to our analysis of EDMs and the spin-independent cross section. The CP phases of these parameters are described as $Ml, $M2 , $M3 , $At), respectively, where each phases of a quantity X are defined by X = |X|ei*X. Note that some of those CP phases are unphysical. It is known that there is a rephasing degree of freedom in the MSSM. Actually, all the physical quantities are described by the following combinations of the parameters, By using the rephasing degree of freedom, without loss of generality, we can take the basis of CP phases as $M3 = 0. We also take $At = 0 for simplicity. In general, the CP phase $At also significantly contributes to the predictions of the EDMs. However, in our benchmark parameter set given in Eqs. (11.3) - (11.8), the contribution from $At is strongly suppressed because the mass splitting between two stops is small. Therefore we scan the following four parameters, 11.3 Numerical analysis In calculations of dark matter thermal relic density and the Higgs mass, we use micrOMEGAs 4.3.5 [11] with CPsuperH2.3 [12]. The Higgs mass is almost fixed to be 125 GeV in our benchmark point. When we scattered the parameters, we pick up the parameter sets which reproduce the correct DM relic abundance and the correct Higgs mass. Then we calculate the electron EDM, the neutron EDM, and the mercury EDM. We also discuss the scattering cross section for the direct detection experiments. Since the sfermions are too heavy to contribute to the EDMs via one-loop diagrams, the two-loop Barr-Zee diagrams provide dominant contributions unless Wino, stop, and sbottom masses are heavy enough to be decoupled. In Fig. 11.1, we show our numerical results. We can see the M1 dependence by comparing the left panels and the right panels where M1 = 1 TeV and 2 TeV, respectively. It is easily seen that larger M1 weaken the constraint from EDM experiments. The Bino mass M1 is approximately identified to be the mass of the dark matter neutralino. Then for larger M1, heavy Higgs bosons and Higgsinos become heavier, and the contributions to the EDMs become smaller. We also discuss the ^ and $M2 dependence of the EDMs. The left panels in Fig. 11.1 shows the electron EDM, the mercury EDM, and the neutron EDM with $Ml = 0. The shaded regions are already excluded by the current upper bound on the EDMs. We find the combination of the electron EDM and the mercury EDM exclude the arg(MiMJ) , arg(MiA^) , arg(^Mi) , arg(^At) , (i,j = 1,2,3) . (11.9) (IM1I, $ (11.10) large region of the parameter space. Both ^ and cannot be large. We also find that the electron EDM strongly depends on . On the other hand, dependence of the mercury EDM and the neutron EDM are milder. Fig. 11.2 displays the dependence. Taking into account the constraint from the mercury EDM, we find that the mercury EDM and the neutron EDM are almost independent of . On the other hand, the dependence of the electron EDM on is mild but visible. From these figures, one can see that the neutron and the mercury EDMs are sensitive to and also weakly depend on . On the other hand, the electron EDM is sensitive to + and weakly depend on . Most of the parameter space in Figs. 11.1 and 11.2 are within the future prospects of the electron EDM and the neutron EDM. In Summer of 2018, the constraint on the electron EDM is updated to be |de/e| <1.1 x 10-29e cm by ACME collaboration[13]. With this new constraint, the allowed regions in Figs. 11.1 and 11.2 become very thin stripes. Thus the correlation among the EDMs in future experiments provide a strong hint to explore the CP phases in the SUSY breaking sector. Let us discuss DM-nucleon scattering cross section. Since we consider the Higgs funnel scenario, the DM neutralino couples to neutral scalar bosons. Through these couplings, the DM neutralino and nucleon interact with each other. Though the couplings are rather small in the scenario, the couplings lead to a significant size of the spin-independent cross section and it will be within future prospects of the DM direct detection experiments. In Figure 11.3, the and ^ dependence of ctsi is shown. In this figure, the parameter choice is the same as in Fig. 11.1. Figure 11.4 displays the and ^ dependence of o$i with the same parameter choice as in Fig. 11.2. The spin-independent cross section is found to be smaller than the current upper bound [14-16] in all the region of the parameter space. However is is within the future prospects of the DARWIN [17], the DarkSide-20k [18], and the LZ [19]. Note that the scattering cross section depends on + and the dependence is not significant. Fig. 11.1. The EDMs for tan ( = 30 and = 0°. The left (right) panels are for Mi = 1 TeV (Mi = 2 TeV). The contours in the top, the center, and the bottom panels are those of the electron EDM, the mercury EDM, and the neutron EDM, respectively. The dashed lines show the negative values. The red and blue shaded regions are excluded by the electron EDM and the mercury EDM, respectively. The figures are taken from Ref. [1]. Fig. 11.2. The EDMs for Mi = 1 TeV and tan P = 30. In the left (right) panels, <£m2 = 0° (30°). The shadings and contours are the same as in Fig. 11.1. The figures are taken from Ref. [1]. Fig. 11.3. The DM-nucleon scattering cross sections for tan p = 30 and = 0°. The left (right) panel is for |Mi | = 1 TeV (2 TeV). The shadings are the same as in Fig. 11.1. The figures are taken from Ref. [1]. Fig. 11.4. The DM-nucleon scattering cross sections for Mi = 1 TeV and tan p = 30. The left (right) panel is for |^m2 | = 0° (30°). The shadings are the same as in Fig. 11.1. The figures are taken from Ref. [1]. 11.4 Summary In this talk, we have considered the MSSM with CP phases, and we have focused on a DM scenario where the Bino-like neutralino is a DM whose annihilation cross section is enhanced enough through heavy Higgs bosons exchange so that the observed relic abundance of the DM can be explained. In this benchmark scenario, we have examined several EDMs and the spin-independent cross section for DM scattering with nuclei. We have shown that the EDMs are very powerful tool to explore the parameter space in the MSSM with CP phases even when most of the SUSY particles are very heavy. Acknowledgements This work was supported by JSPS KAKENHI Grant Number 17H05408 and Ko- gakuin University Grant for the project research. References 1. T. Abe, N. Omoto, O. Seto and T. Shindou, Phys. Rev. D 98 (2018) no.7, 075029. 2. N. Nagata and S. Shirai, JHEP 1501, 029 (2015) [arXiv:1410.4549 [hep-ph]]. 3. M. Drees and M. M. Nojiri, Phys. Rev. D 47, 376 (1993) [hep-ph/9207234]. 4. H. Baer and M. Brhlik, Phys. Rev. D 53, 597 (1996) [hep-ph/9508321]. 5. H. Baer and M. Brhlik, Phys. Rev. D 57, 567 (1998) [hep-ph/9706509]. 6. V. D. Barger and C. Kao, Phys. Rev. D 57, 3131 (1998) [hep-ph/9704403]. 7. J. R. Ellis, T. Falk, G. Ganis, K. A. Olive and M. Srednicki, Phys. Lett. B 510, 236 (2001) [hep-ph/0102098]. 8. G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. 267, 195 (1996) [hep-ph/9506380]. 9. B. C. Allanach et al, Comput. Phys. Commun. 180, 8 (2009) [arXiv:0801.0045 [hep-ph]]. 10. P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. 594 (2016) A13 doi:10.1051/0004-6361/201525830 [arXiv:1502.01589 [astro-ph.CO]]. 11. D. Barducci, G. Belanger, J. Bernon, F. Boudjema, J. Da Silva, S. Kraml, U. Laa and A. Pukhov, Comput. Phys. Commun. 222, 327 (2018) [arXiv:1606.03834 [hep-ph]]. 12. J. S. Lee, M. Carena, J. Ellis, A. Pilaftsis and C. E. M. Wagner, Comput. Phys. Commun. 184,1220 (2013) [arXiv:1208.2212 [hep-ph]]. 13. V. Andreev et al. [ACME Collaboration], Nature 562 (2018) no.7727, 355. doi:10.1038/s41586-018-0599-8 14. D. S. Akerib et al. [LUX Collaboration], Phys. Rev. Lett. 118 (2017) no.2, 021303 doi:10.1103/PhysRevLett.118.021303 [arXiv:1608.07648 [astro-ph.CO]]. 15. E. Aprile et al. [XENON Collaboration], Phys. Rev. Lett. 119 (2017) no.18, 181301 doi:10.1103/PhysRevLett.119.181301 [arXiv:1705.06655 [astro-ph.CO]]. 16. X. Cui et al. [PandaX-II Collaboration], Phys. Rev. Lett. 119 (2017) no.18, 181302 doi:10.1103/PhysRevLett.119.181302 [arXiv:1708.06917 [astro-ph.CO]]. 17. J. Aalbers et al. [DARWIN Collaboration], JCAP 1611, 017 (2016) [arXiv:1606.07001 [astro-ph.IM]]. 18. C. E. Aalseth et al, Eur. Phys. J. Plus 133, no. 3,131 (2018) [arXiv:1707.08145 [physics.ins-det]]. 19. D. S. Akerib et al. [LUX-ZEPLIN Collaboration], arXiv:1802.06039 [astro-ph.IM]. Discussion Section The discussion section is reserved for those open problems presented and discussed during the workshop, that might start new collaboration among participants or at least stimulate participants to start to think about possible solutions of particular open problems in a different way, or to invite new collaborators on the problems, or there was not enough time for discussions and will hopefully be discussed in the next Bled workshop. Since the time between the workshop and the deadline for contributions for the proceedings is very short and includes for most of participants also their holidays, it is not so easy to prepare there presentations or besides their presentations at the workshop also the common contributions to the discussion section. However, the discussions, even if not presented as a contribution to this section, influenced participants' contributions, published in the main section. Contributions in this section might not be yet pedagogically enough written, although they even might be innovative and correspondingly valuable indeed. As it is happening every year also this year quite a lot of started discussions have not succeeded to appear in this proceedings. Organizers hope that they will be developed enough to appear among the next year talks, or will just stimulate the works of the participants. There are seven contributions in this section this year. One contribution discusses shortly a possible influence of the "dark atom"s on the expanding universe, offering the explanations for some puzzles in the experiments trying to detect dark matter. "Dark atom" is the "atom", which contains a double electromagnetically charged "baryon" made of three stable ill quarks, decoupled from the three observed families of quarks. A bound state of an ordinary He nucleus with such a "baryon" would made the "dark atom". The elaboration of this idea looks very interesting. There is the contribution, which is pointing out that inconsistency between the theoretical predictions and the experimental data is not necessarily a signal for new physics, since it can just be due to the higher order corrections not included in the theoretical evaluations. The authors, discussing several cases, conclude that there might not be yet any experimental data, which could be interpreted as a signal of new physics beyond the standard model. And yet we all hope that the new data, either the cosmological ones or, hopefully, also the LHC or of other experiments ones, will confirm the theory beyond the standard models, like it is the spin-charge-family theory, since there are (so many) assumptions in the standard model, which ought to have explanations. The contribution presenting the Dirac operators Ya and Sab, and of the operators Ya and the corresponding Sab, determining in the spin-charge-family theory — first the family members quantum numbers of fermions, and the second the family quantum numbers of fermions — was stimulated by participants of this year workshop. In the basic states in (3 + 1) (out of d = (13 + 1)), the matrices have the dimension 16 x 16. The contribution is to make it easier for the reader to recognize the differences between the quantum numbers describing the family members and those describing families. The contribution discussing some representations of the second quantisable integer spin fermions is meant to recognize better the differences between the fermions with the internal degrees of freedom described in Clifford space (with the spins and charges in fundamental representations of the groups, the subgroups of the Lorentz group) and the fermions with the internal degrees of freedom described in Grassmann space (with the spins and charges in the adjoint representations of the subgroups of the Lorentz group), presented in this proceedings in the talks section. If consisting of the integer spins fermions (only), nuclei, atoms, molecules ... would in such an universe be of completely different kind. Nature has obviously "made a choice" of the Clifford space. There is the contribution, which represents the improvement of the by the author proposed model with the broken SU(3) gauged family symmetry. It reports on the parameter space region, in which all the results are in agreement with so far observed data. The mass of the SU(2)L weak singlet vector-like D quark, proposed in this theory, may be of the orden of 10 TeV. There are two contributions in which the author constructs, while recognizing the correspondence between the Clifford algebra states as represented in the spin-charge-family theory and the binary codes, the geometrical model with closed packed cells of two different shapes representing quarks and leptons with their observed charges. Author tries to extract out of these cells, recognizing different possible symmetries, even forces among these constituents. Although the author has almost incredible recognitions, yet it is very questionable what one can learn out of such a model, especially when one would like to look beyond the standard model to understand the origin of properties of fermion and boson fields, and in the author's case, what does determine assumptions and parameters of his model. All discussion contributions are arranged alphabetically with respect to the authors' names. Diskusije Ta razdelek je namenjen odprtim vprašanjem, o katerih smo med delavnico razpravljali in bodo morda privedli do novih sodelovanj med udeleZenci, ali pa so pripravili udeleZence, da razmislijo o moZnih rešitvah odprtih vprašanj na drugačne načine, ali pa bodo k sodelovanju pritegnili katerega od udeleZencev, ali pa ni bilo dovolj casa za diskusijo na doloceno temo in je upati, da bo prišla na vrsto na naslednji blejski delavnici. Ker je cas med delavnico in rokom za oddajo prispevkov zelo kratek, vmes pa so poletne pocitnice, je zelo tezko pripraviti prispevek in se tezje poleg prispevka, v katerem vsak udeleženec predstavi lastno delo, pripraviti še prispevek k temu razdelku. Tako se precejšen del diskusijne bo pojavil v letosnjem zborniku. So pa gotovo vplivale na prispevek marsikaterega udeleženca. Nekateri prispevki se morda niso dovolj pedagosško napisani, so pa vseeno lahko inovativni in zato dragoceni. Organizatorji upamo, da bodo te diskusije do prihodnje delavnice dozorele do oblike, da jih bo mogocše na njej predstaviti. Letos je v tem razdelku sedem prispevkov. Eden prispevek obravnava na kratko motnost vpliva "temnih atomov" na razvoj vesolja in ponudi razlago nekaterih ugank v poskusih, ki naj bi merili temno snov. "Temni atom" je atom, ki vsebuje "barion" z dvojnim elektromagnetnim nabojem in ga tvorijo trije stabilni kvarki U. Ti so neodvisni od treh ze poznanih druzin kvarkov. Vezano stanje helijevega jedra s takim "barionom" bi tvorilo "temni atom". Obravava te ideje se zdi zanimiva. Avtorja pokazeta, da neujemanje teoreticnih napovedi z meritvami se ne pomeni nujno, da je to signal za novo teorijo, ki preseke standardni model elek-trosšibke interakcije, ker so neujemanja lahko tudi posledica tega, da pri teoreticšnih izracunih niso vkljuceni popravki dovoljvisokih redov. Obravnavata vec primerov in skleneta, da po njuno doslej še ni meritev, ki bi jih ne bilo mogoce pojasniti s standardnim modelom. In vendar vsi upamo, da bodo bodisi kozmoloske meritve bodisi meritve na pospesevalniku LHC ali na drugih pospeševalnikih kmalu potrdile pravilnost teorij(e), kot je denimo teorija spinov-nabojev-druzin, saj standardni model s svojimi 30 privzetki nima razlage za vse te privzetke. Prispevek, ki obravnava Diracove operatorje Ya in ustrezne Sab, ter Ya in ustrezne Sab, ki dolocajo v teoriji spinov-nabojev-druzin prvi spine in naboje fermionov, drugi družinska kvantna stevila fermionov, so spodbudile razprave udeležencev letošnje delavnice. Matrike imajo na baznih stanjih v prostoru (3 + 1) (ki je vključen v d =(13 + 1)) dimenzijo 16 x 16. Namen prispevka je olajšati bralcem, da prepoznajo razliko med kvantnimi števili, ki opišejo člane družin in kvantnimi števili, ki opišejo družine. V razdelku je tudi prispevek, ki razpravlja o nekaterih upodobitvah fermionov s celostevilčnim spinom v drugi kvantizaciji, kar najpomaga bolje razumeti razlike med fermioni, katerih notranje prostostne stopnje opišemo v Cliffordovem prostoru (spini in naboji so v tem primeru v fundamentalnih upodobitvah grup, ki so podgrupe Lorentzove grupe), ter fermioni, katerih notranje prostostne stopnje so opisane v Grassmannovem prostoru (spini in naboji so tem primeru v adjungiranih upodobitvah podgrup Lorentzove grupe). Diskusija je povezana s prispevkom v razdelku predavanj v tem zborniku in obravnava primer, v katerem bi imeli fermioni le celostevilcni spin. Taka izbira bi vodila do popolnoma drugacnih jeder, atomov, molekul____Narava je ocitno izbrala Cliffordov prostor. Prispevek, v katerem avtor predlaga model z zlomljeno druzinsko simetrijo SU(3), obravnava obmocje parametrov, ki zagotovi ujemanje modela z izmerjenimi podatki. Maso napovedanega novega kvarka oceni na ~ 10 TeV. Razdelek vsebuje dva prispevka, v katerih postavi avtor, upoštevajoc zveze med stanji Cliffordove algebre kot jih predstavi teorija spinov-nabojev-družin in binarnimi kodami, geometrijski model, v katerem so kvarki in leptoni ter njihovi naboji predstavljeni s tesno zlozzenimi celicami dveh oblik. Iz tega geometrijskega modela poskusša avtor izpeljati z uposštevanjem mozšnih simetrij, ki jih ponudi model, lastnosti in celo sile med sestavnimi delci, to je kvarki in leptoni. Avtorjev pristop je neverjetno domiselen, saj vse lastnosti osovnih delcev in polj pripisše geometrijskim lastnostim modela. Vprasanje pa je, kajse lahko naucimo iz takega modela, kjer parametre modela doloca geometrija, zlasti, ce zelimo razumeti od kod lastnosti fermionskih in bozonskih poljter v avtorjevem primeru, kajdoloca parametre modela. Prispevki v tej sekciji so, tako kot prispevki v glavnem delu, urejeni po abecednem redu priimkov avtorjev. Bled Workshops in Physics Vol. 19, No. 2 A Proceedings to the 21 st Workshop What Comes Beyond ... (p. 261) Bled, Slovenia, June 23-July 1, 2018 12 On Triple-periodic Electrical Charge Distribution as a Model of Physical Vacuum and Fundamental Particles E.G. Dmitrieff * Irkutsk State University, Russia Abstract. In this study we consider triple-periodical electrical charge distributions with the pattern similar to the Weaire-Phelan structure. According to it, the space is splitted to opposite-charged cells separated with electrically neutral border. Possible configurations obtained as results of exchanges of these cells appear to have properties that can be corresponded to the quantum numbers of known fundamental particles. We find it promising to use models of this kind, aiming to infer the axioms and constants of the Standard Model from the emergent geometrical properties of the distribution. Povzetek. Prispevek obravnava trojne periodične porazdelitve električnih nabojev, ki imajo vzorec podoben Weaire-Phelan strukturam. V modelu je prostor razdeljen na celice z nasprotnimi naboji, ki jih loci elektricno nevtralna meja. Konfiguracije, ki sledijo z izmenjavo teh celic, imajo lastnosti, ki jih avtor poveze s kvantnimi stevili kvarkov in leptonov. Avtor meni, da ti modeli omogocijo izpeljavo privzetkov in konstant standardnega modela. Keywords: Particle model,Weaire-Phelan tessellation 12.1 Introduction The spin-charge-family theory presented in [1], [2], [8], [9] offers reasonable explanations for the phenomena of the Standard Model of the fundamental particles. Originating from Clifford algebra, it comes to the binary internal degrees of freedom, explaining properties of existing fundamental particles and predicting existence of extra fermion families. In turn, we reproduce particle properties starting with binary code model. As we have shown in [7], Boolean models designed for fundamental particles can reproduce most of their properties, including charges (electrical, color, weak and hyper-charge), lepton- and baryon numbers, fermion flavor and family membership, and boson spin magnitude. The particles are represented as combinations or codes of symbols carrying one of two possible values, so these models are binary. * E-mail: eliadmitrieff@gmail.com Developing these models, we started with well-known linear codes, that consist of binary digits (bits) with usual values either 1 or 0. Then, in order to reduce the amount of information carried by the code, we abandoned the linear structure in favor of spatial one. Also we have symmetrized and normalized the values carried by bits, using + 6 and -6 instead of 1 and 0. These values could be directly interpreted as electrical charge in units of electron charge e. Using spatial combination of eight symbols of this kind, we managed to represent all known fundamental particles. Also, analyzing unused combinations, we proposed existence of new scalar particle forming the vacuum condensate. It could be represented by this combination that is repeated periodically, filling the space as a tessellation. Since the tessellation can be chiral, the space filled with small alternating charged regions, comparing to simple empty space, has an advantage of offering possible explanation for difference between right- and left-handed particles in respect of the vacuum. Different particle codes, substituting vacuum codes in the tessellation, violate the periodicity with different ways. We suppose that it may be used to infer associated rest energies (masses) instead of postulating them. Treating vacuum expectation value as Coulomb potential between neighboring opposite-charged "bits" [11], we estimated that the distance between them should be on scale of« 10-21 m. Being inspired by idea of vacuum domains [3], we suppose that the interpretation of these "bits" as domains can explain the problem of their observations absence. As asserted originally by Zeldovich with co-authors, the vacuum domains should appear as consequence of symmetry break in the phase transition. In our models, they do exist but have the correlation radius on sub-particle scale instead of cosmological one. This should happen in case the 2-order phase transition is not yet complete but just approaches its critical point. Having a model with some spatial distribution of charged bits, or vacuum domains, we recognize that it is necessary to find out the pattern of this distribution which is consistent with other observable properties of vacuum and particles, including their symmetry, mass spectrum, propagation, interactions and so on. After checking simple (NaCl-like) and volume-centered (CsCl-like) cubic lattices, we found out that the A15 (Nb3 Sn-like) lattice, or Weaire-Phelan structure, has some advantages allowing it to be the possible vacuum- and particle model. 12.2 Overview of the original Weaire-Phelan tessellation The original Weaire-Phelan structure is described in [12]. It is a foam of equal-volumed cells separated by thin walls. Among other structures, having the same cell volume, this one has the minimal (known at the present time) inter-cell wall area, so it is a candidate solution for the Kelvin problem [14]. There is evidence of self-assembling of this tessellation driven by minimization of the surface energy [13]. Cells forming the Weaire-Phelan structure have almost flat faces and just slightly curved edges, thus they can be closely approximated by irregular polyhe- dra. It is nesessary to use two kinds of them - dodecahedra (D) and tetrakaideca-hedra (T)*. Unlike dodecahedra, the tetrakaidecahedra have three possible orientations in respect of the three Cartesian axes. The cells of both kinds can be included in the tessellation in two ways, so they became chiral. Eight cells, differing in kind, chirality, and orientation, form one translation unit. These translation units, in turn, form simple cubic grid. Assuming the size of translation unit to be 1 = 4A in each dimension (where A is a scale factor, and 4 is used to get most of coordinates integer), we get the unit volume Vu = 13 = 64A3, and cell volume Vc = 8 Vu = 8A3 (remembering that all cells are equal-volumed). Having the coordinate axes perpendicular to the hexagonal faces of the tetrakaidecahedra, and associating the origin with the center of one of dodec-ahedra, one can obtain coordinates of the centers of all other cells: D TX Ty (0,0,0) (0,2,1) (1,0,2) (2,1,0) (2,2,2) (0,2,3) (3,0,2) (2,3,0) These coordinates are expressed in units of À and derermined up to 4À, meaning that one can obtain coordinates of each cell by adding of "even" vector Ve = (4nx, 4ny, 4nz) À, n G Z. (12.1) Further, we omit the scale factor À where it shouldn't cause misunderstanding. Here we chose the R and L mark of the chirality by the arbitrary choice. There are four symmetry axis C3 defined by equations ±x = ±y = ±z. Since the structure does not possess reflection symmetry, it is chiral, so there are two mirror-reflected structures. For instance, after reflecting in the plane x = y the chirality is reversed and the coordinates are changed as the following: D Tx Ty Tz (0,0,0) (2,0,1) (0,1,2) (1,2,0) (2,2,2) (2,0,3) (0,3,2) (3,2,0) After performing the shift (move) of the whole infinite structure with the "odd" vector Vo = VE + (±2, ±2, ±2)À, (12.2) 1 The dodecahedron is a pyrithohedron with twelve equal pentagonal faces, possessing terrahedral symmetry Th, and the tetrakaidecahedron is truncated hexagonal trapezohe-dron posessing rotoreflextion symmetry C3h, with two hexagonal faces, four large and eight small pentagonal faces. Tz i.e. for the half-size of the translation unit (2%/3A), in direction of C3 axis, we get the original structure again: D Tx Ty Tz (2,2,2) (2,0,3) (2,3,0) (1,2,0) (0,0,0) (o,2,l) (2,1,0) (l,0,2) Thus, the structure possesses global PS symmetry, where P is parity (particularly, exchange of any two coordinates) and S is shift along C3 axis on the half of translation unit size (or, generally, on the odd vector). It also means that despite of mirror asymmetry of each finite part, there is only one infinite Weaire-Phelan structure, which is either right- or left-handed depending on the choice of origin. It can be also considered as two overlapped chiral structures consisting of the same elements but shifted in respect of each other with the odd vector (12.2). 12.3 Dual-charged Weaire-Phelan structure To use the Weaire-Phelan structure as a spatial version of binary-code model, we need to assume that each cell carries electrical charge with magnitude of 6. Since the space containing no particles is electrically neutral, the counts of positive and negative cells in any volume > I3 should be the same. Any change of cell charge, that can be from + 6 to " 1, or back, would cause the total electric charge to change on ± 1. Thus, all the particles in this model will have discrete charges with step of 3, that is according to experiments. So the existence of particles with charges, for instance, of ± 2, is impossible. In general, the charge inside cells can be distributed being determined by physical law acting on this scale, for instance: • all the charge can be concentrated in cell centers, in point-size sub-particles (partons or rishons); • the charge can be distributed smoothly inside cells around their centers, falling to zero on the inter-cell borders; • the charges of opposite sign can be concentrated on both sides of the walls between opposite-charged cells, and also can be smoothly distributed along them. In the following subsections we consider these simplified assumptions of the charge distribution. We assume that the basic "vacuum" alteration of charged cells in the tessellation should fulfill the following requirements: • each translation unit should be electrically neutral, and • cells with opposite chirality should also be opposite-charged. So, we assume positive charge of cells of one chirality and negative for another. However, at this stage we do not recognize any natural rule that would define the absolute chirality. So, there are 23 = 8 choices of Tt charges and also 2 choices of D. We make this choice as shown in the following table: Cell type Charge Coordinates Plane Dr - (0,0,0) x + y + z = 4n Tri + (0,2,1), (1,0,2), (2,1,0) x + y + z = 4n + 3 Dl + (2,2,2) x + y + z = 4n + 2 Tli - (0,2,3), (3,0,2), (2,3,0) x + y + z = 4n + 1 In the last column of the table we show the equations of planes that contain all the cell centers of particular type. Making the choice of charge sign for Tt, we break the symmetry between C3 axis, so one of them becomes dedicated. Also, making this choice for D cell charge breaks symmetry between opposite handednesses. So there are two possible dual-charged Weaire-Phelan structures. That corresponds to the principal possibility of physical vacuum with reversed chirality. 12.4 Cell Centers approximation Here we abstract from the details of spatial distribution of the electric charge, and suppose it is just concentrated somewhere in the vicinity of the cell centers. We do so to simplify the charge calculation, replacing the integration of the charge density in the volume of interest with counting the number of centers of positive and negative cells falling into it. Since the coordinates of the cell centers are integers (i.e., proportional to the scale factor A), they can lay on the certain planes only, between which, in this approximation, there is nothing. 12.4.1 CPS symmetry The following set of grids (Fig. 12.1) illustrates the placement of positive and negative cells' centers, as black and white circles, respectively, in the cubic translation unit of size 4 x 4 x 4 starting with its left bottom front corner from the origin of reference frame. Centers of D-cells are marked with double-border. The first grid is the cross-section for plane z = 0, the second one is for plane z = 1 and so on. The plane z = 4 is the same as z = 0 due to the periodicity. Considering the translation unit cube that is shifted with the "even" vector (2n + 1 )(2,2,2), for instance (-2, -2, -2), i.e. performing S operation, we get the scheme on the Fig. 12.2 (the first grid is plane z = — 2 and so on). After reflecting in the plane x = y (P operation) we get the scheme on the Fig. 12.3. One can ensure that this shift operation (S) followed by reflection (P) has the same result as the charge inversion (C). So these three operations being applied ; = 0 4 = 0 f. —( 1 1—tj p—1 —( —f - 0 1 2 3 4 x Fig. 12.1. The placement positive- and negative-charged cell centers in the translation unit 2 3 y r-< ^33J _-4. Due to the CPS symmetry, we also have T rr 9 > 1 qenv(Z< ^3J _+ 4; T rr 9 > 1 q env (Z > J _+ 4 for the environment of the negative-charged small T-triangle at Z _ 9/%/3. (12.6) (12.7) 12.4.6 Handedness change as Exchange of D triangles Although D-sublattice has no influence on the total charge of the small T+triangle's environment qenv (12.5), exchanges in it can redistribute the electric charge between rear (Z < -==) and front (Z > -==) half-spaces because it is asymmetric in respect of the plane (Z = 3/%/3). We examine such exchanges whether they can be used to represent the particle's handedness that also does not influence on its charge. Following the model that assumes the charge is located closely to the cell centers, we must conclude that D triangles just before and after small triangle at Z = 3/%/3 have the charges 2 1 Iq2 = qD(Z = —: )=+ f 1 (12.8) lq4 = qD(Z = ^ )=- 2, and in case they exchange, the charge of 1 will redistribute from rear half-space to the front one4. However, considering the case when the charge of cells is not concentrated in their centers, being instead distributed on radii comparable to the inter-centers distance (« 2... %/5 « 2.236), we recognize that the charge of cells in plane Z = 2/%/3 would not reside just before the plane Z = 3/%/3. It is so because the offset between these planes is significantly less than the inter-center distance: 1/%/3 « 0.577 < 2 (Fig. 12.8), and is comparable with the distribution radius. That is why one should assert that some part of q2 would reside after the plane Z = 3/%/3, and some part of q4 would, in turn, reside before it (see Fig. 12.8). To be the representation of the reversed handedness, the D-exchange operation should redistribute only the half of the charge (12.8). This requirement is fulfilled in case a quarter of the charge of each D-cell in triangles is distributed on the other side of the plane Z = 3/%/3 that is located at the 1/\/3 of its center. So we can use this condition to obtain more realistic rule of the charge distribution rather than simple charged point in the cell center. Now we use the halved values of (12.8), that are equal to qenv by the magnitude, so they can effectively compensate them: "D'(i="3 >=- 4, (12.9) In this case, 3 3 2 11 (Z < —) = qlnvIZ < —) + qD*lZ = —) = -4 + 1 = 0; I3 f f 4 1 1 (12.10) '(Z> "3 ] = q^nv|Z> "3 )+qD*(Z="/3 }=-4-1=-2, Such an exchange also can be considered as a rotation of a spatial hexagon containing all six cell centers of the both D-triangles, with the angle of 60° in any direction q q Fig. 12.8. Visually overlapping cells of two D triangles with Z = 2/V3 and Z = 4/V3 in polyhedral approximation and after exchange between D-triangles at Z = 2/%/3 and 4/%/3 they would turn into 3 3 2 1 1 1 qenv(Z < ^) = q!nv(Z < ^) + qD*(Z = ^) = -4 - 4 = -2; 3 3 4 11 (12.11) qenv(Z > ) = q!nv(Z > ^3) + q°*(Z = ^3) = -4 + 4 = 0. So we can use them to represent weak isospin and weak hypercharge for "down" particles: 31 T3down = qenv(Z < ^3) = 1 (Iq1 + Iq2) (12.12) 31 YWown/2 = Iq3 + qenv(Z > ^3) = Iq3 + 2 (Zq4 + Iq5) (12.13) At Z = 9/\/3, the small T-triangle and its neighborhood are inverted in respect to Z = 3/%/3 due to the CPS symmetry, so original qenv(Z < ^3) = qTnv(Z < ^3) + qD*(Z = ^3) = + 4 - 4 = 0; 9 > T „ 9 . D*,r 10 , 11 1 q env (Z > ^3) = qlnv (Z > ^3) + qD*(Z = ^3) = + 4 + 4 = + 2 would turn after D-exchange into (12.14) 9 9 8 1 1 1 qenv(Z < ^3) = q!nv(Z < ) + q°*(Z = ^3) = + 4 + 4 = + ^ 9 9 10 1 1 qenv(Z > ^3) = q!nv(Z > ^3) + qD*(Z = ^) = +1 - 4 = 0, (12.15) and both the values qenv(Z < -73) and £q9+qenv(Z > -93) would, again, coincide with weak isospin T3 and weak hypercharge YW/2 for "up" fermions, respectively: 9 1 T3up = qenv(Z < —) = 2C^qz + Zq8) (12.16) YWP/2 = Iq9 + qenv(Z> —) = Iq9 + 1 (Iqio + Iqn) (12.17) 1 So the exchange between D triangles (or, that is the same, rotation of the distorted D hexagon) can be used as a model representing switching between two handednesses. 12.4.7 Down fermions as Inversions in small T+triangle Inverting charges of cells it the small T+triangle q3, namely of q (2,1,0), q (2,1,0) and q(2,1,0) in (x,y,z) reference frame5, one can get eight possible cases (Table 12.2). The total electric charge Q, that changes with steps of ± 1 according to the count of inverted cells, coincides with the electric charge of eight "down"6 fermions. T3 := q«3) q310 q302 q321 £q3 q(>3) YW 2 : q(>3) Q symbol + + + +1/2 0 0 - + + dR1 + — + s* +1/6 —1/3 —1/3 dR2 2 + + — > © nx 00 2 x@ H dR3 + — — uf1 0 — + + So —1/6 —1/2 —2/3 —2/3 uf2 uf3 — — — §0 —1/2 —1 —1 1- Table 12.2. Eight cases of inversions in the small T-triangle at Z = 3/V^ with original (unchanged) D-triangles at Z = 2/\/3 and 4/\/3, associated with weak-uncharged "down" fermions The original unchanged state with Q = 0 is the vacuum state, so it takes place of the left-handed anti-neutrino, that, according to experiments, does not exist. In 5 In the (£,, Z,u) reference frame they are q(V2, V3, 0), q(-—, V3, -—), q(--j, V3, -g)). 6 We consider anti-"up" fermions as "down" ones, and vice versa. The "up" particles as well as "up" (anti-"down") anti-particles have the electric charge greater by 1 then the charge of corresponding "down" particles or anti-particles. T3 := q(<3) q310 q302 q321 £q3 q (>3) YW 2 :_ q(>3) Q symbol + + + +1/2 +1/2 0 Vr - + + dL1 + - + s* +1/6 +1/6 —1/3 dL2 2x °9) YW/2 := q(>9) Q + + + •I +1/2 + 1 +1 1+ - + + uR1 + — + •2 +1/6 +2/3 +2/3 uR2 1 v • 2 *©• © + + — •s 2 ^T > > UR3 + — — ® dL1 0 — + + —1/6 +1/2 +1/3 +1/3 dL2 dL3 — — — o°o —1/2 0 0 ED + + + •I +1/2 +1/2 +1 — + + uL1 1 2 X®»% + + + + •2 •S +1/6 > +1/6 +2/3 uL2 uL3 + — — @ dR1 +1/2 — + + —1/6 0 —1/6 +1/3 dR2 dc3 dR — — — —1/2 —1/2 0 VL Table 12.4. Eight cases of inversions in the small T-triangle at Z = 9/v^3, repeated twice with original and exchanged D-triangles at Z = 8/V3 and 10/associated with "up" fermions Since D cell has 6 equal-charged and also 6 opposite-charged neighbors, the inversion does not affect the area (AS = 0) and AEo = 0. (12.19) In contrast, among 14 neighbors of T cell six ones are equal-charged but there are eight opposite-charged ones. Both opposite-charged neighbors that become equal-charged ones in an inversion, are separated with the hexagonal faces with area S6. So AEt = 2ApS6. (12.20) Assuming the energy density for wall between equal-charged cells is greater than for opposite-charged ones, Ap > 0 and AE > 0. In case of inversions of two neighboring cells, there is an additional effect caused by their common face. In case two neighbor cells exchange their charge (thus, they are D and T touching each other with large pentagonal face S5L or two T touching each other with small pentagonal face S5s or hexagonal one S6) the common face remains separating opposite-charged cells, instead of being turned into separating equal-charged cells, so energy effect is negative: aeq^T = -2ApS5L, aeT ^T5 =-2ApS5s, (12.21) aeT ^T6 = -2ApS6. In case of two neighbor cells inverting in the same direction, the additional effect of the common face is opposite, i.e. positive: aeQ^T = 2ApS5L, aeT ^T5 = 2ApS5s, (12.22) aeT ^T6 = 2ApS6. Note that numerical values of the faces' areas (in units of A2) are such that S5L is almost equal to the arithmetic mean of S6 and S5s: S5L S5s S6 S6 + S5s - 2S5L Now we can build the simple hierarchical seesaw model of mass based on addition and subtraction of energy effects. • Since D exchanges have AE = 0, massless particles like photon and neutrino must be associated with D-only exchanges. • Following our 8-bit model [7], associate W+ boson with five defects combination shown on Fig. 12.9W. Note that it is colorless and has correct electric charge Q = +1. The affected area of these defects is 1.77477, 1.15338, 2.41260, so 0.0164. (12.23) Fig. 12.9. Models of W+ and Z0 bosons in polyhedral approximation ASW = 6 (S6 + S5L) = 25.12422. (12.24) Using experimental value of mW = 80.385GeV we get Ap = "mW « 3.1995 GeV/A2. (12.25) AS W • Following the same way, we associate Z0 boson with neutral six T defect configuration shown on Fig. 12.9Z. Using the same Ap value, we get mZ = 12ApS6 « 92.629 GeV. (12.26) • The Higgs boson having, accordingly to 8-bit model, the defects structure similar to Z boson but with one additional D defect pair (Fig. 12.9H), must have one of D cells isolated the same way as W has, to get the appropriate mass: mH = Ap(12S6 + 6S5L) « 126.699 GeV. (12.27) Fig. 12.10. Models of y photon and H0 bosons in polyhedral approximation • For charged lepton we suppose the structure of small-T-triangle inversion combined with eight inversions of D cells providing the compensation (Fig.12.11).This mechanism does not follow the pattern used in 8-bit model for fermion families representation7, but it offers effective mass reduction below GeV scale. ml = Ap(6S6 " 12S5s + 6S5L) « 0.315 GeV. (12.28) • The zero-charged compensating "frame" consisting from D cells could be associated with massless neutrino (Fig.12.11v). • Although the exchange between two or more stacked T cells has the positive energetic effect, its magnitude does not depend on the stack length, and originates just from the non-compensated ends of the stack that has the color charge due to their asymmetry. So it can be associated with the gluon thread terminated with quarks. 7 the latter involves additional T-D exchange. Fig. 12.11. Models of charged lepton with compensated mass, and massless neutrino in polyhedral approximation 12.6 Analytical approximation of charge distribution In addition to the Polyhedral and Cell-Center approximations we consider an approximation of the structure by the triple-periodical analytical function of electrical charge density distribution. The electrical charge of the cell concentrated at its center (x0,y0, z0) can be expressed analytically using the 5-function: q = 6 6(x — x0,y — y0,z — z0) dx dy dz (12.29) The delta function can be considered as the spherically-symmetrical Gaussian distribution with zero deviation: §(x,y,z) = lim 6(x,y,z, c); 5(x,y,z, c) = x2+y2+z2 2a2 (12.30) (12.31) As we have shown in section 12.4.6, the model explaining the weak isospin T3 = 0 for right-handed fermions and T3 = ±1/2 for left-handed ones by the charge exchange between D-triangles at Z = 2/V3 and 4/V3, requires one quarter of the charge of each D cell to reside behind the section plane located at the distance of 1/\/3 from the cell center: -1/V3 bco +CO p(x,y,z) dx dy dz = 4 (12.32) x=—co —co z=—co Assuming charge density p(x,y,z) to be the Gaussian distribution (12.31), and solving the equation 1 -1/V3 (c\/2n)3 x2 +y 2 + z2 1 2 a2 dx dy dz = — » 4 (12.33) x=—CO —CO z=—CO e e e 280 E.G. Dmitrieff numerically, we found ct « 0.87377. Soliton model To construct the charge distribution in the analytical form, we can use, instead of each cell, some spherical-symmetrical function, which decreases quite rapidly on distance from its center, i.e. soliton. We consider the soliton function as normalized error function Pi = ± 6 exp 6CTv2n (x - Xi)2 + (y - yi)2 + (z - Zi)2 2ct2 (12.34) representing positive or negative charged cell with the center at (xt,yt, zi). The charge density in the particular point is calculated as a sum of contributions of all the cells in the model: P = X Pi (12.35) i One can manage the position and charge of each individual cell, so this model should be flexible. On another hand, it requires extensive computation to calculate each point. Triple-periodic trigonometric function Since the most interesting application of this model is to represent the only one or several defects being surrounded by the "pure" vacuum, we looked for the periodic function that has the same symmetry as the dual-charged Weaire-Phelan structure considered above. It is intended to represent the pure vacuum avoiding calculating of plenty periodically allocated solitons. At first, we consider the real function that has zero surface close to the Schwartz P minimal surface [15]. Po = cos x + cos y + cos z, (12.36) or, equivalently, p0 = ^ cos xi. (12.37) i xn yn zn Po = cos — + cos — + cos —, (12.38) 2À 2À 2À It has minimum in points (2nnx, 2nny,2nnz) = 2n(nx,ny,nz) and maximum in n(2nx + 1,2ny + 1,2nz + 1 ) since ^ = - sin xi = 0 ^ xi = nni, (12.39) oxi and dx? =- cos xi. (12.40) The last equation also means that Apo = -Po, (12.41) so po is eigenfunction of the Laplasian, with eigenvalue -1. The translation unit with nx = ny = nz = 0 is a cube with xi G [-n; n]. So, p0 has one minimum in (0,0,0) and one maximum in n (2,2,2). As the second step, we consider the surface p0 = 0. Its saddle points are the same with the T cell center points. So we can add the function with extremals at these points, namely at centers of D cells: pxz = 1 sin y (1 - cos x)(1 + cos z) (12.42) Pyx = 1 sinz(1 - cosy)(1 + cosx) (12.43) Pzy = 1 siny (1 - cosz)(1 + cosy) (12.44) pxy = 1 sinz(1 - cosx)(1 + cosy) (12.45) pyz = 1 sin y(1 - cos y)(1 + cos z) (12.46) pzx = 1 siny (1 - cosz)(1 + cosx) (12.47) PR = Pxy + Pyz + Pzx (12.48) PL = Pyx + Pzy + Pxz (12.49) We construct right and left vacuum electric charge density as P0R = Po + PR (12.50) P0L = Po + PL. (12.51) Note that Pxz (12.42) and other Pij can be rewritten in the following way: Pxz = 1 (sin y + sin y cos z - sin y cos x - sin y cos x cos z), (12.52) so pr and pl can be represented as sums of four functions listed below, which accumulate summands of four particular types, that occur in (12.42). Introducing "Schwartz P"- like distribution Pe = cos(x - 9) + cos(y - 9) + cos(z - 9), (12.53) right and left gyroid-like distributions Gr = cos x sin y + cos y sin z + cos z sin x, (12.54) Gl = cos x sin z + cos y sin x + cos z sin y, (12.55) and "layers-with-holes" distribution H = cos x sin y cos z + cos y sin z cos x + cos z sin x cos y, (12.56) we can express pR through them: Por = 4 [p«/2 + Gl - Gr - H] - 3Po. (12.57) Since G and H are also eigenfunctions of the Laplasian A: AG = -2G; AH = -3H, (12.58) one can find the scalar electric potential: divgrad cpoR = A^or = 4npoR, (12.59) 1 1 9or = Po - 77— 12n 16n Pn/2 + ^gl - 1gr - (12.60) Combining triple-periodical trigonometric equation for the vacuum state with doubled opposite-charged soliton located in particular cell centers one can obtain a model representing one or more particles surrounded by the vacuum. 12.7 Discussion 12.7.1 Two-dimension model Consider the surface of zero potential (12.60): 11 13 14\ (k910 + k1112 2V3 a/2 . / n nn \ -sin--\-- 4V3 ,13 14^ (13.7) 3 sn6 + t) XV3 6{±7;±27! 13.2.2 Model with alternation of charged binary triangles Note that a charged binary triangle cannot be electrically neutral. Nevertheless, a pair of opposite-charged binary triangles, or, generally, any even number of them can hold zero electric charge. _ 1 2 Fig. 13.2. The isometric projection of the state cube along the T4 axis Consider a big system containing sufficiently large amount of opposite-charged binary triangles, half of them consisting of three positive-charged elements, and half of three negative-charged ones, arranged in alternating pattern. The whole system is electrically neutral. Each triangle in this system, either having charge of + 2 or _2, is surrounded by opposite-charged environment with the same magnitude: qenv = t4. (13.8) We found out that in this system, where positive- and negative-charged triangles have different own places due to their alternation, the additional degree of freedom emerges for any single triangle. For instance, the negative-charged triangle with q = t4 = — 1 in its own place must be effectively neutralized by its environment and therefore must be indistinguishable from the background. But the same triangle in the place of positive-charged one should be treated as having effective charge of —1 that emerges as a sum of the negative triangle charge and the negative charge of the environment surrounding the place where the positive triangle should be: Q = T4 + qenv = _ 1 +(— =_1. (13.9) So the state space for the charged binary triangle that participates in the neutral alternation of such triangles, must reflect this emergent binary degree of freedom. The state space becomes four-dimensional, splitting each original state to the doublet with the triple magnitude 1 in comparison to original 1 (Fig. 13.3). One of the states shifts up in charge with +1 while another one shifts down, with _ i 2 . One can ensure that among these 16 states there are neutral and integer-and fractional-charged ones with step of 3 so the effective charges coincide with charges of known fundamental fermions and anti-fermions belonging to one family. -1 ±0 +1 Q Fig. 13.3. Four-dimensional state hyper-rectangle space for the single charged binary triangle in the environment of neutral tessellation of alternating ± 1 -charged binary triangles, labeled with symbols of corresponding fermion family members The degree of freedom emerging from implementing the tessellation instead of isolated triangle manifests the physical sense of the isospin, or the weak charge, connecting corresponding up and down particles. Following the observations mentioned above, we suppose that one should search for geometrical structure containing equilateral triangles, aiming to obtain suitable model for the fundamental particles. It must be chiral to represent hand-edness and also must possess some additional degrees of freedom to be able to represent fermion families and fundamental bosons. 13.3 Calculation of electrical charge and Weaire-Phelan tessellation We consider a graph for calculating the electric charge Q from values kab [1], that are the doubled eigenvalues (13.3) of Lorentz transformations infinitesimal generators Sab (13.1) [1], [2]. The graph is constructed aiming to fetch all the data required for the calculation from the charges of cells in the dual-charged Weaire-Phelan tessellation [5]. In the Spin-Charge-Family theory, as well as in the Standard Model, the electric charge of a particle is calculated as a sum of the third projection of its SU(2)i weak charge t13 and the hypercharge Y: Q = t13 + Y. Since the weak charge operator is defined as (13.10) 1 = 1 (s58 - S67, S57 + S68, S56 - S78^ (13.11) +1 + 2 3 3 and each Sab has two eigenvalues, namely 2kab, where kab charge is expressed through kab in the following way: 11 4 - 4' Therefore it can be of one of three different values: t13 = -k56 — -k/s. ±1, the weak (13.12) k56 k78 t13 -1 -1 0 1 1 0 -1 1 —1/2 1 -1 1/2 In turn, the hypercharge is the sum of SU(2)n charge t23 and U(1) "fermion charge" t4: Y = t23 + t4, where = 1 (S58 + S67, S57 - S68, S56 + S78^ and 3 After transition to the eigenvalues, = -! (S910 + S1112 + S13 ). T23 = -k56 + 7k/8, 1 1 4 4 So and, finally, 4 K K K T =—7 K910 — 7 k11 12 — 7 k13 14. b b b Y = 1k56 + 4k78 — 1k9 10 — b kii 12 — gkl3 14, Q = 1k56 — 4k78 + 4k56 + 1k78 — 1k9 10 — bk11 12 — 1k13 14. (13.13) (13.14) (13.15) (13.16) (13.17) (13.18) (13.19) One can build the following graph illustrating how the electrical charge is calculated, where the arcs show the data dependence between nodes: Q 2 4 The value of k78 is included in equation (13.19) twice, with opposite signs, so it has no influence on the total charge Q, and the equation can be simplified: Q = 2k56 6k9 10 6ki112 6k13 14. (13.20) The corresponding simplified calculation graph is the following: Q 6 k9 10 6 k11 12 -i k1314 In this form, the graph is equivalent to the charge calculation in our 4-bit model presented in [3]: Q = L C - q2 i=0 (13.21) with the following correspondence: (13.22) 2k56 = -(q2 - 1 ) 6k910 =-3 (C0 - 2] 6k1112 =-3 (C1 - 2] 6k1314 =-3 (C2 - 2 The ci G {0; 1} are three bits of the color code and q2 G {0; 1} is the most significant bit of the electrical charge code in the ones' complement convention. After splitting the node 2 k56 into three nodes 6 k56, 1 k56, and 1 k56, the graph becomes equivalent to our 6-bit model [3]: Q 1 k(1) 6 k5 6 1 k(2) 6 k5 6 1 k(3) 6 k5 6 6 k9 10 22 Q = L bC + L bT3, 3 14 i=0 i=0 where symbols b? are produced from ct by scaling and shifting down: 1 bc = "3- G {0;1;2};bc G -6; 11 (13.23) (13.24) The symbols bt3 are produced from q2 by splitting it into three parts, scaling and shifting up: = 6 - f -{o;i;2}^-66} (i3.25) Note that we do not mean an increase in the number of degrees of freedom as a consequence of splitting nodes, at least while considering members of one fermion family, since all the three subnodes are assumed keeping the same values that are equal to value of splitted node.. Both these graphs have the following advantage in relation to the original one: they allow interpretation of particle's electrical charge as a simple sum of values of all the nodes2 due to its tree-form and arcs meaning addition only. The last one also has an advantage of equal magnitude of nodes3. To get these advantages in the original graph, we transform it the following way, getting rid of two loops and the subtracting arc. To do so, we assume that there are two different subnodes behind 4k5 6 and two others behind 1 k7 8, always keeping equal values in the first case, and opposite values in the second one. After transformation the graph becomes the following: 4 k7 8 1 k(2) 4 k7 8 1 k(2) 4 k5 6 1 4 k Then we double the factors in all the nodes for k56 and k78, introducing compensating nodes that divide the corresponding values back. That makes these nodes ready to split on three sub-nodes with the factor of 6. Arguments in favor of division in half for nodes k5 6 and k7 8 are different and given below and are discussed in detail in [5]. 2 or integration of charge density in continuous models 3 The choice of positive or negative eigenvalues is made while choosing the initial vacuum state corresponding to the right neutrino, and it can be changed to eliminate minus signs. -Q i k(1) 2 k5 6 1 k(2) 2 k5 6 — 1 k9 10 — 1 k11 12 — 6k1314 At the last step, we split each node k5 6, k7 8 to three sub-nodes. Then we establish correspondence between these nodes and cells in the dual-charged Weaire-Phelan tessellation: Q 1 k(1) 2 k5 6 /1 k(2'1)N '6 k5 6 1 k(2'2) 6 k5 6 1 k52,3) \ 6 k5 6 I V Zqi \ -1 k(1) 2 k7 8 (-1 k(2,1) 6 k7 8 -6 k(2'2) 6 k7 8 V-6 k72'3) \ 6 k7 8 J ^ Iq2 \ k9 10 1 kn 12 6k1314y ® ® ® 0 iq. 1 k(2) 2 k7 8 /1 k(2,1) '6 k7 8 6 k7 8 \ 6 k7 8 J ^ Iq4 \ Iq» ® ® r ® iq9 ® ®/ '1 k(2) 2 k5 6 /1 k(2'1)N '6 k5 6 6 k5 6 \ 6 k5 6 ! x Iq5 \ ® ® X X 2 4 T The last two rows contain the corresponding triangles of charged cells, residing in sequential Z-planes in the tessellation. These triangles are listed in the Table 13.1 (it is borrowed from [5]). The first row contains cells triangles in planes from 1 to 5 and represents down fermions; in turn, the charge-inversed and mirror-reflected triangles in planes from 7 to 11 represent corresponding up particles. Note that cells in planes 1 and 5, 7 and 11, that are the data sources for the k5 6 nodes, do not have any degrees of freedom and carry the negative charge for down particles and positive for up particles. Since all six cells are equal in charge and they must represent the qenv = ±2, i.e. charge of the environment for cells in planes 3 or 9, their charge value is divided in half. kind, charge shape size shape description 0 D- © - Axial D- 1 T - o 0 o VÎ4 Large T - triangle counterclockwise ® ® 2 D+ ® V32 D+ triangle u-down 3 T+ V6 Small T+triangle (£,-right) 4 D- © © © V32 D- triangle u-up 5 T - o u o VT4 Large T - triangle clockwise 6 D+ ® - Axial D+ 7 T+ • • • VT4 Large T + triangle clockwise 8 D- © V32 D- triangle u-down 9 T - V6 Small T - triangle (£,-left) ® V32 10 D+ ® ® D+ triangle u-up 11 T+ • • • VT4 Large T + triangle counterclockwise 12 D- © - Axial D- Table 13.1. Shapes of cell center placements in twelve different planes The cells in planes 2 and 4, 8 and 10 provide the data for the k7 8 nodes and each couple of triplets has just one degree of freedom, that represent the exchange between triangles in the coupled plane. The exchanged electrical charge of 2 moves from one side of the plane 3 or 9 to another, representing the change of handedness and adjustment of weak charge and hyper-charge. Since the cells in planes 2 and 4, 8 and 10 appear to have finite size, they partially overlap in projection to zeta axis, providing only half of charge is exchanged. The three cells in plane 3, and three cells in plane 9 provide data for k910, kii 12, and k1314. Each of them keeps its degree of freedom, so there are eight combinations for small triangle in these planes, corresponding to eight down- and eight up particles or antiparticles.They are listed in Tables 13.2 and 13.3. Note that in our approach the corresponding down and up particles with the same color have in this representation the opposite projections to the t38 axis. It is so because they are mirror reflections of each other due to the P operation between them (C operation is not applied because we list all 8 combinations of charge for both cases, q3 and q9, in the same order). The spin-charge-family theory, in contrast, provides equal t38 values in this case [2]. 13.3.1 On k numbers without influence on total electric charge The value of k7 8 has no contribution to the total electric charge. As we have shown above, it can be considered as existing of mutually compensating cells of opposite charges. Also we note that the expression (13.19) can be expanded by including II ^ "f q§10 q102 q021 x4 = x" = 34,5 Y = 13,4,5 Q = xl3 +Y symbol = xJS = Zq3t x^ + + + • * +1 12 0 0 ED 0 0 — + + 0 1 vT + — + O* +1 /6 -1 /3 -1/3 + 1 /2 1 2 M P a 3 Cft Table 13.3. Eight cases of inversions in the small T-triangle at £ = 9/\/3, repeated twice with original and exchanged D-triangles at £ = 8/Viand 10/V3, associated with "up" fermions additional terms, arbitrary in magnitude, that cancel each other. Since they have no influence on the electrical charge, they can't be determined from the charge analyse. We suppose that the expression of the electrical charge can also contain the last real eigenvalue, k1 2: Q = ak112)-ak12' + 4k56-4kz8 + ^55+^k/s-^910-^kn 12-6^314, (13.26) where a is a factor that can be equal to 1/6. It allows to associate k^, k^ with the "axial" D cells at Z = 0 and 6/%/3 for down fermions, and 6/%/3 and 12/%/3 for up ones. In the spin-charge theory the value of k03 is dependent on values of other kab since the equation k03 = - iknk56 k7sk910kn 12^314 (13.27) is fulfilled for each fermion combination in [1], [2]. In our opinion, it is connected with the fact that the seven binary values of kab generate only 26 = 64 combinations. For one family there are only six independent degrees of freedom represented by kab, so since there are seven of them, one (in our case, k03) should be expressed through six others. Thus, there is no degree of freedom connected with k03 and there is no corresponding cell in the Weaire-Phelan structure, so the value of spin always can be computed based on other data 13.27. Totally, we have the following correspondence between values of kab in the Spin-Charge-Family theory and charges associated with cells of dual-charged Weaire-Phelan model: k12 =3q-3 - 3q+3 k5 6 = Iq-2 +Iq+2 k/s = Iq-1 - Iq+1 k9 10 = 6qijk (13.28) knu = 6qjki k13 14 = 6qkij k0 3 = -ik12k56k78k9 10k11 12k13 14 It is provided in relative form, for both up and down particles. The lower index counting the Z-plane number relative to the plane of the small T-triangle (that is 3 for down or 9 for up fermions), and the upper index counts x,y,z coordinates of three individual cells in the triangle; the I sign means sum of these three cells. 13.4 Conclusion We presented here our approach to the particle and vacuum modelling. It is, being applied to one fermion family, reproduces the same quantum numbers as those obtained in the spin-charge-family theory. The advantage of spatial tessellation model, on our opinion, is the lower dimension count, so it can fit in the usual spacetime and be more demonstrative. Also it provides native CPS symmetry and emergent weak charge. We suppose that one can find out the appropriate 3-or 4-dimensional spatial model that would, keeping the shown advantages, also represent and explain fermion families and also fundamental bosons, basing on 8-bit code model [3]. References 1. N.S. Mankoc Borštnik: Can spin-charge-family theory explain baryon number non conservation? arXiv:1409.7791v3 - 24 February 2015. 2. N.S. Mankoc Borstnik: Fermions and Bosons in the Expanding Universe by the Spin-charge-family theory, in: N.S. Mankoc Borstnik, H.B.F. Nielsen, D. Lukman: Proceedings to the 20th Workshop 'What Comes Beyond the Standard Models', Bled, July 9-17 2017. 3. E.G. Dmitrieff: Experience in modeling properties of fundamental particles using binary codes, in: N.S. Mankoc Borstnik, H.B.F. Nielsen, D. Lukman: Proceedings to the 19th Workshop 'What Comes Beyond the Standard Models', Bled, 11. - 19. July 2016. 4. D.Weaire, R.Phelan, A counter-example to Kelvin's conjecture on minimal surfaces, Phil. Mag. Lett., (1994) 69:107-110, doi:10.1080/09500839408241577 5. E.G.Dmitrieff: On triple-periodic electrical charge distribution as a model of physical vacuum and fundamental particles, in N.S. Mankoc Borstnik, H.B.F. Nielsen, D. Lukman: Proceedings to the 21th Workshop 'What Comes Beyond the Standard Models', Bled, 23. - 29. June 2018 Bled Workshops in Physics Vol. 19, No. 2 JLV Proceedings to the 21 st Workshop What Comes Beyond ... (p. 299) Bled, Slovenia, June 23-July 1, 2018 14 Ko - Ko, Do - Do in a Local SU(3) Family Symmetry A. Hernandez-Galeana * Departamento de Física, ESFM - Instituto Politécnico Nacional. U. P. "Adolfo Lopez Mateos". C. P. 07738, Ciudad de Mexico, Mexico. Abstract. Within a broken SU(3) gauged family symmetry, we report the analysis of AF = 2 processes induced by the tree level exchange of the new massive horizontal gauge bosons, which introduce flavor-changing couplings. We provide a parameter space region where this framework can accommodate the hierarchical spectrum of quark masses and mixing and simultaneously suppress within current experimental limits the contributions to Ko — Ko and Do — Do mixing. In addition we find out that the mass of the SU(2)L weak singlet vector-like D quark introduced in this BSM, may be of the orden of 10 TeV. Povzetek. Avtor v okviru svojega predloga teorije z zlomljeno družinsko simetrijo SU(3) analizira procese tipa AF = 2, ki jih inducira izmenjava novih masivnih horizontalnih umeritvenih bozonov na drevesnem nivoju, kar privede do sklopitev, ki spremenijo okus. Najde območje v prostoru parametrov, ki dovoljuje izmerjeni masni spekter kvarkov ter njihovo mesalno matriko, pri tem pa so prispevki mesanja Ko — Ko in D o — D1o pod trenutnimi eksperimentalnimi mejami. Maso napovedanega kvarka D, ki je sibki singlet vektorskega tipa SU(2)l, oceni na ~ 10 TeV. Keywords: Quark and lepton masses and mixing, Flavor symmetry, AF = 2 Processes. PACS: 14.60.Pq, 12.15.Ff, 12.60.-i 14.1 Introduction Flavor physics and rare processes play an important role to test any Beyond Standard Model(BSM) physics proposal, and hence, it is crucial to explore the possibility to suppress properly these type of flavor violating processes. Within the framework of a vector-like gauged SU (3) family symmetry model[1,2], we study the contribution to AF = 2 processes[3]-[6] in neutral mesons at tree level exchange diagrams mediated by the gauge bosons with masses of the order of some TeV's, corresponding to the lower scale of the SU(3) family symmetry breaking. * E-mail: albino@esfm.ipn.mx The reported analysis is performed in a scenario where light fermions obtain masses from radiative corrections mediated by the massive bosons associated to the broken SU(3) family symmetry, while the heavy fermions; top and bottom quarks and tau lepton become massive from tree level See-saw mechanisms. Previous theories addressing the problem of quark and lepton masses and mixing with spontaneously broken SU(3) gauge symmetry of generations include the ones with chiral local SU(3)H family symmetry as well as other SU(3) family symmetries. See for instance [7]-[14] and references therein. 14.2 SU(3) family symmetry model The model is based on the gauge symmetry G = SU(3)f SU(3)c < SU(2)l << U(1 )Y (14.1) where SU(3) is a completely vector-like and universal gauged family symmetry. That is, the corresponding gauge bosons couple equally to Left and Right Handed ordinary Quarks and Leptons, with gH, gs, g and g' the corresponding coupling constants. The content of fermions assumes the standard model quarks and leptons: ¥ = (3,3,2,3 )L , = (3,1,2, —1 )l (14.2) = (3,3,1, 3)R , ^(3,3,1, —2)R , ¥0 = (3,1,1, — 2)r (14.3) where the last entry is the hypercharge Y, with the electric charge defined by Q = T3L + 2 Y. The model includes two types of extra fermions: Right Handed Neutrinos: = (3,1,1,0)R, introduced to cancel anomalies [7], and a new family of SU(2)L weak singlet vector-like fermions: Vector like quarks U0,UR = (1,3,1, 3) and DL,DR = (1,3,1, — 2), Vector Like electrons: EL,ER = (1,1,1,—2), and New Sterile Neutrinos: N°,NR = (1,1,1,0). The particle content and gauge symmetry assignments are summarized in Table 14.1. Notice that all SU(3) non-singlet fields transform as the fundamental representation under the SU(3) symmetry. 14.3 SU(3) family symmetry breaking To implement the SSB of SU(3), we introduce two flavon scalar fields: KA nt = (3,1,1,0)= (n02 J , i = 1,2 (14.4) W3/ SU(3) SU(3)c SU(2)l U(1)y cq 3 3 2 1 3 Cur 3 3 1 4 3 CdR 3 3 1 2 - 3 co 3 1 2 -1 3 1 1 -2 Cr 3 1 1 0 3 1 2 -1 od 3 1 2 +1 ni 3 1 1 0 Ul,r 1 3 1 4 3 DL,r 1 3 1 2 - 3 E°,r 1 1 1 -2 nl,r 1 1 1 0 Table 14.1. Particle content and charges under the gauge symmetry with the "Vacuum ExpectationValues" (VEV's): T = (A1}0,0) , : ^(Yi+Yr + y2+y2-) + ^(Z1 + f + 2zï 53) 2 „2 A 2 2 Y2 : ^(Y+Y- + Y+Y-) + gHA2Z32 :z2 v2~ Z2 The "Spontaneous Symmetry Breaking" (SSB) of SU(3) occurs in two stages SU(3) x Gsm ^ SU(2) ? x GSM —1 GSM FCNC ? Notice that the hierarchy of scales A2 > Ai yield an "approximate SU(2) global symmetry" in the spectrum of SU(2) gauge boson masses. Therefore, neglecting tiny contributions from electroweak symmetry breaking, we obtain the gauge boson mass terms. (m2 + m2) y+y- + m2 y+y- + m2 y+y- + -m? z? + 2 M2 = 1M2 + 4M2 Z2 + -(M2)^ Z? Z2 (14.7) 2 3 M2 = ^ 2 'V3' (14.8) M2 M Ml 73 M2 M2+4M 3+4M2 73 Table 14.2. Zi — Z2 mixing mass matrix Diagonalization of the Z1 — Z2 squared mass matrix yield the eigenvalues M- = ; ^m2 + m2 — ^(m2 — m2)2 + m?m2 m+ = 2 (m2 + m2 + ^(m2 — m?)2 + m?m2 3 2 and finally (14.9) (14.10) Z2 Z2 (m? + m2) y+y- + m2 y+y- + m2 y+y- + m- — + m+ z+ . where cos § sin §\ (Z - sin § cos §/ \Z . . , V3_Mi_ cos § sin § = —----' - 4 y'M? + M2(M2 — M2) (14.11) (14.12) (14.13) Z Z 2 Z Z 2 3 2 14.4 Electroweak symmetry breaking For electroweak symmetry breaking we introduce two triplets of SU(2)L Higgs doublets, namely; with the VEV's where Ou =(3,1,2,-1) W>N (Ou> = | (®U> (o^ = _L fvui Od = (3,1,2, +1), '(®d>N (Od> = | (®d> | , ,(®3d>, = ¿C-V (14.14) (14.15) (14.16) The contributions from (Ou> and (O d> generate the W and Zo SM gauge boson masses (vU + vdd) W+W- + -i9-^-1 (vU + vdd) ZO + tiny contribution to the SU(3) gauge boson masses and mixing with Zo , (14.17) (14.18) (14.19) vU = vL + vL + v3u , v3 = v-d + v-d + v3d. So, if mw = -gv we may write v = \jvU + vd « 246 GeV. 14.5 Fermion masses 14.5.1 Dirac See-saw mechanisms The scalars and fermion content allow for quarks the gauge invariant Yukawa couplings Hu Ou UR + hiu ^Ur ni UL + Mu UL UR + h.c (14.20) Hd Od DR + hid ^3r ni DL + Md DL DR + h.c (14.21) MU , Md are free mass parameters and Hu, Hd , hiu, hid are Yukawa coupling constants. When the involved scalar fields acquire VEV's, we get in the gauge basis rt = (e°, m°,t°, E°)l,r, the mass terms + h.c, where M° = ( 0 0 0hv^ 0 0 0hv2 0 0 0hv3 Vh At h2A2 0 M J ( 0 0 0 a,\ 0 0 0 a2 0 0 0 a3 \b1 b2 0 Mj (14.22) M° is diagonalized by applying a biunitary transformation = V° R xl,r. V°TM° VR = Diag(0,0,-A3,A4) (14.23) V° 1 M°M° 1 V° = V°' M° 1 M° V° = Diag(0,0,A2,A4) , (14.24) where A3 and A4 are the nonzero eigenvalues, A4 being the fourth heavy fermion mass, and A3 of the order of the top, bottom and tau mass for u, d and e fermions, respectively. We see from Eqs.(14.23,14.24) that from tree level the See-saw mechanism yields two massless eigenvalues associated to the light fermions: 14.6 One loop contribution to fermion masses The one loop diagram of Fig. 1 gives the generic contribution to the mass term mij e°Le°R, < Vk > < > Fig. 14.1. Generic one loop diagram contribution to the mass term my e°Le°R 2 mij = cy^ Y mO (VO)ik(VR)jkf(My,m£) , «h = , (14.25) n i— k=3,4 My being the mass of the gauge boson, cy is a factor coupling constant, Eq.(14.6), 2 2 mO = -A3 and mO = A4, and f(x,y) = x2-y2 ln yr, k=3,4 X mk (VL°)ik(VR)jkf(MY,mk) = F(My) , (14.26) A24 - A23 M2 m2 M2 M2 i = 1,2,3 , j = 1,2, and F(My) = M2_TA2 ln M - M2_TA2 ln M. Adding up all possible the one loop diagramss, we get the contribution ^R + h.c., my -a2 LmI M° = /Du D12 0 0\ D21 D22 0 0 D31 D32 D33 0 V 0 0 0 0/ oh n (14.27) Dl1 = mil ^ + + Fm) + 1 M-22^1 D12 = M-12( — ^ — Fm) D21 = M-21 (--12 — Fm) D31 = ^31 l—^ + -f2 ) D33 = 2 (W1F2 + M-22 F3 ) gH D22 = 2 M-11 F1 + 3 ^22FZ2 D32 = ^32( — -f2 + Fm) «h = , F1 = f(myi ) , F2 = f(my2 ) , F3 = f(my3 ) 4 n Fz1 = cos2 ^ F(M_) + sin2 ^ F(M+) Fz2 = sin2 ^ F(M_) + cos2 ^ F(M+) cos ^ sin ^ Fm =-2-3 [F(M+) — F(M_) ] . (14.28) (14.29) (14.30) (14.31) FZl , FZ2 are the contributions from the diagrams mediated by the Z1 , Z2 gauge bosons, Fm comes from the Z1 — Z2 mixing diagrams, with M1, M2, M_, M+ the horizontal boson masses, Eqs.(7-11), at bj M at bj = ,2 ,2 = A3 Ca Cp , A2 — A3 (14.32) ca = cos a, cp = cos |3, sa = sin a, sp = sin |3 are the mixing angles from the diagonalization of Mo. Therefore, up to one loop corrections the fermion masses are ^°M° ^R + ^°M° ^R = xl M Xr , (14.33) where = Vf,R Xl,r, and M= Diag(0,0, —A3,A4)+ V° 1 M° V° maybe written as: ( mn m12 m21 m22 M = Cp m13 Cp m23 sp m13 \ sp m23 Ca m31 Ca m32 (—A3 + CaCp m33) CaSp m.33 Vsa m31 Sa m32 SaCp m33 (A4 + SaSp m.33)/ (14.34) The diagonalization of M, Eq.(14.34) gives the physical masses for u and d quarks, e charged leptons and v Dirac neutrino masses. Using a new biunitary transformationxl,r = V^R YL,R; Xl M Xr =^lVR1' M V1R1)Vr, with Yl,rt = (f1, f2, f3, F)l,r the mass eigenfields, that is VL1)TMMt VL1' = V(1)TMt M vR1' = Diag(m?,m2,m|,M2) , (14.35) ml = m;|, mj = m2, m| = mT and Mj = M| for charged leptons. So, the rotations from massless to mass fermions eigenfields in this scenario reads ^L = VLo VL1' ¥l and ^R = Vr° vR1' ¥r (14.36) 14.6.1 Quark Mixing Matrix Vckm We recall that vector like quarks are SU(2)L weak singlets, and hence the in- teraction of L-handed up and down quarks; f£L = (uL,cL,tL)L and fdL = (dL, sL, bo)L, to the W charged gauge boson is ^2 f"LuLY2fdLW+2 = (VCKM)4x4 Y^dL W+2 , (14.37) where the non-unitary quark mixing matrix Vckm of dimension 4 x 4 is (VCKM)4x4 = TO V^x^ (VLL V^^ (14.38) 14.7 Numerical results for quark masses and mixing As an example of the possible spectrum of quark masses and mixing from this scenario, we show up the following fit of parameters at the MZ scale [15] Using the input values for the horizontal boson masses, Eq.(8), and the coupling constant of the SU(3) symmetry: M1 = 3.3 x 103 TeV , M2 = 3.3 x 105 TeV , — = 0.05 , (14.39) n we write the tree level Mq, and up to one loop corrections Mq quark mass matrices, as well as the corresponding mass eigenvalues and mixing: d-quarks: Tree level see-saw mass matrix: 0 0 906.643 0 0 5984.81 0 0 8139.76 MeV, \3.00124 x 106 —670943. 0 9.10502 x 106/ (14.40) the mass matrix up to one loop corrections: / —5.64571 —11.0583 46.8646 15.829 \ Md = -29.9051 —39.4588 —11.5894 —3.91444 40.9245 —30.3588 —2859.86 130.424 V0.0409246 —0.0303588 0.386143 9.61036 x 106/ MeV , (14.41) the d-quark mass eigenvalues (md ,ms,mb,Mo) = (2.97549, 51.0, 2860.72, 9.61036 x 106 ) MeV. and the product of mixing matrices: (14.42) VdT = V°r V. (1). dL dL { 0.981831 0.17522 -0.0728363 0.0000922X — i 0.183881 0.783786 —0.593184 0.0005976 0.0468496 —0.5958 —0.801765 0.0008133 V—0.0000187 —6.6982 x 10-10 0.0010134 0.999999 ) (14.43) VdR = V°R Vd1^!: / 0.146421 0.577678 0.803005 \—0.0056951 —i -0.175219 —0.783785 0.595801 -9.0660 x 10- — 0.922135 0.312291. \ 0.217014 —0.0698145 0.0142936 4.3164 x 10-9 0.319949 0.947418 (14.44) u-quarks: MU = 0 0 673649. \ 0 0 5.57857 x 106 0 0 7.8041 x 106 \4.10528 x 108 —4.1775 x 107 0 1.92243 x 1010/ MeV, (14.45) Mu = /—0.47816 —0.551837 5.4868 0.117774 \ —3.21341 602.954 4467.75 95.9001 4.51209 1368.75 —173107. 714.009 \0.00225605 0.684377 16.632 1.92287 x 1010/ MeV , (14.46) 8 the u-quark mass eigenvalues (mu,mc,mt,MU) = (1.37677, 638.055 , 173170, 1.92287 x 1010)MeV (14.47) and the product of mixing matrices: VuL = VUL VUV: { 0.996356 0.0468431 -0.0010006 -0.829224 -0.0852899 0.556949 -0.0712817 0.0000350\ -0.558915 0.0002900 -0.826155 0.0004057 \ 0 0.0000128 0.0004998 1. VuR = V°R VuR (1). 0.0003359 0.0032952 0.999995 0.0934631 -0.995394 0.0213497 \ 0.995617 0.0934386 -0.0021725 -0.0033122 0.0000265 0 \—1.4066 x 10-8 0.0001676 0.0214593 0.99977 ) (14.48) (14.49) and the quark mixing matrix VCKM: 0.974441 0.224564 — i 0.0059177 \6.3092 x 10-8 0.224613 -0.973557 0.0416636 8.2754 x 10-6 0.0035948 0.0000219 0.041928 -0.0000382 0.999114 -0.0010126 0.0004999 5.0666 x 10-7/ (14.50) 14.8 AF = 2 Processes in Neutral Mesons Here we study the tree level FCNC interactions that contribute to Ko — K0, Do — Do mixing via Z1 , Y± exchange from the depicted diagram in Fig. 2. K0 d s Fig. 14.2. Generic tree level exchange contribution to Ko gauge bosons. K0 d Ko from the SU(3) horizontal i i Y^ Tiy2 The Z1 , Y± (Y± = 2) gauge bosons become massive at the second stage of the SU(3) symmetry breaking, and have flavor changing couplings in both left-and right-handed fermions, and then contribute the AS = 2 effective operators s Oll = (aLY^SL)(aLY^SL ) , ORR = (dRY^sR)(aRY^sR) (14.51) olr = (aLY^SL)(aRY^sR) (14.52) The SU(3) couplings to fermions, Eq.(14.6), when written in the mass basis yield the gauge couplings Lint,Z1 = (CLZ1 d_LY^SL + Cr z, ÎrY^Sr) Z1^ (14.53) Lint,Yi = ^H (CLY1 d_LY^SL + CRyi d~RY^sr) Y] ^ (14.54) Lint,Y2 = ^H (Cl y| d~LY^SL + Cr y| dVy^sr) i Y2 ^ (14.55) with the coefficients Cl Zi = Lu L12 — L31 L32 , Cr Zi = R11 R12 — R31 R32 ClyI = L12 L31 + L11 L32 , CryI = R12 R31 + R11 R32 (14.56) CLYf = (L12 L31 — L11 L32) , CR Yf = (R12 R31 — R11 R32) where VL,R == V° R vL^, and Lij = VLij , Rij = VRij. For each gauge boson, the effective four-fermion hamiltonian at the scale of the gauge boson mass is Hzi = ^ (CL Zi oll + 2 Cl zi Cr Zi olr + CR Zi orr) (14.57) Zi 2 hyI = 4^2 (C2l Yi oll + 2 Cl yI Cr yI olr + CR yI orr) (14.58) 2 = -^ (CL Y22 oll + 2 cl ^2 cr y, olr + CR Y22 orr) (14.59) with myi = MY2 = Mi. Therefore, the total four-fermion hamiltonian %su(2) = HZl + hyi + HY2 can be written as HSU(2) = 4MH2 (CLZi + CLyi - CL Y2 )OLL + (CRZl + CRy, + CRY2 )ORR +2(CLZi CRZ, + CLY] CR Y.| - CLY2 CR Y2 )OLR g2 1 1 + T"(- ^/[2 ) [Clzi Oll + CRz, Orr + 2ClZi Crz, Olr)] (14.60) Zi 1 From the coefficients in eq.(14.56) we obtain: CLZi + CL Y1 - CLY2 = SL , CRZi + CR Y.1 - CR Y2 = SR , , (14.61) CL,Zi CR,Zi + CL,Y] CR,Y1 - CL,Y2 CR,Y2 = SL SR + 2(L11 R31 - L31 R11 )(L32 R12 - L12 R32) , (14.62) and we can write hsu(2) = 4Mj2 [SL oll + SR orr + 5lr olr (14.63) + ^ (m^ - M ) [(L11L12 - L31L32)2 oll + (R11 R12 - R31R32)2 orr Zi 1 +2(Ln L12 - L31 L32)(R11 R12 - R31R32) olr)] with sl = L11 L12 + L31 L32 , sr = R11 R12 + R31 R32 (14.64) slr = 2(sl sr + 2(Ln R31 - L31 Rn)(L32 R12 - L12 R32)) (14.65) The reported parameter space region in section 7 generate MZi « M1 with quite good approximation, and then the dominant contribution to neutral meson mixing comes from the four-fermion Hamiltonian in eq.(14.63). The suppression of the generic meson mixing couplings (q~iLY^PL,R qj)2 come out as follows 14.8.1 Ko — K° meson mixing 9h/ 1 1 SL = 0.144124 , gHTiL = 32594.5 TeV SR = 0.452775 , gMl = 10375.2 TeV Mi — |6rI (14.66) = 0.361261 , gH Mi = 13003.4 TeV "2 V |6LR 1 14 Ko - K° , Do - D° in a Local SU(3) Family Symmetry 311 14.8.2 Do — Do meson mixing 5l = -0.000829741 , aHMl = 5.66157 x 106 TeV L ' Is L | 5r = -0.00328084 , aHMl = 1.43184 x 106 TeV (14.67) JWlrÎ = 0.456165 , ^ = 10298.1 TeV v 1 LKI ' ^a/ ISlrI 2 These numerical values are within the suppression required for BSM contributions reported for instance in the review "CKM Quark - Mixing Matrix" in PDG2018[16]. 14.9 Conclusions Horizontal gauge bosons from the local SU(3) family symmetry introduce flavor changing couplings, and in particular mediate AF = 2 processes at tree level. We reported the analytic and numerical contribution to K° - K° and D° - D° meson mixing from tree level exchange diagrams mediated by the SU(2) horizontal gauge bosons Zï ,Y±. We provided in section 7 a particular parameter space region where this scenario can accommodate the hierarchy spectrum of quark masses and mixing, and simultaneously suppress properly the AS = 2 and AC = 2 processes. 2 Acknowledgements It is my pleasure to thank the organizers N.S. Mankoc-Borstnik, H.B. Nielsen, M. Y. Khlopov, and participants for the stimulating Workshop at Bled, Slovenia. The author is grateful for the warm hospitality at the APC Laboratory, Paris, France, during sabbatical staying. This work was partially supported by the "Instituto Politecnico Nacional", (Grants from EDI and COFAA) in Mexico. 14.10 Appendix: Diagonalization of the generic Dirac See-saw mass matrix M° = /0 0 0 ai\ 0 0 0 a2 0 0 0 a3 \0b2 b3 cj (14.68) The tree level Mo 4 x 4 See-saw mass matrix is diagonalized by a biunitary transformation ^0 = VR xl and ^R = VR Xr. The diagonalization of MoMoT (MoTM0) yield the nonzero eigenvalues A2 = 2 (B - VB2 - 4D) , A2 = ^B + VB2 - 4D) (14.69) and rotation mixing angles cos a: cos ß _ /A4 — a2 A4 — A2 /A4 — b2 A4 — A3 sin a _ sin ß = la2 — A2 A4 — A3 lb2 — A3 A4 — A3 (14.70) B = a2 + b2 + c2 _ A2 + A4 D = a2b2 _ A3A4 (14.71) a2 _ a? + a2 + a3 b2 = b? + b2 + b2 (14.72) The rotation matrices V£, VR admit several parametrizations related to the two zero mass eigenstates, for instance vLo ( Ci —si S2 si C2 Ca Si C2 Sa\ 0 C2 S2 Ca S2 Sa —Si —Ci S2 Ci C2 Ca Ci C2 Sc 0 0 Sa Ca V o VR /10 0 0 \ 0 Cr Sr Cß Sr Sß 0 —Sr Cr Cß Cr Sß \0 0 —Sß Cß ) (14.73) a? + a 3 bn = b 2 + b3 an + a 2 b _*/ bn + b2 (14.74) Si ai _ a3 an ' an S2 a2 _ an a2a ^ ' Cr _ b3 (14.75) References 1. A. Hernandez-Galeana, Rev. Mex. Fis. Vol. 50(5), (2004) 522. hep-ph/0406315. 2. A. Hernandez-Galeana, Bled Workshops in Physics, (ISSN:1580-4992), Vol. 17, No. 2, (2016) Pag. 36; arXiv:1612.07388[hep-ph]; Vol. 16, No. 2, (2015) Pag. 47; arXiv:1602.08212[hep-ph]; Vol. 15, No. 2, (2014) Pag. 93; arXiv:1412.6708[hep-ph]; Vol. 14, No. 2, (2013) Pag. 82; arXiv:1312.3403[hep-ph]; Vol. 13, No. 2, (2012) Pag. 28; arXiv:1212.4571[hep-ph]; Vol. 12, No. 2, (2011) Pag. 41; arXiv:1111.7286[hep-ph]; Vol. 11, No. 2, (2010) Pag. 60; arXiv:1012.0224[hep-ph]; Bled Workshops in Physics,Vol. 10, No. 2, (2009) Pag. 67; arXiv:0912.4532[hep-ph]; 3. E. Golowich, J. Hewett, S. Pakvasa, and A. Petrov, Phys. Rev. D 76, 095009 (2007). 4. E. Golowich, J. Hewett, S. Pakvasa, and A. Petrov, Phys. Rev. D 79,114030 (2009). 5. M. Kirk, A. Lenz, and T. Rauh, arXiv:1711.02100[hep-ph]; T. Jubb, M. Kirk, A. Lenz, and G. Tetlalmatzi-Xolocotzi, arXiv:1603.07770[hep-ph]; 6. C. Bobeth, A. J. Buras, A. Celis, and M. Junk, arXiv:1703.04753[hep-ph]; A. J. Buras, arXiv:1611.06206[hep-ph]; arXiv:1609.05711[hep-ph]; 7. T. Yanagida, Phys. Rev. D 20, 2986 (1979). an= a= Sr= 8. Z. Berezhiani and M. Yu.Khlopov: Theory of broken gauge symmetry of families, Sov.J.Nucl.Phys. 51, 739 (1990). 9. Z. Berezhiani and M. Yu.Khlopov: Physical and astrophysical consequences of family symmetry breaking, Sov.J.Nucl.Phys. 51, 935 (1990). 10. J.L. Chkareuli, C.D. Froggatt, and H.B. Nielsen, Nucl. Phys. B 626, 307 (2002). 11. Z.G. Berezhiani: The weak mixing angles in gauge models with horizontal symmetry: A new approach to quark and lepton masses, Phys. Lett. B 129, 99 (1983). 12. T. Appelquist, Y. Bai and M. Piai: SU(3) Family Gauge Symmetry and the Axion, Phys. Rev. D 75, 073005 (2007). 13. T. Appelquist, Y. Bai and M. Piai: Neutrinos and SU(3) family gauge symmetry, Phys. Rev. D 74, 076001 (2006). 14. T. Appelquist, Y. Bai and M. Piai: Quark mass ratios and mixing angles from SU(3) family gauge symmetry, Phys. Lett. B 637, 245 (2006). 15. Zhi-zhong Xing, He Zhang and Shun Zhou, Phys. Rev. D 86, 013013 (2012). 16. M. Tanabashi et al. (Particle Data Group), Phys. Rev. D98, 030001 (2018). 17. See PDG review: CKM Quark-Mixing Matrix, by A. Ceccucci (CERN), Z. Ligeti (LBNL) and Y. Sakai (KEK). Bled Workshops in Physics Vol. 19, No. 2 A Proceedings to the 21 st Workshop What Comes Beyond ... (p. 314) Bled, Slovenia, June 23-July 1, 2018 15 Beyond the Standard Models of Particle Physics and Cosmology M.Yu. Khlopov * Institute of Physics, Southern Federal University Stachki 194, Rostov on Don 344090, Russia Abstract. The modern Standard cosmological model of inflationary Unvierse and baryosyn-thesis deeply involves particle theory beyond the Standard model (BSM). Inevitably, models of BSM physics lead to cosmological scenarios beyond the Standard cosmological paradigm. Scenarios of dark atom cosmology in the context of puzzles of direct and indirect dark matter searches, of clusters of massive primordial black holes as the source of gravitational wave signals and of antimatter globular cluster as the source of cosmic antihelium are discussed. Povzetek. V standardni kozmoloski model inflacijskega vesolja in tvorbe barionov vključi avtor tudi teorijo osnovnih delcev in polj, kar razsiri standardni model. Avtor obravnava model "temnih atomov", to je atomov, ki vsebujejo fermione družine z veliko maso. Predstavi prispevek temnih atomov v experimentih, ki naj bi detektirali temno snov, vlogo temnih atomov kopic masivnih prvotnih crnih lukenj, ki sevajo gravitacijske valove ter v globularnih kopicah antisnovi, ki naj bi bil izvor antihelija v vesolju. Keywords: cosmoparticle physics, inflation, baryosynthesis, dark matter, dark atoms, clusters of massive primordial black holes, antimatter, double charged particles, nuclear reactions, nucleosynthesis PACS: 12.60.-i; 95.35.+d; 14.80.-j; 21.90.+f; 36.10.-k; 98.80.-k; 98.80.Cq; 98.80.Ft; 04.70.-s; 15.1 Introduction The basis of the modern Standard cosmological paradigm, involving inflation, baryosynthesis and dark matter as its neccessary basic elements, is related to new physics predicted in theory beyond the Standard model (BSM) of elementary particles (see e.g. Ref. [1] for review and reference). However, BSM models, reproducing the necessary basic elements of the modern cosmology, inevitably contain additional model dependent consequences that lead beyond the Standard cosmological scenario [2]. Methods of cosmoparticle physics, studying fundamental relationship of cosmology and particle physics in the combination of its physical, astrophysical and * E-mail: khlopov@apc.in2p3.fr cosmological signatures, involve such model dependent cosmological predictions to probe models of BSM physics and cosmological scenarios, based on them. [3-5]. Here we show that BSM physics leads to cosmological scenarios accomplished by nontrivial deviations from the Standard cosmological model that deserve special interest in the context of the recent experimental progress. We address a possibility of existence of stable double charged particles O bound with primordial helium in neutral nuclear interacting O-helium dark atoms (Section 15.2) and consider advantages of this scenario to resolve puzzles of direct and indirect dark matter searches, as well as the open problems of OHe interaction with matter. We show that BSM physics of inflationary models that naturally leads to strong primordial inhomogeneities and to clusters of massive primordial black holes, in particular, is possibly reflected in the gravitational wave signal from massive black hole coalescence (Section 15.3). We discuss in Section 15.4 existence of antimatter stars in our Galaxy, originated from nonhomogeneous baryosynthe-sis in baryon asymmetrical Universe and reflected in cosmic antihelium fluxes, possibly detected by AMS02 [6,7]. 15.2 Dark atom physics and cosmology In the simplest case physics of dark matter is reduced to prediction by BSM model of new neutral elementary weakly interacting massive particle (WIMP). This type of prediction is beyond the standard model of elementary particles, but fits perfectly well the standard cosmological LambdaCDM paradigm. Super-symmetric (SUSY) models, predicting WIMP candidates, seemed to support this simple approach to dark matter physics. However negative results of experimental underground WIMP searches, as well as of collider searches for SUSY particles appeal to other possible BSM solutions for the dark matter problem. Possibly, SUSY physics and cosmology corresponds to superhigh energy scales as discussed in this Volume in [8]. In fact, the necessary conditions for dark matter candidates to be stable, satisfy the measured dark matter density and be decoupled from plasma and radiation at least before the beginning of matter dominated stage in no case demand these particle candidates to be weakly or superweakly interacting. Even nuclear interacting particles can play the same role due to decoupling of the gas of such particles from plasma and radiation before the end of radiation dominated stage. It gives rise to models of dark matter in the form of Strongly Interacting Massive Particles (SIMPs) [9-14]. By definition dark matter should be 'dark', nonluminous, what seem to favor neutral elementary particles. However ordinary atomic matter is neutral but it is composite and consists of electriclly charged particles (nuclei and electrons). In the same way O-helium dark atoms represent a specific example of composite SIMPs, in which hypothetical double charged O -- particles are bound with primordial helium nuclei by ordinary Coulomb force [15-20]. 15.2.1 OHe and O-nuclearites The main problem for hypothetical stable charged particles is their absence in the matter. If they do exist, they should be bound with ordinary matter and form anomalous isotopes. Severe experimental constraints on such isotopes, on anomalous hydrogen especially, seem to exclude a possibility for stable charged particles. However, if there exist stable particles with charge -2 in excess over corresponding particles with charge +2, such negatively charged particles are captured by primordial helium and form neutral OHe dark atom. There are various models, in which such stable -2 charged particles O are predicted [15-20]. Moreover, if these particles possess electroweak SU(2) gauge charges, their excess can be equilibrated by electroweak sphaleron transitions with baryon excess, as it is the case in Walking Technicolor models [17]. The general analysis of the bound states of single O with nuclei was developed in a simple model [21-23]. For small nuclei the Coulomb binding energy is like in hydrogen atom and is given by Eb = lz2ZOa2Amp. (15.1) For large nuclei O is inside nuclear radius and the harmonic oscillator approximation is valid for the estimation of the binding energy Eb — 3 ( ZZ°a _ 1 ( ZZOa )1/2). lb = 2_ R(AmpRJ ' J. (15.2) Here Z is the charge of nucleus, A is its atomic number, R is radius of nucleus, ZO = 2 is the charge of O , mp is the proton mass and a = 1/137 is the fine structure constant. In the case of OHe ZZO aAmp R < 1, what proves its Bohr-atom-like structure (see [19,20] for review and references). However, the radius of Bohr orbit in these "atoms" [15,17] ro ~ 1/(ZOZHeamHe) ~ 2 • 10-13 cm is of the order the size of He nucleus. Therefore the corresponding correction to the binding energy due to non-point-like charge distribution in He nucleus is significant. O particles are either elementary lepton-like states, or clusters of heavy U quarks with charge -2/3 UUU, which have strongly suppressed QCD interaction. In the contrary to ordinary atoms OHe has heavy lepton-like core and nuclear interacting shell. If multiple O are captured by a heavy nucleus, the corresponding neutral bound system can acquire the form of O-nuclearites, in which negative charge of O is compensated by posistive charge of protons in the nucleus [24]. The energy of such a O-nuclearite is given by [24] £ — _16MeV- A d3r(np _ 2no)V d3r ^+. (153) Here the first term is the volume energy of the atomic nucleus with atomic number A, next two terms describe the electromagnetic energy, and fO — d3r Pf,q 2 2 " (15.4) n2 2mO is the kinetic energy of the O-fermions of the mass mO; V = —e^ is the potential well for the electron in the field of the positive charge (e > 0, ^ > 0) and on the other hand it is the potential well also for the protons in the field of the negative charge of O-particles. The most energetically favorable O-particle distribution inside the nucleus is that follows the proton one, fully compensating the Coulomb field. Thereby O-particles, if their number were NO > A/4, would be re-distributed to minimize the energy, and finally the density of O inside the atomic nucleus becomes nO = np/2 = (n0/2) 9(r — R) for O-nuclearite, that corresponds to V = const for r < R. Excessive O-particles are pushed out. 15.2.2 Cosmoparticle physics of OHe model After the Standard Big Bang Nucleosynthesis (SBBN) O charged particles capture 4He nuclei in neutral OHe "atoms" [15]. For the mass of O mO ~ 1 TeV, O abundance is much smaller than helium abundamce, so that He is in excess in such capture, making the abundance of frozen out free O exponentially small. The cosmological scenario of OHe Universe involves only one parameter of new physics — the mass of O . Such a scenario is insensitive to the properties of O (except for its mass), since the main features of the OHe dark atoms are determined by their nuclear interacting helium shell. Before the end of radiation domination stage the rate of expansion exceeds the rate of energy and momentum transfer from plasma to OHe gas and the latter decouples from plasma and radiation. Then OHe starts to dominate at the Matter Dominated stage, playing the role of Warmer than Cold Dark Matter in the process of Large Scale Structure formation[15,19]. This feature is due to conversion of small scale fluctuations in acoustic waves before OHe decoupling and to their corresponding suppression. However, the suppression of such fluctuations is not as strong as the free streaming suppression for few keV dark matter particles in Warm Dark matter models. In terrestrial matter OHe dark atoms are slowed down and cannot cause significant nuclear recoil in the underground detectors, making them elusive for detection based on nuclear recoil. The positive results of DAMA experiments (see [25] for review and references) can find in this scenario a nontrivial explanation due to a low energy radiative capture of OHe by intermediate mass nuclei [19,1,20]. This explains the negative results of the XENON100 and LUX experiments. The rate of this capture is proportional to the temperature: this leads to a suppression of this effect in cryogenic detectors, such as CDMS. OHe collisions in the central part of the Galaxy lead to OHe excitations, and de-excitations with pair production in E0 transitions can explain the excess of the positron-annihilation line, observed by INTEGRAL in the galactic bulge [1,20,26,27]. Due to the large uncertainty of DM distribution in the galactic bulge this interpretation of the INTEGRAL data is possible in a wide range of masses of O-helium with the minimal required central density of O-helium dark matter at mO = 1.25 TeV. For smaller or larger values of mo one needs larger central density to provide effective excitation of O-helium in collisions. Current analysis favors lowest values of central dark matter density, making possible O-helium explanation for this excess only for a narrow window around this minimal value. In a two-component dark atom model, based on Walking Technicolor, a sparse WIMP-like component of atom-like state, made of positive and negative doubly charged techniparticles, is present together with the dominant OHe dark atoms. Decays of doubly positive charged techniparticles to pairs of same-sign leptons can explain [28] the excess of high-energy cosmic-ray positrons, found in PAMELA and AMS02 experiments[29-32]. Since even pure lepton decay channels are inevitably accompanied by gamma radiation the important constraint on this model follows from the measurement of cosmic gamma ray background in FERMI/LAT experiment[33]. The multi-parameter analysis of decaying dark atom constituent model determines the maximal model independent value of the mass of decaying +2 charge particle, at which this explanation is possible mO < 1TeV. One should take into account that even in this range hypothesis on decaying composite dark matter, distributed in the galactic halo, can lead according to [34] to gamma ray flux exceeding the measurement by FERMI/LAT. It can make more attractive interpretation of these data by an astrophysical pulsar local source[35] or by some local source of dark matter annihilation or decay. Experimental probes for OHe dark matter at the LHC strongly differ from the usual way of search for dark matter at acelerators, involving missed energy and momentum detection. Pending on the nature of the double charge constituents it may be search for new stable U-hadrons (heavy stable hadrons that appear in the result of production of UU pair) or search for stable double charged lepton-like particles. In the first case there are applicable constraints from the search for supersymmetric R-hadrons, having similar experimental signatures and giving the minimal mass for UUU close to 3 TeV. It excludes OHe interpretation of the cosmic positron anomalies in terms of heavy quark cluster constituents of OHe. The possibility to interpret cosmic positron anomalies in terms of OHe cos-tituents that appear in the experiments as stable lepton-like double charged particles is also close to complete test. The ATLAS and CMS collaborations at the LHC are searching for the double charged particles since 2011 [36-38]. The most stringent results achieved so far exclude the existence of such particles up to their mass of 680 GeV. This value was obtained by both ATLAS and CMS collaborations independently. It is expected that if these two collaborations combine their independently gathered statistics of LHC Run 2 (2015-2018), the lower mass limit of double charged particles could reach the level of about 1.3 TeV. It will make search for exotic long-living double charged particles an experimentum crucis for interpretation of low and high energy positron anomalies by composite dark matter [39,40]. The successful and self-consistent OHe scenario implies the existence of dipole Coulomb barrier, arising in OHe-nuclear interaction and supporting dominance of elastic OHe-nuclear scattering. This problem of nuclear physics of OHe remains the main open question of composite dark matter, which implies correct quantum mechanical solution [41]. The lack of such a barrier and essential contribution of inelastic OHe-nucleus processes seem to lead to inevitable overproduction of anomalous isotopes [42]. The advantages of the qualitative picture of OHe scenario appeal to increase the efforts to solve this problem. 15.3 Primordial massive black hole clusters The standard cosmological model considers homogeneous and isotropic Universe as the result of inflation. The observed celestial objects and strong inhomogeneities are evolved from small primordial density fluctuations that are also originated from small fluctuations of the inflaton field. It seems that there is no room for strong primordial inhomogeneities in this picture. Moreover, the existence of large scale inhomogeneities at the scales ^ 100Mpc is excluded by the measured isotropy of CMB. However, BSM physics, predicting new fields and mechanisms of symmetry breaking, adds new elements in this simple scenario that provide the existence of strong primordial inhomogeneities. Such predictions are compatible with the observed global homogeneity and isotropy of the Universe, if the strongly inho-mogeneous component i with (Sp/p)i ~ 1 contributes into the total density pt°t whithin the observed level of the large scale density fluctuations (Sp/p) = S0 C 1. It implies either large scale inhomogeneities, suppressed by the small contribution of the component i into the total density pi/pt°t < S0, or inhomogeneities at small scales. A simple example of an axion-like model with U(1) symmetry broken spontaneously and then explicitly illustrates these two possible forms of strong primordial inhomogeneities. In this model spontaneous U(1) symmetry breaking is induced by the vacuum expectation value If the first phase transion takes place after inflation at T = f and f ^ A, the potential Eq. (15.7) doesn't influence continuous degeneracy of vacua on 9 and string network is formed, which is converted in a walls-surrounded-by-strings network, separating regions with discrete vacuum degeneracy 9vac + 0,2n,... after the second phase transition at T = A. The vacuum structure network is unstable and decays, but the energy density distribution of 9 field oscillations is strongly inhomogeneous and retains the large scale structure of this network, as it was shown in the example of axion models in [43-45]. To fit the observational constraints on the inhomogeneity at large scales the contribution into the total density of such structure, called archioles, should be suppressed. It causes serious (15.5) (15.6) (15.7) problem for CDM models, in which the dominant form of dark matter is explained by axions [43-45]. If the first phase transition takes place at the inflational stage and f ^ A, as it was considered in [46], there appears a valley relative to values of phase in the field potential in this period. Fluctuations of the phase 0 along this valley, being of the order of A0 ~ H/(2nf) (here H is the Hubble parameter at inflational stage) change in the course of inflation its initial value within the regions of smaller size. Owing to such fluctuations, for the fixed value of 06o in the period of inflation with e-folding N = 60 corresponding to the part of the Universe within the modern cosmological horizon, strong deviations from this value appear at smaller scales, corresponding to later periods of inflation with N < 60. If 06o < n, the fluctuations can move the value of 0N to 0N > n in some regions of the Universe. After reheating, when the Universe cools down to temperature T = A the phase transition to the true vacuum states, corresponding to the minima of Veb takes place. For 0N < n the minimum of Veb is reached at 0vac = 0, whereas in the regions with 0N > n the true vacuum state corresponds to 0vac = 2n. For 06o < n in the bulk of the volume within the modern cosmological horizon 0vac = 0. However, within this volume there appear regions with 0vac = 2n. These regions are surrounded by massive domain walls, formed at the border between the two vacua. Since regions with 0vac = 2n are confined, the domain walls are closed. After their size equals the horizon, closed walls can collapse into black holes. The mass range of formed BHs is constrained by fundamental parameters of the model f and A. The maximal BH mass is determined by the condition that the wall does not dominate locally before it enters the cosmological horizon. Otherwise, local wall dominance leads to a superluminal a 4. Povzetek. V teoriji spinov-nabojev-druzin je v d-razsezznem prostoru 2d Cliffordovih vektorjev, ki opisujejo notranje prostostne stopnje fermionov, to je njihove druzine in clane druzin. Ker imamo dve vrsti Cliffordovih objektov, ki so v tej teoriji definirani kot Ya in Ya [2-7], ima vsak vektor dve vrsti indeksov. Operatorji Sab = 4 (YaYb —YbYa) dolocajov d = (3 +1)-razseznem prostoru spin in vse naboje kvarkov in leptonov, S ab = 4 (Y aYb — YbYa) pa kvantna stevila njihovih druzin. V tem prispevku je baza v d = (3 + 1) Cliffordovem prostoru izbrana tako, da matricne upodobitve operatorjev Ya in generatorjev Lorent-zovih transformacij Sab v notranjem prostoru sovpadajo z Diracovimi matrikami za vsako druzinsko kvantno stevilo, doloceno s i>ab. V prispevku ne upostevamo zahtev druge kvantizacije [8], ki zmanjšajo število stanj z 2d na 2 d-1 druzin s po 2d-1 clani. Vendar velja v teoriji spinov-nabojev-druzin to le za d = 2(2n + 1), kjer je d > 4. * This contribution is written to help readers of the Bled proceedings and participants at future Bled Workshops "What Comes Beyond the Standard Models" to understand the difference between the Dirac Ya matrices and the Ya matrices, which are all defined in 2d space and used in the spin-charge-family theory to describe families and family members [2-7]. Keywords:Dirac matices, Clifford algebra,Kaluza-Klein theories, Higher dimensional spaces, Beyond the standard model, Lepton and quark families PACS:04.50.-h, 04.50.Cd, 11.30.Ly 16.1 Introduction In the spin-charge-family theory there are in d-dimensional space two kinds of operators, Ya and Ya, which operate on 2d Clifford vectors, describing internal degrees of freedom of fermions; Ya determine family quantum numbers, Ya determine family members. Due to these two kinds of the Clifford algebra objects each vector carries two kinds of indexes [2-7]. Operators jYa Yb determine in d = (3 + 1) space the spin and all the charges of quarks and leptons, -jYa Yb determine families of quarks and leptons. Here only basis in d = (3 + 1) Clifford space is discussed, which in the spin-charge-family theory is only a part of d = (13 + 1). The basis is chosen in a way that the matrix representation of the Ya matrices and of the generators of the Lorentz transformations in internal space Sab = 4(YaYb — YbYa) coincide for each family quantum number, determined with Sab = 4(YaYb — YbYa), with Dirac matrices. This contribution is written to help the reader of the proceedings of Bled workshops "What comes beyond the standard models" to realize the differences between the Dirac matrices (operators) Ya and the operators Ya [2]. We do not take here into account the second quantization requirements [8], which reduce the number of states from 2d to 2 d -1 families of 2 d -1 family members each, since these requirements concern the states in d = 2(2n + 1), and not at all the particular subspace, in our case d = (3 + 1). We use in this contribution 2d vectors in Clifford space, expressible with Ya with the properties {Ya,Yb}+ = 2nab . (16.1) A general vector can correspondingly be written as d B = ^ aa,a2...ak Ya Ya2 ...Yak I^LC >, a* < ai+1 , (16.2) k=0 where |^L > is the vacuum state. We arrange these vectors as products of nilpotents and projectors ab 1 naa (k): = 1 (Ya + VYb), ab 1 i [k]: = 2 (1 + kYaYb), (16.3) where k2 = naanbb, their Hermitian conjugate values are ab ^ ab ab ^ ab (k) = naa (-k), [k] =[k], (16.4) and that they all are eigenstates of the Cartan subalgebra of the generators of the 4 ( Lorentz transformations Sab = 4 (YaYb = YabYa) in this internal space g03 s12 s56 • • • Sd_1 d (16.5) with the eigenvalues ab 1 ab ab 1 ab Sab (k) = 2k (k), Sab [k]= 2k [k] . (16.6) We find in this Clifford algebra space two kinds of the Clifford algebra objects, besides Ya also Ya [2-7], which anticommute with Ya (Ya,Y~b}+ = 0, {Ya, Yb}+ = I 2nab, for a, b e {0,1,2,3,5, ••• , d}, (16.7) for any d, even or odd. I is the unit element in the Clifford algebra. One of the authors (N.S.M.B.) recognized these two possibilities in Grassmann space [2]. But one can as well as understand the appearance of the two kinds of the Clifford algebra object by recognizing Ya B |^o >: = ( ao Ya + aa, Ya Yai + aa,a2 Ya Yai Ya2 + ••• + aa,...ad Ya Yai ••• Yad ) l^oe >, Ya B |^o >:= (iaoYa - iaa, Yai Ya + iaa,a2 Yai Ya2 Ya + ••• + i (-1)d aa,...ad Yai ••• Yad Ya )l^o > . (16.8) The nilpotents and projectors oof Eq. (16.3) are the eigenstates also of the generators of the Cartan subalgebra S03,S 12,S56, ••• ,Sd-1 d , (16.9) with the eigenvalues ab k ab ab k ab Sab (k) = 2 (k), Sab [k] = -k [k] . (16.10) One finds the relations ab ab ab ab ab ab ab ab Ya (k)= naa [-k], Yb (k)= -ik [-k], Ya M=(-k), Yb M= -iknaa (-k), ab ab ab ab ab ab ab ab Y~a (k) = -inaa [k], Y~b (k) = -k [k], Y~a [k]= i (k), Y~b [k]= -knaa (k) . (16.11) We discuss in what follows the representations of the operators Ya, Ya, Sab and Sab only in d = (3 + 1). In Ref. [8], as well as in this proceedings, the second quantization in Clifford and in Grassmann space is discussed. There the restrictions on the choices of products of nilpotents and projectors, which can be recognized as independent states in the Clifford space, and yet allow the second quantization, is analyzed. The restrictions reduce, as noticed above, the number of states from 2d to 2d -1 families with 2d -1 family members each. All the states of this contribution appear as a part of states (included as factors) already in d = (5 + 1 ). In what follows we shall notpay attention on these limitations. We only present matrices of the operators ya, Ya, Sab and S ab for all possible states. 16.2 Basis in d = (3 + 1) There are 24 = 16 basic states in d = (3 + 1). We make a choice of products of nilpotents and projectors, which are eigenstates of the Cartan subalgebra operators as presented in Eqs. (16.6,16.10). The family members are reachable by Sab, or by Ya representing twice two vectors of definite handedness r(d) in d = (3 + 1) r(d) : = (i)d/2 n (VnaaYa), if d = 2n. (16.12) a Each vector carries also the family handedness r(d) : = (i)d/2 n (Vn^Ya), if d = 2n. (16.13) a In what follows we first define the basic states and then represent all the operators — Ya, Sab, Ya, Sab, r(d) (= -4iS03S12 in d = 4), r(d) (= -4iS03S12 in d = 4) — as 16 x 16 matrices in this basis. We see that the operators have a 4 x 4 diagonal or off diagonal or partly diagonal and partly off diagonal substructure. Let us start with the definition of the basic states, presented in Table 16.1. As seen in Table 16.1 Ya change handedness. Sab, which do not belong to Cartan subalgebra, generate all the states of one representation of particular hand-edness, Eq. (16.12), and particular family quantum number. Sab, which do not belong to Cartan subalgebra, transform a family member of one family into the same family member of another family, Ya change the family quantum number as well as the handedness r (3+1', Eq. (16.13). Dirac matrices Ya and Sab do not distinguish among the families, they "see" all the families in the same way and correspondingly "see" only four states — instead of 4x four states. The operators Ya and Sab are correspondingly 4 x 4 matrices. Let us define, to simplify the notation, the unit 4 x 4 submatrix and the submatrix with all the matrix elements equal to zero as follows 1 = (01) , « = (00). (16-) We also use (2 x 2) Pauli matrices: = (00) , ^ = (0 o) , -3 = (1 -1). (16.15) d = 4 % Y0 % y1 % Y2 Y3 % Y0 % Y1 % Y2 % Y3 S03 S12 S03 S12 r 3 + 1 i 3 + 1 (+i)( + ) i 1 2 i 2 1 2 1 1 i 1 i 2 1 2 1 1 [ - i](+) i 1 2 i 2 1 2 1 (+i)[-] -i%4 -i%4 -i%4 i 2 1 2 i 2 1 2 1 [+i](+) i 2 1 2 i 2 1 2 1 -i%2 i 1 2 i 2 1 2 1 (-i)(+) i 2 1 2 i 2 1 2 [+i][-] -i%4 i 2 1 2 i 2 1 2 % (+i)[+] -i%4 i 2 1 2 i 2 1 2 1 i 1 2 i 2 1 2 1 [ - i][+] i 2 1 2 i 2 1 2 (+i)(-) -i%4 i 2 1 2 i 2 1 2 [+i][+] i 2 1 2 i 2 1 2 1 1 -i%2 -i%2 i 2 1 2 i 2 1 2 1 1 (-i)[+] i 2 1 2 i 2 1 2 1 [+i](-) -i%4 i%8 i%4 i 2 1 2 i 2 1 2 1 Table 16.1. In this table 2d = 16 vectors, describing internal space of fermions in d = (3 +1), are presented. Each vector carries the family member quantum number — determined by S03 and S12, Eqs. (16.6) — and the family quantum number — determined by Si03 and Si12, Eq. (16.10). Looking in Table 16.1 one easily finds the matrix representations for y0, y1 , Y2 and y3 Y Y = 0 a1 a01 a0 0 0 0 0 0 a1 a1 0 0 0 0 0 0 a1 a1 0 0 0 0 0 0 a1 a1 0 0 a1 a1 0 0 0 0 \ 0 0 -a1 1 0 0 0 Y 0 Y 0 0 -a2 0 0 0 0 a3 .0 0 0 0 0 a2 -a2 0 0 0 a3 -a3 0 0 0 -a1 a1 0 0 0 0 0 a2 -a2 0 0 0 0 0 a3 -a3 0 0 0 a1 1 a 0 0 a3 a3 0 (16.16) (16.17) (16.18) (16.19) a 0 0 (j 0 2 a 0 One sees as well the 4 x 4 substructure along the diagonal of 16 x 16 matrices. The representations of the ya, these do not appear in the Dirac case, manifest the off diagonal structure as follows / Y = -iff3 0 0 iff3 0 iff 0 0 iff3 0 0 0 iff3 0 0 -iff3 \ -iff3 0 0 iff3 0 (16.20) Y = Y Y iff3 0 0 iff3 0 0 iff3 0 0 iff3 0 iff3 0 0 iff3 ff3 0 0 ff3 3 V iff3 0 0 iff 0 0 0 iff3 0 0 iff3 0 0 0 iff 0 0 ff3 0 0 0 0 0 0 iff3 0 0 -iff3 0 -ff3 0 0 iff3 0 0 -iff 0 / (16.21) (16.22) (16.23) Matrices Sab have again the 4 x 4 substructure along the diagonal structure, as expected, manifesting the repetition of the Dirac 4 x 4 matrices, since the Dirac Sab do not distinguish among families. A c s01 = 0 - 2 0 0 0 \ 2 ff 0 (16.24) rff1/ S02 = 0 0 0 rff2 0 (16.25) 0 0 0 0 0 0 0 0 3 0 3 0 ff 0 3 0 ff ff3 0 0 3 0 d 0 0 0 0 0 0 ff -T ff 2 0 0 2 0 o 2 0 0 ff 2 0 0 2 0 ff 2 0 2 0 ff 2 2 0 cr 2 2 0 - 2 ff 2 0 cr 2 0 d 2 2 0 d ( S03 = S12 = S13 = S23 = 0 0 0 /1 a2 0 1 a2 0 0 0 2 a 0 0 0 0 i a3 o - 2 0 0 0 \ (16.26) 0 (16.27) 1 a3/ 0 0 0 a2 0 (16.28) r3+1 = —4iS03S12 = 03 12 0 1 0 0 2 a -1 a1 0 0 0- 1 a1 0 1 a1 0 0 2 10 0 -1 0 0 0 \ 0 10 0 -1 0 0 0 0 10 0 -1 0 V 0 0 0 10 0 -1 ) (16.29) / The operators Sab have again off diagonal 4 x 4 substructure, S12, which are diagonal. S01 = S02 = S03 = ( 0 0 0 —2 A 0 0 — i 1 2 1 0 0 — i 1 2 1 0 0 v - i 1 2 1 0 0 0 0 0 0 1 0 0 11 2 1 0 0 1 2 10 0 I—2 10 0 0 /21 0 0 0 0 — i 1 2 1 0 0 0 0 i 1 2 0 0 0 0- 2 V (16.30) except S03 and (16.31) (16.32) (16.33) 3 0 CF 0 0 3 0 ^ o 2 0 0 3 D 2 3 0 a 2 i 0 D 2 0 0 i a 3 0 a 2 0 1 a3 0 2 U3 0 2 i-a3 0 2 1 a3 0 2 1 a3 0 2 o3 0 0 2 0 1 a2 0 2 U2 0 2 1 a2 0 2 o2 0 2 1 a2 0 2 1 0 o 2 0 0 i-a1 0 2 1 a1 2 0 S12 = S13 = S23 = p 3 + 1 = -4iS03S12 a 1 0 0 0 0 11 2 1 0 0 0 0- 11 2 0 0 0 0 -1V 0 0 0 - 21\ 0 0 i 1 2 1 0 0 - i 1 2 1 0 0 V21 0 0 0 0 0 0 -1 0 0 11 2 1 0 0 11 2 1 0 0 V-2 10 0 0 /1 0 0 0\ 0 -10 0 0 0 -1 0 0 0 0 1 (16.34) (16.35) (16.36) (16.37) References 1. P.A.M. Dirac, "The Quantum Theory of the Electron", Proc. Roy. Soc. (London) A117 (1928) 610. 2. N.S. Mankoc Borstnik, "Spinor and vector representations in four dimensional Grassmann space", J. Math. Phys. 34, 3731-3745 (1993). 3. N.S. Mankoc Borstnik, "Can spin-charge-family theory explain baryon number non conservation?", Phys. Rev. D 91 (2015) 6, 065004 ID: 0703013. doi:10.1103; [arxiv:1409.7791, arXiv:1502.06786v1]. 4. N.S. Mankoc Borstnik, "Spin-charge-family theory is offering next step in understanding elementary particles and fields and correspondingly universe", Proceedings to the Conference on Cosmology, Gravitational Waves and Particles, IARD conferences, Ljubljana, 6-9 June 2016, The 10th Biennial Conference on Classical and Quantum Relativistic Dynamics of articles and Fields, J. Phys.: Conf. Ser. 845 012017 [arXiv:1607.01618v2]. 5. N.S. Mankoc Borstnik, "The explanation for the origin of the higgs scalar and for the Yukawa couplings by the spin-charge-family theory", J. of Mod. Phys. 6 (2015) 2244-2274. 6. N.S. Mankoc Borstnik, H.B.F. Nielsen, "How to generate spinor representations in any dimension in terms of projection operators", J. of Math. Phys. 43 (2002) 5782, [hep-th/0111257]. 7. N.S. Mankoc Borstnik, H.B.F. Nielsen, "How to generate families of spinors", J. of Math. Phys. 44 4817 (2003) [hep-th/0303224]. "Why nature made a choice of Clifford and not Grassmann coordinates", Proceedings to the 20th Workshop "What comes beyond the standard models", Bled, 9-17 of July, 2017, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA ZaloZnistvo, Ljubljana, December 2017, p. 89-120 [arXiv:1802.05554v1v2]. D. Lukman and N.S. Mankoc Borstnik, "Representations in Grassmann space and fermion degrees of freedom", [arXiv:1805.06318 ]. 8. Bled Workshops in Physics Vol. 19, No. 2 A Proceedings to the 21 st Workshop What Comes Beyond ... (p. 335) Bled, Slovenia, June 23-July 1, 2018 17 Properties of Fermions With Integer Spin Described in Grassmann Space * D. Lukman and N.S. Mankoc Borštnik1 1 Department of Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia Abstract. In Ref. [1] one of the authors (N.S.M.B.) study the second quantization of fermions with integer spin while describing the internal degrees of freedom of fermions in Grassmann space. In this contribution we study the representations in Grassmann space of the groups SO(5,1), SO(3,1), SU(3) x U(1), and SO(4), which are of particular interest as the subgroups of the group SO(13,1). The second quantized integer spin fermions, appearing in Grassmann space, not observed so far, could be an alternative choice to the half integer spin fermions, appearing in Clifford space. The spin-charge-family theory, using two kinds of Clifford operators — ya and ya — for the description of spins and charges (frst) and family quantum numbers (second), offers the explanation for not only the appearance of femilies but also for all the properties of quarks and leptons, the gauge fields, scalar fields and others [2-5]. In both cases the gauge fields in d > (13 +1) — the spin connections (of the two kinds in Clifford case and of one kind in Grassmann case) and the vielbeins faa — determine in d — (3 + 1) scalars, those with the space index a —(5,6, • • • , d), and gauge fields, those with the space index a — (0,1,2,3). While states of the Lorentz group and all its subgroups (in any dimension) are in Clifford space in the fundamental representations of the groups, with the family degrees of freedom included [2,3,1], states in Grassmann space manifest with respect to the Lorentz group adjoint representations, allowing no families. Povzetek. V clanku [1], ki uporabi za opis notranjih prostostnih stopenj fermionov Grass-mannov prostor, predstavi eden od avtorjev (N.S.M.B.) drugo kvantizacijo fermionov s celostevilskimi spini. Prispevek predstavi lastnosti upodobitev grup SO(5,1 ), SO(3,1 ), SU(3) x U(1) in SO(4) v Grassmannovem prostoru. Te grupe so posebej zanimive kot podgrupe grupe SO(13,1 ). Kreacijski in anihilacijski operatorji, ki ustreZejo komutacijskim relacijam za fermione, nosijo v Grassmannovem prostoru celostevilcni spin. Fermioni s celostevilcnim spinom ponudijo alternativni opis fermionom v Cliffordovem prostoru, ki nosijo polstevilcni spin. Opazeni so le fermioni s polstevilcnim spinom. Teorija spinov-nabojev-druzin, ki uporabi dve vrsti operatorjev y v Cliffordovem prostoru — ya in ya — prvega za opis spina in vseh nabojev in drugega za opis druzinskega kvantnega stevila, ponuja razlago ne samo za pojav druzin, ampak tudi pojasni vse lastnosti kvarkov in leptonov, umeritvenih polj, skalarnih polj in drugo [2-5]. Umeritvena polja v d > (13 +1 ) — spinske povezave œaba (dveh vrst v Cliffordovem primeru in ene vrste v Grassmannovem primeru) in "vielbeini" faa — določajo v obeh primerih v d = (3 + 1 ) skalarje, ce nosijo prostorski indeks a = (5,6, • • • , d), ter umeritvena polja, kadar imajo prostorski indeks * This contribution developed during the discussions at the 20th — Bled, 09-17 of July, 2017 — and 21st — Bled, 23 of June to 1 of July — Workshops "What Comes Beyond the Standard Models", Bled, 09-17 of July, 2017. a = (0,1,2,3). Stanja Lorentzove grupe in vseh njenih podgrup so za poljubno dimenzijo v Cliffordovem prostoru v fundamentalni upodobitvi in vključujejo družinske prostostne stopnje [2,3,1], v Grassmannovem prostoru pa so glede na Lorentzovo grupo v adjungirani upodobitvi in ne dopusčajo družin. Keywords: Spinor representations in Grassmann space, Second quantization of fermion fields in Grassmann space, Higher dimensional spaces, Kaluza-Klein theories, Beyond the standard model PACS:04.50.-Cd, 11.10.Kk, 11.25.Mj, 11.30.Hv, 12.10.-g, 12.60.-i 17.1 Introduction In Ref. [2] the representations in Grassmann and in Clifford space were discussed. In Ref. ([1] and the references therein) the second quantization procedure in both spaces — in Clifford space and in Grassmann space — were discussed in order to try to understand "why nature made a choice of Clifford rather than Grassmann space" during the expansion of our universe, although in both spaces the creation operators bj and the annihilation operators bj exist fulfilling the anticommutation relations required for fermions [1] {bi,6j}+|^o > = 5ij l^o >, {6i,6j}+|^o > = 0 |^o >, {6{,6]}+|^o > = 0 |^o >, bjl^o > = > > = 0|^o >. (17.1) |^o > is the vacuum state. We use |^o >= |1 >. The creation operators can be expressed in both spaces by products of eigen-states of the Cartan subalgebra, Eq. (17.33), of the Lorentz algebra, Eqs. (17.3,17.11). Starting with one state (Ref. [1]) all the other states of the same representation are reachable by the generators of the Lorentz transformations (which do not belong to the Cartan subalgebra), with Sab presented in Eq. (17.32) in Grassmann space and with either Sab or Sab, Eq. (17.34), in Clifford space. But while there are in Clifford case two kinds of the generators of the Lorentz transformations — Sab and Sab, the first transforming members of one family among themselves, and the second transforming one member of a particular family into the same member of other families — there is in Grassmann space only one kind of the Lorentz generators — Sab. Correspondingly are all the states in Clifford space, which can be second quantized as products of nilpotents and projectors [9,10,1], reachable with one of the two kinds of the operators Sab and Sab, while different representations are in Grassmann space disconnected. On the other hand the vacuum state is in Grassmann case simple — |^o >= |1 > — while in Clifford case is the sum of products of projectors, Eq. (17.17). In Grassmann space states are in the adjoint representations with respect to the Lorentz group, while states in Clifford space belong to the fundamental representations with respect to both generators, Sab and Sab, or they are singlets. Correspondingly are properties of fermions, described with the spin-charge-family theory [3,4,6,5,8,7], which uses the Clifford space to describe fermion degrees of freedom, in agreement with the observations, offering explanation for all the assumptions of the standard model (with families included) and also other observed phenomena. In Grassmann case the spins manifest, for example, in the case of SO(6) or SO(5,1) decuplets or singlets — triplets and singlets in Clifford case, Table 17.2 — while with respect to the subgroups SU(3) and U(1) of SO(6) the states belong to either singlets, or triplets or sextets, Tables 17.3,17.4 — triplets and singlets in the Clifford case. In what follows we discuss representations, manifesting as charges and spins of fermions, of subgroups of SO(13,1), when internal degrees of freedom of fermions are described in Grassmann space and compare properties of these representations with the properties of the corresponding representations appearing in Clifford space. We assume, as in the spin-charge-family theory, that both spaces, the internal and the ordinary space, have d = 2(2n + 1 )-dimensions, n is positive integer, d > 14 and that all the degrees of freedom of fermions and bosons originate in d = 2(2n + 1), in which fermions interact with gravity only. After the break of the starting symmetry SO (13,1) into SO(7,1)xSU(3)xU(1), and further to SO(3,1) x SU(2) x SU(2) x SU(3) x U(1), fermions manifest in d = (3 + 1) the spin and the corresponding charges and interact with the gauge fields, which are indeed the spin connections with the space index m = (0,1,2,3), originating in d = (13,1) [7]. Also scalar fields originate in gravity: Those spin connections with the space index a = (5,6,7,8) determine masses of fermions, those with the space index a = (9,10,...,14) contribute to particle/antiparticle asymmetry in our universe [4]. We pay attention on fermion fields, the creation and annihilation operators of which fulfill the anticommutation relations of Eq. (17.1). 17.1.1 Creation and annihilation operators in Grassmann space In Grassmann d = 2(2n + 1)-dimensional space the creation and annihilation operators follow from the starting two creation and annihilation operators, both with an odd Grassmann character, since those with an even Grassmann character do not obey the anticommutation relations of Eq. (17.1) [1] = (-—=)d (00 - 03)(01 + i02)(05 + i06) ••• (0d-1 + i0d), i2 £ei i 1 i-d r 3 .9 d d bl = ( —2* 2 ( 90d-1 - i"aed * •••(- "ae3*, b®2t = (-—=)d (e0 + e3)(01 + ie2)(e5 + ie6)••• (ed-1 + ied), v2 = (^*d (led-r - ^* • • • ^ + w*. (17.2) All the creation operators are products of the eigenstates of the Cartan subalgebra operators,Eq. (17.33) S ab fna (ea ± eeb) = (ea ± eeb), e e = 1 , for naa = 1 , e = i, for naa = -1 , Sab(eaeb ± eeced) =0, Scd(eaeb ± eeced) =0. (17.3) The two creation operators, b®11" and b®21", if applied on the vacuum state, form the starting two states and of the two representations, respectively. The vacuum state is chosen to be the simplest one [1] — |$0 >= |1 >. The rest of creation operators of each of the two groups, be1t and be2t , follow from the starting one by the application of the generators of the Lorentz transformations in Grassmann space Sab, Eq. (17.32), which do not belong to the Cartan subalgebra, Eq. (17.33), of the Lorentz algebra. They generate either > of the first group or > of the second group. Annihilation operators b®1 and b®2 follow from the creation ones by the Hermitian conjugation [1], when taking into account the assumption (ea)f _^naa = 3ea -ipeanaa: from where it follows )f = naa ea, 3ea ea^t _ (Pea) -inaaec (17.4) (17.5) The annihilation operators b®1 and b®2 annihilate states > and |$2 >, respectively. The application of S01 on b®11, for example, transforms this creation operator into b®1t _ (-j)d-1 (e0 e3 + ie1e2) (e5 + ie6) • •• (ed-1 - ied). Correspondingly its Hermitian conjugate annihilation operator is equal to 6®1 _ (-1)d-1 ( i d )... (_9__9__i_9__) i aed> ae3 ae0 i ae2 ae1 3 aed -1 a®4) • • •( ae3 ae0 i ae2 ae1 All the states are normalized with respect to the integral over the Grassmann coordinate space [2] < $ta|$b > _ dd-1xddea w < <|e >< e|$b >_ sab 5tj, 3 aek w _ + ek), (17.6) where w is a weight function, defining the scalar product < |$b >, and require that [2] we {dea,eb}+ _ o, dde e0e1•••ed _ 1, dea _ o, deaea _ 1, dde _ ded...de0, (17.7) with ec = nac. There are 1 in each of these two groups of creation operators of an odd 2 2 2 Grassmann character in d = 2(2n + 1)-dimensional space. The rest of creation operators (and the corresponding annihilation operators) would have rather opposite Grassmann character than the ones studied so far: like a. e0e1 for the creation operator and [g|r gfo ] for the corresponding annihilation operatorin d = (1 +1) (since{e0e1, gfr gfo}+ gives (1 +(1 + 1)e0e1 gfr gfo )),and like b. (e0Te3)(e1 ±ie2) for creation operator and [(gfr Tigfr)(gfo T gfr)] for the annihilation operator, or e0e3e1 e2 for the creation operator and [gf- gfy gfr gfo ] for the annihilation operator in d = (3 + 1) (since, let say, {2 (e0 — e3)(e1 + ie2), 1 (afr —idf)(df — df)}+ gives (1+ 4(1+1)(e0 — e3)(e1 + ie2)(afr —idf)(afo — gfr) and equivalently for other cases), but applied on a vacuum states some of them still fulfill some of the relations of Eq. (17.1), but not all (like {2 (e0 — e3)(e1 + ie2), 2(e0 + e3)(e1 — ie2)}+ = ie0e1e2e3, while it should be zero2). Let us add that, like in Clifford case, one can simplify the scalar product in Grassmann case by recognizing that the scalar product is equal to Sab 5tj < $a|e >< e|$b > = Sab Stj, (17.8) without integration over the Grassmann coordinates. Let us manifest this in the case of d = (1 + 1):< 1|^(gfo — gfr)^(e0 — e1)11 >= 1, |1 > is the normalized vacuum state, < 111 >= 1. It is true in all dimensions, what can easily be understood for all the states, which are defined by the creation operators on the vacuum state |1 >, |^b >= b{|1 >, fulfilling the anticommutation relations of Eq. (17.1). 17.1.2 Creation and annihilation operators in Clifford space There are two kinds of Clifford objects [2], ([3] and Refs. therein), Ya and Ya, both fulfilling the anticommutation relations {Ya,Yb}+ = 2nab = {Y a,Y b}+ , {Ya,Y b}+ = 0. (17.9) Both Clifford algebra objects are expressible with ea and -dfr [2,1], ([3] and Refs. therein) Ya = (ea + ), 3ea Ya = i (ea — ^), 3ea ea = 1 (Ya — iYa), 7\ 1 ^ = 1 (Ya + iYa), (17.10) 3ea 2 from where it follows: (-a)f = Yanaa, (Ya)f = Yanaa, YaYa = naa, Ya(Ya)f = 1, Y aYa = naa, Y a(Y a)f = 1. Correspondingly we can use either Ya or Ya instead of 0a to span the internal space of fermions. Since both, ya and Ya, are expressible with 9a and the derivatives with respect to 0a, the norm of vectors in Clifford space can be defined by the same integral as in Grassmann space, Eq.(17.6), or we can simplify the scalar product (as in the Grassmann case, Eq. (17.8) by introducing the Clifford vacuum state |^oc >, Eq. (17.17), instead of |1 > in Grassmann case. We make use of Ya to span the vector space. As in the case of Grassmann space we require that the basic states are eigenstates of the Cartan subalgebra operators of Sab and Sab, Eq. (17.33). ab 1 naa abt ab (k): = 2(Ya + n^), (k) = naa (-k), ab 1 i ab1 ab [k]: = ^(1 + -yV), [k]=[k], ab 1 ab ab 1 ab Sab (k) = ^k (k), Sab [k]= ^k [k], ab 1 ab ab 1 ab Sab (k) = ^k (k), Sab [k]= -2k [k], (17.11) ab ab with k2 = naanbb .To calculate S ab (k) and S ab [k] we use [10,9] the relation on any Clifford algebra object A as follows (Y~aA = iR(A)AYa)|^oc >, (17.12) where A is any Clifford algebra object and (-)(A) = —1, if A is an odd Clifford algebra object and (—)(A) = 1, if A is an even Clifford algebra object, |^oc > is the vacuum state, replacing the vacuum state |^o >= |1 >, used in Grassmann case, with the one of Eq. (17.17), in accordance with the relation of Eqs. (17.10,17.6, 17.7), Ref. [1]. We can define now the creation and annihilation operators in Clifford space so that they fulfill the requirements of Eq. (17.1). We write the starting creation operator and its Hermitian conjugate one (in accordance with Eq. (17.11) and Eq.(17.33)) in 2(2n + 1 )-dimensional space as follows [1] 03 12 56 d-1 d 61* = (+i)(+)(+) ••• (+) , d-1 d 56 12 03 61 =(—)••• (—)(—)(—i) . (17.13) The starting creation operator 61*, when applied on the vacuum state |^oc >, defines the starting family member of the starting "family". The corresponding starting annihilation operator is its Hermitian conjugated one, Eq. (17.11). All the other creation operators of the same family can be obtained by the application of the generators of the Lorentz transformations Sab, Eq. (17.34), which do not belong to the Cartan subalgebra of SO(2(2n + 1) — 1,1), Eq. (17.33). 61* = b1t|^°c > are normalized, where |^°c > is the normalized vacuum state, < ^°c|^°c >= 1. The creation operators creating different "families" with respect to the starting "family", Eq. (17.13), can be obtained from the starting one by the application of Sab, Eq. (17.34), which do not belong to the Cartan subalgebra of SO(2(2n + 1 ) — 1,1), Eq. (17.33). They all keep the "family member" quantum number unchanged. bTat « S ab ••• Sef b!f. (17.15) Correspondingly we can define (up to the proportionality factor) any creation operator for any "family" and any "family member" with the application of Sab and Sab [1] bat « S ab ••• S efSmn ••• Sprb 1 t « Smn ••• Sprb 1 tSab ••• Sef . (17.16) All the corresponding annihilation operators follow from the creation ones by the Hermitian conjugation. There are 2 d _ 1 x 2 d _ 1 creation operators of an odd Clifford character and the same number of annihilation operators, which fulfill the anticommutation relations of Eq. (17.1) on the vacuum state |^°c > with 2d_1 summands |^°c >= 03 12 56 d_1 d 03 12 56 d_1 d 03 12 56 d_1 d a([—i][—][—] ••• [—] +[+i][+][—] ••• [—] +[+i][—][+] ••• [—] + ••• )|0>, 1 •v — -- « for d = 2(2n + 1), (17.17) n is a positive integer. For a chosen a = , d the vacuum is normalized: v 2 d-1 < ^°c|^°c > = 1. It is proven in Ref. [1] that the creation and annihilation operators fulfill the anticommutation relations required for fermions, Eq. (17.1). 17.2 Properties of representations of the Lorentz group SO(2(2n + 1 )) and of subgroups in Grassmann and in Clifford space The purpose of this contribution is to compare properties of the representations of the Lorentz group SO(2(2n + 1)), n > 3, when for the description of the internal degrees of freedom of fermions either i. Grassmann space or ii. Clifford space is used. The spin-charge-family theory ([6,5,3,4,8,7,11] and the references therein) namely predicts that all the properties of the observed either quarks and leptons or vector gauge fields or scalar gauge fields originate in d > (13 + 1 ), in which massless fermions interact with the gravitational field only — with its spin connections and vielbeins. However, both — Clifford space and Grassmann space — allow second quantized states, the creation and annihilation operators of which fulfill the anticommutation relations for fermions of Eq. (17.1). But while Clifford space offers the description of spins, charges and families of fermions in d = (3 + 1), all in the fundamental representations of the Lorentz group SO (13,1) and the subgroups of the Lorentz group, in agreement with the observations, the representations of the Lorentz group are in Grassmann space the adjoint ones, in disagreement with what we observe. We compare properties of the representations in Grassmann case with those in Clifford case to be able to better understand "the choice of nature in the expanding universe, making use of the Clifford degrees of freedom", rather than Grassmann degrees of freedom. In introduction we briefly reviewed properties of creation and annihilation operators in both spaces, presented in Ref. [1] (and the references therein). We pay attention on spaces with d = 2(2n + 1) of ordinary coordinates and d = 2(2n + 1) internal coordinates, either of Clifford or of Grassmann character. i. In Clifford case there are 2 2 -1 creation operators of an odd Clifford character, creating "family members" when applied on the vacuum state. We choose them to be eigenstates of the Cartan subalgebra operators, Eq.(17.33), of the Lorentz algebra. All the members can be reached from any of the creation operators by the application of Sab, Eq. (17.34). Each "family member" appears in 22-1 "families", again of an odd Clifford character, since the corresponding creation operators are reachable by Sab, Eq. (17.34), which are Clifford even objects. There are correspondingly 2 2 -1 • 2 2 -1 creation and the same number (2 2 -1 • 2 2 -1) of annihilation operators. Also the annihilation operators, annihilating states of 2 t -1 "family members" in 2 2-1 "families", have an odd Clifford character, since they are Hermitian conjugate to the creation ones. The rest of 2- 2 2 -1 • 2 2 -1 members of the Lorentz representations have an even Clifford character, what means that the corresponding creation and annihilation operators can not fulfill the anticommutation relations required for fermions, Eq. (17.1). Among these 22-1 products of projectors determine the vacuum state, Eq. (17.17). ii. In Grassmann case there are -¿7x7 operators of an odd Grassmann 2 ■ 2 ■ character, which form the creation operators, fulfilling with the corresponding annihilation operators the requirements of Eq. (17.1). All the creation operators are chosen to be products of the eigenstates of the Cartan subalgebra Sab, Eq. (17.33). The corresponding annihilation operators are the Hermitian conjugated values of the creation operators, Eqs. (17.4,17.5,17.2). The creation operators form, when applied on the simple vacuum state |^o >= |1 >, two independent groups of states. The members of each of the two groups are reachable from any member of a group by the application of Sab, Eq. (17.32). All the states of any of the two decuplets are orthonormalized. We comment in what follows the representations in d = (13 + 1) in Clifford and in Grassmann case. In spin-charge family theory there are breaks of the starting symmetry from SO(13,1) to SO(3,1) x SU(2) x SU(3) x U(1) in steps, which lead to the so far observed quarks and leptons, gauge and scalar fields and gravity. One of the authors (N.S.M.B.), together with H.B. Nielsen, defined the discrete symmetry operators for Kaluza-Klein theories for spinors in Clifford space [19]. In Ref. [1] the same authors define the discrete symmetry operators in the case that for the description of fermion degrees of freedom Grassmann space is used. Here we comment symmetries in both spaces for some of subgroups of the SO(13,1) group, as well as the appearance of the Dirac sea. 17.2.1 Equations of motion in Grassmann and Clifford space We define [1] the action in Grassmann space, for which we require — similarly as in Clifford case — that the action for a free massless object 9 , 1 A = 2 < ddx dde w - 29°-— ) - (eapa + naaeatpaM, (17.18) is Lorentz invariant. The corresponding equation of motion is 1 7\ 7\ 2[(1 - 200âëô) 0a + - 200^) 0a)t] > = 0> (m9) Pa = i gfr, leading to the Klein-Gordon equation {(1 - 200 )0aPa}t 0bpb|^f > = papa|^f >= 0. (17.20) In the Clifford case the action for massless fermions is well known A = ddx - (^Y° Yapah.c., (17.21) leading to the equations of motion YaPal^a > = 0, (17.22) which fulfill also the Klein-Gordon equation YaPaYbPbl^f > = papaic >= 0. (17.23) 17.2.2 Discrete symmetries in Grassmann and Clifford space We follow also here Ref. [1] and the references therein. We distinguish in d-dimensional space two kinds of dicsrete operators C, P and T operators with respect to the internal space which we use. In the Clifford case [19], when the whole d-space is treated equivalently, we have Ch = n Ya K, TH = Y° n Ya KIx0 , PHd-1) = Y° I« , Yaea yaen Ixxa = —xa , Ixoxa = (—x°, x), I«x = —x, I«3xa = (x°, —x1, —x2, —x3,x5,x6,...,xd). (17.24) The product ^ Ya is meant in the ascending order in ya. In the Grassmann case we correspondingly define cg = n Yg K, tg = yg n Yg KIx TgeaTa with yG defined as ygeKy" -p(d-l)___0 ' G = yG Ix (17.25) yG = (1 - 29ana^), Oöa (17.26) while Ix, Ix3 is defined in Eq. (17.24). Let be noticed, that since yG (= —inaa YaYa) is always real as there is YaiYa, while Ya is either real or imaginary, we use in Eq. (17.25) Ya to make a choice of appropriate yQ. In what follows we shall use the notation as in Eq. (17.25). We define, according to Ref. [1] (and the references therein) in both cases — Clifford Grassmann case — the operator "emptying" [6,5] (arxiv:1312.1541) the Dirac sea, so that operation of "emptyingN" after the charge conjugation CH in the Clifford case and "emptyingG" after the charge conjugation CG in the Grassmann case (both transform the state put on the top of either the Clifford or the Grassmann Dirac sea into the corresponding negative energy state) creates the anti-particle state to the starting particle state, both put on the top of the Dirac sea and both solving the Weyl equation, either in the Clifford case, Eq. (17.22), or in the Grassmann case, Eq. (17.19), for free massless fermions "emptyingN" = Ya K in Clifford space, "emptyingG" = ^ yG K in Grassmann space, (17.27) although we must keep in mind that indeed the anti-particle state is a hole in the Dirac sea from the Fock space point of view. The operator "emptying" is bringing the single particle operator Ch in the Clifford case and CG in the Grassmann case into the operator on the Fock space in each of the two cases. Then the anti-particle state creation operator — [¥p] — to the corresponding particle state creation operator — can be obtained also as follows ¥fa[¥p] |vac > = CH¥p[¥p] |vac >= Y^X) (Ch V*)) d(d-1)x |vac > , Ch = "emptyingN" • CH in both cases. The operators CH and CG (17.28) Ch = "emptyingN" • CH , cg = "emptyingng" • cg , (17.29) operating on ¥p (x) transforms the positive energy spinor state (which solves the corresponding Weyl equation for a massless free fermion) put on the top of the Dirac sea into the positive energy anti-fermion state, which again solves the corresponding Weyl equation for a massless free anti-fermion put on the top of the Dirac sea. Let us point out that either the operator "emptyingN" or the operator "emptyingNG" transforms the single particle operator either CH or CG into the operator operating in the Fock space. We use the Grassmann even, Hermitian and real operators yG, Eq. (17.26), to define discrete symmetry in Grassmann space, first we did in ((d + 1) — 1) space, Eq. (17.25), now we do in (3 + 1) space, Eq. (17.30), as it is done in [19] in the Clifford case. In the Grassmann case we do this in analogy with the operators in the Clifford case [19] m 'x6x8...xa Cng = n Ym KI yjie^y™ Tng = Yg H KIx0 Ix5x7...xa-1 , y^eay™ Png-i)=Yg n YgIx , s = 5 Cng = YG >Ix6x8...xa > yG eny! d Cng^nq1' = Yg n Yg IX3 Ix6x8...xa ) TGe3Ys,s=5 CNGTNGPNd-1' = n YG IxK. (17.30) YGe3Ya 17.2.3 Representations in Grassmann and in Clifford space in d = (13 + 1) In the spin-charge-family theory the starting dimension of space must be > (13 + 1), in order that the theory manifests in d = (3 + 1) all the observed properties of quarks and leptons, gauge and scalar fields (explaining the appearance of higgs and the Yukawa couplings), offering as well the explanations for the observations in cosmology. Let us therefore comment properties of representations in both spaces when d = (13 + 1), if we analyze one group of "family members" of one of families in Clifford space, and one of the two representations of 2 d! 2 a!a!• a. Let us start with Clifford space [3,5,4,6,13,12,2]. Each "family" representation has 2a-1 = 64 "family members". If we analyze this representation with respect to the subgroups SO(3,1), (SU(2) x SU(2)) of SO(4) and (SU(3)x U(1)) of SO(6) of the Lorentz group SO(13,1), we find that the representations have quantum numbers of all the so far observed quarks and leptons and antiquarks and antileptons, all with spin up and spin down, as well as of the left and right handedness, with the right handed neutrino included as the member of this representation. Let us make a choice of the "family", which follows by the application of S15 on the "family", for which the creation operator of the right-handed neutrino 03 12 56 78 9 10 11 12 13 14 with spin 2 would be (+i) (+) | (+) (+) || (+) (+) (+) . (The corresponding 13 14 11 12 9 10 78 56 12 annihilation operator of this creation operator is (—) (—) (—) || (—)(—) | (—) 03 (—i)). In Table 6.3 (see pages 112-113 in this volume) presented creation operators for all the "family members" of this family follow by the application of Sab 03 12 56 78 9 10 11 12 13 14 on S15 (+i) (+) | (+) (+) || (+) (+) (+) . (The annihilation operator of S15 03 12 56 78 9 10 11 12 13 14 13 14 11 12 9 10 78 56 12 03 (+i) (+) I (-6) (-8) || (+) (+) (+) is [—][—](—) || (—8) [+] | [+] (—i).) This is the representation of Table 6.3 (see pages 112-113 in this volume), in which all the 'family members" of one "family" are classified with respect to the subgroups SO(3,1) x SU(2) x SU(2) x SU(3) x U(1). The vacuum state on which the creation operators, represented in the third column, apply is defined in Eq. (17.17). All the creation operators of all the states are of an odd Clifford character, fulfilling together with the annihilation operators (which have as well the equivalent odd Clifford character, since the Hermitian conjugation do not change the Clifford character) the requirements of Eq. (17.1). Since the Clifford even operators Sab and S ab do not change the Clifford character, all the creation and annihilation operators, obtained by products of Sab or S ab or both, fulfill the requirements of Eq. (17.1). We recognize in Table 6.3 (see pages 112-113 in this volume) that quarks distinguish from leptons only in the SO(6) part of the creation operators. Quarks belong to the colour (SU(3)) triplet carrying the "fermion" (U(1)) quantum number t4 = g, antiquarks belong to the colour antitriplet, carrying the "fermion" quantum number t4 = — 1. Leptons belong to the colour (SU(3)) singlet, carrying the "fermion" (U(1)) quantum number t4 = — 2, while antileptons belong to the colour antisinglet, carrying the "fermion" quantum number t4 = 1. Let us also comment that the oddness and evenness of part of states in the subgroups of the SO (13,1) group change: While quarks and leptons have in the part of SO(6) an odd Clifford character, have antiquarks and antileptons in this part an even odd Clifford character. Correspondingly the Clifford character changes in the rest of subgroups. Families are generated by Sab applying on any one of the "family members". Again all the "family members" of this "family" follow by the application of all Sab (not belonging to Cartan subalgebra). The spontaneous break of symmetry from SO (13,1) to SO (7,1) xSU(3) xU(1), Refs. [3-5], makes in the spin-charge-family theory all the families, generated by Smt and Sst, [m = (0,1,2,3), s = (5,6,7,8),t = (9,10,11,12,13,14)], massive of the scale of > 1016 GeV [14-16]. Correspondingly there are only eight families of quarks and leptons, which split into two groups of four families, both manifesting the symmetry SU(2) x SU(2) xU(1). (The fourth of the lower four families is predicted to be observed at the LHC, the stable of the upper four families contributes to the dark matter [17].) In the spin-charge-family theory fermions interact with only gravity, which manifests after the break of the starting symmetry in d = (3 + 1) as all the known vector gauge fields, ordinary gravity and the higgs and the Yukawa couplings [7,3- 5,11]. There are scalar fields which bring masses to family members. The theory explains not only all the assumptions of the standard model with the appearance of families, the vector gauge fields and the scalar fields, it also explains appearance of the dark matter [17], matter/antimatter asymmetry [4] and other phenomena, like the miraculous cancellation of the triangle anomalies in the standard model [8]. b. We compare representations of SO(13,1) in Clifford space with those in Grassmann space. We have no "family" quantum numbers in Grassmann space. We only have two groups of creation operators, defining — when applied on the vacuum state |1 > — 2 ¿fdr equal in d = (13 + 1) to 1716 members in each of 2 2 ' 2 ' the two groups in comparison in Clifford case with 64 "family members" in one "family" and 64 "families", which the breaks of symmetry reduce to 8 "families", making all the (64 — 8) "families" massive and correspondingly not observable at low energies ([5,14] and the references therein). Since the 1716 members are hard to be mastered, let us look therefore at each subgroup — SU(3) x U(1), SO(3,1) and SU(2) x SU(2) of SO(13,1) —separately. Let us correspondingly analyze the subgroups: SO (6) from the point of view of the two subgroups SU(3) x U(1), and SO(7,1) from the point of view of the two subgroups SO(3,1) x SO(4), and let us also analyze SO(4) as SU(2) x SU(2). 17.2.4 Examples of second quantizable states in Grassmann and in Clifford space We compare properties of representations in Grassmann and in Clifford space for several choices of subgroups of SO (13,1) in the case that in both spaces creation and annihilation operators fulfill requirements of Eq. (17.1), that is that both kinds of states can be second quantized. Let us again point out that in Grassmann case fermions carry integer spins, while in Clifford case they carry half integer spin. States in Grassmann and in Clifford space for d = (5 +1) We study properties of representations of the subgroup SO (5,1) (of the group SO (13,1)), in Clifford and in Grassmann space, requiring that states can be in both spaces second quantized, fulfilling therefore Eq. (17.1). a. In Clifford space there are 2 d -1, each with 2 d -1 family members, that is 4 families, each with 4 members. All these sixteen states are of an odd Clifford character, since all can be obtained by products of Sab, Sab or both from an Clifford odd staring states and are correspondingly second quantizable as required in Eq. (17.1). All the states are the eigenstates of the Cartan subalgebra of the Lorentz algebra in Clifford space, Eq. (17.33), solving the Weyl equation for free massless spinors in Clifford space, Eq. (17.22). The four familes, with four members each, are presented in Table 17.1. All of these 16 states are reachable from the first one in each of the four families by Sab, or by S ab if aplied on any family member. Each of these four families have positive and negative energy solutions, as presented in [19], in Table I.. We present in Table 17.1 only states of a positive energy, that is states above the Dirac sea. The antiparticle states are reachable from the particle states by the application of the operator Cn Pj/-1) = Y°Y5Ix3 Ix6, keeping the spin 2, while changing the charge from 1 to — 2. All the states above the Dirac sea are indeed the hole in the Dirac sea, as explained in Ref. [19]. S°3 S12 S56 S °3 S12 S56 °3 12 56 °3 12 56 [—i][—](+) °3 12 56 °3 12 56 (+i)[—][—] i 2 i 2 i 2 i i 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 i 2 i 2 i 2 i 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 °3 12 56 [+i][+](+) °3 12 56 (-i)(-)(+) °3 12 56 (—i)[+][—] °3 12 56 [+i](—)[—] °3 12 56 i 2 i 2 i 2 i 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 i 2 i 2 i 2 i 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 %4I1 [+i](+)[+] °3 12 56 °3 12 56 °3 12 56 [+i][—](—) °3 12 56 i 2 i 2 i 2 i 1 2 1 2 1 2 1 1 2 1 2 1 2 1 2 i 2 i 2 i 2 i 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 %3V %4V (+i)[+][+] °3 12 56 [—i](-)[+] °3 12 56 [—i][+](—) °3 12 56 (+i)(-)(-) i 2 i 2 i 2 i 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 i 2 i 2 i 2 i 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 Table 17.1. The four families, each with four members. For the choice p a = (p0,0, 0,p3,0,0) have the first and the second member the space part equal to e-i|p |x +i|p |x and e-i|p0|x0-i|p3|x3, representing the particles with spin up and down, respectively. The third and the fourth member represent the antiparticle states, with the space part equal to e-i|p0|x°-i|p3|x3 and e-i|p0|x0+i|p3|x3, with the spin up and down respectively. The antipar-ticle states follow from the particle state by the application of Cn P' = Y°Y5IX3 Ix6. The charge of the particle states is 2, for antiparticle states — 2. b.0 In Grassmann space there are -¿7x7 second quantizable states as required 2 ' 2 ' in Eq. (17.1), forming in d = (5 + 1) two decuplets — each with 2 ¿ddr states — 2 '2 ' all are the eigenstates of the Cartan subalgebra of the Lorentz algebra in (internal) Grassmann space. All the states of one (anyone of the two) decuplets are reachable by the application of the operators Sab on a starting state. The two decouplets are presented in Table 17.2 Let us first find the solution of the equations of motion for free massless fermions, Eq. (17.19), with the momentum pa = (p°,p1 ,p2,p3,0,0). One obtains for % = a(e° — e3)(e1 + ie2)(e5 + ie6) +p(e°e3 + ie1e2)(e5 + ie6)+ y(0° + (17.31) e3)(e1 - ie2)(e5 + ie6) the solution R _ 2y(p1 - ip2) _ 2y(p0 + p3) _ -2a(p0 - p3) _ -2a(p1 + ip2) 1 (p0 - p3) (p1 + ip2) (p1 - ip2) (p0 + p3) (p0)2 _(pY + (p2)2 + (p3)2 , _ 2(p0 - p3) ^ _ (p0 - p3)2 . -a (p1 - ip2) ' -a (p1 - ip2)2 One has for p0 _ |p01 the positive energy solution, describing a fermion above the "Dirac sea", and for p0 _ — |p01 the negative energy solution, describing a fermion in the "Dirac sea". The "charge" of the "fermion" is 1. Similarly one finds the solution for the other three states with the negative "charge" -1 , again with the positive and negative energy. The space part of the "fermion" state is for "spin up" equal to e-i|p |x +ipx, for his antiparticle for the same internal spin e-i|p |x -ipx. The discrete symmetry operator CNG PNG-1 ', which is in our case equal to YqYgIx3 Ix6, transforms the first state in Table 17.2 into the sixth, the second state into the fifth, the third state into the fourth, keeping the same spin while changing the "charge" of the superposition of the three states ^Ip. Both superposition of states, Eq. (17.31) represent the positive energy states put on the top of the "Dirac" sea, the first describing a particle with "charge" 1 and the second superposition of the second three states ^Ia, describing the antiparticle with the"charge" -1. We namely apply Cm pfe1' °n —P [—?°s] by applying Cm pN-15 on —?°s as follows: Cngrn%-1) —p[—PH (Cng PNdG-1))-1 _ — aNG[Cn g PNiG-1)vP°S].One recognizes that it is Cng pug1 5 —P°S _ —i°S (Table 17.2), which must be put on the top of the "Dirac" sea, representing the hole in the particular state in the "Dirac" sea, which solves the corresponding equation of motion for the negative energy. Properties of SO(6) in Grassmann and in Clifford space when SO(6) is embedded into SO(13,1) a. Let us first repeat properties of the SO(6) part of the SO(13,1) representation of 64 "family members" in Clifford space, presented in Table 6.3 (see pages 112-113 in this volume). As seen in Table 6.3 (see pages 112- d 9 10 1112 13 14 113 in this volume) there are one quadruplet (2d-1 _ 4) — ((+) [-] [-] , 910 1112 1314 910 1112 1314 910 1112 1314 [-] (+) [-] , [-] [-] (+), (+) (+) (+) ), representing quarks and leptons 910 1112 1314 910 1112 1314 910 1112 1314 — and one antiquadruplet — ([-] (+) (+), (+) [-] (+), (+) (+) [-] , 9 10 1112 13 14 [-] [-] [-] ), representing antiquarks and antileptons, which both belong to the 64th-plet, if SO(6) is embedded into SO(13,1). The creation operators (and correspondingly their annihilation operators) have for 32 members (representing quarks and leptons) the SO(6) part of an odd Clifford character (and can be correspondingly second quantized (by itselves [1] or) together with the rest of space, manifesting SO(7,1) (since it has an even Clifford character). The rest of 32 creation operators (representing antiquarks and antileptons) has in the SO(6) part an even Clifford character and correspondingly in the rest of the Clifford space in SO(7,1) an odd Clifford character. I decuplet S S12 S 1 (tfU - tf3)^1 + itf2)(tf5 + itfb) i 1 1 2 (tfUtf3 + itf^2)^5 + itfb) 0 0 1 3 (tfU + tf3)^1 - itf2)(tf5 + itfb) -i -1 1 4 (tfU - tf3)^1 -itf2)(tf5 - itfb) i -1 -1 5 (tfUtf3 - itf^2)^5 - itfb) 0 0 -1 6 (tfU + tf3)^1 +itf2)(tf5 - itfb) -i 1 -1 7 (tfU - tf3)^2 + tf5tfb) i 0 0 8 (tfU + tf3)^2 - tf5tfb) -i 0 0 9 (tfUtf3 + itf^Xtf1 + itf2) 0 1 0 10 (tfUtf3 - ¿tf5tfb)(tf1 - itf2) 0 -1 0 II decuplet SU3 S12 S5b 1 (tfU + tf3)(tf1 + itf2)(tf5 + itfb) -i 1 1 2 (tfUtf3 - ¿tf1tf2)(tf5 + itfb) 0 0 1 3 (tfU - tf3)(tf1 - itf2)(tf5 + itfb) i -1 1 4 (tfU + tf3)(tf1 -itf2)(tf5 - itfb) -i -1 -1 5 (tfUtf3 + ¿tf1tf2)(tf5 - itfb) 0 0 -1 6 (tfU - tf3)(tf1 +itf2)(tf5 - itfb) i 1 -1 7 (tfU + tf3)(tf1tf2 + tf5tfb) -i 0 0 8 (tfU - tf3)(tf1tf2 - tf5tfb) i 0 0 9 (tfUtf3 - ¿tf5tfb)(tf1 + itf2) 0 1 0 10 (tfUtf3 + ¿tf5tfb)(tf1 - itf2) 0 -1 0 cgn cqnp>gj-1) cgnpgn^ 4 \ Table 17.2. The creation operators of the decuplet and the antidecuplet of the orthogonal group SO(5,1) in Grassmann space are presented. Applying on the vacuum state >= |1 > the creation operators form eigenstates of the Cartan subalgebra, Eq. (17.33), (S03,S12, S56). The states within each decuplet are reachable from any member by Sa . The product of the discrete operators Cng (= OYg Ix6x8 xd, denoted as C in the last column) PNG ' (= Yg nf=5 Yg Ix3 ) transforms, for example, into ^6, ^2 into ^5 and ^3 into ^4. Solutions of the Weyl equation, Eq. (17.19), with the negative energies belong to the "Grassmann sea", with the positive energy to the particles and antiparticles. Also the application of the discrete operators CGN, Eq. (17.30) and CNG Png-1', Eq. (17.30) is demonstrated. Let us discuss the case with the quadruplet of SO (6) with an odd Clifford character. From the point of view of the subgroups SU(3) (the colour subgroup) and U(1) (the U(1) subgroup carrying the "fermion" quantum number), the quadruplet consists of one SU(3) singlet with the "fermion" quantum number — 2 and one triplet with the "fermion" quantum number 6. The Clifford even SO(7,1) part of SO (13,1) define together with the Clifford odd SO (6) part the quantum numbers of the right handed quarks and leptons and of the left handed quarks and leptons of the standard model, the left handed weak charged and the right handed weak chargeless. In the same representation of SO (13,1) there is also one antiquadruplet, which has the even Clifford character of SO (6) part and the odd Clifford character in the SO(7,1) part of the SO(13,1). The antiquadruplet of the SO(6) part consists of one SU(3) antisinglet with the "fermion" quantum number 1 and one antitriplet with the "fermion" quantum number — 1. The SO(7,1) x SO(6) antiquadruplet of SO(13,1) carries quantum numbers of left handed weak chargeless antiquarks and antileptons and of the right handed weak charged antiquarks and antileptons of the standard model. Both, quarks and leptons and antiquarks and antileptons, belong to the same representation of SO (13,1), explaining the miraculous cancellation of the triangle anomalies in the standard model without connecting by hand the handedness and the charges of quarks and leptons [8], as it must be done in the SO (10) models. b. In Grassmann space there are one (2 d^r _ 10) decuplet representation 2 2 ' 2 ' of SO(6) and one antidecuplet, both presented in Table 17.3. To be able to second quantize the theory, the whole representation must be Grassmann odd. Both decuplets in Table 17.3 have an odd Grassmann character, what means that products of eigenstates of the Cartan subalgebra in the rest of Grassmann space must be of an Grassmann even character to be second quantizable. Both decuplets would, however, appear in the same representation of SO(13,1), and one can expect also decuplets of an even Grassmann character, if SO(6) is embedded into SO(13,1) 1. With respect to SU(3) x U(1) subgroups of the group SO(6) the decuplet manifests as one singlet, one triplet and one sextet, while the antidecuplet manifests as one antisinglet, one antitriplet and one antisextet. All the corresponding quantum numbers of either the Cartan subalgebra operators or of the corresponding diagonal operators of the SU(3) or U(1) subgroups are presented in Table 17.3. While in Clifford case the representations of SO(6), if the group SO(6) is embedded into SO (13,1), are defining an Clifford odd quadruplet and an Clifford even antiquadruplet, the representations in Grassmann case define one decuplet and one antidecuplet, both of the same Grassmann character, the odd one in our case. The two quadruplets in Clifford case manifest with respect to the subgroups SU(3) and U(1) as a triplet and a singlet, and as an antitriplet and an antisinglet, respectively. In Grassmann case the two decuplets manifest with respect to the subgroups SU(3) and U(1) as a (triplet, singlet, sextet) and as an (antitriplet, antisinglet, antisextet), respectively. The corresponding multiplets are presented in Table 17.4. The "fermion" quantum number t4 has for either singlets or triplets in Grassmann space, Table 17.4, twice the value of the corresponding singlets and triplets in Clifford space, Table 6.3 (see pages 112-113 in this volume): (—1, +1) in 1 This can easily be understood, if we look at the subgroups of the group SO(6). i. Let us look at the subgroup SO(2). There are two creation operators of an odd Grassmann character, in this case (09 — i010) and (09 + i010). Both appear in either decuplet or in antidecuplet — together with 09010 with an even Grassmann character — multiplied by the part appearing from the rest of space d _ (11,12,13,14). But if SO(2) is not embedded in SO(6), then the two states, corresponding to the creation operators, (e9 ^ ie10), belong to different representations, and so is e9e10. ii. Similarly we see, if we consider the subgroup SO(4) of the group SO(6). All six states, (e9 + ie10) • (e11 + ie12), (e9—ie10) • (e11 — ie12), (e9e10+ene12), (e9+ie10) • (e11 — ie12), (e9—ie10) • (e11 + ie12), (e9e10 — e11 e12), appear in the decuplet and in the antidecuplet, multiplied with the part appearing from the rest of space, in this case in d _ (13,14), if SO(4) is embedded in SO(6). But, in d _ 4 space there are two decoupled groups of three states [2]: [(e9 + ie10) • (e11 + ie12), (e9e10 + e11e12), (e9 — ie10) • (e11 — ie12)] and [(e9 — ie10) • (e11 + ie12), (e9e10 — e11e12), (e9 + ie10) • (e11 — ie12)]. Neither of these six members could be second quantized in d _ 4 alone. I decuplet s910 s11 l2 s1314 T4 t33 t38 1 (e9 + ielü)(eM + iel2)(el3 + iel4) 1 1 1 -1 0 0 2 (e9 + ie10)(e11e12 + el3el4) 1 0 0 l 3 +2 + 2/3 3 (e9 + ielu)(e11 -iel2)(el3-ie14) 1 -1 -1 +3 +1 4 (e9el0 + e11e12)(e13 + ie14) 0 0 1 l 3 0 l a/3 5 (e9 -ielU)(eM -iel2)(el3 + iel4) -1 -1 -1 +3 0 2 /3 6 (en + iel2 )(e9el0 + el3el4) 0 1 0 l 3 l 2 + 2/3 7 (e9 -iel0)(eM + iel2)(el3-iel4) -1 1 -1 +3 -1 8 (e9el0 -ellel2)(el3-iel4) 0 0 -1 +3 0 + Y1 9 (e9el0-el3el4)(en -iel2) 0 -1 0 +3 +2 l 2/3 10 (e9-iel0)(ellel2-el3el4) -1 0 0 +3 l 2 l 2/3 II decuplet s910 s11 l2 s1314 T4 t33 T38 1 (e9 -iel0)(en -iel2)(el3 -iel4) -1 -1 -1 +1 0 0 2 (e9 - iel0)(ellel2 + el3el4) -1 0 0 +3 l 2 l 2//3 3 (e9 -ie10)(e11 + iel2)(el3 + iel4) -1 1 1 l 3 -1 l y/3 4 (e9el0 + ellel2)(el3 - iel4) 0 0 -1 +3 0 + 5 (e9 +ie10)(e11 + iel2)(el3 - iel4) 1 1 -1 l 3 0 + ^ 6 (eM -iel2)(e9el0 + el3el4) 0 -1 0 +3 +2 l 2/3 7 (e9 +ie10)(e11 - iel2)(el3 + iel4) 1 -1 1 l 3 +1 l /3 8 (e9el0 - ellel2)(el3 + iel4) 0 0 1 l 3 0 l //;3 9 (e9el0 - el3el4)(eM + iel2) 0 1 0 l 3 l 2 + 2/3 10 (e9 + iel0)(ellel2 - el3el4) 1 0 0 l 3 +2 + 2/3 Table 17.3. The creation operators of the decuplet and the antidecuplet of the orthogonal group SO(6) in Grassmann space are presented. Applying on the vacuum state |^0 >= |1 > the creation operators form eigenstates of the Cartan subalgebra, Eq. (17.33), (S910, S11 12, S1314). The states within each decouplet are reachable from any member by Sab. The quantum numbers (t33, t38) and T4 of the subgroups SU(3) and U(1) of the group SO(6) are also presented, Eq. (17.38). Grassmann case to be compared with (—2, + 2) in Clifford case and (+1, —2) in Grassmann case to be compared with (+1, — 1) in Clifford case. When SO(6) is embedded into SO(13,1), the SO (6) representations of either even or odd Grassmann character contribute to both of the decupled, 1716 states of SO(13,1) representations contribute, provided that the SO(8) content has the opposite Grassmann character than the SO (6) content. The product of both representations must be Grassmann odd in order that the corresponding creation and annihilation operators fulfill the required anticommutation relations for fermions, Eq. (17.1). Properties of the subgroups SO (3,1) and SO (4) of the group SO (8) in Grassmann and in Clifford space, when SO(8) is embedded into SO(13,1) a. Let us again repeat first properties of the SO(3,1) and SO(4) parts of the SO (13,1) representation of 64 "family members" in Clifford space, presented in Table 6.3 (see pages 112-113 in this volume). As seen in Table 6.3 (see pages 112-113 in I t4 t33 t38 singlet (e9 + ie10)(e11 + ie12)(e13 + ie14) -1 0 0 triplet 1 (e9 + iel0)(e"e12 + el3e14) 1 3 +2 + 2T3 2 (e9e10 + e11e12)(e13 + ie14) 1 3 0 1 3 (e11 + ie12)(e9e10 + e13e14) 1 3 1 2 + 2T3 sextet 1 (e9 + ie10)(eM -ie12)(e13-ie14) 1 3 +1 + 4 2 (e9 -ie10)(e11 -ie12)(e13 + ie14) 1 3 0 2 3 (e9 -ie10)(e11 + ie12)(e13-ie14) 1 3 -1 + ^ 4 (e9e10 -e11e12)(e13-ie14) 1 3 0 +T* 5 (e9e10 -e13e14)(e11 -ie12) 1 3 +2 1 2T3 6 (e9 - ie10)(e11e12 - e13e14) 1 3 1 2 1 2T3 II t4 t33 t38 antisinglet (e9 - ie10)(e11 - ie12)(e13 - ie14) +1 0 0 antitriplet 1 (e9 - ie10)(e11e12 + e13e14) +3 1 2 1 2T3 2 (e9e10 + e11e12)(e13 - ie14) +3 0 + Y3 3 (e11 - ie12)(e9e10 + e13e14) +3 +2 1 2T3 antisextet 1 (e9 - ie10)(e11 + ie12)(e13 + ie14) 1 3 -1 1 T3 2 (e9 + ie10)(e11 + ie12)(e13 - ie14) 1 3 0 +? 3 (e9 + ie10)(e11 - ie12)(e13 + ie14) 1 3 +1 1 T3 4 (e9e10 - e11e12)(e13 + ie14) 1 3 0 1 T3 5 (e9e10 - e13e14)(e11 + ie12) 1 3 1 2 +_L_ + 2T3 6 (e9 + ie10)(e11e12 - e13e14) 1 3 +2 + _L_ + 2T3 Table 17.4. The creation operators in Grassmann space of the decuplet of Table 17.3 are arranged with respect to the SU(3) and U(1) subgroups of the group SO(6) into a singlet, a triplet and a sextet. The corresponding antidecuplet manifests as an antisinglet, an antitriplet and an antisextet. t33 = 2 (S910 - S11 12), t38 = ^ (S910 + S11 12 - 2S13 14), t4 = 1iC910 | c1112 i c1314i.cab ;(nQ 3 nb 3 \ - 3 (S + S + S ), S = i(e gdb -e gea). Fig. 17.1. Representations of the subgroups SU(3) and U(1) of the group SO(6) in Grassmann space for two Grassmann odd representations of Table 17.4 are presented. On the abscissa axis and on the ordinate axis the values of the two diagonal operators, t33 and t38 of the coulour (SU(3)) subgroup are presented, respectively, with full circles. On the third axis the values of the subgroup of the "fermion number" U(1) is presented with the open circles, the same for all the representations of each multiplet. There are one singlet, one triplet and one sextet on the left hand side and one antisinglet, one antitriplet and one antisextet on the right hand side. this volume) there are four octets and four antioctets of SO (8). All four octets, having an even Clifford character and forming 32 states when embedded into SO(13,1), are the same for either quarks or for leptons, they distinguish only in the SO(6) part (of an Clifford odd character) of the SO(13,1) group, that is in the colour (SU(3)) part and the "fermion quantum number" (U(1)) part. Also the four antioctets, having an odd Clifford character, are all the same for the 32 family members of antiquarks and antileptons, they again distinguish only in the Clifford even SO (6) part of SO(13,1), that is in the anticolour (SU(3)) part and the "fermion quantum number" (U(1)) part. The 64th-plet of creation operators has an odd Clifford character either for quarks and leptons or for antiquarks and antileptons — correspondingly have an odd Clifford character also their annihilation operators — and can be second quantized [1]. Let us analyze first the octet (2 2 -1 = 8), which is the same for all 32 members of quarks and leptons. The octet has an even Clifford character. All the right 56 78 handed uR-quarks and vR-leptons have the SO(4) part of SO(8) equal to [+] (+), 56 78 while their left handed partners have the SO(4) part of SO(8) equal to [+] [-]. All the right handed dR-quarks and eR-leptons have the SO(4) part of SO(8) equal 56 78 to (-) [-], while their left handed partners have the SO(4) part of SO (8) equal 56 78 to (-) (+)]. The left handed quarks and leptons are doublets with respect to t1 and singlets with t2, while the right handed quarks and leptons are singlets with respect to t1 and doublets with t2 . The left and right handed quarks and lepton belong with respect to the SO(3,1) group to either left handed or the right handed spinor representations, respectively. b. In Grassmann space the SO (8) group of an odd Grassmann character has 1 4841 = 35 creation operators in each of the two groups and the same number of annihilation operators, obtained from the creation operators by Hermitian conjugation, Eq. (17.4). The corresponding states, created by the creation operators on the vacuum state >, can be therefore second quantized. But if embedded the group SO(8) into the group SO(13,1) the subgroup SO(6) must have an even Grassmann character in oder that the states in SO (13,1) can be second quantized according to Eq. (17.1). According to what we learned in the case of the group SO(6), each of the two independent representations of the group SO (13,1) of an odd Grassmann character must include either the even SO (7,1) part and the odd SO(6) part or the odd SO (7,1) part and the even SO(6) part. To the even SO(7,1) representation either the odd SO (3,1) and the odd SO(4) parts contribute or both must be of the Grassmann even character. In the case that the SO (7,1) part has an odd Grassmann character (in this case the SO(6) has an even Grassmann character) then one of the two parts SO(3,1) and SO(4) must be odd and the other even. 17.3 Concluding remarks We learned in this contribution that although either Grassmann or Clifford space offer the second quantizable description of the internal degrees of freedom of fermions (Eq. (17.1)), the Clifford space offers more: It offers not only the description of all the "family members", explaining all the degrees of freedom of the observed quarks and leptons and antiquark and antileptons, but also the explanation for the appearance of families. The interaction of fermions with the gravity fields — the vielbeins and the spin connections — in the 2(2n + 1 )-dimensional space can be achieved, as suggested by the spin-charge-family theory ([5,4] and references therein), by replacing the momentum pa in the Lagrange density function for a free particle by the covariant momentum, equally appropriate for both representations. In Grassmann space we have: poa = faa p0a, with p0a = pa — 1 SabHaba, where faa is the vielbein in d = 2(2n + 1 )-dimensional space and Haba is the spin connection field of the Lorentz generators Sab. In Clifford space we have equivalently: p0a = faa p0a, poa = pa — 2 SabWaba — 1S ab CD aba. Since Sab = Sab + S ab we find that when no fermions are present either Haba or caba or CD aba are uniquely expressible by vielbeins faa ([5,4] and references therein). It might be that "our universe made a choice between the Clifford and the Grassmann algebra" when breaking the starting symmetry by making condensates of fermions, since that for breaking symmetries Clifford space offers better opportunity". 17.4 Appendix: Useful relations in Grassmann and Clifford space The generator of the Lorentz transformation in Grassmann space is defined as follows [2] Sab = (eap0b — ebp0a) = sab + Sab, {Sab, Scd}— = 0, (17.32) where Sab and S ab are the corresponding two generators of the Lorentz transformations in the Clifford space, forming orthogonal representations with respect to each other. We make a choice of the Cartan subalgebra of the Lorentz algebra as follows S03 s12 s56 • • • Sd-1 d S03 s12 S56 • • • sd— 1 d S03 S12 S56 • • • S d—1 d if d = 2n. (17.33) We find the infinitesimal generators of the Lorentz transformations in Clifford space Sab = ^(YaYb — YbYa), Sabt = naanbbSab , S ab = 4 (Y aYb — Y bYa), S abt = naanbbS ab, (17.34) where Ya and Ya are defined in Eq. (17.10). The commutation relations for either Sab or Sab or Sab, Sab = Sab + Sab, are {Sab,Scd}_ = 0, {Sab, Scd}- = i(nadSbc + nbcSad - nacsbd - nbdsac), {Sab, S cd}_ = i(nadSbc + nbcSad - nacSbd - nbdSac). (17.35) The infinitesimal generators of the two invariant subgroups of the group SO(3,1 ) can be expressed as follows 1 N ±(= N (L,R)) := _(S23 ±iS0l,S31 ± iS ,S12 ± iS ). (17.36) The infinitesimal generators of the two invariant subgroups of the group SO (4) are expressible with Sab, (a, b) = (5,6,7,8) as follows T1 : = 1 (S58 - S67, S57 + S68, S56 - S78), T2 : = 1 (S58 + S67, S57 - S68, S56 + S78), (17.37) while the generators of the SU(3) and U(1) subgroups of the group SO(6) can be expressed by Sab, (a,b) = (9,10,11,12,13,14) T3 := 1 {S912 2 1 c9 14 S10 11 s9 11 + S10 12 S9 10 1112 ;10 13 c9 13 S11 13 + S1214 , S9 13 + S (S S 10 14 c11 14 _ s12 13 2S13 14 ,S1 9 10 + s11 12 )} , T4 := -1 (S 910 + S11 12 + S1314 ). (17.38) The hyper charge Y can be defined as Y = t23 + t4. The equivalent expressions for the "family" charges, expressed by S ab follow if in Eqs. (17.36 -17.38) Sab are replaced by Sab. The breaks of the symmetries, manifesting in Eqs. (17.36,17.37,17.38), are in the spin-charge-family theory caused by the condensate and the nonzero vacuum expectation values (constant values) of the scalar fields carrying the space index (7,8) (Refs. [5,3] and the references therein). The space breaks first to SO(7,1) xSU(3) x U(1)n and then further to SO(3,1) x SU(2)i x U(1)i xSU(3) x U(1)n, what explains the connections between the weak and the hyper charges and the handedness of spinors. Let ius present some useful relations [3] ab ab ab naa [-k], (-k)(-k)= 0, ab ab ab 0, [-k][-k]=[-k], ab ab ab -k), (-k) [-k]= 0, ab ab ab 0, [-k](-k)=(-k) . (17.39) ab ab (k)(k) = 0, ab ab (k)(-k)= naa ab [k], ab ab (-k)(k) ab ab [k] [k] II k[ a ]b ab ab [k][-k]= 0, ab ab [-k][k]= ab ab (k) [k] = 0, ab ab [k](k)= ab (akb) , ab ab (-k)[k]= ab ab (k) [-k] ab = (k), ab ab [k](-k): = 0, ab ab [-k](k)= Acknowledgment The author N.S.M.B. thanks Department of Physics, FMF, University of Ljubljana, Society of Mathematicians, Physicists and Astronomers of Slovenia, for supporting the research on the spin-charge-family theory by offering the room and computer facilities and Matjaž Breskvar of Beyond Semiconductor for donations, in particular for the annual workshops entitled "What comes beyond the standard models". References 1. N. S. Mankoc Borstnik, H.B. Nielsen, "Why nature made a choice of Clifford and not Grassmann coordinates", Proceedings to the 20th Workshop "What comes beyond the standard models", Bled, 9-17 of July, 2017, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Založnistvo, Ljubljana, December 2017, p. 89-120 [arXiv:1802.05554v1v2] (arXiv:1806.01629 whole proceedings). 2. N. Mankoc Borstnik, "Spinor and vector representations in four dimensional Grassmann space", J. of Math. Phys. 34 (1993), 3731-3745. 3. N.S. Mankoc Borstnik, "Spin-charge-family theory is offering next step in understanding elementary particles and fields and correspondingly universe", Proceedings to the Conference on Cosmology, Gravitational Waves and Particles, IARD conferences, Ljubljana, 6-9 June 2016, The 10th Biennial Conference on Classical and Quantum Rela-tivistic Dynamics of Particles and Fields, J. Phys.: Conf. Ser. 845 012017 [arXiv:1409.4981, arXiv:1607.01618v2]. 4. N.S. Mankoc Borstnik, "Matter-antimatter asymmetry in the spin-charge-family theory", Phys. Rev. D 91 065004 (2015) [arxiv:1409.7791]. 5. N.S. Mankoc Borstnik, "The explanation for the origin of the Higgs scalar and for the Yukawa couplings by the spin-charge-family theory", J.ofMod. Physics 6 (2015) 2244-2274, http://dx.org./10.4236/jmp.2015.615230 [http://arxiv.org/abs/1409.4981]. 6. N.S. Mankoc Borstnik N S, "The spin-charge-family theory is explaining the origin of families, of the Higgs and the Yukawa couplings", J. of Modern Phys. 4 (2013) 823 [arxiv:1312.1542]. 7. N.S. Mankoc Borstnik, D. Lukman, "Vector and scalar gauge fields with respect to d = (3 + 1) in Kaluza-Klein theories and in the spin-charge-family theory", Eur. Phys. J. C 77 (2017) 231. 8. N.S. Mankoc Borstnik, H.B.F. Nielsen, "The spin-charge-family theory offers understanding of the triangle anomalies cancellation in the standard model", Fortschritte der Physik, Progress of Physics (2017) 1700046. 9. N.S. Mankoc Borstnik, H.B.F. Nielsen, J. of Math. Phys. 43, 5782 (2002) [hep-th/0111257]. 10. N.S. Mankoc Borstnik, H.B.F. Nielsen, J. of Math. Phys. 44 4817 (2003) [hep-th/0303224]. 11. N.S. Mankoc Borstnik, "The spin-charge-family theory explains why the scalar Higgs carries the weak charge ± 1 and the hyper charge ^ 1", Proceedings to the 17th Workshop "What comes beyond the standard models", Bled, 20-28 of July, 2014, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Založnistvo, Ljubljana December 2014, p.163-82 [ arXiv:1502.06786v1] [http://arxiv.org/abs/1409.4981]. 12. A. Borstnik Bracic, N.S. Mankoc Borstnik,"The approach Unifying Spins and Charges and Its Predictions", Proceedings to the Euroconference on Symmetries Beyond the Standard Model", Portorož, July 12 - 17, 2003, Ed. by N.S. Mankoc Borstnik, H.B. Nielsen, C. Froggatt, D. Lukman, DMFA Založnistvo, Ljubljana December 2003, p. 31-57, hep-ph/0401043, hep-ph/0401055. 13. A. Borštnik Bracic and N.S. Mankoc Borštnik, "Origin of families of fermions and their mass matrices", Phys. Rev. D 74, 073013 (2006) [hep-ph/0301029; hep-ph/9905357, p. 52-57; hep-ph/0512062, p.17-31; hep-ph/0401043, p. 31-57]. 14. D. Lukman, N.S. Mankoc Borstnik and H.B. Nielsen, "An effective two dimensionality cases bring a new hope to the Kaluza-Klein-like theories", New J. Phys. 13:103027, 2011. 15. D. Lukman and N.S. Mankoc Borštnik, "Spinor states on a curved infinite disc with nonzero spin-connection fields", J. Phys. A: Math. Theor. 45:465401, 2012 [arxiv:1205.1714, arxiv:1312.541, hep-ph/0412208 p.64-84]. 16. D. Lukman, N.S. Mankoc Borstnik and H.B. Nielsen, "Families of spinors in d = (1 + 5) with a zweibein and two kinds of spin connection fields on an almost S2 ", Proceedings to the 15th Workshop "What comes beyond the standard models", Bled, 9-19 of July, 2012, Ed. N.S. Mankoc Borštnik, H.B. Nielsen, D. Lukman, DMFA ZaloZništvo, Ljubljana December 2012,157-166, arxiv:1302.4305. 17. G. Bregar and N.S. Mankoc Borstnik, "Does dark matter consist of baryons of new stable family quarks?", Phys. Rev. D 80, 083534 (2009) 1-16. 18. A. Borstnik Bracic, N. S. Mankoc Borstnik, "On the origin of families of fermions and their mass matrices", hep-ph/0512062, Phys Rev. D 74 073013-28 (2006). 19. N.S. Mankoc Borstnik, H.B. Nielsen, "Discrete symmetries in the Kaluza-Klein-like theories", Jour. of High Energy Phys. 04 (2014) 165 [http://arxiv.org/abs/1212.2362v3]. Bled Workshops in Physics Vol. 19, No. 2 JLV Proceedings to the 21 st Workshop What Comes Beyond ... (p. 359) Bled, Slovenia, June 23-July 1, 2018 18 Could Experimental Anomalies Reflect Non-perturbative Effects? H.B. Nielsen1 and C.D. Froggatt2 1 Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, Copenhagen 0, Denmark 2 Glasgow University, Department of Physics and Astronomy Glasgow University, Glasgow G12 8QQ, Scotland, UK Abstract. We investigate whether some of the rather few anomalies, in the sense of deviations from the Standard Model, could be explained as due to non-perturbative effects caused by the top-Yukawa-coupling being of order unity (in a sense to be discussed briefly in this article). The main achievement of our non-perturbative rule or model is to relate the deviations of ratios between B-meson decay rates for flavour universality violation for neutral currents to the deviations for the charged current flavour universality violations. In fact the anomaly in the ratio R(D *) for a charged current with T and its neutrino relative to the rate with the |i and its neutrino is being related in our model for non-perturbative effects to an analogous effect in a neutral current B-meson decay. It is suggested that the ratio of the anomalous amplitudes contributing to these two combinations of decay processes are to very first approximation given by the squared mass ratio of the heaviest lepton involved in the two ratios, which by their deviation from the Standard Model prediction signal lack of flavour universality. The muon g — 2 anomaly also fits well in our non-perturbative model. But we have to mutilate the model somewhat in order to avoid a far too large anomaly prediction for, say Bs-Bs, particle - antiparticle mixing. Povzetek. Avtorja v prispevku raziskujeta, ali lahko odstopanja od napovedi Standardnega modela pojasnita z neperturbativnimi efekti, ki se pojavijo, ker so Yukawine sklopitve za top kvark reda ena (v smislu razloZenem v prispevku). PoveZeta odstopanja med dosedanjimi napovedmi razmerij razpadnih stanj B mezonov za krsitve univerzalnosti tokov za nevtralne in za nabite tokove in rezultati meritev. Odstopanje v razmerju R(D*) za nabite tokove za delec T in njegov nevtrino in za delec | in njegov nevtrino je povezano z analognimi odstopanji v primeru razpadov nevtralnih mezonov B. Predlagata, da je razmerje anomalnih amplitud, ki prispevajo k tem dvem kombinacijam razpadnih procesov, v prvem priblizku dano s kvadratom razmerij mas najtezjih leptonov v teh razpadih. Odstopanje od napovedi Standardnega modela nakazuje odvisnost od okusa (flavor). Model je uporabljiv tudi za odstopanja med poskusi in racuni za vrednost g — 2 za mione, denimo za mesanje Bs-Bs, ce model popačita in se tako izogneta velikim odstopanjem. Keywords: Decay rate anomalies, non-perturbative effects, flavor universality 18.1 Introduction Are the Tensions in LHCb etc data due to Non-perturbative Effects in the Pure Standard Model ? The Standard Model works surprisingly well for LHC physics: Almost no new physics, and at least nothing truly statistically significant! However there is a small number of tensions in the data with a few standard deviations significance: Small lepton universality violating deviations [1-3], say. The present proposal is that even these small tensions are not due to genuine new physics, but rather to effects forgotten because of the systematic use of perturbation theory except for the QCD-sector; i.e. the tensions should be non-perturbative effects. Ratio Rk* of mm versus ee for B —> K*U, anomalous. -i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—r 1.0 I--0.8 0.6 0.4 0.2 0.0 t LHCb J_I_I_I_I_l_ • LHCb ▲ BiP T CDHMV ' ■ EOS • flav.io ' • JC j_i_i_i_i_i_i_i_i_i_i_i_i_i_i_i_i_i_i_i_l 0 12 3 4 5 6 q2 [GeV2/c4] Ratio Rk of i\i to ee Ratio for B+ —> K+ll decay, anomalous for separate q2?. -•-LHCb — BaBar Belle T 15 20 q2 [GeV2/c4] 18 Could Experimental Anomalies Reflect Non-perturbative Effects? 361 Ratio tvt versus for B —» D*v + lepton, an anomaly BiiBar badrotiic tag PRD 8S (2013) 072012 H—■—m 0.332 ± 0.IH2 ± 0.0IS IJ oik hadionic tag PRD 92 (2015) 072014 ii » ■ 0,293 ±0.038 ±0,015 liL-lk- SL lag PRD 94 (2016) 072007 ■-•-- 0.302+0.030 + 0.011 i BlIIc 1 -prong PRL1 IS (2017) 211801 1 1 ■ • 11 0.270 + 0.035 + 0.027 i LHCb muonic PRL 115(2015) 1118flf t—i-* 1 1 0.336 ±0.027 ±0.03o * ; LHCb 3-prong LHCb-PAPER-2017-017 >-*—I—■ 0.291 ±0.01 9 ±0.029 LHCb average j 0,310 ±0.016 ±0,022 T Fiijlcr ct al. (SM) PRD 85 (2012) 094G25 « 0,252 ±0,003 _I_I_I_I_L_I_I_L_Ll_I_I_I_I_I_I_I_ 0.1 0.2 0.3 0.4 R(D*) Ratio R(J/V) of tvt versus also in B —» J/¥ + v + lepton an anomaly. LHCb R(J/x[i) LHCb-PAPER-2017-035 0.71 ± 0.17± 0.18 1—1-*-1—1 SM predictions PLB 452 (1999) 129 arXiv:hep-ph/0211021 PRD 73 (2006) 054024 PRD 74 (2006) 074008 Range 0.25 - 0.28 1 1 1 1 1 1 1 1 1 1 1 1 1 -0.5 0 0.5 1 The two Deviations from SM at LHCb: In the following table we summarize the two deviations from the Standard Model at LHCb and compare our prediction for the ratio of the corresponding anomaly amplitudes with the data. Channel Branch. fraction "R" Ratio Deviation relative Anomaly-amplitude B -> K>+|-neutral c current 10-6 exp. 0.66 SM 1.00 -34 % —0.3W10-6/2 = —1.7 * 10-4 = —1.7 * 10-3V% B -> D*TVt charged current 2% exp. 0.31 SM 0.25 +24 % 0.24V0.02/2 =0.017 =0.17V% Ratio 2 * 104 —102 Pred. ratio - °.4 * (mt)2 115 In the table we perform a very crude estimate of the ratio of the anomalous contributions to the amplitude of the two decay processes B —» D*tvt (which is a charged current process) relative to the anomalous contribution for B —» K*|+|-(which is a neutral current one). It is based on a few very crude but we think reasonable assumptions in our model: • Since our non-perturbative anomalous prediction is strongly increasing with the mass of the charged lepton involved, we of course blame practically the whole anomaly on the decay rate for the process involved in the ratio revealing the deviation from lepton universality, which has the biggest mass. In R(D*) for instance it is the t channel that has the anomaly, while for R(K*) which is a ratio between a |i and an e channel it is the |i channel that carries the anomaly. • We make the approximation that the channels all have the same phase space -which means ignoring the differences between the masses of the particles in the final state of the decays (compared roughly to the B-meson mass). This also implies that, in this approximation, we can simply talk about the amplitude for going into the single final state for each of the considered channels of decay. This allows us to use the normalization of simply writing the amplitude of a decay measured in square roots of %, and simply in this notation have the decay fraction to a channel be the square numerically of the added up amplitudes. The columns of the table denote the following • The first column represents the decay channel, corresponding to the two different ratios revealing the violation of flavour universality (for leptons), which has the heavier lepton in the decay. These decay channels are thus, according to our assumption, the ones that are (most) anomalous in our model. We shall neglect the anomaly in the other decay channels in the ratios. • The next column gives the branching fraction of these two channels thought to be carrying an anomaly. • The next - 3rd - column now gives both the experimental ratio and the Standard Model predictions for the ratio associated with the channels lined up in column 1. That is to say for the first row, or rather the one associated with B —» K*|+|-, we talk about the ratio R(K*) being the ratio of this decay rate to the corresponding one with the muons replaced by electrons. Similarly the second of the genuine rows refers to the ratio of the decay listed in first column divided by the corresponding one with the lepton replaced by the lighter lepton, in this case thus B —» D* pv^. • The relative deviation between experiment and Standard Model is calculated in the next - the 4th - column. In our philosophy this also gives the relative deviation between the size of the decay in column 1 experimentally relative to the Standard Model. Thus the anomalous probability contribution is the product of this relative percentage and the rate as in column 2. • Finally in the last column we identify the deviation corresponding to the anomaly with 2 times the amplitude - meaning the square root - of the rate (from column 2) multiplied with the "anomalous amplitude". It is then the latter that is presented in the last column. Finally the result of interest is that we estimate the ratio of the anomalous amplitudes for the anomalous parts of the decay amplitudes of the two "rows". It is this ratio we have a chance to predict, because as a ratio it means that our parameter K gets divided out. The calculation in the table, which we can at the moment hope to confront with our model, is an order of magnitude one meaning that neither factors 2 or n etc nor even the sign are our under control so far. It might seem that just substituting a mu-coupling by a tau-coupling would only change the anomalous amplitude by a very well-defined real positive ratio given by the masses actually very precisely. However, in our comparison, we have it interfere with the Standard Model amplitude for two very different processes from the Standard Model point of view. So to get even the sign one would need the relative sign of these Standard Model amplitudes, something that would be quite a complicated task. We hope to come back to this exercise of calculating the relative sign of the Standard Model amplitudes, so as to make possible a sign prediction for our model about the sign of the ratio of the two anomalies which we studied. 0.4 some order of unity number in the last entry in the table. In fact the order unity factor 0.4 in our predicted ratio is given in our non-perturbative model by Vtb Vts292 = 0.4. (18.1) Vbcg2 The numerically more significant factor is the ratio 9t _ mT _ 17772 _ 9T _ mT _ 10572 _ 283. (182) The numerical coincidence, that should suggest the truth of our non-perturbative effect idea, is: (R(D*)lexp/R(D*)|sM - WB(B —> D*tvt) ^ mT (R(K*)|exp/R(K*)lsM - 1)^/B(B —> ) ~ m2 Here the "R" ratios are defined as: R(K-)- (18.4) R(D')- raW. (185) Note that these "R " ratios test the lepton universality, the numerator and the denominator only deviating by the flavour of the lepton pair produced. But in R(D*)it is the ratio T-pair over ^-pair, while R(K*) is for ^-pair over e-pair. Decays into channels only deviating by "hadronic details" support such models as e.g. our "non-perturbative" model. That is to say the approximate equalities RSiP — 0.75 « 0.66 — R&XP, (18.6) R(K)|sm R(K*)sm R(J/^)lexp — 2^124 — R(D*)exp (187) R(J/^)Ism — 2.3 ~1.24 — R(DiexM (187) confirm that the anomaly is approximately the same for different hadronic developments with the same underlying weak process behind, thus supporting an e.g. non-perturbative effect, or a new physics at the weak scale. Have now to build arguments that the lepton pair needs to couple twice with its Higgs Yukawa coupling to the strongly interacting particles/sector. We imagine there is some coupling gt which is so strong that very complicated diagrams involving it become relevant. But somehow we hope to argue that the leptons only get interacting with the bunch of "new strong" interaction particles via two Higgs couplings in the processes we looked at with the anomalies. Also an agreement for the anomaly in the anomalous magnetic moment for the muon, a^ — (g — 2)/2|^. We get a correction to the anomalous magnetic moment for the muon in our non-perturbative model, using an overall fitting constant K for the non-perturbative effects (to be explained later): e g (a J full — a^lperturbative) * - ~ K* < ^Higgs > (-) . (18.8) 9t With our fitted value K ~ 5G1V2, we get | | 246GeV * 0.105GeV _ 1 a^fuii - a^perturbattve ~ 5GeV2 * 17003 _ 1 * 10 to be compared with the anomaly found experimentally 2.7 * 10 9. 18.2 Strong Coupling Except for aS the strongest coupling in Standard Model is the Top Yukawa Coupling gt. The Coupling on the Border between Weak and Strong Interactions for Particle with Only One Component is g ~ 4n. Taking very crudely by a "dimensional argument" d^p ~ 1 (by dimensional argument) and the borderline coupling gb°rder to have the increase factor by adding a loop of g2 d4q (2n)4|q|4 in the propagators) we get gborder 1 (ignoring the mass squares = 4n. n2 (18.9) Another crude estimate of the border coupling corresponds to taking the Rydberg constant — 2 a2mec 4nh to be of the order of the mass-energy mec2: — mec implying a2 — 4n for c — h — 1. meaning e — 4 (4n)3 « 6 (18.10) (18.11) (18.12) (18.13) Size of Borderline Coupling and Number of "Components" If there were e.g. a color quantum number taking N values for the particle type encircling the loop, then there would be N various loops for each one. According to our philosophy of the increase factor by inserting a loop gborderN d4q (2n)4|q|4 1 (18.14) then the N-dependence of the borderline coupling between perturbative and non-perturbative regimes would be gborder (18.15) For say 16 "Components" Borderline Coupling ~ 1.5 to 3 Very crudely counting particle and antiparticle also as different "components" and counting together both the Higgs with its 4 real components and the top with its 3*2*2=12 components, we get in total for the particles interacting via the top Yukawa coupling gt 12+4 = 16 components. Thus the borderline value for gt becomes gt border ~ (6 to 4n)/vT6 = 1.5 to 3. (18.16) Experimentally gt exp = 0.935 (18.17) 18.3 Procedure Very High Order Diagrams Likely to be Important Diagrams with Almost Only Top-Yukawa Couplings of High Order Could be Significant and give the Anomalies about to be Statistically Significant "Ten- sions". L can be both left top and left bottom, strange, d R right can be only top. H can be both eaten Higgs and the "radial" observed Higgs Suggested Procedure of Model We imagine a lot of Feynman diagrams - that shall be summed up of course -each with almost only the top-Yukawa coupling gt in it, and only a few external lines/propagators of other types (like muon say). Then the rules/assumptions of our non-perturbative model are as follows: • The sum over the many diagrams with only gt (from which we modify a bit by putting external lines on) is supposed to give just one overall factor K, which we must fit. • When we use an external L line as a left bottom, strange or d quark line, we include a Vtb, Vts, or Vtd mixing angle factor • Other couplings than gt needed must give rise to the extra factors being these couplings, compared to the gt they replace. • Propagators for W, Higgs, top,... are similar order of magnitudewise, and we ignore the differences in our crude rule. From the Physics involving Rather Heavy Particles the Result of the Non-perturbative Effects should be Effective Lagrangian Terms of Unrenormaliz-able Dimensionality. The rather high mass of the particles, like the top quark and Higgs partticle, involved in the diagrams developing non-perturbative effects suggests these effects at the relatively low energies involved, in B-meson decay say, should be described by an effective field theory. The effective terms which have an operator dimension like in renormalizable theory are already present in the Standard Model. Thus such non-perturbative effects contributing to terms with dimension less than or equal to [GeV4] would just be absorbed into these terms already present in the Standard Model. We can only realistically hope to measure terms not of this renormalizable type, because otherwise we would need some knowledge about the bare couplings not coming from the usual measurements: Denoting say leptons and quark fields by and ^ and the bosons as W^, Z^ and effective field theory terms that might result from non-perturbative effects could have e.g. the forms (PL is left handed Y5 projector) : of renormalizable theory dimension [GeV4] %bYvPL'%s'4yYvPL'%|i : Dimension [GeV6], so not renormalizable. Example of an Effective Lagrangian Density Coefficient Estimated in Our Non-perturbative Scheme: Say we want the coefficient to the term of the form 'b(x)Yv's(x)'|(x)Yv'|(x), which can represent that a bottom quark b described by 'b (x) becomes a strange quark s described by's by a "neutral current exchange" and the production of a muon antimuon pair produced by the operator 'i(x)Yv'i(x). Then we need a non-perturbative diagram with the four external particles corresponding to b—>s, p and p.. In fact it shall be a series of diagrams with an arbitrary number of gt vertices and associated with tL, tR and Higgs, but as few as possible other - and therefore smaller - couplings (except we might include the strong QCD couplings). If the b and the s are taken to be of the left handed helicity, bL and sL, we are really interested in the coefficient to the effective term 'b(x)YvPL's(x)'|(x)Yv'|(x). (18.18) We can interpret it, that the weak SU(2) partners of the left handed top-components tL , which are also allowed in the bulk of our diagrams, are already present with amplitudes Vtb and Vts respectively for the left handed bL and sL. So they do not "cost" extra coupling factors except for these CKM matrix elements, Vtb and Vts. Ignoring the propagators and thereby the masses, we have in the bulk diagram perfect formal conservation of weak charge SU(2), and thus the two left handed quarks b and s being doublets cannot couple to only one Higgs. We must have two external Higgs bosons coupling to the muon-antimuon pair. The muon cannot be interpreted as being already there in the bulk diagram and must instead be coupled, as we already argued, to two Higgs-bosons. This causes the applicable type of diagram to include a factor g| - or if we want to consider it a replacement of gt couplings by analogous g /s, it must include a factor ("gt) . S0 the coefficient to the b —» s,p, p transition operator (18.18) becomes ''coefficient to c->sp p" = K * Vtb Vts g1^ . (18.19) Here K is an overall constant depending on the non-perturbative part of the calculation, which we cannot do. Thus we must fit via this overall factor K, while gt, and g^ are the Yukawa couplings to the Higgs of the top quark and the muon respectively. Vtb and Vts are the mixing matrix elements. Another Example: b —» c, t, vt; Charged Current Process The coefficient to the "non-renormalizable" charged current simulating effective field theory term pL^Vt (18.20) becomes similarly K * Vtb(VtbVbc"+ " VtsVSc"+ " VtdVdc^92 — ) . (18.21) V9t 9t/ Here 92 is the weak SU(2) gauge theory coupling, and as before: K is the overall non-perturbative constant, Vqq / the mixing matrix elements, and gt, gT the respective Yukawa Higgs couplings. Order of magnitudewise we only care for the dominant one of the three mixing matrix element products. Fitting our overall constant K: With the notation 4G e2 Heff = -4^VtbVts — Y-(CiOi + ci°i) + h.c. (18.22) and 09'5 = (sY^PL(R]b)(ÎY^l), (18.23) the fit of the "new physics" NP in the coefficient C9 to the effective term O9, which we considered is about C9 « -1.3. (18.24) ■ ejtcl □ JUÏCL □ *imi 1 j~"j r: lil+ I if».' Kax'I ibn L m. a« -0,15 -0.10 -0.05 0.00 0.05 0.10 0.15 cP FIG, 1: i'it ta [G? '. C» " ) 1 ua^tbc oi?«; Jurg>rc«s] bin j jùl B K'y'ji' ûliisita-'-jua, lù^uifli- wl",]l B -i- Xy, ¿1 Xrfi+)!" , n i ftJî. -t Thil dathwl jYmtiviiT.I iiudud;; boch Laij^c;- and low-dcc:âJ bina, v.'h^icus tic oeillet; iaûLJil} outÉ uie jilJj Uib 1-G (rfV* bici iii B -?■ ft^HfTTflKlfi. Thil rtripii — (0,0) cyir-riprtr rk tn fchf: S\T vhJujbo toe tjs WUeoil toiHitiartb C^c — (—0.2S, I.'J7) it jj,, = fitiV. The conventional VtbVts factors in Heff = -VtbVs iln Z(Ci0i + Ci°i) + h-c-are just the same as in our formula for the non-perturbative effect coefficient ''coefficient to c->sp. H" = K * Vtb Vt^. Thus we should fit to (g^\2 4Gf e2 Gf K * =--* C9 =--J= a * C9 V gt / V2 16n a/2 = 1.1663787(6) x 10-5GeV-2/(V2 * 137.037) * (-1.3) = -6.01847886 * 10-8GeV-2 * (-1.3). Since (j^) 2 = (0.1056583745/172.44)2 = 3.77 * 10-7, we get from fitting the 09 coefficient „ 6.018 * 10-8GeV-2 , „ „ K = 3.77* 10-7 *1.3 (18.25) = 0.21 GeV-2 (18.26) = (18.27) Embarrassingly Huge Overall Constant K ~ 4 G1eV2 for the Non-perturbative Effect. Imagine that the non-perturbative effect in reality is the effect of some loop with, or just the effect of, a bound state formed from the top-quarks and the Higgs. If consisting, as we usually speculate, of 6 top + 6 anti-top quarks its constituent mass would be 12mt = 2.1 TeV. So, even if we did not count suppression from there being a loop say, an order of magnitude K ~ 4 Tev2 would have been rather expected. But now, if we have about 12 constituents in the bound state, a top-quark or a Higgs would couple to such a bound state with a total coupling of the order of 12gt. Very optimistically a diagram with four external lines would have four such factors and the resulting K would be enhanced by a factor (12gt)4 « 20000 which would bring K ~ TriVT up to K ~ 200 GeV2. If the bound state mass were say 750 GeV rather than 2.1 TeV, a reduction by a factor (2.1/.75)2 of the above speculated value 200GeV2 would be argued for. Then we might say that we could understand if K were of order of magnitude 20 GeV2, but the fitted value K ~ 4 2 still seems to be a bit - a factor 5 - bigger than we would even speculate optimistically. But of course the point is that it is too hard to compute or even speculate on the overall strength K, so that we must rather trust a fit to the data. Our Prediction for the Ratio of the anomalous Charged Current B—» Xctvt to the anomalous Neutral Current B—» Xsp.^ amplitudes The ratio of the experimentally found quite separate anomalies measured in their rates/branching ratios is "Anomalous rate B —» Xctvt" . .. . . - = (—J1 * 104 "Anomalous rate B —» while the ratio of the normal rates is: BR(B -> XctvtJ 2% BR(B->Xspv^) 2 * 10-6 corresponding to an amplitude ratio: = 1* 104 Ai^XeTV.) /= ! * lo2_ A(B^Xspv2) V 2 * 10-6 By accident it does not matter whether the anomalies come by interference - as we think they do - or by just adding to the rate. In any case it is needed experimentally that the ratio of the two anomalous parts of the amplitude must be -100: Ann(m">Xe ^) = 100- (18.28) Ann (B->Xsp. p) Is that then what our model predicts? Our prediction for the ratio of the anomalous parts of the amplitudes is: Ann(B -> XcTVt) _ K * Vtb (Vtb Vbc'' + " VtsVsc'' + " VtdVdc) (f gt^ Aan(B ^ XsPP) k * VtbVts (gL VtbVbc 92 9 * 2 „2 T Vts 9u9t m2 « 1 * 0.4 * mT m2 17772 = 0-4 * = 115. (18.29) Very good agreement with experiment! Dominant Anomaly in B+ —» K+t+t-Our prediction for the branching ratio for B+ —> K+t+ t-: The anomaly amplitude is enhanced by the factor m^/m2 compared to the B+ —} K+ p+ p- anomaly amplitude and therefore dominates the usual SM amplitude. So the branching ratio value for B+ —» K+T+T-is: Branching ratio For SM For our anomaly Experiment. - 2 x 10-7 - 3 x 10-4 < 2.25 x 10-3 18.4 g minus 2 There is a small deviation from experiment in the perturbative Standard Model prediction for the anomalous magnetic moment for the muon. The non-perturbative contribution of our model is illustrated in the following diagram, which is followed by a list of comments on it. The muon anomalous magnetic moment term in the effective Lagrangian density a,W> |oMFvp(x)YvYp^ (x) = a^p.(x)Fvp(x)YvYVM (18.30) makes a transition between the chirality left to right or opposite. (Contrary to simple electromagnetic coupling making it left to left or right to right.) • Thus we need to couple the muon line series an odd number of times to Higgs in our non-perturbative contribution. • Only one Higgs exchanged would just give a renormalization of the Higgs propagator, and would thus already be included in the Standard Model calculation and not count as an anomalous term for the anomalous magnetic moment. • This contribution must then, because we ignore the propagator masses in it, have a Higgs-line couple to vacuum via the expectation value < H >_ 246 GeV, so that it conserves weak isospin. • These remarks give the factor (ggf) < H >. • When we use our speculated non-perturbative effect, we have the "overall" factorK~ 4Gew. for the Finally we get a non-perturbative contribution to a^ — (g 2)/2|| muon. a||full a||perturbative 246GeV*0.105GeV = 1.3 * 10-9. (18.31) 4GeV2 * 17003 This is to be compared with the anomaly found experimentally 2.7 * 10-9. 18.5 Mixing The mixing of mesons and their antiparticles such as Bs mixing with Bs is a problem, as was pointed out by a member of the audience when HBN gave a talk about this work in Tallinn. The problem is that at first sight it looks as though we have, according to our rule above, just a few mixing angles suppressing the transition from say Bs to Bs. This is very analogous to the way we got the b to s transition, using that both s and b can for the left handed case be considered to be in the doublet with the left handed top and thus indeed participating significantly in the diagrams supposed to be of very high order and still important. However this is not quite true, because the quarks that have to be converted in the mixing process for pseudoscalar mesons - which are w.r.t. strong interactions stable ones, so that mixing experiments can be practically performed - are both right handed and left handed. 18.5.1 Formal If we take completely formally our rules as set up, including the rule of neglecting propagators and thereby especially the masses of the quarks and leptons in the strong diagram, then a right handed quark of electric charge 2/3 (like the top) can, by interaction with a Higgs-doublet, only be converted into the left handed one of the same flavour or the weak isodoublet partner of this left handed one of the same flavour. This weak isodoublet partner is a superposition of all three flavours of the quark with the other electric charge than the starting right quark. This superposition carries in principle the signal of the flavour of the starting right handed quark. If we ignore the masses and only have it interact via the Higgses in the supposed to dominate diagrams, this superposition can only go back to the right handed quark of just the same flavour as from the start. In this way the "right flavour" has become formally a conserved quantum number, as long as we exclude other interactions than in our rule. Only if there is transition into a right handed quark of the other charge, i.e. charge -1/3, will another set of Yukawa-couplings (namely the -1/3 charge ones) come into the game and more complicated flavour changes become possible. The value of K = (4 to 5)Gev2 we found, by fitting flavour universality violations, would give us a non-renormalizable Lagrangian term for say top-quark scattering, which would not be suppressed, ~ 5Gey2 tMY^M * t(x)y^t(x). (18.32) This is quite absurd, if you think of using it up to a cut-off scale of say the order of A ~ 0.5 TeV or a "lattice scale" of the order a ~ 0 51TeV. We would in fact like to argue that you cannot use perturbation theory for such a coupling unless for K * t(x)y^t(x) * t(x)Y|ot(x) (18.33) one has K/a2 < 1. (18.34) Too Strong (Effective) Coupling Term gets Absurd/not Perturbatively Applicable, when K/a2 > 1 for dim =6 /_x \ 7 cut off scale a This figure is supposed to make clear the absurdity in the too strong coupling regime, which does not at least crudely obey K/a2 < 1. The figure is based on the assumption that inside the interacting particles (in our example top quarks) we have some structure or fields with which they interact with the other particle, and now illustrates how one particle passes into the field or matter belonging to the other one. Then the idea is to estimate the phase rotation of the amplitude of the scattering, i.e. after the passage. For the case that the particles did indeed pass within the distance a, we can argue dimensionally that the phase rotation 5 must be of the order 5 « K/a2 (18.35) -1 q2 or if there is some suppression factor such as e.g. "suppression" = : K 5 « ^-: jr~2. (18.36) suppression a2 Now the important point is that such a phase rotation 5 only makes sense modulo 2n. So it cannot be expected to give any sensible result when it becomes very big compared to 2n. First the point is that you simply cannot "see" the difference in various sizes once the 2n is past. Realistically, we would physically rather imagine that interference between slightly different passage ways of the one particle through the field or matter around the other one would get relative to 2n rather big phase differences, so that strong (destructive) interference would take place. Spoiled by such interference it seems unavoidable that, seen from outside, the end result would be an effective coupling looking much smaller than the a priori one K/"suppression". Therefore we would like to conclude that the very strong coupling, not obeying our requirement K/("suppression"a2) < 1 is not realistic in practice. Basically the strong interaction would cause further interactions or make different details in the interaction come out of phase. Thus the effective resulting interaction would be brought back to a size obeying the upper limit, which we suggest. A slightly different way to think of this "strong couplings killing themselves down" to only of order unity, would be to notice that passing a region with too strong interactions would cause reflection. So the particle would never come through but rather get reflected on the surface. In this way the interaction would be reduced to a size compatible with only the surface regions being used in the effective interaction as seen form outside. This is illustrated in the figure by the track of a particle turning around and going out again. If there is not a correction factor reducing the K to be sensible, we cannot take it seriously, but must correct it down: On this figure we now illustrate what we shall effectively do in our model, so as to take into account that the absurdly strong couplings cannot be taken seriously. From the rule of our non-perturbative model one starts from our fitted constant K and then has to put various factors such as g^/gt to some powers etc. so that one at the end divide by a "suppression" - a suppression factor. In the figure to give an idea of what we shall do, this suppression factor "suppression" is plotted as the abscissa. As the ordinate is plotted the effective field theory term coupling coefficient. If we did not modify our model this effective field theory coupling would of course just be K/"suppression" and that is represented by the skew straight line, simply with slope —1 in the logarithmic plot. If the suppression factor is sufficiently big, perturbation theory on top of our non-perturbative effect is still o.k. and we can take the result seriously. If, however, the suppression factor for some effective field theory interaction, we look for, turns out so small that the effective coupling becomes bigger than the limit, we should cut the coupling down to agree with the limit. This is indicated by the red arrow on the figure. So in reality we shall use the kinky curve given on this figure which for small "suppression" is flat, but for large enough "suppression" kinks into the —1 slope straight curve piece. In order that we can claim the success of our main result on the ratio of the anomalous amplitudes fort the two B-meson anomalies, it is crucial that they both fall in the region with the skew part of the curve. I.e. that suppression is enough. 18.5.2 Conservations In order to put forward a little better the problems with making contributions to meson anti-meson mixing in our scheme, we shall think of a certain truncated Standard Model: extra fac- tors In the region of our new strong interaction it is only right and left top quarks and the Higgs doublet, which are present. We must though consider the left top to also include a certain superposition of bottom, strange and down left quarks, namely the one that is in a doublet with the left top. At least for pedagogical reasons, but also really logically, we are allowed to use as a strictly speaking more accurate model a restriction of the Standard Model which also includes the three important particles for the new strong sector: the right top, the Higgs doublet and the doublet containing the left top. Let us indeed for our study, pedagogically or logically, choose the model with all the quarks and for that matter also the leptons, both right and left, and the Higgs doublet. However, we do not let into this restricted model the gauge bosons, so there is no transverse W nor transverse Z. (Only the longitudinal components in the form of eaten Higgses are let in). This sub-model contains all the components that are crucial for the non-perturbative effects. So it is in principle "better" than the only new strong interaction approximation. Now let us contemplate the conserved quantities of this "better" restriction of the Standard Model, and let us in the spirit of our proposed rule of ignoring the propagators or at least their masses, take all the quarks and leptons to be massless except for vacuum expectation values for the Higgs. But the Higgs vacuum expectation is assumed to be small on the mass scale we have in mind, so we indeed ignore the masses in the propagators, even for the Higgs, which has a mass of a similar order of magnitude. In this our "better" restricted Standard Model the weak isospin is only a global SU(2) symmetry, as is also the electric charge. We can without problems use a different flavour basis for the T3 = 1/2 and the T3 = -1/2 quarks, as one in fact does in practice. In such a notation then all the flavours get totally conserved. Roughly speaking: We switched off the weak interactions and then the flavours are conserved. It should though be borne in mind that our restricted sub-model of the Standard Model only had the transverse weak gauge bosons switched off, while the longitudinal components in the form of eaten Higgs components are still included. But this is then at first very promising for the mixing of the various pseu-doscalar mesons with their antiparticles in our model. Namely in first approximation, in which we could claim that we only need the just constructed restricted Standard Model, we can say that flavour changing is totally forbidden. Without flavour changing we can have no meson anti-meson mixing and thus our non-perturbative sector cannot produce any contribution to the mixing in this first approximation. 18.5.3 Problem However, there still seems to be a problem: The Standard Model contribution to meson anti meson mixing already has in amplitude two W-exchanges - as are needed for the flavour violation. Now the experimental method of measuring mixing is very sensitive and we cannot rely on the anomalous contribution from our non-perturbative model being negligible even if decorated with two W-propagators. We could therefore expect a non-negligible anomalous contribution basically simulating the Standard model term, but letting the two top quark propagators present in the Standard Model main term for the mixing interact via our non-perturbative effect. This would mean crudely some usual top-propagators, being of the order 1/mt each, if counted as fermion propagators, would in our anomalous term be replaced - following dimensionality rules - by a top-quark scattering effective coupling proportional to our K parameter with associated suppression factors. However, for top quark scattering we have in our model no further suppression and thus we simply get a K replacing the factor 1/m2 from the Standard Model perturbatively. Our estimating of the correction factor to the full contribution from the Standard Model would then be of the order m2K = ^g3- ~ 5000. This prediction would of course be catastrophic for the hope that our model could be right. There is certainly no place for an extra mixing even of the same order as the Standard Model mixing, let alone 5000 times as much. Now, however, although formally correct according to our rules, such an estimate is physically rather crazy. We must realistically expect that the effectively "new physics", due to the non-perturbative effects, has to do with say some bound state or some little clump of a new vacuum or whatever, which only truly comes into play when the interacting particles come sufficiently close to each other that the bound state or a couple of them say could be exchanged between them. Such bound state would presumably already have been observed if it were not of mass of the order of say the by now disappearing F(750) digamma. Let as say that, since no such bound state or replacement for it has been seen, a mass of the order of 1 TeV at least should be estimated. We would then say that we have an effective field theory and may take the scale | for it to be of the order of 1 TeV. 18.5.4 Coupling's Maximum Now we then want to argue that when we consider an effective field theory at a scale |i = a-1, where a is the typical length for the scale of phenomena considered, there must be an upper bound of what the effective field theory coupling G on some vertex such as G%1 can physically be. Here the ... just stands for some Y-matrices or the like. In fact we want to argue that order of magnitudewise we must have I2G = G/a2 APC Colloquiums), APC Seminars (Previous APC Seminars) and Events (Previous Events), Calendar of the past and future VIA events (All events) and VIA Forum (Forum). In the upper right angle there are links to Google search engine (Search in site) and to contact information (Contacts). The announcement of the next VIA lecture and VIA online transmission of APC Colloquium occupy the main part of the homepage with Next regular Lecture's 11 1er 7, 2018 116h ■ 17h Paris Lecture by Fiorella Burgio \ i Title of lecture: "Neutron star universal relations with microscopic equations of state" Language of lecture: w institute & Country: INFN Section Catania, Catania, Italy Previous Lecture 11 0BL Lecture by Lecture by Bill IpP "Significant Excess of Electron-Like Events See All presentations GW170814: A three-detector observation of gravitational waves from a, binary black hole The latest T2K neutrino oscillation results First Dark Matter Search Results from the XENON1T Experiment sciences cosmovia.org help Beyondthe standard model in the News HowtouseVIA Astroparticle physics Applications Cosmology Facebook Gravitational waves experiments Partners of VIA Astrophysics Contact Neutrinos Ai [Put Ä- ¿gfo UNIVERSITÉ \m W' Surs. W DE GENÈVE Fig. 19.1. The home page of VIA site the record of the most recent VIA events below. In the announced time of the event (VIA lecture or transmitted APC Colloquium) it is sufficient to click on "to participate" on the announcement and to Enter as Guest (printing your name) in the corresponding Virtual room. The Calendar shows the program of future VIA lectures and events. The right column on the VIA homepage lists the announcements of the regularly up-dated hot news of Astroparticle physics and related areas. In 2010 special COSMOVIA tours were undertaken in Switzerland (Geneva), Belgium (Brussels, Liege) and Italy (Turin, Pisa, Bari, Lecce) in order to test stability of VIA online transmissions from different parts of Europe. Positive results of these tests have proved the stability of VIA system and stimulated this practice at XIII Bled Workshop. The records of the videoconferences at the XIII Bled Workshop are available on VIA site [17]. Since 2011 VIA facility was used for the tasks of the Paris Center of Cos-mological Physics (PCCP), chaired by G. Smoot, for the public programme "The two infinities" conveyed by J.L.Robert and for effective support a participation at distance at meetings of the Double Chooz collaboration. In the latter case, the experimentalists, being at shift, took part in the collaboration meeting in such a virtual way. The simplicity of VIA facility for ordinary users was demonstrated at XIV Bled Workshop in 2011. Videoconferences at this Workshop had no special technical support except for WiFi Internet connection and ordinary laptops with their internal webcams and microphones. This test has proved the ability to use VIA facility at any place with at least decent Internet connection. Of course the quality of records is not as good in this case as with the use of special equipment, but still it is sufficient to support fruitful scientific discussion as can be illustrated by the record of VIA presentation "New physics and its experimental probes" given by John Ellis from his office in CERN (see the records in [18]). In 2012 VIA facility, regularly used for programs of VIA lectures and transmission of APC Colloquiums, has extended its applications to support M.Khlopov's talk at distance at Astrophysics seminar in Moscow, videoconference in PCCP, participation at distance in APC-Hamburg-Oxford network meeting as well as to provide online transmissions from the lectures at Science Festival 2012 in University Paris7. VIA communication has effectively resolved the problem of referee's attendance at the defence of PhD thesis by Mariana Vargas in APC. The referees made their reports and participated in discussion in the regime of VIA videoconference. In 2012 VIA facility was first used for online transmissions from the Science Festival in the University Paris 7. This tradition was continued in 2013, when the transmissions of meetings at Journees nationales du Developpement Logiciel (JDEV2013) at Ecole Politechnique (Paris) were organized [20]. In 2013 VIA lecture by Prof. Martin Pohl was one of the first places at which the first hand information on the first results of AMS02 experiment was presented [19]. In 2014 the 100th anniversary of one of the foundators of Cosmoparticle physics, Ya. B. Zeldovich, was celebrated. With the use of VIA M.Khlopov could contribute the programme of the "Subatomic particles, Nucleons, Atoms, Universe: Processes and Structure International conference in honor of Ya. B. Zeldovich 100th Anniversary" (Minsk, Belarus) by his talk "Cosmoparticle physics: the Universe as a laboratory of elementary particles" [21] and the programme of "Conference YaB-100, dedicated to 100 Anniversary of Yakov Borisovich Zeldovich" (Moscow, Russia) by his talk "Cosmology and particle physics" [22]. In 2015 VIA facility supported the talk at distance at All Moscow Astrophysi-cal seminar "Cosmoparticle physics of dark matter and structures in the Universe" by Maxim Yu. Khlopov and the work of the Section "Dark matter" of the International Conference on Particle Physics and Astrophysics (Moscow, 5-10 October 2015). Though the conference room was situated in Milan Hotel in Moscow all the presentations at this Section were given at distance (by Rita Bernabei from Rome, Italy; by Juan Jose Gomez-Cadenas, Paterna, University of Valencia, Spain and by Dmitri Semikoz, Martin Bucher and Maxim Khlopov from Paris) and its work was chaired by M.Khlopov from Paris [27]. In the end of 2015 M. Khlopov gave his distant talk "Dark atoms of dark matter" at the Conference "Progress of Russian Astronomy in 2015", held in Sternberg Astronomical Institute of Moscow State University. In 2016 distant online talks at St. Petersburg Workshop "Dark Ages and White Nights (Spectroscopy of the CMB)" by Khatri Rishi (TIFR, India) "The information hidden in the CMB spectral distortions in Planck data and beyond", E. Kholupenko (Ioffe Institute, Russia) "On recombination dynamics of hydrogen and helium", Jens Chluba (Jodrell Bank Centre for Astrophysics, UK) "Primordial recombination lines of hydrogen and helium", M. Yu. Khlopov (APC and MEPHI, France and Russia)"Nonstandard cosmological scenarios" and P. de Bernardis (La Sapiensa University, Italy) "Balloon techniques for CMB spectrum research" were given with the use of VIA system [28]. At the defense of PhD thesis by F. Gregis VIA facility made possible for his referee in California not only to attend at distance at the presentation of the thesis but also to take part in its successive jury evaluation. Since 2018 VIA facility is used for collaborative work on studies of various forms of dark matter in the framework of the project of Russian Science Foundation based on Southern Federal University (Rostov on Don). In September 2018 VIA supported online transmission of 17 presentations at the Commemoration day for Patrick Fleury, held in APC [29]. The discussion of questions that were put forward in the interactive VIA events is continued and extended on VIA Forum. Presently activated in En-glish,French and Russian with trivial extension to other languages, the Forum represents a first step on the way to multi-lingual character of VIA complex and its activity. Discussions in English on Forum are arranged along the following directions: beyond the standard model, astroparticle physics, cosmology, gravitational wave experiments, astrophysics, neutrinos. After each VIA lecture its pdf presentation together with link to its record and information on the discussion during it are put in the corresponding post, which offers a platform to continue discussion in replies to this post. 19.2.2 VIA e-learning, OOC and MOOC One of the interesting forms of VIA activity is the educational work at distance. For the last eleven years M.Khlopov's course "Introduction to cosmoparticle physics" is given in the form of VIA videoconferences and the records of these lectures and their ppt presentations are put in the corresponding directory of the Forum [23]. Having attended the VIA course of lectures in order to be admitted to exam students should put on Forum a post with their small thesis. In this thesis students are proposed to chose some BSM model and to study the cosmological scenario based on this chosen model. The list of possible topics for such thesis is proposed to students, but they are also invited to chose themselves any topic of their own on possible links between cosmology and particle physics. Professor's comments and proposed corrections are put in a Post reply so that students should continuously present on Forum improved versions of work until it is accepted as admission for student to pass exam. The record of videoconference with the oral exam is also put in the corresponding directory of Forum. Such procedure provides completely transparent way of evaluation of students' knowledge at distance. In 2018 the test has started for possible application of VIA facility to remote supervision of student's scientific practice. The formulation of task and discussion of porgress on work are recorded and put in the corresponding directory on Forum together with the versions of student's report on the work progress. Since 2014 the second semester of the course on Cosmoparticle physics is given in English and converted in an Open Online Course. It was aimed to develop VIA system as a possible accomplishment for Massive Online Open Courses (MOOC) activity [24]. In 2016 not only students from Moscow, but also from France and Sri Lanka attended this course. In 2017 students from Moscow were accompanied by participants from France, Italy, Sri Lanka and India [25]. The students pretending to evaluation of their knowledge must write their small thesis, present it and, being admitted to exam, pass it in English. The restricted number of online connections to videoconferences with VIA lectures is compensated by the wide-world access to their records on VIA Forum and in the context of MOOC VIA Forum and videoconferencing system can be used for individual online work with advanced participants. Indeed Google Analytics shows that since 2008 VIA site was visited by more than 242 thousand visitors from 153 countries, covering all the continents by its geography (Fig. 19.2). According to this statistics more than half of these visitors continued to enter VIA site after the first visit. Still the form of individual educational work makes VIA facility most appropriate for PhD courses and it is planned to be involved in the International PhD program on Fundamental Physics, which can be started on the basis of Russian-French collaborative agreement. In 2017 the test for the ability of VIA to support fully distant education and evaluation of students (as well as for work on PhD thesis and its distant defense) was undertaken. Steve Branchu from France, who attended the Open Online Course and presented on Forum his small thesis has passed exam at distance. The whole procedure, starting from a stochastic choice of number of examination ticket, answers to ticket questions, discussion by professors in the absence of student and announcement of result of exam to him was recorded and put on VIA Forum [26]. ^ - V Fig. 19.2. Geography of VIA site visits according to Google Analytics 19.2.3 Organisation of VIA events and meetings First tests of VIA system, described in [5,7-9], involved various systems of videoconferencing. They included skype, VRVS, EVO, WEBEX, marratech and adobe Connect. In the result of these tests the adobe Connect system was chosen and properly acquired. Its advantages are: relatively easy use for participants, a possibility to make presentation in a video contact between presenter and audience, a possibility to make high quality records, to use a whiteboard tools for discussions, the option to open desktop and to work online with texts in any format. Initially the amount of connections to the virtual room at VIA lectures and discussions usually didn't exceed 20. However, the sensational character of the exciting news on superluminal propagation of neutrinos acquired the number of participants, exceeding this allowed upper limit at the talk "OPERA versus Maxwell and Einstein" given by John Ellis from CERN. The complete record of this talk and is available on VIA website [30]. For the first time the problem of necessity in extension of this limit was put forward and it was resolved by creation of a virtual "infinity room", which can host any reasonable amount of participants. Starting from 2013 this room became the only main virtual VIA room, but for specific events, like Collaboration meetings or transmissions from science festivals, special virtual rooms can be created. This solution strongly reduces the price of the licence for the use of the adobeConnect videoconferencing, retaining a possibility for creation of new rooms with the only limit to one administrating Host for all of them. The ppt or pdf file of presentation is uploaded in the system in advance and then demonstrated in the central window. Video images of presenter and participants appear in the right window, while in the lower left window the list of all the attendees is given. To protect the quality of sound and record, the participants are required to switch out their microphones during presentation and to use the upper left Chat window for immediate comments and urgent questions. The Chat window can be also used by participants, having no microphone, for questions and comments during Discussion. The interactive form of VIA lectures provides oral discussion, comments and questions during the lecture. Participant should use in this case a "raise hand" option, so that presenter gets signal to switch out his microphone and let the participant to speak. In the end of presentation the central window can be used for a whiteboard utility as well as the whole structure of windows can be changed, e.g. by making full screen the window with the images of participants of discussion. Regular activity of VIA as a part of APC includes online transmissions of all the APC Colloquiums and of some topical APC Seminars, which may be of interest for a wide audience. Online transmissions are arranged in the manner, most convenient for presenters, prepared to give their talk in the conference room in a normal way, projecting slides from their laptop on the screen. Having uploaded in advance these slides in the VIA system, VIA operator, sitting in the conference room, changes them following presenter, directing simultaneously webcam on the presenter and the audience. 19.3 VIA Sessions at XXI Bled Workshop VIA sessions of XXI Bled Workshop continued the tradition coming back to the first experience at XI Bled Workshop [7] and developed at XII, XIII, XIV, XV, XVI, XVII, XVIII, XIX and XX Bled Workshops [8-16]. They became a regular part of the Bled Workshop's program. In the course of XXI Bled Workshop, the list of open questions was stipulated, which was proposed for wide discussion with the use of VIA facility. The list of these questions was put on VIA Forum (see [31]) and all the participants of VIA sessions were invited to address them during VIA discussions. During the XXI Bled Workshop the announcement of VIA sessions was put on VIA home page, giving an open access to the videoconferences at VIA sessions. Though the experience of previous Workshops principally confirmed a possibility to provide effective interactive online VIA videoconferences even in the absence of any special equipment and qualified personnel at place, VIA Sessions were directed at XXI Workshop by M.Khlopov at place. Only laptop with microphone and webcam together with WiFi Internet connection was proved to support not only attendance, but also VIA presentations and discussions. In the framework of the program of XXI Bled Workshop, S. Ketov, gave his talk "Starobinsky Inflation in Gravity and Supergravity" (Fig. 19.3), from Japan, while his co-author M.Khlopov continued the talk in Bled. VIA session also included discussion of searches for new physics at the LHC with participation at distance by A.Romaniouk from CERN. It provided an additional demonstration of the ability of VIA to support the creative non-formal atmosphere of Bled Workshops (see records in [32]). The talks "Theories for initial conditions" by Holger B. Nielsen(Fig. 19.4) "Experimental consequences of spin-charge family theory" by Norma Mankoc-Borstnik (Fig. 19.5) were given at Bled, inviting distant participants to join the discussion. Fig. 19.3. VIA talk "Starobinsky Inflation in Gravity and Supergravity" by S. Ketov from Japan at XXI Bled Workshop Fig. 19.4. VIA talk by Holger B. Nielsen "Theories for initial conditions" at XXI Bled Workshop Fig. 19.5. VIA talk "Experimental consequences of spin-charge family theory" by Norma Mankoc-Borstnik at XXI Bled Workshop The records of all these lectures and discussions can be found in VIA library [32]. 19.4 Conclusions The Scientific-Educational complex of Virtual Institute of Astroparticle physics provides regular communication between different groups and scientists, working in different scientific fields and parts of the world, the first-hand information on the newest scientific results, as well as support for various educational programs at distance. This activity would easily allow finding mutual interest and organizing task forces for different scientific topics of astroparticle physics and related topics. It can help in the elaboration of strategy of experimental particle, nuclear, astro-physical and cosmological studies as well as in proper analysis of experimental data. It can provide young talented people from all over the world to get the highest level education, come in direct interactive contact with the world known scientists and to find their place in the fundamental research. These educational aspects of VIA activity is now being evolved in a specific tool for International PhD programme for Fundamental physics. VIA applications can go far beyond the particular tasks of astroparticle physics and give rise to an interactive system of mass media communications. VIA sessions became a natural part of a program of Bled Workshops, maintaining the platform of discussions of physics beyond the Standard Model for distant participants from all the world. This discussion can continue in posts and post replies on VIA Forum. The experience of VIA applications at Bled Workshops plays important role in the development of VIA facility as an effective tool of e-science and e-learning. Acknowledgements The initial step of creation of VIA was supported by ASPERA. I am grateful to P.Binetruy, J.Ellis and S.Katsanevas for permanent stimulating support, to J.C. Hamilton for support in VIA integration in the structure of APC laboratory, to K.Belotsky, A.Kirillov, M.Laletin and K.Shibaev for assistance in educational VIA program, to A.Mayorov, A.Romaniouk and E.Soldatov for fruitful collaboration, to M.Pohl, C. Kouvaris, J.-R.Cudell, C. Giunti, G. Cella, G. Fogli and F. DePaolis for cooperation in the tests of VIA online transmissions in Switzerland, Belgium and Italy and to D.Rouable for help in technical realization and support of VIA complex. The work was supported by grant of Russian Science Foundation (project N-18-12-00213). I express my gratitude to N.S. Mankoc Borstnik, D. Lukman and all Organizers of Bled Workshop for cooperation in the organization of VIA Sessions at XXI Bled Workshop. References 1. http://www.aspera-eu.org/ 2. http://www.appec.org/ 3. M.Yu. Khlopov: Cosmoparticle physics, World Scientific, New York -London-Hong Kong - Singapore, 1999. 4. M.Yu. Khlopov: Fundamentals of Cosmic Particle Physics, CISP-Springer, Cambridge, 2012. 5. M. Y. Khlopov, Project of Virtual Institute of Astroparticle Physics, arXiv:0801.0376 [astro-ph]. 6. http://viavca.in2p3.fr/site.html 7. M. Y. Khlopov, Scientific-educational complex - virtual institute of astroparticle physics, Bled Workshops in Physics 9 (2008) 81-86. 8. M. Y. Khlopov, Virtual Institute of Astroparticle Physics at Bled Workshop, Bled Workshops in Physics 10 (2009) 177-181. 9. M. Y. Khlopov, VIA Presentation, Bled Workshops in Physics 11 (2010) 225-232. 10. M. Y. Khlopov, VIA Discussions at XIV Bled Workshop, Bled Workshops in Physics 12 (2011) 233-239. 11. M. Y. .Khlopov, Virtual Institute of astroparticle physics: Science and education online, Bled Workshops in Physics 13 (2012) 183-189. 12. M. Y. .Khlopov, Virtual Institute of Astroparticle physics in online discussion of physics beyond the Standard model, Bled Workshops in Physics 14 (2013) 223-231. 13. M. Y. .Khlopov, Virtual Institute of Astroparticle physics and "What comes beyond the Standard model?" in Bled, Bled Workshops in Physics 15 (2014) 285-293. 14. M. Y. .Khlopov, Virtual Institute of Astroparticle physics and discussions at XVIII Bled Workshop, Bled Workshops in Physics 16 (2015) 177-188. 15. M. Y. .Khlopov, Virtual Institute of Astroparticle Physics — Scientific-Educational Platform for Physics Beyond the Standard Model Bled Workshops in Physics 17 (2016) 221-231. 16. M. Y. .Khlopov: Scientific-Educational Platform of Virtual Institute of Astroparticle Physics and Studies of Physics Beyond the Standard Model. Bled Workshops in Physics 18 (2017) 273-283. 17. In http://viavca.in2p3.fr/ Previous - Conferences - XIII Bled Workshop 18. In http://viavca.in2p3.fr/ Previous - Conferences - XIV Bled Workshop 19. In http://viavca.in2p3.fr/ Previous - Lectures - Martin Pohl 20. In http://viavca.in2p3.fr/ Previous - Events - JDEV 2013 21. In http://viavca.in2p3.fr/ Previous - Conferences - Subatomic particles, Nucleons, Atoms, Universe: Processes and Structure International conference in honor of Ya. B. Zeldovich 100th Anniversary 22. In http://viavca.in2p3.fr/ Previous - Conferences - Conference YaB-100, dedicated to 100 Anniversary of Yakov Borisovich Zeldovich 23. In http://viavca.in2p3.fr/ Forum - Discussion in Russian - Courses on Cosmoparticle physics 24. In http://viavca.in2p3.fr/ Forum - Education - From VIA to MOOC 25. In http://viavca.in2p3.fr/ Forum - Education - Lectures of Open Online VIA Course 2017 26. In http://viavca.in2p3.fr/ Forum - Education - Small thesis and exam of Steve Branchu 27. http://viavca.in2p3.fr/ Previous - Conferences - The International Conference on Particle Physics and Astrophysics 28. http://viavca.in2p3.fr/ Previous - Conferences - Dark Ages and White Nights (Spec-troscopy of the CMB) 29. http://viavca.in2p3.fr/ Previous - Events - Commemoration day for Patrick Fleury. 30. In http://viavca.in2p3.fr/ Previous - Lectures - John Ellis 31. In http://viavca.in2p3.fr/ Forum - CONFERENCES BEYOND THE STANDARD MODEL - XXI Bled Workshop "What comes beyond the Standard model?" 32. In http://viavca.in2p3.fr/ Previous - Conferences - XXI Bled Workshop "What comes beyond the Standard model?" Poem by Astri Kleppe Bled Workshops in Physics Vol. 19, No. 2 A Proceedings to the 21 st Workshop What Comes Beyond ... (p. 397) Bled, Slovenia, June 23-July 1, 2018 20 June Astri Kleppe Tonight they all spring into blossom, Lilacs, Bird Cherry with strands of willow, weaving their way to the beginning; How it all was meant to be. Birdsong, sound of running steps, of tea and cups; Soon the larvae will be heading for the bird cherry, enfolding it in silvery cocoons; and soon warm nights of August will bring darkness, for Orion to be seen. But in this night of early June it's all in ecstasy, in vigil for the blossom; No one sleeps, the birds, the flies are all awake. Who are you then, I asked the Lilacs We are strangers here, the answer came, and we belong to no one But tell me who you are, I begged And flower clusters sprinkled over me, a waterfall ofpetals, We just arrived, tonight, what more is there to say? 398 Astri Kleppe ii A loose dream from a corner of the universe is driving towards us, our island; Clouds awakening of granite and cadavers, blue and earth; A boomerang towards The Milky Way, its icy stars and howling wolves, and tightly curved around the little heat from our own speed. We are but animals of auguries, ofhope and salt; and though great dreams ofLeibniz, Alan Guth and Hubble led us, it was other tokens that the Universe imagined, ofanother kind; And suddenly this otherness sticks out: a tree With roots in galaxies and whispers, in galactic summer, leaves that dance in morning breeze and drizzle, with a scent of seed and clover, pregnant visions, over paths through rain-gray grass. Beware of those who tread on dew and stop under that tree. So slowly night is turning, in this space of the improbable, a darkness where a tree can grow from nothing, rise with flower buds and day. It was in June, an early morning in a sea before the blue of skies, before all grass and oxygen; but still a sea, a meadow, all its flowers. 20 June 399 They gave us this, a place so wonderful, our garden with its tender plants and trees of darkest mold with branches stretching out to heaven, all the blue. And over time we built our house, with kitchen windows with a view of oaks, and in the winter nights the dog lay by the oven. And from stone and dust and clay we built our roads, from sand and metal the first street light, zippers and TV. The dogs got leashes, we got blogs, so marvelous our garden, and so tall the skyscrapers; In street dust and the growing noise we saw the progress, in the dying trees the halved diversity of everything. Do you remember all these woods? So vast and wild, the rugged mountains roaring over trees and oceans. O, the ocean! Waving waves around the boat, like fingers through a golden field, the time when the first crops were grown. This was our Earth, with seas and meadows, timothy, all life born within this triangle where ice and clouds meet seas and rivers. It was ours. We had all this, the grass, the water, sky and clouds. Our garden was complete. iv The elements are four, they said, it's Fire, Earth and Air and Water. Air is made of eyes, the Earth is made of red and green, and Water shapes our dreams. Only Fire, that's the Sun, is still unstained. No smog is registered in solar atmospheres, no drought, no poison or depletion of the soil. Fire alone is pure. It burns, it is; It shall not want. Three things, they said to me, are good things. Four is still OK, and five is not so bad, but six is already too much, and seven is pure waste. But seven, I protested, is a lovely number, seven good things is far more than three. Three or seven, never mind, they said. Our only worry is when letters are forgotten or when too many are added; And in the early morning flocks of black deer ran beneath us, dark clouds of pure music on the newgrown grass. Blejske Delavnice Iz Fizike, Letnik 19, št. 2, ISSN 1580-4992 Bled Workshops in Physics, Vol. 19, No. 2 Zbornik 21. delavnice 'What Comes Beyond the Standard Models', Bled, 23. -29. junij2018 Proceedings to the 21st workshop 'What Comes Beyond the Standard Models', Bled, June 23.-29., 2018 Uredili Norma Susana Mankoc Borstnik, Holger Bech Nielsen in Dragan Lukman Izid publikacije je finančno podprla Javna agencija za raziskovalno dejavnost RS iz sredstev drZavnega proracuna iz naslova razpisa za sofinanciranje domacih znanstvenih periodicnih publikacij Brezplacni izvod za udelezzence Tehnicni urednik Matjaz Zaversnik Zalozilo: DMFA - zaloznistvo, Jadranska 19,1000 Ljubljana, Slovenija Natisnila tiskarna Itagraf v nakladi 100 izvodov Publikacija DMFA stevilka 2077