Blejske delavnice iz fizike
Bled Workshops in Physics
ISSN 1580-4992
Letnik 15, St. 2
VOL. 15, NO. 2
Proceedings to the 17th Workshop
What Comes Beyond the Standard Models
Bled, July 20-28, 2014
Edited by
Norma Susana Mankoc Borstnik Holger Bech Nielsen Dragan Lukman
DMFA - zaloZniStvo Ljubljana, december 2014
The 17th Workshop What Comes Beyond the Standard Models,
20.- 28. July 2014, 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 / BS Storitve d.o.o. (MatjaZ Breskvar)
Scientific Committee
John Ellis, CERN David Gross, KITP Roman Jackiw, MIT
Organizing Committee
Norma Susana Mankoi BorStnik 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 17th 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
>	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
>
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	Dark Atoms and Their Decaying Constituents
K. Belotsky, M. Khlopov and M. Laletin................................. 1
2	Dark Matter Particles in the Galactic Halo
R. Bernabei......................................................... 10
3	New Experimental Data for the Quarks Mixing Matrix are in Better Agreement with the Spin-charge-family Theory Predictions
G.	Bregar and N.S. Mankoc Borštnik ................................... 20
4	Emergent SUSY Theories: QED, SM & GUT
J.L. Chkareuli....................................................... 46
5	Some Potential Problems of OHe Composite Dark Matter
J.R. Cudell, M. Khlopov and Q. Wallemacq.............................. 66
6	Notoph-Graviton-Photon Coupling
V. V. Dvoeglazov .................................................... 75
7	General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family Symmetry with Sterile Neutrinos
A. Hernandez-Galeana............................................... 93
8	Gravitational Interactions and Fine-Structure Constant
U.D. Jentschura, J.H. Noble and I. Nandori..............................115
9	Can Spin-charge-family Theory Explain Baryon Number Non-conservation? N.S. Mankoc Borštnik................................................123
10	The Spin-charge-family Theory Explains Why the Scalar Higgs Carries the Weak Charge ± \ and the Hyper Charge ^ 1
N.S. Mankoc Borštnik................................................163
11	Deriving Veneziano Model in a Novel String Field Theory Solving String Theory by Liberating Right and Left Movers
H.B.	Nielsen and M. Ninomiya........................................183
Discussion Section..................................................221
12	Properties of Families of Spinors in d = (5 + 1) with Zweibein of an Almost S2 and Two Kinds of Spin Connection Fields, Allowing Massless and Massive Solutions in d = (3 + 1)
D. Lukman and N.S. Mankoc Borštnik..................................223
13	Influence from Future, Arguments
H.B.Nielsen ........................................................239
14	Deriving Diffeomorphism Symmetry
H.B. Nielsen and A. Kleppe...........................................267
Virtual Institute of Astroparticle Physics Presentation...................283
15	Virtual Institute of Astroparticle physics and "What comes beyond the Standard model?" in Bled
M.Yu. Khlopov......................................................285
Preface
The series of 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. It is the seventeenth workshop which took place this year in the picturesque town of Bled by the lake of the same name, surrounded by beautiful mountains and offering pleasant walks and mountaineering. 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. 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, analysing 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 showed up very clearly. The ideas therefore seem to develop in these years considerably faster than they would without our workshops. In the seventeen years of our workshops the organizers, together with the participants, are trying to answer several open questions of the elementary particle physics and cosmology. Experiments have made large steps in this time. Among the most notable and might be also among the most important ones was two years ago the LHC confirmation that the scalar field, the Higgs, is like other fermionic and bosonic fields - just a field. And yet it is a very unusual field: A boson with the fractional weak and hyper charges. Do we have the explanation for that? Can we explain the origin of families and Yukawa couplings? Can we understand the origin of the matter-antimatter asymmetry? Can we explain all the assumptions of the standard Model?
The evolution of the universe and the dynamics of it on all levels, from the elementary particles to the matter, can be understood only, if we have the theory behind, which explains the observations and predicts new phenomena. Should we design theories and models in steps, each one more or less adapted for explaining a new experimental observation? Or can we suggest the theory which answers several (all?) open questions at the same time? Do the laws of Nature emerge at the low energy regime in the way we observe them? Or are the laws of Nature simple and elegant on all scales, while the observation of systems of many degrees of freedom do not show up the laws behind in a transparent way?
Can it happen that at the LHC no new fields, scalars, vectors or fermions will be observed, so that there will be no sign which will help to make a trustable step beyond the standard model?
If trusting the spin-charge-family theory, predicting the fourth family, coupled to the observed three families, and several scalar fields, this is not possible. This theory offers the explanation for all the assumptions of the standard model. It answers the question why does the Higgs's scalar (in this theory a superposition of several scalar fields) carry the weak charge and the hyper charge equal to (± 1, T 2), respectively. Since these scalar fields carry also quantum numbers of the family groups, the theory offers as well the explanation for the origin of the Yukawa couplings. Predicting the second group of four families, the lowest of these four families might explain the origin of the dark matter.
There are colour triplet scalars in this theory, transforming antileptons and antiquarks into quarks and back, transforming antimatter into matter and back. The scalar condensate of two right handed neutrinos with the family quantum numbers of the upper four families breaks the matter-antimatter symmetry. This could, hopefully, explain the observed matter-antimatter asymmetry. What is beautiful in this theory is, that a simple and elegant starting action has (only) all the needed degrees of freedom to explain in the low energy regime all the properties of quarks and leptons and of the vector and scalar gauge fields, predicting even how will more accurate measurements of the mixing matrices among the observed families of fermions change the values. The experiments might observe, due to this theory, that the space time is more than 3+1 dimensional.
There are inventive new predictions in this proceeding, in the main part and also in the Discussion section: Like the one that the local gauge symmetries appear due to the spontaneous break of the Lorentz invariance and supersymmetry, and there are interesting trials to built the origin of all the gauge fields in a common theory within the theory of higher spins, there are trials to explain emergency of the space and the influence of the future on the past.
There are attempts to extend the standard model with new fields of fermionic and bosonic origin to explain mass matrices and correspondingly masses and mixing matrices of quarks and leptons. Although it is not easy to see, to which extent different theories overlap when describing the same physical phenomena, yet our discussions helped to clarify many points a lot.
There is the work of new string theory, made of non interacting equal scalar objects, which reproduces the Veneziano scattering amplitude. These objects, which do not recognize "their own identity" when they meet with another group of equal
objects and the scattering matrix in the Hilbert space of which is equal to unity, manifest all the properties of strings, with the tension included. It is the experimental contribution presenting the dark matter observations of DAMA/LIBRA through more than a decade of years. It includes also explanation why other experiments have not confirmed the DAMA/LIBRA experiments yet. There are works trying to explain the direct observations from different laboratories with the idea that new particles, like OHe, exist and form the dark matter. There were works in this year workshop with very promising ideas and developments of the ideas, the authors of which have not succeeded to send in time their contributions. And there were very discerning discussions among the participants, for which there has been not enough of time to matured the discussions into the written contribution.
Since the time to prepare the proceedings is indeed very short, three months if
vacations are not counted, authors did not have a time to polish their contributions carefully enough.
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. Questions and answers as well as lectures enabled by M.Yu. Khlopov via Virtual Institute of Astroparticle Physics (www.cosmovia.org) 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 www.cosmovia.org in Previous - Conferences. The four talks delivered by: R.Bernabei (Dark matter particles in the galactic halo), M. Laletin and M.Yu. Khlopov (Dark Atoms and their Decaying Constituents), N.S. Mankoc Borštnik (The Spin-Charge-Family theory offers the explanation for the assumptions of the Standard model, for the Dark matter, for the Matter-antimatter asymmetry), H.B.F. Nielsen (Novel string field theory), can be accessed directly at http://viavca.in2p3.fr/what_comes_beyond_the_standard_models_xvii.html Most of the talks can be found on the workshop homepage http://bsm.fmf.uni-lj.si/.
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.
The workshops take place in the house gifted to the Society of Mathematicians, Physicists and Astronomers of Slovenia by the Slovenian mathematician Josip Plemelj, well known to the participants by his work in complex algebra.
Norma Mankoč Borštnik, Holger Bech Nielsen, Maxim Y. Khlopov, (the Organizing comittee)
Norma Mankoč BorStnik, Holger Bech Nielsen, Dragan Lukman, (the Editors)
Ljubljana, December 2014
1 Predgovor (Preface in Slovenian Language)
Serija delavnic "Kako preseči oba standardna modela, kozmološkega in elek-trošibkega" ("What Comes Beyond the Standard Models?") se je začela leta 1998 z idejo Norme in Holgerja, da bi organizirali delavnice, v katerih bi udeleZenci v izčrpnih diskusijah kritično soočili različne ideje in teorije. Letos smo imeli sedemnajsto delavničo v mestu Bled ob slikovitem jezeru, kjer prijetni sprehodi in pohodi na čudovite gore, ki kipijo nad mestom, ponujajo prilozznosti in vzpodbudo za diskusije.
K našim zelo odprtim, prijateljskim, dolgim in zahtevnim diskusijam, polnim iskrivega sodelovanja, je prispevalo veliko fizikov in čelo nekaj matematikov. Večina predlogov teorij in modelov, predstavljenih in diskutiranih na naših blejskih delavničah, išče odgovore na vprasanja, ki jih v fizikalni skupnosti sprejeta in s številnimi poskusi potrjena standardni model osnovnih fermionskih in bo-zonskih polj ter kozmološki standardni model pusčata odprta. Ceprav je večina udelezenčev teoretičnih fizikov, mnogi z lastnimi idejami kako narediti naslednji korak onkraj sprejetih modelov in teorij, so še posebej dobrodosli predstavniki eksperimentalnih laboratorijev, ki nam pomagajo v odprtih diskusijah razjasniti resnično sporočilo meritev in katere napovedi lahko poskusi najzanesljiveje preverijo.
Organizatorji moramo priznati, da smo se na blejskih delavničah v (dolgih) predstavitvah (z odmori in nadaljevanji čšez večš dni), ki so jim sledile zelo podrobne diskusije, naučili veliko, morda več kot večina udelezenčev. Upamo in verjamemo, da so veliko odnesli tudi sštudentje in večšina udelezšenčev. Velikokrat so se predavanja spremenila v zelo pedagoske 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 sibke točke predlogov. Zdi se, da so se ideje v teh letih razvijale bistveno hitreje, zahvaljujoč prav tem delavničam.
V teh sedemnajstih letih delavnič smo organizatorji skupajz udelezenči poskusili odgovoriti na marsikatero odprto vprasanje v fiziki osnovnih delčev in kozmologiji. Na vsakoletnem napovedniku nasše delavniče objavimo zbirko odprtih vprasšanj, na katera bi udelezenči utegnili predlagati resitve. V sedemnajstih letih so eksperimenti napravili velike korake. Med najpomembnejšimi dosezki je potrditev LHC, da je skalarno pole, Higgsov deleč, prav tako polje kot ostala fermionska in bo-zonska polja. In vendar je to skalarno polje zelo nenavadno polje: Je bozon s polovičnim šibkim in hiper nabojem. Ali to razumemo? Ali lahko pojasnimo izvor druzin in Yukawinih sklopitev? Znamo pojasniti nesimetrijo med snovjo in antisnovjo v vesolju? Znamo razločiti privzetke standardnega modela?
Dinamiko vesolja na vseh nivojih, od osnovnih delcev do snovi, lahko razumemo samo, ce ponudimo teorijo, ki opaZenja razloZi in napove nova spoznanja. Je prava pot pri postavljanju teorijta, da prilagodimo teorijo eksperimentalnim spoznanjem po korakih? Ali lahko ponudimo teorijo, ki odgovori na mnoga (morda vsa) doslej odprta vprasanja?
Ali se naravni zakoni manifestirajo le pri nizkih energijah? Ali pa so preprosti in elegantni na vseh nivojih, le da jih pri nizkih energijah opazujemo v sistemih z velikim stevilom sodelujočih, kjer jih je tezko razspoznati?
Kaj pa, če na LHC ne bodo izmerili nobenega novega polja, ne skalarja, ne vektorja, ne fermiona in ne bo ponudil eksperiment nobenega napotka, kako izbrati naslednji korak od standardnega modela?
Ce ima teorija spina, naboja in družin prav, potem bodo na LHC izmerili četrto druzino, ki je z ze opazenimi sklopljena ter verjetno se kakšnega od dveh tripletov in treh singletov, ki jih teorija napoveduje. Teorija razlozi vse predpostavke standardnega modela, pojasni izvor druzin kvarkov in leptonov, vektorskih umeritvenih poljin izvor Higgsovega skalarja ter Yukawinih sklopitev. Razlozi tudi, zakajima Higgsov skalar polstevilčen sibki in hiper naboj. To lastnost imata oba tripleta (ki nosita tudi druzinska kvantna števila) in trije singleti skalarnih polj, ki poskrbijo za masne matrike kvarkov in leptonov ter razlozijo Yukawine sklopitve. Teorija napove še eno skupino štirih druzin kvarkov in leptonov. Družina z najnizjo energijo razlozi nastanek temne snovi.
Teorija napove skupino skalarnih polj, ki so barvni tripleti. Povzročijo rojstvo kvarkov iz antileptonov in antikvarkov, s tem pa tudi rojstvo nukleonov. Reak-čije tečejo tudi v obratni smeri. Kondenzat iz dveh desnoročnih nevtrinov z družinskimi kvantnimi stevili zgornjih štirih druzin zlomi simetrijo med obem reakčijama in ponudi odgovor na vprasšanje, kaj je vzrok presezšku snovi nad antisnovjo v opazljivem delu vesolja.
Teorija je lepa in elegantna, ker ponudi preprosta začšetna akčija vsa fermionska (druzšine kvarkov in leptonov z opazšenimi lastnostmi) in bozonska (vektorska umeritvena polja in skalarna polja) polja, ki so ze opazena, direktno ali indirektno, napove nove fermione, nova skalarna polja in napove, kako se bodo spreminjali matrični elementi mesalne matrike pri boljnatančnih meritvah. Novi eksperimenti bodo prejali slejdokazali, če ima teorija ,,prav", da je dimenzija prostora-časa več kot le (3 + 1).
V zborniku so nove zelo preroske teorije: Lokalne umeritvene simetrije se pojavijo kot odziv sistema na spontano zlomitev Lorentzove invarianče, pa tudi super-simetrije. Zanimiva je prav gotovo tudi teorija, ki gradi na spinih večjih kot 2, da razlozi vsa umeritvena polja. Zbornik vsebuje tudi prispevek, ki razsiri standardni model z novimi fermionskimi in bozonskimi polji, da bi pojasnil masne matrike kvarkov in leptonov, ter s tem njihove mase in mesalne matrike. Je tudi delo, nova teorija strune, ki jo sestavljajo enaki skalarni objekti, ki med seboj sploh ne interagirajo. In vendar reprodučira teorija Venezianovo sipalno amplitudo. Ti objekti, ki „pozabijo", ko srečajo drugo struno iz enakih objektov, kateri struni so pripadali in katerih sipalna matrika v njihovem Hilbertovem prostoru je identiteta, manifestirajo vse lastnosti teorije bozonskih strun, vklučno z napetostjo strune. Fermionska inačiča nove strune je v delu.
Vselej smo veseli eksperimentalnih prispevkov, ki tokrat predstavi vec kot desetletje meritev temne snovi na DAMA/LIBRA. Pojasni tudi, zakajjih ostali eksperimenti se niso potrdili.
Predstavljeni sta tudi deli, ki poskušata razloZiti obstoječe neskladje med različnimi laboratotiji, ki merijo direktno interakcijo z delci temne snovi. Avtorji študirajo interakcijo novih delcev, kot je OHe, z jedri v merilnih aparaturah. Je tudi nekaj del, ki so bila predstavljena na delavnici, ki pa jih avtorji v tako kratkem casu niso uspeli pripraviti za zbornik. Je tudi kar precejšnje stevilo zelo obetavnih diskusij, kijih avtorji prav tako niso utegnili pravocasno pripraviti. 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. Vprasanja in odgovori ter tudi predavanja, ki jih je v zadnjih letih omogocil M.Yu. Khlopov preko Virtual Institute of Astroparticle Physics (www.cosmovia.org, APC, Pariz), so v izcrpnih diskusijah pomagali razcistiti marsikatero dilemo.
Bralec najde zapise vseh predavanj, objavljenih preko "cosmovia" od leta 2009, na www.cosmovia.org v povezavi Previous - Conferences. Stiri letosnja predavanja: R. Bernabei (Dark matter particles in the galactic halo), M. Laletin in M.Yu. Khlopov (Dark Atoms and their Decaying Constituents), N.S. Mankoc Boretnik (The Spin-Charge-Family theory offers the explanation for the assumptions of the Standard model, for the Dark matter, for the Matter-antimatter asymmetry) in H.B.F. Nielsen (Novel string field theory), so dostopna na
http://viavca.in2p3.fr/what_comes_beyond_the_standard_models_xvii.html Vecino predavanjnajde bralec na spletni strani delavnice na http://bsm.fmf.uni-lj.si/.
Naj zakljucšimo ta predgovor s prisrcšno in toplo zahvalo vsem udelezšencem, prisotnim na Bledu osebno ali preko videokonferenc, za njihova predavanja in se posebno za zelo plodne diskusije in odlicšno vzdusšje.
Delavnica poteka v hisi, ki jo je Društvu matematikov, fizikov in astronomov Slovenije zapustil v last slovenski matematik Josip Plemelj, udelezencem delavnic, ki prihajajo iz razlicnih koncev sveta, dobro poznan po svojem delu v kompleksni algebri.
Norma Mankoč Borštnik, Holger Bech Nielsen, Maxim Y. Khlopov, (Organizacijski odbor)
Norma MankoC Borštnik, Holger Bech Nielsen, Dragan Lukman, (uredniki)
Ljubljana, grudna (decembra) 2014
Talk Section
All talk contributions are arranged alphabetically with respect to the authors' names.
Bled Workshops in Physics Vol. 15, No. 2
A
Proceedings to the 17th Workshop What Comes Beyond ... (p. 1) Bled, Slovenia, july 20-28, 2014

1 Dark Atoms and Their Decaying Constituents
K. Belotsky1'2, M. Khlopov1'2'3 and M. Laletin1
1	National Research Nuclear University "MEPHI" (Moscow Engineering Physics Institute), 115409 Moscow, Russia
2	Centre for Cosmoparticle Physics "Cosmion", 115409 Moscow, Russia
3	APC laboratory 10, rue Alice Domon et Leonie Duquet, 75205 Paris Cedex 13, France
Abstract. The nonbaryonic dark matter of the Universe might consist of new stable charged species, bound by ordinary Coulomb interactions in various forms of heavy neutral "dark atoms". The existing models offer natural implementations for the dominant and subdominant forms of dark atom components. In the framework of Walking Technicolor the charge asymmetric excess of both stable negatively doubly charged technilepton Z and metastable but longliving positively doubly charged technibaryon UU++ can be generated in the early Universe together with the observed baryon asymmetry. If the excess of Z exceeds by several orders of magnitude the excess of UU, dark matter might consist dominantly by HeZ dark atoms of nuclear interacting O-helium (OHe) bound state of Z with primordial helium. This dominant dark matter component causes negligible nuclear recoil in underground experiments, but can explain positive results of DAMA/Nal and DAMA/LIBRA experiments by annual modulations of few keV energy release in radiative capture of OHe by sodium. However, a sufficiently small subdominant component of WIMP-like objects UUZ can also form. Making up a small fraction of dark matter, it can also evade the severe constraints on WIMPs from underground detectors. Although sparse, this subdominant component can lead to observable effects, since leptonic decays of technibaryons UU give rise to two positively charged leptons contrary to the pairs of opposite charge leptons created in decays of neutral particles. We show that decays of UU++ —> e+ e+, |+ |+, t+t+ of the subdominant UUZ component of dark matter, can explain the observed high energy positron excess in the cosmic rays if the fraction of UUZ is ~ 10-6 of the total dark matter density, the mass of UU++ about 1 TeV and the lifetime about 1020 s. Optimizing fit of recent AMS-02 data by model parameters, the predicted mass range of such long-living double charge particle is challenging for its search at the
Povzetek. Avtorji predpostavijo, da sestavlja nebarionsko temno snov v vesolju nova, stabilna, nabita vrsta snovi, katere delce v obliki težkih nevtralnih "temnih atomov" veze običajna Coulombska interakcija. V obstoječih modelih, denimo v "Walking Technicolor", najdejo potrditev za tako domnevo: V zgodnjem vesolju lahko nastane presežek stabilnih tehnileptonov Z ž nabojem —2 in metastabilnih vendar dovolj dolgoživih tehnibarionov UU++ ž nabojem +2, ki pojasnijo barionsko-antibarionsko asimetrijo. Ce je presežžek delcev Z za vec velikostnih redov vecji od presežka delcev UU, sestavljajo temno snov pretežno temni atomi HeZ O-helija (OHe), ki je vežano stanje delca Z in jedra obicajnega helija. Ce je OHe prevladujoci del temne snovi, je njegov prispevek k odrivu na jedrih merilnih aparatur, ki so postavljene pod žemljo, žanemarljiv, vendar lahko pojasni poskuse na DAMA/NaI in DAMA/LIBRA, ki ižmerita periodicno letno modulacijo, v obmocju nekaj keV energije.
LHC.
Opazljiv pa je, pravijo avtorji, tudi majhen delež UUZ, ki tudi sestavlja temno snov, ker nastaneta pri leptonskem razpadu tehnibarionov UU dva pozitivno nabita leptona, pri razpadu nevtralnih delcev pa nastaneta dva nasprotno nabita leptona. Avtorji pokažejo, da lahko razpadi UU++ —> e+ e+, |+ |+, t+t+ UUZ pojasnijo izmerjeni presezek pozitronov v kozmicnih zarkih, ce je masa UU++ priblizno 1 TeV, njihov prispevek h gostoti temne snovi ~ 10-6 in zivljenska doba okrog 1020 s. Avtorji napovedujejo, da bodo te delce izmerili tudi na LHC.
1.1 Introduction
Dark atoms offer an interesting possibility to solve the puzzles of dark matter searches. It turns out that even stable electrically charged particles can exist hidden in such atoms, bound by ordinary Coulomb interactions (see [1-3] and references therein). Stable particles with charge -1 are excluded due to overproduction of anomalous isotopes. However, there doesn't appear such an evident contradiction for negatively doubly charged particles.
There exist several types of particle models where heavy stable -2 charged species, O , are predicted:
(a)	AC-leptons, predicted as an extension of the Standard Model, based on the approach of almost-commutative geometry [4-7].
(b)	Technileptons and anti-technibaryons in the framework of Walking Technicolor (WTC) [8-14].
(c)	stable "heavy quark clusters" lilIJU formed by anti-U quark of 4th generation [4,15-19]
(d)	and, finally, stable charged clusters U5U5U5 of (anti)quarks U5 of 5th family can follow from the approach, unifying spins and charges[20].
All these models also predict corresponding +2 charge particles. If these positively charged particles remain free in the early Universe, they can recombine with ordinary electrons in anomalous helium, which is strongly constrained in terrestrial matter. Therefore a cosmological scenario should provide a mechanism which suppresses anomalous helium. There are two possible mechanisms than can provide a suppression:
(i)	The abundance of anomalous helium in the Galaxy may be significant, but in terrestrial matter a recombination mechanism could suppress this abundance below experimental upper limits [4,6]. The existence of a new U(1) gauge symmetry, causing new Coulomb-like long range interactions between charged dark matter particles, is crucial for this mechanism. This leads inevitably to the existence of dark radiation in the form of hidden photons.
(ii)	Free positively charged particles are already suppressed in the early Universe and the abundance of anomalous helium in the Galaxy is negligible [1,16].
These two possibilities correspond to two different cosmological scenarios of dark atoms. The first one is realized in the scenario with AC leptons, forming neutral AC atoms [6]. The second assumes a charge asymmetry of the O which forms the atom-like states with primordial helium [1,16].
If new stable species belong to non-trivial representations of the SU(2) elec-troweak group, sphaleron transitions at high temperatures can provide the relation between baryon asymmetry and excess of -2 charge stable species, as it was demonstrated in the case of WTC [8,21-23].
After it is formed in the Standard Big Bang Nucleosynthesis (BBN), 4 He screens the O charged particles in composite (4He++O ) OHe "atoms" [16]. In all the models of OHe, O behaves either as a lepton or as a specific "heavy quark cluster" with strongly suppressed hadronic interactions. The cosmological scenario of the 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. In terrestrial matter such dark matter species are slowed down and cannot cause significant nuclear recoil in the underground detectors, making them elusive in direct WIMP search experiments (where detection is based on nuclear recoil) such as CDMS, XENON100 and LUX. The positive results of DAMA and possibly CRESST and CoGeNT experiments (see [24] 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 [1-3]. 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 [2,3,21,25].
One should note that the nuclear physics of OHe is in the course of development and its basic element for a successful and self-consistent OHe dark matter scenario is related to the existence of a dipole Coulomb barrier, arising in the process of OHe-nucleus interaction and providing the dominance of elastic collisions of OHe with nuclei. This problem is the main open question of composite dark matter, which implies correct quantum mechanical solution [26]. The lack of such a barrier and essential contribution of inelastic OHe-nucleus processes seem to lead to inevitable overproduction of anomalous isotopes [27].
It has been shown [8,21-23,28] that a two-component dark atom scenario is also possible and can be naturally realized in the framework of a WTC model, in which both stable double charged technilepton Z , playing the role of O , and positively double charged technibaryon UU are predicted. Along with the dominant Z abundance, a much smaller excess of positively doubly charged techniparticles UU can be created. These positively charged particles are hidden in WIMP-like atoms, being bound to Z . In the framework of WTC such positively charged techniparticles can be metastable, with a dominant decay channel to a pair of positively charged leptons. We have shown in [28] that even a 10-6 fraction of such positively charged techniparticles with a mass of 1 TeV or less and a lifetime of 1020 s, decaying to e+e+, |j.+ |j.+, and t+t+ can explain the observed excess of cosmic ray positrons, being compatible with the observed gamma-ray background.
The anomalous excess of high-energy positrons in cosmic rays was first observed by PAMELA [29] and was later confirmed by AMS-02 [30]. These results
generated widespread interest, since the corresponding effect cannot be explained by positrons of only secondary origin and requires primary positron sources, e.g. annihilations or decays of dark matter particles. Recently AMS-02 collaboration has reported new results on positron and electron fluxes in cosmic rays [31] and positron fraction [32]. These measurements cover the energy ranges 0.5 to 700 GeV for electrons and 0.5 to 500 GeV for positrons and provides important information on the origins of primary positrons in the cosmic rays. In particular, new results show, for the first time, that above ~ 200 GeV the positron fraction no longer exhibits an increase with energy. The possibility to explain the cosmic positron excess by the decays of UU particles, comprising the tiny WIMP-component of dark matter in the considered scenario, was discussed in detail in [28]. Here we estimate the optimal values of model parameters by achieving the best agreement with AMS-02 data on cosmic positron flux and FERMI-LAT data on diffuse gamma-ray flux [33].
1.2 Cosmic positron excess and fit to the latest AMS-02 data
In the considering scenario the metastable UU particles, which together with Z forms the subdominant component of dark matter, decays as
UU -> e+e+
in principle with different branching ratios. All decay modes give directly or through cascades positrons and gamma photons. The latter are hereafter referred to as final state radiation (FSR). The positron flux at the top of the Earth's atmosphere can be estimated as
F(E) =
c nloc 1 4n t PE2
m/2
^NQ(A(Eo,E))dEo,	(1.1)
dEo
where n[oc = £ • (0.3 GeV/cm3)mUU is the local number density of UU particles with £ = 10-6, dN/dE0 is the number of positrons produced in a single decay (obtained using Pythia 6.4 [34]), P - 10-16 s-1GeV-1 and
(A - h)2(2À + 4) 2h(A2 - r2) Q = 1--2A5-n(A - h)--3A3— n(A - R), (1.2)
(see [28] for details).
E
Below ~ 10 GeV behavior of positrons is affected by solar modulation. This effect can be in principle taken into account, using the force field model with two different parameters ^ both for electrons and positrons, which can be easily adjusted in order to fit the data points at low energies. However, the effects of solar modulations are insignificant at the energies above ~ 20 GeV and thus for our analysis we consider the positron spectrum from 20 to 500 GeV. The positron background component was taken from [35].
Any scenario that provides positron excess is also constrained by other observational data, mainly from the data on cosmic antiprotons and gamma-radiation from our halo (diffuse gamma-background) and other galaxies and clusters. If dark matter does not produce antiprotons, then the diffuse gamma-ray background gives the most stringent and model-independent constraints. For the FSR photons produced by UU decays in our Galaxy, the flux arriving at the Earth is given by
f(g) m — n'oc 1
fFSR( EJ —
t 4nahobs
nM dlda • ^, (1.3)
nioc	dE
where we use an isothermal number density distribution = (5 kp5 kp+)8+rkpc) ' r and I are the distances from the Galactic center and the Earth respectively. We obtain the averaged flux over the solid angle AHobs corresponding to |b| > 10°, 0 < l< 360°.
Out of our Galaxy, decays of UU homogeneously distributed over the Universe should also contribute to the observed gamma-ray flux. For FSR photons this contribution can be estimated as
n(u)m - c (nmod)
FpSR(EJ — 4n t
dNdt — c(nmod) x dE	4nT
fmm(1100, m + 1) dN	H-1 ,dz
x	-dr(Eo = +1)) /o m°d f==, (1.4)
Jo	dE0	Vha + am(z + 1)3
where z = 1100 čorresponds to the rečombination epočh, (nmod) is the čurrent čosmologičal number density of UU, H-^ = 3 tmod V^a ln-1	is the
inverse value of the Hubble parameter with tmod being the age of the Universe. nA and am = 1 — nA are respečtively the čurrent vačuum and matter relative densities.
Contribution into gamma-ray flux indučed by sčattering off bačkground elečtromagnetič radiations of elečtrons and positrons from dečays is small at high energy tail of spečtrum, where observation data put the strongest čonstraint [28], and is not taken into aččount here.
The best-fit parameter values were obtained by fitting the čurve, given by Eq.(1.1), to the AMS-02 data points in the least squares sense. Sinče the same parameters define the predičted gamma-ray flux in order not to čontradičt the FERMI-LAT data we extend the fitting pročedure, involving several lowest points in the diffuse gamma-bačkground spečtrum above - 100 GeV to be fit. For eačh čhoiče of muu from 700 to 1400 GeV we have evaluated the best-fit values of the lifetime t and three brančhing ratios Bre, Br^ and BrT = 1 — Bre — Br^. To čhoose the sčenario, whičh is most čonsistent with the experimental data, a x2 statističal test was used. The best fit (x2/n.d.f. = 0.57) čorresponds to the following parameter values: M = 900 GeV, t = 4.59 1020 s, Bre = 0.195, Br^ = 0.129, BrT = 0.676. Positron flux, positron fračtion and gamma-ray flux in the best-fit čase are shown in Figs. 1.1,1.2 and 1.3 respečtively.
obs
Positron tixix from DM decavs. compared to experimental data -1-1-1-1-1-1-1-1-1-
□ PAMELA ----Background
_l_I_I........I_I_I
20	100	500
E (GeV)
Fig. 1.1. Positron flux from UU decays compared to PAMELA and AMS-02 data
Fig. 1.2. Positron excess due to UU decays compared to PAMELA and AMS-02 data
Diffuse gamma background
M - 900 GeV, t =4.5928-1020s
• Feriui-LAT
10°	lo'	10*	io'
E (GeV)
Fig. 1.3. Gamma-ray flux from UU decays in the Galaxy (|b| > 100) compared to the Fermi/LAT data on diffuse gamma-background
1.3 Conclusions
Being the reflection of fundamental particle symmetry beyond the Standard model, the set of stable particles - dark matter candidates - can hardly be reduced to one single species [2]. It makes natural to consider multi-component dark matter and one can hardly expect that various components put equal or comparable contribution into the total density. The situation with dominance of one component coexisting with some other subdominant components doesn't seem too exotic in this case.
Dark matter solution for the puzzles of dark matter searches can involve the form of neutral OHe dark atoms made of stable heavy doubly charged particles and primordial He nuclei bound by ordinary Coulomb interactions. This scenario can be realized in the framework of Minimal Walking Technicolor, in which an exact relation between the dark matter density and baryon asymmetry can be naturally obtained. Strict conservation of technilepton charge together with approximate conservation of technibaryon charge results in the prediction of two types of doubly charged species with strongly unequal excess - dominant negatively charged technileptons Z and a strongly subdominant component of technibaryons UU++, bound with Z in a sparse component of WIMP-like dark atoms (Z UU++). Direct searches for WIMPs put severe constraints on the presence of this component. However we have demonstrated in [28] that the existence of a metastable positively doubly charged techniparticle, forming this tiny subdominant WIMP-like dark atom component and satisfying the direct WIMP searches constraints, can play an important role in the indirect effects of dark matter. We found that decays of such positively charged constituents of WIMP-
like dark atoms to the leptons e+e+, |+|+,t+t+ can explain the observed excess of high energy cosmic ray positrons, while being compatible with the observed gamma-ray background. These decays are naturally facilitated by GUT scale interactions. The best fit of the data takes place for a mass of this doubly charged particle of 1 TeV or below making it accessible in the next run of LHC. Our refined analysis of the best fit description of the recent data of the AMS-02 experiment, presented here, can provide a crucial test for the decaying dark atom hypothesis in the experimental searches for stable doubly charged lepton-like particles at the LHC.
Acknowledgements
We thank Chris Kouvaris for collaboration in development of the presented approach. The work by M.Kh. and M.L. on initial cosmological conditions was supported by the Ministry of Education and Science of Russian Federation, project 3.472.2014/K and the work of K.B., M.Kh. and M.L. on the forms of dark matter was supported by grant RFBR 14-22-03048.
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Bled Workshops in Physics Vol. 15, No. 2
A
Proceedings to the 17th Workshop What Comes Beyond ... (p. 10) Bled, Slovenia, July 20-28, 2014

2 Dark Matter Particles in the Galactic Halo
R. Bernabei
Dip. di Fisica, Universita di Roma "Tor Vergata" and INFN-Tor Vergata, I-00133 Rome, Italy
Abstract. The DAMA/LIBRA-phase1 and the former DAMA/NaI data (cumulative exposure 1.33 ton x yr, corresponding to 14 annual cycles) give evidence at 9.3 a C.L. for the presence of Dark Matter (DM) particles in the galactic halo, on the basis of the exploited model independent DM annual modulation signature by using highly radio-pure NaI(Tl) target. Results and comparisons will be shortly addressed as well as perspectives of the presently running DAMA/LIBRA-phase2. Finally, some arguments arisen in the discussion section of this workshop are mentioned in the Appendix.
Povzetek. Avtorica, ki je postavila in vodi laboratorij DAMA/NaI in DAMA/LIBRA za merjenje delcev, ki tvorijo temno snov, poroca o rezultatih poskusov, ki tecejo ze štirinajsto leto. Poroca o tem, kako in do koliksne mere so uspeli izločiti nečistoče in poznane izvore signalov na merilni aparaturi. Poroca o odvisnosti stevila izmerjenih delcev temne snovi v odvisnosti od relativne hitrosti Zemlje okoli Sonca glede na hitrost Sonca okoli centra galaksije. Prikaze napredek v zanesljivosti meritev v odvisnosti od let ter izrazitost letne modulacije meritev. Komentira dosedanje rezultate drugih laboratorijev in vzroke, zaradi katerih drugi njihovih meritev se niso potrdili. Napove drugi ciklus meritev ter odgovori na vprasanja, ki so jih postavili udelezzenci delavnice.
2.1 Introduction
About 80 years of experimental observations and theoretical arguments have pointed out that a large fraction of the Universe is composed by Dark Matter particles 1 .
The presently running DAMA/LIBRA 250 kg of full sensitive target-mass) [1-9] experiment, as well as the former DAMA/NaI (~ 100 kg of full sensitive
1 For completeness, it is worth recalling that some efforts to find alternative explanations to Dark Matter have been proposed such as MOdified Gravity Theory (MOG) and MOdified Newtonian Dynamics (MOND); they hypothesize that the theory of gravity is incomplete and that a new gravitational theory might explain the experimental observations. MOND modifies the law of motion for very small accelerations, while MOG modifies the Einstein's theory of gravitation to account for an hypothetical fifth fundamental force in addition to the gravitational, electromagnetic, strong and weak ones. However, e.g.: i) there is no general underlying principle; ii) they are generally unable to account for all small and large scale observations; iii) they fail to reproduce accurately the Bullet Cluster; iv) generally they require some amount of DM particles as seeds for the structure formation.
target-mass) [10-16], has the main aim to investigate the presence of DM particles in the galactic halo by exploiting the model independent DM annual modulation signature (originally suggested in Ref. [17]).
As a consequence of the Earth's revolution around the Sun, which is moving in the Galaxy with respect to the Local Standard of Rest towards the star Vega near the constellation of Hercules, the Earth should be crossed by a larger flux of DM particles around ~ 2 June and by a smaller one around ~ 2 December. In the former case the Earth orbital velocity is summed to the one of the solar system with respect to the Galaxy, while in the latter the two velocities are subtracted2. This DM annual modulation signature is very distinctive since the effect induced by DM particles must simultaneously satisfy all the following requirements: the rate must contain a component modulated according to a cosine function (1) with one year period (2) and a phase that peaks roughly ~ 2 June (3); this modulation must only be found in a well-defined low energy range, where DM particle induced events can be present (4); it must apply only to those events in which just one detector of many (9 in DAMA/NaI and 25 in DAMA/LIBRA) actually "fires" (single-hit events), since the DM particle multi-interaction probability is negligible (5); the modulation amplitude in the region of maximal sensitivity must be ~ 7% for usually adopted halo distributions (6), but it can be larger (even up to ~ 30%) in case of some possible scenarios such as e.g. those in Ref. [18,19]. Thus, this signature is model independent and very effective; moreover, the developed highly radio-pure NaI(Tl) target-detectors [1] and the adopted procedures assure sensitivity to a wide range of DM candidates (both inducing nuclear recoils and/or electromagnetic radiation), interaction types and astrophysical scenarios.
In particular, the experimental observable in DAMA experiments is the modulated component of the signal in NaI(Tl) target and not the constant part of it as in other approaches as those by CDMS, Xenon, etc., where in addition e.g.: i) different target materials are used; ii) the sensitivity is mainly restricted to candidates inducing just nuclear recoils; iii) many (by the fact largely uncertain) selections/subtractions of detectors and of data and (highly uncertain) extrapolations of detectors' features are applied.
The DM annual modulation signature might be mimicked only by systematic effects or side reactions able to account for the whole observed modulation amplitude and to simultaneously satisfy all the requirements given above. No one is available or suggested by anyone over more than a decade [1,2,5,6,8,13,9].
It is also worth noting that the DM annual modulation signature acts itself as a strong background reduction as pointed out since the early paper by Ref. [17], and especially when all the above peculiarities can be experimentally verified in suitable dedicated set-ups as it is the case of the DAMA experiments.
2 Thus, the DM annual modulation signature has a different origin and peculiarities than the seasons on the Earth and than effects correlated with seasons (consider the expected value of the phase as well as other requirements listed below).
2.2 The DAMA results
The total exposure of DAMA/LIBRA-phasel is: 1.04 ton x yr in seven annual cycles; when including also that of the first generation DAMA/Nal experiment it is 1.33 ton x yr, corresponding to 14 annual cycles. The variance of the cosine during the DAMA/LIBRA-phasel data taking is 0.518, showing that the set-up has been operational evenly throughout the years [2,6].
Many independent data analyses have been carried out [2,6] and all of them confirm the presence of a peculiar annual modulation in the single-hit scintillation events in the 2-6 keV energy interval, which - in agreement with the requirements of the DM signature - is absent in other parts of the energy spectrum and in the multiple-hit scintillation events in the same 2-6 keV energy interval (this latter condition correspond to have "switched off the beam" of DM particles). All the analyses and details can be found in the literature given above. In particular, Fig. 2.1 shows the time behaviour of the experimental residual rates of the single-hit scintillation events for DAMA/Nal [13] and DAMA/LIBRA-phase1 [2,6] cumulatively in the (2-6) keV energy interval. The data points present the experimental errors as vertical bars and the associated time bin width as horizontal bars. The superimposed curve is the cosinusoidal function A cos ^(t — t0) with a period T = = 1 yr, a phase t0 = 152.5 day (June 2nd) and modulation amplitude, A, equal to the central value obtained by best fit on the data points. The dashed vertical lines correspond to the maximum expected for the DM signal, while the dotted vertical lines correspond to the expected minimum. The major upgrades are also pointed out.
In order to continuously monitor the running conditions, several pieces of information are acquired with the production data and quantitatively analysed. In particular, all the time behaviours of the running parameters, acquired with the production data, have been investigated: the modulation amplitudes obtained for each annual cycle when fitting the time behaviours of the parameters including a cosine modulation with the same phase and period as for DM particles are well compatible with zero. In particular, no modulation has been found in any possible source of systematics or side reactions; thus, cautious upper limits (90% C.L.) on possible contributions to the DAMA/LIBRA measured modulation amplitude have been derived (see e.g. [2]). It is worth noting that they do not quantitatively account for the measured modulation amplitudes, and are not able to simultaneously satisfy all the many requirements of the signature. Similar analyses have also been carried out for the DAMA/NaI data[13].
No other experimental result has been verified over so long time so accurately and with various significant upgrades of the set-ups.
For completeness I mention that sometimes naive statements were put forwards as the fact that in nature several phenomena may show some kind of periodicity. The point is whether they could mimic the annual modulation signature in DAMA/LIBRA (and former DAMA/NaI), i.e. whether they could quantitatively account for the observed modulation amplitude and also simultaneously satisfy all the requirements of the DM annual modulation signature. The same is also for side reactions. This has already been deeply investigated in Ref. [1,2]
DAMA/NaI & DAMA/LIBRA experiments main upgrades and improvements
PHASE2
Time (day)
On 2003 DAMA/LIBRA has begun first operations
July 2000 new DAQ and new electronic chain installed (MULTIPLEXER removed, now one TD channel for each detector):
(i)	TD VXI Tektronix;
(ii)	Digital Unix DAQ system;
(iii)	GPIB-CAMAC.
July 2002 DAMA/NaI data taking completed
Sept.-Oct. 2008 01 DAMA/LIBRA upgrade: ! one detector recovered by replacing a
broken PMT " a new optimization of some PMTs and HVs performed
® all the TD replaced with new ones (U1063A Acqiris 8-bit 1GS/s DC270 High-Speed cPCI Digitizers)
$ a new DAQ with optical read-out installed.
The second DAMA/LIBRA upgrade in Fall 2010: Replacement of all the PMTs with higher Q.E. ones from dedicated developments (+new preamp in Fall 2012 and other developments in progress) DAMA/LI BRA-phase2 in data taking
Fig. 2.1. Experimental residual rate of the single-hit scintillation events measured by DAMA/NaI and DAMA/LIBRA-phasel in the (2-6) keV energy interval as a function of the time. The data points present the experimental errors as vertical bars and the associated time bin width as horizontal bars; see text. As always in DAMA results, the given rate is already corrected for the overall efficiency. The major upgrades of the experiment are also pointed out.
and references therein; the arguments and the quantitative conclusions, presented there, also apply to the entire DAMA/LIBRA-phasel data. Additional arguments can be found in Ref. [5,6,8,9]. In particular, Ref. [9] further outlines in a simple and intuitive way why neutrons (of whatever origin), muons and solar neutrinos cannot give any significant contribution to the DAMA annual modulation results and - in addition - can never mimic the DM annual modulation signature since some of its specific requirements fail. Table 2.1 summarizes the safety upper limits on the contributions (if any) 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 also reported there. As seen in Table 2.1, they are all negligible and they cannot give any significant contribution to the observed modulation amplitude; in addition, neutrons, muons and solar neutrinos are not a competing background when the DM annual modulation signature is investigated since they cannot mimic this signature. For details see Ref. [9] and references therein.
Source	(neutrons cm 2 s 1 )	nk	tk	Rû,k Ak = Rû,knk (cpd/kg/keV) (cpd/kg/keV)		Ak/smp
thermal n (10-2 - 10"1 eV) SLOW neutrons epithermaln (eV-keV)	1.08 x 10-6 2 x 10-6	- 0 however C 0.1 - 0 however — 0.1	-	<8 x 10-6 <3 x 10-3	C 8 x 10-7 C 3 x 10-4	C 7 x 10-5 C 0.03
fission, ( a, n) —> n (1-10 MeV)	- 0.9 x 10-'	- 0 however — 0.1		<6 x 10-4	C 6 x 10-5	C5 x 10-3
—> n from rock FAST (>10 MeV) neutrons —> n from Pb shield (> 10 MeV)	-	3 x 10-' -	6 x 10-'	0.0129 0.0129	end of June end of June	-C 7 x 10-4 -C 1.4 x 10-3	C 9 x 10-6 C 2 x 10-5	C 8 x 10-4 C 1.6 x 10-3
v —> n (few MeV)	- 3 x 10-1û	0.03342*	Jan. 4th*	C 7 x 10-5	C 2 x 10-6	C 2 x 10-4
direct direct v	®û"J - 20 p. m-2d-1 ®ûv) - 6 x 101û v cm-2s-	0.0129 0.03342*	end of June Jan. 4th*	- 10-7 - 10-5	- 10-9 3 x 10-7	- 10-7 3 x 10-5
Table 2.1. Summary of the contributions to the total neutron flux at LNGS; the value, the relative modulation amplitude, and the phase of each component is reported. It is also reported the counting rate in DAMA/LIBRA for single-hit events, in the (2 — 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, Smp — 0.0112 cpd/kg/keV [2]. As can be seen, they are all negligible and they cannot give any significant contribution to the observed modulation amplitude. In addition, neutrons, muons and solar neutrinos are not a competing background when the DM annual modulation signature is investigated since in no case they can mimic this signature. For details see Ref. [9] 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.
In conclusion, DAMA gives a model-independent evidence - at 9.3ct C.L. over 14 independent annual cycles - for the presence of DM particles in the galactic halo.
2.2.1 On comparisons
No direct model independent comparison is possible in the field when different target materials and/or approaches are used; the same is for the strongly model dependent indirect searches3.
3 It should be noted that the rising behaviour of the positron flux reported in Ref. [20,21] does not give any intrinsic evidence for production due to DM annihilation; this may arise only when a particular model of the competing background is assumed as e.g. the GALPROP code. But other more complete models exist which do not support any significant excess evidence. Moreover, an interpretation in terms of DM particle annihilation would require the assumption of: i) a very large boost factor (~ 400) of the density; ii) to boost the annihilation cross section through an assumed new interaction type; iii) to
In order to perform corollary investigations on the nature of the DM particles, model-dependent analyses are necessary4. Thus, many theoretical and experimental parameters and models are possible (see e.g. in [2,6,22,23]) and many hypotheses must also be exploited, while specific experimental and theoretical assumptions are generally adopted in the field assuming a single arbitrary scenario without accounting neither for existing uncertainties nor for alternative possible scenarios, interaction types, etc.
The obtained DAMA 9.3 ct C.L. model independent evidence is compatible with a wide set of scenarios regarding the nature of the DM candidate and related astrophysical, nuclear and particle Physics. For examples some scenarios and parameters are discussed e.g. in Ref. [10,11,13,2,6,22,23]. Further large literature is available on the topics (see for example in the bibliography of Ref. [6]). By the fact, both the negative results and all the possible positive hints are largely compatible with the DAMA model-independent DM annual modulation results in various scenarios considering also the existing experimental and theoretical uncertainties; the same holds for the strongly model dependent indirect approaches.
It is also worthwhile to further recall that these DAMA experiments are not only sensitive to DM particles with spin-independent coupling inducing just nuclear recoils, but also to other couplings and to other DM candidates as those giving rise to part or all the signal in electromagnetic form. Finally, scenarios exist in which other kind of targets/approaches are disfavoured or even blind.
2.3 DAMA/LIBRA-phase2 and perspectives
An important upgrade has started at end of 2010 replacing all the PMTs with new ones having higher Quantum Efficiency; details on the developments and on the reached performances in the operative conditions are reported in Ref. [4]. They have allowed us to lower the software energy threshold of the experiment to 1 keV and to improve also other features as e.g. the energy resolution [4].
Since the fulfillment of this upgrade and after some optimization periods, DAMA/LIBRA-phase2 is continuously running in order e.g.: (1) to increase the experimental sensitivity thanks to the lower software energy threshold; (2) to improve the corollary investigation on the nature of the DM particle and related astrophysical, nuclear and particle physics arguments; (3) to investigate other
adjust the propagation parameters; iv) to consider extra-source (subhalos, IMBHs); v) to consider only a leptophilic candidate to justify the absence of any excess in the antiproton spectrum. Finally, other well known sources can account for a similar positron fraction: pulsars, supernova explosions near the Earth, SNR.
4 For completeness, it is worth recalling that it does not exist any approach to investigate the nature of the candidate in the direct and indirect DM searches, which can offer this information independently on assumed astrophysical, nuclear and particle Physics scenarios. On the other hand, searches for new particles beyond the Standard Model of particle Physics at accelerators cannot credit by themselves that a certain particle is in the halo as a solution or the only solution for DM particles, and - in addition - DM candidates and scenarios (even for the neutralino) exist which cannot be investigated there.
signal features and second order effects. This requires long and dedicated work for reliable collection and analysis of very large exposures.
In the future DAMA/LIBRA will also continue its study on several other rare processes as also the former DAMA/NaI apparatus did.
Finally, further future improvements of the DAMA/LIBRA set-up to increase the sensitivity (possible DAMA/LIBRA-phase3) and the developments towards the possible DAMA/1ton (1 ton full sensitive mass on the contrary of other kind of detectors), we proposed in 1996, are considered at some extent. For the first case developments of new further radiopurer PMTs with high quantum efficiency are starting, while in the second case it would be necessary to overcome the present problems regarding: i) the supplying, selection and purifications of a large number of high quality NaI and, mainly, TlI powders; ii) the availability of equipments and competence for reliable measurements of small trace contaminants in ppt or lower region; iii) the creation of updated protocols for growing, handling and maintaining the crystals; iv) the availability of large Kyropoulos equipments with suitable platinum crucibles; v) etc.. At present, due to the change of rules for provisions of strategical materials, the large costs and the lost of some equipments and competence also at industry level, a satisfactory development appears quite difficult.
2.4 Conclusions
The data of DAMA/LIBRA-phase1 have further confirmed the presence of a peculiar annual modulation of the single-hit events in the (2-6) keV energy region satisfying all the many requirements of the DM annual modulation signature; the cumulative exposure by the former DAMA/NaI and DAMA/LIBRA-phase1 is 1.33 ton x yr (orders of magnitude larger than those typically released in the field).
As required by the DM annual modulation signature: 1) the single-hit events show a clear cosine-like modulation as expected for the DM signal; 2) the measured period is equal to (0.998 ± 0.002) yr well compatible with the 1 yr period as expected for the DM signal; 3) the measured phase (144 ± 7) days is compatible with ~ 152.5 days as expected for the DM signal; 4) the modulation is present only in the low energy (2-6) 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) of the single-hit events in the (2-6) keV energy interval is: (0.0112 ± 0.0012) cpd/kg/keV (9.3 ct C.L.). No systematic or side processes able to simultaneously satisfy all the many peculiarities of the signature and to account for the whole measured modulation amplitude is available.
DAMA/LIBRA-phase2 is continuously running in its new configuration with a lower software energy threshold aiming to improve the knowledge on corollary aspects regarding the signal and on second order effects as discussed e.g. in Ref. [6,8].
Few comments on model-dependent comparisons have also been addressed here.
Acknowledgments
It is a pleasure to thank all my DAMA čollaborators who effečtively dedičated their efforts to this experimental ačtivity and the čolleagues in this Workshop for the interesting topičs we have disčussed, for the question sečtion, and for the pleasant srientifk environment.
Appendix: Questions & Answers
This sečtion shortly summarizes some of the topičs extensively disčussed at the Workshop, where the time dedičated to disčussions and the interest in deeply understanding the topičs were rather large.
Question 1: may you comment about the ratio of the measured dark matter particles modulation amplitude to the total signal: the Sm/S0 ratio?
Answer 1: the measured čounting rate in the čumulative energy spečtrum is about 1 čpd/kg/keV in the lowest energy bins; this is the sum of the bačk-ground čontribution and of the čonstant part of the signal S0. As disčussed e.g. in TAUP2011 [24], the bačkground in the 2-4 keV energy region is estimated to be not lower than about 0.75 čpd/kg/keV; this gives an upper limit on S0 of about 0.25 čpd/kg/keV. Thus, the Sm/S0 ratio is equal or larger than about 0.01/0.25 ~ 4 %.
Question 2: may you comment on the quenching factors, on their dependence on the type of the particles, and on some typical examples of extreme properties?
Answer 2: The quenčhing fačtor values play a role only when čorollary model-dependent analyses for DM čandidates indučing just nučlear rečoils are čarried out, in order to derive the energy sčale in terms of nučlear rečoil energy.
As is widely known, the quenčhing fačtor is a spečifič property of the employed detečtor and not a general quantity universal for a given material. For example, in liquid noble-gas detečtors, it depends - among others - on the level of trače čontaminants whičh čan vary in time and from one liquefačtion pročess to another, on the čryogenič mičrosčopič čonditions, etč.. In bolometers it depends for instanče on spečifič properties, trače čontaminants, čryogenič čonditions, etč. of eačh spečifič detečtor, while generally it is assumed exačtly equal to unity (the maximum possible value). The quenčhing fačtors in sčintillators depend, for example, on the dopant čončentration, on the growing method/pročedures, on residual trače čontaminants, etč., and are expečted to be energy dependent. Thus, all these aspečts are already by themselves relevant sources of unčertainties when interpreting whatever result in terms of DM čandidates indučing just nučlear rečoils. Similar arguments have been addressed e.g. in Ref. [2,3,13,15,25].
Question 3: May you comment under which extreme conditions your experiment is successful and comment what can at most the experiment which does not fulfil one of the conditions or more than one of them at most can "see"?
Answer 3: The full description and potentiality of the DAMA/LIBRA set-up have been discussed in details in Refs. [1,2,4] and references therein. Obviously all the set-up specific features and adopted procedures contribute to the possibility to point out the signal through the model independent DM annual modulation signature. The absence/difference of one of them would limit whatever else result.
Question 4: May you comment about muons?
Answer 4: An extensive discussion on this topics can be found in the dedicated Ref. [5,9], where its has been quantitatively demonstrated (see also Table 2.1 in this paper) that - for many reasons (and just one would suffice) - muons cannot play (directly or indirectly) any role in the DAMA annual modulation effect.
Question 5: May you comment about neutrinos?
Answer 5: The contribution from solar, atmospheric,.. neutrinos is obviously negligible; a quantitative discussion can be found in Ref. [9] (see also Table 2.1 in this paper).
Question 6: May you comment about the operating temperature of your measuring apparatus?
Answer 6: The DAMA set-ups operate at environmental temperature maintained stable by suitable and redundant air-conditioning system (2 independent devices for redundancy); moreover, the Cu housings of the detectors are in direct contact with the multi-ton metallic shield, thus there is a huge heat capacity 106 cal/0C). In addition, the operating temperature of the detectors is continuously monitored and analysed as the production data. A discussion on temperature in operating condition can be found e.g. in Ref. [2,6].
Question 7: May you comment about the Snowmass plots and its meaning?
Answer 7: The recent plot from Snowmass and that in Ref. [26] about the "status of the Dark Matter search" do not point out at all the real status of Dark Matter searches since e.g.: i) Dark Matter has wider possibilities than WIMPs inducing just nuclear recoil with spin-independent interaction under single (largely arbitrary) set of assumptions; ii) neither the uncertainties for existing experimental and theoretical aspects nor alternative possible assumptions are accounted for; iii) they do not include possible systematic errors affecting the data (such as e.g. "extrapolations" of energy threshold, of energy resolution and of efficiencies, quenching factors values, convolution with poor energy resolution, correction for non-uniformity of the detector, multiple subtractions/selection of detectors and/or data, assumptions on quantities related to halo model, form factors, scaling laws, etc.); iv) the DAMA implications - even adopting the many arbitrary assumptions considered there - appear incorrect, for example the S0 prior is not accounted for, etc., etc.. The perspectives as well appear incorrect/too optimistic.
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Bled Workshops in Physics Vol. 15, No. 2
A
Proceedings to the 17th Workshop What Comes Beyond ... (p. 20) Bled, Slovenia, July 20-28, 2014

3 New Experimental Data for the Quarks Mixing Matrix are in Better Agreement with the
Spin-charge-family Theory Predictions
G. Bregar and N.S. Mankoc Borštnik
Department of Physics, FMF, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
Abstract. The spin-charge-family theory [1-14] predicts before the electroweak break four - rather than the coupled and observed three - massless families of quarks and leptons. Mass matrices of all the family members demonstrate in this proposal the same symmetry, determined by the scalar fields: There are two SU(2) triplets, the gauge fields of the family groups, and the three singlets, the gauge fields of the three charges (Q, Q' and Y') distinguishing among family members - all with the quantum numbers of the standard model scalar Higgs with respect to the weak and the hyper charge [13]: ±2 and + \, respectively. Respecting by the spin-charge-family theory proposed symmetry of mass matrices and simplifying the study by assuming that mass matrices are hermitian and real and mixing matrices real, we fit the six free parameters of each family member mass matrix to the experimental data of twice three measured masses of quarks and to the measured quarks mixing matrix elements, within the experimental accuracy. Since any 3 x 3 sub matrix of the 4 x 4 matrix (either unitary or orthogonal) determine the whole 4 x 4 matrix uniquely, we are able to predict the properties of the fourth family members provided that the experimental data for the 3 x 3 sub matrix are enough accurate, which is not yet the case. However, new experimental data [15] fit better to the required symmetry of mass matrices than the old data [16]. The obtained mass matrices are very closed to the democratic ones.
Povzetek. Teorija spinov-nabojev-družin [1-14] napoveduje stiri in ne le tri opažene družine kvarkov in leptonov. Simetrija masnih matrik je v tej teoriji enaka za vse clane družine. Določajo jo skalarna polja: Dve tripletni upodobitvi grupe SU(2), ki določata družinska kvantna stevila in tri singletne upodobitve grup nabojev (Q, Q' in Y'). Vsa skalarna polja, to je obe tripletni in vsa tri singletna polja, nosijo tudi šibki in hiper naboj kot ju za Higgsov skalarni delec privzame standardni model [13]: ali (2 in — 2), ali pa (— 2 in 2). Prva vrednost v oklepaju velja za sibki in druga za hiper naboj. Avtorja v prispevku upostevata simetrijo masnih matrik, ki jo za stiri druzine predlaga teorija spinov-nabojev-družin. Parametre masnih matrik dolocita iz izmerjenih podatkov. Izracun masnih matrik in mesalnih matrik poenostavita s privzetkom, da so masne metrike hermitske in realne. Mesalne matrike so tedaj ortogonalne. Vsaka matrika ima v tem primeru sest prostih parametrov. Matricni elementi vsake unitarne matrike nxn so enolicno doloceni z z matricnimi elementi podmatrike (n — 1 )x(n — 1), ce je n > 4, pri ortogonalnih matrikah pa za vsak n. Ker pa eksperimentalni podatki nosijo napako, je mesala matrika 4x4 lahko samo priblizno dolocena. Tedaj lahko iz eksperimentalnih podatkov za 2 krat po 3 mase in iz mesalne matrike dolocimo lastnosti cetrte druzine v okviru eksperimentalne natancnosti. Izmerjeni matricni elementi mesalne matrike nosijo preveliko napako, da bi avtorja lahko napovedala mase cetrte druzine bolj
natančno kot da so blizu 1 TeV. Vzpodbudno pa je, da se novi eksperimentalni podatki za meSalno matriko kvarkov bolje ujemajo z zahtevano simetrijo masnih matrik kot stari. Avtorja ugotavljata, da so masne matrike zelo blizu edinkam (demokratičnim matrikam).
3.1 Introduction
There are several attempts in the literature to reconstruct mass matrices of quarks and leptons out of the observed masses and mixing matrices and correspondingly to learn more about properties of the fermion families [17-28]. The most popular is the n x n matrix, close to the democratic one, predicting that (n — 1) families must be very light in comparison with the nth one. Most of attempts treat neutrinos differently than the other family members, relying on the Majorana part, the Dirac part and the "sea-saw" mechanism. Most often are the number of families taken to be equal to the number of the so far observed families, while symmetries of mass matrices are chosen in several different ways [29-31]. Also possibilities with four families are discussed [32-34].
In this paper we follow the spin-charge-family theory [1-14], which predicts four families of quarks and leptons and the symmetries of their mass matrices, the same for all the family members.
The mass matrix of each family member is in the spin-charge-family theory determined by the scalar fields, which carry besides by the standard model required weak and hyper charges [13] (± 1 and ^ 1, respectively) also the additional charges: There are two SU(2) triplets, the gauge fields of the family groups, and three singlets, the gauge fields of the three charges (Q, Q' and Y'), which distinguish among family members. These scalar fields cause, after getting nonzero vacuum expectation values [13], the electroweak break. Assuming that the contributions of all the scalar (and in loop corrections also of other) fields to mass matrices of fermions are real and symmetric, we are left with the following symmetry of mass matrices
(— a — a e	d b \
e — a2 — a b d d	b a2 — a e
\b	d	eai — a/
the same for all the family members a G {u, d, v, e}. In appendix 3.5.1 the evaluation of this mass matrix is presented and the symmetry commented. The symmetry of the mass matrix Eq.(3.1) is kept in all loop corrections.
A change of phases of the left handed and the right handed basis - there are (2n — 1 ) free choices - manifests in a change of phases of mass matrices.
The differences in the properties of the family members originate in the different charges of the family members and correspondingly in the different couplings to the corresponding scalar and gauge fields.
We fit (sect. 3.3.1) the mass matrix (Eq. (3.1)) with 6 free parameters of any family member to the so far observed properties of quarks and leptons within the experimental accuracy. That is: For a pair of either quarks or leptons, we fit twice 6 free parameters of the two mass matrices to twice three so far measured masses and to the corresponding mixing matrix.
Ma =
(3.1)
Since we have the same number of free parameters (6 parameters determine in the spin-charge-family theory the mass matrix of any family member after the mass matrices are assumed to be real) as there are measured quantities for either quarks or leptons (two times 3 masses and 6 angles of the orthogonal mixing matrix under the simplification that the mixing matrix is real and hermitian), we should predict the fourth family masses and the missing mixing matrix elements (VUld4, Vu4dl, i € (1,2,3)) uniquely, provided that the measured quantities are accurate. The n — 1 sub matrix of any unitary matrix determines the unitary matrix uniquely for n > 4. The experimental inaccuracy, in particular for leptons and also for some of the matrix elements of the mixing matrix of quarks, is too large to be able to estimate the fourth family masses better than very roughly even for quarks. Yet we found out that our fitting to the experimental data for quarks are better when using the new experimental data for the quarks mixing matrix [15] than the old ones [16], which mainly differ in the second and the third diagonal values. This might be a signal that the spin-charge-family theory is the right step beyond the standard model (if taking into account also other predictions of this theory [1-14]), although we assume in this calculations the real mass matrices (Eq. (3.1)) and the orthogonal mixing matrices.
We treat all the family members, the quarks and the leptons, equivalently, as required by the spin-charge-family theory. We take into account the estimations of the influence of the fourth family masses to the mesons decays of the refs. [43], making also our own estimations (pretty roughly so far, this work is not presented in this paper) 1.
We can say that the so far obtained data do not contradict the prediction of the spin-charge-family theory that there are four coupled families of quarks and leptons, the mass matrices of which manifest the symmetry determined by the family groups - the same for all the family members, quarks and leptons. The mass matrices are quite close to the "democratic" ones, in particular for leptons.
Since the mass matrices offer an insight into the properties of the scalar fields, which determine mass matrices (together with other fields), manifesting effectively as the observed Higgs and the Yukawa couplings, we hope to learn about the properties of these scalar fields also from the mass matrices of quarks and leptons.
In sect. 3.2 the procedure to fit free parameters of mass matrices (Eq. (3.1) to the experimental data is discussed.
We comment our studies in sect. 3.4.
1 M.I.Vysotsky and A.Lenz comment in their papers [43] that the fourth family is excluded provided that one assumes the standard model with one scalar field (the scalar Higgs) while extending the number of families from three to four when, in loop corrections, evaluating the decay properties of the scalar Higgs. We have, however, several scalars: Two times three triplets with respect to the family quantum numbers and three singlets, which distinguish among the family members [13], all with the quantum numbers of the scalar Higgs with respect to the weak and hyper charge. These scalar fields determine all the masses and the mixing matrices of quarks and leptons and of the weak gauge fields, what in the standard model is achieved by the choice of the scalar Higgs properties and the Yukawa couplings. Our rough estimations of the decay properties of mesons show that he fourth family quarks might have masses close to 1 TeV.
In appendix 3.5 we offer a very brief introduction into the spin-charge-family theory, which the reader, accepting the proposed symmetry of mass matrices without knowing the origin of this symmetry, can skip. In Appendix neutrino the old results [11] for leptons are presented *What follows must be carefully checked and corrected. It must go to the discussion section.
In appendix 3.5 we offer a very brief introduction into the spin-charge-family theory, which the reader, accepting the proposed symmetry of mass matrices without knowing the origin of this symmetry, can skip.
3.2 Procedure used to fit free parameters of mass matrices to experimental data
This part repeats in many points the ref. [11]
Matrices, following from the spin-charge-family theory, might not be hermitian (appendix 3.7). We, however, simplify our study, presented in this paper, by assuming that the mass matrix for any family member, that is for quarks and Oleptons, is real and symmetric. We take the simplest phases up to signs, which depend on the choice of phases of the basic states, as discussed in appendices 3.5.12.
The matrix elements of mass matrices, with the loop corrections in all orders taken into account, manifesting the symmetry of Eq. (3.1), are in this paper taken as free parameters. Due to this symmetry, required by the family quantum numbers of the scalar fields [13], there 6 parameters having (n — 1) • (2 — 2)/2) complex phases. Assuming, to simplify the calculations, that mass matrices are real, there are correspondingly 6 free real parameter for the mass matrix for u and d quarks and for v and e leptons.
Let us first briefly overview properties of mixing matrices, a more detailed explanation of which can be found in subsection 3.2.1 of this section.
Let Ma, a denotes the family member (a = u, d, v, e), be the mass matrix in the massless basis (with all loop corrections taken into account). Let Vap = SaSPt, where a represents either the u-quark and (3 the d-quark, or a represents the v-lepton and ( the e-lepton, denotes a (in general unitary) mixing matrix of a particular pair: the quarks one or the leptons one.
For n x n matrix (n = 4 in our case) it follows:
i.	If a known sub matrix (n — 1) x (n — 1) of an unitary matrix n x n with n > 4 is extended to the whole unitary matrix n x n, the n2 unitarity conditions determine (2(2(n — 1) + 1)) real unknowns completely. If the sub matrix (n — 1) x (n — 1) of an unitary matrix is made unitary by itself, then we loose the information of the last row and last column.
ii.	If the mixing matrix is assumed to be orthogonal, then the (n — 1) x (n — 1) sub matrix contains all the information about the n x n orthogonal matrix to which it belongs and the n(n + 1 )/2 conditions determine the 2(n — 1) + 1 real unknowns
2 In the ref. [9] we made a similar assumption, except that we allow that the symmetry on the tree level of mass matrices might be changed in loop corrections. We got in that study dependence of mass matrices and correspondingly mixing matrices for quarks on masses of the fourth family.
completely for any n.
If the sub matrix of the orthogonal matrix is made orthogonal by itself, then we loose all the information of the last row and last column.
We make in this paper, to simplify the present study, several assumptions [39], as it has been already written in the introduction. In what follows we present the procedure used in our study and repeat the assumptions.
1.	If the mass matrix Mf is hermitian, then the unitary matrices Sf and Tf, introduced in appendix 3.7 to diagonalize a non hermitian mass matrix, differ only in phase factors depending on phases of basic vectors and manifesting in two diagonal matrices, Ff S and Ff T, corresponding to the left handed and the right handed basis, respectively. For hermitian mass matrices we therefore have: Tf = Sf FfSFfT t. By changing phases of basic vectors we can change phases of (2n — 1) matrix elements.
2.	We take the diagonal matrices Mf and the mixing matrices Vap from the available experimental data. The mass matrices Mf in Eq. (3.1) have, if they are hermitian and real, 6 free real parameters (af, af, a", bf, ef, df), a = (u, d, v, e).
3.	We limit the number of free parameters of the mass matrix of each family member a by taking into account n relations among free parameters, in our case n = 4, determined by the invariants
ia = — Z <, if = Z <mf,
i=1,4	i>j = 1,4
If = — Z mf mf mf, I? = mf mf mf mi ,
i>j>k=1,4
a _ u, d, v, e,	(3.2)
which are expressions appearing at powers of Af, Af + Af I1 + Af I2+ Af I3+ Af I4 = 0, in the eigenvalue equation. The invariants are fixed, within the experimental accuracy of the data, by the observed masses of quarks and leptons and by the fourth family mass, if we make a choice of it. for a chosen mf. Correspondingly there are (6 — 4) free real parameters left for each mass matrix, after a choice is made for the mass of the fourth family member.
4.	The diagonalizing matrices Sf and S3, each depending on the reduced number of free parameters, are for real and symmetric mass matrices orthogonal. They follow from the procedure
Mf_Sf Mf Tf t Tf_Sf Ff SFf t t
Mf _ (mf, mf, mf, mf),	(3.3)
provided that Sf and S3 fit the experimentally observed mixing matrices Vf 3 within the experimental accuracy and that Mf and M3 manifest the symmetry presented in Eq. (3.1). We keep the symmetry of the mass matrices accurate. One can proceed in two ways.
A. : S3 _ Vf3Sf , B. : Sf _ VfPS3 , A. : Vf3 Sf M3 SftVfp _ M3 , B. : VfP S3 Mf S3tVf3 _ Mf .(3.4)
In the case A. one obtains from Eq. (3.3), after requiring that the mass matrix Ma has the desired symmetry, the matrix Sa and the mass matrix Ma (= SaM£ Sat), from where we get the mass matrix M3 = V^p SaM3 SatVa|3. In case B. one obtains equivalently the matrix S3, from where we get Ma (= V«p S3 Ma S3tVa3). We use both ways iteratively taking into account the experimental accuracy of masses and mixing matrices.
5.	Under the assumption of the present study that the mass matrices are real and symmetric, the orthogonal diagonalizing matrices Sa and S3 form the orthogonal mixing matrix Vap, which depends on at most 6 (= n(n—1 ^) free real parameters (appendix 3.7). Since, due to what we have explained at the beginning of this section, the experimentally measured matrix elements of the 3 x 3 sub matrix of the 4 x 4 mixing matrix (if not made orthogonal by itself) determine (within the experimental accuracy) the 4 x 4 mixing matrix, also the fourth family masses are determined, again within the experimental accuracy. We must not forget, however, that the assumption of the real and symmetric mass matrices, leading to orthogonal mixing matrices, might not be an acceptable simplification, since we do know that the 3 x 3 sub matrix of the mixing matrix has one complex phase, while the unitary 4 x 4 has three complex phases. (In the next step of study, with hopefully more accurate experimental data, we shall relax conditions on hermiticity of mass matrices and correspondingly on orthogonality of mixing matrices.) We expect that too large experimental inaccuracy leave the fourth family masses in the present study quite undetermined, in particular for leptons.
6.	We study quarks and leptons equivalently. The difference among family members originate on the tree level in the eigenvalues of the operators (Qa, Q a, Y/a), which in loop corrections together with other contributors in all orders contribute to all mass matrix elements and cause the difference among family members 3.
Let us conclude. If the mass matrix of a family member obeys the symmetry required by the spin-charge-family theory, which in a simplified version (as it is taken in this study) is real and symmetric, the matrix elements of the mixing matrices of quarks and leptons are correspondingly real, each of them with n(n2-1 ^ free parameters. These six parameters of each mixing matrix are, within the experimental inaccuracy, determined by the three times three experimentally determined sub matrix. After taking into account three so far measured masses of each family member, the six parameters of each mass matrix reduce to three. Twice three free parameters are within the experimental accuracy correspondingly determined by the 3 x 3 sub matrix of the mixing matrix. The fourth family masses are correspondingly determined - within the experimental accuracy.
Since neither the measured masses nor the measured mixing matrices are determined accurately enough to reproduce the 4 x 4 mixing matrices, we can in the best case expect that the masses and mixing matrix elements of the fourth family will be determined only within some quite large intervals.
3 There are also Majorana like terms contributing in higher order loop corrections [7] which might strongly influence in particular the neutrino mass matrix.
3.2.1 Submatrices and their extensions to unitary and orthogonal matrices
In this part well known properties of n x n matrices, extended from (n — 1) x (n-1) submatrices are discussed. We make a short overview of the properties, needed in this paper, although all which will be presented here, is the knowledge on the level of text books.
Any n x n complex matrix has 2n2 free parameters. The n + 2n(n — 1 )/2 unitarity requirements reduce the number of free parameters to n2 (= 2n2 — (n + 2n(n — 1 )/2)). Let us assume a (n — 1) x (n — 1) known sub matrix of the unitary matrix. The sub matrix can be extended to the unitary matrix by (2 x [2(n — 1) +1 ]) real parameters of the last column and last row. The n2 unitarity conditions on the whole matrix reduce the number of unknowns to (2(2n — 1) — n2). For n = 4 and higher the (n — 1) x (n — 1) sub matrix contains all the information about the unitary n x n matrix.
The ref. [37] proposes a possible extension of an (n — 1) x (n — 1) unitary matrix V(n-1 )(n-i) into n x n unitary matrices Vnn.
The choice of phases of the left and the right basic states which determine the unitary matrix (like this is the case with the mixing matrices of quarks and leptons) reduces the number of free parameters for (2n — 1). Correspondingly is the number of free parameters of such an unitary matrix equal to n2 — (2n — 1), which manifests in 2n(n — 1) real parameters and 2(n — 1)(n — 2) (= n2 — 2n(n — 1) — (2n — 1)) phases (which determine the number of complex parameters).
Any real n x n matrix has n2 free parameters which the 1 n(n +1) orthogonality conditions reduce to 2 n(n—1). The (n—1) x (n—1) sub matrix of this orthogonal matrix can be extended to this n x n orthogonal matrix with [2(n — 1) + 1] real parameters. The 2n(n + 1) orthogonality conditions reduce these [2(n — 1) + 1] free parameters to (2n — 1 — 2n(n +1)), which means that the (n — 1) x (n — 1) sub matrix of an n x n orthogonal matrix determine properties of its n x n orthogonal matrix completely. Any (n — 1 ) x (n — 1 ) sub matrix of an orthogonal matrix contains all the information about the whole matrix for any n. Making the sub matrix of the orthogonal matrix orthogonal by itself one looses the information about the n x n orthogonal matrix.
3.2.2 Free parameters of mass matrices after taken into account invariants
It is useful for numerical evaluation purposes to take into account for each family member its mass matrix invariants (sect. 3.2), expressible with three within the experimental accuracy known masses, while we keep the fourth one as a free parameter. We shall make a choice of aa = 4 If (Eqs. (3.1, 3.6)) instead of the fourth family mass.
We shall skip in this section the family member index a and introduce new parameters as follows
T, f A ,	A	ai + a2	ai — a2
a, b, f = d + e, g = d — e, q =-=— , r =-=— . (3.5)
After choosing as a free parameter a = 41 (Eq. (3.6)), which is indeed the fourth family mass - summed together with the three known (from the experiment)
masses in Ii - the four invariants of Eq. (3.2) reduce the number of free parameters to 2. The four invariants therefore relate six parameters leaving three of them undetermined. There are for each pair of family members the measured mixing matrix elements, assumed in this paper to be orthogonal and correspondingly determined by six parameters, which then fixes these two times 3 parameters. The (accurately enough) measured 3 x 3 sub matrix of the (assumed to be orthogonal) 4 x 4 mixing matrix namely determines these 6 parameters within the experimental accuracy.
Using the starting relation among the invariants It, i G (1,2,3,4) and replacing new parameters (a, b, f, g, q, r) from Eq. (3.5) we obtain
Ii
a = 4 ,
I2 = -I2 + 6a2 - q2 - r2 - 2b2 = f2 + g2 , 1
2b1
I3 = (I3 - 2aI2 + 4a2) = f2 - g2 ,
I4 = I4 - aI3 + a2I2 - 3a4
= 4(q2 - r2)2 + (q2 + r2)b2 + 1 (q2 - r2) • (±) • [±] 2gf
+ b2(f2 + g2) + 4(2gf)2 .	(3.6)
We eliminate, using the first two equations, the parameters f and g, expressing them as functions of 12 and 13, which depend, for a particular family member, on the three known masses, the parameter a and the three parameters r, q and b. We are left with the four free parameters (a, b, q, r) and the below relation among these parameters
{-2(q4 + r4) + (-2b2 + 1 (-I2 + 6a2 - 2b2))(q2 + r2)
+ (I4 - 4((-I2 + 6a2 - 2b2)2 + I32) + b2(-I2 + 6a2 - 2b2))}2
= -4(q2 - r2)2((-I2 + 6a2 - 2b2 - (q2 + r2))2 - I32),	(3.7)
which reduces the number of free parameters to 3. These 3 free parameters must be determined, together with the corresponding three parameters of the partner, from the measured mixing matrix.
We eliminate one of the 4 free parameters in Eq. (3.7) by solving the cubic equation for, let us make a choice, q2
aq6 + pq4 + yq2 + 6 = 0.	(3.8)
Parameter (a, p, y, 6) depend on the 3 free remaining parameters (a, b, r) and the three, within experimental accuracy, known masses.
To reduce the number of free parameters from the starting 6 in Eq. (3.1) to the 3 left after taking into account invariants of each mass matrix, we look for the solution of Eq (3.8) for all allowed values for (a, b, r). We make a choice for a in the interval of (amtn, amax), determined by the requirement that a, which solves
the equations, is a real number. Allowing only real values for parameters f and g we end up with the equation
-I2 + 6a2 - 2b2 - (q2 + r2) > |13 + ^ 2al21,	(3.9)
which determines the maximal positive b for q = 0 = r and also the minimal positive value for b. For each value of the parameter a the interval (bmin, bmax), as well as the interval (rmin = 0, rmax), follow when taking into account experimental values for the three lower masses.
Trying to fit the free parameters to the experimental values of the 3 x 3 sub matrix to the mixing matrix we minimize the uncertainty defined in Eq. (3.10)
/ VUidj exp VUidj cal \2
OV ,,
(i,j) = 1 idj exp 6Vuldj = |Vuidj exp - Vui dj cal |,	(3.10)
O Vui dj exp
where expressions ovu d. ex stay for the experimental uncertainties, presented in Eqs. (3.11,3.12). Ui j exp
3.3 Numerical results
Taking into account the assumptions and the procedure explained in sect. 3.2 and in the ref. [39] we are looking for the 4 x 4 in this paper taken to be real and symmetric mass matrices for quarks and leptons, obeying the symmetry of Eq. (3.1) and manifesting observed properties - masses and mixing matrices - of the so far observed three families of quarks in agreement with the experimental limits for the appearance of the fourth family masses and mixing matrix elements to the lower three families, as presented in the refs. [16,15,43]. We also take into account our so far made rough estimations of possible contributions of the fourth family members to the decay of mesons. More detailed estimations are in progress. The results for leptons, presented in Appendix 3.6 are the old ones, taking from [11]. They are added only for the comparison.
We hope that we shall be able to learn from the mass matrices of quarks and leptons also about the properties of the scalar fields, which cause masses of quarks and leptons, manifesting effectively so far as the measured Higgs and Yukawa couplings.
We take the 3 x 3 measured mixing matrices for quarks and leptons and the measured masses, all with the experimental inaccuracy. We extend the measured nine mixing matrix elements for each pair to the corresponding 4 x 4 mass matrix, by taking into account the unitarity of the 4 x 4 matrix, in our case indeed the orthogonality of the 4 x 4 matrices. We then look for twice 4 x 4 mass matrices with the symmetry of Eq. (3.1), and correspondingly for the fourth family masses, for quarks and leptons.
We perform the calculations for quarks with the old [16] and new [15] experimental data for the quarks mixing matrix, to see, whether or not the more accurate values fit better into by the spin-charge-family theory predicted symmetry of mass matrices (Eq. (3.1)). We present in appendix 3.6 also one trial for the lepton mass matrices. Since the experimental data for the mixing matrix and masses are for leptons known so inaccurate, the results do not tell much.
To test the predicting power of our model in dependence of the experimental inaccuracy of masses and mixing matrices, we compare the calculated mass matrices for quarks, obtained when choosing different values for the fourth family masses, among themselves and with the experimental data, the old [16] ones and the new [15] ones.
3.3.1 Numerical results for the observed quarks with mass matrices obeying Eq. (3.1)
We take for the quarks masses the experimental values [16], recalculated to the Z boson mass scale. We take two kinds of the experimental data for the quark mixing matrices, the older data from [16] and the last data [15], with the experimentally declared inaccuracies for the so far measured 3 x 3 mixing matrix. We assume, as suggested by the spin-charge-family theory, that these nine matrix elements belong to the 4 x 4 unitary mixing matrix. We take into account the experimentally allowed values for the fourth family masses and other limitations, presented in refs. [43,3234]. We have made also our own rough estimations for the limitations which follow from the meson decays to which the fourth family members participate. Our estimations are still in progress.
A lot of effort was put into the numerical procedure to be sure as much as one can, that we fit the parameters of mass matrices to the experimental values within the experimental inaccuracy, in the best way, that is with the smallest errors.
It is expected that the inaccuracy, mainly due to the quarks mixing matrix, masses do not influence the results so strongly, does not allow to tell much about the fourth family masses. Yet, what we have learned not only supports the predicted symmetry of the spin-charge-family theory, but also predicts to what values will the more accurately measured matrix elements of the 3 x 3 sub matrix of the 4 x 4 mixing matrix move.
Let us admit that from the so far obtained results we are not yet able to predict the fourth family quarks mass accurately enough, although the results show that the most trustable might be results pushing the fourth family quarks to 1 TeV or above.
The results manifest that the mass matrices are very close to the democratic ones, which is, as expected, more and more the case the higher might be the fourth family masses, and it is true for quarks and leptons.
The calculated 4 x 4 mixing matrix predicts, in dependence of the fourth family masses, not only the fourth family matrix elements of the mixing matrix, but also the direction in which will the matrix elements of the 3 x 3 sub matrix move in the future more accurate measurements - under the assumption that the spin-charge-family theory is offering the right next step beyond the standard model.
In this paper we do not take yet into account the complex phases of the mass matrix elements and correspondingly of the mixing matrices. Sooner or latter we ought to do that.
We present below two types of the experimental values for the quarks 3 x 3 mixing matrix, taken as the sub matrix of the 4 x 4 matrix, the older experimental data [16] and the newer experimental data [15].
We start with the older experimental data [16]
|Vud| =
/0.97425 ± 0.00022 0.2252 ± 0.0009 0.00415 ± 0.00049 |VU1 d41\ 0.230 ± 0.011 1.006 ± 0.023 0.0409 ± 0.0011 |VU2d41 0.0084 ± 0.0006 0.0429 ± 0.0026 0.89 ± 0.07 |VU3d41
V |Vu4d, | |Vu4d2 I |Vu4 d3 I |Vu4d4 \J
(3.11)
and then repeat all the calculations also with the new experimental data [15]
|Vud| =
/0.97425 ± 0.00022 0.2253 ± 0.0008 0.00413 ± 0.00049 |VU1 d4|\ 0.225 ± 0.008 0.986 ± 0.016 0.0411 ± 0.0013 |VU2d41 0.0084 ± 0.0006 0.0400 ± 0.0027 1.021 ± 0.032 |VU3d41
V |Vu4d, |	|Vu4d2 |	|Vu4 d3 |	|Vu4d4 \J
(3.12)
The matrix elements of the 4 x 4 quark mixing matrix will be determine in the numerical procedure, which searches for the best fit of the two quarks mass matrices free parameters presented in Eq. (3.1) to the experimental data, taking into account the experimental inaccuracy and unitarity of the mixing matrix, ensuring as much as possible, the best fit.
Let us notice that in the new experimental data differ slightly from the old ones only in the two diagonal matrix elements, Vcs = Vu2d2 and Vtb = Vu3d3, appearing in new data with smaller inaccuracy. The corresponding fourth family mixing matrix elements (|VUid41 and |VU4dj |) are accordingly in both cases determined from the unitarity condition for the 4 x 4 mixing matrix through the fitting procedure, as also all the other matrix elements of the mixing matrix are.
Using first the old experimental data we predict the direction in which new more accurately measured matrix elements should move and then check if this is happening with the new experimental data.
Then we use new experimental data, repeat the procedure in look at what are the new results predicting.
For the quark masses at the energy scale of MZ we take
MU/MeV/c2 = (1.3 + 0.50 — 0.42,619 ± 84,172 000. ± 760.,
mU4 = 700 000., 1 200 000.), Md/MeV/c2 = (2.90 + 1.24 — 1.19,55 + 16 — 15,2900. ± 90.,
md4 = 700 000., 1 200 000.).	(3.13)
We found that the results are not influenced much if changing the masses within the experimental uncertainties.
Experimental values for leptons as well as the obtained mass matrices are presented in appendix 3.6.
Following the procedure explained in sect. 3.2 we look for the mass matrices for the u-quarks and the d-quarks by requiring that the mass matrices reproduce experimental data while manifesting symmetry of Eq. (3.1), predicted by the spin-charge-family theory.
We look for several properties of the obtained mass matrices:
i.	We test the influence of the experimentally declared inaccuracy of the 3 x 3 sub matrices of the 4 x 4 mixing matrices and of the twice 3 measured masses on the prediction of the fourth family masses.
ii.	We look for how do the old and the new matrix elements of the measured mixing matrix influence the accuracy with which the experimental data are reproduced in the procedure which takes into account the symmetry of mass matrices.
iii.	We look for how different choices for the masses of the fourth family members limit the inaccuracy of particular matrix elements of the mixing matrices or the inaccuracy of the three lower masses of family members.
iv.	We test how close to the democratic mass matrix are the obtained mass matrices in dependence of the fourth family masses.
v.	We look for the predictions of the 4 x 4 mass matrices with the symmetry presented in Eq. (3.1).
The numerical procedure, used in this contribution, is designed for quarks and leptons. We present in this paper the results for quarks. The results for leptons, presented in appendix 3.6 is only to manifest the general properties of leptons, since the experimental data for leptons are far too non accurate to lead to trustable predictions.
In the next subsection 3.3.1 the numerical results are presented for the 4 x 4 mass matrices of the u-quarks and the d-quarks as they follow from the by the
spin-charge-family theory required symmetry after fitting the experimental data.
Mass matrices for quarks In order to test whether or not our results have some experimental support, we use two kinds of the experimental values for the quark mixing matrix, presented in Eqs. (3.11, 3.12), respectively, for several values of the fourth family quark masses.
Searching for mass matrices with the symmetry of Eq. (3.1) to determine the interval for the fourth family quark masses in dependence of the values of the mixing matrix elements within the experimental inaccuracy, we repeat the numerical procedure for data with several values of masses of the fourth family quarks. Here we present results for two of them: for mu4 = 700 GeV = md4 and for mu4 = 1 200 GeV = md4.
We present below the results for the two experimental matrix elements [16,15] for the quark mixing matrix, first for the data [16] and then for the data [15], in both cases first for mu4 = 700 GeV = md4 and then for mu4 = 1 200 GeV = md4.
Having results from the fitting procedure when used the old experimental data for the quark mixing matrix, we look for the predictions, which the calculated 3 x 3 matrix elements of the 4 x 4 mixing matrix obeying the symmetry of Eq. (3.1) offer, and then check to which extend the predictions agree with new experimental data.
Then we repeat calculations with new experimental data.
Results for the mass matrices of the two quarks family members, fitted to the mixing matrix elements presented in the ref. [16]. The fit offers the smallest common deviation (Eq. 3.10)) of the sum of all the average values of the nine matrix elements of the 3 x 3 sub matrix. The masses of quarks and the mixing matrix resulting from diagonalizing the two best fitted mass matrices are also presented.
1. Here mu4 = 700 GeV md4 = 700 GeV is chosen.
Mu =
Md =
/227623. 131877. 132239. 217653.\ 131877. 222116. 217653. 132239. 132239. 217653. 214195. 131877. \217653. 132239. 131877. 208687./
/175797. 174263. 174288. 175710.\ 174263. 175666. 175710. 174288. 174288. 175710. 175813. 174263. \175710. 174288. 174263. 175682./
(3.14)
Vud =
(3.15)
/-0.97423 0.22531 -0.003 0.01021 \ 0.22526 0.97338 -0.042 0.0016 -0.00663 -0.04197 -0.9991 -0.0004 0.00959 -0.00388 -0.0003 0.99995
The corresponding absolute values for the deviations from the average experimental values (Eq.(3.10)) are
0.091 0.117 2.339^ 6Vud = ( 0.431 1.418 1.348 2.951 0.358 1.559 >
(3.16)
The corresponding total absolute average deviation Eq. (3.10) is 4.5579.
The two mass matrices correspond to the diagonal masses
M^/MeV/c2 = (1.3,620.0,172 000., 700 000.), Md /MeV/c2 = (2.88508,55.024,2 899.99,700000.).
(3.17)
2. In the next case mu4 = 1 200 GeV and md4 = 1 200 GeV are chosen, again fitting the old [16] experimental for quark mixing matrix elements.
Mu =
Md =
/351916. 256894. 257204. 342714.\ 256894. 344411. 342714. 257204. 257204. 342714. 341900. 256894. \342714. 257204. 256894. 334395./
300783. 299263. 299288. 300709.\ 299263. 300623. 300709. 299288. 299288. 300709. 300856. 299263. 300709. 299288. 299263. 300696.
(3.18)
Vud =
/-0.97425 0.22536 -0.00301 0.00474 \ 0.22534 0.97336 -0.04239 0.00212 0.00663 -0.04198 -0.9991 -0.00021 0.00414 -0.00315 -0.00011 0.99999
— i
(3.19)
The corresponding values for the deviations from the average experimental value of the matrix elements of the 3x3 sub matrix are
/0.003 0.226 2.335^ 5Vud = I 0.424 1.419 1.357 \2.949 0.355 1.559;
The corresponding total average deviation Eq. (3.10) is 4.5595.
The two mass matrices correspond to the diagonal masses
M^/MeV/c2 = (1.3,620.0,172000., 700000.), Md/MeV/c2 = (2.9,55.0,2 900.0,700 000.).
(3.20)
(3.21)
Let us notice, that while the mass matrices of the u and the d quarks change for a factor of« 1.5, becoming more "democratic" (that is the matrix elements become more and more equal), when changing the fourth family masses from 700 GeV to 1 200 GeV, the mixing matrix elements of the 3 x 3 sub matrix do not change a lot (Eqs.(3.15, 3.19)).
Let us now see what does our calculations say. We first make comparison for the old [16] (expo) mixing matrix with the calculated ones when the fourth family quark masses are 700 GeV, and 1 200 GeV. Results are presented in Eq. (3.22)
/expo 0.97425 ± 0.00022 0.2252 ± 0.0009 0.00415 ± 0.00049\
|V(ud)old 1 =
oldi	0.97423	0.22531	0.003
old2	0.97425	0.22536	0.00301
expo	0.230 ± 0.011	1.006 ± 0.023	0.0409 ± 0.0011
old1	0.22526	0.97338	0.042
old2	0.22534	0.97336	0.04239
expo	0.0084 ± 0.0006	0.0429 ± 0.0026	0.89 ± 0.07
oldi	0.00663	0.04197	0.9991
^old2	0.00663	0.04198	0.9991
/
(3.22)
The calculated mixing predicts:
i.	The matrix element VUl dl should almost not change, VUl d2 may slightly rise, and (Vu2d3 and Vu3d3) will also rise.
ii.	The matrix elements (VU1 d3, VU2d,, VU2d2, VU3d,, VU3d2) should lower. Checking the new experimental values one sees that the prediction was in all the cases in agreement with those new experimental data which were done with better accuracy.
let us repeat the calculations with new experimental data [15] to see how will the new data influence the mass matrices and the mixing matrix elements.
Results for the mass matrices of the two quarks family members, fitted to the new mixing matrix elements [15], which lead to the smallest common deviation for the sum of all the average values of the nine matrix elements of the 3 x 3 sub matrix, are presented, together with the masses of quarks and the mixing matrix resulting from diagonalizing the two mass matrices. Again the fourth quark masses are first (mu4 = 700 GeV, md4 = 700 GeV) and then (mu4 = 1 200 GeV, md4 = 1 200 GeV) 1. Here mu4 = 700 GeV and md4 = 700 GeV is chosen.
Mu =
Md =
/226521. 131887. 132192. 217715.\ 131887. 219347. 217715. 132192. 132192. 217715. 216964. 131887. \217715. 132192. 131887. 209790./
/175776. 174263. 174288. 175709.\ 174263. 175622. 175709. 174288. 174288. 175709. 175857. 174263. \175709. 174288. 174263. 175703./
(3.23)
Vud =
/—0.97423 0.22539 -0.00299 0.00776 \ 0.22534 0.97335 -0.04245 0.00349 0.00667 -0.04203 -0.99909 -0.00038 V 0.00677 -0.00517 -0.00020 0.99996 /
(3.24)
The corresponding values Eq. (3.10) for the deviations from the average experimental values are
0.074 0.109 2.339^ 6Vud = I 0.043 0.791 1.032 2.291 0.753 0.685
(3.25)
The corresponding total absolute average deviation Eq. (3.10) is 4.07154.
The two mass matrices correspond to the diagonal masses
M^/MeV/c2 = (1.3,620.0,172000., 700000.), Md/MeV/c2 = (2.9,55.0,2 900.0,700 000.).	(3.26)
2. Here mU4 = 1 200 GeV md4 = 1 200 GeV is chosen.
Mu =
Md =
/354761. 256877. 257353. 342539.\ 256877. 350107. 342539. 257353. 257353. 342539. 336204. 256877. V342539. 257353. 256877. 331550./
/300835. 299263. 299288. 300710.\ 299263. 300714. 300710. 299288. 299288. 300710. 300765. 299263. V300710. 299288. 299263. 300644./
(3.27)
Vud =
( 0.97423 0.22538 0.00299 0.00793 \ -0.22531 0.97336 0.04248 -0.00002 0.00667 -0.04206 0.99909 -0.00024 0.00773 -0.00178 0.00022 0.99997 )
(3.28)
The corresponding values for the deviations from the average experimental value for each matrix element are
/ 0.07 0.097 2.329\ 5Vud = I 0.038 0.79 1.061 I .	(3.29)
\2.889 0.762 0.685/
The corresponding total average deviation Eq. (3.10) is 4.0724. The two mass matrices correspond to the diagonal masses
M^/MeV/c2 = (1.3,620.0,172 000., 1 200 000.), Md/MeV/c2 = (2.88508,55.024,2 899.99,1 200 000.). (3.30)
Again we notice that the mass matrices of the u and the d quarks change for a factor of k 1.5 when the masses of the fourth family members grow from 700 GeV to 1 200 GeV. The mass matrices become more "democratic". The mixing matrix elements of the 3 x 3 sub matrix do not change a lot (Eqs.(3.24, 3.28)) with the masses of the fourth family quarks, but they do agree better with the newer [15] than with the older [16] experimental values.
Let us now compare the old [16] (expo) and the new [15] (expn) mixing matrix elements of the 3 x 3 sub matrix with the calculated ones for either the old [16] or for the new [15] experimental values, fitting them to the mass matrices of Eq. (3.1), in both cases for mu4 = md4 = 700 GeV and for mu4 = md4 = 1 200 GeV, to see whether we can learn something out of this comparison.
We present below the old data (expo), the new data (expn) and both calculated values, each for mu4 = md4 = 700 GeV (oldi, new]) and mu4 = md4 = 1 200
GeV (old2, new2), putting together all these values in the same matrix.
|V(ud)| =
( expo	0.97425 ± 0.00022 0.2252 ± 0.0009 0.00415 ± 0.00049\		
expn	0.97425 ± 0.00022 0.2253 ± 0.0008 0.00413 ± 0.00049		
oldi	0.97423	0.22531	0.003
old2	0.97425	0.22536	0.00301
newi	0.97423	0.22531	0.00299
new2	0.97423	0.22538	0.00299
expo	0.230 ± 0.011	1.006 ± 0.023	0.0409 ± 0.0011
expn	0.225 ± 0.008	0.986 ± 0.016	0.0411 ± 0.0013
oldi	0.22526	0.97338	0.042
old2	0.22534	0.97336	0.04239
newi	0.22534	0.97335	0.04245
new2	0.22531	0.97336	0.04248
expo	0.0084 ± 0.0006	0.0429 ± 0.0026	0.89 ± 0.07
expn	0.0084 ± 0.0006	0.0400 ± 0.0027	1.021 ± 0.032
oldi	0.00663	0.04197	0.9991
old2	0.00663	0.04198	0.9991
newi	0.00667	0.04203	0.99909
^new2	0.00667	0.04206	0.99909
(3.31)
Comparing the above results and the results for mass matrices and 4 x 4 mixing matrices one finds:
i.	The old and new experimental data differ mainly in the diagonal matrix elements.
ii.	The old and new experimental data lead in the fitting procedure to quite similar 3 x 3 sub matrix, while their influence on the fourth family matrix elements are stronger.
iii.	The fourth family masses change the mass matrices considerably, while their influence on the 3 x 3 sub matrix of the 4 x 4 mixing matrix is much weaker.
iv.	The prediction (Eq. (3.22)) of the calculated mixing matrix elements, obtained by fitting the symmetry of the mass matrices (Eq. (3.1)) to the experimental data [16], was confirmed by improved experimental data [15]. In all cases are the calculated 3 x 3 matrix elements closer to the new experimental values than to the old experimental values.
v.	Calculations with new experimental data predict: We expect (Eq. (3.31)) that more accurate experiments will bring a slightly smaller values for (VU1 dl, VU1 d3, Vu3d3), smaller (V^d2, V^d,), (VUl d2, V^d,) will slightly grow and (V-^ds) Vu3d2 will grow.
vi.	The matrix elements Vui d4 and Vu4 di change considerably with the mass of the fourth family members, and they differ quite a lot also when using new instead of the old experimental data for the mixing matrix.
vii.	Fitting the free parameters of the mass matrices to the new experimental data [15] gives smaller parameter ct (Eq. sigma) than when fitting old experimental data [16]: 4.07154 with respect to 4.5579 for the masses 700 GeV and 4.0724 with respect to 4.5595 for the masses 1 200 GeV, while with the mass ct does not really
change. Only very accurate mixing matrix elements would allow to determine fourth family quarks masses more accurately. Since the choice of the fourth family quark masses does not appreciable influence either the fitting procedure or the obtained 3 x 3 mixing matrix, and also not the accuracy of the masses of the three lower families, it is difficult to predict the interval for the masses of the fourth family members. For the masses of the fourth family quarks to be close or above 1 TeV speak more other experimental data, like decays of mesons.
An estimation how trustable is the numerical procedure, used to fit free parameters of the quarks mass matrices to the experimental data, can be made by comparing the results for the mixing matrix for several choices of the fourth family masses. The fitting procedure shows up that the 3 x 3 mixing matrix does not change appreciable, even not for much lower masses from 300 GeV up.
We can conclude: Requiring that the experimental data respect the symmetry of the mass matrices (Eq. (3.1)) (suggested by the spin-charge-family theory) the prediction can be made for the change of the matrix elements of the 3 x 3 sub matrix in future experiments. The masses of the fourth family members are more difficult to predict, since the accuracy of the experimental data for the quark masses and in particular for the mixing matrix should be extremely high to really limit the fourth family masses. For a known fourth family masses the fourth family matrix elements of the mixing matrix are accurate. For masses of the fourth family quarks to be close or above 1 TeV speak more other experimental data, like decays of mesons.
3.4 Discussions and conclusions
One of the most important open questions in the elementary particle physics is: Where do the family originate? Explaining the origin of families would answer the question about the number of families possibly observable at the low energy regime, about the origin of the scalar field(s) and Yukawa couplings and would also explain differences in the fermions properties - the differences in masses and mixing matrices among family members - quarks and leptons.
Assuming that the prediction of the spin-charge-family theory that there are four rather than so far observed three coupled families, the mass matrices of which demonstrate in the massless basis the SU(2) x SU(2) (each of two SU(2) is a subgroup of its own SO(4)) symmetry of Eq. (3.1), the same for all the family members - the quarks and the leptons - and simplifying the numerical procedure by the assumption that the mass matrices are symmetric and real and correspondingly the mixing matrices orthogonal, we fit the free parameters of the quarks mass matrices (6 for u-quarks and 6 for d-quarks to twice three masses of quarks and to the mixing matrix 4 x 4, extracted from the 3 x 3 sub matrix elements, fitted to 6 parameters of the orthogonal matrix) to the experimental data. Every unitary n x n matrix is for n > 4, through the unitary conditions, uniquely determined by the 3 x 3 sub matrix.
The numerical procedure, explained in this paper, to fit free parameters to the experimental data within the experimental inaccuracy of masses and in particular of the mixing matrix is very tough. The accurate mixing matrix elements and
masses would completely determine the fourth family masses. The experimental inaccuracies are too large to tell the trustable mass interval, within which the fourth family masses of quarks lie.
In this paper we are not yet able to tell the mass intervals for the fourth family quarks. But since the matrix elements of the 3 x 3 sub matrix depend very weakly on the fourth family masses, the calculated matrix (from the experimental data under the assumption that the mass matrices manifest the symmetry of Eq. (3.1)) offer the prediction to what values will more accurate measurements move the present experimental data. We checked this prediction by performing calculations with the old matrix elements [16] and then test the prediction on the new ones [15]. The results are presented in Eq. (3.22). Repeating calculations with the new matrix elements for several masses of the fourth family quarks we predict further change of the 3 x 3 sub matrix elements, presented in Eq. (3.31).
We expect: More accurate experiments will bring a slightly smaller values for (Vud, Vub, Vtb), smaller (Vcs, Vtd), (Vus, Vcd) will slightly grow and (Vcb) Vts will grow.
The fourth family mixing matrix elements depend, as expected, strongly on the fourth family masses. For chosen masses of the fourth family members their matrix elements can be quite accurately predicted (Eqs. (3.24, 3.28)).
Mass matrices are quite close to the "democratic" ones not only for leptons but also for quarks. With the growing fourth family masses the "democracy" in matrix elements grow (Eqs. (3.23, 3.23)), as expected.
Although we have not study complex mass matrices, we do not expect that the presented results would change considerably after taking into account the complex phases of mass matrices and correspondingly also of the mixing matrices. We estimate the accuracy of our calculations by comparing the results of the calculated 3 x 3 matrix elements for the interval of the fourth family masses, from 300 GeV to 1 200 GeV. It look very trustable, offering for all these masses only slowly changing matrix elements.
We are concluding: Requiring that the experimental data respect the symmetry of the mass matrices (Eq. (3.1)) (suggested by the spin-charge-family theory) the prediction is made for the change of the matrix elements of the 3 x 3 sub matrix in future more accurate experiments. More (much more) accurate measured 3 x 3 sub matrix elements in future will determine, following the spin-charge-family theory, the fourth family masses and the fourth family matrix elements. However, even with the present experimental data our calculations, respecting the symmetry of the mass matrices (Eq. (3.1)) offer the prediction for the direction to which will more accurately measured matrix elements move.
Since the symmetry of the mass matrices are determined in the spin-charge-family theory by two triplet (with respect to the family charges) and tree singlet (with respect to the family members charges (Q, Q', Y)) scalar fields [13,14], all with the weak and the hyper charges as assumed in the standard model for the scalar fields, we hope to learn from the properties of the mass matrices and the corresponding mixing matrices more about these scalar fields.
3.5 APPENDIX: A brief presentation of the spin-charge-family theory
We present in this section a very brief introduction into the spin-charge family theory [1-14]. The reader can skip this appendix taking by the spin-charge family theory required symmetry of mass matrices of Eq. (3.1) as an input to the study of properties of the 4 x 4 mass matrices - with the parameters which depend on charges of the family members - and can come to this part of the paper, if and when would like to learn where do families and scalar fields possibly originate from.
Let us start by directing attention of the reader to one of the most open questions in the elementary particle physics and cosmology: Why do we have families, where do they originate and correspondingly where do scalar fields, manifesting as Higgs and Yukawa couplings, originate? The spin-charge-family theory is offering a possible explanation for the origin of families and scalar fields, and in addition for the so far observed charges and the corresponding gauge fields.
There are, namely, two (only two) kinds of the Clifford algebra objects: One kind, the Dirac Ya, takes care of the spin in d = (3 + 1), while the spin in d > 4 (rather than the total angular momentum) manifests in d = (3 + 1) in the low energy regime as the charges. In this part the spin-charge family theory is like the Kaluza-Klein theory, unifying spin (in the low energy regime, otherwise the total angular momentum) and charges, and offering a possible answer to the question about the origin of the so far observed charges and correspondingly also about the so far observed gauge fields. The second kind of the Clifford algebra objects, forming the equivalent representations with respect to the Dirac kind, recognized by one of the authors (SNMB), is responsible for the appearance of families of fermions.
There are correspondingly also two kinds of gauge fields, which appear to manifest in d = (3 + 1) as the so far observed vector gauge fields (the number of -obviously non yet observed - gauge fields grows with the dimension) and as the scalar gauge fields. The scalar fields are responsible, after gaining nonzero vacuum expectation values, for the appearance of masses of fermions and gauge bosons. They manifest as the so far observed Higgs [36] and the Yukawa couplings.
All the properties of fermions and bosons in the low energy regime originate in the spin-charge-family theory in a simple starting action for massless fields in d = [1 + (d - 1)]. Fermions interact with the vielbeins f«a and correspondingly with the two kinds of the spin connection fields: with dabc = fac daba which are the gauge fields of Sab = 4 (yayb - ybya) and with d>abc = fac daba which are the gauge fields of Sab = 4 (YaYb - YbYa). a, p,... is the Einstein index and a, b,... is the flat index. The starting action is the simplest one
S =
ddx E Lf +
ddx E (aR + aR), Lf = 1 (1 YaPoa1) + h.c.
P0a = faa poa + 2^ (P«, Ef«a}-, P0a = Pa - 2sabdaba - 2 sabdaba,(3.32)
R = 2 {fa[afßb] (^aba,ß - ^caa D%ß)} + h.c. , R = 1 fa[afßb] (CDaba,ß - CDcaaCDcbß) + h.C. .	(3.33)
E = det(eaa) and eaafßa = 6ß. Fermions, coupled to the vielbeins and the two kinds of the spin connection fields, manifest (after several breaks of the starting symmetries) before the electroweak break four massless families of quarks and leptons, the left handed fermions are weak charged and the right handed ones are weak chargeless. The vielbeins and the two kinds of the spin connection fields manifest effectively as the observed gauge fields and (those with the scalar indices in d = (1 + 3)) as several scalar fields. The mass matrices of the four family members (quarks and leptons) are after the electroweak break expressible on a tree level by the vacuum expectation values of the two kinds of the spin connection fields and the corresponding vielbeins with the scalar indices ([1,2,6,7,12,13]):
i.	One kind originates in the scalar fields cDabc, manifesting as the two SU(2) triplets- A* l i,i = (1,2,3) ,s = (7,8); A]1 ,i = (1,2,3) ,s = (7,8); - and one singlet - A4 , s = (7,8) - contributing equally to all the family members.
ii.	The second kind originates in the scalar fields Dabc, manifesting as three singlets - Aq , Aq , AY', s = (7,8) - contributing the same values to all the families and distinguishing among family members. Q and Q' are the quantum numbers from the standard model, Y' originates in the second SU(2) (a kind of a right handed "weak") charge.
All the scalar fields manifest, transforming the right handed quarks and lep-tons into the corresponding left handed ones 4 and contributing also to the masses of the weak bosons, as doublets with respect to the weak charge. Loop corrections, to which all the scalar and also gauge vector fields contribute coherently, change contributions of the off-diagonal and diagonal elements appearing on the tree level, keeping the tree level symmetry of mass matrices unchanged 5.
3.5.1 Mass matrices on the tree level and beyond which manifest SU(2) X SU(2) symmetry
Let us make a choice of a massless basis i = (1,2,3,4), for a particular family memeber a. And let us take into account the two kinds of the operators, which transform the basis vectors into one another
NL ,i =(1,2,3), < ,i =(1,2,3),	(3.34)
4	It is the term y0ys , where with s = (7, 8) denotes any of the scalar fields, which transforms the right handed fermions into the corresponding left handed partner [1,7,2,6,12-14]. This mass term originates in tJj yap0a^ of the action Eq.(3.32), with a = s = (7,8) and pos = (pff - 2Sabd>aba - 2 Sstdstff).
5	It can be seen that all the loop corrections keep the starting symmetry of the mass matrices unchanged. We have also started [7,42] with the evaluation of the loop corrections to the tree level values. This estimation has been done so far [42] only up to the first order and partly to the second order.
with the properties
NL (^1,^2,^3,^4) = 1 (-^1,^2, -^3,^4), N + (^1 ,^2,^3,^4) = (^2, 0,^4, 0), IN - (^1 ,^2,^3,^4) = (0 ,^1, 0,^3),
Tf3 (^1,^2,^3,^4) = 1 (-^1, -^2,^3,^4) ,
f+ (^1,^2,^3,^4) = (^3,^4, 0, 0) ,
f- (^1,^2,^3,^4) = ( 0, 0,^1,^2).	(3.35)
This is indeed what the two SU(2) operators in the spin-charge-family theory do. The gauge scalar fields of these operators determine, together with the corresponding coupling constants, the off diagonal and diagonal matrix elements on the tree level. In addition to these two kinds of SU(2) scalars there are three U(1) scalars, which distinguish among the family members, contributing on the tree level the same diagonal matrix elements for all the families. In loop corrections in all orders the symmetry of mass matrices remains unchanged, while the three U(1) scalars, contributing coherently with the two kinds of SU(2) scalars and all the massive fields to all the matrix elements, manifest in off diagonal elements as well. All the scalars are doublets with respect to the weak charge, contributing to the weak and the hypercharge of the fermions so that they transform the right handed members into the left handed onces.
With the above (Eq. (3.35) presented choices of phases of the left and the right handed basic states in the massless basis the mass matrices of all the family members manifest the symmetry, presented in Eq. (3.1). One easily checks that a change of the phases of the left and the right handed members, there are (2n - 1) possibilities, causes changes in phases of matrix elements in Eq. (3.1).
3.6 APPENDIX: Mass matrices for leptons
We evaluate 3 x 3 matrix elements from the data [16]
-17
7.05 • 10-17 < A(m21/MeV/c2)2 < 8.34 • 10 2.07 • 10-15 < A(m(3i),(32)/MeV/c2)2 < 2.75 • 10-15 , 0.25 < sin2 612 < 0.37, 0.36 < sin2 623 < 0.67, sin2 613 < 0.035(0.056), sin2 2613 = 0.098 ± 0.013,
(3.36)
which means that n - 10 < 023 < ? + 10, 54 - 10 < 012 < n + 10, 013 < f|. This reflects in the lepton mixing matrix Vve = Sv Se t
IVvel =
(0.8224 0.5200 0.1552 |VVie41\ 0.3249 0.7239 0.6014 |VV2e41 0.4455 0.4498 0.7704 | Vv3e41 V I Vv4C,I I Vv4C2 I |Vv4e3l I ^4 I/
(3.37)
determining for each assumed value for any mixing matrix element within the experimentally allowed inaccuracy the corresponding fourth family mixing matrix elements (|VVie41 and |VV4ej |) from the unitarity condition for the 4 x 4 mixing matrix. The masses of leptons are taken from [15,16] while we take the fourth family masses as free parameters, checking how much does the experimental inaccuracy influence a possible prediction for the fourth family leptons masses and how does this prediction agree with experimentally allowed values [15,16,43] for the fourth family lepton masses.
Md/MeV/c2 = (1 • 10-9, 9 • 10-9, 5 • 10-8, mV4 > 90000.), Md/MeV/c2 = (0.486570161 ± 0.000000 042, 102.718 135 9 ± 0.000 009 2, 1746.24 ± 0.20, me4 > 102 000 )
(3.38)
3.6.1 Numerical results for leptons
We present here the old results [11] for leptons, manifesting properties of the lepton mass matrices. These results are less informative than those for quarks, since the experimental results are for leptons mixing matrix much less accurate than in the case of quarks and also masses are known less accurately.
We make a choice of the fourth family masses and take the mixing matrix elements from the old experimental data [16] We have
Mv =
( 14021. 14 968. 14 968.
14	968.
15	979. 15 979.
14	968.
15	979. 15 979
14 021.\ 14 968. 14 968.
\—14021. -14 968. -14 968. 14 021. / / 28933. 30057. 29762. —27207A
Me =
30 057. 29 762. \—27207.
which leads to the mixing matrix V-
32009. 31 958. 29 762.
31 958. 32009. -30057.
-29 762. -30057. 28 933.
Vvel =
0.82363 0.54671 0.50263 0.58049
-1


0.15082 0. 0.64062 0. 0.26268 0.60344 0.75290 0. 0. 0. 0. 1.
and the masses
(3.39)
(3.40)
M^/MeV/c2 = (5 • 10-9 ,1 • 10-8 ,4.9 • 10-8 ,60 000.) Md/MeV/c2 = (0.510999, 105.658, 1 776.82, 120 000)
(3.41)
We did not adapt lepton masses to Zm mass scale. Zeros (0.) for the matrix elements concerning the fourth family members means that the values are less than 10-5 and 1. means that the difference from 1 occurs on the sixth digit at most.
We notice:
i.	The required symmetry, Eq. (3.1), is kept exactly.
ii.	The mass matrices of leptons are very close to the "democratic" matrix.
iii.	The mixing matrix elements among the first three and the fourth family members are very small, what is due to our choice, since the matrix elements of the 3 x 3 sub matrix of the 4 x 4 unitary matrix, predicted by the spin-charge-family theory are very inaccurately known.
3.7 APPENDIX: Properties of non Hermitian mass matrices
This pedagogic presentation of well known properties of non Hermitian matrices can be found in many textbooks, for example [44]. We repeat this topic here only to make our discussions transparent.
Let us take a non Hermitian mass matrix Ma as it follows from the spin-charge-family theory, a denotes a family member (index ± used in the main text is dropped).
We always can diagonalize a non Hermitian Ma with two unitary matrices,
Sa (Sat Sa = I) and Ta (Tat Ta = I)
Sa t Ma Ta = = (mf ... ma ...m£).	(3.42)
The proof is added below.
Changing phases of the basic states, those of the left handed one and those of the right handed one, the new unitary matrices S/a = Sa FaS and T/a = Ta FaT change the phase of the elements of diagonalized mass matrices M J
S/a t Ma T /a = F^s M£ FaT =
di.ag(mfei(*"S-*"T) .. .ma e1^-^) ,.. .mi	)),
Fas = diag(e-i*"S, ... , e-i^S ,... , e-i^S), FaT = diag(e-i*"T, ... , e-i^"T ,... , e-i^T).	(3.43)
In the case that the mass matrix is Hermitian Ta can be replaced by Sa, but only up to phases originating in the phases of the two basis, the left handed one and the right handed one, since they remain independent.
One can diagonalize the non Hermitian mass matrices in two ways, that is either one diagonalizes MaMa t or MatMa
(SatMaTa)(SatMaTa)t = SatMaMatSa = M^ , (SatMaTa)t(SatMaTa) = TatMatMaTa = M^ ,
Mas = Mas, Ma^ = MaT.	(3.44)
One can prove that Mas = MaT• The proof proceeds as follows. Let us define two Hermitian (Ha, H") and two unitary matrices (U", Hf)
Ha = SaMaSSa t, Ha = TaMaTTa t,
Usa = HiH Ma, Ua = Ha-1 Ma t,	(3.45)
It is easy to show that Hi?1 = Ha, H^1 = Hf, U£ U^1 = I and U^ U^1 = I. Then it follows
s«t h^ sa = M^S = M^t = Sat Ma U£-1 Sa = Sat MaTa , Tat H^ Ta = M^T = M^-t = Tat Ma 1U£-1 Ta = Tat Mat Sa, (3.46)
where we recognized U^-1 Sa = Ta and U^-1 Ta = Sa. Taking into account
Eq. (3.43) the starting basis can be chosen so, that all diagonal masses are real and
positive.
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Bled Workshops in Physics Vol. 15, No. 2
A
Proceedings to the 17th Workshop What Comes Beyond ... (p. 46) Bled, Slovenia, July 20-28, 2014

4 Emergent SUSY Theories: QED, SM & GUT
J.L. Chkareuli*
Center for Elementary Particle Physics, Ilia State University 0162 Tbilisi, Georgia
Abstract. It might be expected that only global symmetries are fundamental symmetries of Nature, whereas local symmetries and associated massless gauge fields could solely emerge due to spontaneous breaking of underlying spacetime symmetries involved, such as relativistic invariance and supersymmetry. This breaking, taken in the form of the nonlinear a-model type pattern for vector fields or superfields, puts essential restrictions on geometrical degrees of freedom of a physical field system that makes it to adjust itself in such a way that its global internal symmetry G turns into the local symmetry Gloc • Remarkably, this emergence process may naturally be triggered by spontaneously broken supersymmetry, as is illustrated in detail by an example of a general supersymmetric QED model which is then extended to electroweak models and grand unified theories. Among others, the U(1) x SU(2) symmetrical Standard Model and flipped SU(5) GUT appear preferable to emerge at high energies.
Povzetek. Avtor dokazuje, da so samo globalne simetrije temeljne simetrije narave, lokalne simetrije in njim pridruZena umeritvena polja pa se pojavljajo samo zaradi spontane zlomitve simetrije prostora-casa, kot sta relativistična invarianca in supersimetrija. Avtor uporabi nelinearni model a za obravnavo sistema z relativisticno simetrijo in supersimetrijo. Ugotavlja, da zlomitev relativisticne invariance ter supersimetrije bistveno zozzi geometrijske prostostne stopnje vektorskih in superpolja, zato se ta odzove s spremembo globalne notranje simetrije G v lokalno simetrijo Gloc. Spontano zlomljena supersimetrija lahko naravno sprozzi ta proces porajanja. Avtor ta spontani prehod ilustrira na primeru splosnega modela supersimetricne kvantne elektordinamike. Ta model razsiri tudi na elektrosibki standardni model ter na teorije, imenovane "velike teorije poenotenja" (GUT). Pri visokih energijah se pri takih zlomitvah simetrij pojavita prav standardni model in ,,prekucnjen" (flipped) model SU(5).
4.1 Introduction
We all believe that internal gauge symmetries form the basis of modern particle physics being most successfully realized within the celebrated Standard Model (SM) of quarks and leptons and their fundamental strong, weak and electromagnetic interactions. At the same time, local gauge invariance, contrary to a global symmetry case, may look like a cumbersome geometrical input rather than a "true" physical principle, especially in the framework of an effective quantum field theory (QFT) becoming, presumably, irrelevant at very high energies. In this connection,
* E-mail: j.chkareuli@iliauni.edu.ge
one could wonder whether there is any basic dynamical reason that necessitates gauge invariance and the associated masslessness of gauge fields as some emergent phenomenon arising from a more profound level of dynamics. By analogy with a dynamical origin of massless scalar particle excitations, which is very well understood in terms of spontaneously broken global internal symmetries [1], one could think that the origin of massless gauge fields as vector Nambu-Goldstone (NG) bosons are related to the spontaneous Lorentz invariance violation (SLIV) that is the minimal spacetime global symmetry underlying particle physics. This well-known approach providing a viable alternative to quantum electrodynamics [2], gravity [3] and Yang-Mills theories [4] has a long history started over fifty years ago, though has been significantly revised in the recent years [5-8].
4.1.1 An emergence conjecture
Directly or indirectly, the approach mentioned includes several key points which in a conventional QFT framework may be formulated nowadays in the following way (see [9] and comprehensive references therein):
•	Only global symmetries are fundamental symmetries of Nature. Local symmetries and associated massless gauge vector (tensor) fields could only emerge due to some phase transition producing them as appropriate Nambu-Goldstone modes,
•	The underlying Lorentz invariance is proposed to be spontaneously broken since only spacetime symmetry breaking could basically provide an existence of vector (tensor) emerging modes which mediate all interactions involved,
•	The theory itself is proposed to be "physically" viable in the sense that any appropriate initial value condition (IVC), which determines the subsequent dynamical evolution of a physical field system, is uniquely satisfied. This means in turn that an interacting field system can not be superfluously restricted in the number of physical degrees of freedom in order to remain physical,
•	Together, they naturally lead to the gauge symmetry emergence (GSE) conjecture which I will follow throughout the paper: Let there be given an interacting field system containing some vector field (or vector field multiplet) A^ together with fermion (^), scalar (§) and other matter fields in an arbitrary relativistically invariant Lagrangian	...) which possesses only global Abelian or non-Abelian internal symmetry G. Suppose that an underlying relativistic invariance of this field system is spontaneously broken in terms of the "length-fixing" covariant constraint put on vector fields,
A^A^ = n2M2	(4.1)
(where M stands for the proposed SLIV scale, while n^ is a properly-oriented unit Lorentz vector, n2 = n^n^ = ±1). If this constraint is preserved under the time development given by the field equations of motion, then in order to be protected from further reduction in degrees of freedom this system will modify its global symmetry G into a local symmetry Gloc, that will in turn convert the vector field constraint itself into a gauge condition thus virtually resulting in gauge invariant and Lorentz invariant theory.
To see how technically a global internal symmetry may be converted into a local one, let us consider in some detail the question of consistency of a possible constraint for a general 4-vector field A^ with its equation of motion in an Abelian symmetry case, G = U(1). In the presence of the SLIV constraint C(A) = A^A^ - n2M2 = 0 (4.1), it follows that the equations of motion can no longer be independent. The important point is that, in general, the time development would not preserve the constraint. So the parameters in the starting Lagrangian have to be adjusted in such a way that effectively we have one less equation of motion for the vector field A^ not to be superfluously restricted. This means that there should be some relationship given by a functional equation F(C = 0; ea,e^, ...) = 0 between all the vector and matter field Eulerians involved1 which are individually satisfied on the mass shell. According to Noether's second theorem [10] such a relationship gives rise to an emergence of local symmetry for the field system considered provided that the functional F satisfies the same symmetry requirements of Lorentz and translational invariance, as well as all the global internal symmetry requirements, as the general starting Lagrangian does.
In this way, the nonlinear SLIV condition (4.1), due to which true vacuum in the theory is chosen and massless gauge fields are generated, may provide a dynamical setting for all underlying internal symmetries involved through the GSE conjecture [9]. One might think that the length-fixing vector field constraint (4.1) itself first introduced by Nambu in a conventional QED framework [11] (for some extensions and generalizations, see also [12-17]) does not especially stand out in the present context. Actually, it seems that the GSE conjecture might be equally formulated for any type of covariant constraint, say for the spin-1 vector field condition,	= 0 [18]. However, as is generally argued in [9], the
SLIV constraint (4.1) appears to be the only one whose application leads to a full conversion of an internal global symmetry G into a local symmetry Gloc that forces a given field system to remain always physical. Other constraints could only lead to partial gauge invariance being broken by some terms in an emerging theory.
Based upon the SLIV constraint (4.1), the starting vector field A^ may be expanded around the true vacuum configuration in the theory,
A^ = a^ + n^M2 - n2a2 , n^ = 0 (a2 = a^a^) ,	(4.2)
which means that it develops the vacuum expectation value (VEV) (A^) = n^M. Meanwhile, its a^ components which are orthogonal to the Lorentz violating direction n^ describe a massless vector NG boson being an eventual gauge field (photon) candidate.
4.1.2 Gauge invariance versus spontaneous Lorentz violation
One can see that the gauge theory framework, be it taken from the outset or emerged, makes in turn spontaneous Lorentz violation to be physically unobserv-
1 The field Eulerians (EA, E^, ...) are determined, as usual, (EA)^ = 3L/3A^ — 3V[3L/3(3VA^)], and so forth.
able both in Abelian and non-Abelian symmetry case. In substance, the essential part of the SLIV pattern (4.2), due to which the vector field A^(x) develops the VEV M, may itself be treated as a pure gauge transformation with a gauge function linear in coordinates, d(x) = n^x^M. This is what one could refer to as the generic non-observability of SLIV in gauge invariant theories. I shall call it the "inactive" SLIV in contrast to the "active" SLIV case where physical Lorentz invariance could effectively occur. From the present standpoint, the only way for an active SLIV to occur would be if emergent gauge symmetries presented above were slightly broken at small distances. This could inevitably happen, for example, in a partially gauge invariant theory which might appear if the considered field system could become "a little unphysical" at distances being presumably controlled by quantum gravity [19]. One may think that quantum gravity could in principle hinder the setting of the required IVC in the appropriate Cauchy problem (thus admitting a superfluous restriction of vector fields) due to the occurrence of some gauge-noninvariant high-order operators near the Planck scale. As a consequence, through special dispersion relations appearing for matter and gauge fields, one is led a new class of phenomena which could be of distinctive observational interest in particle physics and astrophysics. They include a significant change in the Greizen-Zatsepin-Kouzmin cutoff for ultra-high energy cosmic-ray nucleons, stability of high-energy pions and W bosons, modification of nucleon beta decays, and some others just in the presently accessible energy area in cosmic ray physics [19] (for many phenomenological aspects, see pioneering works [20,21]).
4.1.3 SUSY profile of emergent theories
The role of Lorentz invariance may change, and its spontaneous violation may not be the only reason why massless photons and other gauge fields could dynamically appear, if spacetime symmetry is further enlarged. In this connection, special interest is related to supersymmetry which has made a serious impact on particle physics in the last decades (though has not been yet discovered). Actually, as we will see, the situation is changed dramatically in the SUSY inspired emergent gauge theories. In sharp contrast to non-SUSY analogs, it appears that the spontaneous Lorentz violation caused by an arbitrary potential of vector superfield V(x, 6, 6) never goes any further than some nonlinear gauge condition put on its vector field component A^(x) associated with a photon or any other gauge field. Remarkably, this condition coincides, as we shall see below, with the SLIV constraint (4.1) given above in the GSE conjecture. This allows to think that physical Lorentz invariance is somewhat protected by SUSY, thus only requiring the "condensation" of the gauge degree of freedom in the vector field A^. The point is, however, that even in the case when SLIV is not physical it inevitably leads to the generation of massless photons as vector NG bosons provided that SUSY itself is spontaneously broken. In this sense, a generic trigger for massless photons to dynamically emerge happens to be spontaneously broken supersymmetry rather than physically manifested Lorentz noninvariance.
While there are many papers in the literature on Lorentz noninvariant extensions of supersymmetric models (for some interesting ideas, see [22,23] and
references therein), an emergent gauge theory in a SUSY context has only recently been introduced [9,24]. Actually, the situation was shown to be seriously changed in a SUSY context which certainly disfavors some emergent models considered above. It appears that, while the constraint-based models of an inactive SLIV successfully matches supersymmetry, the composite and potential-based models of an active SLIV leading to physical Lorentz violation cannot be conceptually realized in the SUSY context. The reason is that, in contrast to an ordinary vector field theory where all kinds of polynomial terms	(n = 1,2,...) can be included
into the Lagrangian in a Lorentz invariant way, SUSY theories only admit the bilinear mass term A^A^ in the vector field potential energy. As a result, without a stabilizing high-linear (at least quartic) vector field terms, the potential-based SLIV never occurs in SUSY theories. The same could be said about composite models [2-4] as well: a fundamental Lagrangian with multi-fermi current-current interactions can not be constructed from any matter chiral superfields. So, all the models mentioned above, but the constraint-based models determined by the GSE conjecture (4.1), are ruled out in the SUSY framework and, therefore, between the two basic SLIV versions, active and inactive, SUSY unambiguously chooses the inactive SLIV case.
4.1.4 Outline of the paper
The paper is organized in the following way. In the next section 2 I consider supersymmetric QED model extended by an arbitrary polynomial potential of massive vector superfield that breaks gauge invariance in the SUSY invariant phase. However, the requirement of vacuum stability in such class of models makes both supersymmetry and Lorentz invariance to become spontaneously broken. As a consequence, the massless photino and photon appear as the corresponding Nambu-Goldstone zero modes in an emergent SUSY QED, and also a special gauge invariance is simultaneously generated. Due to this invariance all observable relativistically noninvariant effects appear to be completely cancelled out and physical Lorentz invariance is recovered. Further in section 3, all basic arguments developed in SUSY QED are generalized successively to the Standard Model and Grand Unified Theories (GUTs). For definiteness, I focus on the U(1) x SU(N) symmetrical theories. Such a split group form is dictated by the fact that in the pure non-Abelian symmetry case one only has the SUSY invariant phase in the theory that makes it inappropriate for an outgrowth of an emergence process. As possible realistic realizations, the Standard Model case with the electroweak U(1) x SU(2) symmetry and flipped SU(5) GUT including some immediate applications are briefly discussed. And finally in section 4, I summarize the main results and conclude.
The present talk is complimentary to my last year talk in Bled [25]. Some more detail can also be found in the recent extended paper [9].
4.2 Emergent SUSY theories: a QED primer
In contrast to attempts simply probing physical Lorentz noninvariance through some SM extensions [8,20] with hypothetical external vector (tensor) field back-
grounds originated around the Planck scale, we will principally focus here on a spontaneous Lorentz violation in an ordinary Standard Model framework itself. Particularly, we will try to extend an emergent SM with electroweak bosons appearing as massless vector NG modes to their supersymmetric analogs [9,24]. Such theories seem to open a new avenue for exploring the origin of gauge symmetries. Indeed, as I discussed at the previous workshop [25], the emergent SUSY theories, in contrast to the non-SUSY ones, could naturally have some clear observational signature. Actually, we have seen above that ordinary emergent gauge theories are physically indistinguishable from the conventional ones unless gauge invariance becomes broken being caused by some high-dimension couplings. Meanwhile, their SUSY counterparts - supersymmetric QED, SM and GUT - can be experimentally verified in another way. The point is that they generically emerge only if supersymmetry is spontaneously broken in a visible sector in order to ensure stability of the underlying theory. Therefore, the verification of emergent theories is now related to an inevitable emergence of a goldstino-like photino state in the SUSY particle spectrum at low energies, while physical Lorentz invariance may be still left intact.
4.2.1 Spontaneous SUSY violation
Since gauge invariance is not generically assumed in an emergent approach, all possible gauge-noninvariant couplings could in principle occur in the theory in a pre-emergent phase. The most essential couplings, as I discussed earlier [25], appear to be the vector field self-interaction terms triggering an emergence process in non-SUSY theories. Starting from this standpoint, I consider a conventional su-persymmetric QED being similarly extended by an arbitrary polynomial potential of a general vector superfield V(x, 6,6) which in the standard parametrization [26] has a form
V(x, 6, 6) = C + i6x - i6x + 266S - ^S*
26
-6o^6A^ + i666A/ - i666A/ + ^6666D/,	(4.3)
where its vector field component A^ is usually associated with a photon. Note that, apart from an ordinary photino field A and an auxiliary D field, the superfield (4.3) contains in general some additional degrees of freedom in terms of the dynamical C and x fields and nondynamical complex scalar field S (I have used the brief notations, A/ = A + 2ct^X and D/ = D + 192C with ct^ = (1,—) and o^ = (1, )). The corresponding Lagrangian can be written as
L = lsqed + 2d2 + X bkVklo	(4.4)
k=1
where, besides a standard SUSY QED part, new potential terms are presented in the sum by corresponding D-term expansions Vk|D of the vector superfield (4.3) into the component fields (bk are some constants). It can readily be checked that the first term in this expansion is the known Fayet-Iliopoulos D-term, while other
terms only contain bilinear, trilinear and quartic combination of the superfield components A^, S, A and x, respectively.
Actually, the higher-degree terms only appear for the scalar field component C(x). Expressing them all in terms of the C field polynomial
P(C) = X ybkCk-1 (x)	(4.5)
2
k=1
and its first three derivatives
9P	92P	93P
P' = — P'' = d P P'" = d P	(4 6)
Pc " 9C ' Pc " 9C2 ' Pc " 9C3	(6)
one has for the whole Lagrangian L
L = Lsqed + ¿D2 + P(D + ¿92C
+P'c ( ¿SS* - xA' - XA' - ^A^
1 (i_„ i	,,_„ A 1
+ 2PC (jXXS - ¿XXS* - X^XA^J + 8PC'(XXXX) . (4.7)
As one can see, extra degrees of freedom related to the C and x component fields in a general vector superfield V(x, 6, 6) appear through the potential terms in (4.7) rather than from the properly constructed supersymmetric field strengths, as appear for the vector field A^ and its gaugino companion À.
Note that all terms in the sum in (4.4) except Fayet-Iliopoulos D-term explicitly break gauge invariance. However, as we will see later in this section, the special gauge invariance constrained by some gauge condition will be recovered in the Lagrangian in the broken SUSY phase. Furthermore, as is seen from (4.7), the vector field A^ may only appear with bilinear mass term in the polynomially extended superfield Lagrangian (4.4) in sharp contrast to the non-SUSY theory case where, apart from the vector field mass term, some high-linear stabilizing terms necessarily appear in a similar polynomially extended Lagrangian. This means in turn that physical Lorentz invariance is still preserved. Actually, only supersymmetry appears to be spontaneously broken in the theory.
Indeed, varying the Lagrangian L with respect to the D field we come to
D = -P(C)	(4.8)
that finally gives the following potential energy for the field system considered
1 i
U(C) = 2[P(C)]2 .	(4.9)
The potential (4.9) may lead to spontaneous SUSY breaking in the visible sector provided that the polynomial P (4.5) has no real roots, while its first derivative has,
P = 0 , P'c = 0.	(4.10)
This requires P(C) to be an even degree polynomial with properly chosen coefficients bk in (4.5) that will force its derivative PC to have at least one root, C = C0, in which the potential (4.9) is minimized. Therefore, supersymmetry is spontaneously broken and the C field acquires the VEV
(C) = Co , PC(Co)= 0 .	(4.11)
As an immediate consequence, that one can readily see from the Lagrangian L (4.7), a massless photino A being Goldstone fermion in the broken SUSY phase make all the other component fields in the superfield V(x, 6,6) including the photon to also become massless. However, the question then arises whether this masslessness of the photon will be stable against radiative corrections since gauge invariance is explicitly broken in the Lagrangian (4.7). I show below that it could be the case if the vector superfield V(x, 6,6) would appear properly constrained.
4.2.2 Instability of superfield polynomial potential
Let us first analyze possible vacuum configurations for the superfield components in the polynomially extended QED case taken above. In general, besides the "standard" potential energy expression (4.9) determined solely by the scalar field component C(x) of the vector superfield (4.3), one also has to consider other field component contributions into the potential energy. A possible extension of the potential energy (4.9) seems to appear only due to the pure bosonic field contributions, namely due to couplings of the vector and auxiliary scalar fields, A^ and S, in (4.7)
It 1
Utot = ¿P2 + 2PC (A^ - SS*)	(4.12)
rather than due to the potential terms containing the superfield fermionic components. It can be immediately seen that these new couplings in (4.12) can make the potential unstable since the vector and scalar fields mentioned may in general develop any arbitrary VEVs. This happens, as emphasized above, due the fact that their bilinear term contributions are not properly compensated by appropriate four-linear field terms which are generically absent in a SUSY theory context.
4.2.3 Stabilization of vacuum by constraining vector superfield
The only possible way to stabilize the theory seems to seek the proper constraints on the superfield component fields (C, A^, S) themselves rather than on their expectation values. This will be done again through some invariant Lagrange multiplier coupling simply adding its D term to the above Lagrangian (4.4, 4.7)
1 i
Ltot = L + ^A(V- Co)2|d ,	(4.13)
where A(x, 6,6) is some auxiliary vector superfield, while C0 is the constant background value of the C field which minimizes the potential U (4.9). Accordingly, the potential vanishes for the supersymmetric minimum or acquires some positive
value corresponding to the SUSY breaking minimum (4.10) in the visible sector. I shall consider both cases simultaneously using the same notation Co for either of the background values of the C field.
Writing down the Lagrange multiplier D term in (4.13) through the component fields
CA, XA, sa, Aa, A;, = Aa + 2^Xa, Da = Da + 1a2cA (4.14)
and varying the whole Lagrangian (4.13) with respect to these fields one finds the constraints which appear to put on the V superfield components [25]
C = Co, X = 0, A^A^ = SS*.	(4.15)
They also determine the corresponding D-term (4.8), D = —P(C0), for the spontaneously broken supersymmetry. As usual, I only take a solution with initial values for all fields (and their momenta) chosen so as to restrict the phase space to vanishing values of the multiplier component fields (4.14). This will provide a ghost-free theory with a positive Hamiltonian.
Finally, implementing the constraints (4.15) into the total Lagrangian Ltot (4.13,4.7) through the Lagrange multiplier terms for component fields, we come to the emergent SUSY QED appearing in the broken SUSY phase
DC
Lem = Lsqed + P(C)D + da(C — Co)2 — ca (A^ — SS*) . (4.16)
The last two term with the component multiplier functions CA and DA of the auxiliary superfield A (4.14) provide the vacuum stability condition of the theory. In essence, one does not need now to postulate from the outset gauge invariance for the physical SUSY QED Lagrangian Lsqed. Rather, one can derive it following the GSE conjecture (section 1.1) specified for Abelian theory. Indeed, due to the constraints (4.15), the Lagrangian Lsqed is only allowed to have a conventional gauge invariant form
1	-It
Lsqed = — ^F^ + iAo^A + ^D2	(4.17)
Thus, for the constrained vector superfield involved
v(x, e, e) = C0 + 2ees — 2ees* — eo-^eA^ + ieeeA — ieeeA + leeeeD, (4.18)
we have the almost standard SUSY QED Lagrangian with the same states - a photon, a photino and an auxiliary scalar D field - in its gauge supermultiplet, while another auxiliary complex scalar field S gets only involved in the vector field constraint in (4.15). The linear (Fayet-Iliopoulos) D-term with the effective coupling constant P(C0) in (4.16) shows that supersymmetry in the theory is spontaneously broken due to which the D field acquires the VEV, D = —P(C0). Taking the nondynamical S field in the constraint (4.15) to be some constant background field we come to the SLIV constraint (4.1) underlying the GSE conjecture. As is seen from this constraint in (4.16), one may only have the time-like SLIV in a SUSY
framework but never the space-like one. There also may be a light-like SLIV, if the S field vanishes2. So, any possible choice for the S field corresponds to the particular gauge choice for the vector field A^ in an otherwise gauge invariant theory. So, the massless photon appearing first as a companion of a massless photino (being a Goldstone fermion in the visible broken SUSY phase) remains massless due to this recovering gauge invariance in the emergent SUSY QED. At the same time, the "built-in" nonlinear gauge condition in (4.16) allows to treat the photon as a vector Goldstone boson induced by an inactive SLIV.
4.3 On emergent SUSY Standard Models and GUTs
4.3.1 Potential of Abelian and non-Abelian vector superfields
Now, we extend our discussion to the non-Abelian internal symmetry case given by some group G with generators tp
[tp,tq] = ifpqrtr , Tr(tptq) = Spq (p,q,r = 0,1,..., Y - 1) (4.19)
where fPqT stand structure constants, while Y is a dimension of the G group. This case may correspond in general to some Grand Unified Theory which includes the Standard Model and its possible extensions. For definiteness, I will be further focused on the U(1) x SU(N) symmetrical theories, though any other non-Abelian group in place of SU(N) is also admissible. Such a split group form is dictated by the fact that in the pure non-Abelian symmetry case supersymmetry does not get spontaneously broken in a visible sector that makes it inappropriate for an outgrowth of an emergence process3. So, the theory now contains the Abelian vector superfield V, as is given in (4.3), and non-Abelian superfield multiplet Vp
yp(x, e, e) = cp + iexp - iexp + ^eesp - ^ees**
+ ieeeX^ - ieeeA'p + 1eeeeD/p, (4.20)
where its vector field components A£ are usually associated with an adjoint gauge field multiplet, (A^j = (Aptp j (i, j, k = 1,2,..., N ; p, q, r = 1,2,..., N2 -1). Note that, apart from the conventional gaugino multiplet Ap and the auxiliary fields Dp, the superfield Vp contains in general the additional degrees of freedom in terms of the dynamical scalar and fermion field multiplets Cp and xp and nondynamical complex scalar field Sp. Note that for the non-Abelian superfield components I use hereafter the bold symbols and take again the brief notations, A'p = Ap + 2 and D p = Dp + 232Cp.
2	Indeed, this case, first mentioned in [11], may also mean spontaneous Lorentz violation with a nonzero VEV < A^ > = (M, 0, 0, M) and Goldstone modes A1,2 and (A0 + A3 )/2 -M. The "effective" Higgs mode (A0 - A3 )/2 can be then expressed through Goldstone modes so as the light-like condition A^A^ = 0 to be satisfied.
3	In principle, SUSY may be spontaneously broken in the visible sector even in the pure non-Abelian symmetry case provided that the vector superfield potential includes some essential high-dimension couplings.
Augmenting the SUSY and U(1) x SU(N) invariant GUT by some polynomial potential of vector superfields V and Vp one comes to
11
L = Lsgut + ^D2 +1 DpDp + [£V + biV3/3 + b2V(VV) + ba(VVV)/3]D (4.21)
where £ and bi,2,3 stand for coupling constants, and the last term in (4.21) contains products of the Abelian superfield V and the adjoint SU(N) superfield multiplet VJ = (Vptp) •. The round brackets denote hereafter traces for the superfield V|
(VV...) = Tr(VV...)	(4.22)
and its field components (see below). For simplicity, we restricted ourselves to the third degree superfield terms in the Lagrangian L to eventually have a theory at a renormalizible level. Furthermore, I have only taken the odd power superfield terms that provides, as we see below, an additional discrete symmetry of the potential with respect to the scalar field components in the V and Vp superfields
C^ - C, Cp-> - Cp.	(4.23)
Finally, eliminating the auxiliary D and Dp fields in the Lagrangian L we come to the total potential for all superfield bosonic field components written in terms of traces mentioned above (4.22)
Utot = U(C, C) + 1b! C(A^ - SaSa) + 2 b2C[(A^) - (SaSa)j
+1b2[A^(A^C) - Sa(SaC)] + 2b3[(A^C) - (SaSaC)] . (4.24)
Note that the potential terms depending only on scalar fields C and Cj = (Cata)j are collected in
U (C, C) = 1 [£+bi C2 + b2(CC)]2 +1 [b2C2(CC)+ b2b3C(CCC) + ^(CCCC)] 8	2	4
(4.25)
and complex scalar fields Sa and S£ (a = 1,2) are now taken in the real field basis like as
Si =(S + S*)/2, S2 = (S - S*)/2i,	(4.26)
an so on. One can see that all these terms are invariant under the discrete symmetry (4.23), whereas the vector field couplings in the total potential Utot (4.24) break it. However, they vanish when the V and Vp superfields are properly constrained that we actually confirm in the next section.
Let us consider first the pure scalar field potential U (4.25). The corresponding extremum conditions for C and Ca fields are,
UC = bi(£ + bi C2)C + b2(bi - 2b2)C(CC) = 0, Tr(UC t) = 3b2C(CC)+ b3(CCC) = 0,	(4.27)
respectively. As shows the second partial derivative test, the simplest solution to the above equations
Co = 0 , C| = 0	(4.28)
provides, under conditions put on the potential parameters,
£, b >0 , b2 > 0 or £, bi <0 , b2 < 0	(4.29)
its global minimum
1
U(C, C)msn = ^£2 .	(4.30)
This minimum corresponds to the broken SUSY phase with the unbroken internal symmetry U (1) x SU (N) that is just what one would want to trigger an emergence process. This minimum appears in fact due to the Fayet-Iliopoulos linear term in the superfield polynomial in (4.21). As can easily be confirmed, in absence of this term, namely, for £ = 0 and any arbitrary values of all other parameters, there is only the SUSY symmetrical solution with unbroken internal symmetry
u (c, omr=0.	(4.31)
Interestingly, the symmetrical solution corresponding to the global minimum (4.31) may appear for the nonzero parameter £ as well
C0±] = ±V—£/bi, Cj = 0	(4.32)
provided that
£bi <0 .	(4.33)
However, as we saw in the QED case, in the unbroken SUSY case one comes to the trivial constant superfield when all factual constraints are included into consideration [25] and, therefore, this case is in general of little interest.
4.3.2 Constrained vector supermultiplets
Let us now take the vector fields A^ and into consideration that immediately reveals that, in contrast to the pure scalar field part (4.25), U(C, C), the vector field couplings in the total potential (4.24) make it unstable. This happens, as was emphasized before, due the fact that bilinear term VEV contributions of the vector fields A^ and A£, as well as the auxiliary scalar fields Sa and S£, are not properly compensated by appropriate four-linear field terms which are generically absent in a supersymmetric theory framework.
Again, as in the supersymmetric QED case considered above, the only possible way to stabilize the ground state (4.28,4.29,4.30) seems to seek the proper constraints on the superfields component fields (C, Cp; A^, Ap; Sa, S£) themselves rather than on their expectation values. Provided that such constraints are physically realizable, the required vacuum will be automatically stabilized. This will be done again through some invariant Lagrange multiplier couplings simply adding their D terms to the above Lagrangian (4.21)
11 Ltot = L + ^A(V — C0)2|d + ^(VV)Id ,	(4.34)
where A(x, 6,6) and n(x, 6,6) are auxiliary vector superfields. Note that C0 presented in the first multiplier coupling is just the constant background value of the
C field for which the potential part U(C, C) in (4.24) vanishes as appears for the supersymmetric minimum (4.31) or has some nonzero value corresponding to the SUSY breaking minimum (4.30) in the visible sector.
I will consider both cases simultaneously using the same notation C0 for either of the potential minimizing values of the C field. The second multiplier coupling in (4.34) provides, as we will soon see, the vanishing background value for the non-Abelian scalar field, Ca = 0, due to which the underlying internal symmetry U(1) x SU(N) is left intact in both unbroken and broken SUSY phase. The Lagrange multiplier terms presented in (4.34) have in fact the simplest possible form that leads to some nontrivial constrained superfields V(x, 6, 6) and Vp (x, 6, 6). Writing down their invariant D terms through the component fields one finds the precisely the same expression as in the SUSY QED [25] case for the Abelian superfield V and the slightly modified one for the non-Abelian superfield Va
n(vv )|d = Cn
CD' + ( ISS* - xA' - xA' - 1A^
+ xn [2CA' + i(xS* + i^xA^)] + Xn [2CA' - i(xS - ix^A^)]

+ ^Sn ( CS* + -x^l + 2SM CS - 2xx
1
+ 2A£(CA^ - x^x) + 2Ain(Cx) + 2An(Cx) + ¿D^CC) (4.35)
where the pairly grouped field bold symbols mean hereafter the SU(N) scalar products of the component field multiplets (for instance, CD' = CpD'p, and so forth) and
Cn, xn, Sn, Aft, Af, = An + 2^Xn, D^ = Dn + 1d2Cn (4.36)
are the component fields of the Lagrange multiplier superfield n(x, 6, 6) in the standard parametrization (4.20).
Varying the total Lagrangian (4.34) with respect to the component fields of both multipliers, (4.14) and (4.36), and properly combining their equations of motion we find the constraints which appear to put on the V and Va superfields components [9]
C = Co, x = 0, A^ = SaSa,
Cp = 0, xp = 0, (A^) = (SaSa) , a = 1,2 .	(4.37)
As before in the SUSY QED case, one may only have the time-like SLIV in a supersymmetric U(1) x SU(N) framework but never the space-like one (there also may be a light-like SLIV, if the S and S fields vanish). Also note that we only take the solution with initial values for all fields (and their momenta) chosen so as to restrict the phase space to vanishing values of the multiplier component fields (4.14) and (4.36) that will provide a ghost-free theory with a positive Hamiltonian. Again, apart from the constraints (4.37), one has the equations of motion for all fields involved in the basic superfields V(x, 6,6) and Vp (x, 6, 6). With vanishing
multiplier component fields (4.14) and (4.36), as was proposed above, these equations appear in fact as extra constraints on components of the V and Vp superfields. Indeed, equations of motion for the S a, x and C fields, on the one hand hand, and for the S£, xp and Cp fields, on the other, are obtained by the corresponding variations of the total Lagrangian Ltot (4.34) including the potential (4.24). They are turned out to be, respectively,
SaCo = 0 , ACo = 0 , (£ + b1C2)Co = 0 ,
SaCo = 0 , ApCo = 0, b2[A^j - SaSa|] + b3[(A^)j - (S«Sa) j] =(4.38)
where the basic constraints (4.37) emerging at the potential U(C, C) extremum point (Co, CP = 0) have been also used for both broken and unbroken SUSY case. Note also that the equations for gauginos A and Ap in (4.38) are received by variation of the potential terms in (4.21) containing fermion field couplings
U = b1 C(xA/ + XA) + b2C[(xA/) + (XA)]
+2b2 [X(A/C) + X(A/C) + A/(xC) + A/(XC)]
+b3(xA/C) + (XA/C)] .	(4.39)
One can immediately see now that all equations in (4.38) but the last equation system turn to trivial identities in the broken SUSY case (4.28) in which the corresponding C field value appears to be identically vanished, Co = 0. In the unbroken SUSY case (4.32), this field value is definitely nonzero, Co = -^/b1, and the situation is radically changed. Indeed, as follows from the equations (4.38), the auxiliary fields S(x) and Sp, as well as the gaugino fields A(x) and Ap(x) have to be identically vanished. This causes in turn that the gauge vector fields field A^ and A£ should also be vanished according to the basic constraints (4.37). So, we have to conclude, as in the SUSY QED case, that the unbroken SUSY fails to provide stability of the potential (4.12) even by constraining the superfields V and Vp and, therefore, only the spontaneously broken SUSY case could in principle lead to a physically meaningful emergent theory.
4.3.3 Broken SUSY phase: an emergent U (1) x SU (N) theory
With the constraints (4.37) providing vacuum stability for the total Lagrangian Ltot (4.34) we eventually come to the emergent theory with a local U(1) x SU(N) symmetry that appears in the broken SUSY phase (4.28). Actually, implementing these constraints into the Lagrangian through the Lagrange multiplier terms for component fields one has
1	d	C
Lem = Lsgut + + "¡p(C - Co)2 - (A^ - SS*)
+D1 (CC)- (A^ - SS*)	(4.40)
with the multiplier component functions Cp and Dp of the auxiliary superfield A (4.14) and component functions Cn and Dn of the auxiliary superfield n (4.36)
presented in the Lagrangian (4.34). Again, with these constraints and the GSE conjecture (section 1.1) specified for non-Abelian theories, one does not need to postulate gauge invariance for the physical SUSY GUT Lagrangian LSGUT from the outset. Instead, one can derive it starting from an arbitrary relativistically invariant theory. Indeed, even if the Lagrangian LSGUT is initially taken to only possess the global U(1) x SU(N) symmetry it will tend to uniquely acquire a standard gauge invariant form
1	- 1 T
lsgut = -	+ iAa^A + ^D2
- 1 FppvFPv + iApaPDpAp + 1 DpDp	(4.41)
where the conventional gauge field strengths for both U(1) and SU(N) part and terms with proper covariant derivatives for gaugino fields Ap necessarily appear [9]. Again as in the pure Abelian case, for the respectively constrained vector su-perfields V and Vp we come in fact to a conventional SUSY GUT Lagrangian with a standard gauge supermultiplet containing gauge bosons Ap and Ap, gauginos A and Ap, and auxiliary scalar D and Dp fields, whereas other auxiliary scalar fields Sa and Sp get solely involved in the Lagrange multiplier terms (4.41). Actually, the only remnant of the polynomial potential of vector superfields V and Vp (4.21) survived in the emergent theory (4.40) appears to be the Fayet-Iliopoulos D-term which shows that supersymmetry in the theory is indeed spontaneously broken and the D field acquires the VEV, D = -2£,.
Let us show now that this theory is in essence gauge invariant and the constraints (4.37) on the field space appearing due to the Lagrange multiplier terms in (4.34) are consistent with supersymmetry. Namely, as was argued in [25] (see also [9]), though constrained vector superfield (4.18) in QED is not strictly compatible with the linear superspace version of SUSY transformations, its supermultiplet structure can be restored by appropriate supergauge transformations. Following the same argumentation, one can see that similar transformations keep invariant the constraints (4.37) put on the vector fields Ap and Ap. Leaving aside the U(1) sector considered in [25] in significant details, I will now focus on the SU(N) symmetry case with the constrained superfield Vp transformed as
vp^ vp + 2 (n - n*)p	(4.42)
The essential part of this transformation which directly acts on the vector field constraint
ApApp = SpS*p	(4.43)
has the form
Vp -> Vp + i66Fp - ^99Fp - 6ape9p^p	(4.44)
where the real and complex scalar field components, and Fp, in a chiral super-field parameter Op are properly activated. As a result, the corresponding vector and scalar component fields, Ap and S£, in the constrained supermultiplet Vp transform as
AP -> ap = Ap - 9p^p, Sp -> sp = Sp + Fp .	(4.45)
One can readily see that our basic Lagrangian Lem (4.40) being gauge invariant and containing no the auxiliary scalar fields Sp is automatically invariant under either of these two transformations individually. In contrast, the supplementary vector field constraint (4.43), though it is also turned out to be invariant under supergauge transformations (4.45), but only if they act jointly. Indeed, for any choice of the scalar in (4.45) there can always be found such a scalar Fa (and vice versa) that the constraint remains invariant. In other words, the vector field constraint is invariant under supergauge transformations (4.45) but not invariant under an ordinary gauge transformation. As a result, in contrast to the Wess-Zumino case, the supergauge fixing in our case will also lead to the ordinary gauge fixing. We will use this supergauge freedom to reduce the scalar field bilinear SpS*p to some constant background value and find a final equation for the gauge function (x). It is convenient to come to real field basis (4.26) for scalar fields S£ and F£ (a = 1,2), and choose the parameter fields F£ as
Fa = raep(M + f), ras£ = 0, r2a = 1, epep = 1	(4.46)
so that the old S£ fields in (4.45) are related to the new ones s£ in the following
way
Sa = sa - raep(M + f), rasp = 0, SPSP = spsp + (M + f)2. (4.47)
where M is a new mass parameter, f (x) is some Higgs field like function, ra is again the two-component unit "vector" chosen to be orthogonal to the scalar s£, while ep is the unit SU(N) adjoint vector. This parametrization for the old fields S£ formally looks as if they develop the VEV, (S£) = —raep M, due to which the related SO(2) x SU(N) symmetry would be spontaneously violated and corresponding zero modes in terms of the new fields s£ could be consequently produced (indeed, they they never appear in the theory). Eventually, for an appropriate choice of the Higgs field like function f (x) in (4.47)
f = — M + VM2 — sasa	(4.48)
we come in (4.43) to the condition
ApAp^ = M2 .	(4.49)
leading, as in the QED U(1) symmetry case [25], exclusively to the time-like SLIV.
Remarkably, thanks to a generic high symmetry of the constraint (4.49) one can apply the emergence conjecture with dynamically produced massless gauge modes to any non-Abelian internal symmetry case as well, though SLIV itself could produce only one zero vector mode. The point is that although we only propose Lorentz invariance SO(1,3) and internal symmetry U(1) x SU(N) of the Lagrangian Lem (4.40), the emerged constraint (4.49) possesses in fact a much higher accidental symmetry SO(Y, 3Y) determined by the dimension Y = N2 — 1 of the SU(N) adjoint representation to which the vector fields Ap belong4. This
4 Actually, a total symmetry even higher if one keeps in mind both constraints (4.1) and (4.49) put on the vector fields A^ and A£, respectively. As long as they are independent the related total symmetry is in fact SO(1,3) x SO(Y, 3Y) until it starts breaking.
symmetry is indeed spontaneously broken at a scale M leading exclusively to the time-like SLIV case, as is determined by the positive sign in the SUSY SLIV constraint (4.49). The emerging pseudo-Goldstone vector bosons may be in fact considered as candidates for non-Abelian gauge fields which together with the true vector Goldstone boson entirely complete the adjoint multiplet of the internal symmetry group SU(N). Remarkably, they remain strictly massless being protected by the simultaneously generated non-Abelian gauge invariance. When expressed in these zero modes, the theory look essentially nonlinear and contains many Lorentz and CPT violating couplings. However, as in the SUSY QED case, they do not lead to physical SLIV effects which due to simultaneously generated gauge invariance appear to be strictly cancelled out.
As in the pure QED case, one can calculate the gauge functions (x) comparing the relation between the old and new vector fields in (4.45) with a conventional SLIV parametrization for non-Abelian vector fields [9]
= a£ + npVM2 - n2a2 , np= 0 (a2 = ap(4.50) They are expressed through the non-Abelian Goldstone and pseudo-Goldstone
modes
= ep
d(n^)VM2 - n2a2 .	(4.51)
Here n^ is the unit Lorentz vector being analogous to the vector introduced in the Abelian case (4.2), which is now oriented in Minkowskian spacetime so as to be "parallel" to the vacuum unit np matrix. This matrix can be taken in the
"two-vector" form
np = n^ep , epep = 1	(4.52)
where ep is the unit SU(N) group vector belonging to its adjoint representation.
x
4.3.4 Some immediate outcomes
Quite remarkably, an obligatory split symmetry form U(1) x SU(N) (or U(1) x G, in general) of plausible emergent theories which could exist beyond the prototype QED case, leads us to the standard electroweak theory with an U(1) x SU(2) symmetry as the simplest possibility. The potential of type (4.21) written for the corresponding superfields requires spontaneous SUSY breaking in the visible sector to avoid the vacuum instability in the theory. Eventually, this requires the SLIV type constraints to be put on the hypercharge and weak isospin vector fields, respectively,
B^ = M2 , WPWP^ = M2 (p = 1,2,3).	(4.53)
These constraints are independent from each other and possess, as was generally argued above, the total symmetry SO(1,3) x SO(3,9) which is much higher than the actual Lorentz invariance and electroweak U(1) x SU(2) symmetry in the theory. Thanks to this fact, one Goldstone and three pseudo-Goldstone zero vector modes b^ and are generated to eventually complete the gauge multiplet of
the Standard Model
B| = b| + n^M2 - b^b1 , n^ = 0 ,
W = w| + n^e^M2 - w^wqi, n^wpi = 0	(4.54)
where the unit vectors n^ and ep are defined in accordance with a rectangular unit matrix n| taken in the two-vector form (4.52). The true vector Goldstone boson appear to be some superposition of the zero modes b^ and w|. This superposition is in fact determined by the conventional Higgs doublet in the model since just through the Higgs field couplings these modes are only mixed [19]. Thus, when the electroweak symmetry gets spontaneously broken an accidental degeneracy related to the total symmetry of constraints mentioned above is lifted. As a consequence, the vector pseudo-Goldstones acquire masses and only photon, being the true vector Goldstone boson in the model, is left massless. In this sense, there is not much difference for a photon in emergent QED and SM: it emerges as a true vector Goldstone boson in both frameworks.
Going beyond the Standard Model we unavoidably come to the flipped SU(5) GUT [27] as a minimal and in fact distinguished possibility. Indeed, the U(1) symmetry part being mandatory for emergent theories now naturally appears as a linear combination of a conventional electroweak hypercharge and another hypercharge belonging to the standard SU(5). The flipped SU(5) GUT has several advantages over the standard SU(5) one: the doublet-triplet splitting problem is resolved with use of only minimal Higgs representations and protons are naturally long lived, neutrinos are necessarily massive, and supersymmetric hybrid inflation can easily be implemented successfully. Also in string theory, the flipped SU(5) model is of significant interest for a variety of reasons. In essence, the above-mentioned natural solution to the doublet-triplet splitting problem without using large GUT representations is in the remarkable conformity with string theories where such representations are typically unavailable. Also, in weakly coupled heterotic models, the flipped SU(5) allows to achieve gauge coupling unification at the string scale 10i7 GeV if some extra vector-like particles are added. They are normally taken to transform in the 10 and 10 representations, that is easy to engineer in string theory.
So, supersymmetric emergent theories look attractive both theoretically and phenomenologically whether they are considered at low energies in terms of the Standard Model or at high energies as the flipped SU(5) GUTs being inspired by superstrings.
4.4 Summary
As we have seen above, spontaneous Lorentz violation in a vector field theory framework may be active as in the composite and potential-based models leading to physical Lorentz violation, or inactive as in the constraint-based models resulting in the nonlinear gauge choice in an otherwise Lorentz invariant theory. Remarkably, between these two basic SLIV versions SUSY unambiguously chooses
the inactive SLIV case. Indeed, SUSY theories only admit the bilinear mass term in the vector field potential energy. As a result, without a stabilizing quartic vector field terms, the physical spontaneous Lorentz violation never occurs in SUSY theories. Hence it follows that the composite and potential-based SLIV models can in no way be realized in the SUSY context. This may have far-going consequences in that supergravity and superstring theories could also disfavor such models in general.
Though, even in the case when SLIV is not physical it inevitably leads to the generation of massless photons as vector NG bosons provided that SUSY itself is spontaneously broken. In this sense, a generic trigger for massless photons to dynamically emerge happens to be spontaneously broken supersymmetry rather than physically manifested Lorentz noninvariance. To see how this idea might work we have considered supersymmetric QED model extended by an arbitrary polynomial potential of a general vector superfield that induces spontaneous SUSY violation in the visible sector, and gauge invariance gets broken as well. Nevertheless, the special gauge invariance is in fact recovered in the broken SUSY phase that universally protects the photon masslessness.
All basic arguments developed in SUSY QED were then generalized to Standard Model and Grand Unified Theories. Remarkably, thanks to a generic high symmetry of the length-fixing SLIV constraint (4.49) put on the vector fields the emergence conjecture with dynamically produced massless gauge modes can be applied to any non-Abelian global internal symmetry case due to which it gets converted into to the local one. For definiteness, we have focused above on the U(1) x SU(N) symmetrical theories. Such a split group form is dictated by the fact that in the pure non-Abelian symmetry case one only has the SUSY invariant phase in the theory that would make it inappropriate for an outgrowth of an emergence process. As we briefly discussed, supersymmetric emergent theories look attractive both theoretically and phenomenologically whether they are considered at low energies in terms of the Standard Model or at high energies as the flipped SU(5) GUTs inspired by superstrings.
However, their most generic manifestations, as I discussed here in Bled about a year ago [25] (for more details, see also [9]), is related to a spontaneous SUSY violation in the visible sector that seems to open a new avenue for exploring the origin of gauge symmetries. Indeed, the photino emerging due to this violation will be then mixed with another goldstino which stems from a spontaneous SUSY violation in the hidden sector. Eventually, it largely turns into light pseudo-goldstino whose physics seems to be of special interest. Such pseudo-Goldstone photinos might appear typically as the eV scale stable LSP or the electroweak scale long-lived NLSP, being accompanied by a very light gravitinos in both cases. Their observation could shed some light on an emergence nature of gauge symmetries.
Acknowledgments
Discussions with the participants of the Workshop "What Comes Beyond the Standard Models?" (21-28 July 2014, Bled, Slovenia) were very useful. This work
is partially supported by Georgian National Science Foundation (Contracts No. 31/89 and No. DI/12/6-200/13).
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Bled Workshops in Physics Vol. 15, No. 2
A
Proceedings to the 17th Workshop What Comes Beyond ... (p. 66) Bled, Slovenia, July 20-28, 2014

5 Some Potential Problems of OHe Composite Dark Matter
J.R. Cudella'*, M. Khlopovb'** and Q. Wallemacqa'***
a IFPA, Dep. AGO, Universite de Liege, Sart Tilman, 4000 Liege, Belgium b National Research Nuclear University "MEPHI" (Moscow Engineering Physics Institute), 115409 Moscow, Russia
Centre for Cosmoparticle Physics "Cosmion" 115409 Moscow, Russia
APC laboratory 10, rue Alice Domon et Lonie Duquet 75205 Paris Cedex 13, France
Abstract. Among composite-dark-matter scenarios, one of the simplest and most predictive is that of O-helium (OHe) dark atoms, in which a lepton-like doubly charged particle O is bound with a primordial helium nucleus, and is the main constituent of dark matter. This model liberates the physics of dark matter from many unknown features of new physics, and it demands a deep understanding of the details of known nuclear and atomic physics, which are still somewhat unclear in the case of nuclear interacting "atomic" shells. So far the model has relied on the dominance of elastic scattering of OHe with the matter. In view of the uncertainty in our understanding of OHe interaction with nuclei we study the opposite scenario, in which inelastic nuclear reactions dominate the OHe interactions with nuclei. We show that in this case all the OHe atoms bind with extra He nuclei, forming doubly charged O-beryllium ions, which behave like anomalous helium, causing potential problems with overabundance of anomalous isotopes in terrestrial matter.
Povzetek. Avtorji obravnavajo model, v katerem sestavljajo temno snov atomi O-helija (OHe), v katerih se novi lepton O z dvojnim nabojem veže z jedrom helija. Sila med jedrom in leptonom je tedaj elektromagnetna. Kljub preprostosti modela pa je izračun lastnosti takega atoma pri elasticnem in neelasticnem sipanju na obicajni snovi zahteven. Avtorji studirajo v tem prispevku neelasticno sipanje teh atomov na obicajni snovi s predpostavko, da je to dominanten prispevek temne snovi. Pokazzejo, da se tedaj pri sipanju OHe na helijevih jedrih veze OHe s helijem v O-berilij z dvema elektromagnetnima nabojema. Avtorji pridejo do zakljucka, da bi lahko tak model napovedal preveliko gostoto anomalnih izotopov na Zemlji.
5.1 Introduction
Direct searches for dark matter have produced surprising results. Since the DAMA collaboration observed a signal, several other collaborations seem to confirm an observation, while many others clearly rule out any detection. The current
* E-Mail: JR.Cudell@ulg.ac.be
** E-mail: Khlopov@apc.univ-paris7.fr
*** E-mail: Quentin.Wallemacq@ulg.ac.be
experimental situation is reviewed in [1]. This apparent contradiction comes from the analysis of the data under the assumption that nuclear recoil is the source of the signal.
Starting from 2006 it was proposed [2-6] that the signal may be due to a different source: if dark matter can bind to normal matter, the observations could come from radiative capture of thermalized dark matter, and could depend on the detector composition and temperature. This scenario naturally comes from the consideration of composite dark matter. Indeed, one can imagine that dark matter is the result of the existence of heavy negatively charged particles that bind to primordial nuclei.
Cosmological considerations imply that such candidates for dark matter should consist of negatively doubly-charged heavy 1 TeV) particles, which we call O , coupled to primordial helium. Lepton-like technibaryons, technileptons, AC-leptons or clusters of three heavy anti-U-quarks of 4th or 5th generation with strongly suppressed hadronic interactions are examples of such O particles (see [3-6] for a review and for references).
It was first assumed that the effective potential between OHe and a normal nucleus would have a barrier, preventing He and/or O from falling into the nucleus, allowing only one bound state, and diminishing considerably the interactions of OHe. Under these conditions elastic collisions dominate in OHe interactions with matter, and lead to a successful OHe scenario. The cosmological and astrophysical effects of such composite dark matter (dark atoms of OHe) are dominantly related to the helium shell of OHe and involve only one parameter of new physics — the mass of O . The positive results of the DAMA/NaI and DAMA/LIBRA experiments are explained by the annual modulations of the rate of radiative capture of OHe by sodium nuclei. Such radiative capture is possible only for intermediate-mass nuclei: this explains the negative results of the XENON100 experiment. 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 [5-10]. 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 atom and the decays of doubly positive charged techniparticles to pairs of same-sign leptons can explain the excess of high-energy cosmic-ray positrons, found in PAMELA and AMS02 experiments [11].
These astroparticle data can be fitted, avoiding many astrophysical uncertainties of WIMP models, for a mass of O ~ 1 TeV, which stimulates searches for stable doubly charged lepton-like particles at the LHC as a test of the composite-dark-matter scenario.
In this paper, we want to explore the opposite scenarion, in which OHe dark matter interacts strongly with normal matter: OHe is neutral, but a priori it has an unshielded nuclear attraction to matter nuclei. We first study some effects of inelastic collisions of OHe in the early Universe and in the terrestrial matter
and find that such collisions strongly increase the formation of charged nuclear species with O bound in them. Recombination of such charged species with electrons (even if it is partial) leads to the formation of atoms (or ions) of anomalous isotopes of helium and heavier elements. The atomic size of such atoms (or ions) of anomalous isotopes strongly suppresses their mobility in the terrestrial matter, making them stop near the surface, where anomalous superheavy nuclei are strongly constrained by the experimental searches. In Section 5.2 we study effect of inelastic processes during the period of Big Bang Nucleosynthesis and show that if these processes are not suppressed all the OHe atoms capture additional He nuclei, forming a doubly charged ion of O-beryllium (OBe). In Section 5.3 we briefly examine the problems of an OBe-dominated universe and show that, because the mobility of the anomalous isotopes is greatly suppressed even if they recombine with only one electron, their drift to the center of the Earth is strongly slowed down, and their abundance increases near the terrestrial surface and in the World Ocean with the danger of their overabundance. We stress the importance of solving the open questions of OHe nuclear physics in the Conclusion.
5.2 Inelastic processes with OHe in the early Universe
As soon as all the OHe is formed in the early Universe, inelastic processes between OHe and OHe itself and between OHe and the primordial He take place and start consuming the available OHe. The two relevant reactions are:
Note that in these reactions the addition of a He nucleus to the bound OHe system will result in merging the two He nuclei into 8 Be, since in the presence of O , 8 Be becomes stable: we calculated, as in Ref. [13], that the energy of OBe is 2.9 MeV smaller than that of OHe+He. The temperature T0 at which OHe forms depends on its binding energy, which has been accurately evaluated as 1.175 MeV in Ref. [13], and corresponds approximately to T0 = 50 keV. As the cosmological time t is related to the temperature through t(s) ~ T2 (MeV), processes (5.1) and (5.2) start at a time t0 ~ 0 052 = 400 s after the Big Bang and continue until helium freezes out at t* ~ 10 min = 600 s.
During these 200 s, the OHe atoms are consumed at a rate:
where nOHe and nHe are the number densities of OHe and He, H = 2t is the expansion rate of the Universe during the radiation-dominated era, c and c2 are the cross sections of processes (5.1) and (5.2) respectively and v1 and v2 are the OHe-OHe and OHe-He mean relative velocities. The first term in the right-hand side of equation (5.3) corresponds to the dilution in an expanding universe. The number of helium nuclei per comoving volume is assumed to be unaffected by
OHe + OHe -> O2Be OHe + He -> OBe
(5.1)
(5.2)
- = -3HnOHe - n.OHe°"1 V1 - nOHenHe°2V2,
(5.3)
reaction (5.2) since the abundance of helium is more than an order of magnitude higher than that of OHe, so that the only effect on nHe is due to the expansion:
dnHe =-3HnHe,	(5.4)
from which it follows that:
UHe(t)= j) Z '	(5.5)
where nHe is the number density of He at t = t0 (In the following, we shall use a superscript 0 to denote quantities taken at the time of the decoupling of OHe, t = to).
To take into account the effect of the expansion and calculate the decrease of the fraction of free OHe atoms due to their inelastic reactions, we study the ratio f of the number density of OHe atoms to the number desity of He nuclei, f = -—p. From (5.3) and (5.4), its evolution is given by:
df
— =-nHef (civi f + 0-2V2)	(5.6)
The capture cross sections ci and 02 are of the order of the geometrical ones:
ai « 4n (2rOHe)2 ,	(5.7)
02 « 4n (rOHe + rHe) ,	(5.8)
where rOHe is the Bohr radius of an OHe atom and rHe is the radius of a He nucleus. As both of them are approximately equal to 2 fm, oi « o2 « 64n 10-26 cm2. As the OHe and He species are in thermal equilibrium with the plasma at temperature T, the mean relative velocities v1 and v2 are obtained from the Maxwell-Boltzmann velocity distributions of OHe and He and are given by:
vi = >	(5.9)
V nm
v2 = \\	(5.10)
v
where m = mOHe/2 and ~ mHe are the reduced masses of the OHe-OHe and OHe-He systems. mOHe = 1000 GeV is the mass of an OHe atom, and mHe = 3.73 GeV that of a He nucleus. Given the time dependence of the temperature during the radiation-dominated era, Tt1/2 = T0tJ/2, one can use it to express the velocities (5.9) and (5.10) as functions of time and insert the resulting expressions together with (5.5) in equation (5.6) and get:
1 r	-I
- =-Yt7^f (af + P),	(5.11)
with
°"1
vw'
(5.12)
p = VJL,	(5.13)
vm2
0 .7/4 /8T0
y = nHeto Y — •	(5.14)
The solution of (5.11) corresponding to the initial condition f(t0) = f0 is given
by:
f(t) =-j--( 3/4 pfo XX-•	(5.15)
exp (f Py (t-3/4 - t"3/4JJ (afo + p) - a
The number density of He nuclei at the time of OHe formation, nHe, can be found from its value nHe today (In the following, the superscript 1 will denote quantities at the present time). Helium nuclei represent nowadays approximately 10% of all baryons, which have an energy density pB of about 5% of the critical density pC: nHe — C.lng = 0.1 mpB — 0.1 x 0.05P-, where mp is the mass of the proton. The present critical density is measured to be p^ = 5.67 x 10-6mp /cm3, so that nHe — 2.8 x 10-8 cm-3. As it was assumed that the He number density was not affected by reaction (5.2), the only effect between t0 and now has been a dilution due to the expansion, and hence nHe « « T3, where a is the scale factor. Knowing that the temperature of the CMB today is T1 = 2.7 K= 2.33 x 10-7 keV,
this gives nHe = nHe (T7) 3 — 2.8 x 10~^2.33^010_^3 — 2.8 x 1017 cm-3.
At the time of OHe formation, all the O particles were in the form of OHe, i.e. the number density of O at t = t0, nO, was equal to that of OHe, n0He. Between t0 and today, O particles may have been bound in different structures, but they have not been created or destroyed, so that their number density has only been diluted by the expansion in the same way as that of He nuclei, so that the ratio of the number density of O particles to the number density of He nuclei remains unchanged: nO = —0- Therefore, the initial fraction f0 of OHe atoms can
be calculated from present quantities: f0 = -—p- = —0 = —r. nO is obtained from the fact that O-- saturates the dark matter energy density, which represents about 25% of the critical density: nO — 0.25p — 1.3 x 10-9 cm-3, where m0 = 1 TeV is the mass of O . With the previously calculated value of nHe, this gives f0 — 0.05.
We can now insert the numerical values into Eq. 5.15 and get the fraction of OHe atoms at the time of helium freeze-out t = t* = 600 s:
f(t*) — 5 x 10-6133 < f0,	(5.16)
meaning that no OHe survives reactions (5.1) and (5.2). More precisely, most of the OHe atoms have captured He nuclei via process (5.2) and are now in the form of OBe. Indeed, the majority of the suppression of f comes from the exponential term present in (5.15), evaluated to be e14127. The large argument of the exponential
a =
represents the number N2 of reactions (5.2) that happened between t0 and t*, per OHe atom:
N2 =
n-He(t)a-2V2(t)dt
8T0t,
0L0
1/2
0 .3/2 = nHet3 C2„ nn
0 +7/4 /8T0 CT 2 nHet0 '
4
4ßY
1
3/4
3/4
t
to
t7/4
dt

3/4

3/4
where we have used (5.5), (5.10) and Tt1/2 = T0t(!/2 to pass from the first to the second line and the definitions (5.13) and (5.14) for the last line.
Therefore, the realization of the scenario of an OHe Universe implies a very strong suppression of reaction (5.2), corresponding to N2 C 1. Such a suppression needs the development of a strong dipole Coulomb barrier in OHe-He interaction. The existence of this barrier and its effect is one of the most important open problems of the OHe model.
t
t
o
t

=*=
0
0
5.3 Problems of OBe "dark" matter
Due to Coulomb repulsion further helium capture by OBe is suppressed and one should expect that dark matter is mostly made of doubly charged OBe, which recombines with electrons in the period of recombination of helium at the temperature Tod = 2 eV, before the beginning of matter dominance at TRM = 1 eV. It makes anomalous helium the dominant form of dark matter in this scenario. After recombination the OBe gas decouples from the plasma and from radiation and can play the role of a specific Warmer than Cold dark matter, since the adiabatic damping slightly suppresses density fluctuations at scales smaller than the scale of the horizon in the period of He recombination. The total mass of the OBe gas within the horizon in that period is given by analogy with the case of OHe [2,5] by
Mod = T^mpi(m1 )2 « 2 • 1050 g = 1O17M0,	(5.17)
' od	T od
where M0 is the solar mass.
At momentum values of interest, one finds that elastic cross sections are significantly enhanced from their geometrical estimate. In the following, we shall use the estimate of Ref. [14], based on a compilation of results from general quantum mechanical scattering and from detailed quantum computations of hydrogen scattering [15]:
d = 4n(Kr0)2, k = 3 - 10,	(5.18)
with larger values of k at low momentum.
For a size of OBe atoms equal to that of helium r0 = 3 • 10-9 cm and one obtains an elastic scattering cross section on light elements of the order of c « 10-15 —10-14 cm2. It makes this "dark matter" follow the ordinary baryonic matter in the process of galaxy formation, and makes it collisional on the scale of galaxies. This causes problems with the explanation of the observations of halo shapes [16]. Presence of OBe in stars can also influence nuclear processes, in particular helium burning in the red giants. The processes in stars can lead to the capture by OBe of additional nuclei, thus creating anomalous isotopes of elements with higher Z. OBe atoms can also be ionized in the Galaxy, but in the following we shall assume that neutral OBe atoms are the dominant part of this "dark matter" on Earth, considering also that slowing down anomalous nuclei in the atmosphere leads to ionization and their neutralisation through electron capture.
Falling down on Earth OBe atoms are slowed down and due to the atomic cross section of their collisions have a very low mobility. After they fall down to the terrestrial surface, the OBe atoms are further slowed down by their elastic collisions with matter. They drift, sinking down towards the center of the Earth with velocity
V = — < 2.710-11 cm/ s « 270 fm/ s.	(5.19)
nov
Here n is the number density of terrestrial atoms, cv is the rate of atomic collisions, taken at room temperature, and g = 980 cm/s2. We assimilated the crust of the Earth as made of SiO2, and got the number density to be n = 0.27 1023 molecules/cm3. Using (5.18), and taking the geometrical radius to be that of SiO2, i.e. r0 « 2 A, we obtained c > 4.5 10 14 cm2, and for collisions on SiO2 v « 3 104cm/s.
The OBe abundance in the Earth is determined by the equilibrium between the in-falling and down-drifting fluxes. The in-falling O-helium flux from dark matter halo is given by [4]
f=nn ^+^
where Vh is the speed of the Solar System (220 km/s), VE the speed of the Earth (29.5 km/s) and n0 = 3 • 10-4 cm-3 is the assumed local density of OBe dark matter (for an OBe of mass 1 TeV). Furthermore, for simplicity, we didn't take into account the annual modulation of the incoming flux and take |Vh + VE| = u « 300 km/ s.
The equilibrium concentration of OBe, which is established in the matter consisting of atoms with number density n, is given by [4]
noE = ,	(5.20)
and the ratio of anomalous helium isotopes to the total amount of SiO2 is given by
noE 2n • Fcv . , , -i /\ — 9
roE =-=- > 3.1 10 ,	(5.21)
oE	n	g
being independent of the atomic number density of the matter. Note that the migration rate (and the dilution) considered here is of larger than that observed at the Oklo site for heavy elements [18].
The upper limits on the anomalous helium abundance are very stringent [17] roE < 10-19, and our rough estimate is ten orders of magnitude too large. Together with other problems of OBe Universe stipulated above, this rules out the OBe scenario.
5.4 Conclusion
The advantages of the OHe composite-dark-matter scenario is that it is minimally related to the parameters of new physics and is dominantly based on the effects of known atomic and nuclear physics. However, the full quantum treatment of this problem turns out to be rather complicated and remains an open.
We have considered here the scenario in which such a barrier does not appear. This leads to a significant role of inelastic reaction of OHe, and strongly modifies the main features of the OHe scenario. In the period of Big Bang Nucleosynthesis, when OHe is formed, it captures an additional He nucleus, so that the dominant' form of dark matter becomes charged, recombining with electrons in anomalous isotopes of helium and heavier elements. Over-abundance of anomalous isotopes in terrestrial matter seems to be unavoidable in this case.
This makes the full solution of OHe nuclear physics, started in [12], vital. The answer to the possibility of the creation of a dipole Coulomb barrier in OHe interaction with nuclei is crucial. Without that barrier one gets no suppression of inelastic reactions, in which O binds with nuclei. These charged species form atoms (or ions) with atomic cross sections, and that strongly suppresses their mobility in terrestrial matter, leading to their storage and over-abundance near the Earth's surface and oceans.
Hence, the model cannot work if no repulsive interaction appears at some distance between OHe and the nucleus, and the solution to this open question of OHe nuclear physics is vital for the composite-dark-matter OHe scenario.
Acknowledgements
The research of J.R.C. and Q.W. was supported by the Fonds de la Recherche Scientique - FNRS under grant 4.4501.05. Q.W. is also supported by the Fonds de la Recherche Scientifique - FNRS as a research Fellow. The work by M.Kh. on initial cosmological conditions was supported by the Ministry of Education and Science of Russian Federation, project 3.472.2014/K and his work on the forms of dark matter was supported by grant RFBR 14-22-03048.
References
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2.	M.Y. Khlopov and C. Kouvaris, Phys. Rev. D 78 (2008) 065040 [arXiv:0806.1191 [astro-ph]];
3.	M.Y. Khlopov, Pisma Zh. Eksp. Teor. Fiz. 83 (2006) 3 [JETP Lett. 83 (2006) 1] [astro-ph/0511796]; D. Fargion, M.Y. Khlopov and C. A. Stephan, Class. Quant. Grav. 23 (2006) 7305 [astro-ph/0511789]; M.Y. Khlopov and C.A. Stephan, [astro-ph/0603187]; M.Y. Khlopov and N.S. Mankoc Borstnik, ckwBWP 11 (2010) 177.
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Bled Workshops in Physics Vol. 15, No. 2
A
Proceedings to the 17th Workshop What Comes Beyond ... (p. 75) Bled, Slovenia, July 20-28, 2014

6 Notoph-Graviton-Photon Coupling
V.V. Dvoeglazov*
UAF, Universidad de Zacatecas
Apartado Postal 636, Suc. 3 Cruces, Zacatecas 98068, ZAC., Mexico
Abstract. In the sixties Ogievetskii and Polubarinov proposed the concept of notoph, whose helicity properties are complementary to those of photon. Later, Kalb and Ramond (and others) developed this theoretical concept. And, at the present times it is widely accepted. We analyze the quantum theory of antisymmetric tensor fields with taking into account mass dimensions of notoph and photon. It appears to be possible to describe both photon and notoph degrees of freedom on the basis of the modified Bargmann-Wigner formalism for the symmetric second-rank spinor.
Next, we proceed to derive equations for the symmetric tensor of the second rank on the basis of the Bargmann-Wigner formalism in a straightforward way. The symmetric multispinor of the fourth rank is used. It is constructed out of the Dirac 4-spinors. Due to serious problems with the interpretation of the results obtained on using the standard procedure we generalize it, and we obtain the spin-2 relativistic equations, which are consistent with the general relativity. The importance of the 4-vector field (and its gauge part) is pointed out.
Thus, we present the full theory which contains the photon, the notoph (the Kalb-Ramond field) and the graviton. The relations of this theory with the higher spin theories are established. In fact, we deduced the gravitational field equations from relativistic quantum mechanics. The relations of this theory with scalar-tensor theories of gravitation and f(R) are discussed. We estimate possible interactions, fermion-notoph, graviton-notoph, photon-notoph, and we conclude that they will be probably seen in experiments in the next few years.
Povzetek. V sestdesetih letih sta Ogievetskii in Polubarinov predlagala delec z lastnostmi, komplementarnimi fotonu. Poimenovala sta ga notof. Lastnosti delca notof sta kasneje studirala in dopolnila tudi Kalb in Ramond ter drugi. Zdaj lastnosti tega delca studirajo stevilni teoretiki.
Avtor prispevka analizira kvantno teorijo antisimetricnih tenzorskih polj in dopusti, da nosita foton in notof od nic razlicno maso. Izkazalo se je, da nekoliko spremenjen formalizem za simetricne spinorje ranga 2 avtorjev Bargmanna-Wignerja opise tudi foton in notof.
Uspe mu preprosta izpeljeva enacbe gibanja za simetricni tenzor ranga 2 iz Bargmann-Wignerjevega formalizma. Uporabi simetricni multispinor ranga stiri, ki ga izvede iz Diracovih stiri spinorjev. Standardni postopek posplosi, ker so tezave z interpretacijo. Posplositev ga pripelje do relativisticne enacbe za spin 2, ki je skladna s splosno teorijo relativnosti. Avtor poudari pomen polja 4-vektorjev (in njegovega umeritvenega dela).
Avtor predstavi celovito teorijo, ki vsebuje foton, notof (Kalb-Ramondovo polje) in graviton. Osvetli relacijo te teorije s teorijami z visjimi spini. Iz relativisticne kvantne
* E-mail: valeri@fisica.uaz.edu.mx
mehanike izpelje enacbe za gravitacijsko polje. Diskutira o povezavi te teorije s skalarno-tenzorskimi in teorijami gravitacije f (R) , oceni interakcije med fermionom in notofom, med gravitonom in notofom ter med fotonom in notofom ter napove, da bodo najbrz opazili te interakcije pri poskusih v prihodnjih nekaj letih.
6.1 Introduction
In the series of the papers [1-5], cf. with Refs. [6-8], we tried to find connection between the theory of the quantized antisymmetric tensor (AST) field of the second rank (and that of the corresponding 4-vector field) with the 2 (2s + 1) Weinberg-Tucker-Hammer formalism [9,10].
Several previously published works [11-16], introduced the concept of the notoph (the Kalb-Ramond field) which is constructed on the basis of the antisymmetric tensor "potentials". It represents itself the non-trivial spin-0 field. The well-known textbooks [17-20] did not discuss the problems, whether the massless quantized AST field and the quantized 4-vector field are transverse or longitudinal fields (in the sense if the helicity h = ±1 or h = 0)? can the electromagnetic potential be a 4-vector in a quantized theory? how should the massless limit be taken? and many other fundamental problems of the physics of bosons. In my opinion, the most rigorous works are refs. [22,9,23,21], but it is not easy to extract corresponding answers even from them.
First of all, we note that 1) "...In natural units (c = h = 1) ... a lagrangian density, since the action is dimensionless, has dimension of [energy]4"; 2) One can always renormalize the lagrangian density and "one can obtain the same equations of motion... by substituting L —» (1/MN)L, where M is an arbitrary energy scale", cf. [2]; 3) the right physical dimension of the field strength tensor Fpv is [energy]2; "the transformation Fpv —» (1/2M)Fpv [which was regarded in Ref. [5]]... requires a more detailed study ... [because] the transformation above changes its physical dimension: it is not a simple normalization transformation". Furthermore, in the first papers on the notoph [12-14]1 the authors used the normalization of the 4-vector Fp field2 to [energy]2 and, hence, the antisymmetric tensor "potentials" Apv, to [energy]1. We try to discuss these problems on the basis of the generalized Bargmann-Wigner formalism [22]. Thus, the Proca and Maxwell formalisms are generalized, see, e. g., Ref. [24].
In the Sections 6.3 and 6.4 we consider the spin-2 equations. The general scheme for derivation of higher-spin equations has been given in [22]. A field of the rest mass m and the spin s > 1 is represented by a completely symmetric multispinor of rank 2s. The particular cases s = 1 and s = | have been considered in the textbooks, e. g., Ref. [17]. The spin-2 case can also be of some interest because we can believe that the essential features of the gravitational field are obtained from transverse components of the (2,0) © (0,2) representation of the Lorentz group. Nevertheless, questions of the redundant components of the higher-spin relativistic equations are not yet understood in detail [25].
1	It is also known as the longitudinal Kalb-Ramond field, but the consideration of Ogievet-skil and Polubarinov permits to study the m —> 0 procedure.
2	It is well known that it is related to the third-rank antisymmetric field tensor.
In the last Sections we discuss the questions of interactions.
6.2 Photon-Notoph Equations
For spin 1 we start from3
[YapPaPp + Apapa + Bm2]¥ = 0,	(6.1)
where p^ = —and Yap are the Barut-Muzinich-Williams covariantly - defined 6 x 6 matrices, Y.^ Yp.^ = 0. The determinant of tyappapp + Apapa + Bm2] is of the 12th order in p If we are interested in solutions with E2 — p2 = m2, c = h = 1, they can be obtained on using the constraints in the above equation:
B = 1, T^T = 1.	(6.2)
A + 1 ' A - 1 We may also have the tachyonic solutions, etc. The particular cases are:
•	A = 0, B = 1 ^ we have the Weinberg's equation for s = 1 with 3 solutions
E = p2 + m2, 3 solutions E = -\Jp2 + m2, 3 solutions E = + \Jp2 - m2 and 3 solutions E = -\Jp2 - m2. Tachyonic solutions have been reformulated in various ways, for instance, as the ones leading to the spontaneous symmetry breaking, and to the non-zero quantum vacuum.
•	A = 1, B = 2 ^ we have the Tucker-Hammer equation for s = 1. The solutions are with E = p2 + m2 only.
Thus, the addition of the Klein-Gordon equation to (6.1) may change the physical content even on the free level.
What are the corresponding equations for the antisymmetric tensor field? They can be the Proca equations in the massive case, and the Maxwell equations in the massless case. We have shown in Refs. [1,2] that one can obtain four different equations for antisymmetric tensor fields from the Weinberg's 2(2s + 1 )-component formalism. First of all, we note that ¥ is, in fact, bivector, Et = -iF4t, Bi = 1 etjkFjk,, or Et = -2 etjkFjk, Bt = -iF4t, or their combinations. One can separate the four cases:
(E + iB\
•	¥(I) = i e iB J, P = -1, where Et and Bt are the components of the tensor.
B - iE
•	¥(n) = i - iE j, P = +1, where Et, Bt are the components of the tensor.
•	¥(III) = ¥(I), but (!) Et and Bt are the corresponding vector and axial-vector components of the dual tensor F|V.
•	¥<IV) = ¥(II), where Et and Bt are the components of the dual tensor F|V.
3 In the classic works on this formalism the authors worked in the Euclidean metrics. However, there is no any problem to write the equations and other formulas in the pseudo-Euclidean metrics accustomed today; just change the sign of p^p^, and other products.
The mappings of the WTH equations are:
9«9,F2 - + ^y19,9,?^ - ^F^ = 0,	(6.3)
9 9 F(II) - 9r9 F(n' - A + 1 9 9 F(II) + Bm2F(II) - 0	(64) d d F ("I' d d F (III) A + 1 d d F (III) + Bm2F (III) - 0
2 2	(6.5)
9«9,FiIV' - 9P9,F,V' + Ay19,9,FaV' - BBm2Fav' - 0. (6.6)
In the Tucker-Hammer case (A - 1, B - 2) we can recover the Proca theory from (6.3):
3«9,F,p - 3p3,F,a - m2Fap ,	(6.7)
(Av — mlr3«Fav should be substituted in F,v — 9,AV - 9VA,, and the result is multiplied by m2).
We also noted that the massless limit of this theory does not coincide with the Maxwell theory in some cases, while it contains the latter as a particular case. In [3,5,30] we showed that it is possible to define various massless limits for the Duffin-Kemmer-Proca theory. Another one is the Ogievetskii-Polubarinov notoph (which is also called the Kalb-Ramond field), Ref. [12] in the US literature. The transverse components of the AST field can be removed from the corresponding Lagrangian by means of the "new gauge transformation" A,v —» A,v + 3,AV -3VA,, with the vector gauge function A,. The second (II) case is
3«3,F,p - 3p3,F,a - [9,9, - m2]Fap.	(6.8)
So, on the mass shell we have [9,9, - m2]Fap - 0, and, hence,
3«9,F,p - 3p3,F,a - 0,	(6.9)
which rather corresponds to the Maxwell-like case. However, if we calculate dispersion relations for the second case, Eq. (6.9), it appears that the equation has solutions even if m - 0.
Now we are interested in the parity-violating equations for antisymmetric tensor fields. We investigate the most general mapping of the Weinberg-TuckerHammer formulation to the antisymetric tensor field formulation too. Instead of ^(I-IV' we shall try to use now
¥(A) - (B+iB) - 1^A(I)+^r5 ^(ii) . (6.10)
As a result, the equation for the AST fields is
1	A	B
3«9,F,p - 3p3,F,a - 1 (3,3,)Fap + [-(9,9,) + ^m2]Fap . (6.11)
6 Notoph-Graviton-Photon Coupling 79 , ^(A) _ /	\ _ _i^(A), and the equation is unchanged. The
Of course, .
, E
different choice is
¥(b) _ ( EB+_iBJ = i+lV) _ ¥(ij) . (6.12)
Thus, one has
1	A	B
-	= 2)F«p + [a(3^3^) - ^m2]?ap . (6.13)
Of course, one can also use the dual tensor (E1 = — 2 e^Fand B1 = —iF41) and obtain analogous equations:
1 _	r A,_ _ , B
3«3^F|p _ 3p3|F= 2ap + [_y (3^3^) + ]Fap ,
1	~ A	B
3«3^F|p _ 3p3^F_ 2(3^3^)Fap + [a(3^3^) _ ym^Fap .
(6.14)
(6.15)
They are connected with (6.11,6.13) by the dual transformations.
The states corresponding to the new functions ¥(A), ¥(B) etc are not the parity eigenstates. So, it is not surprising that we have Fap and its dual Fap in the same equations. In total we have already eight equations. One can also consider the most general case
■^(W) _ I aF4i + bF4i + ceijkFjk + de^F(6 16) _ VeF4i + fF4i + geijkFjk + heijkFjky .	(. )
So, we shall have dynamical equations for Fap and Fap with additional parameters a, b, c, d,... G C. We have a lot of antisymmetric tensor fields here. However,
•	the covariant form preserves if there are some restrictions on the parameters, only. Alternatively, we have some additional terms of 34 or V2;
•	both F|v and F |v are present in the equations;
•	under the definite choice of a, b, c, d... the equations can be reduced to the above equations for the tensor and its dual:
~	C3	C4	~
H|iv _ C1 F|iv + c2f|iv + ~2 e^v«pF«p + ~2 e|ivapFap ')	(6.17)
•	the parity properties of ¥(W) are very complicated.
Anther way of constructing the equations of high-spin particles has been given in [22,17].4 Bargmann and Wigner claimed explicitly that they constructed
4 On can also obtain the s = 0 Kemmer equations on using the Bargmann-Wigner procedure. One should use the antisymmetric second-rank multispinor in this case.
(2s + 1) states.5 Below we present the standard Bargmann-Wigner formalism for the spin s = 1 (and turn to the standard pseudo-Euclidean metric):
[iy^ - m]ap = 0,	(6.18)
[iy^ - m]yp = 0,	(6.19)
If one has with6
= (Y^+ (a^R)apF^ ,	(6.20)
R = -If-e) - = 0 o>	(6.25)
in the spinorial representation of Y-matrices, we obtain the Duffin-Kemmer-Proca equations:
3aF^ = yA^,	(6.26)
2mF^v = - .	(6.27)
In order to obtain these equations one should add the equations (6.18,6.19) and compare functional coefficients at the corresponding commutators, see Ref. [17]. After the corresponding re-normalization A^ —» 2mA^ (or F^v —» (1/2m)F^v), we obtain the standard textbook set:
3aF^ = m2A,,	(6.28)
F^v = Av - 3vA^.	(6.29)
It gives the equation (6.7) for the antisymmetric tensor field. Of course, one can investigate other sets of equations with different normalization of the F^v and A^ fields. Are all these sets of equations equivalent? As we see, to answer this question is not trivial. It was argued that the physical normalization is such that in the massless limit the zero-momentum field functions should vanish in the momentum representation (there are no massless particles at rest). Moreover, we advocate the following approach: the massless limit can and must be taken in the end of all calculations only, i. e., for physical quantities.
How can one obtain other equations following from the Weinberg-TuckerHammer approach? The recipe for the third equation is simple: use, instead of (a^vR)F^v, another symmetric matrix (y5^^vR)F^v.
1 The Weinberg-Tucker-Hammer theory has essentially 2(2s + 1) components. ' The reflection operator R has the properties
Rt = -R , R1" = R = R-1 ,	(6.21)
R-1 y5R = (Y5)t ,	(6.22)
R-1 y^R = -(y^)T ,	(6.23)
R-1 a^R = -(o-HT .	(6.24)
After taking into account the above observations let us repeat the procedure of derivation of the Proca equations from the Bargmann-Wigner equations for a symmetric second-rank spinor. However, we now use
^{«p) = (Y^R)ap(c amA| + CfF^) + (a^YR)ap(cAm(y )ppA|v + CpIppF^v),
(6.30)
with the same R and 0 as above. Matrices y1 are again chosen in the Weyl (spino-rial) representation, i.e., y5 is assumed to be diagonal. Constants ct are some numerical dimensionless coefficients. The properties of the reflection operator R are necessary for the expansion (6.30) to be possible in such a form, i.e., in order to have the y^R, (t|vR and y5c|vR to be symmetric matrices.
The substitution of the above expansion into the Bargmann-Wigner equations, Ref. [17], gives us the new Proca-like equations:
Cam(9^Av - ) + Cf (9|FV - 9VF|) = icAm2eaP|^Aap + 2mcFF|(6.31)
Cam2A^ + cfmF| = icAme^vap9vAap + 2c?avF|v .	(6.32)
In the case ca = 1, cF = 1 and cf = cA = 0 they are reduced to the ordinary Proca equations.7 In the general case we obtain dynamical equations which connect the photon, the notoph and their potentials. The divergent (in m 0) parts of field functions and those of dynamical variables should be removed by corresponding gauge (or Kalb-Ramond gauge) transformations. It is well known that the notoph massless field is considered to be the pure longitudinal field after one takes into account	= 0. Apart from these dynamical equations we can
obtain a number of constraints by means of the subtraction of the equations of the Bargmann-Wigner system (instead of the addition as for (6.31,6.32)). They read
mca9|A| + cf9f = 0,	(6.33)
mcA9aAa| + 2cFeaPv|3aFpv = 0,	(6.34)
that suggests F|V - imA|V and f1 - mA1, as in [12].
Thus, after the suitable choice of the dimensionless coefficients ct the La-grangian density for the photon-notoph field can be proposed:
L = LProca + LNotoph = -if^f1 -	+
8	4
2 2 m2 ,, m2
+ ^A^A1 + —A|VA|V ,	(6.35)
The limit m —» 0 may be taken for dynamical variables, in the end of calculations only.
7 We still note that the division by m in the first equation is not the well-defined operation in the case if someone is interested in the subsequent limiting procedure m —> 0. Probably, in order to avoid this obscure point one may wish to write the Dirac equations in the form [(iYl3|)/m - I] ^(x) = 0, which follows straightforwardly in the derivation of the Dirac equation on the basis of the Ryder relation [7] and the Wigner rules for the boosts of the field functions from the zero-momentum frame.
Furthermore, it is logical to introduce the normalization scalar field p(x), and consider the expansion:
= (y1R)ap(pA^) + (a|VR)«pF|v .	(6.36)
Then, we arrive at the following set
2mF^v = p(9|Av - 3VA|) + (3|p)Av - (9vP)A^,	(6.37)
= m(PA|),	(6.38)
which in the case of the constant scalar field p = 2m can also be reduced to the system of the Proca equations. The additional constraints are
(a^p)A^ + p(9^)= 0,	(6.39)
9|F|V = 0.	(6.40)
At the moment it is not yet obvious, how can we account for other equations in the (1,0) © (0,1) representation, e.g. [7b], rigorously. For instance, one can wish to seek the generalization of the Proca equations on the basis of the introduction of two mass parameters m1 and m2. But, when we apply the BW procedure to the Dirac equations we cannot obtain new physical content. Another equation in the (1/2,0) © (0,1/2) representation was discussed in Ref. [26]. It has the form:
[iY^n - m1 - Y5m2] ¥(x) = 0 .	(6.41)
The Bargmann-Wigner procedure for this system of equations (which include the Y5 matrix in the mass term) yields:
2m! F|V + 2im2F^v = p(9^Av - 9VA^) + (91p)Av - (9»A^, (6.42)
mi 2
= mi (pA|oJ,	(6.43)
with the constraints
(3|p)A| + p(9|A|) = 0,	(6.44)
im.2 ,
T
9vF|v = (pA|).	(6.45)
In general, we can now use the four different mass parameters in the equations which are analogous to (6.18,6.19). However, the equality of mass factors8 (mi ' = mi2) and m^1' = m^2') is obtained as necessary conditions in the process of calculations in the system of the Dirac-like equations.
In fact, the results of this paper develop the old results of Ref. [12]. According to [12, Eqs.(9,10)] we proceed in constructing the "potentials" for the notoph as follows:9
A|v(p)= N k1](p)42)(p)-e«1)(p)e|2)(p)l .	(6.46)
8	Here, the superscripts (1) and (2) refers to the first and the second equations, respectively, in the modified Bargmann-Wigner system.
9	The notation is that of Ref. [12] here.
We use explicit forms for the polarization vectors (e.g., Refs. [21] and [5, formu-las(15a,b)]) boosted to the momentum p:
£"(0>+,) = - Ti
M , i
w
, e"(0,0) =
M
0
0 ,
'e"(0'-,) = TI
0 ,
—i 0
and (pi = pi/|p|, y = Ep/m), Ref. [21, p.68] or Ref. [19, p.108],
e"(p,a) = L" v(p)ev(0,a),	_
L0o(p)= Y, Li o(p)= L0 i(p) = piVY2—,, Li k(p) = Sik + (y — ,)pipk .
(6.47)
(6.48)
(6.49)
(6.50)
N, the normalization factor, should be taken into account for possible analyses of propagators and massless limits. After substitutions in the definition (6.46) one obtains
iN2
A"v(p) = — m
0 p2
-pi
0
-p2 0
m
PrPl
pi
m + Prpi m + po + m
0
po +m
— P2P3 po +m
0 P1P3 po+m
P2P3 po+m P1P3 po +m 0
(6.51)
i.e., it coincides with the longitudinal components of the antisymmetric tensor obtained in Refs. [7a,Eqs.(2.14,2.17)] and [5, Eqs.(17b,18b)] within the normalization and different forms of the spin basis. The A"V(p) potential reduces to zero in the limiting case (m —» 0) under appropriate choice of the normalization N = ma, a > ,/i .If N = y/m this reduction of the non-transverse state occurs if a s = , particle moves along with the third axis OZ.10 It is also useful to compare Eq. (6.51) with the formula (B2) in Ref. [8] in order to think about correct procedures for taking the massless limits.
Next, the Tam-Happer experiments [27] on two laser beams interaction did not find satisfactory explanation in the framework of the ordinary QED (at least, their explanation is complicated by huge technical calculations). On the other hand, in Ref. [28] a very interesting model has been proposed. It is based on gauging the Dirac field on using the coordinate-dependent parameters a"V (x) in
^(x)-> V(x') = n^(x) , a = exp
-a"va"v (x)
(6.52)
and, thus, the second "photon" was introduced. The compensating 24-component (in general) field B",vA reduces to the 4-vector field as follows (the notation of [28] is used here):
B",va = ^"vAaMx) .	(6.53)
1 But, even in this case we cannot have a good behaviour of the 4-vector fields/potentials
in the massless limit in the instant form of the relativistic dynamics, cf. [8].
As readily seen, after comparison of these formulas with those of Refs. [12-14], the second photon is nothing more than the Ogievetskii-Polubarinov notoph within the normalization. Parity properties are dependent not only on the explicit forms of the momentum-space field functions of the (1/2,1/2) representation, but also on the properties of corresponding creation/annihilation operators. Helicity properties depend on the normalization.
6.3 The Standard Bargmann-Wigner Formalism Applied for Spin 2
In this Section we use the commonly-accepted procedure for the derivation of higher-spin equations [22]. We begin with the equations for the 4-rank symmetric spinor:
[iyp9p - m]aa,	= 0,	(6.54)
[iyp3p - m]pp,¥ap/y6 = 0,	(6.55)
[iyp9p - m]yy; ¥apr6 = 0,	(6.56)
[iyp9p - m]66, ¥apys' = 0.	(6.57)
The massless limit (if one needs) should be taken in the end of all calculations.
We proceed expanding the field function in the set of symmetric matrices (as in the spin-1 case, cf. Ref. [5]). In the beginning let us use the first two indices:11
^[«flyS = (YpR)aP^p6 + (Cpv^ap^ .	(6.58)
We would like to write the corresponding equations for functions and in the form:
2
=-^6 ,	(6.59)
wpv =
= 2m
9p¥;6 - 9V
(6.60)
Constraints (1/m)9p¥p6 = 0 and (1/m)epvap Qp^f = 0 can be regarded as the consequence of Eqs. (6.59,6.60).
Next, we present the vector-spinor and tensor-spinor functions as
^iVs) = (YKR)ysGK p + (aKTR)T6FKTp,	(6.61)
^ = (TKR)y6TK pv + (aKTR)T6RKTpv ,	(6.62)
i. e., using the symmetric matrix coefficients in indices y and 6. Hence, the total function is
V{ap}{T6} = (yPrW(yKr)T6gK P + (YpR)ap(aKTR)T6FKTp+ + (cPVrW(yKr)T6tK pv + (cpvR)ap(cKTR)T6RKTpv , (6.63)
The matrix R can be related to the CP operation in the (1/2,0) © (0,1/2) representation.
6 Notoph-Graviton-Photon Coupling 85 and the resulting tensor equations are:
m^TK mv =— GK v ,	(6.64)
^R^ = — fktv ,	(6.65)
tk mv = Jm [3^gk v — 3vgk m] ,	(6.66)
R-^ = 2m — .	(6.67)
The constraints are re-written to
m^GK M = 0, ¿V^ = 0,	(6.68)
-eap^3aTK pV = 0, -eap^3aRK/v = 0.	(6.69)
m	m
However, we need to make symmetrization over these two sets of indices {ap} and {y6}- The total symmetry can be ensured if one contracts the function V{ap}{y5} with antisymmetric matrices R-Y, (R-1Y5)pY and (R-1Y5YA)py, and equate all these contractions to zero (similar to the s = 3/2 case considered in Ref. [17, p. 44]. We obtain additional constraints on the tensor field functions:
G^ = 0, G^] = 0, gkm =	V ,	(6.70)
1
2:
fk/ = F^ = 0, e^F^ = 0,	(6.71)
T V = T ^ = 0, = 0,	(6.72)
fkt^ = tm'kt , e^F^ + tk,t^)= 0,	(6.73)
rkv^ = rvk^v = rkv^ = rvkv^ = r,mvmv = 0,	(6.74)
e^ap(gpKR^T,va — gpTRva,^)= 0 e^R^v = 0.	(6.75)
Thus, we encountered with the well-known difficulty of the theory of spin-2 particles in the Minkowski space. We explicitly showed that all field functions
become to be equal to zero. Such a situation cannot be considered as a satisfactory one (because it does not give us any physical information), and it can be corrected in several ways.12
6.4 The Generalized Bargmann-Wigner Formalism for Spin 2
We shall modify the formalism in the spirit of Ref. [30]. The field function (6.58) is now presented as
V{ap)y5 = ai (ymR)ap^5 + «2 (a^R) ap^MV + «3 (y" ^vr) ap^V , (6.76)
12 The reader can compare our results of this Section with those of Ref. [29]. I became aware about their consideration from Dr. D. V. Ahluwalia (personal communications, May 5, 1998). I consider their discussion of the standard formalism in the Sections I and II, as insufficient.
86 V. V. Dvoeglazov with
V{yS) = ßi(YKR)y6GK { + ß2(aKTR)y6FKT^ + ß3(Y5aKTR)y6FKT^ , (6.77) = ß4(YKR)y6TK {V + ß5(aKTR)y6RKT^v + ß6(Y5aKTR)y5RKT{V,(6.78) ^s) = ß7(YKR)y6RK {V + ß8(aKTR)y6öKT{V + ß9(Y5aKTR)y5DKT^v(6.79) Hence, the function V{aß){y6) can be expressed as a sum of nine terms:
^{aß){yS) = ai ßl (Y{R)aß(YKR)y6GK { + ai ß2 (Y{R) aß (^R^sF^ { + + aiß3(Y{R)aß(Y5tfKTR)y6RKT{ ++a2 ß4 (^vR) aß (YKR)ysTK {V +
+ a2ßs(a,v R)aß(aKTR)y6RKT{V + a2ß6(a{vR)aß(Y5^KTR)y6RKT{V + + a3ßz(Y5^{vR)aß(YKR)y6TK {V + a3 ß8 (Y5 aß R)y6D KT{V +
+ a3ß9(YXvR)aß(Y5tfKTR)y6DKT{V .	(6.80)
The corresponding dynamical equations are given by13
2aßL9vTk {V + ^e{vaß9vTK,aß = aißiGK {,	(6.81)
m	m
2a2ß5 d R {V + ia2 ß6 e d Raß,{v + ia3ß8 e{vaßd D
-"vKkt +--eaßKTÖv^	+--e OvDKT,aß —
m	m	m
—	e{vaßeA6KTDASaß = aiß2FKT{ + ^ eaßKTFaß'{,	(6.82) 2a2ß4TK {V + ia3ß7eaß{vRK,aß = ^^(3{GK V — 9VGK {), (6.83) 2a2ß5RKT{V + ia3ß8eaß{VD KT,aß + ia2ß6£aßKTRaß'{V
—	a3ß9 eaß{ve DÄS _
2 e	eA6KTD aß —
_ alß2(3{Fktv — 9vFkt{) + ^aimeaßKT(3{Raß'V — 9VRaß'{). (6.84)
The essential constraints are:
aißiG{{ _ 0, ai ßi G[K{] _ 0,	(6.85)
2iai ß2Fa{{ + ai ß3eKTW _ 0,	(6.86)
2iai ß3Ra{{ + ai ß2eKTaFKT,{ _ 0,	(6.87)
2ia2ß4T{ {a — a3ß7eKTaRK,T{ _ 0,	(6.88)
2ia3ß7R{ {a — a2ß4eKTaTK,T{ _ 0,	(6.89)
13 All indices in this formula are already pure vectorial and have nothing to do with previous notation. The coefficients a and p may, in general, carry some dimension.
ie
||VKT
a2p6R<T,|v + a3psD kt,|v
+ 2a2psR|V|v + 2a3p9D|V|v = 0,
(6.90)
ie|VKT [a2 p5Rkt,|v + «3 P9 D kt,|v] + 2a2 p6R^V + 2^3 P8D ^ = 0, (6.91)
2ia2 p5RP||a + 2ia3p9Dp||a + a2p6evaAp RA||V + «3p8evaApD = 0,
(6.92)
2ia p2FA|| - 2ia2p4T| |A + a!p3eKT|ARKT,| + a3p/eKT|ARK,T| = 0, (6.93)
2ia p3RA|| - 2ia3p7R| |A + aip2eKT|AFKT,i + a2p4eKT|AT<,T| = 0, (6.94)
ai pi (2Ga a - gA aG1 2a2p5(2RA||a + 2Ra||A + gA aR|V|V) +
+ 2a3p9(2DA||a + 2Da||A + gA aD|V|v)
+ 2ia3 p8(eKa|VD KA|V - eKT|AD KT,|a) -
-	2ia2p6(eKa|VRKA|v - eKT|ARKT,|a) = 0,	(6.95)
2a3p8(2D A||a + 2D a||A + gA aD - 2a2p6(2RA||a + 2Ra||A +
+ gA aRlV) ++2ia3p9(eKa|VDKA|v - eKT|ADKT,|a) -
-	2ia2p5(eKa|VRka- eKT|ARKT,|a) = 0,	(6.96)
ai p2(Fap,A - 2FpA,a + F^ gAa - Fa|^ gAp) -
-	a2p4(TA,ap - 2Tp,Aa + T^ |agAp - T^ |pgAa) +
+ 2ai p3(eKTapRKTA + 2eAKaPR</ + 2e|Ka^RA K,^) -
-	2a3p7(e|vapRA + 2evAaPR| ^ + 2e|KaPR<,|A) = 0.	(6.97)
They are the results of contractions of the field function (6.80) with six antisymmetric matrices, as above. Furthermore, one should recover the relations (6.70-6.75) in the particular case when a3 = p3 = p6 = p9 = 0 and ai = a2 = pi = p2 = p4 = p5 = p7 = p8 = 1 .
As a discussion, we note that in such a framework we have physical content because only certain combinations of field functions can be equal to zero. In general, the fields Fkt1,Rkti, T< |V, Rk |V, and Rkt|v, Rkt|v, Dkt|v, D<t|v can correspond to different physical states and the equations above describe couplings one state with another.
Furthermore, from the set of equations (6.81-6.84) one obtains the second-order equation for the symmetric traceless tensor of the second rank (a = 0, pi = 0):
—~2 [3v3^gk v — 9v3vgk *] = gk *.	(6.98)
m2
After the contraction in indices k and this equation is reduced to
3*G* a = Fa ,	(6.99)
3aFa = 0,	(6.100)
m2
i. e., to the equations connecting the analogue of the energy-momentum tensor and the analogue of the 4-vector potential (the additional notoph field as opposed to the Logunov theory?). As we showed in our recent work [30] the longitudinal potential may have importance in the construction of electromagnetism (see also the works on the notoph and notivarg concept [31]). Moreover, according to the Weinberg theorem [9] for massless particles it is the gauge part of the 4-vector potential - 3*x, which is the physical field. The case, when the longitudinal potential is equated to zero, can be considered as a particular case only. This case may be relevant to some physical situation but hardly to be considered as a basis for unification. Further investigations may provide additional foundations to "surprising" similarities of gravitational and electromagnetic equations in the low-velocity limit, Refs. [32-34,36].
6.5 Interactions with Fermions
The possibility of terms as a ■ [A x A*] appears to be related to the matters of chiral interactions [38,39]. As we are now convinced, the Dirac field operator can be always presented as a superposition of the self- and anti-self charge conjugate field operators (cf. Ref. [37]). The anti-self charge conjugate part can give the self charge conjugate part after multiplying by the y5 matrix, and vice versa. We derived14
[iy*D* — m]^f = 0,	(6.102)
15
or15
[iy*D* — m]^^ = 0.	(6.104)
14 The anti-self charge conjugate field function can also be used. The equation has then the form:
[iy*D* + mH2a = 0.	(6.101)
15 The self charge conjugate field function also can be used. The equation has the form:
[iy*D* + mj^i = 0.	(6.103)
As readily seen, in the cases of alternative choices we have opposite charges in the terms of the type a • [A x A*] and in the mass terms.
Both equations lead to the terms of interaction such as o • [A x A*] provided that the 4-vector potential is considered as a complex function(al). In fact, from (6.102) we have:
io^V^xi - m4>i = 0,	(6.105)
i&^V^i - mxi = 0.	(6.106)
And, from (6.104) we have
(6.107)
(6.108)
The meanings of o^ and &^ are obvious from the definition of y matrices. The derivatives are defined:
D^ = 9^ - ieY5C^ + eB^ , V^ = 9^ - ieA^,	(6.109)
and A^ = C^ + iB^. Thus, relations with the magnetic monopoles can also be established.
¿From the above system we extract the terms as ±e2 o1 At A*, which lead to the discussed terms [38,39].16 Furthermore, one can come to the same conclusions not applying to the constraints on the creation/annihilation operators (which we have previously chosen for clarity and simplicity in Ref. [39]). It is possible to work with self/anti-self charge conjugate fields and the Majorana anzatzen. Thus, in the considered cases it is the y5 transformation which distinguishes various field configurations (helicity, self/anti-self charge conjugate properties etc) in the coordinate representation.
6.6 Boson Interactions
The most general relativistic-invariant Lagrangian for the symmetric 2nd-rank tensor is
L-= -ai (9aGaA)(9pGPA) - a2(9aGPA)(9aGpA)
- aa(9aGPA)(9pGaA) + m2Ga|3GaP .	(6.110)
It leads to the equation
[a292 + m2] + (ai + aa)9{^ (9aG a|v}) = 0.	(6.111)
In the case a2 = 1 >0 and ai + a3 = -1 it coincides with Eq. (6.98). There is no any problem to obtain the dynamical invariants for the fields of the spin 2 from the above Lagrangian. The mass dimension of is [energy]i.
16 I am grateful to Prof. S. Esposito for the e-mail communications (1997-98) on the alternative proof of the considered interaction. We would like to note that the terms of the type o • [A x A*] can be reduced to ( o • V) V, where V is the scalar potential.
ic^X2 - m^2 = 0, i &- mx2 = 0.
We now present possible relativistic interactions of the symemtric 2nd-rank tensor. They should be the following ones:
Lift ~ G|vF|Fv ,	(6.112)
Li-t - (3lG|v)Fv ,	(6.113)
Li-t - G|v(9|Fv).	(6.114)
The term - (9|Gaa)F| vanishes due to the constraint of tracelessness. Obviously, these interactions cannot be obtained from the free Lagrangian (6.110) just by the covariantization of the derivative 9^ —» 9^ + gF^.
It is also interesting to note that thanks to the possible terms
V (F) = p1 (F|F^) + p2(F|F|)(FvFv)	(6.115)
we can give the mass to the G00 component of the spin-2 field. This is due to the possibility of the Higgs spontaneous symmetry breaking [40]
(v + 9cxM\
=
g1
g2
3
(6.116)
V g3 /
with v being the vacuum expectation value, v2 = (F^F1) = -p1 /2p2 > 0. Other degrees of freedom of the 4-vector field are removed since they can be interpreted as the Goldstone bosons. It was stated that "for any continuous symmetry which does not preserve the ground state, there is a massless degree of freedom which decouples at low energies. This mode is called the Goldstone (or Nambu-Goldstone) particle for the symmetry". As usual, the Higgs mechanism and the Goldstone modes should be important in giving masses to the three vector bosons.17 As one can easily see, this expression does not permit an arbitrary phase for F1, which is possible only if the 4-vector would be the complex one.
Next, due to the Lagrangian interaction of fermions with notoph are of the order e2 since the beginning (as opposed to the interaction with the 4-vector potential A| ), it is more difficult to observe it. However, as far as I know the theoretical precision calculus in QED (the Lande factor, the anomalous magnetic moment, the hyperfine splittings in positronium and muonium, and the decay rate of o-Ps and p-Ps) are about the order corresponding to the 4th-5th loops, where the difference may appear with the experiments [41,42].
17 It is interesting to note the following statement (given without references in wikipedia.org): "In general, the phonon is effectively the Nambu-Goldstone boson for spontaneously broken Galilean/Lorentz symmetry. However, in contrast to the case of internal symmetry breaking, when spacetime symmetries are broken, the order parameter need not be a scalar field, but may be a tensor field, and the corresponding independent massless modes may now be fewer than the number of spontaneously broken generators, because the Goldstone modes may now be linearly dependent among themselves: e.g., the Goldstone modes for some generators might be expressed as gradients of Goldstone modes for other broken generators."
6.7 Conclusions
We considered the Bargmann-Wigner formalism to derive the equations for the AST field and for the symmetric tensor of the 2nd rank. We introduced additional scalar normalization field in the Bargmann-Wigner formalism in order to take into account possible physical significance of the Ogievetskil-Polubarinov-Kalb-Ramond modes. We introduced the additional symmetric matrix in the Bargmann-Wigner expansion (y5o"vR) in order to take into account the dual fields. The consideration is similar to Ref. [43].
Furthermore, we discussed the interactions of notoph, photon and graviton (and, probably, notivarg18). For instance, the interaction notoph-graviton may give the mass to spin-2 particles in the way which is similar to the spontaneous-symmetry-breaking Higgs formalism.
Acknowledgements
I am grateful to the referee of "International Journal of Modern Physics" and "Foundation of Physics", whose advice of the mass dimensions (normalizations) of the fields was very useful. I acknowledge insightful discussions with participants of recent conferences on Symmetries and the Standard Model. I am thankful to organizers and El Colegio Nacional for partial financial support during the QTS-8. I am thankful to the UAEM for partial financial support during the INRACOM13.
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Bled Workshops in Physics Vol. 15, No. 2
A
Proceedings to the 17th Workshop What Comes Beyond ... (p. 93) Bled, Slovenia, July 20-28, 2014

7 General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family Symmetry with Sterile Neutrinos
A. Hernández-Galeana*
Departamento de Física, Escuela Superior de Física y Matemáticas, I.P.N., U. P. "Adolfo Lopez Mateos". C. P. 07738, Mexico, D.F., Mexico
Abstract. Within the framework of a local SU(3) family symmetry model, we report a general analysis of the mechanism for neutrino mass generation and mixing, including light sterile neutrinos. In this scenario, ordinary heavy fermions, top and bottom quarks and tau lepton, become massive at tree level from Dirac See-saw mechanisms implemented by the introduction of a new set of SU(2) L weak singlet vector-like fermions, U, D, E, N, with N a sterile neutrino. Right-handed and the NL,R sterile neutrinos allow the implementation of a 8 x 8 general Majorana neutrino mass matrix with four or five massless neutrinos at tree level. Hence, light fermions, including light neutrinos get masses from radiative corrections mediated by the massive SU(3) gauge bosons. We report the corresponding Majorana neutrino mass matrix up to one loop. Previous numerical analysis of the free parameters show out solutions for quarks and charged lepton masses within a parameter space region where the vector-like fermion masses MU , MD , Me, and the SU(3) family gauge boson masses lie in the low energy region of 0(1 — 20) TeV, with light neutrinos within the correct order of square neutrino mass differences: m| — mf ra 7 x 10-5 eV2, mj — mf ra 2 x 10-3 eV2, and at least one sterile neutrino of the order ra 0.5 eV. A more precise fit of the parameters is still needed to account also for the quark and lepton mixing.
Povzetek. Avtor pojasnjuje pojav družin pri leptonih tako, da uporabi za opis družin model z lokalno simetrijo SU(3). Trem družinam kvarkov in leptonov doda se družinski triplet desnorocnih nevtrinov, ki nosi samo druzinski naboj, levorocni in desnorocni U in prav tak D kvark, ki nosijo poleg barve le hiper naboj, levorocni in desnorocni nevtrino, ki ne nosita nobenega naboja, ter levorocni in desnorocni elektron s hipernabojem (—2). Vsi ti novi delci so masivni. Novi fermioni poskrbijo na drevesnem nivoju samo za maso tretje druzine kvarkov in leptonov. Lahkim fermionom, tudi lahkim nevtrinom, priskrbijo maso popravki v naslednjih redih pri interakciji z masivnimi bozoni, ki nosijo druzinsko kvantno stevilo. Avtor izračuna masno matriko 8x8 za Majoranine nevtrine do prvega reda. Proste parametre modela doloci z izmerjenimi masami in mesalnimi matrikami. Po dosedanjih izracunih so primerne vrednosti za mase fermionov MU , MD , ME in za maso druzinskega tripleta umeritvenega bozona v intervalu 0(1 — 20) TeV, za izmerjene masne razlike lahkih nevtrinov m22 — mi 2 ra 7 x 10-5 eV2, m32 — mi 2 ra 2 x 10-3 eV2 lahko avtor poskrbi s se vsaj enim sterilnim nevtrinom, ki ima maso ra 0.5 eV. Avtor pricakuje, da bo z bolj natancnimi izracuni lahko s pomocjo tega modela pojasnil mesalne matrike kvarkov in leptonov.
* E-mail: albino@esfm.ipn.mx
7.1	Introduction
Although the standard picture with three light flavor neutrinos has been successful to describe the neutrino oscillation data. On the other hand, there have been recent hints from the LSND and MiniBooNe short-baseline neutrino oscillation experiments[1,2] on the possible existence of at least one light sterile neutrino in the eV scale, which mix with the active neutrinos. On the other hand, an explanation of the strong hierarchy of quark and charged lepton masses is still a big challenge in particle physics. This hierarchy have suggested to many model building theorists that light fermion masses could be generated from radiative corrections, while those of the top and bottom quarks and the tau lepton are generated at tree level. This may be understood as the breaking of a symmetry among families , a horizontal symmetry.
In this report we update the general features of a "Beyond the Standard Model"(BSM) proposal which introduces a SU(3) [3] gauged family symmetry 1 commuting with the Standard Model group. Previous reports[4] within this scenario showed that this model has the features and particle content to account for a realistic spectrum of charged fermion masses and quark mixing. This BSM model introduce a hierarchical mass generation mechanism in which the light fermions obtain masses through one loop radiative corrections, mediated by the massive bosons associated to the SU(3) family symmetry that is spontaneously broken, while the masses for the top and bottom quarks as well as for the tau lepton, are generated at tree level from "Dirac See-saw"[5] mechanisms implemented by the introduction of a new generation of SU(2)L weak singlets vector-like fermions.
The SU(3) family symmetry model allows one to address the problem of quark and lepton masses and mixing, including active and light sterile neutrinos.
7.2	SU(3) flavor symmetry model 7.2.1 Fermion content
Before "Electroweak Symmetry Breaking"(EWSB) all ordinary, "Standard Model"(SM) fermions remain massless, and the global symmetry in this limit of all quarks and leptons massless, including R-handed neutrinos, is:
SU(3)qL <g> SU(3)uR <8> SU(3)dR <g> SU(3)1l <g> SU(3)Vr <8> SU(3)eR (7.1)
D SU(3)qL+uR + dR + lL + eR + vR = SU(3)	(7.2)
We define the gauge group symmetry G = SU(3) < GSM, where Eq.(7.2) defines the SU(3) gauged family symmetry, and GSM = SU(3)C < SU(2)L < U(1)Y is the "Standard Model" gauge group, with gH, gs, g and g' the corresponding
1 See [3,4] and references therein for some SU(3) family symmetry models.
coupling constants. The content of fermions assumes the ordinary quarks and leptons assigned under G as:
^ = (3,3,2,1 )L , = (3,3,1,4 )R , ^ = (3,3,1, - 3 )r
= (3,1,2, -1 )l , C = (3,1,1, -2)r ,
where the last entry corresponds to the hypercharge Y, and the electric charge is defined by Q = T3L + 2 Y. The model also includes two types of extra fermions:
•	Right handed neutrinos ¥V = (3,1,1,0)R required to cancel anomalies[6], and
•	the SU(2)l singlet vector-like fermions:
UL,R = (1,3,1,4) , Dl,r = (1,3,1, - 2)	(7.3)
N0,r = (1,1,1,0) , E0,r = (1,1,1, -2),	(7.4)
which conserve the previous anomaly cancellation. The transformation of these vector-like fermions allows the mass invariant mass terms
Mu tJLL UR + Md DL DR + Me E° ER + h.c. ,	(7.5)
and
mD INL NR + mL INL (NL)C + mR NR (NR)C + h.c	(7.6)
These SU (2) L weak singlets vector-like fermions have been introduced to give masses at tree level only to the third family of known fermions through Dirac See-saw mechanisms. MU , MD , ME play a crucial role to implement a hierarchical spectrum for quarks and charged lepton masses and mixing, meanwhile mD , mL , mR play a similar role for neutrino masses and lepton mixing, all together with the radiative corrections.
7.3 SU(3) family symmetry breaking
The corresponding SU(3) gauge bosons are defined through their couplings to fermions as
iLi-t = gH (f?Y|f? - W°°) Z| + 2VH3 (f?Y|f? + f0Y|f2o - 2flY|f3°) Z|
+ (f°Y|f° Y+ + f°Y|f° Y+ + f°Y|f° Y+ + h.c.) (7.7)
f° = u°, d°, e°, v° , f° = c°,s°,^°,v| and f° = t0,b°,T°,v°. To implement a hierarchical spectrum for charged fermion masses, and simultaneously to achieve
the SSB of SU(3), we introduce the flavon scalar fields: ni = (3,1,1,0), i = 1, i, 3, transforming as the fundamental representation under SU(3) and being standard model singlets, with the "Vacuum Expectation Values" (VEV's):
(m)T = (At,0,0) , <n2>T = (0,A2,0) , (na)T = (0,0,As) .
(7.8)
Actually, let us point out here that only two scalar flavons in the fundamental representation are needed to completely break down the SU(3) symmetry. The most convenient way to accomplish the spontaneous breaking of the SU(3) family symmetry is under current study. Thus, the contribution to the horizontal gauge boson masses from Eq.(7.8) read
^ (Y+Y- + Y+ Y-) + ^(Z2 + f + iZi )
m n2 n3
9h, A1 2
9H2 A2
H (Y+Yr + Y3+ Y-) +
9h2 a2
(z2+Z2 - 2zi a? )
V3J Z2 )
as)
^ (Y+Y- + Y+ Y-) + gHH3 ASzS2
Therefore, neglecting tiny contributions from electroweak symmetry breaking, we obtain the gauge boson mass terms
(M2 + m2) Y+Y- + (M2 + M3) Y+Y- + (M2 + M3) Y+Y-
2 \
12 _i_ 1\/12!
1 -, -, -, 1 m2 + m2 + 4m2 i 1 i -, 2 +1 (m? + m2) z2 + 2' + M2 + 3 Z2 + 1 (M? - m2)^ zt Z2 (7.9)
3
V3
M? =
gH, a?
m2 = "H2
gH, a2
M23 =
gH A3
(7.10)
Z2
Z? Z2
m2 + M2 ^tm
M, —M, M, +M, +4M3
75	3
Table 7.1. Zt — Z2 mixing mass matrix
From the diagonalization of the Z1 — Z2 squared mass matrix, we obtain the
eigenvalues
m- = 2 (m? + m2 + m3 — ^(m2 — m?)2 + (m2 — m?)(m3 — M2)^j m+ = 3 ('? + '2 + m3 + \J(m2 — m?)2 + (m3 — m2)(m2 — m2)^
4
2
2
Z
Z2
2 Z+
my, Y+Y- + My2 y+y- + My3 y+y- + Ml — + M+
2
where
(7.11)
mY = M2 + M2 , MY2 = M1 + M2 , MY3 = M2 + M3 (7.12)
cos 9 — sin 9 A f Z
sin 9 cos 9
Z
(7.13)
A • A ^	M1 - M2
cos 9 sin 9 = —----' 2	-,
4 ^/(M2 - M2)2 + (M3 - M2)(M2 - M2)
with the hierarchy M', M2 ^ MW2. Due to the Z' - Z2 mixing we diagonalize the propagators involving Zi and Z2 gauge bosons according to Eq.(7.13):
Zi = cos 9 Z_ — sin 9 Z+ , Z2 = sin 9 Z_ + cos 9 Z
2
(Zi Zi > = cos2 9 (Z_Z_> + sin2 9 (Z+Z+> (Z2Z2) = sin2 9 (Z_Z_> + cos2 9 (Z+Z+)
(Z1Z2) = cos 9 sin9 ((Z_Z_) - (Z+Z+)) 7.4 Electroweak symmetry breaking
Recently ATLAS[7] and CMS[8] at the Large Hadron Collider announced the discovery of a Higgs-like particle, whose properties, couplings to fermions and gauge bosons will determine whether it is the SM Higgs or a member of an extended Higgs sector associated to a BSM theory. The electroweak symmetry breaking in the SU(3) family symmetry model involves the introduction of two triplets of SU(2)l Higgs doublets.
To achieve the spontaneous breaking of the electroweak symmetry to U(1 )q, we introduce the scalars: =(3,1,2, -1) and ® d = (3,1,2, +1), with the VEVs:
w>n (®u> = | (®U>
'(®f>N
(®d> = | (®2> ,(®2>,
(7.14)
/®u\ = -L
<-2> = .^(vO,
(7.15)
I2; sin ^ = 0, cos ^ = 1
contribute to the W and Z boson masses:
(vU + vd) W+W- +	(vU + vd) ZO
vU = vUi + vu2 + vUa z vd = vdi + vd2 + vd3. Henceif we define Mw = 1gvwe may write v = * /v2 + vd « 246 GeV.
7.5 Tree level neutrino masses
Now we describe briefly the procedure to get the masses for ordinary fermions. The analysis for quarks and charged leptons has already discussed in [4]. Here, we introduce the procedure for neutrinos.
7.5.1 Tree level Dirac neutrino masses
With the fields of particles introduced in the model, we may write the Dirac type gauge invariant Yukawa couplings
ho V? NR + hi ni NL + h2n2 NL + hana NL
+ Mo NL NR + h.c (7.16)
ho, h1, h2 and h3 are Yukawa couplings, and Mo a Dirac type, invariant neutrino mass for the sterile neutrinos N? R. After electroweak symmetry breaking, we obtain in the interaction basis	=	No)L,R, the mass terms
ho [vi vOl + v2 v°L + v3 vOj NR + [hiAi vOr + h2A2 v°R + h3A3 vOR NR
+ Mo IN R NR + h.c. (7.17)
7.5.2 Tree level Majorana masses:
Since Nrr, Eq.(7.4), are completely sterile neutrinos, we may also write the left and right handed Majorana type couplings
hLPR ®u(NR)c + mL INR (NR)c + h.c	(7.18)
and
hiR PVni (NR)c + h2RPVn2 (NR)c + h3RPVna (NR)c
+ mR INR (NR)c + h.c , (7.19)
respectively. After spontaneous symmetry breaking, we also get the left handed and right handed Majorana mass terms
7 General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family... 99 hL [vi vOl + V2 v°L + V3 V0L] (NL)c + mL NO (NR)c + h.c.,	(7.20)
+ [hiR Ai vOr + h2R A2 v°R + h3R A3 vOr] (NR)c + mR NR (NR)c + h.c., (7.21)
	(v»l)c	Kl)c	(v»l)c	(NL)C	V»R			NR
V»L	0	0	0	hL Vi	0	0	0	ho Vi
	0	0	0	hL V2	0	0	0	ho V2
	0	0	0	hL V3	0	0	0	ho V3
nl	hL Vi	hL V2	hL V3	mL	hi Ai	h2 A2	h3 A3	mo
(v»r)c	0	0	0	hi Ai	0	0	0	hiR Ai
(V°r)c	0	0	0	h2 A2	0	0	0	h2R A2
(v»r)c	0	0	0	h3 A3	0	0	0	h3R A3
(NR)C	ho vi	ho V2	ho V3	mo	hiR Ai h2R A2 h3R A3			mR
Table 7.2. Tree Level Majorana masses
Thus, in the basis
w»
1 V
= (-
eL
VL'
,v»l ,N» , (v»r)c , (v°R)c , (v»r)c , (NR)c ) ,
(7.22)
the Generic 8 x 8 tree level Majorana mass matrix for neutrinos MV, from Table 7.2, ¥"V MV (^V)c, read
MV =
ML MD
<Mdt MR,
(7.23)
where
ML =
/ 0	0 0 a \
0	0 0 a2
0	0 0 a3
\ai	a2 a3 mL/
MR =
( 0 0 0 ßi\ 0 0 0 ß2 0 0 0 ß3 Vßi ß2 ß3 mR/
(7.24)
O
o
and
MD -
0	0	0 a1
0	0	0 a2
0	0	0 a3
b1	b2	b3 mD
(7.25)
a - hL vt , at - hD vt , bt - hi At , pi - htR At	(7.26)
Diagonalization of mv°', Eq.(7.23), yields four zero eigenvalues, associated to the
neutrino fields:
^ - ^ , ^ ^ + aa=-a3 v,L - aP v°L, (7.27) ap	ap ,L	ap a eL ap a ,L a TL
b2 ° b! °	bi ba ° b2 ba ° bT
bpv°R bpv,R , b^V°R + b^v,R "b
ap - y a2 + a2 , bp - ^bf + b2 , a - ^ a2 + a2 + a3 , b - ^ b2 + b2 + b2 .
Assuming for simplicity i^p - ^^ - = cR, that is
at Hl	pt htR
- - Z— - CL , TT - ~T~ - CR ,
at hD	bt ht
the Characteristic Polynomial for the nonzero eigenvalues of MV reduce to the one of the matrix m43, Eq.(7.29), where
m4
/0 a 0 a \ a mL b mD 0 b 0 p Va mD p mRJ
a - \/a2 + a2 + a3
U4 -
/Ufi U12 U13 Ui4\
U21 U22 U23 U24
U31 U32 U33 U34 VU41 U42 U43 U44J
p - + p2 + P3 .
(7.29)
U4 m4 U4 - Diag(m°,m°,m°,m°) = d4 , m4 - U4 d4 U{	(7.30) Eq.(7.30) impose the constrains
u11 m° + u22 m° + u13 m° + u?4 m° - 0	(7.31)
u31 m° + u22 m° + u33 m° + u34 m° - 0	(7.32)
U11U31 m° + U12U32 m° + U13U33 m° + U14U34 m° - 0,	(7.33)
3 The relation a b - a p would yield five massless neutrinos at tree level.
7 General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family... 101 corresponding to the (m4)n = (m4)33 = (m4)i3 = 0 zero entries, respectively. Therefore, MV is diagonalized by the orthogonal matrix
/ ai ai a3 ap a ap
o ai un ai uuai ui3 afui4\

ai a2 a3 o ap a ap
a
o aiuii aiu^a2ui3aiui
ai
a
a 2 .
a
o a3uii aiui2a3u^a^M
0
0
0
Uo
0 0 0
0 0
00
u2i u22 u23 u24
b2 b1 b3 b,, b^, b^, b^, bp Tbbp3 bb"u3i ""u32 ""u33 "" u34
bl bibs biu3i "bu32 u33 ""u34

bp b bp b
0 -b"P "3u3i "3u32 "3u33 "b3u34
00
u4i
(7.34)
u42 u43 u44 /
(UV)T MV UV = Diag(0,0,0,0,mO,mO,m?,mg)	(7.35)
7.6 One loop neutrino masses
After tree level contributions the fermion global symmetry, Eq.(7.1), is broken
down to
SU(2)qL <g> SU(2)uR < SU(2)dR < SU(2)lL << SU(2)Vr < SU(2)eR . (7.36)
Therefore, in this scenario light neutrinos may get extremely small masses from radiative corrections mediated by the SU(3) heavy gauge bosons.
7.6.1 One loop Dirac Neutrino masses
After the breakdown of the electroweak symmetry, neutrinos may get tiny Dirac mass terms from the generic one loop diagram in Fig. 7.1, The internal fermion line in this diagram represent the tree level see-saw mechanisms, Eqs.(7.16-7.21). The vertices read from the SU(3) family symmetry interaction Lagrangian
iLint = gH (VOYivO -	Z|+^ (v-oy^vO + V°Y|V| - 2v?y^0) Z|
+ (V-0Y|V| Y+ + v-OY|VO Y+ + v-|Y|VO Y3+ + h.c.) (7.37)
The contribution from these diagrams may be written as
Cy-mv(My )ij
n
«h
gH
4n '
mv (My)ij = ^ mk U?kU?k f (My, mg)
k=5,6,7,8
f(My '
m

k) - M2
l Mr.
mo2 lnmo7
oT Y-mk
(7.38)
(7.39)

Y
< nk >
< >
V
iL
Fig. 7.1. Generic one loop diagram contribution to the Dirac mass term m^ v°Lv°R. M — Md, mL, mR
o A
o
	vor	V°R	V?R	NR
V ol	Dv 11	D v 12	Dv13	0
V°L	Dv 21	Dv22	Dv23	0
V ol	Dv 31	Dv32	Dv33	0
N L	0	0	0	0
Table 7.3. One loop Dirac mass terms O1 Dv ^ v?L V°R
mv(My )i,4+j —	Fv(My )
J ab
(7.40)
Fv(My) — U,,U3, mO f(My,mg) + UT2U32 mg f(My,m§)
+ U13U33 mO f(My,m°) + U14U34 mg f(My,mg) (7.41)
Dvii =
mbi ab
1 Fy(Mz, ) + 12Fv(Mz2 ) + Fv,m
+ ■
a2b2 Fv(MYl) + a3b3Fv(MY2 )
ab
ab
D v 12 =
aib2 ab
-4Fv(Mz,) + 12Fv(Mz2 )
D v 13 =
aib3
ab

Fv(Mz2) - Fv
Dv2i =
a2bi ab

4Fv(Mz,) + 12Fv(Mz2 )
Dv22 =
a2b2 ab
1 Fv(Mz, ) + 12Fv(Mz2 ) - Fv,m
1 + 2
aib
1 Fv(MYl) + ^Fv(MY3) ab	1 ab	3
Dv23 =
a2b3 ab
Fv(Mz2) + Fv
Dv 3i =
ab
ab

Fv(Mz2) - Fv
Dv32 =
a3b2 ab
Fv(Mz2) + Fv
D v 33 =	Fv(Mz2 ) + 2
3 ab
aiblFv(MY2 ) + ^Fv(MY3) 2 ab	3
ab
Fv(Mz,) = cos2 ^Fv(M-) + sin2 ^Fv(M+)
Fv(Mz2) = sin2 ^Fv(M_) + cos2 ^Fv(M+)
Fv,m = cos ^ sin ^ [Fv(M-) - Fv (M+)] , 2V3
(7.42)
7.6.2 One loop L-handed Majorana masses
Neutrinos also obtain one loop corrections to L-handed and R-handed Majorana masses from the diagrams of Fig. 7.2 and Fig. 7.3, respectively. A similar procedure as for Dirac Neutrino masses leads to the one loop Majorana mass terms
mv(My )i,j = ^ Gv(My )
(7.43)
Gv(My) = mg u?? f(My, mg ) + mg u?2 f(My,mg)+ mg u?3 f(My,mg)
+ mg u?4 f(My,mg) (7.44)
VjL
Y
< >
< >
v
iL
Fig. 7.2. Generic one loop diagram contribution to the L-handed Majorana mass term mij VPL(vPL)T. M = Md , mL,mR
o
	Vol	V°L		NL
vol	Lvii	Lv 12	Lv 13	0L
V°L	Lv 12	Lv 22	Lv 23	0
	Lv 13	Lv 23	Lv 33	0
NL	0	0	0	0
Table 7.4. One loop L-handed Majorana mass terms OH Lvij vgL (v°L)T
T	—
Lv 11 = "T
1 Gv(Mz, ) + I2Gv(Mz2 ) + Gv,m
T	a2
Lv 22 = —r 2
1 Gv(Mz, ) + I2Gv(Mz2 )— Gv,m
1 a2
Lv 33 =	Gv(Mz2 ) ,
3 -2
-i -2
Lv 12 = -7—
-2
—1 Gv (Mz,) + 112Gv (Mz2 ) + 1 Gv (Mi)
Lv 13 =
-i -3
-
2
6 Gv(Mz2 ) + 2 Gv(M2) — Gv,m
Lv 23 =
-2-3
—1 Gv(Mz2 ) + 1 Gv(M3)+ Gv,m b 2
Gv(Mz,) = cos2 9 Gv(M_) + sin2 9 Gv(M+)
Gv(Mz2) = sin2 9 Gv(M_) + cos2 9 Gv(M+)
Gv,m = ^ cos 9 sin 9 [Gv(M-) — Gv(M+)].
(7.45)
7.6.3 One loop R-handed Majorana masses
mv(My )4+i,4+j = ^bj1 Hv(My)
(7.46)
Hv(My) = mO u31 f(My,m0) + m£ u32 f(My,m§) + mO u33 f(My,m?)
, o ^ 1 o .. 2
o ^ 1 o .. 2
+ mo u34 f (My,mg) (7.47)
2
2
2
-
Y

v
iR
< nk >	< ns >
Fig. 7.3. Generic one loop diagram contribution to the R-handed Majorana mass term
tij V°R(v°Rf
mij v°r(v?r)t. M — Mo,mL,mR
	Vor	V°R	V O R	NR
vor	Rvii	Rv 12	Rv 13	0
V°R	Rv 12	Rv 22	Rv 23	0
V?R	Rv 13	Rv 23	Rv 33	0
NR	0	0	0	0
Table 7.5. One loop R-handed Majorana mass terms OH Rvij v°R (v°R)T
R - b1
1 Hv(Mz, ) + 12Hv(Mz2 )+
Rv22 - b2
1 Hv(Mz, ) + 12Hv(Mz2 )- Hv,m
1 b2
Rv 33 — 3 b3Hv(MZ2 ) ,
Rv 12 —
Rv 13 —
Rv 23 —
bib2 b2
bib3 b2
b2b3 b2

-4Hv(Mz, ) + 12Hv(Mz2 ) + 1 Hv(Mi )
1 Hv(Mz2 ) + 1 Hv(M2)- Hv
4Hv(Mz2 ) + 1 Hv(M3)+ Hv,m b 2
2
Hv(Mz, ) = cos2 ^Hv(M-) + sin2 ^Hv(M+)
Hv(Mz2) = sin2 ^Hv(M-) + cos2 ^Hv(M+)
^v.m —
2V3
cos^ sin^ [Hv(M_) -Hv(M+)] ,
(7.48)
where Fv,m , Gv,m and Hv,m, Eqs.(7.42,7.45,7.48), come from Z1 — Z2 mixing diagram contributions.
Thus, in the basis, Eq.(7.22), we may write the one loop contribution for neutrinos as
M»v —
fLvii	LV 12	LV 13	0	DV 11	D v 12	D v 13	0
Lv 12	LV 22	LV 23	0	DV 21	D v 22	D v 23	0
Lv 13	LV 23	LV 33	0	Dv 31	D v 32	D v 33	0
0	0	0	0	0	0	0	0
Dvii	Dv21	Dv 31	0	Rvii	Rv 12	Rv 13	0
Dv12	D v 22	D v 32	0	Rv 12	Rv 22	Rv 23	0
Dv13	D v 23	D v 33	0	Rv 13	Rv 23	Rv 33	0
V 0	0	0	0	0	0	0	0
aH (7.49)
n
7.6.4 Neutrino mass matrix up to one loop
Finally, we obtain the general symmetric Majorana mass matrix for neutrinos up
to one loop
Mv = (UV)T M|V UV + Diag(0,0,0,0,mO,mO,mO,mO), (7.50) where explicitly
(UV )T MOv UV =
/N11 N12 N13 N14 N15 Ni6 N17 Nis\
N12 N22 N23 N24 N25 N26 N27 N28 N13 N23 N33 N34 N35 N36 N37 N38 N14 N24 N34 N44 N45 N46 N47 N48 N15 N25 N35 N45 N55 N56 N57 N58 Ni6 N26 N36 N46 N56 N66 N67 N68 N17 N27 N37 N47 N57 N67 N77 N78 \Ni8 N28 N38 N48 N58 N68 N78 N88/
(7.51)
Majorana L-handed:
22
N11 = a2 a2
-p-2
(Gzi - Gi )
N12 =
22 -1 -2 -3 r-2 - -1
2 a3
-(Gzi - Gi ) + G2 - Ga - 6Gm]
(7.52)
(7.53)
N22 = -|
l (-2 - -2)2 4
2 /MGzi - Gi) + -2 (G2 - Ga) 2 2 i 2
a2p a
a2	a2 - a2
+^2 (Gi + BGz2 - 4G2) - B 2 2 1 Gm 4 a2	2	a2
(7.54)
Dirac:
N13 =
l
2ap bp ab
{(-1 bi + -2b2)Fi + -aba (-2b2F2 + -ibiFa)
+2aibi-2b2Fzi j (7.55)
a
2
2
Ni4 = 2apbp abl3 {bib2(a2 - a2)Fi + aaba(a2biF2 - aib2Fa)
+-1 -2 (b2 - b2)Fzi + 6-1 -2 bp2 Fm} (7.56)
N23 = 2 jp -b~-3 { —1 —2 (b2 — b2)Fi + -3b3 (-ib2F2 — -2biF3)
+bib2(-2 — — 2)Fz, + bb1b2 —p2 F,} (7.57)
N24 = 1
—p bp —2 b2
1	3
4 (—2 — — 2)(b2 — b2)Fz, + 4
+1 (—3b2 + —p2 bp2)(— 1b1F2 + —2b2F3) + 3—3b3(—2b2 — — 2b2)F
—	1 *")	*") *")	*") —	3 -) -)
—3b3[—1b1— 2b2F1 + t(—! — — 2Kb — b2)Fz, +7 —p bp Fz2]
Majorana R-handed:
N34 =
b2b2
N33 = ^ (Hz, — H1)
b1b2b3 [b^ (Hz, — H1)+ H2 — H3 — bHm]
2b3
bp
(7.58)
(7.59)
N44 = b|
1 (b2 — b2)2
4 2b2 b21 (Hz, — H1) + b2 (H2 — H3)
b2,
bp
+(H1 + 3Hz2 — 4H2)— 3
b2 — b1 b2
m
(7.60)
llO A. Hernández-Galeana Majorana L-handed and Dirac:
N15 = G15 U11 + m13 U31 ; N16 = G15 U12 + m13 U32 (7.61) N17 = G15 U13 + m13 U33 ; N18 = G15 U14 + m13 U34 (7.62)
a1 a2
G15 = — ~-
2 ap a
a2 — a2	a2	(2 a2 — ap J
r1 (Gz, — G1 ] + ay(G3 — G2]+ 2 ( 3 2 pJ Gm
n2	n2	n2
m13 = 2ap1ab2 ^b1 b2(a2 — a^F + a3b3(a2b1^2 — a^FO
+a1 a2(b2 — b2}Fz, + 2a1 a2 (bp2 — 2b2]Fm}
N25 = G25 U11 + m23 U31 ; N26 = G25 U12 + m23 U32 (7.63) N27 = G25 U13 + m23 U33 ; N28 = G25 U14 + m23 U34 (7.64)
G25 = 3
4 ap a4
{ (a2 — -2J2 (Gz, — G1 ] + 2 a2(a3 — (G3 — G2} — ap (Gz2 — G1J
—2 -p(-3 — ap] (Gz2 — G2] + 4 (a2 — a?} (a3 — 2ap] Gm }
1
m23
ap a2 b2
a3 [a1b1a2b2F1 + 1 (af — a2Hbí — b2JFz, + 1ap2(bp2 — 2b2JFz2 ] +1b3(a2 — ap^a^F + a2b2F3J + a3[a2(3b2 — b2J + a2(b2 — 3b2}]Fn
7 General Majorana Neutrino Mass Matrix from a Low Energy SU(3) Family... 111 Dirac and Majorana R-handed:
N35 = m.3! un + H35 U3! , N36 — m3i u12 + H35 U32
N37 = m.31 U13 + H35 U33 , N38 — m3i U14 + H35 U34
m3i = 2bp a2b iaia2(b2 -bi)Fi + a3b3(aib2F2 - a2biF3)
+bib2(a! - a2)Fz, + 2bib2(ap2 - 2a3)Fm}
H35 —
bi b2 2bp b
b2 - b2 b2 -b2-(HZ, - Hi ) + b2 (H3 - H2) + 2
(2b3 - bp) b2
m
N45 — m32 Uii + H45 U3i , N46 — m32 Ui2 + H45 U32
(7.65)
N47 — m.32 Ui3 + H45 U33 , N48 — m32 Ui4 + H45 U34
(7.66)
1
m32
1
bp a2 b2
b3[aibia2b2Fi + 1 (a2 - a2)(b2 - b2)Fz, + ^bp2(ap2 - 2a3)Fz2]
+^a3(b3 - bp2)(aibiF2 + a2b2F$) + b3 [b2(3af - a2) + b2(a2 - 3a2)]F
H45 —
b3
4 bp b4 ,2 \~2\2
{(b2 - b2)2 (Hz, - Hi)+ 2b2 (b3 - bp) (H3 - H2)- bp (HZ2 - Hi)
2 bp (b2 - bp) (Hz2 - H2) + 4 (b2 - b2) (b2 - 2bp) Hm }
— /
Majorana L-handed, Dirac and Majorana R-handed:
N55 = G55 u2! + 2 m.33 U11 U31 + H55 u^	(7.67)
N56 _ G55 U11 U12 + m33 (U11 U32 + U12 U31) + H55 U31 U32	(7.68)
N57 _ G55 U11 U13 + m33 (U11 U33 + U13 U31) + H55 U31 U33	(7.69)
N58 = G55 U11 U14 + m.33 (U11 U34 + U14 U31) + H55 U31 U34	(7.70)
N66 = G55 U22 + 2 m.33 U12 U32 + H55 U22	(7.71)
N67 = G55 U12 U13 + m.33 (U13 U32 + U12 U33) + H55 U32 U33	(7.72)
N68 _ G55 U12 U14 + m33 (U14 U32 + U12 U34) + H55 U32 U34	(7.73)
N77 = G55 u23 + 2 m33 U13 U33 + H55 u23	(7.74)
N78 _ G55 U13 U14 + m33 (U14 U33 + U13 U34) + H55 U33 U34	(7.75)
N88 _ G55 U24 + 2m33 U14 U34 + H55 u24	(7.76)
a.2 a2 , a2 a3 + a2 a2 , (a2 - a2)2 , (2^3 - ag)2 ß G55 _ "a^" G1 + "ö^ G2 + ~a4" G3 +-404-GZl +-1204-gz2
| (a2 - a2)(2aa- ag)
+	/1	Gr
m33
a2 b2
4 (a2 - a2)(b2 - b2)Fz, + 12 >
+a3b3(aibiF2 + a2b2F3) + [a2b2 - a2b2 + a3(b2 - b2) + b2(a2 - a2)]Fm}
aibi a2b2Fi + - (a2 - a2)(b2 - b2)FZ, + ^ (ap2 - 2a3)(bp2 - 2b2)Fz2
y _ b2 b^ , b2 b3 + b2 b2 , (b2 - b2)2 , (2b3 - bP)2 y
«55 _ -tt2 H + H2 + H3 + —71:4— hzi + —tt^— hz2
b4 "" b4 2 b4 3 4b4 Z1 12b4
| (b2 -b2)(2b2 -bp) + -b4- Hm
7.6.5 Quark (Vckm)4x4 and (Vpmns)4x8 mixing matrices
Within this SU(3) family symmetry model, the transformation from massless to physical mass fermion eigenfields for quarks and charged leptons is
^o = Vo y(D ¥l and ^R = VO VR1' ,
and for neutrinos ¥V = UV UV ¥V. Recall now that vector like quarks, Eq.(7.3), are SU(2)l weak singlets, and hence, they do not couple to W boson in the interaction basis. In this way, the interaction of L-handed up and down quarks; fOLT = (uO,cO,tO)L and fdLT = (dO, sO,bO)L, to the W charged gauge boson is
^fuLYfdLW^ = -gL^uL [(Vol VU1L))3x4]T (Vol V^'^ Y^dL ,
(7.77)
g is the SU(2)L gauge coupling. Hence, the non-unitary VCKM of dimension 4 x 4 is identified as
(Vckm)4x4 = [(Vol Vu1l))3x4]t (Vol V^^	(7.78)
Similar analysis of the couplings of active L-handed neutrinos and L-handed charged leptons to W boson, leads to the lepton mixing matrix
(Upmns)4x8 = [(Vol V(l')3x4]t (U Uv)3x8	(7.79)
7.7 Conclusions
We reported an updated and general analysis for the generation of neutrino masses and mixing within the SU(3) family symmetry model. The right handed neutrinos (ve v^ vT)R, and the vector like completely sterile neutrinos NL,R, the flavon scalar fields and their VEV's introduced to break the symmetries: ®u, ® d, ni, n2 and n3, all together, yields a 8 x 8 general Majorana neutrino mass matrix with four or five massless neutrinos at tree level. Therefore, light neutrinos get tiny masses from radiative corrections mediated by the heavy SU(3) gauge bosons. Neutrino masses and mixing are extremely sensitive to the parameter space region, and a global fit for all quark masses and mixing together with neutrino masses and lepton mixing is in progress.
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. This work was partially supported by the "Instituto Politecnico Nacional", (Grants from EDI and COFAA) and "Sistema Nacional de Investigadores" (SNI) in Mexico.
References
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2.	J.M. Conrad, W.C. Louis, and M.H. Shaevitz, arXiv:1306.6494; I. Girardi, A. Meroni, and S.T. Petcov, arXiv:1308.5802; M. Laveder and C. Giunti, arXiv:1310.7478; A. Palazzo, arXiv:1302.1102; O. Yasuda, arXiv:1211.7175; J. Kopp, M. Maltoni and T. Schwetz, Phys. Rev. Lett. 107, (2011) 091801; C. Giunti, arXiv:1111.1069; 1107.1452; F. Halzen, arXiv:1111.0918; Wei-Shu Hou and Fei-Fan Lee, arXiv:1004.2359; O. Yasuda, arXiv:1110.2579; Y.F. Li and Si-shuo Liu, arXiv:1110.5795; B. Bhattacharya, A. M. Tha-lapillil, and C. E. M. Wagner, arXiv:1111.4225; J. Barry, W. Rodejohann and He Zhang, arXiv:1110.6382[hep-ph]; JHEP 1107,(2011)091; F.R. Klinkhamer, arXiv:1111.4931[hep-ph].
3.	A. Hernandez-Galeana, Rev. Mex. Fis. Vol. 50(5), (2004) 522. hep-ph/0406315.
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Bled Workshops in Physics
Vol. 15, No. 2 JLV
Proceedings to the 17th Workshop What Comes Beyond ... (p. 115) Bled, Slovenia, July 20-28, 2014
8 Gravitational Interactions and Fine-Structure Constant
U.D. Jentschura1'*, J.H. Noble1 and I. Nandori2
1 Department of Physics, Missouri University of Science and Technology Rolla, Missouri 2MTA-DE Particle Physics Research Group, P.O.Box 51, H-4001 Debrecen, Hungary
Abstract. Electromagnetic and gravitational central-field problems are studied with rela-tivistic quantum mechanics on curved space-time backgrounds. Corrections to the transition current are identified. Analogies of the gravitational and electromagnetic spectra suggest the definition of a gravitational fine-structure constant. The electromagnetic and gravitational coupling constants enter the Einstein-Hilbert-Maxwell Lagrangian. We postulate that the variational principle holds with regard to a global dilation transformation of the space-time coordinates. The variation suggests is consistent with a functional relationship of the form cxqED « (aG )1/2, where cxqED is the electrodynamic fine-structure constant, and aG its gravitational analogue.
Povzetek. Avtorji obravnavajo Diracov delec v elektromagnem in gravitacijskem centralno simetričnem polju. Poisčejo popravke za emisijo fotona v prisotnosti gravitacijskega polja. Po analogiji s spektrom elektrona v centralnosimetricnem potencialu definirajo tudi konstanto gravitacijske fine strukture. Predpostavijo, da velja variacijsko nacelo za transformacijo koordinat prostor-casa z globalno dilatacijo. Predlagana variacija je skladna s funkcijsko zvezo oblike cxqED « (aG)1/2, kjer je cxqED konstanta elektrodinamicne fine strukture, ki ima gravitacijski analog aG.
8.1 Introduction
If we are ever to gain a better understanding of the relationship of gravitational interactions and electrodynamics in the quantum world, then a very practical approach is to try to solve a number of important example problems in gravitational theory, whose solution is known in electromagnetic theory, to try to generalize the approach to the gravitational analogue, and to compare. In order to proceed, it is not necessarily required to quantize space-time itself [1]. Indeed, the formulation of quantum mechanics on curved-space backgrounds in itself constitutes an interesting problem [2-6]. A priori, one might think that the simple substitution 3/3xT —» Vi is the Schrodinger equation might suffice. Here, 3/3xT is the ith partial derivative with respect to the ith spatial coordinate, whereas Vi is the ith covariant derivative. However, this naive approach is destined to fail; the gravitational theory of Einstein and Hilbert inherently is a relativistic theory, and
* E-mail: ulj@mst.edu
the only way to describe quantum particles on curved space-times is to start from a fully relativistic wave function. The Dirac equation
(iy^i - m) ^(x) = 0	(8.1)
generalizes as follows to a curved space-time background [2-6],
(iy^(x)(3^ - r,J - m) ^(x) = 0.	(8.2)
The Dirac algebra [7-9] needs to be generalized to the local metric g|v (x),
|y^(x),yv(x)} = 2 g|v(x), a|v(x) = 2[y^(x),yv(x)].	(8.3)
The spin connection matrix is given as
^ = -4 gpa(x) ^(x) - rv) tfpv (x),	(8.4)
where repeated indices are summed. Finally, the a and b coefficients belong to the square root of the metric,
Yp(x)= bpa(x)ya , Ya(x) = a%(x) Yp,	(8.5)
where the Ya are the flat-space Dirac matrices, which are preferentially used in the Dirac representation [9-11,1,12]. The Christoffel symbols are ravp = ravp(x).
8.2 Central-Field Problem 8.2.1 Foldy-Wouthuysen Method
The Foldy-Wouthuysen method [13,14] is a standard tool for the extraction of the physical, nonrelativistic degrees of freedom, from a fully relativistic Dirac theory. The general paradigm is as follows: The positive and negative energy solutions of a (generalized) Dirac equation are intertwined in the fully relativistic formalism. One has to separate the upper and lower spinors in the bispinor solution, and in order to do so, one eliminates the "off-diagonal couplings" of the upper and lower spinor components order by order in some perturbative parameters, possibly,
using iterated (unitary) transformations.
For the plain free Dirac Hamiltonian, a standard method exists to all orders in perturbation theory, while for more difficult problems, one manifestly has to resort to a perturbative formalism [13,14]. A suitable expansion parameter in a general case is the particle's momentum operator. Let us consider a space-time metric of the form
g iv = diag (w2 (r ), -v2 (r), -v2 (r), -v2 (r) )
(8.6)
Fig. 8.1. The flat-space photon emission vertex (left figure) is promoted to a curved-space vertex (right figure) in general relativity. The curved background leads to higher-order corrections to the transition current, which are summarized, for the Schwarzschild metric, in Eq. (8.12).
The Schwarzschild metric in isotropic coordinates (see Sec. 43 of Chap. 3 of Ref. [15]), involves the Schwarzschild radius rs,
w = (1 - £ 4r
v _ (1+4r)2
i + i? )-' _ 4r—rs
4r/ 4r + rs
i - 2r-
1 + rs w _ 16 r2 (4r - rs) 2r ' v _ (4r + rs)3
1 - rs.
(8.7)
The Schwarzschild radius reads as rs = 2GM, where G is Newton's gravitational constant, and M is the mass of the planet (or "black hole"). The Hamiltonian or time translation operator is necessarily "noncovariant" in the sense that the time coordinate needs to be singled out. If we insist on using the time translation with respect to the time coordinate dt in the metric ds2 = w2 (r) dt2 — v2 (r)dr2 and bring the Hamiltonian into Hermitian form [see Ref. [16] and Eqs. (9)-(13) of Ref. [10]], then we obtain
hds _ 2 {a.* o - r? )}+fm o - 2r ) '
(8.8)
where a1 = y0 y1 is the Dirac a matrix (we here use the Dirac representation). The Foldy-Wouthuysen transformed Dirac-Schwarzschild Hamiltonian is finally obtained as [10]
Hfw _ flm + 2m - - f^
(8.9)
„ / 3rs
+f - sm
+
3nrs 4m
6(3)(ri+srs _3
3rs I -L 8m
r
The parity-violating terms obtained in Refs. [17,16] are spurious.
118 U.D. Jentschura, J.H. Noble and I. Nandori 8.2.2 Transition Current
As we couple the Dirac-Schwarzschild Hamiltonian (8.8) to an electromagnetic field (see Fig. 8.1), it is clear that the transition current in the interaction Hamiltonian is H;nt = —j • A. takes the form
ji = !{1 — y ,ai exp(ik • r)} .	(8.10)
We now employ the multipole expansion
. ^ 1 . ^ .
ai exp(ik • r) « ai + ai (ik • r) — ^(k • r)2	(8.11)
A unitary transformation with the same generators are used for the Dirac-Schwarzschild Hamiltonian then yields the result [10],
>fw=m—pm—2m (k x»)'+1 {m, ('k-T
—	4 {(k^2,m} + 2m(k r) (kxS)'
—	3/pi r±\ + r±(gxr)i — 1 /(ik.r) {pi rs
4 \ m' r J 2r mr2 2 \ V /' \ m' r 3irs (k x a) 1 {rs	pi}	, „ ,
+ -Xs --L + Z " (ik^r), - .	(8.12)
4r m 4 r	m
This result contains a gravitational kinetic correction, and gravitational corrections to the magnetic coupling, in addition to the known multipole and retardation corrections [14,18].
8.2.3 Spectrum
The bound-state spectrum resulting from the Hamiltonian (8.8) has recently been evaluated as [12],
= —	+ aGmeC2 ((8.13)
(7j + 5) 5£,j+1/2 (7j + 2) 6t,j_i/2 N (j + 1)(2j + 1) n3 j (2j + 1) n3 ) aGmec2 aGmec2 /15 14 k + 3 \ = 2n2 + n3 V8n — 2 |k| (2k + 1)/ ,
where I is the orbital angular momentum, j is the total angular momentum of the bound particle, and k is the (integer) Dirac angular quantum number,
k = 2(£ — j) (j + 1/2) = (—1 )j+*+1/2 (j + .	(8.14)
For a bound electron-proton system, the coupling constant entering the gravitational spectrum given in Eq. (8.13) reads as
ac = Gme mp = 3.21637(39) x 10-42 .	(8.15)
h c
The coupling aG is much larger than for particles bound to macroscopic objects. By contrast, the electrodynamic coupling parameter
e2	1
«QED = -T—r- ~ nvmr	(8.16)
4nhe0c 137.036
is just the fine-structure constant.
8.3 Global Dilation Transformation 8.3.1 Lagrangian
The analogy of the leading (Schrodinger) term in Eq. (8.13) for the nonrelativistic contribution to the bound-state energy (under the replacement aG —» oqed) may encourage us to look for connections of gravitational and electromagnetic interactions on a more global scale, possibly, using scaling transformations [19]. Indeed, the first attempts to unify electromagnetism with gravity are almost 100 years old [20,21]. Let us apply a scaling transformation to the boson and fermion fields,
AAP, AAp, ^^A^,	(8.17)
combined with a transformation of the coordinates,
A-1/2 xp, A-1/2 xp,	(8.18)
and of the metric
AgPv, A-1 gpv,	(8.19)
Under this transformation, the space-time intervals, the integration measure, the Ricci tensor Rpv and the curvature scalar R, transform as follows,
ds2 = gPv dxp dxv = gpv dxp dx^ -> ds2 ,	(8.20a) d4x
d4x = d4x^ —T , det g = det g„v -> A4 det g,	(8.20b) A2
R^v -> A RPv , R = gpv R^^ —> R.	(8.20c)
The Einstein-Maxwell Lagrangian, with a coupling to the fermion terms, is given as
R 1
S =
d4x	- ^
+^(x) ( Vp - e Ap) - m] ^ (x)} .	(8.21)
120 U.D. Jentschura, J.H. Noble and I. Nandori It transforms into
R
S' =
^ V-A4 det g
__A2 f
16nG 4 F
+A2 ^(x) [i A-1/2 (A1/2V^ - eAA^ which can be rearranged into
^(x)}
S'
S " = ^ =
A2
d4x \J-det g
_R__1 F
16nGA2 4
iY^ (V^ - e A1/2 A J - m
^(x)} .
The Lagrangian S " is the same S, but with scaled coupling constants,
G -> A2 G ,
e2 -> Ae2.
(8.22)
(8.23)
(8.24)
This scaling suggests a deeper connection of the coupling constants of electromagnetic and gravitational interactions, which is explored in further detail in Ref. [19].
m
8.3.2 Coupling Constants
If we assume that the scaling (8.24) holds globally, with the current Universe "picking" a value of A, then this scaling might suggest a relationship of the type
aQED < e4 <x A2 <x G .	(8.25)
Indeed, as discussed in Ref. [19], a relationship of the type aQED < %/G is otherwise suggested by string theory; the rough analogy being that gravitational interactions in string theory correspond to "closed" strings while electromagnetic interactions correspond to "open" strings. The product of two "open" string amplitudes is proportional to e2 < g2 < «qed, while the "closed"-string amplitude is proportional to k < gc < %/G. According to Eq. (3.7.17) of Ref. [22], the proportionality
gO < gc ^ «qed < vG	(8.26)
therefore is suggested by string theory. A simple analytic form of the proportionality factor in the relationship aQED < %/G has recently been given in Eq. (8) of Ref. [1]. QED
8.4 Conclusions
We have performed an analysis of the gravitationally coupled Dirac equation in the curved space-time surrounding a central gravitating object, which is described by the (static) Schwarzschild metric. The Foldy-Wouthuysen method leads to gravitational zitterbewegung terms and the gravitational spin-orbit coupling, which is also known as the Fokker precession term. In a curved space-time, the photon
emission vertex receives additional corrections due to the curved background, which can be given, within the multipole expansion and for a conceptually simple background metric (e.g., the Schwarzschild metric), in closed analytic form (at least for the first terms of the multipole and retardation expansion). The gravitational bound states display a certain analogy for the gravitational as compared to the electromagnetic (Schrodinger) central-field problem. Based on this analogy, one may explore possible connections of the gravitational and electromagnetic coupling constants, based on scaling arguments. Such a scaling transformation gives additional support for the relationship aQED <x %/G, which has been suggested by string theory [22].
Acknowledgements
The research has been supported by the National Science Foundation (Grants
PHY-1068547 and PHY-1403973).
References
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4.	M. Soffel, B. MUller, and W. Greiner, Particles in a stationary spherically symmetric gravitational field, J. Phys. A 10 (1977) 551-560.
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6.	H. Gies and S. Lippoldt, Fermions in gravity with local spin-base invariance, Phys. Rev. D 89 (2014) 064040.
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10.	U. D. Jentschura and J. H. Noble, Nonrelativistic Limit of the Dirac-Schwarzschild Hamilto-nian: Gravitational Zitterbewegung and Gravitational Spin-Orbit Coupling, Phys. Rev. A 88 (2013) 022121 [6 pages].
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12.	U. D. Jentschura, Fine-Structure Constant for Gravitational and Scalar Interactions, Phys. Rev. A 90 (2014) 022112.
13.	L. L. Foldy and S. A. Wouthuysen, On the Dirac Theory of Spin 1/2 Particles and Its Non-Relativistic Limit, Phys. Rev. 78 (1950) 29-36.
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16.	Y. N. Obukhov, Spin, Gravity and Inertia, Phys. Rev. Lett. 86 (2001) 192-195.
17.	J. F. Donoghue and B. R. Holstein, Quantum mechanics in curved space, Am. J. Phys. 54 (1986) 827-831.
18.	U. Jentschura and K. Pachucki, Higher-order binding corrections to the Lamb shift of 2P states, Phys. Rev. A 54 (1996) 1853-1861.
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20.	T. Kaluza, Preussische Akademie der Wissenschaften (Berlin), Sitzungsberichte, (1921) 966-972.
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Bled Workshops in Physics Vol. 15, No. 2
A
Proceedings to the 17th Workshop What Comes Beyond ... (p. 123) Bled, Slovenia, July 20-28, 2014

9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation?
N.S. Mankoc Borštnik
Department of Physics, FMF, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
Abstract. The spin-charge-family theory [1-12,14], in which spinors carry besides the Dirac spin also the second kind of the Clifford object, no charges, is a kind of the Kaluza-Klein theories [13]. The Dirac spinors of one Weyl representation in d = (13 + 1) manifest [1,4,3,10,14,15] in d = (3 + 1) at low energies all the properties of quarks and leptons assumed by the standard model. The second kind of spins explains the origin of families. Spinors interact with the vielbeins and the two kinds of the spin connection fields, the gauge fields of the two kinds of the Clifford objects, which manifest in d = (3 + 1) besides the gravity and the known gauge vector fields also several scalar gauge fields. Scalars with the space index s € (7, 8) carry the weak charge and the hyper charge j, ± -, respectively), explaining the origin of the Higgs and the Yukawa couplings. It is demonstrated in this paper that the scalar fields with the space index t € (9,10,..., 14) carry the triplet colour charges, causing transitions of antileptons and antiquarks into quarks and back, enabling the appearance and the decay of baryons. These scalar fields are offering in the presence of the right handed neutrino condensate, which breaks the CP symmetry, the answer to the question about the matter-antimatter asymmetry.
Povzetek. V teoriji spinov-nabojev-družin [1-12,14] nosijo spinorji dve vrsti kvantnih stevil, ki jih določata dve vrsti operatorjev ya: Diracovi operatorji ya in avtoričini Ya, obe sta povezani z mnoZenjem Cliffordovih objektov, ena vrsta z leve, druga z desne. Obe vrsti spina sta neodvisni in tvorita druga drugi ekvivalentne upodobitve. Analiza Lorentzove grupe SO(13,1) s podgrupami te grupe pokaze, da vsebuje ena Weylova upodobitev Dira-covih spinorjev v d = (13 + 1) vse kvarke in leptone (ter antikvarke in antileptone) s kvantnimi stevili kot jih predpise standardni model pred elektrosibko zlomitvijo, le da so desnorocni nevtrini enakopravni partnerji elektronom [1,4,3,10,14,15]. Druga vrsta spina pojasni izvor druzin. Spinorji interagirajo s tetradami in s polji dveh vrst spinskih povezav, ki so umeritvena polja obeh vrst operatorjev gamma. Po zlomitvi simetrij, tedaj pri opa-zljivih nizkih energijah, dolocajo ta polja, skupaj z vektorskimi sveznji, gravitacijo in vsa znana umeritvena vektorska polja. Dolocajo pa tudi skalarna polja. Skalarna polja s prostorskim indexom s = (7, 8) so sibki dubleti (t13 = ^ 1 ,Y = ± -), kar pojasni izvor Higg-sovega skalarnega polja in Yukawinih sklopitev. Skalarna polja s prostorskim indeksom t € (9,10,..., 14) pa so barvni tripleti, ki povzrocajo prehode antileptonov in antikvarkov v kvarke in obratno, kar omogoci nastanek in razpad barionov. Vsa skalarna polja nosijo glede na kvantna stevila, ki jih dolocajo Diracovi ya in druzinski Ya, tudi druzinska in Diracova kvantna stevila v adjungirani upodobitvi grup. Lepota te teorije je, da en sam kon-denzat iz dveh desnorocnih nevtrinov z druzinskimi kvantnimi stevili, ki niso druzinska kvantna stevila spodnjih stirih druzin, zlomi diskretno simetrijo CP in poskrbi za maso
vseh skalarnih polj, ter še neopaženega vektorskega polja. Ta skalarna polja ponujajo v prisotnosti kondenžata desnoročnih nevtrinov, ki žlomi simetrijo CP, odgovor na vprasanje kako je v nasem vesolju nastala opažljiva asimetrija med snovjo in antisnovjo. Skalarna polja s prostorskim indexom s = (7, 8) žlomijo ž neničelno vakuumsko pričakovano vrednostjo se sibki in hipernaboj, in spremenijo tudi lastno maso, ter tako pojasnijo vse privžetke standardnega modela. Ker teorija napoveduje dve ločeni gruči po stiri družine kvarkov in leptonov, pojasni stabilna od žgornjih sirih družin ižvor temne snovi. Teorija pa napoveduje tudi, da bodo na LHC ižmerili četrto k trem že opaženim družinam, ižmerili pa bodo tudi več skalarnih polj.
9.1 Introduction
The spin-charge-family [1-12,14] theory is offering, as a kind of the Kaluža-Klein like theories, the explanation for the čharges of quarks and leptons (right handed neutrinos are in this theory the regular members of a family) and antiquarks and antileptons [15,16], and for the existenče of the čorresponding gauge večtor fields. The theory explains, by using besides the Dirač kind of the Clifford algebra obječts also the sečond kind of the Clifford algebra obječts (there are only two kinds [57,3,17,18,20,19], assočiated with the left and the right multipličation of any Clifford obječt), the origin of families of quarks and leptons and čorrespondingly the origin of the sčalar gauge fields čausing the elečtroweak break. These sčalar fields are responsible, after gaining nonžero vačuum expečtation values, for the masses and mixing matričes of quarks and leptons [9-11] and for the masses of the weak večtor gauge fields. They manifest, čarrying the weak čharge and the hyper čharge equal to (± 2, T 2, respečtively) [14], as the Higgs field and the Yukawa čouplings of the standard model.
The spin-charge-family theory predičts two dečoupled groups of four families [3,4,9-11]: The fourth of the lower group will be measured at the LHC [10], while the lowest of the upper four families čonstitutes the dark matter [12].
This theory also predičts the existenče of the sčalar fields whičh čarry the triplet čolour čharges. All the sčalars fields čarry the fračtional quantum numbers with respečt to the sčalar index s > 5, either the ones of SU(2) or the ones of SU(3), while they are with respečt to other groups in the adjoint representations. Neither these sčalar fields nor the sčalars čausing the elečtroweak break are the supersymmetrič sčalar partners of the quarks and leptons, sinče they do not čarry all the čharges of a family member.
These sčalar fields with the triplet čolour čharges čause transitions of antileptons into quarks and antiquarks into quarks and bačk, offering, in the presenče of the čondensate of the two right handed neutrinos with the family quantum numbers belonging to the upper four families whičh breaks the CP symmetry, the explanation for the matter-antimatter asymmetry. This is the topič of the present paper.
Let me point out that the spin-charge-family theory overlaps in many points with other unifying theories [26-31], sinče all the unifying groups čan be seen as the subgroups of the large enough orthogonal groups, with family groups inčluded. But there are also many differenčes. While the theories built on čhosen
groups must for their choice propose the Lagrange densities designed for these groups and representations (which means that there must be a theory behind this effective Lagrange densities), the spin-charge-family theory starts with a very simple action, from where all the properties of spinors and the gauge vector and scalar fields follow, provided that the breaks of symmetries occur.
Consequently this theory differs from other unifying theories in the degrees of freedom of spinors and scalar and vector gauge fields which show up on different levels of the break of symmetries, in the unification scheme, in the family degrees of freedom and correspondingly also in the evolution of our universe.
It will be demonstrated in this paper that one condensate of two right handed neutrinos makes all the scalar gauge fields and all the vector gauge fields massive on the scale of the appearance of the condensate, except the vector gauge fields which appear in the standard model action before the electroweak break as massless fields. The scalar gauge fields, which cause the electroweak break while gaining nonzero vacuum expectation values and changing their masses, then explain masses of quarks and leptons and of the weak bosons.
It is an extremely encouraging fact, that one scalar condensate and the nonzero vacuum expectation values of some scalar fields, those with the weak and the hyper charge equal to by the standard model required charges for the Higgs's scalar, can bring the simple starting action in d =(13 +1) to manifest in d = (3 +1) in the low energy regime the observed phenomena of fermions and bosons, explaining the assumptions of the standard model and can possibly answer also the open questions, like the ones of the appearance of family members, of families, of the dark matter and of the matter-antimatter asymmetry.
The paper leaves, however, many a question connected with the break of symmetries open. Although the scales of breaks of symmetries can roughly be estimated, for the trustworthy predictions a careful study of the properties of fermions and bosons in the expanding universe is needed. It stays to be checked under which conditions in the expanding universe, the starting fields (fermions with the two kinds of spins and the corresponding vielbeins and the two kind of the spin connection fields) after the spontaneous breaks manifest in the low energy regime the observed phenomena. This is a very demanding study, a first simple step of which was done in the refs. [12,22]. The present paper is a step towards understanding the matter-antimatter asymmetry within the spin-charge-family theory.
In the subsection 9.1.1 I present the action and the assumptions of the spin-charge-family theory, with the comments added.
In sections (9.2,9.4,9.5,9.3) the properties of the scalar and vector gauge fields and of the condensate are discussed. In appendices the discrete symmetries of the spin-charge-family theory and the technique used for representing spinors, with the one Weyl representation of SO(13,1) and the families in SO (7,1) included, is briefly presented. The final discussions are presented in sect. 9.7.
9.1.1 The action of the spin-charge-family theory and the assumptions
In this subsection all the assumptions of the spin-charge-family theory are presented and commented. This subsection follows to some extend a similar subsection of
the ref. [14].
i.	The space-time is d(= (13 +1)) dimensional. Spinors carry besides the internal degrees of freedom, determined by the Dirac ya's operators, also the second kind of the Clifford algebra operators [5-7,4], called ya's.
ii.	In the simple action [3,1] fermions ^ carry in d = (13 + 1) only two kinds of spins, no charges, and interact correspondingly with only the two kinds of the spin connection gauge fields, daba and cd aba, and the vielbeins, faa.
S = | ddx E Lf +
ddxE (aR + aR),
Lf = 2 (^yapoa^) + h.c.,
poa = faapoa + 2^ {Pa,Efaa}-, POa = Pa — 2Sabdaba — 2SSab &aba,
R = 2 (fa[afpb] (daba,p — dcaa d%p)} + h.C. ,
R = 2 fa[af«b] (ddaba,P — CDcaadd%p) + h.C. .	(9.1)
Here 1 fa[af«b] = faafPb — fabfPa. sab and Sab are generators (Eqs.(9.5, 9.37, 9.37) of the groups SO(13,1) and SO(13,1), respectively, expressible by ya and y a.
iii.	The manifold M(13+1) breaks first into M(7+1) times M(6) (which manifests as SU(3) xU(1)), affecting both internal degrees of freedom, SO(13 + 1) and SO(13 + 1). After this break there are 2((7+1 )/2-1) massless families, all the rest families get heavy masses 2.
Both internal degrees of freedom, the ordinary SO(13 + 1) one (where ya determine spins and charges of spinors) and the SO (13 +1) (where ya determine family quantum numbers), break simultaneously with the manifolds.
iv.	There are additional breaks of symmetry: The manifold M(7+1' breaks further
1	faa are inverted vielbeins to eaa with the properties eaafab = Sab, eaafPa = 6«, E = det(eaa). Latin indices a, b,.., m, n,.., s, t,.. denote a tangent space (a flat index), while Greek indices a, |3,.., |a, v, ..a, 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,..), indices from the bottom of the alphabets indicate the compactified dimensions (s, t,.. and a, T,..). We assume the signature r|ab = diag{1, —1, —1, • • • , —1}.
2	A toy model [22,23,15] was studied in d = (5 + 1) with the same action as in Eq.'(9.1). For a particular choice of vielbeins and for a class of spin connection fields the manifold Ms+1 breaks into M(3+1' times an almost S2, while 2((3+1 )/2-1' families stay massless and mass protected. Equivalent assumption, although not yet proved that it really works, is made also in the case that M(13+1) breaks first into M(7+1) x M(6). The study is in progress.
9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 127 into M(3+1)x M(4).
v.	There is a scalar condensate of two right handed neutrinos with the family quantum numbers of the upper four families, bringing masses of the scale above the unification scale, to all the vector and scalar gauge fields, which interact with the condensate.
vi.	There are nonzero vacuum expectation values of the scalar fields with the scalar indices (7,8), which cause the electroweak break and bring masses to the fermions and weak gauge bosons, conserving the electromagnetic and colour charge.
Comments on the assumptions:
i.: There are, as already written above, two (only two) kinds of the Clifford algebra objects. The Dirac one (Eq.(9.35)) (ya) will be used to describe spins of spinors (fermions) in d = (13 + 1), manifesting in d = (3 + 1) the spin and all the fermion charges, the second one (Eq.(9.35)) (ya) will describe families of spinors. The representations of Ya's and ya's are orthogonal to one another 3. There are correspondingly two groups determining internal degrees of freedom of spinors in d = (13 + 1): The Lorentz group SO(13,1) and the group SO(13,1). One Weyl representation of SO (13,1) contains, if analysed [1,3,4,15] with respect to the standard model groups, all the family members, assumed by the standard model, with the right handed neutrinos included (the family members are presented in table 9.3). It contains the left handed weak (SU(2)i) charged and SU(2)n chargeless colour triplet quarks and colourless leptons (neutrinos and electrons), the right handed weakless and SU(2)II charged quarks and leptons, as well as the right handed weak charged and SU(2)II chargeless colour antitriplet antiquarks and (anti)colourless antileptons, and the left handed weakless and SU(2)II charged antiquarks and antileptons. The reader can easily check the properties of the representations of spinors (table 9.3), presented in the "technique" (appendix 9.9) way, if using Eqs. (9.5, 9.8,9.9, 9.11,9.14).
Each family member carries the family quantum numbers, originating in ya's
degrees of freedom. Correspondingly Sab changes the family quantum numbers, leaving the family member quantum number unchanged.
ii.: This starting action enables to represent the standard model as an effective low energy manifestation of the spin-charge-family theory, which explains all the standard model assumptions, with the families included. There are gauge vector fields, massless before the electroweak break: gravity, colour SU(3) octet vector gauge fields, weak SU(2) (it will be named SU(2)I) triplet vector gauge field and "hyper" U(1) (it will be named U(1 )I) singlet vector gauge fields. All are superposition of fac cuaba. There are (eight rather than the observed three) families of quarks and leptons, massless before the electroweak break.
These eight families are indeed two decoupled groups of four families, in the fundamental representations with respect to twice SU(2)x SU(2) groups, the
3 One can learn in Eq. (9.44)of appendix (9.9) that S ab transforms one state of the representation into another state of the same representation, while S ab transforms the state into the state belonging to another representation.
subgroups of SO(3,1) x SO(4) G SO(7,1). The scalar gauge fields, determining the mass matrices of quarks and leptons, carry with respect to the scalar index s G (7,8) the weak and the hyper charge of the scalar Higgs, while they carry if they are the superposition of fas & aba two kinds of the family quantum numbers in the adjoint representations, representing two (orthogonal) groups, each of the group contains two triplets (with respect to SU(2)gO(3 1) x SU(2)so(4)). The scalar fields with the quantum numbers (Q, Q', Y', which are the superposition of fas &aba) are the three singlets, again carrying the weak and the hyper charge of the scalar Higgs. One group of two triplets determine, together with the three singlets, after gaining nonzero expectation values, the Higgs's scalar and the Yukawa couplings of the standard model. The starting action contains also the additional SU(2)n (from SO(4)) vector gauge field and the scalar fields with the space index s G (5,6) and t G (9,10,11,12), as well as the auxiliary vector gauge fields expressible (Eqs. (9.56, 9.55) in the appendix 9.10) with vielbeins. They all remain either auxiliary (if there are no spinor sources manifesting their quantum numbers) or become massive after the appearance of the condensate. iii., iv.: The assumed break from M(13+1' first into M(7+1' times M(6) (manifesting the symmetry SU(3) xU(1 )n) explains why the weak and the hyper charge are connected with the handedness of spinors. In the spinor representation of SO (7,1) there are left handed weak charged quarks and leptons with the hyper charges (6, — 2, respectively) and the right handed weak chargeless quarks with the hyper charge either | or — 3, while the right handed weak chargeless leptons carry the hyper charge either 0 or (—1). A further break from M(7+1' into M(3+1' x M4), manifesting the symmetry SO(3,1) xSU(2)i x SU(2)n xU(1)„x SU(3), explains the observed properties of the family members - the coloured quarks, left handed weak charged and SU(2)II chargeless and right handed weak chargeless and SU(2)ii charged and colourless leptons, again left handed weak charged and SU(2)II chargeless and right handed weak chargeless and SU(2)II charged, quarks with the "spinor" charge 6 and leptons with the "spinor" charge — 2 - and of the observed vector gauge fields and the scalar fields (through Higgs's scalar and Yukawa couplings).
Since the left handed members distinguish from the right handed partners in the weak and the hyper charges, the family members of all the families stay massless and mass protected up to the electroweak break 4. Antiparticles are accessible from particles by the Cn and Pn, as explained in refs. [15,16] and briefly also in the appendix (9.8). This discrete symmetry operator does not contain ya's degrees of freedom. To each family member there corresponds the antimember, with the same family quantum number.
v.: It is a condensate of the two right handed neutrinos with the quantum numbers of the upper four families (table 9.2), appearing in the energy region
4 As long as the left handed family members and their right handed partners carry different conserved charges, they can not behave as massive particles, they are mass protected. It is the appearance of nonzero vacuum expectation values of the scalar fields, carrying the weak and the hyper charge, which cause non conservation of these two charges, which makes the superposition of the left and the right handed family members possible, and breaks the mass protection.
above the unification scale, which makes all the scalar gauge fields (those with the space index (5,6,7,8), as well as those with the space index (9,..., 14)) and the vector gauge fields, manifesting nonzero quantum numbers t4, t23, Q ,Y, t4, t23, Q ,Y,NR (Eqs. (9.8,9.9, 9.11,9.12, 9.13,9.14)) massive.
vi.: At the electroweak break the scalar fields with the space index s = (7,8), triplets with respect to the family index (originating in d>abs, Eq. (9.16)) and the three singlets carrying the charges (Q, Q', Y') (originating in dts's, Eq. (9.15)), all with the weak and the hyper charge equal to (^ 2, ± 1, respectively), get nonzero vacuum expectation values, changing also their masses and breaking the weak and the hyper charge symmetry. These scalars determine mass matrices of twice the four families, as well as the masses of the weak bosons.
All the rest scalar fields keep masses of the condensate scale and are correspondingly unobservable in the low energy regime 5. The fourth family to the observed three ones will (sooner or later) be observed at the LHC. Its properties are under the consideration [10], while the stable of the upper four families is the candidate for the dark matter constituents.
The above assumptions enable that the starting action (Eq. (9.1)) manifests effectively in d = (3 + 1) in the low energy regime fermion and boson fields as assumed by the standard model.
To see this [3,1,4-8,2,9,10,12,14], let us formally rewrite the Lagrange density for a Weyl spinor of (Eq.(9.1)), as follows
Lf =	(pm - X gATAiAmi)^ +
Ai
{Y_ ^YsP0s +
s = 7,8
{ Y_ ^POt ,
t=5,6,9,...,14
POs = Ps - 2SS S ds's"s - ^Sabdabs ,
POt = Pt - 2StVdt't"t - 1Sabdabt ,	(9.2)
where m G (0,1,2,3), s G 7,8, (s', s") G (5,6,7,8), (a,b) (appearing in Sab) run within G (0,1,2,3) and G (5,6,7,8), t G (5,6,9,..., 13,14), (t',t") G (5,6,7,8) and G (9,10,..., 14). ^ represents all family members of all the families. The generators of the charge groups TAi (expressed in Eqs. (9.3), (9.9), (9.11) in terms of Sab) fulfil the commutation relations
„-Ai V" „Ai cab T = 2_ c ab S ,
a,b
{TAi, TBj}_ = i6ABfAijkTAk .	(9.3)
5 Correspondingly d = (13 + 1) manifests in d = (3 + 1) spins and charges as if there would be d = (9 + 1), since the plane (5, 6) and the plane in which the vector t4 lies, are unobservable at low energies.
The spin generators are defined in Eq. (9.8). These group generators determine all the internal degrees of freedom of one family members as seen from the point of view of d = (3 + 1): The colour charge (SU(3) with the generators t3) and the "spinor charge" (U(1 )n) with the generator t4 originating in SO(6), the weak charge SU(2)i with the generators t1 and the second SU(2)n charge with the generators t2 originating in SO(4) (SU(2)II breaks in the presence of the condensate into U(1 )I, defining together with t4 the hyper charge Y (= t23 + t4) and the spin determined by SO(3,1).
The condensate of two right handed neutrinos with the family quantum numbers of the upper four families bring masses (of the unifying scale > 1016 GeV or above) to all the scalar and those vector gauge fields which are not observed at so far measurable energies.
The scalar fields causing, when getting nonzero vacuum expectation values, the electroweak phase transitions changing at the transition also their own masses, bring masses to the eight families and to the weak bosons. We shall comment all these fields in what follows.
The first line of Eq. (9.2) describes [1,3] before the electroweak break the dynamics of eight families of massless fermions in interaction with the massless colour Am, weak Am and hyper A^ (= sin A^5 + cos A^J gauge fields, all are the superposition of cabm 6.
The second line of the same equation (Eq. (9.2)) determines the mass term, which after the electroweak break brings masses to all the family members of the eight families and to the weak bosons. The scalar fields responsible - after getting nonzero vacuum expectation values - for masses of the family members and of the weak bosons are namely included in the second line of Eq. (9.2) as (— 2Ss s"cs <-■ - —
Js's s
1S a b cD a b s, s =€ (7,8), (s ',s") € (5,6,7,8), (a ,b) G (0 ,T ,...,8)) 7. The properties of these scalar fields are discussed in sect. (9.4), where the proof is presented that they all carry the weak charge and the hyper charge as the standard model Higgs's scalar, while they are either triplets with respect to the family quantum numbers or singlets with respect to the charges Q, Q' and Y'. While the two triplets (A], AJN1 L) interact with the lower four families, interact (/A:?, A J"1 R) with the upper four families. These twice two triplets are superposition of ] Sab cDabs, s =€ (7,8), Eq. (9.16). The three singlets (AQ, AQ and Am') are superposition of cds 's"s, Eq.(9.15). They interact with the family members of all the families, "seeing" family members charges.
The third line of Eq. (9.2) represents fermions in interaction with all the rest scalar fields. Scalar fields become massive after interacting with the condensate. Those which do not gain nonzero vacuum expectation values, keep the heavy masses of the order of the scale of the condensate up to low energies. The massive
6	These superposition can easily be found by using Eqs. (9.11, 9.9). They are explicitly written in the ref. [3]. The interaction with the condensate makes the fields Am, Eq. (9.14), Am and Am2 very massive (at the scale of the condensate).
7	To point out that Sab and S ab belong to two different kinds of the Clifford algebra objects are the indices (a, b) are in Sab in this paragraph written as (a, b). Normally only (a, b) will be used for S ab and ¡5ab.
scalars with the space index t G (5,6) transform (table 9.3) uR-quarks into dL-quarks and vR-leptons into eL-leptons and back, as well as UR-antiquarks into dL-antiquarks and back and VR-antileptons into eL-antileptons and back, breaking in the presence of the condensate the Q global symmetry. Those scalar fields with the space index t = (9,10, ••• ,14) transform antileptons into quarks and antiquarks into quarks and back. They are offering in the presence of the scalar condensate breaking the CP symmetry the explanation for the observed matterantimatter asymmetry, as we shall show in sect. 9.2.
Let us write down the part of the fermion action which in the presence of the condensate offers the explanation for the observed matter/antimatter asymmetry.
Lf = V Y0 y' { X [Pt-( 2ss V Ws vt + 1st " Dt/ft
t=(9,10,...14)
+ 1sab CDabt)]}	(9.4)
where (s',s") G (5,6,7,8), (t,t',t") G (9,10,..., 14) and (a,b) G (0,1,2,3) and G (5,6,7,8), in agreement with the assumed breaks in sect. (9.1). Again operators Sab determine family quantum numbers and Sab determine family members quantum numbers. Correspondingly the superposition of the scalar fields cdabt and the superposition of the scalar fields C abt carry the quantum numbers determined by either the superposition of Sab or by the superposition Sab in the adjoint representations, while they carry the colour charge, determined by the space index t G (9,10,..., 14), in the triplet representation of the SU(3) group, as we shall see. Similarly the scalars with the space index s G (7,8) carry the weak and the hyper charge in the doublets representations.
The condensate of two right handed neutrinos with the family quantum numbers of the upper four families carries (table 9.2) t4 = 1, t23 = — 1, t13 = 0, Y = 0, Q = 0, and the family quantum numbers of the upper four families and gives masses to scalar and vector gauge fields with the nonzero corresponding quantum numbers. The only vector gauge fields which stay massless up to the electroweak break are the hyper charge field (A^), the weak charge field (Am) and the colour charge field (Am).
The standard model subgroups of the SO (13 + 1) and SO (13 + 1) groups and the corresponding gauge fields This section follows to large extend the refs. cite-JMP,NscalarsweakY2014. To calculate quantum numbers of one Weyl representation presented in table 9.3 in terms of the generators of the standard model charge groups tai (=2! ab cAlab Sab) one must look for the coefficients cAlab (Eq. (9.3)). Similarly also the spin and the family degrees of freedom can be expressed.
The same coefficients cAlab determine operators which apply on spinors and on vectors. The difference among the three kinds of operators - vector and two kinds of spinor - lies in the difference among Sab, Sab and Sab.
While Sab for spins of spinors is equal to
Sab = 7(Ya Yb - Yb Ya), (Ya ,Yb}+ = 2nab ,	(9.5)
and Sab for families of spinors is equal to
Sab = 4(YaYb - Yb Ya), i-Ya ,Yb}+ = 2nab ,
{Ya ,?b}+ = 0,	(9.6)
one must take, when Sab apply on the spin connections ^bde (= fae ^bda) and •bde (= fae •bda), on either the space index e or the indices (b, d, 13, d), the operator
(Sab)ceAd'.'e'.,g = i(nac6b - nbcsa) Ad...e...g.	(9.7)
This means that the space index (e) of ^bde transforms according to the requirement of Eq. (9.7), and so do b, d and 13, d. Here I used again the notation 13, d to point out that Sab and Sab (= Sa ) are the generators of two independent groups[14].
One finds [1,3-8,2] for the generators of the spin and the charge groups, which are the subgroups of SO(13,1), the expressions:
N±(= N(L,R)):= 2(S23 ± iS01,S31 ± iS02,S12 ± iS03),	(9.8)
where the generators N± determine representations of the two SU(2) invariant subgroups of SO(3,1), the generators T1 and T2,
t1 : = 1 (S58 - S67, S57 + S68, S56 - S78),	(9.9)
t2 : = 1 (S58 + S67, S57 - S68, S56 + S78),	(9.10)
determine representations of the SU(2)jX SU(2)n invariant subgroups of the group SO(4), which is further the subgroup of SO(7,1) (SO(4), SO(3,1) are subgroups of SO(7,1)), and the generators T3, t4 and T4
t3 := 1 {S912 - S1011 ,S911 + S1012, S910 - S11 12, S9 14 - S10 13 S913 + s10 14 5H 14 - g12 13
S11 13 + S1214, -L(S910 + S11 12 - 2S1314)}, V3
t4 := -1 (S910 + S11 12 + S1314),
T4 := -3(S910 + S1112 + S13 14),	(9.11)
determine representations of SU(3) x U(1), originating in SO (6), and of T4 originating in SO(6).
One correspondingly finds the generators of the subgroups of SO(7,1),
Nl,r : = 2(S23 ± iS01, S31 ± iS02, S12 ± iS03),	(9.12)
which determine representations of the two SU(2) invariant subgroups of SO (3,1 ), while
T : = 1 (S58 - S67, S57 + S68, S56 - S78),
T2 : = 2(S58 + S67, S57 - S68, S56 + S78),	(9.13)
determine representations of SU(2)ix SU(2)n of SO(4). Both, SO(3,1) and SO(4), are the subgroups of SO(7,1 ). One further finds [3]
Y	= t4 + t23 , Y' = —t4 tan2 §2 + t23 , Q = t13 + Y, Q' =-Ytan2§ + t13 ,
Y	= t4 + t23 , Y' = -t4 tan2 §2 + t23 , Q = Y + t13 , Q ' = - Ytan2 § 1 + t13 .
(9.14)
The scalar fields, responsible [1-3] - after getting in the electroweak break nonzero vacuum expectation values - for the masses of the family members and of the weak bosons, and presented in the second line of Eq. (9.2), can be expressed in terms of cabc fields and cDabc fields as presented in Eq. (9.15), 9.16).
One can find the below expressions by taking into account Eqs. (9.9,9.11,9.12, 9.13) and Eq. (9.14).
- 1ss 's" ds Vs = -(g23 t23 A23 + g13 t13 A]3 + g4 t4 a4) , g13 t13 A]3 + g23 t23 A23 + g4 t4 A4 = gQ QAQ + gQ' Q'aq' + gY' Y' Aj',
A4 = (D9 10 s + D11 12 s + Di3i4s) ,
A]3 = (D56s - D78s) , A;?3 = (D56s + D78s) ,
aq = sin§1 a]3 + cos§1 Aj , aq ' = cos§1 a]3 - sin§1 Aj , Aj = sin §2 A23 + cos §2 A4 , Aj' = cos §2 A23 - sin §2 A4 ,
(s G (7,8)).	(9.15)
In Eq. (9.15) the coupling constants were explicitly written to see the analogy with the gauge fields in the standard model.
- 2s a b CD a b s = -(t1 A ] + N L A N L + T2 A 2 + N R A N R ) ,
A11 = (cD58s - cD67s, cD57s + cD68s, cD56s - cD78s) , /C1 L = (cD23s + icD01 s, CD3Ts + icD02s, cDl2s + icD03s) , A2 = (cD58s + cD67s, cD57s - cD68s, cD56s + cD78s) ,
ANR = (cD23s - icD01 s, cD31 s - icD02s, cD12s - icD03s) ,
(s G (7,8)).	(9.16)
Scalar fields from Eq. (9.16) couple to the family quantum numbers, while those from Eq. (9.15) distinguish among family members.
Expressions for the vector gauge fields in terms of the spin connection fields and the vielbeins, which correspond to the colour charge (Am), the SU(2)n charge (Amm), the weak charge (SU(2)i) (Am) and the U(1 )n charge originating in SO(6) (Am), can be found by taking into account Eqs. (9.9, 9.11). Equivalently one finds the vector gauge fields in the "tilde" sector. One really can use just the expressions from Eqs. (9.15,9.16), if replacing the scalar index s with the vector index m.
9.2 Properties of scalar and vector gauge fields, contributing to transitions of antileptons into quarks
In this - the main - part of the present paper the properties, quantum numbers and discrete symmetries of those scalar and vector gauge fields appearing in the action (Eqs.(9.1,9.2), 9.4) of the spin-charge-family theory [1-9,12] are studied, which cause transitions of antileptons into quarks and back, and antiquarks into quarks and back.
These scalar gauge fields carry the triplet or antitriplet colour charge (see table 9.1) and the fractional hyper and electromagnetic charge.
The Lagrange densities from Eqs. (9.1, 9.2, 9.4) manifest Cn • Pn invariance (appendix (9.8)). All the vector and the spinor gauge fields are before the appearance of the condensate massless and reactions creating particles from antiparticles and back goes in both directions equivalently Correspondingly there is no matterantimatter asymmetry.
The spin-charge-family theory breaks the matter-antimatter symmetry by the appearance of the condensate (sect. 9.3) and also by nonzero vacuum expectation values of the scalar fields causing the electroweak phase transition (sect. 9.4). I shall show that there is the condensate of two right handed neutrinos which breaks this symmetry, giving masses to all the scalar gauge fields and to all those vector gauge fields which would be in contradiction with the observations.
Let us start by analysing the Lagrange density presented in Eq. (9.4) before the appearance of the condensate. The term y7 2 Ss s" ws s"t in Eq. (9.4) can be rewritten, if taking into account Eq. (9.42), as follows
y7 jss 's" ws vt = XL (©) > 's" w tt, ,
2	+-(7^) 2 s"s"(e)
w tt' : = w tt' = (ws's"t T i^s's"t'),
s"s"(© )	s"s"(±)
tt' tt' 1
:=(±)= 2 (y7 ± Y7 ),
(tt') e ((910), (11 12), (1314)).	(9.17)
tt '
I introduced the notations (©) and w tt' to distinguish among different super-
s"s"(© )
position of states in equations below.
tt ' ;
Using Eqs. (9.9, 9.11) the expression (©) i Ss s" œ tt ' can be further
s"s" (©)
rewritten as follows
tt' 1 ' .
-Ss s œ
s"s" (©)
2- ~ tt '
tt '
(@) {t2+ A2+ + t2- A2- + t23 A2;3' + t1+ A1+, + t1- A1- + t13 A13' },
J L	tt '	tt '	tt ' 1	tt '	tt '	tt ' J *
(©
a20
tt ' (©
A2tt,
(©
A'f±.
tt ' (±
Alt '
(©
(© ) (<© ) (© ) (© ) (©> )
) m i(œ tt ' - œ tt '),
L-1	57(© )	68(© )
=(œ tt'	+ œ tt '
58(© )	67(© )
= (œ tt '	+ œ tt '
56(© )	78 (© )
= œ tt ' -	- œ tt ' )
58(© )	67(© )
= (œ tt '	- œ tt '
56(© )	78 (© )
),
) m i(œ tt ' + œ tt ' ),
L-1	57(© )	68(© )
) .	(9.18)
Equivalently one expresses the term Yt 2 Sab cDabt in Eq. (9.4), by using Eqs. (9.12, 9.13), as
1	tt' 1
Yt ,Sab CD abt = (@) 1 Sab CD tt' =
2	2	ab (®) tt '
(ffi) { T2+ A2+ + T2- A2- + T23 A23' + T1+ A 1+ + T1- A1- + T13 A 13' + L tt' tt ' tt' tt ' tt ' tt'
(®) (®) (®) (®) (®) (®)
NN + Anr+ + NN- Anr- + NNR Anr3 + NN + Anl,+ + NN- aAnl,- + NNL AAnl,3},
R tt '	R tt'	R tt '	L tt '	L tt '	L tt ' ' '
(®) (®) (®) (®) (±) (®)
- nRS	|—i
Att/ = (cd tt' — iCD tt') H i(CD tt' — iCD tt'),
(®)	2 3 (®)	01(ffi) ^	31(®)	02(®)
ANtR'3 = (CD tt' — i CD tt' ) ,
t®)	12 (®)	03 (®)
ANL'± = (CD tt' + icD tt') H i(cD tt' + iCD tt'),
(®)	2 3 (®)	01 (®) L-1	31 (®)	02(®)
aANtR'3 = (CD tt' + i cd tt') ,	(9.19)
(® )	12 (®)	03 (® )
with aA ±, A', A ± and aA 1t3' following from expressions for A ± , A^tf', A ±
(t®®t ') (t®®t ') (t®®t ') (t®®t ') (t®®t ') (t®®t ') (t®®t ')
and A13', respectively, in (Eq.(9.18)), if replacing c tt' by cd tt'.
(®	s"s"(® )	s"s" (® )
There is the additional term in Eq. (9.4): Yt 2 St t" ct t"t. This term can be written with respect to the generators St t" as one colour octet scalar field and one U(1)ii scalar field (Eq. 9.11)
Yt 1 Stt '" œt"t '"t = XL (© ) {T3 ■ A 3tt ' + t4 ■ A4;; ' },
2 +,- (tt') (©) (©)
(tt') G ((910),11 12),1314)).	(9.20)
Taking all above equations (9.17,9.18, 9.19, 9.20) into account Eq. (9.4) can be
rewritten, if we leave out p tt / since in the low energy limit the momentum does
c®)
not play any role, as follows
++'
Lf" = V Y°(—)
+>- (tt')
[t2+ A2+' + t2- A2-' + t23 A23'
tt ' tt ' tt '
c®)	(®)	c®)
+ t1+ A1+' + t1-A1-' + t13 A13'
tt' tt' tt'
c®)	c®)	c®)
+ t2+ A 2+' + t2- A2-' + t23 A23'
tt' tt' tt'
c®)	c®)	c®)
+ t1 + A1+' + t1-A1- + t13 A13'
tt' tt' tt'
c®)	c®)	c®)
+ N + Anr+ + N- Anr- + NR Anr3
R tt '	R tt '	R tt '
c®)	c®)	c®)
+ N + Anl'+ + N- Anl'- + N3 Anl'3
L tt'	L tt '	3 tt'
c®)	c®)	c®)
+ T3i A3ti' + t4 A4tt' ]}(9.21)
ct®®t ) ct®®t )
where (t,t') run in pairs over [(9,10),... (13,14)] and the summation must go
over + and — of tt'.
ct®®t )
Let us calculate now quantum numbers of the scalar and vector gauge fields appearing in Eq. (9.21) by taking into account that the spin of gauge fields is determined according to Eq. (9.7) ((Sab)cdAd."e".9 = i(nacSb — nbc^a) Ad."e".9, for each index (g (d... g)) of a bosonic field Ad".g separately). We must take into account also the relation among S ab and the charges (the relations are, of course, the same for bosons and fermions) (Eqs. (9.8, 9.9, 9.11)).
On table 9.1 properties of the scalar gauge fields appearing in Eq. (9.21) are presented.
The scalar fields with the scalar index s = (9,10, • • • ,14), presented in table 9.1, carry one of the triplet colour charges and the "spinor" charge equal to twice the quark "spinor" charge, or the antitriplet colour charges and the anti "spinor" charge. They carry in addition the quantum numbers of the adjoint representations originating in Sab or in Sab. Although carrying the colour charge in one of the triplet or antitriplet states, these fields can not be interpreted as superpartners of the quarks as required by, let say, the N = 1 supersymmetry. The hyper charges and the electromagnetic charges are namely not those required by the supersymmetric partners to the family members.
Let us have a look what do the scalar fields, appearing in Eq. (9.21) and in table 9.1, do when being applied on the left handed members of the Weyl representation presented on table 9.3, containing quarks and leptons and antiquarks and antileptons [4,21,15]. Let us choose the 57th line of table 9.3, which represents in the spinor technique the left handed positron, e+. If we make, let say, the choice
910
of the term (y° (+) t2B ) A^ (the scalar field A^ is presented in the 7th line in
c®)	c®)
table 9.1 and in the second line of Eq. (9.21)), the family quantum numbers will
field	prop.	T4	t'3	T23	(t33,t38 )	Y	Q	T 4	T '3	T 23	N L	N R
A 1 H A9 10 (® ) A 13 A9 1 0 (© )	scalar scalar	a 3 a 3	H 1 0	0 0	(a H 2VJ) (a H 213)	© 3 © 3	© 3+©1 a 3	0 0	0 0	0 0	0 0	0 0
A'H A11 12 (© ) A 13 A 11 12 (© )	scalar scalar	a 3 a 3	11 0	0 0	(a H 213) (a H 213)	© 3 © 3	© 3+©1 a 3	0 0	0 0	0 0	0 0	0 0
A'H A 13 14 (© ) A 13 A 1 3 14 (© )	scalar scalar	a 3 a 3	11 0	0 0	(0, a13) (0, a13)	© 3 © 3	© 3+©1 a 3	0 0	0 0	0 0	0 0	0 0
?H												
A9 10 (® ) A23 A 9 1 0 (® )	scalar scalar	a 3 a 3	0 0	H1 0	(a H 213) (a H )	© 3+©1 © 3	© 3+©1 a 3	0 0	0 0	0 0	0 0	0 0
A 1± A 910 (© ) A 13 A 9 10 (© )	scalar scalar	a 3 a 3	0 0	0 0	(a H 2ÜJ) (a H )	© 3 © 3	a 3 a 3	0 0	H1 0	0 0	0 0	0 0
2 H												
A Q A 910 ()	scalar	a 3	0	0	(a 2, H )	© 3	a 3	0	0	H1	0	0
A 2a) A 910 (© )	scalar	a 3	0	0	(a 2, H )	© 3	a 3	0	0	0	0	0
A nlH A 910 (H ) A NL3 A 910 (H)	scalar	a 3	0	0	(a H 213)	© 3	a 3	0	0	0	H1	0
	scalar	a 3	0	0	(a H ^)	© 3	a 3	0	0	0	0	0
a nr± A 910 (® ) A NR3 A 910 (© )	scalar	a 3	0	0	(a H 213)	© 3	a 3	0	0	0	0	H1
	scalar	a 3	0	0	(a H ^)	© 3	a 3	0	0	0	0	0
a31 A 9 1 0 (a )	scalar	a 3	0	0	(H 1+ H 2, H 213 )	© 3	a 3	0	0	0	0	0
A 4 A 9 1 0 (a )	scalar	a 3	0	0	(a I, a ) (a 2 , H 213 )	© 3	a 3	0	0	0	0	0
A 3 A m	vector	0	0	0	octet	0	0	0	0	0	0	0
A4 A m	vector	0	0	0	0	0	0	0	0	0	0	0
Table 9.1. Quantum numbers of the scalar gauge fields carrying the space index t = (9,10, • • • ,14), appearing in Eq. (9.21), are presented. To the colour charge of all these scalar fields the space degrees of freedom contribute one of the triplets values. These scalars are with respect to the two SU(2) charges, (T1 and T2), and the two SU(2) charges, (T1 and T2), triplets (that is in the adjoint representations of the corresponding groups), and they all carry twice the "spinor" number (t4) of the quarks. The quantum numbers of the two vector gauge fields, the colour and the U(1 )ii ones, are added.
not be affected and they can be any. The state carries the "spinor" (lepton) number t4 = 1, the weak charge t13 = 0, the second SU(2)n charge t23 = 1 and the colour charge (t33, t38) = (0,0). Correspondingly, its hyper charge (Y(= t4 + t23)) is 1 and the electromagnetic charge Q(= Y + t13) is 1.
910
So, what does the term y0 (+) t2B A2B0 make on this spinor? Making use
(e)
910
of Eqs. (9.44, 9.46, 9.54) of appendix 9.9 one easily finds that operator y0 (+) t2-
03 12 56 78 9 10 11 12 13 14
transforms the left handed positron into (+i) (+) | [-] [-] || (+) (-) (-), which
03	03 910
is dR1, presented on line 3 of table 9.3. Namely, y0 transforms [-i] into (+i), (+)
9 10	9 10	56 78	56 78	56 78
transforms [-] into (+), while t2- (= - (-) (-)) transforms (+) (+) into [-] [-]. The state dR1 carries the "spinor" (quark) number t4 = 1, the weak charge t13 = 0, the second SU(2)n charge t23 = -§ and the colour charge (t33,t38) = (§, §73). Correspondingly its hyper charge is (Y = t4 + t23 =) -1 and the electromagnetic
charge (Q = Y + t13 =) -3. The scalar field A2B0 carries just the needed quantum
(e)
numbers as we can see in the 7th line of table 9.1.
If the antiquark uR2, from the line 43 (it is not presented, but one can very easily construct it) in table 9.3, with the "spinor" charge t4 = -6, the weak charge t13 = 0, the second SU(2)n charge t23 = -1, the colour charge (t33,t38) = (1, -273), the hyper charge Y(= t4 + t23 =) -§ and the electromagnetic charge
Q( = Y + t13 =) -§ submits the A§B0 scalar field, it transforms into uR3 from
3	(e)
the line 17 of table 9.3, carrying the quantum numbers t4 = 1, t13 = 0, t23 = 1, (t33, t38) = (0, -73), Y = § and Q = § . These two quarks, dR1 and uR3 can bind together with uR2 from the 9th line of the same table (at low enough energy, after the electroweak transition, and if they belong in a superposition with the left handed partners to the first family) into the colour chargeless baryon - a proton. This transition is presented in figure 9.1.
The opposite transition at low energies would make the proton decay. Let us look at one more example. The 63th line of table 9.3 represents in the spinor technique the right handed positron, e+. Since we shall again not have a look on a transition, in which scalar fields with the nonzero family quantum numbers are involved, the family quantum number of this positron is not important. The state carries the "spinor" (lepton) number t4 = 1, the weak charge t13 = §, the second SU(2)n charge t23 = 0 and the colour charge (t33, t38) = (0,0). Correspondingly, its hyper charge (Y = t4 + t23) is § and the electromagnetic charge Q = Y + t13 is 1.
910
What does, let say, the term y0 (+) t1b A1b0 (the scalar field A1B0 is pre-
(e)	(e0
sented in the first line of table 9.1) make on e+? Making use of Eqs. (9.44, 9.46, 9.54) of appendix 9.9 one easily finds that the right handed positron transforms
910	03 12 56 78 9 10 1 1 12 13 14
under the application of y0 t1- (+) into [-i] (+) | [-] (+) || (+) (-) (-), which
03	03 910
is dR1 presented on line 5 of table 9.3. Namely, y0 transforms (+i) into [-i], (+)
t 4 =1 ,t 13=0,r 23 = (t 23,t 38) = (0,0)
Y=1,Q=1
t4 = 1 t13 = ' 6
0,t 23 = - 2
(t 33,t 38) = ( ) Y=-3 ,Q=-3

dR
A2B
l a9 10,
( + )
t 4=2x(-6 ),t 13=0,t 23=-1
(t 33 ,-38)=( 2, 2- ) Y=-3 ,q=-3
c2
Ul
t 4=-1 ,t13=0,t 23=-2
(t 33,t 38) = ( 22- ) Y=-3,Q=-2
U
t 4 = 1 ,t 13=0,t 23 = §
(t 33 ,t 38) = (0,--1= )
y = 6 ,q= 3
L
c2 c2 UR -*- UR
T4 = 1 T 13 = 0 T23 = 1 ' 6 '' 0'' 2
(T 33,T 38) = (- 2 , Y=§ >Q= 3
Fig. 9.1. The birth of a proton out of an positron e+, antiquark uL2 and quark (spectator) uR2. The family quantum number can be any.
9 10 9 10	56 78	56 56	56 56
transforms [-] into (+), while t1B (=(-) (+)) transforms (+) [-] into [-] (+). The state dL1 carries the "spinor" (quark) number t4 = 1, the weak charge t13 = - 2, the second SU(2)n charge t23 = 0 and the colour charge (t33,t38) = (1, ). Correspondingly its hyper charge is (Y = t4 + t23 =) 6 and the electromagnetic charge (Q = Y + t13 =) -1. The scalar field AjB0 carries all the needed quantum
numbers, as one can see in figure 9.1.
If the antiquark uR2, from the line 47 in table 9.3 (the reader can easily find
03 12 56 78 9 10 11 12 13 14
the expression (+i) (+) | [-] (+) || (+) (-) [+] ), with the "spinor" charge t4 = -1, the weak charge t13 = -2, the second SU(2)n charge t23 = 0, the colour charge (t33, t38) = (1, -), the hypercharge (Y = t4 + t23 =) -1 and
the electromagnetic charge (Q = Y + t13 =) - 3, submits the A1B0 scalar field,
(®)
03 12 56 78 9 10 11 12
it transforms into uL3 from the line 23 of table 9.3 ([-i] (+) | (+) [-] || [-] (-)
13 14
[+] ), carrying the quantum numbers t4 = 1, t13 = 2, t23 = 0, (t33,t38) = (0, -Ti), Y = 6 and Q = 3. These two quarks, dL1 and uL3, can bind (at low enough energy, when making after the electroweak transition the superposition with the right handed partners) together with uL2 from the 15th line of the same
table, into the colour chargeless baryon - a proton. This transition is presented in figure 9.2.
The opposite transition would make the proton decay.
~2' 2 _33 _38
) = (0,0) y=2 ,q=1
T4 = 6 ,T 13 = - 2 ,T23=0
(T 33,T 38) = ( 2 ^ ) Y=1 ,Q=—3
5+
-R
dL

1B
9 10 (+)
T 4 = 2x( —3 ),T13 = — 1 ,T 23=0
(T
33 _38\_ / 1
) = ( 2 , 213 )
Y=—3 ,Q=—3
u
t 4 = — 1 t13 = — 1 t23=0
i	3 ,i	2 ,'	0
(t33 t38) — ( 1 1 )
(t ,t )=( 2 ,—273 )
y=—1 Q=—2
y = 3 ,q= 3
u
c3
(T 33,t 38) = (0, — ^ )
Y = 6 ,Q= 2
L
T 4= 1 ,t13
1 t 23
3
2
c2 c2 uC2 -►- Uc/
t 4— 1 t13= 1 t 23 =
1 = 3 = 2 = 33 38	1 1
0
3) = (- 1 )
)=( 2,2V3)
Y	=1 Q= 2
Y	= 3 ,Q= 3
Fig. 9.2. The birth of a proton out of a positron e+, antiquark UR2 and quark (spectator) uL2 The family quantum number can be any.
Similar transitions go also with other scalars from Eq. (9.21) and table 9.1. The
A1,t„, A2, t„, ANL and Afields cause transitions among the family members, (+) (+) (+) (+) changing a particular member into the antimember of another colour and of
910 n -
another family. The term y0 (+) N- A91R0 transforms e + into uL1, changing the
c®)
family quantum numbers.
The action from Eqs. (9.1,9.2,9.4) manifests Cn •Pn invariance. All the vector and the spinor gauge fields are massless.
Since no one of the scalar fields from table 9.1 have been observed and also no vector gauge fields like Am, Am and other scalar and vector fields, we shall discuss this topic in sect. 9.5, it must exist a mechanism, which makes the non observed scalar and vector gauge fields massive enough.
Scalar fields from table 9.1 carry the colour and the electromagnetic charge. Therefore their nonzero vacuum expectation values would not be in agreement with the observed phenomena. One, however, notices that all the scalar gauge
fields from table 9.1 and several other scalar and vector gauge fields (see sect. 9.5) couple to the condensate with the nonzero quantum number t4 and t23 and nonzero family quantum numbers.
It is not difficult to recognize that the desired condensate must have spin zero, Y = t4+ t23 = 0, Q = Y + t13 = 0 and t1 = 0 in order that in the low energy limit the spin-charge-family theory would manifest effectively as the standard model.
I make a choice of the two right handed neutrinos of the VIIIth family coupled into a scalar, with t4 = -1, t23 = 1, correspondingly Y = 0, Q = 0 and t1 = 0, and with family quantum numbers (Eqs. (9.13, 9.12)) T4 = — 1, t23 = 1, NR = 1, and correspondingly with Y = T4 + T23 = 0, Q = Y + T13 = 0, and T = 0. The condensate carries the family quantum numbers of the upper four families.
The condensate made out of spinors couples to spinors differently than to antispinors - "anticondensate" would namely carry t4 = 1, and t23 = — 1 - breaking correspondingly the Cn• Pn symmetry: The reactions creating particles from antiparticles are not any longer symmetric to those creating antiparticle from particles.
Such a condensate leaves the hyper field A^ (= sin Am3 + cos A^J (for the choice that sin$2 = cos$2 and g4 = g2, there is no justification for such a choice, Am = (Am + Am)) massless, while it gives masses to A^^ and Am (= (Am — Amm3) for sin = cos $2) and it gives masses also to all the scalar gauge fields from table 9.1, since they all couple to the condensate through t4.
The weak vector gauge fields, A m, the hyper charge vector gauge fields, Am, and the colour vector gauge fields, A m, stay massless.
The scalar fields with the scalar space index s = (7,8) - those which couple to all eight families, those which couple only to the upper and those which couple only to the lower four families - carrying the weak and the hyper charges of the Higgs's scalar - wait for getting nonzero vacuum expectation values to change their masses while causing the electroweak break.
The condensate does what is needed so that in the low energy regime the spin-charge-family manifests as an effective theory which agrees with the standard model to the extend that it is in agreement with the observed phenomena, explaining the standard model assumptions and predicting new fermion and boson fields.
It also may hopefully explain also the observed matter-antimatter asymmetry if the conditions in the expanding universe would be appropriate (9.6). The work needed to check these conditions in the expanding universe within the spin-charge-family theory is very demanding. Although we do have some experience with following the history of the expanding universe [12], this study needs much more efforts, not only in the calculations, but also in understanding the mechanism of appearing the condensate, relations among the velocity of the expansion, the temperature and the dimension of space-time in the period of the appearance of the condensate. This study has not yet been really started.
9.3 Properties of the condensate
In table 9.2 the properties of the condensate of the two right handed neutrinos ( Ivjf111 >! | vJV111 >2 ), one with spin up and another with spin down (table 9.3, line
25 and 26), carrying the family quantum numbers of the VIIIth family (table 9.4 ), are presented. The condensate carries the quantum numbers of SU(2)n, t23 = 1 (Eq. (9.9)), of U(1)n originating in SO(6), t4 = —1 (Eq.9.11), correspondingly Y = 0, Q = 0, and the family quantum numbers (table 9.4) T4 = —1 (Eq. (9.11)), T23 = 1 (Eq. (9.13)), and NR = 1 (Eq. (9.12)). Each of the two neutrinos could belong to a different family of the upper four families. In this case the family quantum numbers of the condensate change.
The condensate is presented in the first line of table 9.2 as a member of a triplet of the group SU(2)n with the generators t2\ Correspondingly the condensate couples to all the vector gauge fields which carry nonzero t2t, t4, T2t, NR and T4. The fields A^, Am and A m stay massless. The condensate couples also to all the scalar gauge fields with the scalar indices s G (5,6,7,8,9,..., 14), since they all carry nonzero either t4 or t23.
state	S03 si2 t13 t23 t4 Y Q	T13 T23 T4 Y Q N L N R
(|v1R >1 |v2R >2)	0 0 0 1 —1 0 0	0 1 —1 0 0 0 1
(|v1R >1 |e2R >2) r^VIII . „VIII . (|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 9.2. The condensate of the two right handed neutrinos vR, with the VIIItH family quantum number, coupled to spin zero and belonging to a triplet with respect to the generators T2i, is presented, together with its two partners. The condensate carries T1 = 0, t23 = 1, t4 = -1 and Q = 0 = Y. The triplet carries T4 = -1, T23 = 1 and NR = 1, NL = 0, Y = 0, Q = 0. The family quantum numbers are presented in table 9.4.
Coupling of the scalar gauge fields to the condensate is proportional to
tt' tt'
(<vVRIIh <vVRiiI)(y0 (@) TAi AAi )%0	(@) TAiA'At' )(|vVRI1 >1 IvVR11 >)
c®)	c®)
<x (AA^' )f (AAi ),
(tt') G [(56), (78), (910),..., (1314)].	(9.22)
The condensate does break the Cn • Pn symmetry. (The "anticondensate" would namely carry t23 = —1 and t4 = 1).
The condensate gives masses to all the scalars from table 9.1, either because they couple to the condensate due to t4 or due to t4 and t23 quantum numbers. It gives masses also to all the scalar fields with s G (5,6,7,8), since they couple to the condensate due to the nonzero t23. The scalar fields with the quantum numbers of the upper four families couple in addition through their family quantum numbers.
The condensate couples also to all the vector gauge fields except to the gauge colour octet field Am, the hyper charge vector fields A^ and the weak charge vector triplet fields Am, since they carry zero t23, t4 and Y quantum numbers.
The spin connection fields, of either "tilde" (S ab) or "nontilde" (Sab) origin, which do not couple to the spinor condensate, are auxiliary fields, expressible with vielbeins fields (abstract (9.10)).
Below the scalar and vector gauge fields are presented, which get masses through the interaction with the condensate.
A , , A23, , A1^ , A13,
tt' > tt > tt' > tt
(© ) (© ) (© ) (© )
A 3
(© )
.2A
A
(© )
, a23 ',
.1
(© )
A , A
tt ' '
(© )
13
tt'
(© )
Anlq anl3 Anr± Anr3, tt' ? tt ' ? tt' ? tt'
(© ) (©) (© ) (© )
(tt') G [(910), (11 12), (1314)],
A
2B
1
23
AY , = ^ (A2
ss ' '	ss	P) ^ ss
(© ) (© ) V2 (© ) (ss') G [(56), (78)],
A2@ AY' _ 1 (a23 a4 ) Am , Am _ (Am — Am) ,
A2 A4 Ä nr Am , Am , Am ,
m G (0,1,2,3).
A4 ' )
ss '
(© )
(9.23)
In expression for Am s = f is chosen, just for simplicity, with no justification so far.
It stays as an open question what does make the right handed neutrinos to form such a condensate in the history of the universe.
Since AAi, s g (5,6) couple to the condensate and get masses, while (by assumption) they do not get nonzero vacuum expectation values during the electroweak break (what changes the masses of the scalar fields A^i, s g (7,8)) the restriction in the sum in Eq. (9.2) is justified.
The scalar fields, causing the birth of baryons, have the triplet colour charges. They resemble the supersymmetric partners of the quarks, but since they do not carry all the quantum numbers of the quarks, they are not.
9.4 Properties of scalar fields which determine mass matrices of fermions
This section is a short overview of the ref. [14].
There are two kinds of the scalar gauge fields, which gain at the electroweak break nonzero vacuum expectation values and determine correspondingly the masses of the families of quarks and leptons and to the masses of gauge weak bosons: The kind originating in d>abs and the kind originating in ^tt's, and d78s, both kinds have the space index s = (7,8) and both carry the weak and the hyper charge as the Higgs's scalar. These scalar fields are presented in the Lagrange density for fermions (Eq. (9.2)) in the second line. The "tilde" kind influences the family quantum numbers of fermions, the "Dirac" kind influences the family members quantum numbers.
The two triplets (A], AJ^1- ) influence the lower four families (the lowest three
already observed), while (AAN) influence the upper four families, the stable
of which constitute the dark matter. Recognizing that f1 A] +N L AY^2 A2
+NR AN = ] Sab cDabs, s = (7,8), one easily finds, taking into account Eqs. (9.12, 9.13), the expressions
A1 = (cD58s - cD67s, CD57s + CD68s, CD56s - CD78s) , AN = (cD 23s + i CD ois, CD 31s + iCD02s,CD 12s + iCD03s) ,
A2 = (cD58s + CD67s, CD57s - CD68s, CD56s + CD78s) , aA nr = (cD 23s - iDD ois, DD 31s - i CD 02s,CD 12s - i CD 03s) ,
s = (7,8),	(9.24)
presented already in Eq. (9.16). Similarly one finds, taking into account Eqs. (9.8, 9.9, 9.11, 9.14), the expressions for AQ, Aj and Aj', presented in Eqs. (9.15).
The scalar fields AQ, Aj and Aj' distinguish among the family members, coupling to the family members quantum numbers through Q (= t13 + Y), Y
(= t23 + t4) and Y' = t23 - tan£2 t4, t4 = -3(S910 + S1112 + S1314). The scalars originating in o>abs and distinguishing among families, couple the family
quantum numbers through (t1 and NL), or through (T2 and NR), all in the adjoint representations of the corresponding groups.
Let us now prove that all the scalar fields with the space (scalar with respect to d = (3 + 1)) index s = (7,8) carry the weak and the hyper charge (t13, Y) equal to either (- 2,1) or to (1, -1). Let us first simplify the notation, using a common name AA for all the scalar fields with the scalar index s = (7,8)
aa1 = (ASQ, aq ', Aj', A 4, A 2, A1, A , A nl ),	(9.25)
and let us rewrite the term Y.s=7 8 1 Ys p0s 1 in Eq. (9.2) as follows Y 1 Ys P0s 1,
s = 7,8
78	78
= 1 { (+) P0 + + (-) P0-} 1, P0± = (p07 T ip08) ,
(P07 T ip08) = (P7 T ip8) - ta1 (AA1 t iA^1)
78 1
(±)= 1 (Y7 ± iY8).	(9.26)
Let us now apply the operators Y, Q, Eq. (9.14), and t13 = 1 (S56 - S78), Eq. (9.9),
on the fields AA1 = (AA1 T iAA1). One finds
(±)
1
2
1
2
Q (aA1 T iAA1)= 0,	(9.27)
T13 (aA1 T iAA1) = ±1 (AA1 t iAA1),
Y (aA1 T iAA1) = T 1 (AA1 T iAA1),
This is, with respect to the weak, the hyper and the electromagnetic charge, just what the standard model assumes for the Higgs' scalars. The proof is complete.
78
One can check also, using Eq. (9.44), that y0 (-) transforms the uR1 from the first line of table 9.3 into uR1 from the seventh line of the same table, or vR from the 25th line into the vL from the 31th line of the same table.
The scalars AA7i obviously bring the weak charge 2 and the hyper charge — 2
to the right handed family members (uR, vR), and the scalars Aa1 bring the weak
charge — 2 and the hyper charge 2 to (dR, eR).
Let us now prove that the scalar fields of Eq. (9.25) are either triplets with respect to the family quantum numbers (NR, NL, T2, T1; Eqs. (9.12,9.13)) or singlets as the gauge fields of Q = t13 + Y, Q' = t13 — Y tan2 and Y' = t23 — tan2 t4. One can prove this by applying T2, T1, NR, NL and Q, Q', Y' on their eigenstates. Let us calculate, as an example, N R and Q on
An78l3 and on Aq8 , taking into
(±) (±)
account Eqs. (9.12, 9.11,9.9, 9.7)
NL An8ls = ± A, INL Anl3 = 0,
(±) (±) (±)
QAQ8 = 0,
(±)
- NL±
A 78n = {(cD 78 + icD 78 ) ffl i (cD 78 + icD 78 )},
■ 78
(±)
78
2 3 (±)	01(±) I—I	31(±)	02 (± )
AN7L3 = (cD 78 + icD 78 )
(±)	12(±)	03(±)
Aq8 = sinA1738 + sin$1 (—)(c 78 + c 78 + c 78 ), (9.28)
(±)	(±)	910(±)	11 12 (±)	13 14 (±)
with Q = S56 + t4 = S56 — 3(S910 + S11 12 + S1314), and with t4 defined in Eq. (9.11)).
Nonzero vacuum expectation values of the scalar fields (Eq. (9.25)), which carry the scalar index s = (7,8), and correspondingly the weak and the hyper charges, break the mass protection mechanism of quarks and leptons of the lower and the upper four families. In the loop corrections contribute to all the matrix elements of mass matrices of any family members besides A^1 and the scalar fields which are the gauge fields of Q, Q', Y' also the vector gauge fields.
The gauge fields of N R and i2 contribute only to masses of the upper four families, while the gauge fields of N L and t1 contribute only to masses of the lower four families. The triplet scalar fields with the scalar index s = (7,8) and the family charges N R and i2 transform any family member belonging to the group of the upper four families into the same family member belonging to another family of the same group of four families, changing the right handed member into the left handed partner, while those triplets with the family charges NL and t1 transform any family member of particular handedness and belonging to the lower four families into its partner of opposite handedness, belonging to another family of the lower four families.
The scalars aQ8 (Eq. (9.28)), aQ (= cos $1 A1738 - sin$iA478 ) and A^g (±) (±) (±) (±) (±) (Eq. (9.23)) contribute to all eight families, distinguishing among the family members and not among the families.
The mass matrix of any family member, belonging to any of the two groups of the four families, manifests - due to the SU(2) (R,R) x SU(2) (n,i) (either (R, II)
or (L, I)) structure of the scalar fields, which are the gauge fields of the NR,R and
T2,1 - the symmetry presented in Eq. (9.29)
Ma =
/-a1 - a e	d b \
e -a2 - a b d d	b a2 - a e
\ b	d	e a1 - a/
(9.29)
In the ref. gn2014 the mass matrices for quarks, which are in the agreement with the experimental data, are presented and predictions made.
PC
9.5 Condensate and nonzero vacuum expectation values of scalar fields make spinors and most of scalar and vector gauge fields massive
Let us shortly overview properties of the scalar and the vector gauge fields after a. two right handed neutrinos (coupled to spin zero and with the family quantum numbers (table 9.4) of the upper four families) make a condensate (table 9.2) at the scale > 1016 GeV and after b. the electroweak break, when the scalar fields with the space index s = (7,8) get nonzero vacuum expectation values.
All the scalar gauge fields AAi, t G (5,6,7,8,9,..., 14) (Eqs. (9.2, 9.21, 9.23), table 9.1) interact with the condensate through the quantum numbers t4 and t23, those with the family quantum numbers of the upper four families interact also through the family quantum numbers T2 or N R, getting masses of the order of the condensate scale (Eq.(9.23)).
At the electroweak break the scalar fields A^i, s G (7,8), from Eqs. (9.25,9.25) get nonzero vacuum expectation values, changing correspondingly their own masses and determining masses of quarks and leptons, as well as of the weak vector gauge fields.
The vector gauge fields A^p, Am', A ^p, A and A £ (Eq. (9.23)) get masses due to the interaction with the condensate through t23 and t4, the first two, and/or also due to the family quantum numbers of the upper four families, the last three, respectively.
The vector gauge fields AAand A^ stay massless up to the electroweak break when the scalar gauge fields, which are weak doublets with the hypercharge making electromagnetic charge Q equal to zero, give masses to the weak bosons
(Am± = 772 (Ami T iA£) and a£ = cos $i A^5 - sin $i A^J, while the electromagnetic vector field (a£ = sin $7 Am3 + cos $7 A^) and the colour vector gauge field stay massless.
At the electroweak break, when the nonzero vacuum expectation values of the scalar fields break the weak and the hypercharge global symmetry, also all the eight families of quarks and leptons get masses. Until the electroweak break the families were mass protected, since the right handed partners distinguished from the left handed ones in the weak and hyper charges, what disabled them to make the superposition manifesting masses.
9.6 Sakharov conditions as seen in view of the spin-charge-family theory
The condensate of the right handed neutrinos, as well as the nonzero vacuum
expectation values of the scalar fields AA7i - if leading to the complex matrix
(±)
elements of the mixing matrices - cause the Cn Pn violation terms, which generate the matter-antimatter asymmetry.
It is the question whether both generators of the matter-antimatter asymmetry - the condensate and the complex phases of the mixing matrices of quarks and leptons (this last alone can not with one complex phase and also very probably not with the three complex phases of the lower four families) - can explain at all the observed matter-antimatter asymmetry of the "ordinary" matter, that is the matter of mostly the first family of quarks and leptons.
The lowest of the upper four families determine the dark matter. For the dark matter any relation among matter and antimatter is so far experimentally allowed.
Both origins of the matter-antimatter asymmetry - the condensate and the nonzero vacuum expectation values of the scalar fields carrying the weak and the hyper charge - (are assumed to) appear spontaneously.
Sakharov [24] states that for the matter-antimatter asymmetry three conditions must be fulfilled:
a.	(Cn and) Cn Pn must not be conserved.
b.	Baryon number non conserving processes must take place.
c.	Thermal non equilibrium must be present not to equilibrate the number of baryons and antibaryons.
Sakharov uses for c. the requirement that CPT must be conserved and that {CPT, H}_ = 0. In a thermal equilibrium the average number of baryons < nB >= Tr(e-PHnB) = Tr(e-PHCPTnB(CPT)-1) = < nB >. Therefore < nB > - < nB >= 0 at the thermal equilibrium and there is no excess of baryons with respect to antibaryons. In the expanding universe, however, the temperature is changing with time. It is needed that the discrete symmetry Cn Pn is broken to break the symmetry between matter and antimatter, if the universe starts with no matterantimatter asymmetry.
The spin-charge-family theory starting action (Eq.(9.1)) is invariant under the Cn Pn symmetry. The scalar fields (Eq.(9.21)) of this theory cause transitions, in which a quark is born out of a positron (figures (9.1, 9.2)) and a quark is born out of antiquark, and back. These reactions go in both directions with the same probability, until the spontaneous break of the Cn Pn symmetry is caused by the appearance of the condensate of the two right handed neutrinos (table 9.2).
But after the appearance of the condensate (and in addition of the appearance of the non zero vacuum expectation values of the scalar fields with the space index s e (7,8)), family members "see" the vacuum differently than the antimembers. And this might explain the matter-antimatter asymmetry. It is also predicting the proton decay.
It is, of course, the question whether both phenomena can at all explain the observed matter-antimatter asymmetry. I agree completely with the referee of this paper that before answering the question whether or not the spin-charge-family theory explains this observed phenomena, one must do a lot of additional work to find out: i. Which is the order of phase transition, which leads to the appearance of the condensate. ii. How strong is the thermal nonequilibrium, which leads to the matter-antimatter asymmetry during the phase transition. iii. How rapid is the appearance of the matter-antimatter asymmetry in comparison with the expansion of the universe. iv. Does the later history of the expanding universe enable that the produced asymmetry survives up to today.
Although we do have some experience with solving the Boltzmann equations for fermions and antifermions [12] to follow the history of the dark matter within the spin-charge-family theory, the study of the history of the universe from the very high temperature to the baryon production within the same theory in order to see the matter-antimatter asymmetry in the present time is much more demanding task. These is under consideration, but still at a very starting point since a lot of things must be understood before starting with the calculations.
What I can conclude is that the spin-charge-family theory does offer the opportunity also for the explanation for the observed matter-antimatter asymmetry.
9.7 Conclusions
The spin-charge-family [1,3-8,2,9,12,14,15] theory is a kind of the Kaluza-Klein theories in d = (13 + 1) but with the families introduced by the second kind of gamma operators - the Ya operators in addition to the Dirac Ya in d = (13 + 1). The theory assumes a simple starting action (Eq. (9.1)) in d = (13 +1). This simple action manifests in the low energy regime, after the breaks of symmetries (subsection 9.1.1), all the degrees of freedom assumed in the standard model, offering the explanation for all the properties of quarks and leptons (right handed neutrinos are in this theory the regular members of each family) and antiquarks and antileptons. The theory explains the existence of the observed gauge vector fields. It explains the origin of the scalar fields (the Higgs and the Yukawa couplings) responsible for the quark and lepton masses and the masses of the weak bosons and carrying the weak and the hyper charge of the standard model Higgs ([14]).
The theory is offering the explanation also for the matter-antimatter asymmetry and for the appearance of the dark matter.
The spin-charge-family theory predicts two decoupled groups of four families [3,4,9,12]: The fourth of the lower group of four families will be measured at the LHC [10] and the lowest of the upper four families constitutes the dark matter [12] and was already seen. It also predicts that there might be several scalar
fields observable at the LHC. The upper four families manifest, due to their high masses, a new "nuclear force" among their baryons.
All these degrees of freedom are already contained in the simple starting action. The scalar fields with the weak and the hyper charges equal to (t 1, ± 1), respectively (section 9.4), have the space index s = (7,8), while they carry in addition to the weak and the hyper charges also the family quantum numbers, originating in Sab (they form two groups of twice SU(2) triplets), or the family members quantum numbers, originating in Sab (they form three singlets with the quantum numbers (Q, Q', Y'))• These scalar fields cause the transitions of the right handed family members into the left handed partners and back. Those with the family quantum numbers cause at the same time transitions among families within each of the two family groups of four families. They all gain in the electroweak break nonzero vacuum expectation values, giving masses to both groups of four families of quarks and leptons and to weak bosons (changing also their own masses).
There are in this theory also the scalar fields with the space index s = (5,6); They carry with respect to this degree of freedom they the weak charge equal to the hyper charge (t 1, T1, respectively). They carry also additional quantum numbers Eq.(9.23) like all the scalar fields: The family quantum numbers, originating in S ab and the family members quantum numbers originating in Sab.
And there are also the scalar fields with the scalar index s = (9,10, • • • ,14). These scalars carry the triplet colour charge with respect to the space index and the additional quantum numbers (table 9.1), originating in family quantum numbers SSab and in family members quantum numbers Sab.
There are no additional scalar gauge fields.
There are the vector gauge fields with respect to d = (3 + 1): Ami, with Ai staying for the groups SU(3) and U(1) (both originating in SO (6) of SO (13,1)), for the groups SU(2)„ and SU(2)i (both originating in SO(4) of SO(7,1)) and for the groups SU(2) x SU(2) (g So(3, 1)), in both sectors, the Sab and Sab ones.
The condensate of the two right handed neutrinos with the family charges of the upper four families (table 9.2) gives masses to all the scalar and vector gauge fields, except to the colour octet vector, the hyper singlet vector and the weak triplet vector gauge fields, to which the condensate does not couple. It gives masses also to all the vector gauge fields to which the condensate couples. Those vector gauge fields of either Sab or SSab origin, which do not couple to the condensate, are expressible with the corresponding vielbeins (appendices 9.55, 9.56). The condensate breaks the Cn Pn symmetry (sections (9.3, 9.8)).
There are no additional vector gauge fields in this theory.
Nonzero vacuum expectation values of the scalar gauge fields with the space index s = (7,8) and the quantum numbers as explained in the fourth paragraph of this section change in the electroweak break their masses, while all the other scalars or vectors either stay massless (the colour octet, the electromagnetic field), or keep the masses of the scale of the condensate. The only before the electroweak massless vector fields, which become at the electroweak break massive, are the heavy bosons.
It is extremely encouraging that the simple starting action of the spin-charge-family offers at low energies the explanations for so many observed phenomena, although the starting assumptions (section 9.1.1) wait to be derived from the initial and boundary conditions of the expanding universe.
This paper is a step towards understanding the matter-antimatter asymmetry within the spin-charge-family theory, predicting also the proton decay. The theory obviously offers the possibility that the scalar gauge fields with the space index s = (9,10, ••• ,14) explain, after the appearance of the condensate, the matterantimatter asymmetry. To prove, however, that this indeed happen, requires the additional study: Following the universe through the phase transitions which breaks the Cn Pn symmetry at the level of the condensate and further through the electroweak phase transition up to today, to check how much of the matterantimatter asymmetry is left. The experience when following the history of the expanding universe to see whether the spin-charge-family theory can explain the dark matter content [12] is of some help. However, answering the question to which extend this theory can explain the observed matter-antimatter asymmetry requires a lot of additional understanding and a lot of work.
Let me conclude with the recognition, pointed out already in the introduction, that the spin-charge-family theory overlaps in many points with other unifying theories [26-31], since all the unifying groups can be recognized as the subgroups of the large enough orthogonal groups, with family groups included. But there are also many differences: The spin-charge-family theory starts with a very simple action, from where all the properties of spinors and the gauge vector and scalar fields follow, provided that the breaks of symmetries occur in the desired way. Consequently it differs from other unifying theories in the degrees of freedom of spinors and scalar and vector gauge fields which show up on different levels of the break of symmetries, in the unification scheme, in the family degrees of freedom and correspondingly also in the evolution of our universe.
9.8 APPENDIX: Discrete symmetry operators [15]
I present here the discrete symmetry operators in the second quantized picture, for the description of which the Dirac sea is used. I follow the reference [15]. The discrete symmetry operators of this reference are designed for the Kaluza-Klein like theories, in which the total angular momentum in higher than (3 + 1) dimensions manifest as charges in d = (3 + 1). The dimension of space-time is even, as it is in the case of the spin-charge-family theory.
3
cn = n Ym r(3+1) k Ix6,xs,...,xd,
3ym,m=0 3
Tn = n Ym r(3+1) KIx0 Ixs)X7)...)Xd-1 ,
OTym,m=1
pn = y0 r(3+1) r(d) Ix3.	(9.30)
9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? 151 The operator of handedness in even d dimensional spaces is defined as
r(d) := (i)d/2 n (Vn^Ya),	(9.31)
a
with products of Ya in ascending order. We choose y0, y1 real, y2 imaginary, y3 real, y5 imaginary, y6 real, alternating imaginary and real up to Yd real. Operators I operate as follows:
Ix0 x0 = —x0;
Ixxa = —xa;
Ix0 xa = (—x0, x);
IxX = —x;
II3 xa = (x0, —x1, —x2, —x3,x5,x6,...,xd);
Ix5,x7,...,xd-1 (x0,x1 ,x2,x3,x5,x6,x7,x8,...,xd-1,xd) = , , ,	(x0,x1 ,x2,x3, —x5 ,x6, —x7,..., —xd-1,xd);
Ix6,x8,...,xd (x0,x1 ,x2,x3,x5,x6,x7,x8,...,xd-1,xd) =
(x0, x1, x2, x3, x5, —x6, x7, —x8,..., xd-1, —xd), d = 2n.
Cn transforms the state, put on the top of the Dirac sea, into the corresponding negative energy state in the Dirac sea.
The operator, it is named [1,16,15] Cn, is needed, which transforms the starting single particle state on the top of the Dirac sea into the negative energy state and then empties this negative energy state. This hole in the Dirac sea is the an-tiparticle state put on the top of the Dirac sea. Both, a particle and its antiparticle state (both put on the top of the Dirac sea), must solve the Weyl equations of motion.
This Cn is defined as a product of the operator [1,16] "emptying", (making transformations into a completely different Fock space)
"emptying" = n Ya K = (—)d+1 n Ya r(d)K,	(9.32)
OTya	3ya
and Cn
d	3
Cn = n Ya K n Ym r(3+1) KIx6,x8,...,xd
OTya,a=0	3ym,m=0
d
= n Ys Ix6,x8,...,xd .	(9.33)
OTys,s=5
We shall need inded only the product of operators CnPn, Tn and CnPn Tn, since either Cn or Pn have in even dimensional spaces with d = 2(2n +1) an odd number of Ya operators, transforming accordingly states from the representation
of one handedness in d = 2(2n + 1) into the Weyl of another handedness.
d
CnP.N = Y0 n Ys Ixs Ix6,x8,...,xd ,
3ys,s=5 d
CnPnTn = n Ya KIx .	(9.34)
3ya,a=o
9.9 APPENDIX: Short presentation of technique [6,18,20]
I make in this appendix a short review of the technique [18,20], initiated and developed [5-8] when proposing the spin-charge-family theory [5,6,8,4,1,2,12,9] assuming that all the internal degrees of freedom of spinors, with family quantum number included, are describable in the space of d-anticommuting (Grassmann) coordinates [6], if the dimension of ordinary space is d. There are two kinds of operators in the Grassmann space, fulfilling the Clifford algebra, which anticom-mute with one another. The technique was further developed in the present shape together with H.B. Nielsen [18,20] by identifying one kind of the Clifford objects with ys's and another kind with Ya's.
The objects Ya and Ya have properties
{Ya,Yb}+ = 2nab, {Ya,Yb}+ = 2nab, , {Ya,Yb}+ = 0,
YaB := i(-)nB BYa |^o >,
B = ao + aaYa + aabYV + • • • + aa, -adYai ... YadMo > (9.35)
for any d, even or odd. I is the unit element in the Clifford algebra. The two kinds of the Clifford algebra objects are connected with the left and the right multiplication of any Clifford algebra objects B. In Eq. (9.35) B is expressed as a polynomial of Ya, (—)nB = +1, —1, when the object B has a Clifford even (+1) or odd (+1) character, respectively. |^o > is a vacuum state on which the operators Ya apply.
If B is a Clifford algebra object, let say a polynomial of Ya, then one finds
(Y aB : = i(-)nB BYa )|^o >,
B = ao + aaoYao + aa,a2YaiYa2 + ••• + aa,...adYai ••• Yad , (9.36)
where |^o > is a vacuum state, defined in Eq. (9.50) and (—)nB is equal to 1 for the term in the polynomial which has an even number of Yb's, and to —1 for the term with an odd number of Yb's.
In this last stage we constructed a spinor basis as products of nilpotents and projections formed as odd and even objects of Ya's, respectively, and chosen to be eigenstates of a Cartan subalgebra of the Lorentz groups defined by Ya's and Ya's.
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 the graphic presentation of basic states, 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 any Clifford algebra object.
The Clifford algebra objects Sab and Sab close the algebra of the Lorentz group
Sab := (i/4)(YaYb - YbYa),
Sab : = (i/4)(YaYb - Ya),
{Sab,Scd}_ =
{Sab,Scd}- = i(nadSbc + nbcSad - nacSbd - nbdSac),
{S ab,S cd}_ = i(nad5Sbc + nbcSad - nacSbd - nbdS ac), (9.37)
We assume the "Hermiticity" property for Ya's and Ya's
Yat = naaYa, Yat = naaYa,	(9.38)
in order that Ya and Ya are compatible with (9.35) and formally unitary, i.e.
Ya t Ya = I and Ya t"Ya = I.
One finds from Eq.(9.38) that (Sab)t = naanbbSab.
Recognizing from Eq.(9.37) that two Clifford algebra objects Sab, Scd with all indices 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,
S03, S12, ••• ,Sd-2d-1,	if	d =(2n + 1) >4,
S03,S12,S56, ••• ,S d-1 d,	if	d = 2n > 4,
S03,S12, ••• ,Sd-2d-1,	if	d =(2n + 1) >4.	(9.39)
The choice for the Cartan subalgebra in d < 4 is straightforward. It is useful to define one of the Casimirs of the Lorentz group - the handedness r ({F, Sab}_ = 0) in any d
r(d) :=(i)d/2 ^ (VnaaYa), if d = 2n,
a
r(d) :=(i)(d-1)/2 ^ (V'naaYa), if d = 2n + 1.	(9.40)
a
One proceeds equivalently for f(d), subtituting Ya's by Ya's. We understand the product of Ya's in the ascending order with respect to the index a: y0y1 • • • Yd .It follows from Eq.(9.38) for any choice of the signature naa that rt = r, r2 = I. 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, H- = 0).
To make the technique simple we introduce the graphic presentation as follows
ab 1	naa	ab 1	i
(k): = ^(Ya + n^Yb), [k]:= ^(1 + kYaYb), +: = 1 (1 + r), -:= 1 (1 - r),	(9.41)
154 N.S. Mankoc Borstnik where k2 = naanbb. It follows then
ab ab	ab ab
Ya = (k) + (—k), Yb = iknaa ((k) — (—k)), k ab ab
Sab = k ([k] — [—k])	(9.42)
One can easily check by taking into account the Clifford algebra relation (Eq.9.35) and the definition of Sab and Sab (Eq.9.37) that if one multiplies from the left hand
ab	ab
side by Sab or Sab the Clifford algebra objects (k) and [k], it follows that
ab 1 ab	ab 1 ab
Sab (k)= 2 k (k), Sab [k]= 1 k [k],
ab 1 ab	ab 1 ab
Sab (k)= 2 k (k), Sab [k]= — 2 k [k],	(9.43)
which means that we get the same objects back multiplied by the constant 2k in
ab	ab
the case of Sab, while Sab multiply (k) by k and [k] by (—k) rather than (k). This
ab	ab
also means that when (k) and [k] act from the left hand side on a vacuum state |^0} the obtained states are the eigenvectors of Sab. We further recognize that Ya
ab	ab	ab	ab	ab	ab
transform (k) into [—k], never to [k], while Ya transform (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
yD" (k) = —inaa [k], YDb (k)= —k [k], yD" [k] = i (k), y1" [k] = —knaa (k) (9.44) From Eq.(9.44) it follows
ab cd	i	ab cd	ab cd i	abcd
Sac (k)(k) = — 2naancc [—k][—k], Sac (k)(k)= 2^aVc [k][k],
ab cd i ab cd	ab cd i ab cd
Sac [k][k] = 2 (—k)(—k), Sac [k][k]= — 2 (k)(k),
ab cd	i	ab cd	ab cd i	ab cd
Sac (k)[k] = — 2^aa [—k](—k), Sac (k)[k]= — 2^aa [k](k),
ab cd i	ab cd	ab cd i	ab cd
Sac [k](k) = 2ncc (—k)[—k], Sac [k](k)= 2ncc (k)[k] .	(9.45)
From Eqs. (9.45) we conclude that Sab generate the equivalent representations with respect to Sab and opposite.
Let us deduce some useful relations
ab ab (k)(k) = 0, ab ab ab
[k] [k] = [k],
ab ab (k)[k] = 0, ab ab ab
(k)[—k] = (k),
ab ab	ab
(k)(—k)= naa [k].
ab ab
[k][—k]= 0,
ab ab ab [k](k)=(k), ab ab [k](—k)= 0,
ab ab	ab	ab ab
(—k)(k)=naa [—k], (—k)(—k)= 0,
ab ab	ab ab ab
[—k][k]= 0,
ab ab ab
(—k)[k]=(—k), ab ab [—k](k)= 0,
[—k][—k]=[—k],
ab ab (—k)[—k]= 0, ab ab ab [—k](—k)=(—k) .
(9.46)
We recognize in the first equation of the first line and the first and the second equation of the second line the demonstration of the nilpotent and the projector
ab	ab
character of the Clifford algebra objects (k) and [k], respectively. Defining
ab
1
ab
(±i)= 2 (Ya T Yb),
1
(±1)= 2 (Ya ± iYb),
one recognizes that
ab ab
(k) (k) = 0,
Recognizing that
ab ab
—k) (k) =
in
ab [k],
ab ab	ab
(k) [k]= i (k),
(9.47)
ab ab
(k) [—k]= 0. (9.48)
ab ^	ab	ab ^ ab
(k) = naa (-k), [k] =[k]:
we define a vacuum state |^0 > so that one finds
ab ^ ab
< (k) (k) >= 1 ,
ab^ ab < [k] [k] >= 1 .
(9.49)
(9.50)
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.
For d even we simply make a starting state as a product of d/2, let us say,
ab
only nilpotents (k), one for each Sab of the Cartan subalgebra elements (Eq.(9.39)), applying it on an (unimportant) vacuum state. For d odd the basic states are products of (d — 1)/2 nilpotents and a factor (1 ± r). Then the generators Sab, which do not belong to the Cartan subalgebra, being applied on the starting state from the left, generate all the members of one Weyl spinor.
0d 12 35 (kod)(ki2)(k35) 0d 12 35
-kod][—ki2](k35)
0d 12 35
-kod](ki2)[—k35]
d-1 d-2 (kd-1 d-2) ^0 d-1 d-2 (kd-i d-2) ^0
d-1 d-2
(kd-
1 d-2) ^0
0d 12 35	d-1 d-2
[-kod](ki2)(k35) • • • [-kd-1 d-2] ^0
od 12 35	d-1 d-2
(kod)[-ki2][-k35] • • • (kd-1 d-2) ^0
i	(9.51)
All the states have the 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 made a choice of the simplest one, taking all phases equal to one.
The above graphic representation demonstrate that for d even all the states of one irreducible Weyl representation of a definite handedness follow from a
ab
starting state, which is, for example, a product of nilpotents (kab), by transforming
ab mn	ab mn
all possible pairs of (kabHW) 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. Both Weyl spinors form a Dirac spinor.
For d odd a Weyl spinor has besides a product of (d — 1)/2 nilpotents or projectors also either the factor o:= 1 (1 + r) or the factor •:= 1 (1 — r). As in the case of d even, all the states of one irreducible Weyl representation of a definite handedness follow from a starting state, which is, for example, a product of (1 + r)
ab	ab mn
and (d — 1)/2 nilpotents (kab), by transforming all possible pairs of (kab)(kmn)
ab mn
into [—kab][—kmn]. But Ya's, being applied from the left hand side, do not change the handedness of the Weyl spinor, since {r,Ya}_ = 0 for d odd. A Dirac and a Weyl spinor are for d odd identical and a "family" has accordingly 2(d-1)/2 members of basic states of a definite handedness.
We shall speak about left handedness when r = — 1 and about right handedness when r = 1 for either d even or odd.
While Sab which do not belong to the Cartan subalgebra (Eq. (9.39)) generate all the states of one representation, generate Sab which do not belong to the Cartan subalgebra(Eq. (9.39)) the states of 2d/2-1 equivalent representations.
Making a choice of the Cartan subalgebra set (Eq.(9.39)) of the algebra Sab and Sab a left handed (r(13,1' = — 1) eigen state 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 9 101 1 1213 14
(+i)(+)l ( + )( + ) II (+) ( — ) ( — ) IW> =
(Y0 — Y3)(Y1 + iY2)I(Y5 + iY6)(Y7 + iY8)II
(Y9 + iY10)(Y11 — iY12)(Y13 — iY14)I^>. (9.52)
This state is an eigen state of all Sab and Sab which are members of the Cartan subalgebra (Eq. (9.39)).
The operators Sab, which do not belong to the Cartan subalgebra (Eq. (9.39)), generate families from the starting uR quark, transforming uR quark from Eq. (9.52) to the uR of another family, keeping all the properties with respect to Sab unchanged. In particular S01 applied on a right handed uR-quark, weak charge-less, with spin up, hyper charge (2/3) and the colour charge (1/2,1/(2%/3)) from Eq. (9.52) 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))
03 12 56 78 91011121314 i 03 12 56 78 91011 121314
S01 (+iH+)I(+)(+) II (+)(—)(—) = — 2 [+i][ + ] I (+)(+) II (+) (—)(—) .
2	(9.53)
9 Can Spin-charge-family Theory Explain Baryon Number Non-conservation? Below some useful relations [4] are presented
03 12	03 12
N± = N+ ± iN+ = - (Ti)(±), N± _ N- ± iN- _(±i)(±),
"+ ± iN+
03 12	03 12
:i± _ f^wii ki±
IN + = — C=Fi)C=±), IN± =(±i)(±),
56 78	56 78
1± - _
T'± _(f)(±)(F) , T 56 78
_(T (±)(F) ,
=1±
(T (t)(T),
56 78
2t _(T) (T)(T) .
(9.54)
I present at the end one Weyl representation of SO (13 +1) and the family quantum numbers of the two groups of four families.
One Weyl representation of SO(13 + 1) contains left handed weak charged and the second SU(2) chargeless coloured quarks and colourless leptons and right handed weak chargeless and the second SU(2) charged quarks and leptons (electrons and neutrinos). It carries also the family quantum numbers, not mentioned in this table. The table is taken from the reference [15].
^t3TTt


r (1,7) =
of quarks
^i >_
= 1, r(6) = -1,
and leptons
03 12
+i) ( + ) I
10 1112 13 14
+ ) ( -) ( -)
2 V3
5 ( +
10 11 12 13 14
+ ) ( -) ( -)
2 V3
12
( + )
10 11 12 13 14
+ ) ( -) ( -)
2 V3
10 11 12 13 14
( -) ( -)
2 V3
12
( + ) I
10 1112 13 14
+ ) ( -) ( -)
2 VJ
+i) [ -] I [
10 1112 13 14
+ ) ( -) ( -)
2 VJ
12
( + ) I
10 1112 13 14
+ ) ( -) ( -)
2 VJ
+i) [ -]
5 ( +
10 1112 13 14
+ ) (-) (-)
2 VJ
12
( + ) I
10 1112 13 14
-] [+] (-)
2 V3
5 ( +
10 1112 13 14
-] [+] (-)
2 V3
1 2	0	1 V3	1
1 2	0	1 V3	1
			
1 2	0	0	1
1 2	0	0	1 2
1 - 2	0	0	1 2
1 - 2	0	0	1 2
0	0	0	1 2
0	0	0	1 2
0	0	0	1 2
0	0	0	1 2
1 2	1 2	1 2 VJ	1 6
1 2	1 2	1 2 VJ	1 6
1 2	1 2	1 2 VJ	1 6
12 5
( + ) I ( +
10 1112 13 14
-] (-) [+]
5 ( +
1112 13 14
(-) [+]
12
( + ) I
10
+ )
11 12
[+]
13 14
[+]
5 ( +
11 12
[+]
13 14
[+]
12
( + )
11 12
[+]
13 14
[+]
11 12
[+]
13 14
[+]
12
( + ) I
11 12
[+]
13 14
[+]
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36	a l1	03 12 56 78 910 1112 1314 (+i) [ —] I [ —] [ —] ll[ —] [+] [+]	-1	1 — 2	1	0	1 —2	1 — 2	1 2 V3	1 — 6	2 — 3	2 — 3
37	id c 1 aR	03 12 56 78 9 10 11 12 13 14 (+i) ( + ) I (+) [ —] II [ —] [+] [+]	1	1 2	-1	1 2	0	1 — 2	1 2 V3	1 — 6	1 — 6	1 3
38	ai1	03 12 56 78 9 10 11 12 13 14 [ —i] [ —] I ( + ) [ —] II [ —] [+] [+]	1	1 2	-1	1 2	0	1 2	1 2 V3	1 6	1 6	1 3
39	aR1	03 12 56 78 9 10 11 12 13 14 (+i) ( + ) I [ —] ( + ) I I [ —] [+] [+]	1	1 2	-1	1 2	0	1 2	1 2 V3	1 6	1 6	2 3
40	aR1	03 12 56 78 9 10 11 12 13 14 [ —i] [ —] I [ —] ( + ) 11 [ —] [+] [+]	1	1 2	-1	1 2	0	1 2	1 2 V3	1 6	1 6	2 3
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49	aL3	03 12 56 78 9 10 11 12 13 14 [ —i] (+) I ( + ) ( + ) II ( + ) [+] ( — )	-1	1 2	1	0	1 2	0	1 ,/3	1 — 6	1 3	1 3
												
57	eL	03 12 56 78 9 10 11 12 13 14 [ —i] (+) I ( + ) ( + ) I I [ —] ( —) ( — )	-1	1 2	1	0	1 2	0	0	1 2	1	1
58	eL	03 12 56 78 9 10 11 12 13 14 (+i) [ —] I ( + ) ( + ) I I [ —] ( —) ( — )	-1	1 — 2	1	0	1 2	0	0	1 2	1	1
59	VL	03 12 56 78 9 10 11 12 13 14 [ —i] ( + ) I [ —] [ —] 11 [ —] ( —) ( —)	-1	1 2	1	0	1 — 2	0	0	1 2	0	0
60	VL	03 12 56 78 9 10 11 12 13 14 (+i) [ —] I [ —] [ —] 11 [ —] ( —) ( —)	-1	1 2	1	0	1 2	0	0	1 2	0	0
61	VR	03 12 56 78 9 10 11 12 13 14 (+i) ( + ) I [ —] ( + ) I I [ —] ( —) ( — )	1	1 2	-1	1 2	0	0	0	1 2	1 2	0
62	VR	03 12 56 78 9 10 11 12 13 14 [ —i] [ —] I [ —] ( + ) 11 [ —] ( —) ( —)	1	1 2	-1	1 2	0	0	0	1 2	1 2	0
63	eR	03 12 56 78 9 10 11 12 13 14 (+i) ( + ) I (+) [ —] I I [ —] ( —) ( — )	1	1 7	-1	1 2	0	0	0	1 2	1 2	1
64	eR	03 12 56 78 9 10 11 12 13 14 [ —i] [ —] I ( + ) [ —] 11 [ —] ( —) ( —)	1	1 2	-1	1 2	0	0	0	1 2	1 2	1
Table 9.3. The left handed (F(13,11 = -1) multiplet of spinors - the members of the SO(13,1) group, manifesting the subgroup SO(7,1) - 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 Appendix 9.9. It contains the left handed (F(3,11 = — 1) weak charged (t13 = ±2) and SU(2)n chargeless (t23 = 0) quarks and the right handed weak chargeless and SU(2)n charged (t23 = ± 1) quarks of three colours (cl = (t33,t38)) with the "spinor" charge (t4 = 1) and the colourless left handed weak charged leptons and the right handed weak chargeless leptons with the "spinor" charge (t4 = — 2). S12 defines the ordinary spin ±2. The vacuum state |vac >fQm, on which the nilpotents and projectors operate, is not shown. The reader can find this Weyl representation also in the refs. [21,3]. Left handed antiquarks and anti leptons are weak chargeless and carry opposite charges.
The eight families of the first member of the eight-plet of quarks from Table 9.3, for example, that is of the right handed u1R quark, are presented in the left column of Table 9.4 [3]. 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<2'1)± 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.
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9.10 APPENDIX: Expressions for the spin connection fields in terms of vielbeins and the spinor sources [14]
The expressions for the spin connection of both kind, daba and d aba in terms of the vielbeins and the spinor sources of both kinds are presented, obtained by the variation of the action Eq.(9.1). The expression for the spin connection daba is taken from the ref. [32].
One notices that if there are no spinor sources, carrying the spinor quantum numbers Sab, then daba is completely determined by the vielbeins.
Equivalently one obtains expressions for the spin connection fields carryin family quantum numbers
Acknowledgments
The author acknowledges funding of the Slovenian Research Agency.
References
1. N.S. Mankoc Borstnik, "Spin-charge-family theory is explaining appearance of families of quarks and leptons, of Higgs and Yukawa couplings", in Proceedings to the 16th Workshop "What comes beyond the standard models", Bled, 14-21 of July, 2013, eds. N.S. Mankoc Borstnik, H.B. Nielsen and D. Lukman (DMFA Zaloznistvo, Ljubljana, December 2013) p.113 [arxiv:1312.1542].
(9.55)
(9.56)
2.	N.S. Mankoc Borštnik, "Do we have the explanation for the Higgs and Yukawa couplings of the standard model", http://arxiv.org/abs/1212.3184v2, (http://arxiv.org/abs/1207.6233), in Proceedings to the 15 th Workshop "What comes beyond the standard models", Bled, 9-19 of July, 2012, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Založnistvo, Ljubljana, December 2012, p.56-71, [arxiv.1302.4305].
3.	N.S. Mankoc Borstnik, J. of Modern Phys. 4 (2013) 823 [arxiv:1312.1542].
4.	A. Borstnik Bracic and N.S. Mankoc Borstnik, Phys. Rev. D 74 (2006) 073013 [hep-ph/0512062, hep-ph/9905357, p. 52-57].
5.	N.S. Mankoc Borstnik, Phys. Lett. B 292 (1992) 25.
6.	N.S. Mankoc Borstnik, J. Math. Phys. 34 (1993) 3731.
7.	N.S. Mankoc Borstnik, Int. J. Theor. Phys. 40 (2001) 315.
8.	N.S. Mankoc Borstnik, Modern Phys. Lett. A 10 (1995) 587.
9.	G. Bregar, M. Breskvar, D. Lukman and N.S. Mankoc Borstnik, New J. of Phys. 10 (2008) 093002.
10.	G. Bregar and N.S. Mankoc Borstnik, "Can we predict the fourth family masses for quarks and leptons?", Proceedings (arxiv:1403.4441) to the 16th Workshop "What comes beyond the standard models", Bled, 14-21 of July, 2013, Ed. N.S. Mankoc Borštnik, H.B. Nielsen, D. Lukman, DMFA Založnistvo, Ljubljana December 2013, p. 31-51, http://arxiv.org/abs/1212.4055.
11.	G. Bregar and N.S. Mankoc Borstnik, "The new experimental data for the quarks mixing matrix are in better agreement with the spin-charge-family theory predictions", Proceedings to the 17th Workshop "What comes beyond the standard models", Bled, 2028 of July, 2014, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA Založnistvo, Ljubljana December 2014.
12.	G. Bregar and N.S. Mankoc Borstnik, Phys. Rev. D 80 (2009) 083534.
13.	The authors of the works presented in An introduction to Kaluza-Klein theories, Ed. by H. C. Lee, World Scientific, Singapore 1983, T. Appelquist, A. Chodos, P.G.O. Freund (Eds.), Modern Kaluza-Klein Theories, Reading, USA: Addison Wesley, 1987.
14.	N.S. Mankoc Borstnik,"The spin-charge-family theory explains why the scalar Higgs carries the weak charge ^ 1 and the hyper charge ± j", to appear in Proceedings to the 17th Workshop "What comes beyond the standard models", Bled, 20-28 of July, 2014, eds. N.S. Mankocš Borsštnik, H.B. Nielsen and D. Lukman (DMFA Založšnisštvo, Ljubljana, December 2014).
15.	N.S. Mankoc Borstnik, H.B. Nielsen, "Discrete symmetries in the Kaluza-Klein-like theories", JHEP 04 (2014) 165 http://arxiv.org/abs/1212.2362v2.
16.	T. Troha, D. Lukman, N.S. Mankoc Borstnik, Int. J. of Mod. Phys. A 29 (2014) 1450124, arxiv:1312.1541.
17.	N.S. Mankoc Borstnik, [arxiv:1212.3184, arxiv:1011.5765].
18.	N.S. Mankoc Borstnik and H.B. Nielsen, J. of Math. Phys. 43 (2002) 5782 [hep-th/0111257].
19.	N.S. Mankoc Borštnik and H.B. Nielsen, Phys. Rev. D 62 (2000) 044010 [hep-th/9911032].
20.	N.S. Mankoc Borstnik and H.B. Nielsen, J. of Math. Phys. 44 (2003) 4817-4827, hep-th/0303224.
21.	A. Borstnik, N.S. Mankoc Borstnik, Proceedings to the Euroconference on Symmetries Beyond the Standard Model, Portorož, July 12 -17, 2003, Ed. Norma Mankoc Borstnik, Holger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFA, Založnistvo, Ljubljana, December 2003, p. 31-57, hep-ph/0401043, hep-ph/0401055.
22.	D. Lukman, N.S. Mankoc Borstnik and H.B. Nielsen, New J. Phys. 13 (2011) 103027.
23.	D. Lukman and N.S. Mankoc Borstnik, J. Phys. A: Math. Theor. 45 (2012) 465401 [arxiv:1205.1714, hep-ph/0412208 p.64-84].
24.	A. D. Sakharov, JETP 5 (1967) 24.
25.	C. Itzykson adn J. Zuber, Quantum Field Theory (McGraw-Hill 1980).
26.	H. Georgi, S. Glashow, Phys. Rev. Lett. 32 (1974) 438; Phys. Rev. Lett. 32 (1974) 438.
27.	H. Georgi, in Proceedings of the American Institue of Physics, ed. Carlson (1974).
28.	H. Fritzsch and P. Minkowski, Ann. Phys. 93 (1975) 193.
29.	M. Gell-Mann, P. Ramond and R. Slansky, Rev. Mod. Phys. 50 (1978) 721.
30.	A. Buras, J. Ellis, M. Gaillard and D.Nanopoulos, Nucl. Phys. B 135 (1978) 66.
31.	Unity of Forces in the Universe, ed. A. Zee (World Scientific,1982).
32.	N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, "An example of Kaluza-Klein-like theories leading after compactification to massless spinors coupled to a gauge field-derivations and proofs", Proceedings to the 7th Workshop "What Comes Beyond the Standard Models", Bled, July 19 - 31, 2004, Ed. by Norma Mankoc Borstnik, Holger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFA ZaloZnistvo, Ljubljana December 2004, p.64-84, [hep-ph/0412208].
Bled Workshops in Physics Vol. 15, No. 2
A
Proceedings to the 17th Workshop What Comes Beyond ... (p. 163) Bled, Slovenia, July 20-28, 2014

10 The Spin-charge-family Theory Explains Why the
N.S. Mankoc Borštnik
Department of Physics, FMF, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana,
Abstract. One Weyl representation of SO(13 + 1) contains [1-7], if analysed with respect to the charge and the spin groups of the standard model, left handed weak (SU(2) i) charged and SU(2)ii chargeless colour triplet quarks and colourless leptons, and right handed weakless and SU(2)II charged quarks and leptons (neutrinos and electrons). In the spin-charge-family theory [1-12] spinors carry also the family quantum numbers, explaining the origin of families and correspondingly the masses of fermions and weak bosons and the origin of the scalar Higgs and Yukawa couplings. I am demonstrating in this paper that all the fields appearing in the simple starting action of the spin-charge-family theory in d = (13 + 1) with the scalar index with respect to d = (3 + 1) and determining masses of quarks and leptons (and correspondingly also of the weak boson fields) carry the weak and the hyper charge required by the standard model for the scalar Higgs.
Povzetek. Ena Weylova upodobitev SO(13 + 1) vsebuje [1-7], ce jo analiziramo glede na grupe nabojev in spinov standardnega modela, levorocne kvarke z barvnim tripletnim nabojem in brezbarvne leptone s sibkim nabojem (SU(2)I), ki nimajo naboja SU(2)II ter desnorocne barvne triplete kvarkov in brezbarvnih leptonov, ki ne nosijo sibkega naboja, nosijo pa naboj SU(2)II. V teoriji spinov-nabojev-druzin [1-12] nosijo spinorji tudi kvantna stevila družin, kar pojasni izvor družin in tudi mase fermionov in sibkih bozonov ter izvor Higgsovega skalarja in Yukawinih sklopitev. V tem prispevku pokazem, da nosijo vsa polja s skalarnim indeksom glede na d = (3 + 1) s = (7, 8), ki nastopajo v enostavni začetni akciji teorije spinov-nabojev-druzin v d = (13 + 1) in dolocajo mase kvarkov in leptonov, s tem pa tudi mase sibkih bozonov, sibki in hiper naboj tak, kot ju zahteva standardni model za skalarno Higgsovo polje. Teorija tako ponudi razlago za izmerjene lastnosti Higgsovega skalarja ter Yukawinih sklopitev.
10.1 Introduction
The standard model assumed and the LHC confirmed the existence of the Higgs scalar - the only so far observed bosons with the charge in the fundamental representation.
I am demonstrating in this paper that the spin-charge-family theory explains the appearance of the scalar fields with the charges of the Higgs scalar fields. There are, namely, in this theory, in its simple starting action in d = (13 + 1), the fields
Scalar Higgs Carries the Weak
Slovenia
with the scalar index with respect to d = (3 + 1), which have the properties of the higgs and explain masses of quarks and leptons together with the Yukawa couplings, and correspondingly also the masses of the weak vector boson fields.
Let me add that all the scalars, that is all the gauge fields of this theory with the space index > 5, have in the starting action the corresponding charges with respect to the scalar index in the fundamental representations: They are either doublets with respect to the two SU(2) groups (the weak SU(2)i and the second SU(2)n, which correspondingly result in the properties of the Higgs scalars with respect to the weak and the hyper charge) or they are colour triplets (the properties of these triplets are discussed in a separate paper [13]). All these scalar fields carry the additional charges (the charges not originating in the space index - the family charges, for example) in adjoint representations.
The referee of this paper stated that it is not at all remarkable that there are the scalar fields which are doublets with respect to the weak charge after starting with so many independent fields.
It is, of course, true that a large enough orthogonal group can contain any desired subgroups. But this is not what the spin-charge-family theory proposes: It starts with an (very simple) action for spinors and the corresponding gauge fields, manifesting very limited properties, and it is not at all self-evident that some of these fields manifest the desired properties in the low energy regime while all the other spinors and vector and scalar gauge fields - unobserved in the low energy regime - get high masses through the interaction with only one scalar condensate, what is happening in the spin-charge-family theory.
On the contrary, it is an extremely encouraging fact that one scalar condensate makes all the vector and the scalar gauge fields appearing in the spin-charge-family theory, except those which are observable at the low energy regime (the gravity, the colour vector gauge field, the weak and the hyper charge vector gauge fields, and the eight families of quarks and leptons, decoupled into two times four families), very massive with respect to the weak scale and correspondingly unobservable in the low energy regime. Several scalar gauge fields, however, which when gaining nonzero vacuum expectation values (changing in this case also their masses) cause the electroweak break, have the weak charge equal to ± 2 and the hyper charge correspondingly ^ 2, as the scalar Higgs in the standard model, while they have all the other quantum numbers in the adjoint representations. All the rest of the scalar fields are colour triplets with respect to the scalar space index.
Those who are proposing unifying theories, must offer for the chosen groups and the chosen representations of these groups also the Lagrange densities, designed for those groups and representations, what calls for the theory beyond those effective actions. I am proposing a simple starting action, out of which - after the breaks of symmetries triggered by boundary conditions in a complicated many body problem - manifests in the low energy regime the observable phenomena.
Let me make in this introduction make a short overview of the spin-charge-family theory [1,2,7,6,3-5,8-12], pointing out that this theory is offering the explanation for the assumptions of the standard model: For the properties of quarks and leptons (right handed neutrinos are in this theory the regular members of a family)
and antiquarks and antileptons, and for the existence of the gauge vector fields 1 of the charges. It is offering also the explanation for the existence of the families of quarks and leptons and correspondingly for the scalar gauge fields, which are responsible for masses of quarks and leptons and of weak gauge fields and for Yukawa couplings.
There are, namely, (only) two kinds [3-5,7,16-18] of the Clifford algebra objects (connected by the left and the right multiplication of any Clifford object): the Dirac Ya's and the second Ya's, respectively. These two Clifford algebra objects (Eq. (9.35)) anticommute, forming the equivalent representations with respect to each other. If using the Dirac Ya's in d =(13 + 1) to describe in d = (3 + 1) the spin and all the charges, then Ya's describe families.
All predictions of the spin-charge-family theory in the low energy regime (after the break of the starting symmetry) follow from the simple starting action (Eq.(10.1)) in d = (13 +1) for spinors carrying two kinds of a spin (no charges) and for the vielbeins and the two kinds of spin connection fields, with which spinors interact.
Let us first tell that one Weyl representation of SO(13,1) contains [1,2,7,6,14], if analysed with respect to the subgroups SO(3,1)x SU(2)ix SU(2)n x SU(3) xU(1), all the family members, required by the standard model, with the right handed neutrinos in addition: It contains the left handed weak (SU (2) I) charged and SU(2)II chargeless colour triplet quarks and colourless leptons (neutrinos and electrons), and right handed weakless and SU(2)II charged quarks and leptons, as well as right handed weak charged and SU(2)II chargeless colour antitriplet antiquarks and (anti)colourless antileptons, and left handed weakless and SU(2)II charged antiquarks and antileptons. The antifermions are reachable from the fermions by the application of the discrete symmetry operator Cn Pn, presented in ref. [14].
The theory accordingly explains how and why is the weak charge connected with the handedness determined by the spin degrees of freedom in d = (3 + 1). One Weyl (one family) representation of spinors of the group SO(13,1) is presented in table 9.3. Each state is written as a product of nilpotents and projectors defined in the "technique" [4,16,18,17], short version of which can be found in appendix 9.9. Quantum numbers of each of the family members, all are presented in table 9.3 together with the quantum numbers, are defined in Eqs. (10.8,10.9,10.10).
The symmetry of both kinds of groups, SO(13,1) and SO (13,1) (are assumed to) break simultaneously, influencing family members and families of spinors, as well as the gauge fields. After the break of symmetries from the manifold M(13+1' to M(7+1) x M(6), which makes all the families, except the 2Z++L_1 ones determined by the group SO(7,1), massive 2, carries each family member the family
1	In this sense the spin-charge-family is the Kaluza-Klein like theory [15].
2	(7 + 1 ) _1
In this paper the break of symmetries in the way that only 2 2 families stay massless, while all the others get high masses of the order above the unifying scale, is just assumed. This assumption, however, is supported by several works on the toy model with the same starting action (Eq. (10.1)) but with d = (5 + 1), ref. [20,23], while the preliminary work on this more complex case is in progress.
quantum numbers, belonging in SO(7,1) to two times SU(2) x SU(2) groups, originating in SO(3,1) and in SO(4), respectively (where SO(n) represent the subgroups the generators of which are expressed by Ya). The families correspondingly decouple to two times four families.
The generators of the corresponding subgroups of the family group SO(7,1), are defined in Eqs. (10.11, 10.12). To each family member of each family the antimember corresponds, accessible from the member by the discrete symmetry operator Cn Pn, which does not depend on Ya's, as explained in the ref. [14,25].
Let us add that since all the charges, with the family charge included, emerge from the spins, correspondingly all the charges are quantized.
Quarks and leptons have the "spinor" quantum number (t4, originating in SO (6), Eq. (10.10)) 1 and — 2, respectively 3, with the sum of both equal to
3 x 1 + (—2)= 0.
The spin-charge-family theory therefore predicts that there are two decoupled groups of four families: The fourth [1,7,6,9] to the already observed three families of quarks and leptons should (sooner or later) be measured at the LHC [11], while the lowest of the upper four families constitute the dark matter [10].
Let me summarize this first part of the introduction with the statement: The spin-charge-family theory is offering the explanation for the assumptions of the standard model, having correspondingly a chance to be the right step beyond the standard model.
This paper presents in section 10.2 that the properties of the scalar field, the weak and the hyper charge of the scalar Higgs, which are in the standard model just assumed to properly "dress" the right handed members by the weak and the hyper charge, appear in the spin-charge-family naturally, offering the explanation for the appearance of the scalar fields, observed so far as the scalar Higgs.
In the subsection of this introductory section the simple starting action of the spin-charge-family theory is presented, as well as all the assumptions made to achieve that the theory manifests at low energies the observed phenomena.
In section 10.3 the resume and conclusions are presented. In the first appendix 9.9 a short review of the technique, used in this paper to manifest properties of the spinor states, as well as the expressions for the two kinds of spin connection fields, in terms of vielbeins and the spinor sources, are added.
In section 10.3 the resume and conclusions are presented. In appendix a short review of the technique, used in this paper to manifest properties of the spinor states, as well as the expressions for the two kinds of spin connection fields, in terms of vielbeins and the spinor sources, are added.
10.1.1 The action of the spin-charge-family theory and the assumptions
Let me present the assumptions on which the theory is built, starting with the (simple) action in d =(13 + 1):
3 In the Pati-Salam model [21] this "spinor" quantum number is named -B^ quantum number and is twice the "spinor" quantum number, for quarks equal to 1 and for leptons to —1.
i. In the simple action [7,1] fermions ^ carry in d = (13 + 1) as the internal degrees of freedom only two kinds of spins, no charges, and interact correspondingly with only the two kinds of the spin connection gauge fields, daba and d aba, and the vielbeins, fa
a
S = ddx E Lf +
ddx E (aR + aR), Lf = 2 (^yapoa^) + h.c.,
P0a = faaP0a + ^ {Pa, Ef%}-, P0a = Pa - 1Sabdaba - 2"Saba,
R = 2 {fa[afPb] (daba,P - dCaa d%p)} + h.C. ,
R = 2 fafb] (ddaba,P - dcaad%p) + h.C. .	(10.1)
Here 4 fa[afpb] = faafPb - fabfPa. Sab and Sab are generators of the groups SO(13,1) and SO (13,1), respectively, expressible by Ya and Ya.
ii.	The manifold M(13+1' breaks first into M(7+1' times M(6) (which manifests as SU(3) xU(1)), affecting both internal degrees of freedom - the one represented by Ya and the one represented by Ya, leading to 2((7+1 )/2-1) massless families, all the rest families get heavy masses 5. Both internal degrees of freedom, the ordinary SO(13 + 1) one (where Ya determine spins and charges of spinors) and the SO(13 + 1) (where Ya determine family quantum numbers), break simultaneously with the manifolds.
iii.	There are additional breaks of symmetry: The manifold M(7+1' breaks further into M(3+1)x M(4).
iv.	There is a scalar condensate of two right handed neutrinos with the family quantum numbers of the upper four families, bringing masses of the scale above the unification scale, to all the vector and scalar gauge fields, which interact with the condensate.
4	faa are inverted vielbeins to eaa with the properties eaafab = 5ab, eaafPa = 6a, E = det(eaa). Latin indices a, b,.., m, n,.., s, t,.. denote a tangent space (a flat index), while Greek indices a, |3,.., |a, v, ..a, 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,..), indices from the bottom of the alphabets indicate the compactified dimensions (s, t,.. and a,T,..). We assume the signature r|ab = diag{1, —1, —1, • • • , -1}.
5	A toy model [23,24,14] was studied in d = (5 + 1) with the same action as in Eq.'(10.1). For a particular choice of vielbeins and for a class of spin connection fields the manifold Ms+1 breaks into M(3+1) times an almost S2, while 2((3+1 )/2 1' families stay massless and mass protected. Equivalent assumption, although not yet proved that it really works, is made also in the case that M(13+1) breaks first into M(7+1) x M(6). The study is in progress.
v. There are nonzero vacuum expectation values of the scalar fields with the scalar indices (7,8), which cause the electroweak break and bring masses to the fermions and weak gauge bosons, conserving the electromagnetic and colour charge.
Comments on the assumptions:
i.:	This starting action enables to represent the standard model as the effective low energy manifestation of the spin-charge-family theory, explaining all the standard model assumptions, with the families included. There are (before the electroweak break all massless) observable gauge fields: gravity, colour (SU(3), from SO(6)) octet vector gauge fields, weak (SU(2)i from SO(4)) triplet vector gauge field and "hyper" (U(1) from SO(6)) singlet vector gauge fields. All are superposition of fac caba. And there are (before the electroweak break all mass-less) observable (eight rather than observed three) families of quarks and leptons. (There are indeed two decoupled groups of four families, in the fundamental representations of twice SU(2) x SU(2) groups, the subgroups of SO(3,1) x SO(4). There are correspondingly the scalar fields with the weak and the hyper charge of the scalar Higgs and with either two kinds of the family quantum numbers in the adjoint representations - they are two times two triplets, emerging from the superposition of fCTscDabCT with s G (7,8), in accordance with twice SU(2) x SU(2) groups, the subgroups of SO(3,1)x SO(4) - or with the quantum numbers (Q, Q', Y') emerging from the superposition of fCTscDabCT. Both determine the Yukawa couplings.) The starting action contains also the additional SU(2)II (from SO (4)) vector gauge field and the scalar fields with the space index s G (5,6) and t G (9,10,11,12), as well as the auxiliary vector gauge fields expressible with vielbeins, which are the superposition of f^mcc ab^. They all remain either auxiliary or become massive after the appearance of the condensate.
ii.,	iii.: The assumed breaks explain why the weak and the hyper charge are connected with the handedness of spinors, manifesting the observed properties of the family members - the quarks and the leptons, left and right handed - and of the observed vector gauge fields. Since the left handed members are weak charged while the right handed are weak chargeless, the family members stay massless and mass protected up to the electroweak break. Antiparticles are accessible from particles by the Cn and Pn, as explained in refs. [14,25]. This discrete symmetry operator does not contain Ya's degrees of freedom. To each family member there corresponds the antimember, with the same family quantum number.
iv.:	It is the condensate of two right handed neutrinos with the quantum numbers of the upper four families, which makes all the scalar gauge fields (with the index (5,6,7,8), as well as those with the index (9,..., 14)) and the vector gauge fields, manifesting nonzero t4, t23, Q ,Y, t4, t23, Q ,Y,NR (Eqs. (10.8,10.9, 10.10,10.11,10.12,10.13)) massive [13].
v.:	At the electroweak break the scalar fields with the space index s = (7,8), originating in cDabs, as well as some superposition of cs/s"s, those which conserve the electromagnetic charge, get nonzero vacuum expectation values, what changes also their masses. They determine mass matrices of twice the four families, as well as the masses of the weak bosons. All the rest scalar fields keep masses of the condensate scale and are correspondingly (so far) unobservable in the low energy
regime 6. The fourth family to the observed three ones will (sooner or later) be observed at the LHC. Its properties are under the consideration [11], while the stable of the upper four families is the candidate for the dark matter.
The above assumptions enable that the starting action (Eq. (10.1)) manifests effectively in d = (3 + 1) in the low energy regime by the standard model required degrees of freedom of fermions and bosons [1,2,7,6,3-5,8-12], that is the quarks and the leptons, left and right handed, the families of quarks and leptons and all the known gauge fields, with (several, explaining the Yukawa couplings) scalar fields included.
To see this let us rewrite formally the action for the Weyl spinor of (Eq.(10.1)) as follows
Lf = JYm(pm - X 9ATAiAmi)^ + A,i
{Y_ JYsP0s J} +
s = 7,8
{ Y_ JYtP0t J},
t=5,6,9,...,14 P0s = Ps — ^S5 s ^s/S"S — ^SabO>abs ,
P0t = pt — 2st'^t't"t — 1sabCDabt ,	(10.2)
where m e (0,1,2,3), s e 7,8, (s', s") e (5,6,7,8), (a,b) (appearing in Sab) run within (0,1,2,3) and (5,6,7,8), t e (5,6,9,..., 13,14).
The first line of Eq. (10.2) determines the kinematics and dynamics of spinor fields in d = (3 + 1), coupled to the vector gauge fields. The generators TAi of the charge groups are expressible in terms of Sab through the complex coefficients cAiab, as presented in Eqs. (10.9,10.10,10.13)
TAi = ^ cAiab Sab ,	(10.3)
a,b
and the commutation relations
{xAi, xBj}_ = i6ABfAijkxAk .	(10.4)
The corresponding vector gauge fields Am1 are expressible with the spin connection fields ^stm, with (s, t) either e (5,6,7,8) or e (9,10,..., 13,14), in agreement with the assumptions ii. and iii.. Before the electroweak break the vector gauge fields appearing in the first line of Eq. (10.2) are all massless: Am carries the colour charge SU(3) (originating in SO(6)), Am carries the weak charge SU(2)i (SU(2)i and SU(2)II are the subgroups of SO (4)) and Am = sin$2Am + cosAm (Y is defined in Eq. (10.13), t4 in Eq. (10.10), the corresponding U(1) group originates in SO(6)), Am is defined in Eq. (10.15), if the scalar space index s is replaced by
6 Correspondingly d = (13 + 1) manifests in d = (3 + 1) spins and charges as if one would start with d = (9 + 1) instead of with d = (13 + 1), since the plane (5, 6) and the plane in
which the vector t4 lies, are unobservable at low energies.
the space vector index m, Am is the third component of the second SU(2)n field Am. The corresponding charges (T3, t1 , Y) are the conserved charges.
Before the appearance of the condensate of the two right handed neutrinos with the quantum numbers of the upper four families (properties of the condensate are presented in table 10.1) at the scale far about the electroweak scale, all the three components of the field A mm are massless. The condensate gives the mass of the order of the scale of the appearance of the condensate to A^ = cos $2 Am3—sin $2 A^, and to all the scalar gauge fields, presented in the second and the third line of Eq. (10.2), leading to A^i,s G (5,6,..., 13,14) and AAi, t G (5,6,..., 13,14).
Vector gauge fields A^, A^ and A^ do not couple to the condensate (table 10.1).
In Eqs. (10.15,10.14) the expressions for the scalars with the scalar index (7,8) in terms of both kinds of the spin connection fields are presented. These scalar fields (the second line in Eq. (10.2)) determine after the electroweak break the mass matrices of the two decoupled groups of four families. Getting nonzero vacuum expectation values they cause the electroweak break, changing also their own masses. These scalar fields determine also the masses of the gauge bosons.
state	S03 S12 T13 T23 T4 Y Q	T13 T23 T4 Y Q N L N R
(|v1R >1 |v2R >2j	0 0 0 1 -1 0 0	0 1 -1 0 0 0 1
ik,V111 ^ loVIII 1 (|v1R >1 |e2R >2 J n^vm ^ „VIII . (|e1R >1 |e2R >2 J	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 10.1. 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 and NR = 1, NL = 0, Y = 0, Q = 0. The family quantum numbers are presented in table 9.4, taken from the ref. [13].
Among the vector gauge fields A^ and A^ and the corresponding vielbeins only one of these three vector gauge fields is the propagating one, while the rest two are the auxiliary fields as one can learn from Eqs. (9.55, 9.56) of the second appendix section 9.10, if taking into account that there is no spinor (fermion) sources with the corresponding quantum numbers. Equivalently, also only one of the three vector gauge fields Am, A^ and the corresponding vielbein field is the propagating field, the other two are the auxiliary fields, as well as only one of the three vector gauge fields Am, Am and the corresponding vielbein field is the propagating field, while Am is massive due to the interaction with the condensate of the two right handed neutrinos through quantum numbers N R, presented in Eqs. (10.8,10.11).
Let me summarize this subsection: The starting action (Eq.(10.1)) of the spin-charge-family theory manifests under the assumptions i.-v. in the low energy regime properties of the standard model, explaining the standard model assumptions: Before the electroweak break all the scalar gauge fields and the vector gauge fields -
except the colour, the weak and the hyper vector fields (and the gravity), which stay massless - are massive, due to the interaction with the scalar condensate of the two right handed neutrinos with the family quantum numbers of the upper four families. There are also the two decoupled massless groups of four families.
At the electroweak break the scalar gauge fields, carrying the scalar space index and keeping the electromagnetic charge conserved and changing their own masses, bring masses to all the fermions and all the gauge fields, except to the gavity, electromagnetic and the colour ones.
Let me comment that in the presence of the spinor fields (as it is the condensate, for example) all three gauge fields - the vielbeins and the two kinds of the spin connection fields - are in general the propagating fields. If there are no spinors present, only one of the three fields is the propagating field, the other two are expressible with the propagating one (as it is well known). In the second appendix 9.10 the expressions for the spin connection fields of both kinds in terms of the vielbeins and the spinor sources are presented, taken from the ref. [20].
The assumed breaks should occur spontaneously, determined by the starting action and the boundary conditions. To prove that this really can happen is a very difficult (many body) problem. Although several studies made so far, for either a toy model in d = (5 + 1) or for the d = (13 + 1) case, support these assumptions, yet several additional studies are needed to justify the assumptions and to clarify further the properties of the scalar and vector gauge fields and of the spinor families, appearing in the starting action. Also the comparison with all the other works made on the unifying theories are needed to see to which extend predictions of this theory coincide with the other theories in the literature, in which sense and what one can learn out of them.
The standard model subgroups of the SO (13 + 1) and of the SO (13 + 1) group and the corresponding gauge fields To calculate quantum numbers of one Weyl representation presented in table 9.3 in terms of the generators of the standard model groups tai (= Y. ab cAlab Sab) one must look for the coefficients cAlab (Eq. (10.4)). The generators tai are the generators of the charge groups. Similarly one expresses also the spin and the family degrees of freedom.
The same coefficients cAlab determine operators which apply on spinors and on vectors. The difference among the three kinds of operators - vectors and two kinds of spinors - lies in Sab.
While Sab for spins of spinors is equal to
Sab = 4(Ya Yb — Yb Ya),	(10.5)
and Sab for families of spinors is equal to
Sab = 4(Ya Yb — Yb Ya),	(10.6)
one must take, when Sab apply on the spin connections ^bde (= fae ^bd«) and cDbde (= fae &bda), on either the space index e or the indices (b, d, b, d), the
operator
(Sab)ceAd."e".g = i(nac§b — nbc&£) Ad."e".g .	(10.7)
This means that the space index (e) of cbde transforms according to the requirement of Eq. (10.7), and so do b, d and 15, d. I used the notation 15, d to point out that Sab and Sab (= S a 11) are generators of two independent groups.
One finds [1,7,6,3-5,8,12] for the generators of the spin and the charge groups, which are the subgroups of SO(13,1), the expressions:
N±(= N(L,R)):= 1 (S23 ± iS01,S31 ± iS02,S12 ± iS03), (10.8)
where the generators N± determine representations of the two SU(2) subgroups of SO(3,1), generators t1 and t2,
T1 : = 1 (S58 — S67, S57 + S68, S56 — S78),
T2 : = 1 (S58 + S67, S57 — S68, S56 + S78),	(10.9)
determine representations of the SU(2)ix SU(2)ii invariant subgroups of the group SO(4), which is further the subgroup of SO(7,1) (SO(4), SO(3,1) are subgroups of SO(7,1)), and the generators t3, t4 and t4
T3 := 1 {S912 — S1011 ,S911 + S1012, S910 — S11 12, S9 14 — S10 13 S913 + S10 14 51' 14 — §12 13
S11 13 + S1214, -L(S910 + S11 12 — 2S1314)}, V3
T4 := —1 (S910 + S11 12 + S1314),
t4 := — 3(S910 + S11 12 + S13 14),	(10.10)
determine representations of SU(3) x U(1), originating in SO (6), and of t4 originating in SO(6).
One correspondingly finds the generators of the subgroups of SO(7,1),
Nl,r : = 2(S23 ± iS01, S31 ± iS02, S12 ± iS03),	(10.11)
which determine representations of the two SU(2) invariant subgroups of SO(3,1), while
T1 : = 1 (S58 — S67, S57 + S68, S56 — S78),
T2 : = 2(S58 + S67, S57 — S68, S56 + S78) ,	(10.12)
determine representations of SU(2)Ix Su(2)h of SO(4). Both, SO(3,1) and SO(4) are the subgroups of SO(7,1).
10 The Spin-charge-family Theory Explains Why the Scalar Higgs Carries... One further finds
Y = t4 + t23, Y' = —t4 tan2 £2 + t23,
Y = t4 + t
.23
Y' =
-T4 tan2 £2 + t23,
Q = t13 + Y,
Q = Y + t13,
13
Q' = —Y tan2 £1 + t Q' = —Y tan2 £1 + t13.
(10.13)
The scalar fields, responsible [1,2,7] - after getting in the electroweak break nonzero vacuum expectation values - for masses of the family members and of the weak bosons, are presented in the second line of Eq. (10.2). These scalar fields
are included in the covariant derivatives as
Ss
2 S ^s 's"s
(a, b), € (0, T,..., 8), where a, b is again used to point out that (a, b) belong in this case to the "tilde" space.
One finds, by taking into account Eqs. (10.11,10.12) and Eq. (10.13), for the choice of the aabs scalar gauge fields, contributing to the electroweak break, the expressions
2Sab^a6s, s € (7,8),
TSab a
abs = — (TT A1 + NL AN + T2 A2 + NR AS
¿2
),
A s = (a58s
'Nr
a67s, a57s + a68s, a56s
a
78s
),
ANr = (a23s + va5 T s, a3 Ts + va52s,
a
T 2s
+ va53s),
A 2 = (a58s + a
~ nr , _
As R = (a23s — va, (s € (7,8)).
a68s, a56s + a78s) ,
5 T s
a
3 T s

va
52s
a
T 2s

va
53s
),
(10.14)
Among aabs, which contribute to the mass matrices of quarks and leptons, one finds when using Eqs. (10.9,10.10,10.13), the expressions
T cs 's 2S
as's"s = —(g23 T23 A23 + g1 3 t 13 A s 3 + g4 t4 A4), 4^4 *4 _ „Q naQ , „Q 'n 'a Q ' 1 J V u Y'
g 13 t1 3 AS 3 + g23 t23 a23 + g4 t4 a4 = gQ QAQ + gQ Q'AQ + gY Y' AY ,
V4
s
1s3 = (a56s — a78s), A;:3
sY
A4 = —(a9i5s + an 12s + ai3i4s), A.13 = (a56s — a78s), A23 = (a56s + a78s),
AQ = sin£1 A.3 + cos£1 AY , AQ' = cos£s A.3 — sin£s AY ,
Asv = cos £2 A23 — sin £2 A4 ,
Y
s
t4
s
(s € (7,8)).
(10.15)
Scalar fields from Eq. (10.14) couple to the family quantum numbers, while those from Eq. (10.15) distinguish among family members. In Eq. (10.15) the coupling constants were explicitly written to see the analogy with the gauge fields in the standard model.
Expressions for the vector gauge fields in terms of the spin connection fields and the vielbeins, which correspond to the colour charge (Am), the SU(2)n charge (Amm), the weak charge (SU(2)i) (Am) and the U(T) charge originating in SO(6)
(A4m), can be found by taking into account Eqs. (10.9,10.10). Equivalently one finds the vector gauge fields in the "tilde" sector. One really can use just the expressions from Eqs. (10.15,10.14), if replacing the scalar index s with the vector index m.
Let me summarize this subsubsection: The expressions for the operators TAi are presented, either in terms of Sab (Eq. (10.5)) or (in this case we name them TAi) in terms of Sab (Eq. (10.6)), valid also in terms of Sab (Eq. (10.7)), affecting correspondingly spinors spin and charges quantum numbers, spinors families quantum numbers and scalar or vector gauge fields, respectively. Also the expressions for those scalar gauge fields, which contribute to the electroweak break by getting nonzero vacuum expectation values, in terms of the corresponding spin connection fields are presented (Eqs.(10.15, 10.14)). When the scalar index s is replaced by the vector index m, the expressions for the vector gauge fields in terms of spin connection fields follow.
10.2 Scalar fields contributing to the electroweak break are weak charge doublets
In this main part of the paper is demonstrated that all the scalar gauge fields with the scalar index s G (7,8), which get nonzero vacuum expectation values causing the electroweak break, carry the weak and the hyper charge as does the scalar Higgs of the standard model.
All the scalars (the gauge fields with the scalar index with respect to d = (3 + 1)) of the action (Eq. 10.1) contribute charges in the fundamental representations: The scalars with the space indices s G (7,8) and s G (5,6) are, with respect to this scalar space degree of freedom, before the appearance of the condensate (table 10.1), the weak (SU(2)^ and the second SU(2)n doublets. After the appearance of the condensate only the weak and the hyper charge Y remain the conserved charges, so that it is the third component of t23, which determines the hyper charge (Y = t23 + t4, Eq. (10.13)) of these scalar fields, since t4 applied on the scalar index of these scalar fields gives zero, according to Eqs. (10.9,10.10,10.7).
The scalars with the space indices s G (9,10,..., 13,14) are, again with respect to this scalar space degree of freedom, colour triplets [13]. There are no additional scalar indices and therefore no additional corresponding scalars with respect to the scalar indices in this theory.
The scalars, however, carry additional quantum numbers Ai, the states of which belong to the adjoint representations with respect to either TAi or TAi. While, to reproduce the low energy phenomena, the scalar fields of all the family quantum numbers are allowed, only those TAi are acceptable, which conserve after the electroweak break the electromagnetic charge. The scalar fields with nonzero vacuum expectation values carrying nonzero weak charge also due to t1 would cause nonconservation of the electromagnetic charge (see the assumption v. and the corresponding comments in subsection 10.1.1).
The colour triplet scalars contribute to transition from antileptons into quarks and antiquarks into quarks and back, unless the scalar condensate of the two right handed neutrions, presented in table 10.1, breaks matter-antimatter symmetry [13]. This condensate breaks also the SU(2)n symmetry, leaving massless (besides
gravity) only the colour, weak and the hyper charge vector gauge fields. Also all the scalar fields get masses through the interaction with the condensate.
When at the electroweak break the scalar fields with the scalar indices s G (7,8) originating either in d>Qbs or in those superposition of ds 's"s which conserve the electromagnetic charge (Eq. (10.16)) get nonzero vacuum expectation values, changing also their own masses, they bring masses to all the massless fermions (spinors), breaking their mass protection, and to weak bosons.
Let us recognize, by taking into account Eq. (9.44) and table 9.3, that y0 ys appearing in {^s=7 8 ^Ysp0s in the second line of Eq. (10.2), transform for either s = 7 or s = 8 the right handed u-quark (uR1), weak chargeless, with the hyper charge Y = | from the first line of table 9.3 to the left handed weak charged u-quark (uL1) with the hyper charge 6 from the seventh line of the same table, or that y0 Ys transform the right handed v-lepton (vR), weak chargeless, with the hyper charge Y = 0 from the 25th line of the same table 9.3 to the left handed weak charged v-lepton (vL) with the hyper charge — 1 from the 31st line of the same table.
Now is the time to prove that the scalar fields with the scalar index s G (7,8) from the second line of Eq. (10.2) and with quantum numbers of Eq. (10.16) really carry the weak and the hyper charge as required by the standard model. I introduce in Eq. (10.16) common notation Aa for all these scalar fields, independently of whether they origin in dQbs - in this case they do not carry the additional weak or hyper charge due to ta - or dQbs fields.
aai ^ ( aq ay ay' A1 ANl A2 ANr )
As d ( As , As , As , As , As , As , As ) ,
ta1 d (Q, Y, Y' = — tan2 ^t4 + t23, f\Nl, T2, Nr) . (10.16)
These scalars, the gauge scalar fields of the generators tai and TAl (Eqs. (10.11, 10.12,10.9,10.10)), are expressible in terms of the spin connection fields (Eqs. (10.14, 10.15)).
One expects that the solutions with nonzero momentum lead to higher masses of the fermion fields in d = (3 + 1) [23,24]. We shall correspondingly pay no attention to the momentum ps, s G (4,8), when having in mind the lowest energy solutions, manifesting at low energies.
Scalars, which do not get nonzero vacuum expectation values, keep masses on the condensate scale.
Let me now, by taking into account Eqs. (10.7,10.9), calculate properties of all scalar fields AA1 of Eq. (10.13).
To do this let us first recognize
t1! = 1 [(S58 — S67) g i (S57 + S68)] ,T13 = 2(S56 — S78),
Y = t23 + t4 , Q = Y + t13 ,
and rewrite the scalar fields A^, which determine masses of fermions and weak bosons in Eq. (10.2), appearing in the second line of Eq. (10.2), as follows (the
176 N.S. Mankoc Borstnik momentum ps is left out)
^ lYs (-tA1 AA1 ) =
s = (7,8),A1
78	78
-1fY°{ (+) tA1 (aA1 - iAA1)+(-) (TA1 (aA1 + iAA1) 1}, 78 1
(±)= ^ (Y7 ± iY8 ),	(10.17)
with the summation over Ai performed, since AA represent the scalar fields (AQ,
Aj, Aj', aA4, A1, A2, ANR and ANl).
Application of the operators Y and t13 on the fields (AA1 t i A^11), leads after using Eq. (10.7) for Sab and expressions for t13 and Y (Eq. (10.17)), to
1 2 1 2
Q (aA1 T iAA1) = 0.	(10.18)
(aA1 t iAA1) = ±1 (AA1 t iAA1),
Y (aA1 T iAA1) = T 1 (AA1 T iAA1),
Since Y and t13 give zero, if applied on the upper indices (Q, Y, Y') of (AQ, Aj and Aj ), as one can read from Eq. (10.15), and since Y and t13 commute with the family quantum numbers, one sees that the scalar fields AA (AQ, Aj, Aj',
A4, AQ, A1, A2, ANR, ANL), rewritten as follows, AA1 = (AA1 T iAA1), are eigen states of t13 and Y having the quantum numbers of the standard model Higgs' scalars.
Let us make the notation
AA1 = (AA1 T iAA1),	(10.19)
(±)
and let us calculate what does the operator x1@ (Eq. (10.17)) make if applied on
AA . Taking into account Eqs. (10.7,10.9) one finds that
(±)
x1ffl AA,1 =(AA1 T iAA1) =: AA1 ,
(±) (±)
t1B aa1 = 0.	(10.20)
(±)
The scalar fields A^ are all massive fields with the masses of the condensate scale
(±)
(table 10.1), while the scalar fields A^ change masses at the electroweak break.
(±)
78
Using Eqs. (9.46, 9.44, 9.54) one finds that y0 (-) AA1 transforms the right
handed uR1 quark from the first line of table 9.3 into the leftt handed uR1 quark from the seventh line of the same table, which can be also interpreted in the
standard model way, namely, that A^ "dress" uR1 giving it the weak and the hyper
,c1
charge of the left handed uR1 quark, while y0 changes handedness. Equivalently
78
happens to vR from the 25th line, which the action of y0 (-) AA7i on it transforms
into the vL from the 31th line, which again can be interpreted in the standard model way: With the action of y0 and the "dressing" of AA7i on vR, transforming it into vl.
78
The action of y0 (+) AA7i transform dR1 from the third line of the same table
(+)
into dL1 from the fifth line of this table, or eR from the 27th line into the eL from the 29th line. One can use in this two cases, knowing the properties of the scalar fields
(Eq. (10.18)), again the standard model interpretation, in which the scalar fields AA7i
(+)
take care of the weak and the hyper charges of the right handed members dR1 and eR by "dressing" them with the appropriate weak and the hyper charges, while y0 changes handedness. In the standard model there is the scalar Higgs and the Yukawa couplings, which take care of fermion and also of the weak boson properties.
In the spin-charge-family theory there are several scalar fields, which determine the mass matrices of the two groups of four families.
When the scalar fields (aq8 ,Ay78 , AY78 , A7 , AN, A278 , AN7R8 ) from
(±)	(J8)	(J8)	(J8)	(J8)	(±)	(pm)
Eq. (10.16) get nonzero vacuum expectation values, they determine mass matrices of family members - of quarks and leptons - of the lower (carrying the family quantum numbers (t1, NL)) and the upper (carrying the family quantum numbers
(T2, N
r)) four families, since they carry the weak and the hyper charge (Eqs. (10.9, 10.10)) which breaks the mass protection mechanism of quarks and leptons.
We clearly see that all the scalars AA 8 have the following properties:
(±)
(t13 , Y) aa8 = ± (^ -1) AA8.	(10.21)
(±) 2 2 (±)
The scalars A^i obviously bring the right quantum numbers to the right handed
(-)
partners (uR, vR), and the scalars A^i give the right quantum numbers to (dR,
(+8)
eR).
The scalar fields A^i are in the spin-charge-family theory triplets with respect to
(±)
the family quantum numbers (N R, N L, T2, T1; Eqs. (10.11,10.12)) or singlets as the gauge fields of Q = t13 + Y, Y = t23 + t4 and Y' = - tan2 $2x4 + t23.
One can prove this by applying t23, t13, NR and NL on their eigen states. Let
us do this for AN8L8 and for AQ8, taking into account Eqs. (10.11), and recognizing
(±) (±)
that A"gL± = A,gg1 g i A,g2 (Eq. (10.7)).
(±) (±) (±)
iN L AN g± = g AN g±, n L A n g3 = 0,
(±) (±) (±)
- ,L±	n
A	= {(&-- 78 +	78 ) g i (&-- 78 +	78 )} ,
(7±8)	23(±)	01(±/Q v 31 (±)	02(±)
A^g3 = (&.. 78 +	.. 78 )
(±)	12(±)	03(±)
QAQ8 = 0,AQQ8 = W 78 -	78 + w	78 + w	78 ) , (10.22)
^ (7±8) ' (7±8) 56(±) 1 9 10(±) 11 12(±) 13 14 (±)	'
with Q = S56 + t4 = S56 - 3(S910 + S1112 + S1314), and with t4 defined in Eq. (10.10)).	3
To masses of the lower four families only the scalar fields, which are the gauge fields of N L and T1 contribute. (To masses of the upper four families only
the gauge fields of NR and T2 contribute.) The three scalar fields AQ8, AY78 and
(7±8) (7±8)
A478 "see" the family members quantum numbers and contribute correspondingly to all the families.
The scalar fields, with the weak and the hyper charge in the fundamental representations (Eq. (10.21)) and the family charges in the adjoint representations, transform any family member of the lower four families into the same family member belonging to one of the lower four families (while those with the family charges of the upper four families transform any family member into the same family member belonging to one of the upper four families).
In loop corrections all the scalar and vector gauge fields which couple to fermions contribute.
The mass matrix of any family member, belonging to any of the two groups of the four families, manifests - due to the SU(2) (l,r) x SU(2) (i,n) (either (L, I) or
(R, II)) structure of the scalar fields, which are the gauge fields of the NR,L and
T2,1 - the symmetry presented in Eq. (10.23) 7.
Ma =
-	-a -	a e	d	b	\
	e	—&2 -	ab	d	
	d	b	a2 -	ae	
V	b	d	e	a1 -	a
(10.23)
Let us summarize this section: It is proven that all the scalar fields with the scalar index s G (7,8), which gain nonzero vacuum expectation values and keep the electromagnetic charge conserved, carry the weak and the hyper charge quantum numbers as required by the standard model for the scalar Higgs (Eq. (10.21)):
7 Since the upper four families interact with the condensate of the two right handed
neutrinos, which carry the family quantum numbers of the upper four families, the symmetry of the mass matrix presented in Eq. (10.23) is the symmetry of the upper four families.
pc
(t13 , Y) AA = ± (2, — 2) AA . These are the only scalar fields in this theory with (±) (±) the quantum numbers of Higgs' field.
These scalar fields carry additional quantum numbers: The family quantum numbers. The nonzero vacuum expectation values of the scalars with the space index s =g (7,8) determine on the tree level the mass matrices of the two groups of four families. While the scalars with the family quantum numbers (1,NL) contribute to mass matrices of the lower four families, contribute those with the family quantum numbers (2, N R) to masses of the upper four families and those with the family members quantum numbers (Q, Y, Y') to any of these two groups of four families. In loop corrections in all orders the mass matrices of the two groups of four families follow.
All the other scalar fields: AAi, s G (5,6) and AA , (t, t') G (9,..., 14) have masses of the order of the condensate scale and contribute to matter-antimatter asymmetry.
10.3 Conclusions
The spin-charge-family [1,2,7,6,3-5,8,12,9,10] theory, a kind of the Kaluza-Klein theories [15] with the families introduced by the second kind of gamma operators (Ya in addition to the Dirac Ya), is offering the explanation for the properties of quarks and leptons (right handed neutrinos are in this theory regular members of each family) and antiquarks and antileptons, for the appearance of the gauge vector fields and of the scalar Higgs and Yukawa couplings. All these are in the standard model just assumed.
The theory offers the explanation why are the weak and hyper charges of fermions connected with their handedness (table 9.3) and where do the scalar fields originate (Eqs. (10.14,10.15)).
It also explains why do the scalar fields carry the weak and the hyper charges
as assumed by the standard model (Eq. (10.18)): (t13 , Y) AAi = ± (1, — 1) AAi ,
(±) 2 2 (±) where t13 denotes the third component of the weak charge, Y the hyper charge, Ai denotes (Q, Y, Y') (originating in the first kind Ya of the Clifford algebra objects) and all the family quantum numbers (originating in the second kind of the Clifford algebra objects Ya) While Ya, through Sab, determine all the spin and the charges of families, determine Ya, through Sab, the family quantum numbers.
The spin-charge-family therefore, starting with the simple action (Eq.(10.1)) in d = (13 + 1) for spinors (carrying two kinds of gamma operators) and interacting with the gravity only (with the vielbeins and the two kinds of the spin connection fields), differs essentially from the unifying theories of Pati and Salam [21], Georgi and Glashow [27] and other SO(10) and SU(n) theories [28], although all these unifying theories are answering some of the open questions of the standard model and accordingly have many things in common - among themselves and with the spin-charge-family theory.
The spin-charge-family theory predicts two decoupled groups of four families [7,6,9,10]: The fourth of the lower group of families will be measured at the LHC [11] and the lowest of the upper four families constitute the dark matter [10].
It also predicts that there will be several scalar fields observed sooner or later at the LHC.
Besides the scalar fields with the space index s G (7,8), which by getting non zero vacuum expectation values cause the electroweak break and take care of massless of fermions and the weak bosons, all the other scalar fields get, through the interaction with the scalar condensate of two right handed neutrinos with the family quantum numbers of the upper four families, masses of the condensate scale. There are also only weak, hyper and the colour vector gauge bosons which stay massless up to the condensate scale, since they do not interact with the condensate. The scalar fields with the scalar space index s =g (9,...,14) are colour triplets with respect to the scalar space index and cause, after interacting with the condensate, matter-antimatter asymmetry [13].
All the scalar fields are in the fundamental representations (Eq. ([13])) with respect to the space index. They resemble the supersymmetry particles, although they are not, since they do not meet all the requirements for the bosonic partners of fermions.
Starting with few assumptions, presented in the introduction 10.1 (i.- iv.), I show that the spin-charge-family theory is not only offering the explanation for the so far measured phenomena, with the origin of the dark matter and the scalar fields included, but offers also the predictions for new families (the fourth to the observed three families will be measured at the LHC, the fifth - the lowest of the upper four families - forming baryons [10] explains the appearance of the dark matter) and new scalar fields (there are two triplets and three singlets: AQ, Aj, Aj', A1, ANL, Eqs. (10.14, 10.15,10.16), which determine properties of the four lower families - the Higgs and the Yukawa couplings of the standard model [2,1]). The theory might be able also to answer questions about the (ordinary, mainly made out of the first family) matter/antimatter asymmetry, which is discussed in a separate paper [13]. The quantum numbers of the condensate, responsible for breaking CP symmetry, are presented in this paper (table 10.1). The same condensate makes massive scalar and vector gauge fields which would otherwise be as massless observed at low energies.
Although the spin-charge-family theory starts in d = (13 + 1) dimensional space with the spin connection fields of two kinds (having the origin in Ya and in Ya) and with the vielbeins - all these look like having a very large number of degrees of freedom - it leads under the assumption that there is a condensate of two right handed neutrinos carrying the quantum numbers of the upper four families and that there are scalar fields, which obtain nonzero vacuum expectation values causing the electroweak break, naturally (what means that all unobserved fields of both origins get masses without additional requirements) at the low energy regime to the observed fermion and vector gauge boson fields.
This paper presents, by explaining that in this theory there are the scalar fields, which carry the quantum numbers of the scalar Higgs scalars and correspondingly offering the explanation for the appearance of the scalar Higgs and the Yukawa couplings, a further step towards understanding the properties of quarks and leptons and in particular of those scalar fields (section 10.2), which determine mass matrices of quarks and leptons.
It stays to be solved, why and how does the condensate of the two right
handed neutrinos with the family quantum numbers of the upper four families
appear and why do scalars, with the weak and the hyper charge required by the
standard model, gain non zero vacuum expectation values.
Acknowledgments
Author acknowledges funding of the Slovenian Research Agency. References
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13.	N.S. Mankoc Borstnik, "Matter-antimatter asymmetry in the spin-charge-family theory", to appear in the Proceedings to the 17th Workshop "What Comes Beyond the Standard Models", Bled, July 20 - 28, 2014, [arxiv:1409.7791].
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Bled Workshops in Physics Vol. 15, No. 2
A
Proceedings to the 17th Workshop What Comes Beyond ... (p. 183) Bled, Slovenia, July 20-28, 2014

11 Deriving Veneziano Model in a Novel String Field Theory Solving String Theory by Liberating Right and Left Movers*
H.B. Nielsena'** and M. Ninomiyab'***
a The Niels Bohr Institute,
Blegdamsvej 17 - 21, Copenhagen 0; Denmark ,
b Okayama Institute for Quantum Physics,
Kyoyama 1-9-1 Kita-ku, Okayama-city 700-0015, Japan
Abstract. Bosonic string theory with the possibility for an arbitrary number of strings —i.e. a string field theory—is formulated by a Hilbert space (a Fock space), which is just that for massless noninteracting scalars. We earlier presented this novel type of string field theory, but now we show that it leads to scattering just given by the Veneziano model amplitude. Generalization to strings with fermion modes would presumably be rather easy. It is characteristic for our formulation /model that: 1) We have thrown away some null set of information compared to usual string field theory, 2)Formulated in terms of our "objects" (= the non-interacting scalars) there is no interaction and essentially no time development(Heisenberg picture), 3) so that the S-matrix is in our Hilbert space given as the unit matrix, S = 1, and 4) the Veneziano scattering amplitude appear as the overlap between the initial and the final state described in terms of the "objects". 5) The integration in the Euler beta function making up the Veneziano model appear from the summation over the number of "objects" from one of the incoming strings which goes into a certain one of the two outgoing strings.
Povzetek. Novo bozonsko teorijo strun sta avtorja formulirala na Hilbertovem (Fockovem) prostoru brezmasnih skalarjev, ki ne interagirajo. Teorija dopusča posplositev na poljubno stevilo strun, tedaj na strunsko teorijo polja. Avtorja v tem prispevku pokazeta, da je njuna sipalna amplituda enaka amplitudi v Venezianovem modelu. Kaze, da je posplositev njune bozonske teorije strun na fermionsko enostavna. Bistveno za njuno teorijo strun je: 1) V primerjavi z običajno strunsko teorijo ima njuna teorija manjse stevilo privzetkov. 2) Njuno struno sestavljajo enaki skalarni "objekti", med katerimi ni nobene interakcije in v bistvu tudi nobenega casovnega razvoja (Heisenbergova slika). 3) Sipalna matrika S je v njunem Hilbertovem prostoru kar enotska matrika, S = 1.4) Venezianova sipalna amplituda sledi iz prekrivanja med zacetnim in koncnim stanjem njunih skalarnih "objektov". 5) Integriranje Eulerjeve beta funkcije Venezianovega modela sledi v njunem primeru iz sestevanja po skalarnih "objektih" ene od prihajajocih strun, ki gre v eno od dveh odhajajocih strun.
* 0IQP-14-10
** E-mail: hbech@nbi.dk, hbechnbi@gmail.com
*** E-mail: msninomiya@gmail.com
11.1 Introduction...
We have already earlier put forward [1-3] ideas towards a novel string field theory (meaning a second quantized theory of strings from string theory [4-7,9]), which of course means the theory[10-18] in which you can describe several strings like one describes several particles at a time in in quantum field theory. It may be understood that we as other string field theories in or theory have a Hilbert space or Fock space the vectors of which describe states of the whole universe in the string theory. Our model is similar to the formulation of Thorn [19] and also use a discretization like ourselves, but we discretize after having separated right and left movers as we shall see below. This Hilbert space, which describes the states of the universe, turns in our model/formulation out to be really surprisingly simple in as far as it is simply the second quantized Fock space of a non-interacting massless scalar 25 +1 dimensional particle theory in the bosonic string case!
Let it be immediately be stated that although our formulation/model is supposed just a rewriting of string theory - and thus in its goal there is nothing new fundamentally - it is definitely new because we throw away compared to usual string theory and usual string field theory as Kakku and Kikkawa's and Witten's a null set of information. The information, which we throw away is the one about how the different pieces of strings hang together. That is to say we rather only keep the information about where in target space time you will find a string and where not. Due to this throwing away of information and other technically doubtful treatment of the string theory by us it is a priori no longer guaranteed that our string field theory appearing as just the non-interacting massless scalar theory in 25+1 dimensions is indeed just a rewriting of string theory. Rather one should see our progresses such as the derivation of the string spectrum [3] in reproducing usual properties of string theory from our model/formulation as tests that indeed our model is in spite of the null set of information thrown away indeed the full string theory.
The major achievement in the present work is also such a test, namely testing that our model/formulation leads to the Veneziano model scattering amplitude for scattering of strings formulated in our novel string field theory.
The particles that formally occurs in the construction of our Hilbert space or Fock space of our model or formulation of string theory we call "even objects" and each such "even object" has in our formulation a kind of momentum variable set J^ ( it is proportional to a contribution to the total momentum of the string to which it belongs). Really this J^ has as some technical details got its longitudinal momentum (in target space time of 25 +1 dimensions) component J+ = J0 + J25 fixed by what corresponds to a gauge choice in the string parameterization to be
J+ = ^,	(11.1)
(We shall below that we end up being driven to also allow J+ = — —a- ) and its infinite momentum frame energy proportional component J- = J0 — J25 is written just by the mathematical expression ensuring the light-likeness
(J^)2 = n^ JJ = 0	(11.2)
of the "even object" momentum-like variable. Thus the only genuine degrees of freedom components of this even object variables Jp are the "transverse" components corresponding to the first 24 components, namely those having = i where i = 1,2,3,..., 24, i.e. Ji. In addition the "even objects" have 24 conjugate momenta ni, conjugate to the Ji's, so that
[ni, Jj] = iSij	(11.3)
for ni and Jj belonging to the same even object of course.
Our Hilbert space for states of the universe corresponds now simply to a set of harmonic oscillators, one for every set of Ji-value combinations (of 24 real numbers), and the creation operator for an "even object" with its Ji's being Ji is denoted a* (J1-). Since there is a calculational relation between the set Ji of the transverse components and the full 26-vector Jp given by adding the equations (11.1,11.2), we could equally well use as the symbol in the creation and annihilation operators Jp as the symbol Ji, and so we have by just allowing both notations a* (Ji) = a* (Jp), where it is understood that the Jp is calculated from the only important transverse components Ji. Similarly the destruction operators are a(Jp) = a(Ji) and we shall think of the Hilbert space describing the states of the Universe (in a string theory world) as having basis vectors of the type
a*(Ji(1))a*(Ji(2)) ••• a*(Ji(L))| 0>.	(11.4)
To tell the truth we though better reveal the little technical detail, that this simple situation with only one type of " even objects" that can exist in the states described by (Ji, ni) is only true for the case of a string theory model with open strings, while we for the case of a string theory with only closed strings must have two kind of even objects that can be put into the 24 or 26=25+1 dimensional (Minkowski) space, one right denoted by R and one left denoted by L. So in the only closed string case we could even naturally consider it that the two types of even objects "live" in two different Minkowski spaces - one R and one L-. The figure 11.1 illustrates these two slightly different cases. The connection between the strings present in a given state of the universe and the even objects corresponding to a set of strings is a priori not completely trivial and has to be described. It is not hundred percent true that the strings consist of even objects, but there is so much about it that there is an actually infinite number of even objects corresponding to each string present. This divergent number of many even objects in a string is given as a function of the small parameter a already mentioned in formula (11.1).
11.2 Correspondence from Strings to Objects
The crux of the matter in the formulation of our string field theory model or formulation is to put forward the rule for how a given string state is translated into a state described in terms of a state of what we call "even objects":
In the case of a theory with only closed strings we shall make use of the solution in the conformal gauge for the 26-position fields Xp(ct, t) in terms of right and left movers. Remember that the time-development of a string in string theory is
Strings
0 o
Fig.la	Fig.lb	Fig.lc
For Theory with Only Closed Strings
common
S Cd
Fig.ld	Fig.le
ForTheories with also open strings Chains of Objects on the Lightcones
Fig. 11.1.
described by letting its timetrack - which of course becomes a two-dimensional surface in the 25+1 dimensional "target space" - be parameterized by the two real variables called ct and t. At first one may think of these parameters as parameterizing the timetrack surface in an arbitrary way and therefore one even has to have an action - the Nambu(-Goto) action - chosen so as to be invariant under reparameterization, meaning that one goes over to a new set of coordinates parameterizing the timetrack (ct',t') = (ct'(ct, t),t'(ct, t)). This requirement of reparametrization invariance fixes up to an overall constant the action to be given by the area of the timetrack surface
Single string action = SNambu « area ■
where we have as usual denoted
'det
(X ^)2 X^ -X
X^ X
(X'^)2 )
2 j^T
OT
X>,T) =	.
OCT
(11.5)
(11.6) (11.7)
For an open string the timetrack is like a band extending in the time direction, while for a closed string the track is topologically like a tube/cylinder also extending roughly in time direction.
Now one usually in steps fix the "gauge" meaning the parameterization, i.e. the choice of a new set of coordinates which we again may call (ct, t) (leaving out the prime on (ct', t')). The first step in the gauge choosing is what is called con-formal gauge choice and corresponds to arranging the coordinate equal constant curves to be orthogonal seen from the external/target space of 25+1 dimensions. Often in literature one works with Euclideanized ct and t as if the string timetrack were a two dimensional Euclidean space, but thinking physically on a true string the space felt by a being living attached onto the string would be a 1+1 dimensional space time with one time dimension and one spatial dimension. For the thinking of the present article and our foregoing works on our novel string field theory we shall take this latter - more physical - point of view that the internal space time is indeed a space-time. We think of t as the time coordinate and of ct as the spatial coordinate along the string.
After having chosen the "conformal gauge" the equation of motion derived from the Nambu action at first simplifies and together with the constraints appearing due to the reparameterization symmetry of the original action we can summarize the equations in the conformal gauge:
□Xr(ct, t) = 0(equation of motion)	(11.8)
(Xr(ct,t))2 - (X'r(ct,t))2 = 0 (constraint)	(11.9)
Xr(ct, t) • X'r(ct, t) = 0(constraint also).	(11.10)
Here the D'Alambertian
□ = 9T - 92 = (9t - 3ct)(9t + 9ct),	(11.11)
and the eqations of motion are easily solved by the ansatz
Xr(ct,t) = xR(t- ct) + xR(t + ct),	(11.12)
which is importance for our novel string field theory in as far as it is actually the T-derivatives of the 26-vectorial functions in the solution XR(t - ct) and XR(t + ct), which are going to be identified as we shall see soon by our "objects". Note immediately, that these right and left mover variables XR and XR only depend on one variable each, namely respectively on tr d=f t - ct and tl d=f t + ct, so that the equations of motion with t conceived of as the time have indeed been solved. The ansatz functions
In terms of th very simple form
ansatz functions XR and XR are more like initial conditions for the solution.
In terms of these initial condition variables XR and XR the constraints take the
(XR(tr))2 _ (XR(t — ct))2 _ O(constraint)	(11.13)
(XR(tl))2 _ (XR(t + ct))2 _ O(constraint)	(11.14)
The overview of description of our object rewriting of the string theory is that we let there be an object for every point in (a period for) the coordinates tr and
188 H.B. Nielsen and M. Ninomiya time X0
Running point on time track X(ct, t).
Fig. 11.2.
tl in the case of only closed strings, and that the objects are closely related to the variables and X{\ For continuity of these variables as functions of respectively tr and tl the images of these functions and XJ1 are - except for fluctuations at least - smooth curves, because of the constraints (11.14) these curves must lie on the lightcone(s).
Since we can also consider the variables tr and tl as c-variables for constant t the periodicity w.r.t. c of the position variables etc. - for in fact both open and closed strings- but at least clearly for the closed strings, comes to imply that the just mentioned images for X£ and XJ1 become closed curves on the light cone.
Two points further are illustrated by the figure 11.3: 1) We "discretize", so that we replace the in principle continuum infinity of tr or tl values by a series of discrete points with a "distance between these points" being proportional to a small quantity a later taken to go to zero. 2) We treat the even numbered and the odd numbered "discretized points" differently, as is on the figure illustrated by them being denoted differently by dots and crosses.
11.2.1 Open String Case
The open string case has a tiny technical complication:
At the boundaries of the open string one has boundary conditions which are translated into our model favorite language of X£ and XJ1 implies
Xj^(t) = xr(t) from boundary at c = 0
X^t) = X£(t — 2n)from the c = n end boundary.,	(11.15)
A cyclically ordered chain of "objects" on th lightcone in 25+1
Fig. 11.3.
where we assumed the notation that the length of the string in a-parameter language is n. These two boundary conditions (11.15) imply that we can identify, for the open string, the XR = XL and have that this common (differentiated)"initial condition variable" must be periodic with the period 2n meaning twice the avariable range corresponding to the string length. Thus in the beginning announced we got the fact that while we for only closed string theories have to distinguish XR and XL, this is no longer needed for an open string.
11.2.2 More Precise Correspondence between Strings and "even objects"
More precisely we shall divide up into "discretized" pieces the a range around a closed string or the tour forward and backward along an open string into, let us say, N pieces. What we really want, is to divide up a period for say in its argument tr (and in the closed case the same for XL, while in the open string case just identify the left and right mover variables because of boundary conditions). The precise way of dividing up could be thought of as dividing in equal steps in the variable, say tr, but there is still some coordinate specification/gauge choice left even after the conformal gauge choice. In fact one can still as such a rudimentary freedom of choosing coordinates select any (increasing) function tR (tr) and any (increasing) function t[ (tl) as a new set of coordinates (having in the background of the mind the identifications t' = t — a' and tl = t + a). By discretizing we replace essentially a variable as tr by an integer valued variable - counted modulo N if N corresponds to the period -, so that the tr value corresponding to the integer I is denoted tr (I). Interpolating we can easily make an approximate sense of even tr defined for non-integer values of I. Thus we formally associate any string with a series something we call "objects"- and which is something only defined in our
model -, which are characterized each by a set of degrees of freedom (as if it were particles): JR and the conjugate variables nR or better only n8 where the i only runs over the transverse coordinates i = 1,2,..., 24. The reader may crudely think of these objects as a kind of partons, but really we simply have to define them by their relation to the XR (and for the closed string case also to the separate XR). To every discretization point on the tr axis say, let us say discretization point number I, we associate an "object" for which the dynamical variables JR(I) are given as
JR(I) = Xr(tr(i + 1/2))- xr(tr(i - 1/2)).	(11.16)
Notice that since the difference between the two argument values tr (I - 1/2) and tr (I + 1/2) is small this definition of JR(I) for a discretization point on the tr-axis in reality means that
dTR dI
Jr(I) « XRdTR,	(11.17)
and so indeed as announced our variables JR assigned to the "objects" are "essentially" the TR-derivative XR of the right mover XR part of the solution.
11.2.3 The Even Odd Detail
Now there is an important technical detail in the setup of our model:
We have the problem that if one shall make creation and annihilation operators for some "objects" in a way analogous to how one in usual quantum field theory have creation and annihilation operators for particles, one shall describe these creation and annihilation operators by having an argument for a set of variable describing the "object", a set of variables which commute with each other. It is indeed well-known that one must in quantum field theory either take the creation and annihilation operators to be functions of the spatial momenta of the particles created/annihilated or one can use instead position variables, and that corresponds to working with the second quantized fields ^ (x). But the usual simple mutual commutation rules for creation and annihilation operators could not be obtained if one would attempt to construct them to correspond to a combination of dynamical variables for the particles that did not commute with each other. What could also a creation operator depending on mutually complementary variables for a single particle correspond to creating ? It could not create a particle with the specified quantum numbers in such a case because that would be against the Heisenberg uncertainty relation. In the corresponding way we must choose whatever variables we let our "objects", to be associated with creation, and annihilation operators depend on be arranged so as to commute. But then we have problem, because does our XR's which are proportional to the object-variables JR commuting? No, they do not commute in as far as the theory of a single starting from the Nambu Lagrancian e.g. in the conformal gauge leads to
[Xr(tr),Xr(tr)] <x 9rv6'(tr - tr).	(11.18)
Thinking discretizing, such a derivative of a delta-function commutator means that in the discretized chain the XR or equivalently JR's which are next neighbors do NOT commute.
So we had to invent a trick to avoid to have to make creation and annihilation operators for "objects" sitting in the chains of "objects" along the variable tr as neighbors in the discretization.
The trick which we have chosen consists in only using in the creation and annihilation operators every second of the by discretization by (11.16) defined "object"-variables JR. That is to say we choose to only construct creation and annihilation operators for those "objects", which in the discretized series of objects for a given string have got an even number I. One could say that we in our model construct our Hilbert space only in terms of such "even objects", and one could almost say only consider these "even objects" as "really existing" in our basic Hilbert space description.
But then we are coming to the problem that we need to a full description of the string states also the "odd objects": what to do about them? We say that when you have a series of the "even objects" on a string, we make the rule to construct in between any two next to neighboring "even objects" (i.e. two "even objects " deviating in number by just 2) an "odd object" from the conjugate momenta n1 say of the neighboring "even objects". (There is another technical detail connected with the + and - components in the infinite momentum frame we have chosen to work with, so we shall avoid discussing conjugate momenta to other than the transverse components J1R - the first 24 components -. Therefore we only consider these first 24 components of conjugate momenta to the JR's). In fact we have to take the following rule for constructing the "odd object" JR components for the "odd object" number I (where I then is an odd integer (modulo the even number
N)),
JR =— na '(nR(I + 1) — nR (I — 1)),	(11.19)
in order to obtain the commutation rule corresponding to the derivative of delta function commutation rule (11.18) discretized.
The reader should check and understand that with this construction of the "odd objects" any quantum state of the string expressed as a state of the variables XR ( and for the closed string also XR) can be expressed as a corresponding quantum state of a set of N/2 ( N must be even) "even objects", because the even object commutation rules
[JR(i),nR(K)] = is1k6iK	(11.20)
corresponds just to the commutation rules for the XR ( and XL). There is though one little technical detail to be studied in later works: The absolute position of the string were differentiated away from our discussion by dotting the XR and XLand correspondingly the formula for the "odd objects" does not make use of the sum over all the "even object" conjugate variables nR around the closed chain. So there is suggestively the possibility of identifying the average position of the string proportional to this sum over all the conjugate to even object variables.
11.2.4 Several Strings
So far we should have now given the prescription for constructing a cyclically ordered chain of "even objects" corresponding to a given quantum state of a single
open string. (If one wants a closed string one shall construct two cyclically ordered chains of "even objects" one for right movers consisting of JR "even objects" and consisting JL left mover "even objects"). Since the commutators were arranged to isomorphic to the discretized XR ( and XL) it should be possible to construct such an "even object" state. It is then of course also trivial and completely analogous to usual quantum field theory construction of a state with N/2 particles to construct a Hilbert space ( Fock space) state for N/2 of our "even objects". Corresponding to a single open string we thus simply have Hilbert space state with the large (divergent in the limit of a —» 0) infinitely many (= N/2) "even objects" sitting approximately in a cyclic chain on the light cone in a 25 +1 dimensional (Minkowskian) JR-space.
But now it is the main point of a model being a string field theory (SFT), that such a model can describe several strings in one Hilbert space state. Once we have made our formulation of one string in our "even object" formulation it is, however, rather trivial to construct states with an arbitrary number of strings. One can just act on the "zero even object" with all the product of creation operators corresponding to the various strings - we want to have in the state to be described -each creating the "even objects" associated with the string in question. So to speak if string number 1 is described by the product
C1|0 > = J¥1(JR(0),JR(2),...,JR(N — 2)) • ni=0,2,4,...,N-2 a*(JR(I))
•ni = 0,2,...,N-2ni=1,2,...,24dJR(I)|0 >,	(11.21)
where ¥1 (JR (0), JR (2),..., JR (N — 2)) is the wave function for the state of the single string 1 described in terms of "even objects", and the string number 2 by an analogous expression, then a state with both string 1 and string 2 (say they are open) is given as
C1 C2I0 > = J"¥1 (JR(0), JR(2),..., JR(N — 2))
ni = 0,2,4,...,N-2at(JR(I))ni=0,2,...,N-2ni=1,2,...,24dJR(I)
¥2(JR(0),J1r(2),...,J1r(N — 2))ni=0,2,4,...,N-2a* (JR (I))
ni = 0,2,...,N-2ni=1,2,...,24dJR(I)|0 > .	(11.22)
Luckily the commutation of the creation operators for "even objects" makes it unnecessary to specify any order in which the creation operator products corresponding to different strings have to be written.
We thus have a scheme for constructing Hilbert space states - in the Hilbert space which is really that of massless scalar "even objects" conceived of as particles in an ordinary Fock space - corresponding to any number of strings wanted. In this sense we have a string field theory.
11.2.5 Final Bit of Gauge Choice
As already mentioned the choice of parameterization (often called gauge choice) were not finished by the conformal gauge, since we could still transform the vari-
ables tr and tl by replacing them by some increasing function of themselves. In the "infinite momentum frame gauge" the choice is to fix this freedom by requiring the density of P+ = P5 + P25 (longitudinal momentum we can say) momentum is constantly measured in say tr = t — ct or say in ct. When we discretize as described above and take it that the tr distance between neighboring "objects" along the TR-axis should be the same all along, then this gauge choice comes to mean that each object gets the same P+ momentum. We can therefore describe this gauge choice - which is essentially the usual one in infinite momentum frame, just discretized our way - by saying that we impose some fixed small value for the + component of our "object"(-variables)
aa'
J+ = —— in our first attempt (later problem comes).	(11.23)
With this gauge choice we have made the number of objects N and thus of "even objects" N/2 proportional to the P+ = P5 + P25 component of the 26-momentum of the string in question. So e.g. the conservation of this component of momentum corresponds to the conservation of the number of say "even objects". After this choice of gauge extremely little is left to be chosen for the reparameterization: you can still for the closed string shift the starting point called ct = 0, but that is all. Corresponding to this extremely little reparameterization left unfixed you can still cyclically shift along the topological circles on which the objects of a string sits, and that turn out due to the possibility for adding a constant to t also to be true for the open string. The objects corresponding to a cycle for a string are cyclically order but the starting point is still an unchosen gauge ambiguity. To an open string we have one such loop or cycle, and to a closed one we have two.
11.3 Comparing Our String Field Theory to Other Ones
It should be stressed that our "novel" string field theory really is novel/new, since it deviates from earlier ones like Kakku and Kikkawa or Wittens string field theories in important ways even if some calculations should soon turn out similar:
• 1. The information kept in our formalism is not the full one kept by the theories
by Kaku Kikkawa or by Witten, but deviates by having relative to these other
string field theories thrown out - actually only a null set of - information. It is the information about how the strings hang together, that is thrown out. We could say that we - Ninomiya and Nielsen - only in our rewritten string states keep track of where in the space time you may see a piece of string, but not of how one piece hangs together with another piece. If a couple of strings cross each other there is a point in target space wherein four pieces of string meet, two belonging to each of the crossing strings. In usual string field theories, such as Kaku and Kikkawa [10] and Witten's[11], it is part of the information kept in the Hilbert space vector describing the state of the universe which of these 4 pieces are connected to which. In our formulation, however, this information has been dropped.
•	2. A further consequence of this drop of information is that if two strings scattering by just exchanging tails - as one must think scattering should typically happen classically - then really nothing have to happen at all in our formalism. Indeed it is a second characteristic property of our string field theory model, that in the scattering counted in terms of our "even objects" (which are the ones truly represented in the Hilbert space; the odd ones are just mathematical constructions from the conjugate variables for the "even" ones) nothing happens! The scattering process is not represented in the Hilbert space formalism of ours.
•	3. A consequence of item2. is that the S-matrix gets calculated formally as an overlap of the initial with the final state.
•	4. And this fact is also connected with that the Hilbert space or Fock space of our formulation is the extremely simple free massless scalar Fock space. Actually though there is gauge fixing, that makes the states of the "even objects" even have their J+ components fixed by (11.1). This is contrary to the other string field theories which have much more complicated structures.
•	5. But perhaps the most important distinction for the other string field theories is that we use a description in terms of something quite different from the strings themselves, namely our 'even objects", while the other string field theories have quite clearly all through their formalism the strings one started from. In ours the string has been hit to the extent that we at the end must ask: What happened to the string? The answer is roughly that there is no string sign left in the Hilbert space structure of being only that of free massless scalars. Rather the string in our formalism only finds way into the calculations via the initial and final states put in! That is to say that in our formalism it looks that the whole story of the strings only will appear because there is an extra "stringy" assumption put in about the initial state - and presumably it is necessary even to put it in for the final state - so that the whole string story is not part of the structure of the theory nor of the equation of motion, but rather on an equal level with the cosmological start of the Universe, or the initial conditions of low entropy allowing there to be a second law of thermodynamics. If it should turn out that indeed even extra assumptions about the final states are needed to make our formalism function as a string theory, then one could say that in our formalism an influence from future is required.
With all these deviations from the usual string field theories, one may worry whether our rewriting truly is a rewriting and thus can count as a true string field theory, because does it indeed describe the conventional string theory, or could it be that we had thrown away too much (even though only a null set)?
Because of this possibility that our model does not truly represent string theory at the end it becomes important - also for the purpose of testing if our model is string theory - to check the various wellknown features of string theory. We have not long ago published an article [3] in which we showed that the mass spectrum of the strings in our string field theory became the usual one. This is one such little check that our model/string field theory is on the right track. In the succession of this article we shall concentrate on sketching the calculation of the
scattering amplitude for two ground state strings (tachyons) scattering elastically into two also tachyonic ground state open strings.
Actually it turned out that we were not quite right in the first run, because we only get one term out of three terms that should be present in the correct Veneziano model. This little shock we sought to repair by modifying our gauge fixing condition and allowing "even objects" also with negative J+. As we shall see later we think it reflects a more general problem with infinite momentum frame.
11.4 Yet More Technical Details 11.4.1 The + and — components of JR
Especially if one wants to get an idea about our work [3] checking the spectrum of our strings it is necessary to keep in mind that it is only the components JR for i = 1,2,..., 24, which are simply independent dynamical variables for the "even object". The remaining two components are not independent. Rather:
•	+: The J+ components of actually both even and odd objects are fixed to ± as a remaining gauge choice after the conformal gauge has been used to gauge fix to the largest extend. This would have been the infinite momentum frame choice basically, once we assumed that the distances in say c-variable per object were (put) equal for all the objects. It really means that number of "objects" represent the P+-momentum of the string associated with those objects.
•	—: Next the components J- are fixed from the requirement gotten from the constraints in string theory, namely that (JR)2 = 0. This condition fixes the —component (essentially energy) in terms of the 24 transverse components JR and the gauge fixed J+. Remembering that the "odd objects" are constructed from the even ones by means of (11.19) we can write the —components as :
For even objects: (11.24) J-(even I) = ^i=1'2.....24.(JR)2	(11.25)
For odd I object(constructed): (11.26) J-(0ddI) = n2a' ^-2.....24(nR(I + 1)— nR(I — 1))2 (11.27)
It may be interesting to have in mind that from the point of view of our Hilbert space description with a Fock space only having "even objects", and even those only with their transverse - the 24 components - the odd objects as well as both the + and the - components are just "mathematical constructions" simply put up as mathematicians definitions. In this manner the two of the 26 dimensions are pure "construction"! as well as half the number of objects.
It were basically by means of these "constructions" for a cyclical chain of first even, then filled out by odd ones in between, that we in our previous ar-ticle[3] checked the spectrum of masses. We ran, however, into a slight species doubler problem: Because of our discretization of the tr-variable we were seeking
a spectrum of latticized theory (in one spatial dimension, the tr), and thus we got according to our theorem that one gets species doublers when seeking to make only right mover in fact a species doubler[23]. In order to get rid of that we propose to impose a continuity rule as a postulate.
11.4.2 The Non-Parity Invariant Continuity Rule
The continuity rule which we saw earlier we had to impose to avoid a doubling of the usual string spectrum in our model is actually just the continuity rule, which you would any expect. Crudely it just is that you require the variation of the object JR or JR to vary slowly from object to the next object. So physically it is extremely reasonable to assume this continuity rule. But we assume it - and have to assume it so - for both even and odd "objects", and then because of the antisymmetry of the definition of the odd JR in terms of the conjugate of the even ones, we obtain a condition that is not symmetric under the shift of sign of the object enumeration number I. Intuitively you expect that if a chain of numbers JR say, enumerated by I vary smoothly with I counted in positive direction, then it should also vary smoothly, if we count in the opposite direction. Because of our 'strange" definition of the odd object JR-values, however, the continuity concept we are driven towards does not have this intuitive property of being inversion invariant. Let us in fact write our smooth variation or continuity requirement for three successive "objects" in the chain - with an odd one in the middle say -
JR(i +1) « -na'(nR(i +1)- nR(i -1)) « jr(i -1) (11.28)
Imposing this non-reflection invariant continuity rule not only is a way to at least assume away the species doubler from the lattice, but it also gives an orientation to the tr-variable. For instance when we below shall match wave functions for strings in initial and final states to calculate the overlap, this oriented continuity condition can let us ignore possible overlaps, if the two, to be matched, chains of "objects" are not oriented - in terms of the continuity condition - in the matching way. This rule reduces significantly the possibilities for forming overlap contributions. From a symmetry point of view it may be quite natural that working with only right mover say there should be some asymmetry under reflection.
Thinking, however, on our model as the fundamental theory representing a seeming world with a string theory, it means that this rather strange "continuity principle" not being reflection invariant has somehow to be imposed by the laws of nature. But now as already stated the Hilbert space structure and the dynamics in terms of "even objects" are just the free massless scalar theory, and there is no place for such a reflection non-invariant continuity condition, except in initial and "final state conditions" So in terms of our "even objects" we must have a truly rather funny initial state assumption: The "even objects" sit in chains that are continuous or smooth in our special sense in one direction, but therefore cannot be it in the opposite direction!
Of course in some way this continuity is a description of the continuity of the strings, their hanging together.
11.5 Sketch of Calculation
As one - and perhaps the most important - tests of whether our string field theory in fact leads to the Veneziano model scattering amplitude (at least up to some overall factor, which we shall leave for later works, and modulo a rather short treatment only of the rather important appearance of the Weyl anomaly in 2 dimensions, which happens to be where the dimension of 26 is needed in our calculation). We shall also reduce the troubles of calculation by choosing a very special Lorentz frame, something that would not in principle have mattered provided the theory of ours had been known to be Lorentz invariant. However, since we use infinite momentum frame - which is not manifestly Lorentz invariant - it is in principle dangerous to choose a special frame.
11.5.1 The Veneziano Model to be Derived
Let us shortly - and especially with also a purpose of the extra factor in the integrand, for which we shall need the anomaly for the Weyl symmetry - recall what Veneziano model amplitude we shall derive, if we shall claim that it is a success for supporting that our model/our string field theory is indeed describing string theory of the bosonic 25+1 dimensional type, the most simple string theory having though as a little problem, a tachyon. Since it is the simplest and historically the first to have a Veneziano amplitude for four external particles [20], firstly later we generalized to larger number of external particles [4], we shall start by deriving the Veneziano model for four external particles, although not in the phenomenologically supports case of Veneziano, 3n + Rather we consider here just four external tachyons each having mass square
^ 1
m2 =----(11.29)
a'
where a' is slope of the - before inclusion of loops - assumed "linear Regge trajectories", the leading one of which has the expression
a(t) = a(0) + a' t,	(11.30)
where
a(0)=-a'm2 = 1.	(11.31)
The four point Veneziano model is basically given by the Euler Beta function, which can be defined by the integral
B(x,y) =
ri
zx-1(1 - z)y-1dz	(11.32)
0
being used say for	(11.33)
B(-a(t), —a(s)) =
z-a(t)-1 (1 - z)-a(s)-1 dz.	(11.34)
In writing such four point amplitudes one uses normally the Mandelstam variables
S = (pi + P2)2 = (P3 + P4)2	(11.35)
t =(pi - p4)2 = (P2 - P3)2	(11.36)
U =(pi - p3)2 = (P2 - P4)2	(11.37)
obeying the relation	(11.38)
s + t + u = m2 + m2 + m3 + m4 = 4m2 = -4/a'.	(11.39)
Here the four- or rather 26-momenta pt = pR (i = 1,2,3,4) are the external momenta for the tachyonic string states we consider as scattering states in the simplest case considered here, all four counted the physical way, i.e. with positive energies p0 for the process
1 + 2- >3 + 4,	(11.40)
being considered. Since we consider the case of pure strings without any Chan-Paton factor giving quarks at the ends, the full scattering amplitude becomes a sum over three terms of the betafunction form. In front there is a factor g2 involving the coupling constant g for the string or Veneziano theory being its square g2. We shall, however, postpone the presumably a bit complicated but very interesting question of the overall normalization in our theory to a later article. Thus the full amplitude expected is
A(s,t,u) = g2 {B(-a(s), -a(t))	(11.41)
+B(-a(s),-a(u)) + B(-a(t), -a(u))}	(11.42)
r i
= g2( z-2a 'PlP2-4(1 - z)-a 'pi(-p4)-4dz + (11.43) Jo
i
z-2a 'PlP2-4(1 - z)-a 'pi'(-p3)-4dz +	(11.44)
0
1
2a' p i • ( p 4) 4 (1 - z)-a' pi-(-p3)-4f
o
z-2api(-p4)-4(1 - z)-api(-p3)-4dz). (11.45)
Since we have chosen to set up our model in what deserves to be called infinite momentum frame and to use the gauge that each object carries the same p+ or rather having the fixed value J+ = aa'/2 according to (11.1), our formalism is a
priori highly non-Lorentz invariant, and it almost requires a miracle for it to turn out at the end Lorentz invariant. It is therefore non-trivial and a priori dangerous only as we have chosen in the beginning to compared our model to the Veneziano model in the special case that the four external particles have the same p+ components,
p+ = p+ = p+ = p+	(11.46)
and consequently	(11.47)
N1 = N2 = N3 = N4,	(11.48)
where the (even) integers Nt (i=1,2,3,4) denote the numbers of "objects" attached to the four external particles. This choice of a special coordinate frame leads to a
11 Deriving Veneziano Model in a Novel String Field Theory...	199 simplification of the term without poles in the s-channel:
g2B(—a(t), —a(u)) =	(11.49)
g2 B(—1 — a '(pi — p4)2, —1 — a '(pi — pa)2) =	(11.50)
g2B( —1 + a '(pT! — PT4)2, —1 + a '(pT! — PT3)2) =	(11.51)
g2
-2+a'(pTi-pT4)2 (1 — z)-2+a'(PT1-PT3)2 dz .	(11.52)
Here we have denoted the "transverse" parts - meaning the first 24 components by
pT = {pT}i=!,2,...,24.	(11.53)
The simplification comes about because the +- term in the contraction with the metric in, say, (pi — p4)2 drops out because of our very special frame choice so that (pi — p4) + = 0,and so the (pi — p4)- does not matter, and
(pi — p4)2 = (pT! — p T4)2.	(11.54)
!
z
5
11.5.2 Amplitude in Our Model, Principle of No Interaction!
Whereas in string theory there seems to be an interaction between the strings, it is rather surprising - and a hallmark for our theory - that in the formulation of ours in terms of the object states the S-matrix elements, that shall give the Veneziano amplitude as we shall show, is simply equal to the overlap! That is to say it is calculated as if the genuine S-matrix is just the unit operator. More precisely the S-matrix < 1 + 2|S|3 + 4 >, that shall describe the scattering of say, two incoming open strings 1+2 into two outgoing 3+4 is obtained by writing the states in our formalism - in terms of even "objects" - corresponding or representing the two string state 1+2, say |1 + 2 >eo and also to the two string state 3+4 corresponding state in even object space, say |3 + 4 >eo, and then simply one takes the overlap of these incoming and outgoing states:
< 1 + 2|S|3 + 4 > = <1 + 2|eo|3 + 4 >eo .	(11.55)
Here the subindex eo stands for "even objects" and means the state described in our even object notation. This means that in terms of our string field theory = "even object formulation" a scattering goes on without anything happening (whatever might happen in reality must have been thrown out in the construction of our string field theory model). Symbolically this formula for the S-matrix is shown on the figure 11.4:
11.5.3 Procedure
The main tasks in order to evaluate the scattering amplitude are
• A. First we must evaluate in some useful notation the wave functions for the incoming and outgoing strings - we shall in this article only consider scattering of two open strings coming in and two open strings coming-out, all in the tachyonic ground states.
lO
20 4*\J/
Calculate as if S-matrix were l
Fig. 11.4.
•	B. We must figure out in how many different ways the "even objects" associated to the strings 1 and 2 in the initial state of the scattering can be identified with "even objects" associated with the final state strings.
•	C. For each way of identification of every "even object" in the initial state(i.e. associated with one of the incoming strings/particles) with an "even object" in the final state (i.e. associated with one of these outgoing particles), we have in principle two wave functions for the "even objects" and shall compute the overlap of these two wave functions.
•	D. Then we have to find the total overlap by summing over all the different ways of identifications, considered under B.
•	E. This summation under D. will turn out to be approximated by an integral and we shall indeed see, that it becomes essentially the integration in the Euler Beta function definition thus providing the Veneziano model.
In performing this procedure we make some important approximations and simplifications:
•	a. We shall assume that due to the continuity of the object series it is by far more profitable for obtaining a big overlap contribution to keep as many of the pairs of neighboring "objects" in the initial state, say strings 1 and 2, remain neighbors again in the final state. This means that we assume that the as contributions to the overlap dominating identification - in the sense of B. - are those in which the largest unbroken series of "even objects" go from one initial state string to one of the outgoing strings. This means the most connected or simplest pattern of identification.
In fact the not yet quite confirmed though speculation is, that the successively more and more broken up pattern of identification of initial and final "even objects" will turn out to correspond to higher and higher (unitarity correction) loops in dual models(=Veneziano models). Thus we expect, that considering only the least broken transfer of the "even objects" from the initial to the final strings shall give us the lowest order Veneziano model (the original Veneziano model without unitarity corrections).
•	b. We shall of course use, that we take the limit a —» 0 and correspondingly, that the numbers N1,N2,N3,N4 of "objects" associated with the various strings go to infinity. Thus we can integrate over the number of objects in a chain going some definite way, say from string 1 to string 4.
•	c. To simplify our calculations we choose the very special case of the four strings - the two incoming 1 and 2 and the two outgoing 3 and 4 - all are associated with same number of "even objects" N/2 (and then of course N 'objects" altogether). This assumption is with our letting the number of "objects" be proportional to the P+-component of the 26-momentum of the string in question, i.e. just the choice of Lorentz frame, so as to have all the four external strings/particles have the same P+. So it looks like just being a coordinate choice, but there is the little problem strictly speaking: that our use of infinite momentum frame makes our theory not guaranteed to be Lorentz invariant. Anyway we do it only this non-invariant way in the present article and leave it for later, either to prove Lorentz invariance of our model, or to do it in a more general frame.
•	d. As a further strengthening of point b. above about the chains coming in as long pieces as possible being dominant we remember, that our continuity condition (11.28) was not reflection invariant. It would therefore be extremely little overlap, if we should attempt to identify the "even objects" of a series in the initial state with a series in the final state in the opposite order. That is to say we require, that the orientation in the pieces of series going over as hanging together from initial to final state are kept. Otherwise the contribution is assumed negligible.
Together b. and this item d. means that the dominant contributions come when possibly the longest connected pieces go over from one initial to one final without changing orientation of the piece.
We shall in the following seek a way to progress, that relatively quickly leads to string-theory-like expressions and thinking. But the reader shall have in mind that even, if we shall approach string-theory-like expressions, we have at the outset had a formulation - namely our string field theory - in which at first the stringyness is far from obvious. Rather it seemed that the stringy structure only comes in with the initial and final states, while the structure of our free massless scalar Hilbert space or Fock space is too trivial to contain any sign of being a string theory. It is therefore still interesting to calculate the results of our theory, even if it quickly should go into to run along lines extremely similar to usual string theory.
11.5.4 Construction of Wave Functions for Cyclically Ordered Chains Corresponding to Strings
The wave functions for open strings were in fact investigated in our previous article [3] in as far as the quantum system of N objects forming a cyclically ordered chain corresponding to an open string were resolved into harmonic oscillators and thus a Gaussian wave function were obtained in a high (of order N) dimensional space. The trick we shall use here is to write the wave function of this character by means of a functional integral so reminiscent of the Feynman-Dirac-Wentzel functional
integral for a string propagation already put into the conformai gauge, that we can say that we already managed to "sneak in" the string by this technology.
In fact one considers in single string description functional integrals of the type:
exp(-
(9^(CT1,CT2))2dCT1dCT2)D^,	(11.56)
A
with some boundary conditions along the edge of the region A say in (c1, c2) space, over which the integral in the exponent is performed. We shall for our purpose of making an expression for the wave function in terms of our "even objects" for a string state consider that the region A is taken to be a unit disk and at the edge we imagine putting a series of "even objects" each being assigned a small interval along the circular boundary. Then we identify for example the object JR with the difference of the values of a taken at the two end points of the little interval on the circle surrounding A assigned to the object in question. That is to say for say object number I (here I is even) having as its interval, say, the little region between the points on the circle marked by the angles
0beg = 2n * ^	(11.57)
Send = 2n * ^	(11.58)
we identify (e.g.) the difference
JR (I) id=nt- ^(exp(i6end)) — ^i(exp(i6beg)),	(11.59)
where we have of course taken a new for each of the 24 i-marked components of JR and where we have identified the (c1, c2)- space with the complex plane by considering a function of c1 + ic2.
The idea, which we seek to use here is that - possibly by some minor modifications, which we must state - we should imagine, that we want to construct a prescription for obtaining a wave function of the type ¥(JR (0), JR (2),..., JR (N — 2)) as used in the expression (11.21), describing say the ground state of a string in our formalism by imposing a boundary condition - depending on a set of values for all the "even objects" in a chain - on the functional integral(s). With these boundary conditions imposed at the end the functional integral become the wave function value ¥(JR(0), JR(2),..., JR(N — 2)) for the in the boundary condition used JR(i) values.
Let us before fixing the details immediately reveal that we shall have an extra boundary condition in the center of the disk A at which point we shall cut off an infinitesimally little disk and use the thereby opened boundary conditions to "let in the (transverse components of) the 26-momentum of the string in question". This "letting in" means in principle that we put on the inside of the little circle a series of JR arranged to correspond to string with the right 26-momentum, but due to the smallness of the little circle the details except for this total momentum does not matter. In the figure we illustrate this situation on which we think: The
line ending at the center and ascribed a P symbolize the just mentioned "in-let" in this center. The small tags on the edge of the disk symbolize the attachments of the "even objects", the values of which are used to fix the boundary conditions for the functional integral. Crudely the idea behind this procedure could be considered
We imagine the "objects" sitting along the edge of a disk, over which is defined a y (a1, a2) to be
functionally integrated over:
*>■
The J's of the "objects" are related to the derivatives d^y at the edge.
Fig. 11.5.
to be that we let a (here open) string propagate along during an imaginary time (say an imaginary t), whereby only the lowest mass state survives. The heavier eigenstates of mass decay in amplitude faster that the lightest state by such an imaginary being spent. Thus one gets after infinite imaginary "time" the ground state selected out. Thus investigating the wave function reached after such an infinite imaginary "time" it should turn out being the ground state wave function, and so we should be able to use it as the ¥ we want, if we want the wave function for a ground state string (the tachyon). Then the idea is of course to write the infinite imaginary "time" development by means of Wentzel-Dirac-Feynman path way integration.
Thus we get into our way of presenting the wavefunction
¥(JR (0),JR (2),..., JR (N — 2))
a functional integral with at first having a region, like A, being an infinite half cylinder. The axis along the half infinite cylinder is the imaginary part of the infinite imaginary 'time", while the coordinate around the cylinder is rather the parameter, tr, enumerating the objects in the cyclically ordered chain of objects associated with an open string.
Then the type of functional integral here considered is "essentially" (meaning except for an anomaly becoming very important at the end) invariant under conformal transformations of the region A. Thus ignoring - or seeing that they are not there in the case considered - anomalies we can transform the infinite half cylinder into the unit disk with the little hole in the middle, through which we "let in" the momentum of the string.
Note how the string here comes in (only): We got to a functional integral strongly related to what one usually work with in string theory, just with the purpose of constructing a wave function ¥(JR (0),JR (2),..., JR (N — 2)) describing the string state in terms of our objects. But there is nothing "stringy" in our Hilbert space structure of our object-theory. The string only comes in via this wave function.
But of course it still means, that after we have got this wave function in, we get our calculations being so similar to usual string theory, that we can almost stop our article there, and the string theorist may have exercised the rest so often, that we do not need to repeat. But logically we have to repeat because we are logically doing something else:
There is in our formulation in terms of the 'objects" a priori no strings. We are on the way to see, that after all the strings must be there, because otherwise it would be strange, that we just get the Veneziano amplitude for scattering.
11.5.5 Adjustment of the Details of the Functional Integral
A few details about the functional integral may be good or even rather important to have in mind:
•	I. As long we - as here - just seek to write the exponential for the wave function (which as we know for harmonic oscillators have the Gaussian form - of an exponential of a quadratic expression in the JR(I)'s (even I) -) we could use the old proposal by David Fairlie and one of us (HBN) of evaluating the exponential as the heat production in a resistance constructed as the surface region A as a conducting sheet with specific resistance n(2?)a'. Then one shall identify the boundary conditions by letting the current running out at the interval assigned to a certain "even object" be equal to the JR (I) for that "even object".
•	II. There a is little problem, which we have to solve one way or the other with getting the "continuity condition" (11.28) discussed in 11.4.2. Having fixed only the boundary condition to the "even objects" through their JR (I) but not involving the conjugate variables nR(I) there is of course no way in which the strange non-reflection symmetric continuity condition of our could be imposed. Concerning the classical approximation one may actually find out that one easily can find the classical solution over the complex plane introduced above after the formula (11.59) which reflects the continuity condition as well as you can require for a classical solution by extracting only the analytical part of the saddle point for + ia2).
Indeed one might - and we probably ought to do it - construct a model, in which we use both even and odd JR's on the boundary, in the sense that we assign only half as long intervals on the border for each object - meaning that we replace (11.58) by
(11.60) (11.61)
6beg = 2n * Send = 2n *
I - 1/2 N
I + 1/2 N
and use it for both even I and odd I.
But now what are we to impose for the odd object intervals on the disk border? We want to obtain a wave function V(JR(0), JR(2),..., JR(N — 2)) expressed only as a function of the "even object" JR's, while no nR are accessible among the variables, on which the wave function depends.
However, in functional integrals one can easily extract what corresponds to the conjugate variable; they are so to speak related to the time derivatives, by relations of the type that the conjugate to a variable q in a general Lagrangian theory is given by
On the other hand the continuity condition tells us that approximately the odd JR's can be replaced by their even neighbors. Thus the proposal is being pointed out that we identify the appropriate time derivatives with the values of the neighboring even JR (I)'s. Putting up this proposal is rather easily seen to correspond to, that the boundary condition relating near the boundary to the even object JR's, which we are allowed to use, get decoupled from say the anti-analytic component in So with such a boundary inspired by the non-reflection invariant continuity condition would lead to an arbitrary solution for say the anti-analytical part, while the analytical part would get coupled. We should like to develop this approach in further paper(s), but it may not really be needed.
Instead of seeking to put our continuity condition (11.28) into the functional integral formalism, we here shall use it is as a rule for which pieces of cyclically ordered chains can be identified, and then we shall get only oriented two dimensional surfaces - looking formally like string-surfaces for closed oriented strings although what we are talking about are open strings (but remember that we get the diagrams for open look like the ones say Mandelstam have for closed) -.
• III. Although it is in fact functional integrals like (11.56), that we basically need, it is so that such a functional integral has divergences. These divergences must in principle be cut off. But now it turns out that the cut off necessarily comes to depend on a metric. Therefore we should rather write our functional integral (11.56) as if depending on a metric tensor gap(a1, a2) in the 2-dimensional space time, although it formally would look that there is actually no such dependence on the metric, at least as long as we just scale it up or down by Weyl transformations. This seemingly metric dependent functional integral looks like
where then boundary conditions and region of the (c1, c2)-parameterization must be further specified. The cut off procedure should also be specified; it could for instance be a lattice curt off, a lattice in the ( c1, c2) variables, say. Then the importance of the metric is that you need the metric to describe the lattice spacing. Note though also that formally a scaling of the metric/Weyl transformation
(11.62)
exp( gaP9a^(c1,c2)ygdc1dc2)D^,
(11.63)
gap -> exp2wga|3,
(11.64)
even when the scaling function exp depends on the (c1, c2) does seemingly not change anything, because the determinant g of the two by two matrix g ap just scales with exp so that the square root just compensates for the scaling of the upper index ga|3 metric. The Weyl transformation symmetry is only broken by the cut off (the lattice) depending on gap. It is via this cut off the anomaly can come in.
11.5.6 Overlap Contributions
The crucial step in calculating the Veneziano model amplitude in our model/string field theory is to see what are the possibilities for identifying all the even objects associated with the initial strings/particles 1 and 2 to the ones associated with the final state strings/particles 3 and 4 in a way that to the largest extend keep neighboring (or better next to neighboring, since we only consider the "even objects") "even objects" going into neighboring ones in the same order(same succession).
To simplify the possibilities, we have to consider what we have chosen to assume - basically by appropriate choice of coordinate system - namely that each of the four strings or particles are associated with the same number of "objects". We may remember that by our gauge choice the number of "objects" N associated with say an open string is proportional to the P+ component of its momentum, so that choosing a frame, wherein all the four external particles have equal P+ implies that they have an equal number of associated "objects" also.
Now to keep the "objects" most in the succession they already have in the initial state also in the final state we must let connected pieces of "even objects" pass from say string 1 to string 4. Then the rest of the "even objects" associated with string 1 must go to string 3. Now the "even object" numbers on string 2 that must go to respectively to 3 and to 4 is already fixed for what happened for string 1. Since they have to sit in succession and a cyclic rotation of the cyclically ordered chains is the very last rudiment of gauge choice, there is no physically significant freedom in the identification except for the starting number of how many objects go from 1 to 4.
On the figure it is illustrated how different series of "even objects" from 1 or 2 marked with some signature are refound - with same signature - in 3 or 4. The idea of course is that each of the four series marked by the four different signatures are refound in both initial (1+2) and final (3+4) states, and really are the same. It is understood that the series of "even objects" identified to be in both initial and final states are identified "even object" for "even object" in same succession.
To get the contributions to the overlap - and thereby amplitude - from all the physically different "identification ways" one shall sum over the various values, a non-negative integer number, of "even objects" from 1 that are refound in 4. Since such numbers are of order N - which means it goes to infinity as our cut off parameter a —» 0 - the actual overlap contribution from each separate value of the number summed over varies slowly and smoothly (we may check by our calculation) and we can replace it by an integral over say the fraction of the "even objects" in 1 (i.e. associated with 1) that are identified with 'even objects"
11 Deriving Veneziano Model in a Novel String Field Theory... 207 1 2
3	4
Fig. 11.6.
associated with 4. It is this integration that shall turn out to be the integration in the integration in the Euler Beta function making up the Veneziano model.
11.5.7 The Overlap for One Identification
But before integrating or summing we have to write down the overlap as obtained, if one only includes the possibility of one single "identification" (correspondence between the "even objects" in the initial state 1+2 with them in the final 3+4). This overlap of two states 1+2 and of 3+4 with a fixed "identification" is of course simply the Hilbert product of the two states of the set of N/2 + N/2 "even objects" - at least if one ignores the low probability of two "even objects" in say 1 and 2 being in the same state - so that we calculate it as an inner product in an N/2 + N/2 particle/"even object" system. It becomes an inner product of the form
^ 3+4, ((Jr(0), Jr(2),..., JR(N - 2))|i, (JR(0), JR(2),..., JR(N - 2))|2)
J	with
identification!
xV1+2((Jr(0),Jr(2),...,Jr(N - 2))h, (JR(0)JR(2),...,JR(N - 2))|2) xnM=0,2,...,N-2,k=i,2dJR (I)|k.	(11.65)
Now the crucial point of our technique is that this inner product integration
over the JR-values for all the "even objects" associated with 1 or 2 (and identified with "even objects" in 3 or 4) when the wave functions are written as the functional
integrals, we use, can be interpreted as just gluing functional integral regions together. The point is that the functional integrals basically are just - when cut off
-	integrals over 91 values in all the different "lattice" points along the region A boarder say. At the boarders there is specified linear relations of the 91 values there
-	or rather the derivatives, but they are also linear relations - to the JR(i)'s assigned places on this border. One now has to argue that apart from an overall constant factor we can consider the integration over the JR (I)'s in (11.65) going in as part of the functional integration in a functional integral in which the regions A for the two sides (initial and final) are glued together to one big functional integral. Since the integration over the "even object"-variables JR (I)|k have now been interpreted as part of the functional integration, the new resulting functional integral has no longer any boundary conditions associated with such JR(I)|k's. Rather the 'big' functional resulting - and expressing the overlap for a specific "identification" -only has as boundary conditions the inlets of the external 26-momenta(or rather their transverse components only),P1, P2, P3, P4.
One should notice how this picturing by functional integrals come to look really extremely analogous to gluing together strings. There is though a little deviation from the usual open string theory at first, because we have cyclically ordered chains topologically of form as circles as would be closed strings to represent the open strings. Corresponding to this little deviation we get at first that final contribution from "identification" of "even objects" between final and initial states, becomes conformally equivalent to a Riemann sphere with the four inlets from the four external strings being attached to this Rieman sphere. This is what you would expect for closed string scattering in the usual string theory, but we obtain this for open strings! It turns, however, out that all our four "inlets" -where the momentum boundary conditions are imposed - come by a calculation we shall sketch - to sit on a circle on the Riemann sphere. Thus there is "reflection" symmetry between the two sides of this sphere and mathematically our overlap for the fixed identification come to be equal to a functional integral as usually used for open strings. In this way our model has the possibility of agreeing exactly with usual string theory.
11.5.8 Seeing the Hope
A bit of imagination of how our topologically infinite half cylinders can glue together would reveal, that we could arrange to get them pressed down in a plane but with - we must stress though - in two layers. In such a form we could have arranged that the result would look like a double layer four string bands meeting along intervals with their neighbors but only in one point with their opposite string. In order that we could bring it to look like this, we should put the two incoming strings opposite and the two outgoing strings opposite to each other. This would be the usual string gluing picture for the open strings - just doubled, but that essentially does not matter - for the B(-a(t), -a(u)) term. This means that it is extremely promising that we should obtain this term of the Veneziano model.
But!:
•	1. What happens to the other two terms B(-a(s), -a(t)) and B(-a(s), -a(u)), which we should also have gotten, to get the full Veneziano model?
•	2. We have in principle to check that our model predicts the correct weight factor on the integrand in the Veneziano model. We mean that the integrand, which we obtain does not only have the right dependence on the external momenta, but also the right dependence as a function of the integration variable -which in our model comes from the summation over the different "identifications".
Since in our model this integration comes from the simple summation over "identifications" our model has a very clear rule for what weighting to obtain and one just has to calculate carefully not remembering the anomaly in the functional integral evaluation etc.
•	3. So far we were sloppy about the + and - components, or rather we only started calculating the factors in the integrand coming from the transverse momenta or transverse JR components so far.
11.5.9 Integrand weighting Calculation
If we want to evaluate the integrand meaning the contribution from one specific identification more carefully, we have to be specific about how we for such an "identification" make the construction of the full surface on which the functional integration at the end gets defined. We obtain - using the idea that the overlap integration can be absorbed into the functional integration - that we must glue together four (either infinite half cylinders or) disks corresponding to the four external strings. To be concrete it is easiest to represent the two final state strings 3 and 4 by exteriors of a disk rather than by a disk as we represent the initial strings 1 and 2. The inlets for the final state strings are then at infinity of the Riemann surface, while those of the initial strings are at zero.
Now however, we have two incoming and two outgoing, and so we are forced to work with a complex plane with two layers.
We take say one layer where the complement of the unit disk is put to be the essential disk for string 3, while the other layer belongs then to string 4. Similarly we must have two layers for the initial strings, but now the important point is that the initial and the final ones are to be glued together in a slightly complicated way, depending also on the integration variable, which is essentially given by the number of "even objects" going from 1 to 4 say.
Having settled on giving 3 and 4 each their layer in the complex plane in the complement of the unit disk, we have let these outside unit disk layers be glued to the inside the unit disk ones associated with the two incoming particles 1 and 2 along the unit circle of course. But now the length measured say in angle - or in number of "even objects" proportional to the angle - along which say the layer in the inside assigned to string 1 has to be glued together with complement of disk region layer assigned to string 4 along a piece of circle proportional to the amount/number of "even objects" passing from string 1 to string 4 in the "identification" we consider. along the rest of the unit circle then of course the disk assigned to string 1 is identified with the outside disk layer connected to
string 3. Similarly along the first piece of circle - where 1 is connected to 4 - of course the layer of string 2 shall be connected along the circle to the layer of string 3 (in the outside). Correspondingly along the "rest" of the unit circle the layer assigned to string 2 (inside) is attached (identified with) the layer of string 4. In the figure 11.7 you may see an attempt to give an idea of what to do before having put on the final state strings associated with the complements of the unit disk in their two layers. But on the figure the inner layers are prepared for the gluing together. Now modulo the anomaly - i.e. naively - the functional integrals
Inlet of current/momentum
for 1 and 2 (incoming strings)
Two unit disks lying two layers in the complex plane, seen in perspective, and prepared for being glued to "complements of disks for outgoing 3+4. Green for 2; red for 1
Fig. 11.7.
considered are conformally invariant, so that we are postponing the anomaly allowed to perform a conformal transformation of the combined region - now lying in two layers - of the four disks or complements of disks associated with the four external strings, and the result of the functional integral being proportional to the overlap contribution from the "identification" in question should not be changed.
Since the angle 6 (circle piece length) along which say layer of string 1 is identified with the layer of string 4 is proportional to the number of "even objects" we shall simply integrate to get the an expression proportional to the full overlap (and thus to the Veneziano amplitude hopefully) integrate simply with the measure d6.
At first it looks that we have a little problem by only having wave functions as functions of the 24 transverse coordinates so that seemingly the + and - components of the 26-momenta cannot appear in our hoped for Veneziano model integral. However, luckily for the term we actually obtain B(—a(u), a(t)) we found above
in equation (11.52) that in fact all the terms in the exponents for z and (1 - z) that depend on the external 26-momenta could be arranged to come only from the transverse momenta provided we have made the very special frame choice that the four external particles have the same P+ components. So in this our simplifying the contributions from the + and - components turn out not needed. The point really was that just having the gauge choice and the frame choice arranging the + component for the p1 - p4 and for the p1 - p3 become zero the character of the u and t of having a + component multiplied with a - one made it enough to ignore but the transverse contributions.
11.5.10 The Conformal Transformation
The way to evaluate the contribution to the overlap of |1 + 2 > with |3 + 4 > from one "identification" is to rewrite it into a functional integral the region of which is composed from the four disks or disk complements corresponding to the four external particles/strings. We obtain at first a manifold described as double layered in the Riemann sphere. It has two branch points on the unit circle corresponding to the points where the "even objects" on say 1 shifts from going to 3 to going to 4 (or opposite). Basically we choose to map this doubled layered region by a map with two square root singularities at the two branch points.
11.5.11 Anomaly
The anomaly that gives us an extra factor mutiplying the contribution from a single "identification" is usually written formulated as the trace anomaly
< Ta >= -#fields * 48n ^g * R,	(11.66)
(in the notation of our article with Habara wherein ^gR = -23a3an for the metric tensor of the form gap = exp(2H)n«p,and) where R is the scalar curvature of the metrical space given by the metric tensor (in two dimensions enumerated by a = 1,2;). Here the energy momentum tensor is denoted Ta|3 is indeed for the theory of the field(s) which had been Weyl invariant as it formally looks like, and so the trace TJ = 0 would be zero. The symbol #fields denotes the number of fields ^i; it would in the 26=25+1 theory be 24.
This anomaly can be seen to come in by having in mind that we want to perform a conformal transformation - in fact the one corresponding to the analytical function
f(z) ^zz-1(11.67)
(here we used the notation that the end points of the cut along the unit circle
seperating where sheet 1 connects to 4 from where it connects to the sheet associated with particle 3 were arranged to be exp(iS) and exp(-iS).) and then the anomaly gives rise to corrections to the "naive" result that the functional integral is invariant under a conformal transformation. In fact we may first have in mind that we shall evaluate the functional integral (11.56) with a lattice or other cut
off only depending on the internal geometry so that it only can give variations depending on the metric tensor gaß, which under a conformal transformation only changes its scale locally as under a Weyl transformation (11.64). So what we only need to calculate to obtain the effect of the anomaly is how the overall factor on the metric tensor varies under the conformal transformation, we shall use (11.67). Such a scaling is given by the numerical value of the derivative of the function (here f) representing the conformal transformation,
9f 2
gaß-> Hgaß where then H = | — |2.	(11.68)
This is to be understood, that the metric tensor describing the complex plane metric in the f-plane is exp 2Hgaß when the metric induced from the z-plane usual metric is g aß.
It is easy to see that scaling the metric tensor (locally) with an infinitesimal scaling factor exp 2H with H << 1 leads to a correction to the logarithm of the functional integral by J" A^T^d2a. Since the trace T£ of the energy momentum tensor is only non-zero according to (11.66) where there is a non-zero curvature, and our two layered surface lies mostly in the flat complex plane, we only get contributions to this Weyl transformation local change of scale from the two (singular) branche points exp(iö) and exp(-iö), where the curvature R has delta-function contributions.
We can without any change in value of the functional integral make a formal reparametrization from say the double sheeted complex plane to the single layered
one by means of f = ^Jz-g^p-is) provided one then use after the transformation the transformed metric tensor. With a conformal transformation the transformed metric inherited from the z-plane into the f-plane will only deviate from the flat metric n aß in the f-plane by a Weyl transformation. We know that there only shall be curvature - of delta function type - at those points in the f-plane that are the images of the branch points z = exp(iö) and z = exp(-iö), and so the (Weyl transformed) metric reflecting the metric space from the z-plane into the f-plane, exp(2H)naß, i.e. R = 0 outside these two points f = Oandoo.
This means that the H outside those two points in the f-plane must be a harmonic function of f, meaning the real part of an anlytical function. This outside the two points harmonic function shall though have singularities at the two points on the f-plane (or the corresponding Riemann sphere rather) delivering the delta-function contributions,
R = 4nö(Re(f))ö(Im(f))at f=0 say.,	(11.69)
At the branch points, we have points with the property that going around one of them in the z-plane or system of sheets one get a return angle 6 being 2n more than after the mapping into the f-plane (or Riemann space). Thus the integral over the curvature delivering this extra amount of parallel transport extra shift angle should in an infinitesimal region around the image of a branch point - say the point f = 0 - be 2n. So with a notation with the rule of such an (excess) angle being given as
RVgd2CT = 26	(11.70)
with 6 the extra angle of rotation on return,1 there will in the metric inherited from the z-plane in the f-plane be delta function contribution to the curvature scalar R at the points corresponding to the branch points z = exp(iS) and z = exp(—iS) in the z-plane, and thus f = 0 and f = oo in the f-plane. One can easily see that because there is just 2n extra angle to go around such a branch point in the sheeted z-plane the delta-function contribution becomes e.g. for the f = 0 point
R = 4nS2(f).	(11.72)
If r is the distance to the image of the branch point, say the f = 0 point, so that r = |f|, the solution to R = —23a3aH for this delta function R is a logarithm
of the form
H(r) = ln(r/K).	(11.73)
( Here K is some constant in the sense of not depending on r) That implies that taken at the point r = 0 the H(0) is logarithmically divergent so that the integral to which the anomaly of the logarithm of the correction to the integrand is pro-portioanal becomes divergent. However, we have anyway given up caculating in this article the overall normalization of the Veneziano model, we hope to derive. We shall therefore be satisfied with only calculating the contribution in H that varies with the angle S over which we (finally) integrate. Now the conformal transformation mapping the two-sheeted z-plane into the one-sheeted f-plane is
f(z) = F—xp—|,	(11.74)
and so its logarithmic derivative
if = 1 (__1___1 ) (1175)
fdz 2(z - exp(iS) z - exp(—iS)	( . 5)
and the derivative proper
df = 1 z- exp(iS) * (_1___1_). (11.76)
dz 2 U z — exp(—iS) z — exp(iS) z — exp(—iS)
We are interested in a hopefully finite term in the change in going from the z-plane simple metric to the one in the f-plane, which is the part of
A -f
Hztof = ln(ldzl)	(11.77)
depending on the "integration variable" S. This means that make precise the cut off by saying that we must make the cut off by somehow smoothing out the branch
1 In our notation we have the rule that going around an area and thereby obtaining for for a parallel transported vector on return a rotation by an angle 6, that the integral over this
R^gd2a = 26	(11.71)
point singularity in a fixed way in the z-plane. This means that we perform a regularization by putting into our transformation a fixed distance e in the z-plane marking the distance of z to one of the branch points. That is to say we consider a little circle say of points around the exact branch point counted in the z-plane
zon little circle = e exp(ix) + exp(iS)	(11.78)
(or analogously using exp(-i6) instead of exp(iS).) On this little circle we find that the scaling - Weyl transformation - going from the z-plane to the f-plane using (11.76,11.77)
Hz to f = ln(|df/dz|c1rcie) « ln( _	)	(11.79)
2v/ev/2sm(6)
for the z « exp(i6) case. The 6-dependent part is of course -1 * ln(sin(6)). This is the 6-dependent part of Hz to f which comes into the anomaly correction for the logarithm of the full (product over the 24 values of i of the ) functional integral, according to (11.66) and (11.72) of course with a coefficient proportional to the number of truly present dimensions in the functional integral - which is only the transverse dimensions 24 -.
Thus the 6 dependent part of the anomaly ends up being in the logarithm of the contribution to the overlap from one value of 6 ( meaning one "identification"):
Aanomaly ln integrand = 6 independent + (d - 2) * 48^ * 2 lnsin 6 * (4n + 4n)
d - 2
* lnsin 6 + ...	(11.80)
o
Now we should remember that we have decided in this article to go for the form of the amplitude but have left for further studies the over all normalization of the amplitude. This means that the terms in the logarithm of the integrand of the hopefully to appear Veneziano amplitude which do not depend on the integration variable 6 (which is proportional to the number of (even) objects from string 1 that goes into string 4) but only so on the cut off parameter e are neglected.
Now the conformal mapping (11.74) brings the inlet points for external momenta for the four external particles into the positions sketched on the figure 11.8: Imagining on this figure that one varies the integration variable 6, then the "inlet" points for the two incomming strings 1 and 2 will remain sitting opposite to each other one the unit circle and analogously the two final state string inlet points 3 and 4. So the distances between 1 and 2 or between 3 and 4 are constant as function of 6 and so we can ignore the terms comming proportional to in fact s = -(p1 + p2)2 +... = -(p3 + p4)2 +... from the heat production in the analogue model from the current running co as to depend on the distance between 1 and 2 or analogously between 3 and 4. In a similar way we are allowed with our decission to only keep the 6 dependent terms to ignore terms involving only one of the four inlet points. There are such contributions but they depend on the inlet momentum squared (with a divergent coefficient), but since only on one point the 6-dependence is not there provided we cut off in a 6 independent way of course.
Fig. 11.8.
The divergences form cut off of the anomaly correction connected with originallly - in z-plane - branch points come only in to the extend that they get their cut off small circles scaled to a differnt degree depending on 6.
It is not difficult to see that to seek identification of the 6 dependent terms with the integration variable z ( not to be confused with our complex plane z which is of course something different) in the Veneziano model we must identify
i 1
z = sin2(2 * 6)	(11.81)
1
1 — z = cos2(2 * 6).	(11.82)
Very important for stressing how successful our model/rewritting is to reproduce the Veneziano model integration measure in the z-integration correctly. In our model this integration correponds to the summation over the discrete variable being the number of (even) objects going from string 1 to string 4 and it is proportional to 6, and thus we at first simply the measure of integration is d6. But now to compare with the usual Veneziano formula expressions or our slight rewrittings of it we must of course relate d6 to dz:
dz = cos(^6) sin(^6)d6 <x sin(6)d6 <x Jz(1 — z)d6.	(11.83)
This happens to show that the correction factor from the anomaly needed to just compensate the factor comming from
d6 =	(11.84)
sin(6)
would be just cancelled by the anomaly provided
d - 2
6
= 1.	(11.85)
( Apart from a little calculational mistake above by a factor 4 ) this means that we get precisely the right Veneziano model when the number of transverse dimensions d - 2 = 6 * 1 = shouldbe24. The famous result that the bosonic string must exist in d=26 space time dimesions.
11.6 Our Shock; Only One Term
When we went through the above sketched calculation and arrived at only the one term proportional to B (—a(t), —a(u)), which is the term without poles in the s-channel, it were somewhat unexpected at first. After all we had made up a model essentially written out so as to make it the string theory and thereby the Veneziano model. Then it gave only one out of the three terms it should have given. May be even more strangely, if we imagine investigating crossing symmetry it looks we would get a different term after what particles are incoming and which outgoing. So the term we got is not even properly crossing symmetry invariant. Nevertheless it were very encouraging that we got something so reminiscent of the Veneziano model as simply one of the terms.
We believe we have found a way to get the two missing terms also come out:
In fact we think that it is in a way the infinite momentum frame gauge, which we used, that is the reason for the surprising problem for our model: Really one may say that the infinite momentum frame is a method for avoiding having to think about the vacuum, which in quantum field theories is usually an enormously complicated state. In the infinite momentum frame type calculations you imagine an approximation in which the particles have so high energy that they manage not to "feel" the vacuum. But such an approximation may not be a good one. So we thought it might be best somehow to introduce at least some rudimentary effects of a vacuum even though we want to continue to work with an infinite momentum frame formalism, especially an infinite momentum frame gauge/parameterization choice.
The idea, which we here propose, and which actually seems to help to obtain the lacking two terms in the full Veneziano model amplitude, is to allow not only as we did at first for "objects" with the +components J+ = aa'/2 (a positive number), but also allow "negative objects" having rather their J+ = —aa'/2. At least with inclusion of such negative "objects" you make it at least a possibility to have not totally trivial state with the property of the vacuum of having the "longitudinal" momentum P+ = 0. The vacuum could so to speak consist of a compensating number of usual positive say "even objects" and corresponding number of "negative even objects".
In fact it looks that we with such "negative" "objects" can imagine some of our strings represented by an "extended" cyclically ordered chain(replacement). Hereby we mean that it contains in the "extended" cyclically ordered chain not only usual positive J+ objects, but also one or more series of negative J+ objects,
arranged so that the excess of positive ones over negative ones is proportional to the total P+ component of the 26-momentum for the string in question. With such "extended" cyclically ordered chains representing some open string we obtain the possibility of the negative part of say string 2 annihilating with part of the cyclically ordered chain of string 1. Similarly one of the final state strings could be produced with content in its cyclically ordered chain of some series of negative objects having been produced together with some positive ones in another final state string.
By very similar procedure to the one used above to the term B(-a(u), -a(t)), but now including the negative objects we seem to be able to produce the two missing terms. The detail of the calculation to obtain the full Veneziano amplitude/model will appear soon by the authors [21].
11.7 Conclusion and Outlook
We have in the present article sketched how using our string field theory formalism in which strings are rewritten into be described by states of "even objects" we can obtain the scattering amplitude to be the usual Veneziano model amplitude. It must though be immediately admitted that we at first got only one out of the three terms expected. However, introducing "objects" that can have negative J+-components and can function as a kind of holes for objects, we though believe, that it is promising to obtain the whole Veneziano amplitude. Our model or string field theory has previously been shown [3] to lead to the usual mass square spectrum for strings. In this way we collect increasing evidence that our formalism is indeed another representation of all of string theory.
The way we constructed our formalism working from string theory and only throwing away though a null set of information, it is of course a priori expected, that our formalism should be string theory. In so far there sufficient holes in the "derivation" of our formalism from string theory to be equivalent to the latter, that we still need the more indirect support from rederiving features of string theory such as the Veneziano amplitude from our model.
Our model is a formulation in terms of what we called "objects", and they "sit" in circular "cyclically ordered chains", to an open string is assigned one such circular chain of objects, to a closed string two. The "objects" are supposed to "sit" as smoothly as they can from quantum fluctuations - which put severe constraints though, since the odd numbered "objects" in cyclically ordered chain are not independent dynamical variables, but rather given in terms of (the conjugate variables nR for) the neighboring "even numbered objects" by equation (11.19).
Actually we even stressed that the smoothness or continuity condition because of the dependence of the odd objects on differences of the conjugate momenta of neighboring even ones become non-reflection invariant. That is to say that a cyclically ordered chain being smooth would not remain smooth, if one puts the objects in the opposite order! The crux of the matter is that we have a genuine string field theory in the sense that we construct a state space of Hilbert vectors describing a whole universe in a string theory governed world. Then of course there can in the various states of this Hilbert space exist different numbers of
strings, well this is not hundred percent true, because contrary to other string field theories: our Hilbert space is described in terms of the "even objects" and the number of strings perfectly accurately given once you have a Hilbert space state. The "even objects" can namely be associated to strings in slightly different ways, so that the number of strings only approximately can be derived from a given state; even there are no exact eigenstates for the number of strings. But in practice we believe the approximate access to the number of strings in our description is sufficient. But that the number of strings is not cleanly defined feature of a state in our Hilbert space, is clear from the fact that we have scattering even scattering that change the number of strings, such as if two strings scattered and became three, but that nothing happen in our formalism under a scattering. We just obtained the Veneziano model scattering amplitude as an overlap of initial and final state just corresponding to that nothing happens in the object formulation. In this sense the strings resulting from the scattering must have been there all the time.
One may look at our model as solution of string theory in the sense that we have "even object " description that does not even develop with time so that the "even object" state is more like a system of initial data to a solution of string theory.
11.7.1 Outlook
We foresee that there must be really very much it would be reasonable to do in our formalism, which is in many ways simpler than usual string theory especially than usual string field theory.
Presumably it will be very easy to make the superstring version; if nothing else should work one could in principle bosonize the fermionic modes and then treat the resulting bosons similar to the way we treated in our model of the bosonic modes.
Of course we should really also properly finish getting the Veneziano model calculation remaining details. A special interest might be connected with the overall normalization, which we left completely out here, since our formalism has no obvious candidate for the string coupling g, so the latter should come out from whatever parameters such as our cut off parameter a and a' and possible vacuum characteristic, but we did not use openly vacuum properties in the calculation sketched.
Most interesting might be to use our formalism to obtain a better understanding of the Maldacena conjecture by developing our formalism for the Ads space and then see that the corresponding CFT can also be written by our formalism.
Acknowledgements
One of us (HBN) thanks the Niels Bohr Institute for allowing him to stay as emeritus, and Matjaz Breskvar for economic support to visit the Bled Conference. The other one(M.N.) acknowledges to The Niels Bohr Institute for warmest hospitality extended to him during his stay at NBI. He is supported by the JSPS Grant in Aid for Scientific Research No. 24540293.
References
1.	H. B. Nielsen and M. Ninomiya, An Idea of New String Field Theory - Liberating Right and Left movers -, in Proceedings of the 14th Workshop, "What Comes Beyond the Standard Models" Bled, July 11-21, 2011, eds. N. M. Borstnik, H. B. Nielsen and D. Lukman; arXiv: 1112. 542[hep-th].
2.	H. B. Nielsen and M. Ninomiya, A New Type of String Field Theory, in Proceedings of the 10th Tohwa International Symposium in String Theory", July 3-7, 2001, Fukuoka Japan, AIP conference Proc vol.607, p.185-201; arXiv: hep-th/0111240.v1, Nov.2001
3.	H.B. Nielsen and M. Ninomiya, "A Novel String Field Theory Solving String Theory by Liberating Left and Right Movers" JHEP_202P_05131 v2 (2013)
4.	String references are cited in References [4]-[7], [9]
H. B. Nielsen, A Physical interpretation of the integrand of the n-point Veneziano model, (1969)(Nordita preprint...); a later version "An almost Physical interpretation of the integrand of the n-point Veneziano model" preprint at Niels Bohr Institute; a paper presented at the 15th International Conference on High Energy Physics, Kiev 1970 (see p.445 in Veneziano's talk.)
5.	Y. Nambu, Quark Model and the factorization of the Veneziano amplitude, in Proceedings of the International Conference on Symmetries and Quark Models, June 18-20,1969 ed. by Chand, R. (Gordon and Breach), New York, 269-277; reprinted in Broken Symmetry Selected Papers of Y. Nambu eds. T. Eguchi and K. Nishijima, (World Scientific, Singapore, 1995)258-277.
6.	Y. Nambu, Duality and hadrondynamics, Lecture notes prepared for Copenhagen summer school, 1970; It is reproduced in "Broken Symmetry", Selected Papers of Y. Nambu eds. T. Eguchi, and K. Nishijima, (World Scientific, Singapore, 1995)280.
7.	L. Susskind, Structure of hadrons implied by duality, Phys. Rev. D1(1970)1182-1186; L. Susskind, Dual symmetric theory of hadrons, 1. Nuovo Cimento A69 (1970) 457-496.
8.	For many original papers of early string theory found in
The Birth of String Theory" eds. by Andrea Cappeli, Elena Castellani, Flippo Colomo and Paolo Di Vecchia, Cambridge University Press 2012. Cambridge, New York, Melbourne.
9.	M. B. Green, J. H. Schwarz and E. Witten, Superstring theory, 1-2, Cambridge University Press 1987.
10.	As for bosonic string field theory in the light-cone gauge:
M. Kaku and K. Kikkawa, Phys. Rev. D10(1974)1110; M. Kaku and K. Kikkawa, Phys. Rev. D10(1974)1823; S. Mandelstam, Nucl. Phys. B64(1973)205; E. Cremmer and J.-L. Gervais, Nucl. Phys. B90(1975)410.
11.	Witten type mid-point interaction of covariant bosonic string field theory: for open string
E. Witten, Nucl. Phys. B268(1986)253.
12.	Closed string version of ref.[11] of the Witten theory:
M. Saadi and B. Zwiebach, Ann. Phys. 192(1989)213; T. Kugo, H. Kunitomo and K. Sue-hiro, Phys. Lett. B226(1989)48; T. Kugo and K. Suehiro, Nucl. Phys. B337(1990)434; B. Zwiebach, Nucl. Phys. B390(1993)33.
13.	Joining-Splitting light-cone type: for open string
H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D34(1986)2360.
14.	Closed string of Joining-Splitting light-cone type:
H. Hata, K. Itoh, H. Kunitomo and K. Ogawa, Phys. Rev. D35(1987)1318; H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, Phys. Rev. D35(1987)1356.
15.	Supersymmetrized covariant open Witten theory
E. Witten, Nucl. Phys. B276(1986)291; I. Arefeva and P. Medvedev, Phys. Lett. B212(1988)299; C. Wendt, Nucl. Phys.B314(1989)209; N. Berkovits, Nucl. Phys. B450(1995)90 [Errata, B459(1996)439] hepth/9503099.
16.	Modified cubic theory:
I. Arefeva, P. Medvedev and A. Zubarev, Nucl. Phys. B341(1990)464; C. R. Preitschopf, C. B. Thorn and S. A. Yost, Nucl. Phys. B337(1990)363.
17.	Supersymmetrized Green-Schwarz theory in the light-cone gauge:
M. B. Green and J. H. Schwarz, Nucl. Phys. B218(1983)43; M. B. Green, J. H. Schwarz and L. Brink, Nucl. Phys. B219(1983)437; M. B. Green, and J. H. Schwarz, Nucl. Phys. B243(1984)437; J. Greensite and F. R. Klinkhamer, Nucl. Phys. B281(1987)269; Nucl. Phys. B291(1987)557; Nucl. Phys. B304(1988)108; M. B. Green and N. Seiberg, Nucl. Phys. B299(1988)559; S.-J. Sin, Nucl. Phys. B313(1989)165.
18.	For review articles of string field theory:
W. Taylor and B. Zwiebach, D-branes, Tachyons, and String Field Theory, hep-th/0311017 http://jp.arxiv.org/abs/hep-th/0311017; K. Ohmori, A Review on Tachyon Condensation in Open String Theories, hep-th/0102085 http://jp.arxiv.org/abs/hep-th/0102085.
19.	C. B. Thorn, Derivational model from field theory II, Phys. Rev. D17(1978)1037, and references therein.
20.	G. Veneziano, Construction of a crossing-symmetric, Regge behaved amplitude for linearly rising trajectories, Nuovo Cimento A57 (1968) 190.
21.	H. B. Nielsen and M. Ninomiya, to appear.
22.	For comprehensive reviews of string theory see
e.g. J. Polchinski, String Theory, vol.l-ll, Cambridge University Press 1998.
23.	Appearance of doubling, K. G. Wilson, Confinement of quarks, Phys. Rev. D10 (1974) 2445, see also H. B. Nielsen and M. Ninomiya, Absence of neutrinos on a lattice I - Proof by homotopy theory -.
24.	P. Goddard, J. Goldstone, C. Rebbi, C. B. Thorn, Nucl. Phys. B56 (1973) 109.
25.	J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Advances in Theoretical and Mathematical Physics 2 (1998) 231.
Discussion Section
The discussion section is meant to present in the workshop discussed open problems, which might start a collaboration among participants or at least stimulate them to start to think about possible solutions in a different way. Since the time between the workshop and the deadline for the contributions for the proceedings is very short and includes for most of participants also their holidays, it is not so easy to prepare besides their presentation 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.
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. Consequently this year discussion section has three contributions only.
Most of discussions concerned searches for the theory which would offer a trustable step beyond the standard models and the need to find the overlaps among all possible searches for the theory beyond the standard model. Participants were trying to understand, as deeply as possible, all the assumptions of different approaches: How much have they in common, as well as how far have the assumptions of different models, first of all of the standard model, influenced the experimental results and also how is the choice of the groups of symmetries, through which the universe went in its evolution, connected with smallness of the group representations? And others.
Discussions about the direct dark matter measurements are presented at the end of the talk of Rita Bernabei. She is answering the participants' questions.
The three written contributions concern: a. The toy model following the spin-charge-family theory within d = (5 + 1 ) (instead of (13 + 1 )), from where one can learn how the break of symmetries, global, local and discrete, occur and how do they influence the properties of families and of the scalar and vector gauge fields. b. The idea that the future influences the past and the such an influence is already observed. c. The space can emerge from randomness and diffeomorphism symmetry.
All discussion contributions are arranged alphabetically with respect to the authors' names.
Udeleženci velikega dela diskusijniso uspeli zapisati pravočasno v obliko, da bi ga lahko vključili v letosnji zbornik. Organizatorji upamo, da bodo te diskusije dozorele do oblike, da jih bo mogoče predstaviti na naslednji delavnici. Tako ima ta sekcija samo tri prispevke.
Največ diskusije je teklo okoli vprašanja, kaj mora ponuditi teorija, da ji bo mogoče zaupati, da ponuja pravi naslednji korak k razumevanju lastnosti osnovnih delcev in in evolucije vesolja in razlozši ne le predpostavke teh dveh modelov ampak tudi opazene pojave, kijih modela ne vključujeta. Udelezzenči so poskušali razumeti in razčleniti vse predpostavke različnih predlogov, pa tudi, kako so ti predlogi, predvsem pa standardni model, vplivali na rezultate meritev, ter tudi, denimo, kako je vesolje „izbiralo" grupe simetrijv svoji evolučiji in kako je ta izbor povezan z velikostjo upodobitev grup.
Diskusija o tem, kako skrbno so očiščeni vseh nečistoč njihovi eksperimeni, kako so poskrbeli, da med delče temne snovi ne prištejejo zze poznane delče ter kakšen je vzrok, da ostali eksperimenti (še) niso potrdili njihovih meritev, najde braleč kot dodatek k prispevku Rite Bernabei, v katerem odgovarja vodja laboratorija DAMA/LIBRA na zahtevna vprasanja udelezenčev.
Trije prispevki te sekčije obravnavajo: a. Teorijo spina-nabojev-družin pri predpostavki, da je prostor primarno (5 + 1 )-razsezen in ne (13 + 1 )-razsezen, ki naj pomaga razumeti, kako pride do spontane zlomitve simetrij, globalnih, lokalnih in diskretnih in kako te zlomitve spremenijo lastnosti spinorjev ter lokalnih skalarnih in vektorskih polj. b. Idejo, kako vgraditi v teorijo vpliv prihodnosi na preteklost, ki je opazljiv. č. Kako prostor vznikne iz neurejenosti, ki ji dodamo simetrijo na difeomorfizme.
Prispevki v tejsekčiji so, tako kot prispevki v glavnem delu, urejeni po abečed-nem redu priimkov avtorjev.
Bled Workshops in Physics Vol. 15, No. 2
A
Proceedings to the 17th Workshop What Comes Beyond ... (p. 223) Bled, Slovenia, July 20-28, 2014

12 Properties of Families of Spinors in d = (5 + 1) with Zweibein of an Almost S2 and Two Kinds of Spin Connection Fields, Allowing Massless and Massive Solutions in d = (3 + 1)
D. Lukman and N.S. Mankoc Borštnik
Department of Physics, FMF, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
Abstract. We studied in the refs. [1,2] properties of spinors in a toy model in d = (5 + 1), when M(5+1' breaks to an infinite disc with a zweibein which makes a disc curved on an almost S2 and with a spin connection field which allows on such a sphere only one massless spinor state, as a step towards realistic Kaluza-Klein theories in non compact spaces. In the ref.[3] we allow on S2 two kinds of the spin connection fields, those which are gauge fields of spins in and those which are the gauge fields of the family quantum numbers, both as required for this toy model by the spin-charge-family theory [4,5]. This time we study, by taking into account families of spinors interacting with several spin connection fields, properties of massless and massive solutions of equations of motion, with the discrete symmetries [9,10] (Cn, Pn, TN) included. We also allow nonzero vacuum expectation values of the spin connection fields and study the masses.
Povzetek. Da bi bolje razumeli zlomitve simetrij v teoriji spinov-nabojev-druzin in njhove posledice, studirata avtorja zlomitve na preprostem modelu v prostoru-casu d = (5 + 1). V tem modelu zvije vektorski svezenj neskončen disk v peti in sesti dimenziji v skoraj sfero S2, spinske povezave pa poskrbijo za to, da je v opazljivem prostoru-casu (d=(3+1)) sodo stevilo brezmasnih druzin. Ko dovolita spinskim povezavam, da imajo nenicelno pricakovano vrednost, naboj fermionov (S56) ni vec dobro kvantno stevilo. Druzine pridobijo maso, ki jim jo dolocijo skalarna polja. Studirata tudi diskretne simetrije brezmasnih in masivnih druzin fermionov. Studij ponudi globji vpogled v skalarna polja, ki dolocajo lastnosti fermionov pred in po zlomitvi simetrije naboja. Medtem ko nosijo skalarna polja v teoriji "spini-naboji-druzine", kadar je dimenzija d = (13 + 1), polstevilcni sibki in hipernaboj, tak kot Higgsovo polje v standardnem modelu, pa je v preprostem modelu naboj skalarnih polj celostevilcen.
12.1 Introduction
The spin-charge-family theory [4,5], proposed by one of us (N.S.M.B.), is offering the explanation for the appearance of families of fermions in any dimension. Starting in d = (13 + 1) with a simple action for massless fermions interacting with the gravitational interaction only - that is with the vielbeins and the two kinds of the spin connection fields, the ones originating in the Dirac kind of spin (ya's)
and the others originating in the second kind of the Clifford operators (ya's) - the theory manifests effectively at low energies the observed properties of fermions and bosons, offering the explanation for all the assumptions of the standard model: For the appearance of families, for the appearance of the Higgs's scalar [6] with the weak and the hyper charges (^ 1, ± 1, respectively), for the Yukawa couplings, for the charges of the family members, for the vector gauge fields, for the dark matter content, for the matter-antimatter asymmetry [7].
The theory predicts the fourth family, which will soon be observed at the LHC, and several scalar fields, manifesting in the observed Higgs's scalar [8] and the Yukawa couplings, some superposition of which will also be observed at the LHC.
A simple toy model [1-3], which includes also families in the way proposed by the spin-charge-family theory [4,5]), is expected to help to better understand mechanisms causing the breaks of symmetries needed in the case of d = (13 + 1), where a simple starting action leads in the low energy regime after the breaks to the observable phenomena.
This contribution is a small further step in understanding properties of the families after the breaks of symmetries, caused by the scalar fields which are the gauge fields of the charges of spinors and the scalar fields which are the gauge fields of the family groups. The discrete symmetries of fermions and bosons in the case of only one family are studied already in the ref. [10]. Here the discrete symmetries are studied when the families are taken into account. We allow also that the spin connection fields gain nonzero vacuum expectation values and study solutions of the equations of motion for massive spinors.
We start with massless spinors [1-3,10] in a flat manifold M(5+1), which breaks into M(3+1' times an infinite disc. The vielbein on the disc curves the disc into (almost) a sphere S2
;?) ^=' (ioi,	(12.1)
with
f = 1 + (2P0) = 1 + cos £, x(5) = p cos x(6) = p sin E = f-2.	(12.2)
The angle £ is the ordinary azimuthal angle on a sphere. The last relation follows from ds2 = eSCTesTdxCTdxT = f-2(dp2 + p2d^2). We use indices (s,t) G (5,6) to describe the flat index in the space of an infinite plane, and (ct, t) g ((5), (6)), to describe the Einstein index. Rotations around the axis through the two poles of
a sphere are described by the angle while p = 2p0 y 1-™s|. The volume of
this non compact sphere is finite, equal to V = n (2p0)2. The symmetry of S2 is a symmetry of U(1) group.
We take into account that there are two kinds of the Clifford algebra operators: Beside the Dirac Ya also ya, introduced in [4,5,12]. Correspondingly the covariant
momentum of a spinor on an almost S2 sphere is
P0a = f«a Pa + ^jE iP«, f«aE}- — ^ScdWcda — 2Scd&cda ,
Sab = 4(yV — YV), Sab = 4(YaYb — YbYa),	(12.3)
with E = det(eaa) and with vielbeins faa 1, the gauge fields of the infinitesimal generators of translation, and with the two kinds of the spin connection fields: i. ^aba, the gauge fields of Sab and ii. &aba, the gauge fields of Sab.
We make a choice of the spin connection fields of the two kinds on the infinite disc as follows (assuming that there must be some fermion sources causing these spin connections, the study of such sources of the scalar fields ^sta and &aba are in progress)
e , xCT 1
fCTs' ^sta = iF56 f£st	= — TFiP^,EfCTs'}- est 4Fs6 ,
(Po)2	2E
e / xa	1
fas' ®sta = iF56 f£st	= — TFiPa,Efas'}- Est 4Fs6 ,
(Po)2	2E
1
2El
s = 5,6, d =(5), (6).	(12.4)
We take the starting action in agreement with the spin-charge-family theory for this toy model in d = (5 + 1), that is the action for a massless spinor (Sf) with the covariant momentum p0a from Eq. (12.3) interacting with gravity only and for the vielbein and the two kinds of the spin connection fields (Sb)
s	{p<r , Ef s}— 4Fmn , Fmn F
S = Sb + Sf , Sf =
ddxE Lf
Sb =
ddxE (aR + aR), Lf = Y°ya p0a	(12.5)
The two Riemann scalars, R = Rabcdnacnbd and R = Rabcdnacnbd, are determined by the Riemann tensors
Rabcd = 2 f«[af^b] (^cd|3,a — ^cea^dp) ,
Rabcd = 2 f«[af^b] (&cd|3,a — ^cea^dp) ,	(12.6)
where [a b] means that the anti-symmetrization must be performed over the two indices a and b.
1 faa are inverted vielbeins to eaa with the properties eaafab = 6ab, eaafp a = 6«. Latin indices a, b,.., m, n,.., s, t,.. denote a tangent space (a flat index), while Greek indices a, .., |i, v, ..a, 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,..), indices from the bottom of the alphabets indicate the compactified dimensions (s, t,.. and a, T,..). We assume the signature r|ab = diag{1, —1, —1,..., —1}.
We assume no gravity in d = (3 + 1): = Sm and	= 0 for m, n =
(0,1,2,3), = (0,1,2,3). Accordingly (a,b,...) run in Eq. (12.5) only over s G (5,6). Taking into account the subgroup structure of the operators Smn
N(l,r) = N® = 1 (S23 ± iS01, S31 ± iS02, S12 ± iS03), Nl± — N®@ = N®1 ± iN®2, Nr@ = NeS = Ne1 ± iNe2, (12.7) we can rewrite the 2Scd&cda part of the covariant momentum (Eq. 12.3) as follows

- f Smn^mn± = Y. N®iA®i + Y_ NeiA±i
1 2
ii
— N®BA®B + N®BA®B + N®3A®3 + NeBA±B + NeBAeB + Ne3A±3, &mn± — ®mn5 T i^mn6 .	(12.8)
The notation was used
A®1- — fCTs A®1- — —fCTs{(^23a T i^01a), (&31- T i&02-), (&12- T i^03a)}
= S- 1 {p-, Ef}- 4(F®\F®2,F®3) ,
s A- — 1 s 1
-s 2E lKCT
A®± — 1 (A®1 T iA®2), Ae± — 2 (Ae1 t i Ae2),
F®± — (F23 t F02) — i(±F31 + F01), F®3 — (F12 — iF03),
Fe± — (F23 ± F02) + i(TF31 + F01), Fe3 — (F12 + iF03),
c —((5), (6)), s — (5,6),	(12.9)
with ^abc and &abc defined in Eq. (12.4).
We looked in the ref. [3] for the chiral fermions on this sphere, that is for the fermions of only one handedness in d — (3 + 1) and accordingly mass protected, without including any extra fundamental gauge fields to the action from Eq.(12.5).
In this contribution we study the influence of several spin connection fields on the properties of families, looking for the intervals within which the parameters
of both kinds of the spin connection fields (F56, F56, F®@, F®3, Fe@, Fe3) allow massless solutions of the equation
{Y0YmPm + fY0Ys S-(p0- + {p-, Ef}-)M — 0, with
P0- — p- — 1SSt^st- — 1Sab^ab- ,	(12.10)
for several families of spinors.
We also allow nonzero vacuum expectation values of the scalar (with respect to d — (3 + 1)) gauge fields and study properties of spinors.
The discrete symmetries of the equations of motion and of solutions are studied in sections (12.3.1,12.3).
We look for the properties of spinors and gauge fields, scalars and vectors with respect to d = (3 + 1).
In section 12.2 we present spinor states in "our technique" (see appendix in the ref. [7]). In section 12.2.1 we discuss massless and massive states of families of spinors. In section 12.3.1 we present discrete symmetry operators introduced in the refs. [9,10], in section 12.3 we discuss the properties of spinors and the gauge fields, the zweibein and the two kinds of the spin connection fields, under the discrete symmetry operators.
12.2 Solutions of equations of motion for families of spinors
We first briefly explain, following the refs. [5,1-3], the appearance of families in our toy model, using what is called the technique [12].
There are 2d/2-1 = 4 families in our toy model, each family with 2d/2-1 = 4 members. In the technique [12] the states are defined as a product of nilpotents and projectors
ab	1	ab 1
(±i):= 2 (Ya T Yb), [±i]:= ^ (1 ± YaYb),	for naanbb =-1, ab 1 ab 1
(±):= 2 (Ya ± iYb), [±]:= 2 (1 ± iYaYb), for naVb = 1, (12.11)
which are the eigen vectors of Sab as well as of S ab as follows
ab k ab	ab k ab	ab k ab	ab k ab
Sab (k)= 2 (k), Sab [k]= 2 [k], Sab (k)= 2 (k), Sab [k]= -2 [k] , (12.12)
ab	ab	ab	ab
with the properties that Ya transform (k) into [—k], while Ya transform (k) into [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
Ya (k) = —inaa [k], Y~b (k)= —k [k], Y~a [k]= i (k), Y~b [k]= —knaa (k) (12.13)
After making a choice of the Cartan subalgebra, for which we take: (S03, S12, S56) and (S03, S12, S56), the four spinor families, each with four vectors, which are eigen vectors of the chosen Cartan subalgebra with the eigen values from Eq. (12.12) [3], follow
«Pi1	56 03 12	P111	56 03 12
	= ( + )(+i)(+) ^0,		= ( + )[+i][+] ^0,
«21	56 03 12	p2h	56 03 12
	= ( + )[—i][—] ^0,		= ( + )( —i)(—) ^0
	56 03 12	P111	56 03 12
	=[—][—i](+) ^0,		= [—](—i)[+] ^0,
«21	56 03 12	P211	56 03 12
	= [—](+i)[—] ^0,		= [—][+i]( — ) ^0,
<p1m	56 03 12	^1IV	56 03 12
	= [+][+i]( + ) ^0,		= [+](+i)[+] ^0,
P2ih	56 03 12	^2IV	56 03 12
	= [+](-i)[-] ^0,		= [+][-i](-) ^0,
	56 03 12	2IV	56 03 12
	= (-)(-i)(+-) ^0,		= (-)[-i][+] ^0,
p2ih	56 03 12	2IV ç>2	56 03 12
	= (-)[+i][-] ^0,		= (-)(+i)(-) ^0
(12.14)
where is a vacuum for the spinor state. One can reach from the first member p 11 of the first family the same family member of all the other families by the application of Sab. One can reach all the family members of each family by applying the generators Sab on one of the family member. If we write the operators of handedness in d = (5 + 1) as r(5+1) = yVyVyV (= 23iS03S12S56), in d = (3 + 1) as r(3+1' = —iy0y1 y2y3 (= 22iS03S12) and in the two dimensional space as r(2) = iY5Y6 (= 2S56), we find that all the states of all the families are left handed with respect to r(5+1with the eigen value —1, the first two states of the first family, and correspondingly the first two states of any family, are right handed and the second two states are left handed with respect to r(2), with the eigen values 1 and —1 , respectively, while the first two are left handed and the second two right handed with respect to r(3+1' with the eigen values —1 and 1, respectively.
Having the rotational symmetry around the axis perpendicular to the plane of the fifth and the sixth dimension we require that is the eigen function of the total angular momentum operator M56 = x5p6 — x6p5 + S56 = —ig^ + S56
M5V6) = (n + 1) V6).	(12.15)
Accordingly we write, when taking into account Eq. (12.14), the most general wave function obeying Eq. (12.10) in d = (5 + 1) as
56	56
= N £ (An (+)1 ^ + Bn+1 e1* [-]1 ^l41)] ein*. (12.16)
,(4i)
V)
i=I,II,III,IV
where A^ and B^ depend on xCT, while ^ (+1) and ^ j-1) determine the spin and the coordinate dependent parts of the wave function in d = (3 + 1) in accordance with the definition in Eq.(12.14), for example,
^ = a+ (+i) (+) + ( + [-i] [-],
,.T.	03 12	03 12
^(4)' = a4 [-i] (+) + (3— (+i) [-].	(12.17)
56	56	56 56	56 56
(+)1 = (+), for i = I, II and (+)1 = [+] for i = III, IV, while [-]1 = [-] for i = I, II
56 56
and [-]1 = (-) for i = III, IV. Using ^(6) in Eq. (12.10) and separating dynamics in (1 + 3) and on S2, the following relations follow, from which we recognize the mass
term mI: 0+ (p0 - p3) - §+ (P1 - ip2) = m1, f+ (P0 + P3) - f+ (p1 + ip2) = m1,
a-(P0 +P3) +1-(P1 -ip2) = m1, I-(p0-p3) +1-(p1 -ip2) = m1. (Onenotices
that for massless solutions (m1 = 0) and	for each i = I, II, III, IV,
decouple.)
For a spinor with the momentum pm = (p0,0,0, p3) in d = (3 + 1) the spin
and coordinate dependent parts for four families are: ^ ( +) = a (+i) (+), ^ ( =
03 12	03 12	03 12
a [+i] [+], ^ ( +'n) = a [+i] (+), ^ ( +'V) = a (+i) [+].
Taking the above derivation into account (Eqs. (12.16,12.2,12.4,12.17,12.7,
12.8,12.9)) the equation of motion for spinors follows [3] from the action (12.5)
if {e
1^2S56 [(d + 1
-2Fe
9 dp
1 2Ne E
2S56 9 1 9f
— (id))- df (1 - 2p56 2S p 99 2f 9p
- "" - 2Fe 3 2Ne 3

2F® E 2N
,0 ,5
® E

- 2FeE 2Ne 2F® B 2N
+ y0Y5 m^( 6) = 0.
® E
56
2F56 2S
56

2F® 3 2N® 3)]
( 6)
(12.18)
One easily recognizes that, due to the break of M(5+1) into M(3+1) x an infinite disc, which concerns (by our assumption) both, Sab and Sab sector, there are two times two coupled families: The first and the second, and the third and the fourth, while the first and the second remain decoupled from the third and the fourth. We end up with two decoupled groups of equations of motion [3] (which all depend on the parameters F56 and F56):
i. The equations for the first and the second family
-if{[( dp - n)-(1 - 2F56 - 2F56 - 2Fe3)]An
- fp^ An1} + m Bn +1 = 0,
-if{[( dP + ^)- 2fdP (1 + 2F56 - 2F56 - 2Fe3)] Bn +1

19f
2f 9p
!Bn+1
}+ m An = 0,
(12.19)
-"{[(dp - n)- 2fdP(1 - 2F56 - 2F56+2Fe3)] An - f^F® An} + mBn+1 = 0,


-if{[(dP + )- SSP(1 + 2F56 - 2F56 + 2Fe3)1B"+1 2fd|f 2FeB Bn+1 }+m An' = 0.
ii. The equations for the third and the fourth family 9 n 1 9f
iii
-f [( gp — p) — 2fdfp (1 — 2F56 + 2F56 — 2F®3)] ^
2fdff (—2F®ffl) Anv} + m= 0,
—if{[( dp + ^) — 2f dff (1 + 2F56 + 2F 56 — 2F ®3)] BI+
2fdf (—2F®ffl) BnXi }+m An11 = 0,	(12.20)
—f{[(^ — n) — il1 (1 — 2F56 + 2F 56 + 2F ®3)] Anv 1 9p p 2f9p
(—2F®B) An11} + mBIV+I = 0,
-if{[( dp + ^) — (1 + 2F56 + 2F 56 + 2F ®3)] Bn+i
— ¿dp (—2F®B) Bnl+i } + m Anv = 0.
Let us look for possible normalizable [1,2] massless solutions for each of the two groups in dependence on the parameters which determine the strength of the spin connection fields. Both groups, although depending on different parameters of the spin connection fields, can be treated in an equivalent way. Let us therefore study massless solutions of the first group of equations of motion.
For m = 0 the equations for An and An1 in Eq. (12.19) decouple from those for Bn +1 and Bni+1. We get for massless solutions
AI± = a± pn f2 (1-2F56-2F56) f±VTFeIjI+Feffl"FeB ,
±,/(Fe3)2 + F QfflF + Fe3
An	M	An ,
BI^ = b± p-n-1 f 1 (1+2F56-2F56) f±V(Fe3)2+FefflFeB ,
±,/(FQ3)2 + F QfflF + Fe3
Bn+ = ^-s®-Bn+1,	(12.21)
n is a positive integer. The solutions (A}+, Ani+) and (An , An1 ) are two independent solutions, a general solution is any superposition of these two. Similarly is true for (Bn+1, BI+1).
In the massless case also Ann± decouple from BI+I1±.
One can easily write down massless solutions of the second group of two families, decoupled from the first one, when knowing massless solutions of the
12 Properties of Families of Spinors in d _ (5 + 1 )... 231 first group of families. It follows
AlH± _ a± pn f 2 (1-2F56 + 2F 56) f±V(F ®3)2+F ®fflF ®B ,
+ ,/(F ©3)2 + £ ©fflp ®B + F ®3 aIV± _ V 1 ' __mi±
fffiffl '
RIII± _ b , p-n-1 f 2 (1+2F56+2F 56) f±V(F ®3)2 + F ®fflF ®B
Bn+1 b± p	1	1	)
±,/(F ©3)2 + F ©fflp ®B + F ®3
riv± _ v 1__biii±	(12 22)
Bn+1 _ £ffi ffl Bn+1 ,	(1222)
n is a positive integer, a± and b± are normalization factors.
Requiring that only normalizable (square integrable) solutions are acceptable
2n
Epdp (An*An+Bn*Bn) <00,	(12.23)
0
i G {I, II, III, IV}, one finds that An and Bn are normalizable [1,2] under the following conditions
A^1 : -1 < n < 2(F56 + F56 ± /(F©3)2 + F©HF©ffl ),
Bn11 :	2(F56 - F56 ± /(F©3)2 + F©Hp©ffl ) < n < 1 ,
AniJ,IV :	-1 < n < 2(F56 - F56 ± /(F®3)2 + F®HF®ffl ),
BIII,IV :	2(F56 + F56 (F®3)2 + F®HF®ffl ) < n < 1 . (12.24)
One immediately sees that for F56 = 0 = F56 there is no solution for the zweibein from Eq. (12.2). Let us first assume that F©i = 0; i G {1,2,3}. Eq. (12.24) tells us that the strengths F56, F56 of the spin connection fields (d56a and d56o.) can make a choice between the massless solutions (A^", AnIIJV) and (B^", BnIIJV): For
0<2(F56 + F56) < 1, F56 < F56	(12.25)
there exist four massless left handed solutions with respect to (3 + 1). For
0<2(F56 + F56) < 1, F56 = F56	(12.26)
the only massless solution are the two left handed spinors with respect to (3 + 1)
^(6I,n)m=0 = No f-F56 -F56+1/2 (+6) ^(+I,II).	(12.27)
2
The solutions (Eq.12.27) are the eigen functions of M56 with the eigen value 1/2. Since no right handed massless solutions are allowed, the left handed ones are mass protected. For the particular choice 2(F56 + F56) = 1 the spin connection fields -S56d56CT - S56d56ct compensate the term ^{pCT, Ef}- and the left handed
spinor with respect to d = (1 + 3) becomes a constant with respect to p and To make one of these two states massive, one can try to include terms like F®\
Let us keep Fei = 0 i e {1,2,3} and F56 = F56, while we take Fe3 ,Fe@ non zero. Now it is still true that due to the conditions in Eq. (12.24) there are no massless solutions determined by AIII,IV and BIII,IV. There is now only one massless and mass protected family for F56 = F56. In this case the solutions A— and A0I- are related
A- = N- f2 [1 —2F56 —2F56-2V(F93)2+F60F9B] ,
(J(Fe3 )2 + F efflFeB + Fe3) T A0 = --A- .	(12.28)
There exists, however, one additional massless state, with A0+ related to A0I+ and related to BjI+, which fulfil Eq. (12.24). But since we have left and right handed massless solution present, it is not mass protected any longer.
One can make a choice as well that none of solutions would be massless. According to Eq. (12.38) from sect. 12.3 the equation of motion presented in Eqs. (12.5,12.3) are covariant with respect to the discrete symmetry operator Cn • Pn (Eq. (12.35)), what means that the antiparticle feels the transformed gauge fields and carry the opposite charge with respect to the starting particle.
Let us conclude this section by recognizing that for F= 0 and Fe@ = 0 all the families decouple. There is then the choice of the parameters (F56, F56, F®3, Fe3 ) which determine how many massless and mass protected families exist, if any.
12.2.1 Solutions after the scalar gauge fields gain nonzero vacuum expectation values
Let us now assume that the spin connection fields gain nonzero vacuum expectation values
_ (56)
< ^56± > , ra± ) :=< c56± > ,
(56)
r
- (NRi)
m^ R
rfl4NLi) : = (< CD23± + i£>oi± >, < o>31± + icD02± >,< cD i2± + io>03± >),
= (< CD23± - iCDoi± >,< CD3i± - icD02± >, < CD 12± - icD03± >) ,
),
(12.29)
breaking the charge S56 symmetry, as well as all the "tilde charges" (S56, N (R'L)). Then the equation of motion (12.10) can be rewritten as
56
{Y0Ympom + Y0 X (±) Po±}^ = 0, P0m = Pm — S56 C56m ,
P0± = P± - S56 m±56) - S56 m±56) - ¿ INR m±NRi) - ¿ INL m±NRi(12.30)
i=1 i
One finds that requiring the hermiticity of the equations of motion (Eq. (12.30)) leads to the relations
,(56)
(56) ~ (56)
(56)
,(nr1)
,(nr1) ~(nl1)
-m+ ' = m- ' , m+ ' = m- ' , m+ R ' = m- R ' , m+ L ' = m-NL(12.31)
We also must require, to be consistent with the definition and the Eqs. (12.35,12.36, 12.37,12.38) and Eq. (12.30), that
cnpnm±56) (cnpn)-1 = -m^56) , cnpnm±56) (cnpn)-1 = m^56), cnpn m±NRi) (cnpn)-1 = <NRi), cn •pnm±NLi) (cn •pn)-1 = <NLi). Eq. (12.30) has then the solutions
(12.32)
Nl 1 / (56) _ (56)
m, 2 = 2 (m- — m-

,(NLi)
LiM 2
mN,R = 1 (m-56) + m-56)) ± (m.
(Ntfi) )2
)2,
(12.33)
with the spinor states with no conserved charge S56 any longer
03 12 56 03 12 56
^m(NL) = nNl {(m-NL3) ± JY. (m-NLi) )2 )(+](+)[+] — (-i)(+)(—))
K, ii	K, ii 03 12 56 03 12 56	0
+ (m-NL1) + im-NL2)) ( (+i)[+][+] - [—i][+](—))}e-imx
03 12 56 03 12 56
^(m((NR) = nNr {(m-NR3) T (m-NRi) )2 ) ( [+][+](+) — (—!)[+][—])
03 12 56 03 12 56
+ (m-NR1) — im-NR2)) ((+i)(+)(+) — [—i](+)[—])}e while handedness in the "tilde" sector is conserved.
(12.34)
12.3 Discrete symmetries of spinors and gauge fields of the toy model
In the subsection of this section 12.3.1 the discrete symmetry operators for particles and antiparticles in the second quantized picture are presented, as well as for the gauge fields. This definition for the discrete symmetry operators, as they manifest from the point of view of d = (3 + 1), is designed for all the Kaluza-Klein like theories. At least this way of looking for the appropriate discrete symmetry operators from the point of view of d = (3 + 1) can be helpful in all the Kaluza-Klein cases to find the appropriate discrete symmetry operators in the observable dimensions.
One sees that the operators of discrete symmetries, presented in Eqs. (12.39, 12.41,12.42), do not depend on the family quantum numbers Ya, which means that every particle, described as a member of one family, transforms under the product of the two discrete symmetry operators Cn and Pn, presented in Eq. (12.39) and Eq. (12.42), into the corresponding antiparticle state, which belongs to the same family (carrying the same family quantum numbers).
The discrete symmetry operator Cn •Pn (Eqs. (12.39, 12.42)) is in our case with d = (5 + 1) equal to
Cn •Pn = Y0 Y5 Ix3 Ix6.	(12.35)
It has an even number of Ya's, which guarantees that the operation does not cause the transformation into another Weyl representation in d = (5 + 1), which means that we stay within the Weyl representation from which we started.
Let us check what does this discrete symmetry operator Cn •Pn do when being applied on several operators. One easily finds
Cn •Pn (Y0,Y1,Y2,Y3,Y5,Y6)(CN • Pn)-1 = (—"Y°, Y1, Y2, Y3, —Y5,Y6), Cn •Pn (p0,p1,p2,p3,p5,p6)(Cn •Pn )-1 = (p0, —p1, —p2, —p3,p5, —p6),
56	56
Cn • Pn (±) (Cn • Pn)-1 = (T) ,
Cn • Pn Y~a (Cn • Pn)-1 = Ya , for each a,
Cn • Pn (¿565^, X3,X5,X6),^566(x0, X3, X5, X6)) (Cn • Pn =
( — ¿565 (X0, —X3, X5, —X6), D566 (x0, —X3, X5, —X6)) , Cn • Pn (D565(X0,X3,X5,X6),iD566(X0,X3,X5,X6)) (Cn • Pn)-1 =
(D565(X0, —X3,X5, —X6), —CD566(X0, —X3,X5, —X6)) , (12.36)
where we write cd56s, s = (5,6) to point out that the first two indices belong to the
56s 56
SO(5,1) group. We also use the notation (±)= 2 (y5 ± iY6).
One correspondingly finds, taking into account Eqs. (12.7,12.9)
Cn Pn S56 (Cn Pn )-1 = —S56,
= S56,
= — ¿56m (X0, —X3) ,
= (A® (X5, —X6), — A® (X5, —X6)) ,
= A® (X5, —X6), 56 5
Cn Pn S56 (Cn Pn )-Cn Pn ¿56m (X0, X3) (Cn • Pn )-Cn Pn (A® (x5,x6),a® (x5,x6)) (Cn Pn )-Cn Pn A® (x5,x6) (Cn Pn )-
Cn Pn a±6 (x5,x6) (Cn Pn )-1 = —a^6(x5, —x6) ,	(12.37)
with A±6 = (¿565 T iD566).
From Eqs. (12.36,12.37) it follows
Cn • Pn {Y0Ym (pm — S56 D56m(x0, X3))} (Cn • Pn
= {(—Y0)(—Ym) (pm — (—S56)( —D56m)(x0, —X3))} = {Y0Ym (pm — S56 D56m(x0, — X3))}
56	1 _
Cn -Pn {y° (±) (p± - S56 ^56±(x5,x6)- -Sab &a6±(x5,x6))} (Cn -Pn)-1
56 1 = {y0 (t) (pT - S56 W56T(x5, -x6)- 2Sab &abT(x5, -x6))}.
(12.38)
Taking into account Eq. (12.4) and the equations of (12.38), we see that the equations of motion are covariant with respect to a particle and its antiparticle: A particle and its antiparticle carry the same mass, while the antiparticle carries the opposite charge S56 than the particle and moves in the transformed U(1) field -W56m(x°, -x3) [15].
The equations of motion for our toy model (Eqs. (12.10,12.4), and correspondingly the solutions (Eq. (12.16)) manifest the discrete symmetries Cn■ Pn, 7n and CN■ PN -7N, with the operators presented in Eqs. (12.39,12.42). Both, CN -PN -¥(6) and Cn -Pn¥(6) (12.42) solve the equations of motion, provided that cu56m(x°,x3) is a real field. The field ^56m(x°,x3) transforms under Cn Pn and Cn Pn to -w56m(x°, —x3), like the U(1) field must [15].
The starting action (12.5) and the corresponding Weyl equation (12.10) manifest discrete symmetries Cn■ Pn, 7n and Cn■ Pn '7n from Eqs. (12.39,12.42). Correspondingly all the states with the conserved charges M56 respect this symmetry, transforming particle states into the antiparticle states.
12.3.1 Discrete symmetry operators
To discuss properties of the representations of particle and antiparticle states and of the gauge fields with which spinors interact let us first define the discrete symmetry operators as seen from the point of view of d = (3 + 1) in the second quantized picture as proposed in the ref. [9], where the definition of the discrete symmetries operators for the Kaluza-Klein kind of theories, for the first and the second quantized picture was defined, so that the total angular moments in higher dimensions manifest as charges in d = (3 + 1). The ref. [9] uses the Dirac sea second quantized picture to make presentation transparent.
The ref. [9] proposes the following discrete symmetry operators
3
Cn = n Ym r(3+1) kix6,x8j...,xd,
3ym,m=0 3
7n = n Ym r(3+1) KIx0 Ix5jX7,...jXd-1 ,
PNd-1) = y° r(3+1) r(d) ix3 .	(12.39)
The operator of handedness in even d dimensional spaces is defined as
r(d) := (i)d/2 ^ (Vn^Ya),	(12.40)
a
with products of Ya in ascending order. We choose y°, Y1 real, y2 imaginary, y3 real, y5 imaginary, y6 real, alternating imaginary and real up to Yd real. Operators
236 D. Lukman and N.S. Mankoc Borstnik I operate as follows:
T 0 0
Ixo x = —x ;
Ixxa = —xa;
Ixo xa = (— x0, x);
Ixx = —x;
II3 xa = (x0, —x1, —x2, —x3,x5,x6,...,xd);
Ixs,x7,...,xd-i (x0,x1 ,x2,x3,x5,x6,x7,x8,... ,xd-1 ,xd) =
(x0, x1, x2, x3, —x5, x6, —x7,..., —xd-1,xd);
Ix6,x8,...,xd (x0,x1 ,x2,x3,x5,x6,x7,x8,... ,xd-1 ,xd) =
(x0, x1, x2, x3, x5, —x6, x7, —x8,...,xd-1, —xd), d = 2n.
Cn transforms the state, put on the top of the Dirac sea, into the corresponding negative energy state in the Dirac sea.
We need the operator, we name [11,10,9] it Cn, which transforms the starting single particle state on the top of the Dirac sea into the negative energy state and then empties this negative energy state. This hole in the Dirac sea is the antiparticle state put on the top of the Dirac sea. Both, a particle and its antiparticle state (both put on the top of the Dirac sea), must solve the Weyl equations of motion.
This Cn is defined as a product of the operator [11,10] "emptying", (which is really an useful operator, although it is somewhat difficult to imagine it, since it is making transformations into a completely different Fock space)
"emptying" = ^ Ya K = (—)d1 ^ Ya r(d)K,	(12.41)
OTya	3ya
and Cn
Cn = n Ya K n Ym r(3+1) KIx6,x8,...,xd
OTya,a=0	3ym,m=0
d
= n Ys IxW..,xd .	(12.42)
Kys,s=5
Let us present also the second quantized notation, following the notation in the ref. [9]. Let ¥p [¥p] be the creation operator creating a fermion in the state ¥p and let ¥p (x) be the second quantized field creating a fermion at position x. Then
= (x) ¥p(x)d(d-1)x}|vac > so that the antiparticle state becomes
{Cn =
¥„(x) (Cn¥p(x))d(d-1)x}|vac> .
The antiparticle operator ^JjVp], to the corresponding particle creation operator, can also be written as
^[Vp] |vac > = Cn	|vac >= ^(x) (CnVp(X)) d(d-1)x |vac > ,
CH = "emptying" • CH .	(12.43)
While the discrete symmetry operator Cn has an odd number of Ya operators and correspondingly transforms one Weyl representation in d = (5 + 1) into another Weyl representation in d = (5 + 1), changing the handedness of the representation, stays the operator Cn •Pn within the same Weyl. The same is true for Tn and also for the product Cn •Pn• TN.
12.4 Conclusions and discussions
We make in this contribution a small step further with respect to the refs. [3,10] in understanding the existence of massless and mass protected spinors as well as the massive states in non compact spaces in the presence of families of spinors after breaking symmetries. We take a toy model in M5+1, which breaks into M3+1 x an infinite disc curled into an almost S2 under the influence of the zweibein. Following the spin-charge-family theory we have in this toy model four families. We study properties of families when allowing that besides the spin connection field, which are the gauge field of Sst = 4 (ysYt — YtYs), also the gauge fields of Sst = 4 (YsYt — Y^Ys), determining families, affect the behaviour of spinors.
We simplify our study by assuming the same radial dependence of all the spin connection fields (Eq. (12.4)), while the strengths of the fields (F56, F56, Fmn) are allowed to vary within some intervals.
We found that the choices of the parameters allow within some intervals of parameters (F56, F56, Fmn) four, two or none massless and mass protected spinors.
We allowed the nonzero vacuum expectation values of all the spin connection fields, fCTs' &56o, fos' &56o and fos' &mno, where a = ((5), (6)), s = (5,6), m = ((3, T, 2,3). All indices a belong to the SO(5,1) group, while indices a belong to the SO(5,1) group. The nonzero vacuum expectation values of all the gauge fields causes that the U(1) charge (S56) breaks, as well as also all the family quantum numbers, while the handedness in the "tilde" degrees of freedom keep two groups of families non coupled.
We studied also the discrete symmetries of equations of motion and of solutions, for massless and massive states.
We found: a. Almost S2 or any other shape with the symmetry around the axis, perpendicular to the infinite disc, has the rotational symmetry around this axis. But almost S2 has not the rotational symmetry around the axis which goes through the centre of almost sphere because of the singular point on the southern pole unless we make the translation of the axis. Equivalently the almost torus -infinite disc curled into an almost torus - has no symmetry. b. Even number of families stay massless and mass protected for the intervals of parameters. c. Non zero vacuum expectation values of the scalar gauge fields break all the charges, while the two handedness in the "tilde" sector keeps the two groups of families
separated. d. Let us add that while the weak charge and the hyper charge have
fractional values in the spin-cahreg-family theory in d = (13 + 1), have the scalar
fields in this case of d = (5 + 1) integer valued charges.
References
1.	D. Lukman, N. S. Mankoc Borstnik, H. B. Nielsen, "An effective two dimensionality" cases bring a new hope to the Kaluza-Klein-like theories", http://arxiv.org/abs/1001.4679v5, New J. Phys. 13 (2011) 103027. arXiv:1001.4679v4.
2.	D. Lukman, N. S. Mankoc Borstnik, "Spinor states on a curved infinite disc with nonzero spin-connection fields", http://arxiv.org/abs/1205.1714, J. ofPhys.A: Math. Theor. 45 (2012) 465401 (19pp), doi:10.1088/1751-8113/45/46/465401.
3.	D. Lukman, N. S. Mankoc Borstnik, H. B. Nielsen, "Families of Spinors in d = (1 + 5) with Zweibein and Two Kinds of Spin Connection Fields on an almost S2", Proceedings to the 15 th international workshop "What Comes Beyond the Standard Models", July 9-19, 2012, Ed. N.S. Mankoc Borstnik, H.B. Nielsen, D. Lukman, DMFA ZaloZnistvo, Ljubljana December 2012, p. 163, [arXiv:1012.0224, arXiv:1212.2370].
4.	N. S. Mankoc Borstnik, Phys. Lett. B 292 (1992) 25, J. Math. Phys. 34 (1993) 3731, Int. J. Theor. Phys. 40 (2001) 315, Modern Phys. Lett. A 10 (1995) 587, A. Borstnik, N. S. Mankoc Borstnik, hep-ph/0401043, hep-ph/0401055, hep-ph/0301029, Phys. Rev. D 74 (2006)073013, hep-ph/0512062, G. Bregar, M. Breskvar, D. Lukman, N.S. Mankoc Borstnik, hep-ph/0711.4681.
5.	N.S. Mankoc Borstnik, J. of Modern Phys. 4 (2013) 823 [arxiv:1312.1542], New J. of Phys. 10 (2008) 093002, hep-ph/0606159, hep/ph-07082846, hep-ph/0612250, p.25-50.
6.	N.S. Mankoc Borstnik,"The spin-charge-family theory explains why the scalar Higgs carries the weak charge ^-j and the hyper charge ±-j", this volume, p. 163 [http://arxiv.org/abs/1409.4981].
7.	N.S. Mankoc Borstnik, "Can spin-charge-family theory explain baryon number non conservation?", this volume, p. 123 [arxiv:1409.7791].
8.	G. Bregar, N.S. Mankoc Borstnik, "The new experimental data for the quarks mixing matrix are in better agreement with the spin-charge-family theory predictions", this volume, p. 20.
9.	N.S. Mankoc Borstnik, H.B. Nielsen, "Discrete symmetries in the Kaluza-Klein-like theories", JHEP 04 (2014) 165 http://arxiv.org/abs/1212.2362v2.
10.	T. Troha, D. Lukman, N.S. Mankoc Borstnik, Int. J. of Mod. Phys. A 29 (2014) 1450124, arxiv:1312.1541.
11.	N.S. Mankoc Borstnik, Spin-charge-family theory is explaining appearance of families of quarks and leptons, of Higgs and Yukawa couplings, in Proceedings to the 16th Workshop "What comes beyond the standard models", Bled, 14-21 of July, 2013, eds. N.S. Mankoc Borstnik, H.B. Nielsen and D. Lukman (DMFA ZaloZnistvo, Ljubljana, December 2013) p.113 [arxiv:1312.1542].
12.	N. S. Mankoc Borstnik, H. B. Nielsen, J. of Math. Phys. 43 (2002) 5782, [hep-th/0111257], J. of Math. Phys. 44 (2003) 4817, [hep-th/0303224].
13.	T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin, Math. Phys. 96 (1921) 69, O. Klein, Z.Phys. 37 (1926) 895.
14.	N. S. Mankoc Borstnik, H. B. Nielsen, Phys. Lett. B 633 (2006) 771-775, hep-th/0311037, hep-th/0509101, Phys. Lett. B 644 (2007)198-202, hep-th/0608006, Phys. Lett. B 10 (2008)1016, N. S. Mankoc Borstnik, H. B. Nielsen, D. Lukman, hep-ph/0412208.
15.	C. Itzykson and J. Zuber, Quantum Field Theory (McGraw-Hill 1980).
Bled Workshops in Physics Vol. 15, No. 2
A
Proceedings to the 17th Workshop What Comes Beyond ... (p. 239) Bled, Slovenia, July 20-28, 2014

13 Influence from Future, Arguments
H.B. Nielsen*
The Niels Bohr Institute,
Blegdamsvej 17 - 21, Copenhagen 0; Denmark
Abstract. It is the purpose of the present article to collect arguments for, that there should exist in fact -although not necessarily yet found - some law, which imply an adjustment to special features to occur in the future. In our own "complex action model" we suggest a version in which the "goal" according to which the future is being arranged is to diminish the integral over time and space of the numerical square of the Higgs field. We end by suggesting that optimistically calculated the collected evidences by coincidences runs to that the chance for getting so good agreement by accident would be of the order of only 1 in 30000. In addition we review that the cosmological constant being so small can be considered evidence for some influence backward in time. Anthropic principle may be considered a way of simulating influence backward in time.
Povzetek. Namen tega prispevka je zbrati argumente za trditev, da obstaja nek (najbrž se ne odkrit) naravni zakon, ki dopusča vpliv prihodnosti na dogodke v sedanjosti. Avtor je v "modelu kompleksne akcije", skupaj s sodelavci, predlagal, da je "cilj", ki določa prihodnost, povezan z zmanjsanjem integrala vrednosti kvadrata Higgsovega polja po prostor-casu. Avtor meni, da je antropsko nacelo lahko nacin za simulacijo vpliva nazaj v casu. Zbere nekaj primerov, ki jih uporabi za argument, da smo vpliv prihodnosti na sedanjost in preteklost ze opazili. Predlaga poenoteno sliko enacb gibanja in zacetnih pogojev. Predstavi model s kompleksno akcijo, ter povzame napovedi tega modela. Povzame tudi, kako drugi avtorjevi modeli potrjujejo vpliv prihodnjosti na sedanjost. Tudi majhnost kozmoloske konstante se da pojasniti s vpivom preteklosti na sedanjost.
13.1 Introduction
Since long the present author and various collaborators[1-4] have speculated on possibilities for a physical theory having in it some preorganization in the sense, that there is some law that adjust initial condition and/or coupling constants so as to arrange for special "goals" to occur in the future. In works with K. Nagao[6,8,7,9-11] we sought to calculate, if effects of an imaginary part of the action of the type of the works with M. Nimomiya[3,4] could be so well hidden, that such a model would be viable. One could even say, that it is speculations about, that future could somehow act back on the present and the past. Usually - since Darwin and Wallace - it is considered (essentially) a fundamental law of nature, that this kind of back action does not exist. But is that trustable? In the present article we shall
* E-mail: hbech@nbi.dk, hbechnbi@gmail.com
collect arguments for the opposite, namely that there truly IS such a back action in time.
I would like to seek to make a kind of review of the evidence for such an influence from the future, and use it as an excuse for talking about some relatively recent works[12], some of which may not immediately seem to be relevant, such as my work with Masao[15,13,14] Ninomiya on "A Novel String Field Theory"[13-15]. My real motivation is to look for what the fine tuning problems for the various coupling constants may tell us about the fundamental laws of physics, which we seekto find [12].
In other words we could say, that I want to investigate if retro causation is possible and plan to argue for that there are indications that it is possible.
Although the idea of having retro causation is generally believed not to be true there were at least one proposal for a theory of that kind proposed, namely the theory about electromagnetic radiation being radiated in equal amounts backward and forward in time by Wheeler and Feynman [16].
The work of Feynman and Wheler avoids influence from the future by a discussion of the absorbers of the light emitted backward and forward, a mechanism a priori rather different form the one we use in our complex action model already mentioned; but the quantum mechanics interpretation inspired from the Feynman-Wheeler theory, which is called transactional interpretation and is due to Cramer[17], is the same one as the one supported by our complex action theory.
The plan of this talk about the influence from future will be like this :
•	1)Introduction
•	2) Listing of arguments for influence from future.
•	3) Discussion of Time reversal
•	4)Why should we NOT unite initial state information with equations of motion?
•	5) The finiteness of String Theory may hide in mine and Ninomiyas Novel String Field Theory [13-15] - an influence from the future, and that might be the reason for it being string theory.
•	6) Some fine-tunings as if "God hated the Higgs squared field
•	7) Bennetts and mine argument that at the time the Cosmological Constant must already have hat its value, when densities of energy so low as the present were unknown/did not yet occur.
•	8) The Multiple Point Principle being successful means influence from future.
•	9) If we count optimistically do we have sufficient evidence for a planned universe development?
•	10) At the end we conclude that one must take the possibility seriously.
13.2 Listing of Arguments
Here I should like to list a series of arguments for that there is indeed some adjustment going on to achieve some "goals" we may hope to guess some time:
•	A) Funny that many religious people imagine, that there is a Governor of the world, if the principle preventing such government were truly valid.
•	B) Strange that the laws about the initial conditions and equations of motion behave differently under the CPT-like symmetry (or under time-reversal)
•	C) Cosmological constant were very small compared to the energy density in the beginning; how could it then be selected so small, when it had no significance at that time (argument with D. Bennett).
•	D) Several evidences for anthropic principle, but mostly physicists do not like it. (Personally I would say: The anthropic principle is much like putting in the experimental fact that we humans exist into the theory; putting in experimental results can always help avoiding finetuning problems, so a good theory should be more ambitious than have to include such an input.)
•	E) Multiple Point Principle (almost) successful: Higgs mass, top Yukawa coupling, and Weak scale relative to Planck scale.
•	F) Our Complex Action model with Higgs field square taken to dominate gives[12]:
-	1) n and p+e+antineutrino suppress Higgs field equally much (within errors).
-	2) The "knee" cut the cosmic ray spectrum down close to the effective Higgs threshold.
-	3) Nuclear matter has low binding energy.
-	4) Higgs field in vacuum at lowest Higgs field square.
-	5) Smallness of weak scale/Higgs field relative to fundamental/Planck scale.
•	G)It may be very hard to make an ultraviolet cut off, that does not violate locally in time a little bit. So an ultraviolet meaningful theory may imply influence from future?
•	H) General Relativity allows closed time-like loops...(well known to lead to time machines by worm holes etc.)
•	I) Horowich and Maldacenas influence backward inside the black hole.
•	J) The bad luck of SSC and the - though too little - bad luck of LHC would follow from Higgs machines getting bad luck.
•	K)With large extra dimensions there appear in principle a frame dependence of which moments are earlier than which due to the frame motion in the extra dimension directions.
•	L) Wheeler space time foam and baby universes imply almost unavoidably influence from future, at least small influences from near future. Baby universes make effective coupling constant depending on very far away influences in e.g. Time.
•	M) In String theory in the formulation of Ninomiyas and mine (Novel SFT) the hanging together of "objects to strings, or chains giving strings better, is put in as an initial condition AND IT LOOKS ALSO AS A FINAL STATE CONDITION!
The following arguments are even more theoretical speculation arguments for influence from future:
•	N) When we e.g. Astri Kleppes and mine derivation of space time and locality etc. - seek to derive in Random Dynamics e.g. Feynman path integral we get
the complex action and thus future influence from it. And seeking to derive locality we get left with effective couplings, which much like in baby universe theory depends on, what goes on averaged over all space and time.
•	O) Were the many e-foldings in inflationary organized in order to get a big universe (a miracle) ?
Somewhat aesthetically arguments form the time reversal symmetry should also be mentioned:
•	P) The usual picture: The laws concerning the time development the equations of motion are perfectly invariant under the CPT-symmetry. Nevertheless the initial conditions determining the actual solution to these equations of motion is chosen in a way that makes it look more and more complicated as one progresses forward in time! (This is the law of increasing entropy) Really the mystery is not why finally the world ends up in a state in which one can say almost nothing in a simple way; but we rather should take it that a huge number of states have same probability/ the heat death state. Rather it is the mystery why it ever were in a state that could be described rather simply, the state in early big bang times, with high Hubble expansion rate.
•	Q) And even more mysterious we could claim: Why were the Universe in such a special state in the beginning, but do not also end up in such special and simple state? Initial State Versus Development Laws (equations of motion) seem not to have the same symmetry under time reversal (or say instead CPT) Since Newton we have distinguished between initial state information and the laws for the time development. Seeking the great theory beyond the Standard Models our best hope to progress is to unite some of the information about Nature, which we already have in our literature. One lacking unification is the unification of initial state information and the equations of motion. One little may be indicative trouble is that time reversal or better CPT symmetry is valid for equations of motion but NOT for the initial state information!
13.3 Discussion of Time Reversal-like Symmetry
Let us look now a bit on the problem for the usual point of view and thus the argument for influence from the future Q). What are the possibilities?:
•	1 Possibility) CPT symmetry could be the more fundamental and the asymmetry w.r.t. time direction of the initial state information (we know a lot about the start, but the future gets more and more chaotic) could be due to some sort of spontaneous break down, as e.g. in mine and Ninomiyas complex action model:
In principle the "initial state information" could be put in at any time, but due to some special conditions in a certain time early compared to our era "the actual solution to the equations of motion chosen to be realized (by Nature)" became mainly determined by this certain era early compared our era. This should mean that in that special era the realized solution is arranged to obey some relatively simply rules, e.g. some strongly expanding universe being the rule.
• 2. possibility) The time direction asymmetry might be the more fundamental and the CPT symmetry just some effective result coming out of an a priori time and even CPT noninvariant theory. So the initial state CPT noninvariance were the more fundamental feature, and the CPT symmetry for laws of nature is only some sort of effective or "accidental" symmetry [5]. It is well known that CPT largely follows from Lorentz invariance, so that if it were correct as I have claimed for years, that Lorentz invariance could be a low energy approximation (only for the "poor physicists"), then also CPT would be a low energy limit symmetry.
Taking the first possibility means that you have in principle also the possibility of having some influence from the future, so that our question as to, whether such influence is at all possible, gets answered by yes; but of course the effect may be essentially zero such as the situation of the "spontaneous break down " is realized, since otherwise we should already have observed it so safely, that we would have had to believe it.
If the time dominating the fixation of the solution as in mine and Ninomiyas model becomes a certain time which is earlier than our time - but not necessarily the very first moment (if such one should exist) - there would be an opposite axis for the entropy running on the other side of this special time era(that dominantly fixes the solution being realized). In other words before the solution-determining time-era the entropy would decrease! So in that "before solution dominating era " there would formally be influence from the future. Of course, if we lived in such an era, we would invert our time axis and still say, that entropy grows, except if we get contact theoretically or truly to an era with another entropy development axis.
If the second possibility were realized, we should expect Lorentz non-invariant effects in principle. We should namely expect CPT not to be fundamentally true, but then Lorentz invariance could only with violation of other presumably good assumptions be exact.
If we fundamentally did not have Lorentz invariance it could mean that there were in the "fundamental terminology" beyond the Lorentz invariance appearance perhaps some fundamental frame in which the physics would develop strictly causally, in the sense that it would develop more and more chaotic (i.e. increasing entropy) and without any influence from the future. But logically it could nevertheless be so that in some Lorentz frames moving relative to the fundamental frame there could be influence from future.
13.4 Why Not Unite Initial Conditions and Equations of Motion
In looking for a unified theory of all physics, one often finds the idea of seeking to unify the various simple gauge subgroups of the Standard Model gauge group into some simple gauge group such e.g. SU(5) or groups containing SU(5) as a subgroup, such as S0(10). But since making progress towards finding the "theory of everything" is expected to go via successive unifications, one should also
possibly imagine other types of unifications. Here we then ask: Should we not unify initial-state conditions with equations of motion? This is actually what our already in this article suggested complex action model (see subsection 13.6) would do. It predicts both (something about) the initial conditions and of course the equations of motion from the form of the action (as usual). In this sense one should really guess the form of the complex action so that we can obtain relations between features of the initial state conditions and the equations of motion. We can say that with a Standard Model real part of the action taken as phenomenologically suggested the dimensional arguments used to predict that the most important part of the imaginary part of the action determining (or at least providing some information concerning initial conditions) and ending with that the mass square term for the Higgs field, are results of of such a unification. So in this sense our results from this Higgs-dominated imaginary part can be considered results of a unification of initial state conditions and equations of motion.
Also the Hawking-Hartle no-boundary assumption for their (and others) quantum gravity gives information about initial state conditions, and thus it should be considered a unification of initial conditions and equations of motion.
But now one may have general worries about - this kind of - unifications of initial conditions and equations of motion, unless one allows for the influence from the future:
In fact the time reversal or the CPT-like symmetry leads to that the unified theory presumably should have such a symmetry, at least both in our complex action theory and in the non-boundary theory there IS cpt-like symmetry, except that the whole theory is on the manifold. Therefore it gets very hard not to have also a final state condition. In fact it seems only to be a spontaneous breaking of the symmetry of this type that is likely to solve the phenomenological problem. But then there appears indeed easily some remaining effect of influence form the future.
13.5 String Theory, Regularization Problem, and Our Novel SFT
Only String Theory Seems to Cope with the Cut Off problem in Nice Way!
Presumably the best argument for believing, that (super)String Theory should be the theory of everything(T.O.E.), is that it does NOT HAVE THE USUAL DIVERGENCE PROBLEM. One might wonder how string theory manages to avoid the problem of divergent loops. It is well know that by summing up the infinitely many loops from the various string states the integrand for the loop 26-momentum obtain a damping factor going with an exponential of the square of the loop momentum. Thus the divergence of the usual type got effectively cut off. A related property of the lowest order scattering amplitudes is, that they for large transverse momenta fall off even with an exponential in the square of the transverse momentum. Since String theory has gravity (almost unavoidably) having such wonderful cut off of loops behavior is remarkable good!
13.5.1	Cut off in the Light of Mine and Ninomiyas Novel String Field Theory
Let us now consider the for the success of the string theory in coping with the divergences plaguing the usual quantum field theories so important Gaussian cut off of the large momenta.
As an orientation let us look at the transverse momentum cut off from the point of view of mine and Ninomiyas novel string field theory:
The momentum of an open string say in our formalism is given by a sum over the "contained "objects, each of which has the variables (J, n), i.e. 24 momenta J and their conjugates n, and the total momentum of the open string is proportional to the sum of the even "objects, because the momentum contribution from the odd ones become zero due to their construction as difference of conjugate momenta of the two even neighbors. The scattering is in our SFT-model is simply exchanges of "even objects, while no true interaction takes place, only strings are divided and recollected, so that the "even objects in the initial strings get distributed into various final strings.
So how does the limiting/the strong cutting off of the transverse momenta come about in the optic of our model?
Although there is a divergent number "objects in any string in our novel string field theory, these "objects are sitting in chains with strong negative correlation between the momenta of neighbors (in the chains). So any connected piece of such a chain never reaches momenta much bigger than of the order of one over square root of alpha prime %/a7, except for the momentum assigned the total strings. So if we only split the chains of objects into a few connected pieces we cannot get any combination of the pieces, when recombined to final state strings, to contain big amounts of momenta compared to the alpha prime order of magnitude valueVa7. It is this restriction that means, that we get in Veneziano model the exponential of the squared momentum falling off amplitudes.
The limitation actually exponentially with the square of the momentum in the exponent, i.e. Gaussianly of the amplitude of scattering for large transverse momenta of strings coming out of collisions of strings in our novel string field theory (SFT) is due to the very strong anti-correlation of the momenta of the "objects - crudely functioning as constituents of the strings so that only very limited momenta are statistically found on connected pieces of object-chains. Since this so important - for the momentum cut off (anti)correlation of the "objects on the chains used for strings is put in as INITIAL and even as FINAL STATE conditions in order to describe the strings by means of "object-chains, one can say that in mine and Ninomiyas SFT we have arranged the transverse momentum cut off effectively by the initial or final states having been assumed to have the appropriate (anti)correlations!
13.5.2	The Limitation of Momenta in Loops
For each limited loop order corresponding in our novel SFT to splitting the "cyclically ordered chains of "objects into a limited number of subchains before being recollected into new "cyclically ordered chains forming the final state strings (depending on the order (of loops)) the amount of momentum, that can be sent
out as transverse momentum in a scattering is limited due to the correlations among the "objects (neighboring on the chains). The higher the order though the higher is the effectively allowed order of magnitude of the transverse momentum, corresponding to the well known fact that higher and higher loop order in unitar-ity corrections to the Veneziano model has a slower and slower fall off for large momenta the higher the order (i.e. the larger the number of loops). Roughly this relevant correlation corresponds to the "stringness" in the sense, that it is also this correlation (between neighboring "objects"), that ensures that very small pieces of strings carry only very little momentum. But have in mind, that in OUR theory the hanging together to strings is only put in as initial state (and even final state) conditions. Even the alpha prime a' scale so needed to make a chance of having a cut off effectively is in our model only put in as an initial and final state condition (nothing in the completely trivial and basically non existing dynamics talks about alpha prime!)
So one really in mine and Ninomiyas novel string field theory must ask: String theory cut off, from where does it come?
Generally: When one interacts (locally) with the string in our formalism or in other ones you can only transfer little meaning given by apha prime (inverse square root ) momentum into the scattering. Via Heisenberg uncertainty this is turned into an extension of the strings due to quantum fluctuations. But it is crucial for the effective cut off, that the string hangs piecewise together; if e.g. in mine and Ninomiyas novel SFT you could split the "objects in a way, in which no "objects kept attached to their neighbors almost, then the momentum in the scattering could be much larger, and very likely a divergence problem would reappear.
In fact it is well known that the higher loops one consider in string theory (unitarity corrections to Veneziano model) the slower becomes the coefficient in the Gaussian fall off of the amplitude with the exponential of the square of the transverse momenta. This means that the more pieces the string or in our model the to the strings corresponding "cyclically ordered chains" are cut into and recollected under the scattering, the larger can the transverse momentum become.
If one would attempt to split up the string to be actually built form discretized elements, one would be back in quantum field theory and it would be as hard as usual to avoid divergencies. The continuity of the string or in our novel SFT formulation the cyclically ordered chains is crucial for the achievements w.r.t. avoiding divergencies and keep tranverse momenta low.
13.5.3 Looking for a Cut Off Machinery
Let us now look whereto we are led when we look for a way to make a cut off: Now I would like to speculate as to where we are led to think, if we which to
get sense out of a theory, in e.g. too many dimensions, so that ultraviolet cut off is truly a necessity:
First we could think of modifying geometry or we may seek to keep it:
•	1) Cut offs like lattices which have a discretized geometry.
•	2) Keep e.g. flat geometry or at least a manifold.
In this second case where are we led, if we seek a cut off of the ultraviolet divergencies, but cling to continuous manifold or let us for simplicity say simple Minkowskian geometry (but continuous space and time) ?
If we use point particles with interactions we have no chance to get any form factors to rescue us against the ultraviolet divergencies.(we might though use higher order derivative on the fields in the Lagrangian density, but let us leave that as another possibility). So we are let in the direction, that we must take the particles, with which we want to work, to be composite objects / bound states or rather most importantly extended objects, so that interactions with the various components have the chance to cancel out couplings to very high momentum states (which is what cause the divergencies).
Thus let us at least look towards seeking cut off in direction of bound states:
Let us now think along the line, that we replace the particles, we consider phenomenologically, by bound states or composite structures. That is to say, that looking more deep inside they shall turn out to consist of some "smaller parts "partons say. It is fine that we may then get form factors, since they have the chance to cut off the loop integrals and make them converge.
Now we may talk the language of Bjorken x being the fraction of longitudinal momentum carried by a "parton.
If the partons have non-zero Bjorken x, then you get parton parton scatterings, when the bound states collide and the situation is much like, if the partons really existed and we are back to the point particle play: there will finally result divergencies again.
So if we are looking for avoiding divergencies we are driven in the direction of taken all the Bjorken x = 0. But that then in succession means that collision of only a few partons from one particle(=bound state) with partons in the colliding particle(=bound state) will hardly give any momentum transfer, hardly mean even a scattering.
Once assuming x = 0 for all the partons we will get negligible momentum transfer by just scattering a few partons with each other; that is too much cutting off. The effective way to get some significant scattering to identify with the scattering of the particles(=bound states), we want phenomenologically, is to exchanges from one bound state to another one a large number(infinitely many) partons. This means we are driven towards a picture, in which a scattering is mainly an exchange of some part of one composite particle with part of another one. But none of the constituents (=partons) truly interact. Rather the constituents individually just continue undisturbed as if not interacting at all!
Remark how we got driven towards the picture of String Theory in mine and Ninomiyas novel string field theory: The bound state, we consider should be composed from constituents not interacting at all!
These constituents or partons, we are driven towards, are of course to be identified with the "objects in Ninomiyas and mine novel SFT(= string field theory); precisely these "objects of our theory do not change at all.
So we for the moment think of "Even Objects as Partons:
Does it matter whether we consider our "Objects as constituents or the true string interpretation definition of the "Objects J from discretizing right and left
movers in the string? For this true definition of the "objects" we have to refer to the other article in the present Bled Conference proceedings 2014 on what comes beyond the standard models [15] (starting on p. 184).
Very shortly let us though on the definition of the "Objects say:
Since the "objects are defined as the difference between the values of say the right mover component of X^(ct,t) = X£(t - ct) + X£(t + ct) i.e. as J^I = Xr(tr(I + 1/2))-Xr(tr(I-1/2)) (where tr = t-ct, and we imagine a discretization replacing tr by an integer number I instead and let tr (I ± 1/2) denote the neighboring tr points around the point corresponding to I in the discretization.) at two near to each other values of the ONE relevant variable, it is in fact proportional to the derivative of the right mover component. To reconstruct the position field we both have to integrate (or sum) up and we need both left and right. On open strings boundary conditions causes the left and right mover to be the same. But for open strings they are different.
After we have identified the right with left mover "objects for the open string (as the boundary condition for open string leads to) the objects describing an open string sits topologically in a circle, called by us "a cyclically ordered chain of objects. So the topology of the structure describing the open string by us is a circle and and not as the open string itself an interval. But the momenta of the open string is written as a sum over contributions from the "objects sitting along the cyclically ordered chain (the circle). So as long as one can consider a distribution of momenta to the various "objects, we can consider the "objects constituents (for that momentum distribution purpose at least).
So we might ask: Can we forget the string and only think on Our "Objects ?
If you go over to considering the "objects of our model as constituents of the composite particle(described as the string), you ignore the string as not being the right way of thinking of the same theory.
Contrary to the string point of view, in which the string moves internally as it moves along, the "objects are stale and just do not change (Well, their position is a bit more tricky to consider, so we may think of them as free partons). The "objects fit with the constituents not interacting but just being exchanged en block from bound state to bound state. Pieces of String Time Track per Pair of "Objects with Lightlike Sides Time Track of String from Pieces per Pair of "Objects Lightlike Sides The Very Scattering Moment, Only Exchange of Pieces
Whatever the string may develop mechanically after a collision it is an almost pure exchange of parts that take place at the very collision. At least if the hit is only at ONE POINT of the hitting strings, then from locality nothing can happen at other places in the very first moment. So in the limit of infinitely many constituents (like continuum string) the first moment of a scattering ONLY an exchange of pieces can matter. So, if indeed no parton withx different from 0 is allowed in order to make a good cut off bound state theory, then when first partons hit we can ONLY have exchange of pieces interaction: So in this first moment there is in this sense no true scattering! (Like in mine and Ninomias model).
But there is a need for exchange of pieces
If we have x = 0 bound states, there would without exchange of pieces be no scattering, no essential momentum transfer at all.
Now I say: We are driven in seeking for a cut off to a theory with a system of particles (corresponding to the strings in string theory) being bound states with all partons having Bjorken x = 0, and they scatter only by exchange of pieces. So it is essentially only how one thinks the constituents as distributed between the particles, that change in the scattering.
It is well known that the higher dimensions spacetime has the more severe are the ultraviolet divergencies: High dimensions give ultraviolet divergences.
13.5.4	Rescuing the Species Doubler Problem by Pushing Chiral Charge to Central Station in Extra Dimension
In the Standard Model one has a remarkably tricky cancellation of the chiral
anomalies associated with the (chirally coupled) gauge fields. Non of the fermions in Standard model have their "species doubler (with opposite handedness, but same charge combination). So it should after mine and Ninomiyas no-go theorem be impossible to put the Standard Model on a lattice, or for that matter regularize it in gauge invariant way at all. I.e. No cut off should exist, which can keep gauge invariance. The way Norma Mankoc Borstnik and I attempted to escape this problem were the following:
The way we attempted to escape the no-go theorem was by having infinitely large extra dimensions allowing superfluous fermions to be pushed out to infinity.
Let me look at the nogo theorem problem by thinking of the anomaly telling that the chiral charge is not conserved, but has a lack of conservation correction proportional FF(with some gauge fields put in for the two F's).
13.5.5	Anomaly way of Looking at No-Go Anomaly Requires Pushing out or Fetching in Chiral Fermions
Because of the anomaly we need locally in space-time to be able to obtain extra chiral fermions in spite of them having conservation laws making that impossible in the regularized theory. In Norma Mankoc Borstniks and mine attempt to cope with Wittens no go theorem we propose to have non-compact extra dimensions: Then the superfluous or missing chiral fermions may be pushed out or be brought in from the infinitely far away in the extra dimensions. You almost bring them out to a mysterious central station for pushed out chiral fermions, from where they may reappear in the practical world later or earlier or somewhere else than from where they were pushed out.
With such central station whereto chiral particles are brought in and out to various places or times in the 3+1 dimensional world is to be imagined in the model needed (say Norma Mankoc's and mine), then one may suspect that one easily get times mixed up having such an exchange station for chiral fermions. There namely has to be somehow a control that the total number of chiral fermions of a certain type is conserved in the regularized model. But then how to get the information of the creation seemingly of one at a certain point in the 3+1 space time transfered and brought together with the uses or further creations around space time without endangering the no influence from future principle(which we attempt to attack in this article)?
If really the chiral fermions are fundamentally conserved in the regularization scheme here thought upon as the true theory but just seem not to be because they are pushed out to an in the extra dimensions infinitely far away place, it may seem difficult to keep truly no influence from future from the practical 3+1 dimensional point of view. Would one really could have the number of chiral fermions being added to the central station for such fermions pushed out be kept to net zero without some influence back from the future?
13.6 Some Potential Killings of Our Complex Action Turned Out Supporting It.
Funnily enough I have found a few cases, where seemingly arguments against the validity of the complex action model with its influence from future, actually get turned around and leads to evidence for the influence from future instead, because they turn out rather to show that nature has just some number just finetuned almost to solve the problem.
13.6.1 Short Review of Complex Action Model
Let me here review a bit the main point of the theory of the complex action. A priori it would seem obvious that if we took the action S [history] to be complex rather than as assumed in the usually believed theory, then one would immediately see that effects of non-unitarity and if one used classical calculation one would also expect that otherwise real variables would run complex. In other words at first it would look as if the idea of taking the action complex is phenomenologically so bad that any hope is out unless the imaginary part is extremely small; and so if real and imaginary were about equal in size as one would guess there seems at first to be no chance. But that is according to the calculations or estimations on which we are still working not true! Most convincingly this is seen in a Hamiltonian formalism, in which not so surprisingly a complex action would lead to a non-Hermiten Hamiltonian. In fact the main point is that as long time has past since the start, almost certainly the universe developing by the now assumed non-Hermitean Hamiltonian gets increasing probability for being in those states, which have the largest (eigen)values for the antiHermitean part (divided by i) Hi of the Hamiltonian, if we think of having split it as H = HR + iHI where then HI = 2i (H - H^). If we now have assumed - as we have to assume to avoid that the Wentzel-Dirac-Feynmann-path integral shall not be divergent due to the imaginary part of the action SI [history] going to plus infinity - that there is an upper bound on the antiHermitean part HI or almost equivalently a lower bound on the imaginary part of the action SI, then we argue that the system after long time will arrive to a superposition of states with their (eigen)value for HI close to the assumed upper bound. Once we have argued the system to be in such a state we have the suggestive approximation of HI « "upper bound" and can consider the antiHermitean part HI an approximate c-number and by a time dependent normalization we can completely remove the effect of this antiHermitean part.
This crude argument thus allows us to suppose that after all the antiHermiteamn
part Hi of the Hamiltonian is not important provided we study what happens in a universe, that is already very old compared to some fundamental scale for the theory provided we have just an upper bound on this antiHermitean part. This may not be totally convincing as written, but we have formal formulations and it is essentially correct but in order not to have troubles with the Born rule of quantum mechanics that one shall the probability for measuring a state by using the numerical square of the coefficient to a normalized states one shall a new inner product which we call | q (so that we can write < b| q a >) with the property that w.r.t. this inner product the Hamiltonian H gets normal. Normality means that the antiHermitean part commutes with the Hermitean part i.e. [HR, HI] = 0. (The Q that occurs as an index to the new inner product | q to be used instead of the original inner product | is an operator constructed from the Hamiltonian - using it diagonalization - and then we defined < a|Qb >=< a|Q|b >.)
Even though now we have argued, that one will obtain a time development as if there existed a Hermitean Hamiltonian even, when the true Hamiltonian is not Hermitean, provided one uses the modified inner product |q, there is one very interesting and important effect of the antiHermitean part HI or of the imaginary part SI [history] of the action left: These antiHermitean or imaginary parts determine the initial condition effectively seen! We saw already just above that the antiHermitean part of the Hamiltonian were important for the states into which the likelihood of finding the world got larger and larger as time went on. So effectively in a late stage of the development of the universe it becomes most likely to find that this universe is in a state with a high -i.e. close to the upper bound -value for the (eigen)value of the antiHermitean part HI. This really means that we shall look at the complex action theory as a model unifying the initial conditions with the equations of motion.
Such a unification of course is in principle very wellcome, if one can find it. In the Hamiltonian formalism with a non-Hermitean Hamiltonian one can see that unless one puts the system/world in a state that has absolutely zero component after some eigenvectors of the Hamiltonian, it will go so that as time goes on the various eigenstates in an expansion of the actual state will grow up exponentially with coefficients going as —itAt where At is the for the coefficient relevant eigenvalue of the non-Hermitean Hamiltonian H = HR + iHI. Have in mind that for non-Hermitean Hamiltonian of course the eigenvalues At are typically complex. It is of course the imaginary part of At which gives rise to the time development of the numerical value of a coefficient ct exp —tAt to some eigen vector |At > (even though these eigenvectors are not orthogonal to each other, one could still imagine using them in expansion). Exponentially soon a rather small collection of the eigenstates with the largest - in the sense of most positive - imaginary parts of their At's will soon take over. Thereby a rather specific development of the universe gets selected out and one can understand that the antiHermitean part of the Hamiltonian can have strong influence on which states one at a late stage in time is likely to find such a universe with non-Hermitean Halmiltonian. Thus it is understandable that there can be something in the statement that the theory unites initial condition theory with equation of motion theory.
Our studies have led to that one may distinguish reasonably defendable ways of extracting the information from a quantum theory with a given action - two different ways especially suggestive in the case complex action - namely 1)"with future" and 2)"without future".
13.6.2 Guessing the Standard Model Imaginary Part of the Action
At the present conditions in the Universe - but not at all applicable perhaps in the early times just after a possible big bang say less than 10-12 s say - the Standard Model seems to work perfectly except perhaps in very high energy accelerators and in cosmic radiation. So we should expect that at least the real part SR [history] of the action S [history] should be given well by the action of Standard Model. Now the very natural guess is, that you get the full complex action by just letting all the coefficients of the various terms in the Standard Model action become complex. You might even as the a priori most promising guess think, that the phases are rather random and of order unity, meaning of the order of 1000, except though, that the mass term for the Higgs particle deserves special discussion.
Let us remind about the discussion around the hierarchy or the scale problem for the usual real action Standard Model:
If you imagine a cut off at the Planck scale or some new physics at some GUT scale at almost Planck energy scale, then one has the problem that corrections to the bare Higgs mass square as written in the Lagrangian density mHbaTe in order to obtain from that the measured mass square mHren becomes typically very large, either it is divergent or by means of fixing some unified scale it becomes when renormalized to that scale anyway huge compared to the scale of measured Higgs mass square or the weak scale. So it is a well known finetunig problem how to get the weak scale be small compared to the huge scales involved in the loop calculations even if one renormalizes to some unifying scale. You might keep the corrections smaller by having supersymmetric partners - but the LHC results so far rather show the surprise that such ones are so far not found -. But whatever might be the solution to this problem of how the weak scale became so small say compared to the Planck scale and how to keep it there it might it easily becomes so that the bare mass square mHbare becomes appreciably bigger than the renor-malized one mHren numerically.In the case when some supersymmetric particles exist and makes the mass square correction only logarithmically (divergent) the size of the bare divided by renormalize will though only be "logarithmic", which means not so fantastically big after all. But if the supersymmetric partners do not exist or are very heavy then again the bare mass square will typically be much larger than the renormalized/observed Higgs mass square.
When we now want to guess the size of the imaginary part of the Higgs mass square, the suggested guess is that it should be of the same order as the real one; but now should it be as the real renormalized or as the real bare ? Most likely the loop corrections for the real and for the imaginary parts are completely different and huge, so the question becomes: Would the same mysterious fine tuning, which made the real part mHTen|R = observed/effective Higgs mass square of the renor-malized mass square for the Higgs also function for the imaginary part, so that in
some way - which we may or may not understand - the effective/renormalized (whatever that might exactly mean) imaginary part of the Higgs mass square mHrenli becomes as small as the real renormalized part order of magnitudewise?
Very likely the solution to the finetuning problem (= the scale problem) of why the weak scale is so low compared to the Planck scale say will be solved in a way that will not make also the "renormalized" scale for the imaginary part of the "Higgs mass square" small compared to say the Planck scale. For instance this is the case for our own "solution" to this problem by means of the multiple point principle: This "solution" means, that, if we make the very strong assumption that there is some finetuning fixing the parameters/coupling constants of the theory working in nature in a way restricted so that there becomes several different vacua all having very small energy densities(=dark energies = cosmological constants) (for purposes of the weak scale we just say exactly zero energy densities are assumed in the vacua) we how found a viable picture with strongly bound states of 6 top + 6 anti-top quarks and a set of three different vacua in the Standard Model, in which this requirement leads to an exponentially small value of the weak scale compared the scale of the Higgs field in one of the vacua considered degenerate. In other words with our assumption of vacua with zero energy density (called "multiple point principle" (=MPP)) and some in principle calculable speculation about bound states of quarks and anti quarks the parameters of the standard model need to to take such values that the renormalized Higgs mass square must be very small compared to the scale for the Higgs field in one of the by us assumed vacua. We then add as an extra assumption to our multiple point principle that for one of the vacua the Higgs field present should be of the order of the Planck energy. This latter assumption is already supported by the parameters of the Standard Model if one assumes this Standard Model to be valid up to so high energies (or Higgs fields). It found a support together with the multiple point principle by the Higgs mass found in Nature agreeing with our PREdiction.
But really in our complex action model physics coming out of the real and of the imaginary part of the action are quite different, crudely the real part gives equation of motion and the imaginary the initial conditions, so to expect that some mysterious mechanism make the same finetunin on both is not at all likely. Therefore we shall conclude that it is most likely that there is no finetuning going on to make the effectively observed/"renormalized" imaginary part of the Higgs mass square small compared to say the Planck scale value. If so, then we should expect it to be probably of the order of the Planck scale. Putting into Standard Model extended to have complex action this size of the Higgs mass square imaginary part would mean that considering a process of daily life or of LHC the Higgs mass square term would give contribution to the imaginary part of the action, which are larger than the contributions from the other terms by a factor Mpt/( TeV2) « 1034. This means that we from dimensional arguments think we could argue that the most important term in the imaginary part of the action should be the part from the Higgs mass (square) term.
With this we argued that we under present conditions can approximate the imaginary part of the action Si [history] by only the contribution from the Higgs-
mass-square term
Si [history] « m-Hbareli^HM^Vg^x
2
(13.1)
(the yg is just 4-volume measure inserted to make the formula o.k. in the general relativity case, but really you may use flat space approximation and ignore it). The Higgs field were denoted $H(x) and depends of course on the event coordinate (set)x = (x^j. The integral is, provided we use the "with future"-interpretation of the complex action theory, to be integrated over all space time including both future and past, and then it is this quantity (13.1) which at least in first approximation selects initial conditions or what really happens by letting the true happening history have the minimal value for the imaginary part of the action Si [history] among all the say by equations of motion allowed possible histories. For a crude understanding of our complex action theory one may take it that it predicts roughly that
(more detailed calculations of some predictions may be found in [9-11] and in some of the papers with Ninomiya [3]).
One way of putting forward the idea of the universe initial conditions being arranged in a way governed so as to achieve say small (or preferably numerically large negative) contributions to Si [history] is to call it a "God" (it is only a god in quotes(thanks to Mette H0st)) governing the world so as to seek to minimize the imaginary part of the action SI[history]. In this language our expression (13.1) means that this "God" only cares for the integral over space time of the Higgs field; "He"to day mainly care for Higgs particles and modifications in the Higgs field. Oscillations in the Higgs field meaning physical Higgs particles will obviously make the square of the Higgs field integrated over all space time bigger. So producing Higgses should e.g. be hated and avoided by the "God. (Had the sign been so that it corresponded to "God" loving Higgs bosons instead "He" would have filled more up with Higgs bosons, say an expectation value of the Planck order of magnitude at least).
But if "He hates the Higgs "He" should love the particles suppressing in there neighborhood the Higgs field? And fill the whole Universe with the most favoured ones.
It is for instance the quarks and the charged leptons that are surrounded by a Yukawa potential region in which the Higgs field has an additional Higgs field - the Yukawa potential -, and so a more strong field the bigger the mass or the lepton causing this field. One may easily understand that the Higgs field having in vacuum its well known expectation value < ^h(x) >= 246 GeV is a bit diminished numerically in the Yukawa-potential-region around a quark or a (charged) lepton. Now in principle we do not know whether the square of the Higgs field |^H (x)|2 increases or decreases as one enforce a little region in space(-time) to have a given Higgs field diminished say w.r.t. the usual vacuum Higgs field. Intuitively one would think the square would decrease when the Higgs field itself decreases but there could - and indeed there are - be effects causing it to go oppositely(as have argued for below and in the articles[12]). In any case unless
SI [true history] = SI [any other history].
(13.2)
there is just an extremum of the square < |^H (x)|2 > as a function of the Higgs field itself < $H(x) > in the usual vacuum situation there would be an effect positive or negative upon the imaginary action SI [history] as given by (13.1) from the Yukawa-potential regions around the quarks or (charged)leptons, because the normal Higgs fields a bit suppressed in such Yukawa field neighborhoods.
This means that e.g. the "God" would either love or hate these quarks and charged leptons, and that the more strongly the heavier they and the stronger they therefore couple to the Higgsfield.
This in turn means that e.g. a particle like the neutron with its three valence quarks and further quark pairs inside it will suppress the Higgs field from its usual vacuum value a bit and then depending on the sign of the derivative	>
increase or decrease the imaginary part of the action SI [history], thus the neutron would be respectively hated or loved by "God".
Now in nature one can by weak interactions get a neutron transformed into a proton, an electron and an electron-anti-neutrino. Thus if the "God" loved say the neutron itself more than the proton the electron and the electron-anti-neutrino together we would expect that "He" would have arranged initial conditions - and if "He" were allowed to it also that coupling constants or whatever could help - so as to make there be only neutrons but no protons and electrons etc. We know from astronomy and our own earth neighborhood that there exist both neutrons and protons and electrons (and even neutrinos) in rather large amounts, none of them being truly so much suppressed compared to the other.
At first we may look at this fact there there are both neutrons and protons in the world today as a falsification of the minimization of imaginary part of action ideas!
It becomes in our complex action theory an embarrassing question: Why not only n or only p+e+antineutrino ?
An idea to an attempt to disprove our complex action model with the Higgs
field square integrated as the imaginary part of the action: Why do we not have either?:
1)	Only neutrons n and no protons nor electrons, or
2)	Only protons with their electrons e and antineutrinos, but no neutrons at all.
Either one or the other would probably be favoured and thus by "God be arranged to be realized!
13.6.3 Solution to: Why both protons and neutrons?
Actually this problem of why not only protons( with their electrons) or only neutrons in the world in our complex action model has the "solution":
If the neutron is exactly equally much "loved" as the the proton the electron and the electron-anti-neutrino together -in the sense of contributing the same to the imaginary part of action Si [history], then there would be no reason for "God" to eradicate one of the two types of particles. But this requires a certain relation between the masses of the quarks corrected by their Lorentz contraction factors and the electron mass. But remarkably this
relation is satisfied within calculational accuracy! (light quark masses are rather badly known so the accuracy is not so high)
Basically[12] in order that there shall be no reason to either remove from the world the neutrons nor the combinations of protons and electrons (the neutrinos anyhow contribute much less to the imaginary part than the massive quarks or leptons) we should get just same imaginary part of action contribution from a neutron and from such a combination of proton and electron. In an short time the contribution is estimated as an integral over space of the Higgs field suppression. We here just assume by Taylor expansion in the presumably rather small Higgs field around the quarks and leptons, that any effect will in first approximation be linear in the change in the Higgs field. Now we find small Yukawa-potential regions of size given by the inverse Higgs mass and centered around quark or lepton. A crucial little problem making the estimation a bit less trivial and bit less accurate is, that these regions of significant Yukawa-potentials are Lorentz contracted, because of the non-zero velocity of say the quark it surrounds. (The
electrons most copiously found in our universe have actually very small velocities compared to the light velocity, so for them Lorentz contraction is not important.)
The following the reader should have in mind in order to estimate the contribution to the imaginary part of the action SI [history] under the assumption of the dominant Higgs mass term for a neutron relative a pair of proton and an electron:
•	a Of course - unless a linear term should be lacking - the contribution must go linearly with the Yukawa coupling for the quark or lepton in question. Really the suppression of the Higgs field around a particle - quark or lepton say -must go proportionally to the Higgs Yukawa coupling ( for fixed velocity)
•	b But it will vary with velocity due to the Lorentz contraction of the Higgs-Yukawa effective extension volume, around the particle.
•	c So at the end the effect on the imaginary action SI [history] becomes proportional to
2 _1
m m2 < y >< y 1 > AS I [history] <x gparticie * ""¡Haveraged < -^- (13.3)
E	Eaverage
where m is the mass of the quark say (or lepton) and E its actual kinetic energy including the Einstein energy. The average as the quark flies around in the nucleon say is denoted of its y = E/m is denoted < y >, while the average of the inverse of this same y is denoted < y_1 >. The average kinetic including Einstein energy E is denoted Eaverage. The combination < y >< Y_1 > would in the case of no fluctuations of the actual velocity of the quark be just unity, and thus we may hope that we can estimate this product somewhat more accurately than say its two factors separately.
The various types of quarks have of course the deeper Higgs fields around them the stronger their Higgs Yukawa couplings gparticle. The Higgs field is effectively extended over a range of size given by the Higgs mass but not dependent on the species of quark or lepton in question. The extend of the Yukawa potential rather is over an elliptic region, that is the Lorentz contraction of the spherical Yukawa potential, which is obtained around a resting particle. So the contribution
to the integral of the Higgs field or presumably also over its square over all space from a quark or lepton is proportional to gparticle and to the inverse of E/m where E is the energy and m the mass of the quark or lepton. The Lorentz contraction factor is for Yukawa potentials for quarks due to motion inside nucleons, if we have - as is most copiously the case - resting nucleons. Well, really the speed of the nucleons inside the nuclei is not so negligible again but compared to the speed of quarks inside nucleons it is small.
Does it Pay for "God to make Only Neutrons or No neutrons ? The bigger integrated Yukawa potentials around the quarks and leptons the more the Higgs field is suppressed. The strength of the suppressions is proportional to the Yukawa coupling for particle making the suppression. The extension is roughly like the Lorentz contracted of a sphere forming an ellipsoid given by the Higgs mass(as inverse radius of the sphere).
The proton is almost identical to the neutron except, that one up-quark has been replaced one down-quark.
To keep Universe chargeless a proton should be accompanied by an electron. A neutrino typically runs so fast that its Yukawa potential is much less extended in volume than those of quarks and charged leptons.
13.6.4 Contributions to See Whether Neutrons or Non-neutrons Favored My Prediction from Future Influence
To estimate the contributions coming from a neutron to compared it to that coming from what is its decay products a proton and an electron and even a not so significant electron anti neutrino we need the light quark masses which are not so well determined (and that makes our uncertainty rather large), but let us take
(13.4)
(13.5)
for respectively the up and the down quark masses. One arrives as also sketched here to the relation
jmd - mU = yEqme/"ln"	(13.6)
where we have denoted
"In" =< y >< Y-1 >	(13.7)
because this quantity for light quarks compared to the energy Eaverage tends to be approximately a logarithm. The relation(13.6) is relatively well satisfied, if we take the quark masses (13.5), Eq « 160 MeV and "In" = 2.37.(see my previous article for this crude estimate) In fact then we would get (using me = 0.511 MeV)
R.H.S. = yEqme/'Tn" = 3.81 MeV	(13.8)
L.H.S. = - mU = /T39toV22.75 MeV	(13.9)
= 3.73to4.77 MeV.	(13.10)
mu = 1.7to3.3MeV md = 4.1to5.8MeV
13.7	Fine Tuning Calls for Influence Going Back in Time
One argument, which Don Bennett and myself would give for some influence from the future being called for, is this:
We know the fine tuning problem of why the cosmological constant/dark energy /energy density in the vacuum is so small compared to the energy density given by the most fundamental constants G, c, and h, i.e. the Planck energy density? The ratio of the actual vacuum energy density to the from the dimensional arguments expected value is enormously small. So it is clear that there must have been some enormous fine tuning arranging this enormously small energy density in the vacuum. Now we expect that the vacuum energy density should be constant as time has gone on. So even in a time of say minutes after the start of the universe or Big bang or whatever the vacuum energy should have had the present extremely small value. But now at these early times there were so big energy densities of radiation or matter that the present small vacuum energy density would be very small and insignificant compared to radiation energy density. But when it were at that time so insignificant, how could at that time any physical effect have made a so precisely close to zero as the vacuum energy density to day? So it seems that an influence from the future somehow must have arranged at this early stage already the exceedingly small energy density in vacuum? It is of course because of an argument in the direction of this that is the reason for that, when Weinberg looks through the various explanations for the cosmological constant being so small, then the most promising explanation is to use anthropic principle. The entropic principle, which states that parameters shall be so arranged that humans can come to exist, is namely in reality a method to to arrange a simulated effect of the future influencing the past. By throwing away the scenarios which happen not to allow for humans one has got what functions as a back in time effect.
13.8	Our Multiple Point Principle
There is one very general deduction from such a theory with a principle of minimizing some quantity as we above told that the imaginary part SI [history] would be minimized for the actually realized history. This deduction would be best achieved if we instead of minimizing over histories of the universe minimized over combinations/sets of coupling constants, but since one could imagine some vacuum being selected among several at least the effective coupling constants relevant for the by a quantity like Si [history] selected vacuum would effectively have been determined as if they were adjusted to minimize something (SI) by adjusting the coupling constant combination. The deduction related to is found an article by Ninomiya and myself [18] in the Bled proceedings from 2011. The point is, however, to imagine that the right combination of coupling constants is achieved by asking to obtain the minimum for some quantity - in fact our SI, which we now imagine to depend also on the coupling constants (with an effective vacuum providing such couplings this imagination would be true in our model) - under the restriction that the energy density of the various (local) ground states the vacua should be positive. This assumption of vacuum energy density being positive may
be understandable in our model - as well as phenomenologically supported as a principle - by noting, that if a vacuum gets (appreciably) less energy density than zero, then the usual vacuum becomes unstable against making a transition to this low energy density vacuum. From the point of view of the history being selected such an instability would mean that it would be this vacuum rather than the usual one that got realized and the potential history meant as a history in the "usual" vacuum would no longer be realized; so if this latter history gave a smallest SI that would be a lost achievement if another vacuum tales over. So one should avoid competing vacuum threatening the stability severely for the realized one, or one should presumably preferably think that there are several vacua getting their realization in a turn adjusted to be the most beneficial for the SI being as negative as possible. Also in such a scenario of several vacua coming to exist as time passes on, the transition from to the next should not be too quick, they are to exist for of the order of 13 milliard years. Thus they should be approximately stable and we would obtain an approximate multiple point principle in such a scenario. In any case we already earlier argued that once you have the minimization of something like our SI that just can manage some way to effectively depend on the coupling constants, then the couplings get very likely adjusted to lead to several degenerate vacua, meaning multiple point principle.
Having in mind that this multiple point principle is thus to be considered a deduction from a minimization of some quantity model including future in such a way that it really means influence from the future, we can now look at successes of our multiple point point principle (MPP) as also being evidence for there existing in the laws of nature some influence from the future.
Now I remind the reader that the most impressive confirmation of our multiple point principle were that we - Colin D. Froggatt and myself - PREdicted the Higgs mass[19] many years before the Higgs boson were found to 135 GeV ± 10 GeV ! With the present calculations and top-mass measured our prediction would rather have been 129.4 GeV with an uncertainty now rather down to about ± 1 GeV. So although our prediction is now only 3.4 GeV above the experimental Higgs mass 126 GeV, the deviation compared to the uncertainty may have gone slightly up compared to the old day PREdiction, but we should still consider it a great success for the multiple point principle that the Higgs mass is so close to our prediction!
Historically we - Don Bennett and myself and also in some papers with Colin D. Froggatt - we looked for some way of justifying to fit fine structure constants by phase transition couplings in lattice Yang Mills theories. We worked at that time with what we call Anti-GUT (meaning anti-grand-unification) meaning that we rather than as were most popular to look for simple groups like SU(5) or SO(10) etc. we did not unify in the sense that we used the not at all simple group S(U(2) x U(3)) x • • • x S(U(2) x U(3)) (with Ngen cross product factors), rather meaning that we gave every family of fermions in the Standard Model its own family of also gauge bosons, so that our "anti grand unifying group" were the cross product of one Standard Model gauge group, one for each family (the number Ngen of families not yet known at that time; we had to fit it to the fine structure constants and PREdict it; luckily we PREdicted Ngen = 3). But the problem for
which we needed the multiple point principle were to give an explanation or at least formulate a principle that could imply that the phase transition finestructure constant values were the ones for which Nature should care. But if we somehow had derived that Nature should have a couple (or more) energy density wise degenerate vacua/phases of course if nature really were a lattice Yang Mills theory, then it would mean that Nature should choose the phase transition value of the coupling constant/the finestructure constant.
Once we have suggested to believe in such a multiple point principle in the form of there being many/several energy-wise degenerate vacua, you just have to find one with an appropriate small cosmological constant and you can so to speak transfer that small energy density to other vacua, thus explaining the smallness and even fit the cosmological constant (or the dark energy). Roman Nevzorov, Froggatt and me did such an application in several versions, explaining the cosmological constant[20].
We even managed to make a solution of the scale problem (related to the hierarchy problem) in the sense of using the postulation of the multiple point principle to fix the scale of the weak interactions (compared to the Planck scale, taken as the fundamental scale). This we -Colin D. Froggatt, Larisa Laperashvili and myself - did by speculating up the existence of a further vacuum in which there is a Boson condensate of bound states of 6 top and 6 anti-top quarks. In the spirit of the multiple point principle postulating a further vacuum is somewhat natural, and at least each time we postulate a new vacuum, we get the information out of multiple point principle that this vacuum shall have the same energy density as the other vacua. Thus for each new vacuum we postulate - and take to be degenerate with the other ones - we get one more of the say Standard Model (if that is what we use) determined, because one more relation among them is obtained. Luckily it turns out that we essentially may use this new information to fix the weak energy scale and most importantly:
We get the weak scale out as restriction on between which values of the running top-Yukawa coupling gt(M-) shall be taken on at 1) the high field scale of the second Higgs field effective potential minimum (assumed by us to be essentially the Planck scale) and 2) the weak scale.
Since then the running top Yukawa coupling must "run" between the two predicted values gt (^ = 1818 GeV) and gt (= "weak scale") = 1.02, the ratio of the weak to the supposed more fundamental scale gets predicted to be "exponentially" small! Really the point is that with the rather weak couplings of the Standard Model the 'running" is actually a bit slow as a function of the logarithm of |x. Thus to get a given distance of change in the Yukawa coupling an exponentially big ratio of scales is needed. Actually our prediction of the logarithm of the scale ratio, the scale problem gets very well!
So our multiple point principle is here a great success: both explaining the exponential smallness and giving a good value for its logarithm.
13.9 Do we have Enough Evidence for Influence from Future?
I would like towards the end very optimistically for the hypothesis of there being indeed an influence form the future to give - the relatively optimistic, but still crudely true - numbers for how unlikely it would be that our small coincidences favouring the complex action model with the assumption that the Higgs field square dominates without such a model being true.
Say we look at the coincidence that the "knee" in cosmic radiation spectrum just order of magnitude wise happens to coincide with the threshold for Higgs production. If we say one has studied cosmic rays from some electron volts up to say 1020 electron volts, we could say over 19 orders of magnitude. Then if one finds a knee to coincide within one or two orders of magnitude, it represents a coincidence that should happen by accident only in about 1/10 cases. Similarly looking at the agreement of our formula (13.6) as being that we get inside the right interval of length one MeV for quantities - sides of the equation - being of order of 4 MeV, this is something that should only happen in one out of four cases.
Our argument that the Higgs-field vacuum expectation value should just have gotten that value, that minimizes the squared Higgs field expectation value - we get agreement up to some factor of one or two orders of magnitude - means that our minimization principle led to the right order of magnitude for the weak/Higgs field scale to say a couple of orders of magnitude out of 17 orders of magnitude (taking the Planck scale as the fundamental one). this means again that our influence from future got the right scale among say 17/2 « 10.
We may even count here the smallness of the binding energy in nuclei compared to the separately bigger kinetic and potential energy of the nucleons, say one out of 2 cases accident.
These "numerical" coincidences together would give us a one out of 800 coincidence, which is a factor 4 more than 3 standard deviations. Taking this optimistic estimate seriously we really have more than 3 standard deviation evidence for the influence from future seeking to minimize the Higgs field square (integrated over space time), so as to use it to tune some couplings or the like.
Further to support this complex action with Higgs mass (square) term dominating model for the development of the world being supported we should collect also the evidence coming from the very bad lick of the S.S.C. machine, that would if it had worked according to plans have produced more Higgs bosons than L.H.C. has so far, and the - for our model though too little - bad luck of an explosion in the tunnel, which though were repaired and mainly so far had the effect of making the physicists choose to postpone the running of the L.H.C. with its planned beam energy of 7 TeV against 7 TeV (meaning y/s = 14 TeV) till 2015. Although it now looks that finally it will come to run, we may though consider it, that this caused postponing of the full energy could be a result of our complex action model with Higgs mass term dominance. Together we might consider these after all not so terribly miraculous bad lucks for Higgs producing machines as something that would not be at least the very first expectation without theory predicting it like ours. So we might say e.g. that in at most one out of say 5 cases would so much bad luck hit the Higgs producing machines.
If we combine this estimate with the just counted, we would say that now the Higgs mass square term dominated complex action model has scored a success corresponding to one out of 800 *5 = 4000 cases!
If we add to this counting the evidence coming from say the Higgs mass being PREdicted from our multiple point principle, which also would follow from an influence from the future type theory, and take it that the range for Higgs mass were at first up to 600 GeV or just use the actually Higgs mass to set the scale for Higgs masses the deviation 129.4 — 126 GeV = 3.4 GeV (relative to respectively 600GeV or 126 GeV) means a luck for our multiple point principle as one out of « 200 or one out of « 36 respectively.
If we already have counted the luck of our theory of getting the right weak scale it might no longer be new prediction to use the multiple point principle to predict the top -Yukawa coupling to be 1.02 ± 14% (oterwise this result should give a one out of 7 good luck for our model).
Also it would probably be too much to seek to include as a result of our influence the very remarkable smallness of the cosmological constant because this influence from future type theory in itself does not predict this smallness, although firstly it is very hard to see how such a small cosmological constant could come without an influence from the future and secondly we have works with Roman Nevzorov et al. [20] in which we actually even fit the cosmological well using the multiple point principle(which indeed is consequence of an influence from the future much like the one we discuss here. If we include this cosmological constant as were it prediction it would increase much our measure of the success since even counted only as a success on the logarithmic we could a priori have expect a "Planck energy density value" about 100 orders of magnitude larger. Counting with natural logarithm say we should then say we succeeded as one ot of 100*2.3 = 230.
But even as presumably most fair leaving out the cosmological constant proper as being a success for our model(s), but only taking in the Higgs mass PREdiction from multiple point principle (after replacing the one prediction of MPP by the Higgs or weak scale gotten by adjusting this scale to minimize the squared Higgs field integrated over space and time) we get that good luck for our model is of the order of getting one out of 4000 * 7 « 30000 cases/possibilities correct!
This of course were optimistically counted, but it sounds that one should take possibility of there being effects from the future. especially we did not even in this number include anything from the arguments related to the need for ultraviolet cut off, which especially for gravity may be very hard without a bit of non-locality, thereby allowing the influence from the future sneak in in principle.
13.10 Conclusion and Outlook
We have in the present article looked at a series of arguments for that there should be in the laws of nature some law that makes e.g. the initial conditions or the coupling constants or both be adjusted as if it were with a special purpose (such
as as here suggested to make a certain quantity depending on the history "the imaginary part of the action be minimal).
The main classes of arguments, which I suggested are:
•	Numerical or observational successes of assumptions involving such an influence from the future. This includes:
-	The bad luck of SSC, and if we take it seriously the very minute bad luck of the LHC, both machines (potentially) producing relative to human history exceptionally many Higgs bosons.
-	Our relation relating the light quark mass square difference to the electron mass square and the fraction of energy carried by the quarks in the nucleons. This relation just organizes that the contribution from a neutron and from an electron and a proton (and an electron anti neutrino) together to this imaginary action is same. Thus when this relation - which seems to be fulfilled within errors in nature - happens to be fulfilled there would be no gain in minimizing the imaginary part of the action by neither arranging for more neutrons than for more of its decay products electron + proton (+ anti electron neutrino). The world would potentially be able to exist at a minimum for the imaginary part of the action.
-	analogously I argued that including the effects of virtual top quarks in the vacuum it could within errors be so that the Higgs field square is in fact at a minimum with just the present Higgs expectation value in the vacuum. So indeed the parameters of the Standard Model could have been arranged just so as to minimize the Higgs field square, and that could have led just to the from hierarchy problem consideration rather difficult to accept compared to the Planck scale or Grand Unification scale point of view exceptionally small value Higgs field expectation value.
-	Even the "knee" in the cosmic ray spectrum is so close to the threshold for the severe production of Higgs bosons that we can claim that it is as if it had been arranged to be just like that to make the production of Higgs bosons by the cosmic rays hiding material or planets etc. in the galaxies so small as possible under some restrictions.
The "God" did not quite switch off the cosmic rays above the effective Higgs production threshold, but the "knee" looks like a weak attempt to
do so.
•	We called attention to that cut off methods which are needed to make especially renormalizable gravity theories are very hard if at all possible to conceive of without some non-locality. And then since non-locality really means that influence from future is getting allowed for small distances, also such cut off needs in fact calls strongly for that influence from future cannot be totally avoided. We looked especially as an example on string theory in the recent formulation of Ninomiya and myself. In this model the for the cut off effectivity crucial feature - the "stringiness" one could say - is put in as an initial state -and even as a final state - condition! If instead of or in addition to inclusion of gravity you also want to have more than the experimental number of dimensions 3+1, the need for such cut offs that in turn leads to non-locality and thereby formally admits influence from future gets even stronger.
•	We also mentioned the old worries about that the usually assumed laws of nature for the initial conditions and those for the equations of motion do not have the same CPT ot say just time reversal invariance: The initial conditions usually assumed are only for the initial state, but not for the final state also as a time reversal invariant theory would have to have it. So again some influence from future is called for in order to make the symmetry be a least formally uphold.
•	Although I did not go so deeply into it in the present article, it is of course also one of the arguments for influence future coming into the physical theory that one in general relativity has wormholes and baby universes. very easily leading to time machines. Such time machines namely leads to inconsistencies unless the happenings are finetuned to just make things go in a with the time machine consistent manner. This has been discussed by Novikov.
We will at the end stress that with the lists of arguments in the present article one should at least admit that the absolutely safe belief that there is no influence from the future deserves being investigated and confronted with as much knowledge as we can collect concerning this question. If one truly will uphold this absolutely safe belief that nothing from the future can influence us in any way, there is really no government of the universe - at least no government with any interest in the future fundamentally - then one would have to throw away as bad science/misunderstandings or pure (poetic) invention all stories about the government of God or destinies or the like which may be found in mythology in the holy texts or the like.
At least I hope to have put a little doubt on the validity of this by now in first approximation well working law of nature that future cannot influence anything in past or now and that there is no government of the universe whatsoever.
Instead one could look at it that the strong belief in this no influence from future nor government arranging for the future will turn out to be only something humanity believed in a relatively short historical era from Darwin Wallace Lamark to some day may be next year when a truly bad luck for LHC e.g. would convince humanity that there exists a "God" (here in quotation marks) that hate the Higgs sufficiently to stop a Higgs producing machine before it gets produced too many Higgses!
Acknowledgements
One of us (HBN) thanks the Niels Bohr Institute for allowing him to stay as emeritus, and Matjaz Breskvar for economic support to visit the Bled Conference.
References
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in the Presence of Non-Locality in Physical Theories), talk given by D. L. Bennett at the meeting of the Cross-displiary Initiative at Niels Bohr Institute on September 8, 1995. QLRC-95-2. D. L. Bennett, "Multiple point criticality, nonlocality and fine tuning in fundamental physics: Predictions for gauge coupling constants gives alpha**(-1) = 136.8 +- 9," arXiv:hep-ph/9607341. (CITATION = HEP-PH/9607341;) H. B. Nielsen and C. Froggatt, "Influence from the future," arXiv:hep-ph/9607375. (CITATION = HEP-PH/9607375;)
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Bled Workshops in Physics Vol. 15, No. 2
A
Proceedings to the 17th Workshop What Comes Beyond ... (p. 267) Bled, Slovenia, July 20-28, 2014

14 Deriving Diffeomorphism Symmetry
H.B. Nielsena'* and A. Kleppeb'**
aThe Niels Bohr Institute, Copenhagen, Denmark bSACT, Oslo, Norway
Abstract. In an earlier article, we have "derived" space, as a part of the Random Dynamics project. In order to get locality we need to obtain reparametrization symmetry, or equivalently, diffeomorphism symmetry.
There we sketched a procedure for how to get locality by first obtaining reparametrization symmetry, or equivalently, diffeomorphism symmetry. This is the object of the present
article.
Povzetek. V enem od prejsnjih člankih sta avtorja v okviru projekta Naključne Dinamike "izpeljala" pojavnost prostora. Za izpeljavo lokalnosti postora je potrebno vkljuciti reparame-trizacijsko simetrijo, to je simetrijo na difeomorfizme. V tem prispevku nakazeta avtorja izpeljavo postopka, kako do lokalnosti prostora iz reparametrizacijske simetrije.
14.1 Introduction
In an earlier article [1], we have "derived" space, as a part of the Random Dynamics project [2]. Since we want to have locality, we also need to derive reparametrization symmetry, or more generally, diffeomorphism symmetry [3], essentially ensuring that the choice of coordinates plays no role in the formulation of the physical laws.
We propose that diffeomorphism symmetry comes about as a result of a selection principle, in reality a selection principle for how Nature "chooses" its symmetry groups, a scheme that has been developed by Holger Bech Nielsen and his collaborators [4]. The initial idea was that the small representations of the Standard Model gauge group
is a signature of such a selection principle, singling out groups that have the "smallest" representations.
In the present article we use similar arguments, but instead of taking the
Standard Model group SMG = S(U(2) x U(3)) as the selected group, we consider the combined diffemorphism-and-gauge group
SMG = S(U(2) x U(3))
(14.1)
B = {(A, 9) |A	gD}
(14.2)
* E-mail: hbech@nbi.dk
** E-mail: astri.snofrix@org
where G is the group of all gauge transformations that map the four-dimensional spacetime manifold M on the 12-dimensional manifold of SMG: the Lie group is a manifold,
A : M -> SMG
and D is the group of diffeomorphisms, a diffeomorphism 9 given by a bijective differentiable map
9 : M —» M
14.1.1	An alternative to grand unified models
A major part of the success of the GUT SU(5) model is that the representations of the SU(5) gauge group automatically represent the SU(5) subgroup S(U(2) ( U(3)) with the Standard Model Lie algebra. The GUT SU(5) group thus presents the needed restrictions on the allowed represenstations of the Standard Model algebra. Any successful GUT group, like for example SO(10), reproduces the same restrictions as SU(5) on the representations of the Standard Model Lie algebra, restrictions corresponding to S(U(2) ( U(3)).
Any viable alternative to the GUT scheme must thus supply a prediction not only of the Standard Model Lie algebra, but also of the group structure. There are however many possible scenarios, so unless one has some guiding principle for selection the unificaton group, there isn't much predictive power.
One way to get the Standard Model without a GUT scheme, is by using some selection principle for how Nature selects the Standard Model group. The underlying philosophy is that of Random Dynamics, namely that the fundamental physics is random, and that the observed symmetries are emergent. If only some symmetries emerge, supposedly by accident, but maybe even by some more precise mechanism, then the initially random action could be considered as taking random values for some small region of the value space of the representation of the group, with the transformation properties of the fields or degrees of freedom under the group. The elements of a representation of the group in question then move quite slowly as the group elements themselves vary. (One can vary the group elements much before one varies the fields or matrices of the representation). The slower the representation moves as a function of the variation of the group elements the more likely it is that a symmetry emerges, since displacements inside the group itself corresponding to a small region (over which we assume essentially constancy of the action) become bigger with a slower representation motion rate. A symmetry of the random action is thus more likely to occur when the symmetry is represented by "slowly moving" representation elements (e.g. matrices).
By means of some "goal quantities" we single out the groups that have the largest chance to emerge from a random action model, favouring the experimental gauge group and dimension of spacetime.
14.1.2	Groups and algebras
In Yang Mills theories, only the Lie algebra is important, since two groups G1 and G2 that have the same Lie algebra also have the same Yang Mills system. There
are however many Lie groups with the same algebra. These groups are locally similar, but globally they can be very different, with different representations. For example, SU(2) = SO(3), as for SU(2) we have j = 0,1/2,1,3/2,..., while for SO(3) j = 0,1,2,..., and it is only by studying the representations of Nature like qL, qR, the Higgs and so on, that we can establish which groups are at stake. To a group corresponds
•	The Lie algebra and thus the structure constants fkm and the Yang-Mills
Lagrangian LyM.
•	The system of allowed representations, a given set of representations only being allowed by some Lie groups.
The covering group (for a given Lie algebra) can manage all the representations, so the goal is to find the most choosy group, i.e. the one that allows the fewest representations - which also corresponds to experimental data. Our point of deparure is the Standard Model Lie algebra
S(U(2) x U(3)) - R x SU(2) x SU(3) - U(1) x SU(2) x SU(3)
(14.3)
and since we don't find all its possible representations in Naure, we will concentrate on the Lie group rather than on the Lie algebra. It is so to speak stronger to "predict" the group S(U(2) x U(3)), such that
det
( . .
0 0 0 0 00
0 0
/
= 1,
a group which admits all phenomenological representations, which all obey the rule
y + j3 + 1 "triality" = 0(mod1)
(14.4)
14.2 Skewness
Small representations as one possible selection principle, but another way of singling out Nature's chosen group, is by studying group skewness [5], defined as a lack of symmetry.
Nature seems to select the Lorentz group with the smallest representations; perhaps space moreover prefers those dimensions that give the skewest Lorentz group. The Standard Model group SMG = S(U(2) x U(3)) is very skew, and most probably very "complicated".
There is always the worry that the choice of "goal property" is such that it gets dramatically bigger or smaller with the dimension or some other size parameter of the group. In the case of choosing a skewness measure, this can be dealt with by defining it as
ln(number of outer automorphisms) rank of the group
14.2.1 Inner and outer automorphisms
The degree of skewness is thus a function of the number of outer automorphisms of the group G. An automorphism is an isomorphism of the group onto itself,
p : G -> G	(14.5)
i.e. a correspondance ^ of G with itself respecting the group multiplication, and such that ^ is bijective and $(gh) = ^(g)^(h), g,h G G.
The map p (g) is an inner automorphism if there is an element h G G, such that for all g G G,
p(g) = ph(g) = hgh-1	(14.6)
The group of outer automorphisms O is then defined modulo the inner automorphisms in the sense that in the group of all automorphisms A, we discern the subgroup of inner automorphisms,
Ainn = {phlh gG},	(14.7)
and then define the group of outer automorphisms as
Aout = {O/{ph|h G G}	(14.8)
For the Standard Model group, we have that
•	The automorphisms of R (~ the U(1) factor) are scalings with a factor k = 0.
•	The SU(2) factor has complex conjugation (in the defining representation) as an automorphism, it is however an inner automorphism.
•	For the SU(3), as for all SU(N) algebras with N > 1, complex conjugation is an outer automorphism.
All outer automorphisms of the Standard Model algebra are combinations of these, since an automorphism maps the three invariant subalgebras into three isomorphic invariant subalgebras. There are infinitely many such automorphisms, but the Standard Model algebra together with the set of Standard Model properties (the rule system) is invariant under only one outer automorphism, namely complex conjugation of the SU(3) combined with the U(1) scaling factor k = -1.
Among all algebras of dimensionality up to 12 dimensions, taking quantization rule systems into account, there are four combinations of algebras and rule systems that have no generalized outer automorphisms, namely those with semisimple algebras su(3) and so(3).
14.3 The size of a representation
The other suggested selection principle, namely the size of a representation, was inspired by the fact that after the trivial representation, the lowest-dimensional non-abelian representations in the Standard Model are the remarkably small representations of SU(2) and SU(3).
A probablility argument for the presence of a selection principle can be formulated as follows: look at S(U(2) x U(3)) and count the Lie groups of similarly low rank [6]. It turns out that there are about 28 = 256 groups with low dimension (up to 12, i.e. the dimension of the SMG). Among these about 256 groups, S(U(2) x U(3)) is singled out - most probably by means of some selection principle like the size of the representations.
In order to obtain a more precise formulation of the selection principle, we need to establish what we mean by the "size" of a representation. For this purpose we define a measure for this size in terms of the quadratic Casimir operators, which 'tag' the representations in the sense that they are not defined for the algebra itself, but only for the representations.
A general Casimir invariant is a function f(F) of the Lie group generators Fj which is invariant under the group and commutes with all the generators,
[f(F),Fj]= 0.
The generators Fj of the group constitute a basis for the corresponding Lie algebra
and satisfy the commutation relation [Ft, Fk] = fjkFj, i, k, j = 1,2,----, dG, where
dG is the dimension of the group and f(k are the structure constants by means of which we can construct a Killing metric tensor gkl = fkifThe quadratic Casimir operator
C2 = gklFkFi	(14.9)
is used for measuring the "size" of a representation r. This is done by normalizing the quadratic Casimir of the representation by dividing it with the quadratic Casimir for the adjoint representation, which consists of dG x dG matrices Aj, such that (Aj)k = —j. The metric can thus be written gkl = Tr(AkAl), and in the first approximation, the "size" of the representation r is taken to be
s=(Ca),	(14.10)
where CT and CA are the Casimirs for the representation r and the adjoint representation, respectively.
In the search for the groups with the smallest representations, we thus examine the quadratic Casimir operators, bearing in mind that the quadratic Casimir is well defined only for irreducible representations. Our goal is to show that the combined group B = {(A, cp)|A € G, p € D} has a measure which is smaller than
the Standard Model group measure,
) < (
c I < c^ ,	(14.11)
Ca/ b \ca/smg
since this is a way of necessitating the existence of the group of diffeomor-phisms. When we talk about SMG, it should be noted that we actually have a SMG in each point of spacetime, corresponding to a product of SMG's: SMG x SMG ... x SMG, this product however has the same size measure as SMG itself,
le. (cr/ca)smgxsmg... = (ct/ca)smg.
According to Schur's lemma [7], in an irreducible representation, any operator that commutes with all the generators of the Lie algebra must be a multiple of the identity operator. Therefore Cr = gklFkFl = cr 1,where 1 is the dr x dr identity matrix, and cr is a coefficient which only depends on the representation r, so we have
c 1 c
S = — = —	(14.12)
ca1 Ca
The point is to minimize the relation cp/cA, with some normalization of cA (the
normalization in reality being arbitrary).
We are interested in the SU(N) group, which has the defining representation
i bi
\bN,
and the group elements are U = NxN complex unitary matrices with determinant 1. The matrices U are k 1, and can be written as U = eiF, with infinitesimal generators F. These F's constitute a real vector space with the dimension N2 — 1, i.e. the dimension of SU(N), and we can choose a basis in the F-space, Fi,F2>--->Fn2_i , which can be normalized.
In an irreducible representation
p : G —> (Matrices)
p(g) = p(1)+ ip(Fj)gj
the quadratic Casimir gklp(Fk)p(Fl) is only an eigenvalue, but it represents how intensively p(g) varies, in the sense that a small cp corresponds to a "lazy" p.
The Casimirs thus function as a crude measure for how much the representation matrix varies as a function of the group element it represents. In a lattice context we take the contribution from one plaquette to be the trace of some representation of the group, the most general action Sq is then a linear expansion on traces of all the possible representations of the gauge group, and the traces of the smallest representations supposedly dominate. This domination corresponds to the variation of the action as a function of how the combination of the link variables varies over the gauge group, and if the action varies relatively slowly over the group, it's taken as an indication that it also varies relatively slowly when we vary the gauge group.
So with an action which is dominated by the contributions from small representations, the variation along the gauge variation is presumeably quite small, a situation corresponding to small quadratic Casimir values. This increases the chance that an action which was not perceived as invariant under a gauge transformation, would nevertheless appear as gauge invariant.
To get an intuition of this "smallness" of a representation, consider SU(2) with its quadratic Casimir J2. On an irreducible representation, J2 effectively only takes one value, i.e. it has the same eigenvalue on the whole representation. With
J2 = 9tj JTJj we have a notion of distance, and we can visualise J2 as performing a rotation,
f|a>= j(j + 1 )|a >
where the "smallness" of the representation means that | a > is just slightly rotated,
small I./ I
Fig. 14.1.
By means of the Casimir measure we can thus define a size measure, making it meaningful to say that the representations of the non-abelian parts of the Standard Model are "small". In the abelian case, it is however problematic to establish what we mean by the "size" of a representation. We cannot apply a similar reasoning for the U(1) groups as for non-abelian groups, because in the abelian case we cannot use the dimension of the representation as a measure, since abelian groups always have 1-dimensional representations, so dimension doesn't tell anything. There simply is no Casimir element defined, since for an abelian Lie algebra the Killing form is zero.
What we can do is to consider the ratio of the charges of the representation and refer to a "Quantum of charge", for example the Millikan unit quantum. The unique abelian invariant subgroup in the Standard Model gauge group corresponds to the weak hypercharge. That can however not be used as the Quantum, since the quantum for y/2 is 1/6, while right-handed charged leptons have y/2 = —1, which is 6 times larger than 1/6, and 6 is obviously not the smallest integer after zero.
So instead we consider non-invariant abelian subgroups, and define an abelian representation as small if it has relatively many charges (generators of the Lie algebra) with only the values 0, 1 or -1 measured in the Quantum.
It turns out that the Standard Model has a relatively large set of such charges, so in this perspective, even from the abelian point of view the Standard Model is a model with "small" representations.
14.3.1 The size of a composite group
In order to normalize the measure by means of the Casimir of the adjoint representation, we clearly need that there is a well-defined adjoint representation. In the case of a composed, non-simple group like B = {(A, cp)|A G G, p G D,}, there is however no straightforward definition of the adjoint representation. For B, we therefore must find a way of varying the two adjoint normalizations relative to each other.
In order to achieve this, we seek to establish a (faithful, 1-1) representation r of B on which we define a metric, whereby the image of B becomes a manifold with a metric, allowing us to define a volume.
One way to do this is to establish a (faithful) representation r of B on which we define a metric. Thus the image of B becomes a manifold with a metric, which makes it possible to define a volume. The measure (cf/ca)b is then given as the volume ratio of the two representations, taken to the power 2/dB,
^ J V
2/dB
^CA / b \Vadj
In the representation picture cr - gik, i.e.
(G) CA (r)
9ik = — 9(k)
(14.13)
thus
and in the case of
Vol(G)
Vol(representation r) \ c
CA
G = Gi x G2 x ... x Gk where the Gj are simple, the quantity
(14.14)
(14.15)
_Cr
CA
cr i
CAi
di
cr2
CA2
d2
is a "good quantity".
Crk CAk
dk
d1 +d2 +... + dk
(14.16)
r
d
14.3.2 Competing groups
For an irreducible representation r, consisting of a set of r x r matrices Mj, the second-order index I2 (r) of the representation is defined by
Tr(M;Mj.) = I2(r)Sij	(14.17)
Taking the trace of (14.9), we get for the quadratic Casimir
C2(r)= I2(r)dc/dr	(14.18)
where dr is the dimension of the representation r, and dG is the group dimension. For the defining, fundamental representation N of SU(N) (i.e., in reality the algebra
su(N)) the second-order index is I2 (N) = 1/2, and for the adjoint representation ¡2(A) = N and dA = N2 - 1, which gives fM = NNt1, thus for SU(2)
VCA/ SU(N) 2N
= 3/8.
VCA/SU(2)
There are presumably other candidates, like SO(N), with the fundamental representation consisting of N x N real matrices. One can define higher tensor representations from the defining vector representation N, but there are also additional, double-valued spinor representations, similar to SO(3) - SU(2), generated by direct products of the fundamental spinor.
In the case of the SU(N) group, the faithful representation with the smallest quadratic Casimir, is the fundamental representation N, while for the SO (N) group the picture is much more complicated, as the faithful representation F with the smallest quadratic Casimir might be either the vector representation, or the spinor representation, the spinor representation being the winner for N < 8.
For the vector representation, the second-order index for the SO(N) fundamental and adjoint representations are I2(N) = 2 and I2(A) = 2N — 4, respectively, and the dimension of the adjoint representation dA = N(N — 1 )/2, thus (cx)	= 2 (NN—2), and for the corresponding spinor representation we
/ j spinor	/ j
have l^J^^ 1 = 2 (N-2) NN. Another competitor is sp(2N),with (^J
2N + 1
4(N + 1
SO(N) 2(N—2) 8	VCAZsp(2N)
, thus
, vector	/ \ spinor
cn \	_ N —1	(c^ _ N —1 N
CA J SO(N) 2(N —22)' VCA J SO(N) 2(N—2) 8 cnA	_ N2 —1	^CN A	_ 2N + 1
CW SU(N) 2N2 '	VCW sp(2N) 4(N + 1)
(14.19)
In the search for the groups chosen by Nature, we examine the (cN/cA) for the different groups, but we also worry about possible differences between 3+1 and 4 spacetime dimensions. For example, for dimension d = 3 +1, we have for the Lorentz group SO(3,1) - SL(2, C), while for d = 4, SO(4) - SU(2)R x SU(2)L.For both d = 3 + 1 and d = 4, the Lorentz group however has the same small S,
cn a 2 (1 + 2) 3
S = 1 H) = wrf = 8	(1420)
while for d = 2 and d = 5, the value is bigger.
The Lorentz group SO(d — p,p) in reality comes from a symmetric metric gik = gki. If gki instead had been antisymmetric, we would have symplectic groups, which are not so competitive, as they have bigger (cN /cA). We consider the Lorentz group as a function of the dimension d and of the "geometry", in the sense of the dependence on whether gik is symmetric, antisymmetric or nonexistent. But an antisymmetric gik actually does very poorly, while for d = 3,4 it looks good for symmetric gik; and for the case without metric for d = 2 [8].
Among simple groups, SU(2) has the smallest (cN/cA), but in order to allow SU(3) be let in, some cooperation with SU(2) is necessary, since the Spin(5), which is the covering group of SO(5), in reality seems to outdo SU(3).
SU(3) however has a relatively big center Z3, so if we divide by the group center SU(3) is in good shape, since the SO(5) covering group has a smaller center. For SU(3), the number of elements in the center is 3, while the center of SU(2) merely has 2 elements, and likewise for Spin(5). We thus redefine our measure of representations as
S =( —)-1-2/d	(14.21)
cA [(Number of elements in the center)] '
We in reality consider volumes:
Vol(SU(3))
= 3
Vol(SU(3)/Z3) And with SU(2)/Z2 = SO(3),
Vol(SU(2)) = 2 Vol(SO(3)) '
the quadratic Casimir being a sort of area in the group.
14.4 The group of diffeomorphisms
We define our group B as the combination of the gauge transformations of SMG
and the group of diffeomorphisms.
A diffeomorphism so to say moves a function, by the operation
xM —} xM + nM
The displacement takes place in a given direction, and if we perceive the dif-feomorphisms as vectors over a manifold, then for infinitesimal nM the set of displacements {nM} constitutes a tangent field. The group of diffeomorphisms does not have a (usual) Lie algebra, but we take as the Lie algebra a set of fields {eM} corresponding to the tangents
f(x) = £ e^,	(14.22)
M
which amounts to substituting a manifold with a space of functions on the manifold,
[f1 (x),f2(x)] = [£ £m3m, £ eV3v],	(14.23)
M	v
and then we could take
C =
9uve^(x)ev(x) dx	(14.24)
as a kind of Casimir. There are scarcely any outer automorphisms for the group of diffeomorphisms, and if all the automorphisms for the group of diffeomorphisms are inner, the group of diffeomorphisms is maximally skew. It should however be noted that the group of diffeomorphisms depends on the topology of the space on
which it is operating, for example for R4, the diffeomorphism group has a trivial center.
Even though the Lie algebra for the group of diffeomorphisms is not a usual Lie algebra, the group is still a Lie group. Consider the mappings of a manifold onto itself, M, 9 : x —» x', i.e.
9 : M —> M where 9 is
•	bijective,
•	sufficiently many times continuously differentiable,
•	a group under the group of diffeomorphisms,
then 9 : M —» M is really a "Lie group", which is clear by considering 9 + 69 and take the commutators [69^ 692] = [1 + 69^ 1 + 692] = 0.
One difficulty we meet with respect to the combined group B is that the group D of diffeomorphisms is probably simple, while the group G of gauge transformations is not,
g € G|g : R4 -> SMG,
meaning
g : R4 -> S(U(2) x U(3));g(x) € S(U(2) x U(3)),
and
f, g : R4 -> S(U(2) x U(3));(fg)(x) = f(x) • g(x)
and for a non-simple group we cannot define a straightforward measure like (cT/cA) for the size of representation. There is however one possibility to define a quadratic Casimir replacement, viz.
ln"cN" =
lnCvgd4x	(14.25)
where g = det(gik). The problem is that we cannot really have a metric, since a metric would not be diffeomorphism-symmetric. On the other hand, we don't quite need the metric gik, but only ^g.
14.5 The composite group
Our selected group is B|{(A, 9), A € G, 9 € D}, composed by the group of gauge transformations
G = {A : M -> SMG} and the group of diffeomorphisms
D = {9 : M —> M}
In order to investigate the group structure, we determine the action of the group elements.
Let be a fermion state, and let (A, 9) operate on With the dfinition
(A,9)[^](x) = pk(A(9(x)))[¥k](9(x))	(14.26)
where pk is the representation matrix, we want to determine the properties of the group operation o of B, i.e. of (Ai, p1) o (A2, cp2). First consider
(A,1)MJ(x) = pk(A(x))[yl](x) (1, p)MJ(x) = MJ( p (x)) = [¥k o cp](x),
(14.27)
(14.28)
thus
(A, 1) o (1,p)[¥k](x) = (A,1)[¥k o p](x) = = Pk(A(x))[¥i(p(x))]
Then consider
(1, p) o (A, 1 )[¥k](x) = (1, c)pk(A(x))[¥i(x)] = = pk(A(p(x)))[¥i(p(x))]
which leads to the conclusion that
(A, cp) = (A,1) o (1, cp) (A, p) = (1, cp) o (A, 1)
When investigating (Ai , c i ) o (A2, c 2) we use that
(1,c)lx](x) = (x o cp)(x)
thus
(Ai , cpi ) o (A2, c2)MJ(x) = (Ai, ci) pk (A2 (c2 (x)))[¥i(p2 (x))]
= (Ai,pi )[pk(A2(c2(x)))(¥i o cp2)(x)] = Pm(Ai (pi (x)))pk(A2(c2 o pi (x)))(¥m o cp2 o pi )(x)
(14.29)
which we identify with
(Ai , pi ) o (A2, p2)[^k](x) = (A3, p3)[^k](x) = pm(A3(p3(x)))(¥m o p3(x))
(14.30)
thus
p3 = p2 o pi,	(14.31)
We demand that for all [¥k](x)
pm(A3(p3(x))) = pm(Ai (pi (x)))pk(A2(p2 o pi (x))) which can only be achieved for a faithful representation, and
Ai (pi ) • A2(p3) = A3 (p3)
(where • is the group operation for G) which applies to all x, a special case being p-1 (x). We perform the substitution x —» p-1 (x), thus obtaining
Ai (pi o p-1 (x)) • A2(x) = A3(x)
and with p3 = p2 o p1, we get p1 o p-1 = p-1 and
Ai(p-1 (x)) • A2(x)= A3(x),	(14.32)
thus
(Ai, ) ◦ (A2, 92) = (A3, 93) = (Ai (9-1 ( • )) • A2, 92 0 Pi)	(14.33)
Now Ai (9-1 ( • )) € G, and
(Ai (9-1 ( • ), l)mc(x) = pk(Ai (9-1 (x)))¥i(x),
but exchange of argument in A, i.e. A(x) —> A(p(x)), is an automorphism in G, and in this sense,
Ai(9-1(x)) = [®^2-1 (Ai)](x), is an automorphism in G. The product of two elements of B finally reads
(Ai, 91) 0 (A2, 92) = (A3, 93) = (Ai) • A2, 92 0 91)	(14.34)
With the alternative definition (A, 9)[¥i](x) = pk(A(9-1 (x)))[¥k](9-1 (x)), we moreover get that
(Ai, 91) 0 (A2, 92) = (A3, 93) = (Ai) • A2, 91 0 92)	(14.35)
14.5.1 Subgroups of B
Does B = {(A, 9)} have any subgroups? The relation (14.32) seems to indicate that the gauge group G is a normal (invariant) subgroup of B, which means that for b € B, bGb-1 C G.
With (A,9)[¥l](x) = pk(A(9(x)))[¥k](9(x)) and (14.34), i.e.
A3 = (Ai) ^2, 93 = 92 0 91
we take
(A, 9) 0 (A, 1) 0 (A, 9)-1 = (A, 9) 0 (A, 1) 0 (fc-1 (A), 9-1) =
= (®V(A) • ®^(A) •®-1(A),1) € G
(14.36)
and specifically for (1, 9), we get (1,9) 0 (A, 1) 0 (1,9) 1 = (A, 1), so for these specific representatives (1,9) of the cosets of
{(A, 1 )|A : M ^ G},
(A, 1) is similarity transformation invariant, and we conclude that G is a normal
subgroup of B. This implies that B is not simple, but a semi-direct product group, unless the subgroup D of diffeomorphisms also is normal, i.e.
(A,1)(1,9)(A,1)-1 € D
Again using (14.34), we get that
(A, w) 0 (1,9) 0 (A, w)-1 = (A, w) 0 (1,9) 0 (O"1 (A), w-1) = (O"1 (A) • A-1, 9) € D
(14.37)
thus D is not an invariant subgroup of B, and B is a semidirect product of G and D,
B = G x D	(14.38)
This means that in (14.34) is a group homomorphism ^-1 : D —» Aut(G), where Aut(D) denotes the group of automorphisms of D.
14.6 To evaluate the size of B
We started out from the Standard Model group SMG, which in itself is a compact, 12-dimensional manifold. When we go to the bigger group B encompassing the group of gauge transformations extended with the group of diffeomorphims, we are dealing with an infinite dimensional Lie group. But this does not necessarily have to be so devastating, keeping in mind that the effect of D in B in reality is nothing more than to dislocate the different SMG in the product SMG x SMG .. .xSMG, gabp(Fa)p(Fb). Compared ton SMG, the group B (where also D is included) will still have the same representations.
In deciding on how to measure the size of a representation, we have encountered a set of problems,
•	How to establish a viable 'size' for the U(1) group in SMG.
•	How do we handle the problem with the adjoint representation in the case of B?
•	How do we define cF/cA for a semidirect product?
In spite of all these problems, let us make an attempt to evaluate the difference between the measures for SMG and B, respectively. Represent D by the Lorentz group, taken as SO(3,1) or SO (4), supposing we are in 3+1 or 4 dimensions, and use (cF/cA) = 3/8 (with the corresponding group dimension 6). In accordance with (14.16) we then define a tentative measure for the composite group B = G x D, as
// o// _
SB —
d(SU(2)) SU(2)
(3 )3- (4 )§■ (3 )6'
\ d(SU(3)) — )
Ca/SU(3)
d(SO(4)) SO (4)
Z di
(14.39)
— 0.406213...
where we for Sg have used S
, keeping in mind that the quadratic
>su(2)®su(3)
Casimir for ^ SMG is the same as for SMG itself, and ignored U(1). For SMG alone, we get
// o// _
\ d(SU(2))
cM
Ca/ SU(2)
\ d(SU(3))
fM
Ca/ SU(3)
1
Z di
(14.40)
(3)3.(4)8 " — 0.424320...,
8
so in this crude approach, "Sg < "Sg.
d(SU(2))
// o// _
CF
cW SU(2)
\ d(SU(3)) SU(3)
1
Z
(14.41)
(§)3. (9)8 11 — 0.424320...,
There is however another aspect to this. Let us make no assumption about the dimension N in SO(N), and simply plot the expression (14.39) for "Sg as a function
of N for N < 8,
i
4	6	8	10	12	14
n
Fig. 14.2.
corresponding to a minimal value for the size "Sg at N = 4. These encouraging results, both for the relative smallness of compared to "Sg, and well as the singling out of N = 4, are of course based on a coarse evaluation, which is to be refined with a more precise formulation of the quadratic Casimir for the group B, in order to accomplish a fair comparison between the sizes of the two groups.
14.7 Conclusion
In this article we have taken the first steps in "deriving" diffeomorphism symmetry, which is called for within the framework of the derivation of space. We have discussed different "goal quantities", especially the size of a representation of a group, identified as the size of the quadratic Casimir, which is connected with natural metric on the space of unitary matrices in the representations.
With this "goal quantity" in mind, we argue that diffeomorphism symmetry necessarily comes about, because the size of the bigger group, which is the semidi-rect product of the Standard Model group and the group of diffeomorphisms, is smaller than the size of th Standard Model group.
The next step will be to calculate the Casimirs for the entire group B, and more precisely evaluate the size of B as compared with the size of the Standard Model group.
References
1.	H.B. Nielsen and A. Kleppe, Towards a Derivation of Space, hep-th/1403.14101v, Proceedings to the 16th Workshop "What Comes Beyond the Standard Models?", Bled 14-21 of July 2013, Vol. 14, No. 2, DMFA Zaloznistvo, Ljubljana, Dec. 2013.
2.	H.B. Nielsen, D.L. Bennett and N. Brene, The random dynamics project or from fundamental to human physics, Recent Developments in Quantum Field Theory (1985), ed. J. Ambjorn,B.J. Durhuus,J.L. Petersen, ISBN 0444869786;
H.B. Nielsen and N. Brene, Some remarks on random dynamics, Proc. of the 2nd Nishi-nomiya Yukawa Memorial Symposium on String Theory, Kyoto University, 1987 (Springer, Berlin, 1988);
H.B. Nielsen (Bohr Inst.), Random Dynamics and relations between the number offermion generations and the fine structure constants, NBI-HE-89-01, Jan 1989. 50pp. Talk presented at Zakopane Summer School, May 41 - Jun 10,1988. Published in Acta Phys. Polon. B20:427,1989; "Origin of Symmetries", C. D. Froggatt and H.B. Nielsen, World Scientific (1991);
Random Dynamics Homepage, http://www.nbi.dk/~kleppe/random/qa/qa.html; http://www.nbi.dk/~kleppe/random/qa/rref.html.
3.	D.L. Bennett, H.B. Nielsen Non-locality requires fine tuning and multi-degenerate vacua (2005) arXiv:hep-th/0503080v1.
4.	H. B. Nielsen, "Small Representation Principle", Contribution to 14th Workshop on What Comes Beyond the Standard Models - 14th Bled Workshop 2011 11 - 21 Jul 2011 - Bled, Slovenia / Mankoc Borstnik, Norma Susana (ed.); Nielsen, Holger Bech (ed.); Lukman, Dragan (ed.) 2011 - Published in : Bled Workshops Phys.: 12 (2011) , no. 2 Conference home page CERN library copies; http://bsm.fmf.uni-lj.si/bled2013bsm/Smallrep10.pdf.
5.	H. B. Nielsen and N. Brene, Skewness of the Standard Model - possible implications in P. Nicoletopoulos and J. Orloff "The Gardener of Eden" (In Honour of Robert Brout on his 60 years birthday).
6.	H.B Nielsen, C. Surlykke, S. E. Rugh, Seeking inspiration from the Standard Model in order to go beyond it, hep-th/9407012.
7.	Issai Schur (1905) Neue Begrndung der Theorie der Gruppencharaktere" (New foundation for the theory of group characters), Sitzungsberichte der Kniglich Preuischen Akademie der Wissenschaften zu Berlin, pages 406-432.
8.	H. B. Nielsen, Dimension four wins the same game as the Standard Model group, hep-ph/1304.6051v7.
9.	O'Raifeartaigh, Group Structure of Gauge theories, University Press Cambridge (1986). 10. Harish-Chandra, Representations of a semisimple Lie group on a Banach space,
http://www.ams.org/journals/tran/1953-075-02/S0002-9947-1953-0056610-2/S0002-9947-1953-0056610-2.pdf.
Virtual Institute of Astroparticle Physics Presentation
Bled Workshops in Physics Vol. 15, No. 2
A
Proceedings to the 17th Workshop What Comes Beyond ... (p. 285) Bled, Slovenia, July 20-28, 2014

15 Virtual Institute of Astroparticle physics and "What comes beyond the Standard model?" in Bled
M.Yu. Khlopov1'2'3
1	Centre for Cosmoparticle Physics "Cosmion"
2	National Research Nuclear University "Moscow Engineering Physics Institute", 115409 Moscow, Russia
3	APC laboratory 10, rue Alice Domon et Leonie Duquet 75205 Paris Cedex 13, France
Abstract. Virtual Institute of Astroparticle Physics (VIA) is a unique multi-functional complex of science and education online. VIA website
http://viavca.in2p3.fr/site.html supports participation in conferences and meetings, various forms of collaborative scientific work as well as programs of education at distance, combining online videoconferences with extensive library of records of previous meetings and Discussions on Forum. The VIA facility is regularly effectively used in the programs of Bled Workshops. At XVII Bled Workshop it provided a world-wide discussion of the open questions of physics beyond the standard model.
Povzetek. Virtual Institute of Astroparticle Physics (VIA) predstavlja edinstven vecnamenski kompleks znanosti in izobraževanja na spletu. Njegova domaca stran
http://viavca.in2p3.fr/site.html podpira udelezbo na konferencah in srecanjih, razlizne oblike znantsvenega sodelovanja in programe izobrazevanja na daljavo. Kombinira video konference z obsezno knjiznico digitalnih zapisov prejsnjih srecanj in diskusije na forumu VIA. Moznosti VIA se redno in obsezno uporabljajo v programih Blejskih delavnic. Na sedemnajsti delavnici je omogocil diskusijo o odprtih problemih fizike onkraj obeh standardnih modelov udelezencem sirom sveta.
15.1 Introduction
Studies in astroparticle physics link astrophysics, cosmology, particle and nuclear physics and involve hundreds of scientific groups linked by regional networks (like ASPERA/ApPEC [1,2]) and national centers. The exciting progress in these studies will have impact on the knowledge on the structure of microworld and Universe in their fundamental relationship and on the basic, still unknown, physical laws of Nature (see e.g. [3,4] for review).
Virtual Institute of Astroparticle Physics (VIA) [5] was organized with the aim to play the role of an unifying and coordinating structure for astroparticle physics. Starting from the January of 2008 the activity of the Institute takes place on its website [6] in a form of regular weekly videoconferences with VIA lectures,
covering all the theoretical and experimental activities in astroparticle physics and related topics. The library of records of these lectures, talks and their presentations was accomplished by multi-lingual Forum. In 2008 VIA complex was effectively used for the first time for participation at distance in XI Bled Workshop [7]. Since then VIA videoconferences became a natural part of Bled Workshops' programs, opening the virtual room of discussions to the world-wide audience. Its progress was presented in [8-12]. Here the current state-of-art of VIA complex, integrated since the end of 2009 in the structure of APC Laboratory, is presented in order to clarify the way in which VIA discussion of open questions beyond the standard model took place in the framework of XVII Bled Workshop.
15.2 The structure of VIA complex and forms of its activity 15.2.1 The forms of VIA activity
The structure of VIA complex is illustrated on Fig. 15.1. The home page, presented on this figure, contains the information on VIA activity and menu, linking to directories (along the upper line from left to right): with general information on VIA (About VIA), entrance to VIA virtual rooms (Rooms), the library of records and presentations (Previous) of VIA Lectures (Previous Lectures), records of online transmissions of Conferences(Previous —» Conferences), APC Colloquiums (Previous —> APC Colloquiums), APC Seminars (Previous —> APC Seminars) and Events (Previous —> Events), Calender 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 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 Calender links to 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 [13].
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
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Fig. 15.1. The home page of VIA site
support except for WiFi Internet connection and ordinary laptops with their internal video and audio equipments. 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 [14]).
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 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 [15].
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 Journes nationales du Dveloppement Logiciel (JDEV2013) at Ecole Politechnique (Paris) were organized [16].
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" [17] and the programme of "Conference YaB-100, dedicated to 100 Anniversary of Yakov Borisovich Zeldovich" (Moscow, Russia) by his talk "Cosmology and particle physics" [18].
The discussion of questions that were put forward in the interactive VIA events can be continued and extended on VIA Forum. The Forum is intended to cover the topics: beyond the standard model, astroparticle physics, cosmology, gravitational wave experiments, astrophysics, neutrinos. Presently activated in English,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.
One of the interesting forms of Forum activity is the educational work at distance. For the last five 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 [19]. 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. 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 satisfactory. Then they are admitted to pass their exam. The record of videoconference with their oral exam is also put in the corresponding directory of Forum. Such procedure provides completely transparent way of evaluation of students' knowledge. In 2014 the second part of this course was used for a test of VIA system as a possible supplementary tool for Massive Online Open Courses (MOOC) activity [20]. In the context of MOOC VIA facility can be used for individual online work with advanced students.
15.2.2 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 facility for discussions, the option to open desktop and to work online with texts in any format.
The normal 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 [21]. 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 our 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.
15.3	VIA Sessions at XVII Bled Workshop
VIA sessions of XVII Bled Workshop have developed from the first experience at XI Bled Workshop [7] and their more regular practice at XII, XIII, XIV, XV and XVI Bled Workshops [8-12]. They became a regular part of the Bled Workshop's programme.
In the course of XVII Bled Workshop meeting 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 [22]) and all the participants of VIA sessions were invited to address them during VIA discussions. During the XVII Bled Workshop the test of not only minimal necessary equipment, but either of the use of VIA facility by ordinary users was undertaken. VIA Sessions were supported by personal laptop with WiFi Internet connection only, as well as in 2014 the members of VIA team were physically absent in Bled and all the videoconferences were directed by M.Khlopov at distance. It proved the possibility to provide effective interactive online VIA videoconferences even in the absence of any special equipment and qualified personnel 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 absence of WiFi connection, the 3G connection of iPhone was sufficient for VIA management and presentations.
In the framework of the program of XVII Bled Workshop, M. Khlopov, gave his Introduction at distance to the M.Laletin's talk "Dark Atoms and Their decaying Constituents" (Fig. 15.2). It provided an additional demonstration of the ability of VIA to support the creative non-formal atmosphere of Bled Workshops (see records in [23]).
VIA sessions also included the talks "Novel string field theory solving string theory liberating left and right movers" by Holger Bech Nielsen, "The spin charge family theory offers the explanation for the assumption of the Standard model, for the Dark matter, for the Matter-antimatter asymmetry..., making several predictions" by Norma Mankoc-Borstnik (Fig. 15.3) and"Dark matter particles in the galactic halo" by Rita Bernabei (Fig. 15.4), followed by discussions with distant participants. The records of these lectures and discussions can be found in VIA library [23].
15.4	Conclusions
The Scientific-Educational complex of Virtual Institute of Astroparticle physics provides regular communication between different groups and scientists, working
Fig. 15.2. VIA talk by M.Khlopov served as Introduction to M.Laletin's talk at XVII Bled Workshop
Fig. 15.3. VIA talk by N. Mankoc-Borstnik at XVII Bled Workshop
Fig. 15.4. VIA talk by R.Bernabei at XVII Bled Workshop
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 can provide nontrivial supplementary tool for MOOC. 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. The experience of VIA applications at Bled Workshops plays important role in the development of VIA facility as an effective tool of science and education online.
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 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. I express my gratitude to N.S. Mankoc Borstnik, G.Bregar, D. Lukman, M. Laletin and all Organizers of Bled Workshop for cooperation in the organization of VIA Sessions at XVII 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.	In http://viavca.in2p3.fr/ Previous - Conferences - XIII Bled Workshop
14.	In http://viavca.in2p3.fr/ Previous - Conferences - XIV Bled Workshop
15.	In http://viavca.in2p3.fr/ Previous - Lectures - Martin Pohl
16.	In http://viavca.in2p3.fr/ Previous - Events - JDEV 2013
17.	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
18.	In http://viavca.in2p3.fr/ Previous - Conferences - Conference YaB-100, dedicated to 100 Anniversary of Yakov Borisovich Zeldovich
19.	In http://viavca.in2p3.fr/ Forum - Discussion in Russian - Courses on Cosmoparticle physics
20.	In http://viavca.in2p3.fr/ Forum - Education - From VIA to MOOC
21.	In http://viavca.in2p3.fr/ Previous - Lectures - John Ellis
22.	In http://viavca.in2p3.fr/ Forum - CONFERENCES BEYOND THE STANDARD MODEL - XVII Bled Workshop "What Comes Beyond the Standard Model?"
23.	In http://viavca.in2p3.fr/ Previous - Conferences - XVII Bled Workshop
Blejske Delavnice Iz Fizike, Letnik 15, št. 2, ISSN 1580-4992 Bled Workshops in Physics, Vol. 15, No. 2
Zbornik 17. delavnice 'What Comes Beyond the Standard Models', Bled, 20. -28. julij 2014
Proceedings to the 17th workshop 'What Comes Beyond the Standard Models', Bled, July 20.-28., 2014
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 Birografika Bori v nakladi 100 izvodov
Publikacija DMFA stevilka 1947