BLEJSKEDELAVNICEIZ FIZIKE LETNIK 23, ˇ ST.1 BLED WORKSHOPSIN PHYSICS VOL.23,NO.1 ISSN1580-4992 Proceedings to the 25thWorkshop What Comes Beyond the Standard Models Bled, July 4–10, 2022 [VirtualWorkshop] July 11-12 2022 Edited by Norma Susana Mankoˇc Borˇstnik Holger Bech Nielsen Astri Kleppe DMFA– ZNIˇ ZALOˇSTVO LJUBLJANA, DECEMBER 2022 The 25thWorkshop What Comes Beyond the Standard Models, 4.– 10. July 2022, Bled [VirtualWorkshop, July 11-12, 2022] 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 (Matjaˇz Breskvar) VIA(Virtual Instituteof Astroparticle Physics), Paris MDPI journal “Symmetry”, Basel MDPI journal “Physics”, Basel MDPI journal “Universe””, Basel Scientific Committee John Ellis, King’s College London/CERN Roman Jackiw, MIT Masao Ninomiya, Yukawa Institute for Theoretical Physics, Kyoto University Organizing Committee Norma Susana Mankoˇc Borˇstnik Holger Bech Nielsen MaximYu. Khlopov The Membersof theOrganizing Committeeof the InternationalWorkshop “What Comes Beyond the StandardModels”, Bled, Slovenia, state that the articles published in the Proceedings to the 25thWorkshop “What Comes Beyond the StandardModels”,Bled, SloveniaarerefereedattheWorkshopin intense in-depth discussions. Workshops organized at Bled . What Comes Beyond the StandardModels (June 29–July9, 1998),Vol. 0(1999) No.1 (July 22–31, 1999) (July 17–31, 2000) (July 16–28, 2001),Vol. 2(2001) No.2 (July 14–25, 2002),Vol. 3(2002) No.4 (July 18–28, 2003)Vol. 4(2003) Nos. 2-3 (July 19–31, 2004),Vol. 5(2004) No.2 (July 19–29, 2005),Vol. 6(2005) No.2 (September 16–26, 2006),Vol. 7(2006) No.2 (July 17–27, 2007),Vol. 8(2007) No.2 (July 15–25, 2008),Vol. 9(2008) No.2 (July 14–24, 2009),Vol. 10 (2009) No.2 (July 12–22, 2010),Vol. 11 (2010) No.2 (July 11–21, 2011),Vol. 12 (2011) No.2 (July 9–19, 2012),Vol. 13 (2012) No.2 (July 14–21, 2013),Vol. 14 (2013) No.2 (July 20–28, 2014),Vol. 15 (2014) No.2 (July 11–19, 2015),Vol. 16 (2015) No.2 (July 11–19, 2016),Vol. 17 (2016) No.2 (July 9–17, 2017),Vol. 18 (2017) No.2 (June 23–July1, 2018),Vol. 19 (2018) No.2 (July 6–14, 2019),Vol. 20 (2019) No.2 (July 4–12, 2020),Vol. 21 (2020) No.1 (July 4–12, 2020),Vol. 21 (2020) No.2 (July 1–12, 2021),Vol. 22 (2021) No.1 . Hadrons as Solitons (July 6–17, 1999) . Few-Quark Problems (July 8–15, 2000),Vol. 1(2000) No.1 . Selected Few-BodyProblemsin 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–July6, 2009),Vol. 10 (2009) No.1 . Dressing Hadrons (July 4–11, 2010),Vol. 11 (2010) No.1 . Understanding hadronic spectra (July 3–10, 2011),Vol. 12 (2011) No.1 . Hadronic Resonances (July 1–8, 2012),Vol. 13 (2012) No.1 . Looking into Hadrons (July 7–14, 2013),Vol. 14 (2013) No.1 . Quark Masses and Hadron Spectra (July 6–13, 2014),Vol. 15 (2014) No.1 . Exploring Hadron Resonances (July 5–11, 2015),Vol. 16 (2015) No.1 . Quarks, Hadrons, Matter (July 3–10, 2016),Vol. 17 (2016) No.1 . Advances in Hadronic Resonances (July 2–9, 2017),Vol. 18 (2017) No.1 . Double-charm Baryons and Dimesons (June 17–23, 2018),Vol. 19 (2018) No.1 . Electroweak Processes of Hadrons (July 15–19, 2019),Vol. 20 (2019) No.1 . . Statistical Mechanics of Complex Systems (August 27–September 2, 2000) . Studies of Elementary Steps of Radical Reactions in Atmospheric Chemistry (August 25–28, 2001) Contents Preface in English and Slovenian Language :::::::::. IV 1 New and recent results, and perspectives from DAMA/LIBRA–phase2 R. Bernabei,P. Belli, A. Bussolotti,V. Caracciolo, R. Cerulli, N. Ferrari, A. Leoncini,V. Merlo,F. Montecchia,F. Cappella,A. d’Angelo,A. Incicchitti,A. Mattei, C.J. Dai, X.H. Ma, X.D. Sheng, Z.P.Ye :::::::::::::::::::::::::::. 1 2 Neutrino production by photons scattered on dark matter V. Beylin:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::. 21 3 Elusive anomalies L. Bonora ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::. 31 4 Maximally PreciseTestsof the StandardModel:Eliminationof Perturbative QCD Renormalization Scale and Scheme Ambiguities S. J. Brodsky :::::::::::::::::::::::::::::::::::::::::::::::::::::::. 45 5 Predictions of Additional Baryons and Mesons Paul H. Frampton :::::::::::::::::::::::::::::::::::::::::::::::::::. 75 6 Possibility of Additional Intergalactic and Cosmological Dark Matter Paul H. Frampton :::::::::::::::::::::::::::::::::::::::::::::::::::. 87 7 Near-inflection point inflation and production of dark matter during reheating A. Ghoshal, G. Lambiase, S. Pal, A. Paul, S. Porey :::::::::::::::::::::::. 96 8 Quark masses and mixing froma SU(3) gauge family symmetry A. Hernandez-Galeana ::::::::::::::::::::::::::::::::::::::::::::::. 114 9 Evolution and Possible Forms of Primordial Antimatter and Dark Matter celestial objects MaximYu. Khlopov, O.M. Lecian :::::::::::::::::::::::::::::::::::::. 128 10 The Problem of Particle-Antiparticle in Particle Theory FelixMLev ::::::::::::::::::::::::::::::::::::::::::::::::::::::::. 146 Contents 11 Clifford odd and even objects, offering description of internal space of fermion and boson fields, respectively, open new insight into next step beyond standard model N. S. Mankoˇc Borˇstnik :::::::::::::::::::::::::::::::::::::::::::::::. 162 12 AUnified Solution to the Big Problems of the Standard model R. N. Mohapatra, N. Okada ::::::::::::::::::::::::::::::::::::::::::. 201 13 Dusty Dark Matter Pearls Developed H.B. Nielsen, C. D. Froggatt ::::::::::::::::::::::::::::::::::::::::::. 214 14 What givesa “theoryof Initial Conditions”? H.B. Nielsen, K. Nagao ::::::::::::::::::::::::::::::::::::::::::::::. 228 15 Emergent phenomena in QCD: The holographic perspective GuyF. deT´eramond ::::::::::::::::::::::::::::::::::::::::::::::::. 240 16 Planetary relationship as the new signature from the dark Universe K. Zioutas, V. Anastassopoulos, A Argiriou, G. Cantatore, S. Cetin, A. Gardikiotis, M. Karuza, A. Kryemadhi, M. Maroudas, A. Mastronikolis, K. Ozbozduman,Y.K. Semertzidis,M.Tsagri,I.Tsagris ::::::::::::::::::. 256 17 Abstracts of talks presented at the Workshop and in the Cosmovia forum :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::. 262 Discussion section ::::::::::::::::::::::::::::::::::::::::::. 270 18 Discussion of cosmological acceleration and dark energy FelixMLev ::::::::::::::::::::::::::::::::::::::::::::::::::::::::. 271 19 Clifford odd and even objects in even and odd dimensional spaces N. S. Mankoˇc Borˇstnik :::::::::::::::::::::::::::::::::::::::::::::::. 279 20 Modules over Clifford algebras as a basis for the theory of second quantization of spinors V.V. Monakhov ::::::::::::::::::::::::::::::::::::::::::::::::::::. 290 21 Anew view on cosmology, with non-translational invariant Hamiltonian H. B. Nielsen, M. Ninomiya ::::::::::::::::::::::::::::::::::::::::::. 304 22 Anew Paradigm for the Dark Matter Phenomenon P. Salucci ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::. 315 23 On the construction of artificial empty space Elia Dmitrieff ::::::::::::::::::::::::::::::::::::::::::::::::::::::. 331 Contents III 24 Virtual Institute of Astroparticle physics as the online platform for studies of BSM physics and cosmology MaximYu. Khlopov :::::::::::::::::::::::::::::::::::::::::::::::::. 334 25 Apoem :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::. 348 Preface in English and Slovenian Language This year our series of workshops on ”What Comes Beyond the Standard Models?” took place the twenty-fifth time. The series started in 1998 with the idea of organizinga workshop,in which participants would spend mostof the timein discussions, confronting different approaches and ideas. The picturesque town of Bledby the lakeof the same name, surroundedby beautiful mountains and offering pleasant walks, was chosen to stimulate the discussions. The idea was successful and has developed into an annual workshop, which is taking place every year since 1998.Very open-minded andfruitful discussions have become the trade-mark of our workshop, producing several published works. It takes place in the house of Plemelj, which belongs to the Society of Mathematicians, Physicists and Astronomers of Slovenia. Since the workshop is celebrating its anniversary, 25th, it is an opportunity to look at what has happened in the field of physics of elementary fermion and boson fields all this time. And what new the measurements together with the suggested theories have brought. The technology and computer science has progressed in mean time astonishingly, enabling almost unbelievable measurements in all fields of physics, especially in the physics of fermion and boson fields and in cosmology. The standard model assumptions have been confirmed without offering surprises. The last unobserved field assumed by the standard model as a field, the Higgs’s scalar, detected in June 2012, was confirmed in March 2013. New and new measurements of masses of quarks and leptons and antiquarks and antileptons, of the mixing matrices of quarks and of leptons, of the Higgs’s scalar, of bound states of quarks and leptons, offer new and new data. The waves of the gravitational field were detected in February 2016 and again 2017. If we look at the collection of open questions that we set ourselves at the beginning and continuously supplemented in each workshop, it shows up that we are all the time mostly looking for an answer to the essential question: What is the next step beyond the standard model, which would offer not only the understanding of all the assumed properties for quarks and leptons and all the observed boson fields with the Higgs scalars included, but also for the observed phenomena in cosmology, like it is the understanding of the expansion rate of the universe, of the appearance of the dark matter, of black holes with their (second quantized) quantum nature included, of the necessity of the existence of the dark energy and many others. When trying to understand the quantum nature of fermion and boson fields we are looking for the theory which is anomaly free and possibly renormalizable so that we would be able to predict properties of second quantized fields when proposing measurements. Contents Ifit turnsoutthattheserequirementsleadtothe dimensionofspacetimewhichis higher than (3 + 1) then the questions arise how does it happen that all the dimensions except the (3 + 1)-ones are hidden. How do consequently the internal spaces of fermion and boson fields manifest in d =(3 + 1)?And can sucha theory be free of anomalies and renormalizable? The observationsof the galacticrotation curves are oneof the strongest evidence that it is not only the ordinary matter which determines the properties of the galaxies. The direct measurements of the Dama/Libra experiment with more and more accuracy demonstrates the annual dependence of the number of events. One can hardly accept that these measurements are not connected with the dark matter neededto explaintherotational curvesof galaxiesand behaviourof clouds of galaxies, of whatever origin the dark matter is. The measurements of the Hubble constant state that the universe is expanding and how fast does expend. The cosmological measurements show up that there exist the black holes. In all our annual workshops there have appeared innovative proposals for: i. Explaining the assumptions of the standard model, mostly with new unexplained assumptions. ii. Suggestions of what is the dark matter. iii. Suggestions for what causes inflation. iv. Suggestions what does determine the internal space of fermions andbosons. v. Suggestions how to avoid anomalies and make quantum theories of fields renormalizable. vi. How to suggest experiments which would show the next step beyond the standard model. vii. How does ”Nature make the decision” about breaking of symmetries down to the noticeable ones, if coming from some higher dimension d? viii. Why is the metric of space-time Minkowskian and how is the choice of metric connected with the evolution of our universe(s)? .... And many others. In our proceedings there are papers (many of them later published in journals) discussing these problems and offering suggestions how to solve them. In the last two Covid-19 years the ZOOM workshop replaced our ordinary workshops.It mustbe admitted that the ZOOM meetings can notreplace thereal meetingswhereallthe questionsare welcome evenif answersneedalongtimeto be presented. Talks and discussions in our workshop are not at all talks in the usual way. Each talk as well as discussions lasted several hours, divided in two hours blocks, with a lot of questions, explanations, trials to agree or disagree from the audience or speakers side. Although also on the ZOOM way of presentations several continuations of the same talk were planned and realized, yet the presence in real is much more effective. Thislast,thejubilee workshop,waspartlyin”real”atBledandpartlybyZOOM. The topics presented and discussed in this workshop concern all the above mentioned open problems, illustrated by the question ”How to understand Nature?” We were trying to find the answers in several steps: VI Contents • How to make the next step beyond both standardmodels? o How to come beyond the standard model of the so far observed quarks and leptons and antiquarks and antileptons, appearing in families, and interacting with the electroweak, colour and scalar fields, o How to explain observed cosmological phenomena? • How can we construct the anomalies-free renormalizable theory of all the so far observed fermions and all the so far observed boson fields? o Can this be done as well as for bound states and scattering states of fermion and boson fields? o How do symmetries contribute to bound states? • Can we find the way to treat all the elementary fermion and boson fields in an unique way? o How to find the way to treat fermion and boson fields if space-time is indeed four dimensional? o How does ”Nature make the decision” about breaking of symmetries down to the noticeable ones, if coming from some higher dimension d? o Why is the metric of space-time Minkowskian and how is the choice of metric connected with the evolution of our universe(s)? o What does cause the inflation of the universe? o When does the inflation appear? After the electroweak phase transition? o What does determine the colour scale? • What is our universe made out of besides of the (mostly) first family baryonic matter? o How do black holes contribute to the dark matter? • What is the role of symmetries in Nature? • How to make experiments and how to propose the models so that the data would not be influenced too much by the proposed model? Most of talks are ”unusual” in the sense that they are trying to find out new ways of understanding and describing the observed phenomena. Theproceedingsis divided into two parts.To the first part the invited talks, which appear in time and were refereed, contribute. To the second part, called Discussion section, the contributions are presented, which started to be intensively discussed during the workshop but need more discussions so that the authors of different contributions would agree or disagree, or which seem to the authors of different contributions that they have many commonpoints,expressedinadifferentway, whichmightleadtonew ideasor new conclusions or new collaborations. Some of discussions, started during the workshop, are not appearing in this proceedings and might continue next year and be ready for next proceedings. The organizers are grateful to all the participants for the lively presentations and discussions and the good working atmosphere although most of participants appear virtually, what was lead by Maxim Khlopov. Thereader can find all the talks and soon also the wholeProceedings on the official websiteof theWorkshop: http://bsm.fmf.uni-lj.si/bled2022bsm/presentations.html, Contents VII and on the Cosmovia Forum https://bit.ly/bled2022bsm .. Norma Mankoˇc Borˇstnik, Holger Bech Nielsen, Maxim Khlopov, Astri Kleppe Ljubljana, December 2022 1 Predgovor (Preface in Slovenian Language) To leto je serija delavnic z naslovom ,,Kako preseˇci oba standardna modela, kozmoloˇsibkega” (”What Comes Beyond the StandardModels?”) skega in elektroˇ steklaˇc. Prva delavnicaje stekla leta 1998v ˇzenci ze petindvajsetiˇzelji, da bi udele ˇ v izˇcrpnih diskusijah kritiˇcno sooˇcali razliˇcne ideje in teorije. Slikovito mestece Bled, ob jezeru z enakim imenom, obkroˇ zeno s prijaznimi hribˇcki, nad katerimi kipijo slikovite gore, ki ponujajo prijetne sprehode in pohode, ponujajo prilo ˇ znosti za diskusije. Idejajebilauspeˇsna, razvilasejev vsakoletno delavnico,kiteˇceˇc. ze ptindvajsetiˇ Zelo odprte, prijateljske in uˇcinkovite diskusije so postale ”blagovna znamka” naˇsih delavnic, ideje, ki so se v diskusijah rodile, pa so pogosto botrovale objavljenimˇclankom. Delavnice domujejov Plemljevihiˇsi na Bledu tikob jezeru. Hiˇstvu matematikov, fizikov in astronomov zapustil svetovno priznani so je Druˇ slovenski matematik Jozef Plemelj. Letoˇsnje jubilejno leto ponuja priloˇ znost,da pogledamo,kaj vse seje zgodilov temˇ casu v fiziki osnovnih fermionskih in bozonskih polj in v kozmologiji in kaj so novegav temˇcasu ponudile meritve skupajspredlaganimi teorijami. Tehnologija in raˇcunalniˇstvo sta medtem presenetljivo hitro napredovala in omogoˇcjih fizike, tudi ali ˇ cila skoraj neverjetne meritve na vseh podroˇse posebej v fiziki fermionskihin bozonskihpoljterv kozmologiji. Poskusi so potrdili predpostavke standardnega modela ne da bi prinesli kakrˇ snekoli preseneˇcenje. Zadnje polje, ki ga predpostavi standardni model, Higgsov skalar, ki je bil odkrit junija 2012, so potrdiliv marcu 2013.Vedno bolj natanˇ cne meritve mas kvarkov in leptonov in antikvarkov in antileptonov, meˇsalnih matrik kvarkov in leptonov, mase Higgsovega skalarja, vezanih stanj kvarkov in leptonov, ponujajo novein nove podatke.Valovanje gravitacijskega polja smo zaznali februarja 2016 in spet 2017. ˇ Ce pogledamo zbirko odprtih vpraˇsanj, ki sva si jih s Holgerjem zastavila pred zaˇze, da ves cetkom delavnicinki smojoob vsaki delavnici sproti dopolnjevali,ka ˇ casiˇˇsˇcemo odgovorna bistveno vpraˇsanje:Kajje naslednji korak,kibopresegel standardni model in bo ponudil ne le razlago in razumevanje za vse predpostavljene lastnosti za kvarke in leptone in antikvarke in antileptone in za vsa doslej opa ˇ zena bozonska polja, skupaj s Higgsovim skalarjem, ampak tudi za vse opa ˇsirjenja vesolja, pojav temne zene pojavev vesolju, kotje na primer hitrost ˇ snovi, pomen singularnostiˇcrnih luknenj kot kvantnega skupka fermionovin antifermionov in vseh bozonskih polj, vklju ˇ cno z gravitacijskim poljem v drugi kvantizaciji, kakojez nujnostjo obstoja temne energijein mnogihdrugih opa ˇ zenj. Ko poskuˇsamo razumeti kvantno naravo fermionskihin bozonskih polj,iˇsˇcemo teorijo, ki je brez anomalij in taka, da nam da kon ˇ cne prispevke za predlagane meritve. ˇze,da lahkouresniˇcimote zahtevele,ˇce dovolimo,daimaprostor-ˇ Ce se izkaˇcas veˇc kot le opazljive (3 + 1) razseˇ znosti, potem moramo odgovoriti na vpraˇsanje, zakaj so vse razseˇ znosti razen d =(3 + 1) skrite. Kako se tedajv d =(3 + 1) Contents IX manifestirajo notranji prostori fermionskih in bozonskihpolj? In ali je taka teorija lahko brez anomalij in ponudi konˇcne prispevke za obravane dogodke? Merjenja hitrostirotacije zvezd okoli centra galaksijein gibanja galaksijv jatah so ena najmoˇcnejˇsih dokazov, da ni le obiˇcajna snov, iz katere so zvezde, tista, ki doloˇca lastnosti galaksijinjat galaksij.Tudi neposredne meritve eksperimenta Dama/Libra z vse veˇcjo natanˇcnostjo dokazujejo, da je izmerjeni letni odvisnosti ˇca temna snov, stevila dogodkov potrebno verjeti in da te dogodke povzroˇ kakrˇsenkoliˇ ze je njen izvor. Meritve Hubblove konstante navajajo, kako hitro se vesoljeˇsiri. Kozmoloˇcrne luknje obstajajo. ska merjenjakaˇzejo,da ˇ Naˇ se delavnice so ponudile inovativne predloge: i. Za razlago predpostavk standardnega modela, veˇcinoma z novimi nepojasnjenimi predpostav-kami. ii. Za to,iz ˇca cesaje temna snov. iii. Za pojav in vzrok inflacije vesolja. iv. Za to, kaj doloˇ notranji prostor fermionov in bozonov. v. Kako se izogniti anomalijam in oblikovati kvantne teorije polj, ki jih je mogoˇce renormalizirati. vi. Kako predlagati poskuse,kibi pokazalikajje naslednji korakpo standardnem modelu. vii. Kako se ”Narava odloˇcetnev razse ˇzenih ci” zlomiti simetrije od zaˇznosti d do opaˇ v d=(3+1)? viii. Zakaj je metrika prostora-ˇcasa metrika Minkovskega in kako je izbira metrike povezanaz razvojem naˇsega(ih) vesolja(ij)? ... In mnogo drugih. Vobjavljenih zbornikihinv pozneje objavljenihˇ clankih v revijah so prispevki, ki razpravljajo o teh problemih in ponudijo predloge, kako probleme reˇ siti. Vzadnjih dveh letih Covida-19je delavnicapreko ”ZOOM-a” nadomestilanaˇse obiˇcajne delavnice.Trebaje priznati,da sreˇcanjapreko interneta ne morejo nadomestiti pravih sre ˇsanja dobrodoˇceje za canj, kjer so vsa vpraˇsla, tudi ˇ odgovore potrebno mnogoˇcasa. Pogovori in razprave na naˇcajne diskusije. Razprave sih delavnicah sploh niso obiˇ trajajo po veˇc ur, razdeljene v dvourne bloke, z veliko vpraˇsanji, razlagami, poskusi strinjanja ali nestrinjanja udele ˇ zencev in pojasnjevalca. ˇse delavnice po internetu omogoˇc Ceprav so naˇcile predstavitve del v veˇ nadaljevanjih, je prisotnost v istem prostoru za prave razprave veliko bolj uˇcinkovita. Ta zadnja, jubilejna delavnica, je potekala delno na Bledu in delno po ”ZOOM-u”. Teme, predstavljene in obravnavane na tej delavnici, se nanaˇsajo na vse zgoraj omenjene teme, strnjene v vpraˇ sanje ”kako razumeti Naravo”. Odgovore smo poskuˇsali najti v veˇc korakih: • Kako najti naslednji korak, ki bo odgovoril na odprta vpraˇsanja obeh modelov? o Kako poiskati teorijo, ki bo pojasnila vse privzetke standardnega modela kvarkov in leptonov ter antikvarkov in antileptonov, ki nastopajo v dru ˇ zinah insi izmenjujejo elektromagnetna, ˇ sibka, barvna in skalarna polja? o Kako razlo ˇzene kozmoloˇ ziti doslej opa ˇske pojave? • Kako poiskati renormalizabilno teorijo brez anomalij za vse poznane fermione in njihova umeritvena polja? Contents o Kako poiskatirenormalizabilno teorijobrez anomalij tudi za vezanain sipana stanja fermionov in bozonov? o Kako simetrije prispevajokvezanim stanjem? • Kako obravnavati vsa osnovna fermionska in bozonska polja na ekvivalenten naˇcin? o Kako obravnavati fermionskain bozonska polja, ˇcasres ce je prostor-ˇ ˇzen? stirirazseˇ o Kakose ”Naravaodloˇci”zazlom simetrijodzaˇcetne simetrijev d-razseˇzenih v d=(3+1)? zem prostotu-ˇcasu do opa ˇ o Zakaj je metrika prostora-ˇcasa metrika Minkovskega in kako je izbira metrike povezana z razvojem naˇ sega vesolja? o Kaj povzroˇca inflacijo vesolja? o Kdaj se pojavi inflacija? Po elektroˇsibkem prehodu? o Kaj doloˇca barvno skalo? • Izˇse vesolje polegiz barionov(veˇzine kvarkovin cesa je naˇcinoma iz prve dru ˇ leptonov)? o Kakoˇcrne luknje prispevajoktemni snovi? • Kakˇsna je vloga simetrij v naravi? • Kako narediti poskuse in kako predlagati modele, da ne bi predlagani model preveˇc vplival na izmerjene podatke? Veˇsajo najti novereˇ cina prispevkov je ”nenavadnih” v tem smislu, da poskuˇsitve odprtih problemov. Zbornikje razdeljen na dva dela.Vprvi del so vkljuˇcena vabljenapredavanja,ki so prispela do organizatorjev pravoˇcasno in so bila tudi recenzirana. Vdrugem delu so zbrani prispevki, za katere avtorji menijo, da diskusije niso prinesle odloˇcitve, do katere mere se avtorji strinjajo, ali nestrinjajo s predstavljenimi trditvami, ali pa avtorji menijo, da imajo razliˇ cni pristopi mnogo skupnega ter lahko pripeljejo do novih idej ali novega razumevanja ali celo do sodelovanja. Nekatere od diskusij, ki so se za ˇ cele med delavnico, se v tem zborniku ne pojavijo. Morda se bodo nadaljevale na naslednji delavnici in bodo zapisane v naslednjem zborniku. Organizatorji se iskreno zahvaljujejo vsem sodelujoˇcinkovite cim na delavnici za uˇ predstavitve del, za ˇsje, kljub temu,daje zivahne razprave in dobro delovno vzduˇ veˇzencev sodelovalapreko spleta,kigaje vodil MaximYu. Khlopov. cina udeleˇ Bralec najde vse pogovore in kmalu tudi celoten Zbornik na uradni spletni strani delavnice: http://bsm.fmf.uni-lj.si/bled2022bsm/presentations.html, in na forumu Cosmovia https://bit.ly/bled2022bsm .. Norma Mankoˇc Borˇstnik, Holger Bech Nielsen, Maksim Khlopov, Astri Kleppe Ljubljana, december 2022 Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p.1) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 1 New and recent results, and perspectives from DAMA/LIBRA–phase2 R. Bernabei,P. Belli,A. Bussolotti,V. Caracciolo,R. Cerulli,N. Ferrari,A. Leoncini, V. Merlo,F. Montecchia???1;2 F. Cappella, A. d’Angelo, A. Incicchitti, A. Mattei3;4 C.J. Dai, X.H. Ma, X.D. Sheng, Z.P.Yey5 1 Dip.di Fisica, Universit`adi RomaTorVergata, Rome, Italy 2 INFN, sez. RomaTorVergata, Rome, Italy 3 Dip.di Fisica, Universit`adi RomaLa Sapienza, Rome, Italy 4 INFN, sez. Roma, Rome, Italy 5 Key Laboratoryof Particle Astrophysics IHEP, Chinese Academyof Sciences,Beijing,PR China Abstract. Heretheresults obtainedbyanalysingothertwoannualcyclesof DAMA/LIBRA– phase2 are presented and the long-standing model-independent annual modulation effect measured by DAMA deep underground at the Gran Sasso National Laboratory (LNGS) of the I.N.F.N. with different experimental configurations is summarized. In particular, profiting from a second generation high quantum efficiency photomultipliers and new electronics, the DAMA/LIBRA–phase2 apparatus(' 250kg highly radio-pure NaI(Tl))has allowed the reaching of lower software energy threshold. Including the results of the two new annual cycles, the total exposureof DAMA/LIBRA–phase2 over8annual cyclesis 1.53 ton × yr. The evidenceofa signal that meets all therequirementsof the model independent Dark Matter (DM) annual modulation signature is further confirmed: 11.8 . C.L. in the energyregion(1–6)keV.Inthe energyregion between2and6keV,wheredataarealso available from DAMA/NaI and DAMA/LIBRA–phase1, the achieved C.L. for the full exposure (2.86 ton × yr) is 13.7 ;the modulation amplitude of thesingle-hit scintillation events is: (0:01014 0:00074) cpd/kg/keV,the measured phase is (142:44:2) days and the measured period is (0:99834 ± 0:00067) yr,values all well in agreement with those expected forDM particles.No systematicsorsidereactionabletomimicthe exploitedDM signature (i.e. to account for the whole measured modulation amplitude and to simultaneously satisfy all the requirements of the signature) has been found or suggested by anyone throughout some decades thus far. Povzetek:Avtorji predstavijo rezultate zadnjih in vseh dosedanjih meritev na experimentu DAMA/LIBRA, ki meri letno modulacijo sipanja delcev, za katere zdaj ˇ ze z veliko go- tovostjo menijo, da so lahko samo delci temne snovi. Nacionalni laboratorij Gran Sasso (LNGS) I.N.F.N. se nahaja globoko pod zemljo.Vteh letih so uporabili razliˇ cne konfiguracije in vsebnosti merilcev ter poskrbeli za njihovo ˇcinkovitost.Vposkusu cistost in u ˇ ??? F. Montecchia alsoDip.diIng. Civilee Informatica, Universit‘adi RomaTorVergata, Rome, Italy † Z.P.Ye also Universityof Jinggangshan, Jiangxi, China Authors Suppressed Due to Excessive Length DAMA/LIBRA–phase2(' 250kg visoko radijsko ˇ cistega NaI(Tl)) uporabljajo drugo gen- eracijov fotopomno ˇcinkovitostjo in najsodobnej ˇ zevalkz visoko kvantnouˇso elektroniko, karjimje omogoˇzali energijski prag,do kateregaso meritve ˇ cilo, da so zni ˇse zanesljive. Rezultati novih meritev letne modulacije trkov delcev temne snovi zdelcivmerilni aparaturi, ki so neodvisne od modela, potrjujejo stare meritve temne snovi (1.53 ton × leto) z 11,8 . C.L.(stopnja zanesljivosti)v energijskem obmo ˇcjumed cju (1–6) KeV.Venergijskem obmoˇ (2 -ze s poskusoma DAMA/NaI in DAMA/LIBRA–phase1, 6) KeV, kjer so podatki zbrani ˇ paje C.L. (stopnja zanesljivosti) za polno izpostavljenost (2,86 ton × leto) enaka 13,7 . Am- plituda modulacije scintilacijskih dogodkov single-hit je: (0, 01014 ± 0, 00074) cpd/kg/keV, izmerjena faza je (142, 4 ± 4, 2) dni in izmerjeno obdobje je (0, 99834 ± 0, 00067) na leto. Vsete meritvesov skladuspredpostavko,daso izmerjene dogotke povzroˇ cili delci temne snovi. Noben drug dogodek, v zadnjih desetletjih so jih predlali kar nekaj, ni v skladu z izmerjenimi rezultati. 1.1 Introduction The DAMA/LIBRA [1–23] experiment, as well as the pioneer DAMA/NaI [24– 51], has the main aim to investigate the presence of DM particles in the galactic halo by exploiting the DM annual modulation signature (originally suggested in Ref. [52, 53]). In particular, the developed highly radio-pure NaI(Tl) targetdetectors[ 1,6,9,54] ensuresensitivitytoawide rangeofDM candidates, interaction typesandastrophysical scenarios(seee.g.Refs.[2,14,16–18,25–32,35–42],andin literature). The investigated process is the DM annual modulation signature and related properties; as a consequence of the Earth’s revolution around the Sun, which is moving in the Galaxy with respect to the Local Standardof Rest towards the star Vega near the constellation of Hercules, the Earth should be crossed by a larger flux of DM particles around ' 2June andbya smaller one around ' 2December (in the first case the Earth orbital velocity is summed to that of the solar system withrespecttotheGalaxy,whileintheotheronethetwo velocitiesare subtracted). Thus,thisDMannual modulationsignatureisduetotheEarthmotionwithrespect to the DM particles constituting the Galactic Dark Halo. TheDM annual modulation signatureisvery distinctive sincetheeffect inducedby DM particles must simultaneously satisfy all the followingrequirements: the rate must contain a component modulated according to a cosine function (1) with one year period(2)andaphase thatpeaksroughly ' 2June (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 detectorof many 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% of the constant part of the signal for usually adopted halo distributions (6), but it can be larger in case of some proposed scenarios such as e.g. those in Ref. [55–59] (even up to ' 30%). Thus this signature hasmany peculiaritiesand,in addition,itallowstotestawiderangeof parameters in many possible astrophysical, nuclear and particle physics scenarios. This DM signature might be mimicked only by systematic effects or side reactions able 1 New andrecentresults, and perspectivesfrom DAMA/LIBRA–phase2 to account for the whole observed modulation amplitude and to simultaneously satisfy all therequirements given above. The description of the DAMA/LIBRA set-up and the adopted procedures during thephase1andphase2andotherrelatedargumentshavebeen discussedin details e.g. in Refs. [1–6,19–21, 23]. The radio-purity and details are discussed e.g. in Refs.[1–5,54] andreferences therein. The adoptedproceduresprovide sensitivity to large and low massDM candidates inducing nuclearrecoils and/or electromagnetic signals. The data of the former DAMA/NaI setup and, later, those of the DAMA/LIBRA–phase1 have already given (with high confidence level) positive evidence for the presence of a signal that satisfies all the requirements of the exploitedDM annual modulation signature[2–5,35,36].In particular,attheendof 2010 all the photomultipliers (PMTs) were replaced by a second generation PMTs Hamamatsu R6233MOD, with higher quantum efficiency (Q.E.) and with lower background with respect to those used in phase1, allowing the achievement of the software energythresholdat1keVaswellastheimprovementof some detector’s features such as energy resolution and acceptance efficiency near software energy threshold [6]. The adopted procedure for noise rejection near software energy threshold and the acceptance windows are the same unchanged along all the DAMA/LIBRA–phase2 data taking, throughout the months and the annual cycles. The typical behaviour of the overall efficiency for single-hit events as a function of the energy is also shown in Ref. [6]; the percentage variations of the efficiency follow a gaussian distribution with . = 0.3% and do not show any modulation with period and phase as expected for the DM signal (for a partial data release see Ref. [21]). At the end of 2012 new preamplifiers and special developed trigger modules were installed and the apparatus was equipped with more compact electronic modules [60]. In particular, the sensitive part of DAMA/LIBRA–phase2 set-up is made of 25 highly radio-pure NaI(Tl) crystal scintillators (5-rows by 5-columns matrix) having 9.70 kg mass each one; quantitative analyses of residual contaminants are given in Ref. [1]. In each detector two 10 cm long UV light guides (madeof SuprasilBquartz) act also as optical windows on the two end faces of the crystal, and are coupled to two low background PMTs working in coincidence at single photoelectron level. The detectors are housed in a sealed low-radioactive copperbox installedinthe centerofa low-radioactive Cu/Pb/Cdfoils/ polyethylene/paraffin shield; moreover, about1 m concrete (madefrom the Gran Sassorock material) almost fully surrounds (mostly outside the barrack) this passive shield, acting as a further neutron moderator. The shield is decoupled from the ground by a metallic structure mounted above a concrete basement; a neoprene layer separates the concrete basement and the floor of the laboratory. The space between this basement and the metallic structure is filled by paraffin for several tens cm in height.Athreefold-level sealing system prevents the detectors from contact with the environmental air of the underground laboratory and continuously maintains them in HP (high-purity) Nitrogen atmosphere. The whole installation is under air conditioning to ensure a suitable and stable working temperature.Thehugeheat capacityofthe multi-tonspassive shield(. 106 cal/oC) guarantees further relevant stability of the detectors’ operating temperature. In particular, two independent systems of air conditioning are available for Authors Suppressed Due to Excessive Length redundancy: one cooledby waterrefrigeratedbya dedicated chiller and the other operating with cooling gas.Ahardware/software monitoring systemprovides data on the operating conditions. In particular, several probes are read out and the results are stored with the production data. Moreover, self-controlled computer based processes automatically monitor several parameters, including those from DAQ, and manage the alarms system. All these procedures, already experienced during DAMA/LIBRA–phase1 [1–5], allow us to control and to maintain the running conditions stableata level better than1% alsoin DAMA/LIBRA–phase2(see e.g. Ref. [21, 23]). Duringphase2thelightresponseofthe detectors typically rangesfrom6to10 photoelectrons/keV, depending on the detector. Energy calibration with X-rays/. sourcesareregularly carriedoutinthe samerunning conditiondowntofewkeV (for details see e.g. Ref. [1]); in particular, double coincidences due to internal X-rays from 40K(which is at ppt levels in the crystals) provide (when summing the data over long periods) a calibration point at 3.2 keV close to the software energy threshold. The DAQ system records both single-hit events (where just one of the detectors fires) and multiple-hit events (where more than one detector fires) uptotheMeVregion despitethe optimizationis performedforthe lowest energy. 1.2 Eight DAMA/LIBRA–phase2 annual cycles Table 1.1 summarizes the details of the DAMA/LIBRA–phase2 annual cycles including thelast tworeleased ones. The first cycle was dedicated to commissioning and optimizations towards the achievementof the1keV software energy threshold [6]. On the other hand that cycle having: i) no data before/near Dec. 2, 2010 (the expected minimum of the DM signal); ii) data sets with some set-up modifications; iii) (. - 2)= 0:355 well different from 0.5 (i.e. the detectors were not being operational evenly throughout the year), cannot be used for the annual modulationstudies;however,ithasbeenusedforother purposes[6,13].Thus(see Table 1.1) the considered annual cycles of DAMA/LIBRA–phase2 are eight for an exposure of 1.53 tonyr. The cumulative exposure, when considering also the former DAMA/NaI and DAMA/LIBRA–phase1, is 2.86 tonyr. The total number of events collected for the energy calibrations during the eight annual cycles of DAMA/LIBRA–phase2 is about 1:6 × 108, while about 1:7 × 105 events/keV have been collected for the evaluation of the acceptance window efficiencyfor noiserejection nearthesoftware energythreshold[1,6]. Finally,the duty cycleof the experimentis high, ranging between 76% and 86%: theroutine calibrations and the data collection for the acceptance windows efficiency mainly affect it. 1.2.1 The annual modulation of the residual rate In Fig. 1.1 the time behaviours of the experimental residual rates of the single- hit scintillation events in the (1–3), and (1–6) keV energy intervals are shown 1 New andrecentresults, and perspectivesfrom DAMA/LIBRA–phase2 Table 1.1: Details about the annual cycles of DAMA/LIBRA–phase2. The mean value of the squared cosine is . = hcos2!(t - t0)i and the mean value of the cosine is ß = hcos!(t - t0)i (the averages are taken over the live time of the data taking and t0 = 152:5 day, i.e. June2nd); thus, the variance of the cosine, (. - 2), is ' 0:5 fora detector being operational evenly throughout the year. DAMA/LIBRA–phase2 Period Mass Exposure (. - 2) annual cycle (kg) (kgday) 1 Dec. 23, 2010 – Sept. 9, 2011 commissioning of phase2 2 Nov. 2, 2011 – Sept. 11, 2012 242.5 62917 0.519 3 Oct. 8, 2012 – Sept. 2, 2013 242.5 60586 0.534 4 Sept. 8, 2013 – Sept. 1, 2014 242.5 73792 0.479 5 Sept. 1, 2014 – Sept. 9, 2015 242.5 71180 0.486 6 Sept. 10, 2015 – Aug. 24, 2016 242.5 67527 0.522 7 Sept. 7, 2016 – Sept. 25, 2017 242.5 75135 0.480 8 Sept. 25, 2017 – Aug. 20, 2018 242.5 68759 0.557 9 Aug. 24, 2018 – Oct. 3, 2019 242.5 77213 0.446 DAMA/LIBRA–phase2 Nov. 2, 2011 – Oct. 3, 2019 557109 kgday ' 1.53 tonyr 0.501 DAMA/NaI+DAMA/LIBRA–phase1+DAMA/LIBRA–phase2: 2.86 tonyr for DAMA/LIBRA–phase2. The residual rates are calculated from the measured rate of the single-hit events after subtracting the constant part, as described in Refs.[2–5,35,36].Thenull modulation hypothesisisrejectedatveryhighC.L.by 2 test: 2 = 176 and 202, respectively, over 69 d.o.f. (P = 2.6 × 10-11, andP = 5.6 × 10-15 , respectively). The residuals of the DAMA/NaI data (0.29 ton × yr) are giveninRef.[2,5,35,36], while thoseof DAMA/LIBRA–phase1(1.04ton × yr) in Ref. [2–5]. The former DAMA/LIBRA–phase1 and the new DAMA/LIBRA–phase2 residual rates of the single-hit scintillation events are reported in Fig. 1.2. The energy intervalisfrom2keV, the software energy thresholdof DAMA/LIBRA –phase1,up to6keV.Thenull modulation hypothesisisrejectedatveryhighC.L.by 2 test: 2=d:o:f. = 240/119, corresponding to P-value = 3.5 × 10-10 . The single-hit residual rates of the DAMA/LIBRA–phase2 (Fig. 1.1) have been fitted 2. with the function: A cos !(t - t0), consideringa period T == 1 yr anda phase . t0 = 152:5 day (June2nd)as expectedbytheDM annual modulation signature;this can be repeated for the only case of (2-6) keV energy interval when including also the former DAMA/NaI and DAMA/LIBRA–phase1 data. The goodness of the fits is well supportedbythe 2 test; for example, 2=d:o:f. = 81:6=68, 66:2=68, 130=155 are obtained for the (1–3) keV and (1–6) keV cases of DAMA/LIBRA–phase2, and for the (2–6) keV case of DAMA/NaI, DAMA/LIBRA–phase1 and DAMA/LIBRA– phase2, respectively. The results of the best fits in the different cases are summarizedinTable 1.2.InTable 1.2 also the cases when the period and the phase are kept free in the fitting procedure are shown. The period and the phase are well compatible with expectations for a DM annual modulation signal. In particular, the phase is consistent with about June 2nd and is fully consistent with the value independently determinedby Maximum Likelihood analysis (see later). For com Authors Suppressed Due to Excessive Length 1-3 keV Residuals (cpd/kg/keV) DAMA/LIBRA-phase2 »250 kg (1.53 ton×yr) 1-6 keV Residuals (cpd/kg/keV) DAMA/LIBRA-phase2 »250 kg (1.53 ton×yr) Fig.1.1: Experimentalresidual rateofthe single-hit scintillation events measuredby DAMA/LIBRA–phase2 over eight annual cycles in the (1–3), and (1–6) keV energy intervals as a function of the time. The time scale is maintained the same of the previous DAMA papers for consistency. The data points present the experimental errors as vertical bars and the associated time bin width as horizontal bars. The superimposed curves are the cosinusoidal functional forms A cos !(t - t0) with 2. a period T == 1 yr, a phase t0 = 152:5 day (June2nd)and modulation . amplitudes, A, equal to the central values obtained by best fit on the data points of the entire DAMA/LIBRA–phase2. The dashed vertical lines correspond to the maximum expected for theDM signal (June2nd), while the dotted vertical lines correspond to the minimum. Residuals (cpd/kg/keV) DAMA/LIBRA-phase1 (1.04 ton×yr)Fig. 1.2: Experimental residual rate of the single-hit scintillation events measured by DAMA/LIBRA–phase1 and DAMA/LIBRA–phase2 in the (2–6) keV energy intervals as a function of the time. The superimposed curve is the cosinusoidal 2. functional forms A cos !(t - t0) with a period T == 1 yr, a phase = 152:5 . t0 day (June2nd)and modulation amplitude,A, equal to the central value obtained by best fit on the data points of DAMA/LIBRA–phase1 and DAMA/LIBRA– phase2. For details see Fig. 1.1. pleteness,werecallthataslight energy dependenceofthephasecouldbe expected (see e.g. Refs. [38,58,59,61–63]),providing intriguing information on the natureof Dark Matter candidate and related aspects. 1 New andrecentresults, and perspectivesfrom DAMA/LIBRA–phase2 Table 1.2: Modulation amplitude,A, obtained by fitting the single-hit residual rate of DAMA/LIBRA–phase2,asreportedinFig.1.1,andalso includingtheresidual ratesof the former DAMA/NaI and DAMA/LIBRA–phase1.It was obtainedby 2. fitting the data with the formula: A cos !(t - t0). The period T = and the phase . t0 arekept fixedat1yrandat 152.5day(June2nd),respectively,as expectedbythe DM annual modulation signature, and alternatively kept free. The results are well compatible with expectations for a signal in the DM annual modulation signature. 2. A (cpd/kg/keV) T = . (yr) t0 (days) C.L. DAMA/LIBRA–phase2: 1-3 keV (0.01910.0020) 1.0 152.5 9.7 . 1-6 keV (0.010480.00090) 1.0 152.5 11.6 . 2-6 keV (0.009330.00094) 1.0 152.5 9.9 . 1-3 keV (0.01910.0020) (0.999520.00080) 149.65.9 9.6 . 1-6 keV (0.010580.00090) (0.998820.00065) 144.55.1 11.8 . 2-6 keV (0.009540.00076) (0.998360.00075) 141.15.9 12.6 . DAMA/LIBRA–phase1+phase2: 2-6 keV (0.009410.00076) 1.0 152.5 12.4 . 2-6 keV (0.009590.00076) (0.998350.00069) 142.04.5 12.6 . DAMA/NaI+DAMA/LIBRA–phase1+phase2: 2-6 keV (0.009960.00074) 1.0 152.5 13.4 . 2-6 keV (0.010140.00074) (0.998340.00067) 142.44.2 13.7 . 1.2.2 Absence of background modulation Since the background in the lowest energy region is essentially due to “Compton” electrons, X-rays and/or Auger electrons, muon induced events, etc., which are strictly correlated with the events in the higher energy region of the spectrum, if a modulation detected in the lowest energy region were due to a modulation of the background (rather than to a signal), an equal or larger modulation in the higher energyregions shouldbepresent. Thus, as doneinprevious datareleases, absence of any significant background modulation in the energy spectrum for energy regions not of interest for DM. has also been verified in the present one. In particular,the measured rate integrated above90keV,R90,asafunctionofthetime has been analysed. Fig. 1.3 shows the distribution of the percentage variations of R90 withrespecttothe mean valuesforallthe detectorsin DAMA/LIBRA–phase2. It shows a cumulative gaussian behaviour with . ' 1%, well accounted for by the statistical spread provided by the used sampling time. Moreover, fitting the time behaviourofR90 includingatermwithphaseandperiodasforDM particles, a modulation amplitude AR90 compatible with zero has been found for all the annualcycles(seeTable1.3).Thisalso excludesthepresenceofanybackground modulation in the whole energy spectrum at a level much lower than the effect found in the lowest energy region for the single-hit scintillation events. In fact, otherwise – considering theR90 mean values – a modulation amplitude of order of tens cpd/kg would be present for each annual cycle, that is ' 100 . far away from the measured values. Authors Suppressed Due to Excessive Length (R90 - )/frequency0500100015002000250030003500-0.100.1 Fig.1.3: Distributionofthepercentage variationsofR90 with respect to the mean values for all the detectors in the DAMA/LIBRA–phase2 (histogram); the superimposed curve is a gaussian fit. Table 1.3: Modulation amplitudes,AR90 , obtained by fitting the time behaviour ofR90 in DAMA/LIBRA–phase2, includinga term witha cosine function having phase and period as expected for a DM signal. The obtained amplitudes are compatible with zero, and incompatible(' 100 )with modulation amplitudes of tens cpd/kg. Modulation amplitudes, A(6-14), obtained by fitting the time behaviour of the residual rates of the single-hit scintillation events in the (6–14) keV energy interval. In the fit the phase and the period are at the values expected for a DM signal. The obtained amplitudes are compatible with zero. DAMA/LIBRA–phase2 annual cycle AR90 (cpd/kg) A(6-14) (cpd/kg/keV) 2 (0.120.14) (0.00320.0017) 3 -(0.080.14) (0.00160.0017) 4 (0.070.15) (0.00240.0015) 5 -(0.050.14) -(0.00040.0015) 6 (0.030.13) (0.00010.0015) 7 -(0.090.14) (0.00150.0014) 8 -(0.180.13) -(0.00050.0013) 9 (0.080.14) -(0.00030.0014) 1 New andrecentresults, and perspectivesfrom DAMA/LIBRA–phase2 Similar results are obtained when comparing the single-hit residuals in the (1–6) keV with those in other energy intervals; for example Fig. 1.4 shows the single-hit residualsin the (1–6) keV andin the (10–20) keV energyregions, for the8annual cyclesof DAMA/LIBRA–phase2 asif they were collectedina single annual cycle (i.e. binning in the variable time from the January 1st of each annual cycle). Time Residuals (cpd/kg/keV) -0.02-0.0100.010.02300400500600 Time Residuals (cpd/kg/keV) -0.02-0.0100.010.02300400500600 Fig. 1.4: Experimental single-hit residuals in the (1–6) keV and in the (10–20) keV energy regions for DAMA/LIBRA–phase2 as if they were collected in a single annual cycle (i.e. binning in the variable time from the January 1st of each annual cycle). The data points present the experimental errors as vertical bars and the associated time bin width as horizontal bars. The initial time of the figures is taken at August7th.A clear modulation satisfying all the peculiarities of the DM annual modulation signature is present in the lowest energy interval with A=(0.00956 ± 0.00090) cpd/kg/keV, while it is absent just above: A=(0.0007 ± 0.0005) cpd/kg/keV. Moreover,Table 1.3 shows the modulation amplitudes obtainedby fitting the time behaviour of the residual rates of the single-hit scintillation events in the (6–14) keV energy interval for the DAMA/LIBRA–phase2 annual cycles. In the fit the phase and the period are at the values expected for a DM signal. The obtained amplitudes are compatible with zero. Afurther relevant investigationon DAMA/LIBRA–phase2 data has been per- formedby applyingthe samehardwareand softwareprocedures,usedtoacquire and to analyse the single-hit residual rate, to the multiple-hit one. Since the probability that a DM particle interacts in more than one detector is negligible, a DM signal can be present just in the single-hit residual rate. Thus, the comparison of theresultsof the single-hit events with those of the multiple-hit ones corresponds to compare the cases of DM particles beam-on and beam-off. This procedure also allows an additional test of the background behaviour in the same energy interval where the positive effect is observed. In particular, in Fig. 1.5 the residual rates of the single-hit scintillation events collected during8annual cyclesof DAMA/LIBRA–phase2 arereported, as collected in a single cycle, together with the residual rates of the multiple-hit events, in Authors Suppressed Due to Excessive Length the considered energy intervals. While, as already observed,a clear modulation, satisfying all the peculiarities of the DM annual modulation signature, is present in the single-hit events, the fitted modulation amplitude for the multiple-hit residual rate is well compatible with zero: (0:00030 0:00032) cpd/kg/keV in the (1–6) keV energy region. Thus, again evidence of annual modulation with proper features as required by the DM annual modulation signature is present in the single-hit residuals (events class to which the DM particle induced events belong), while it is absent in the multiple-hit residual rate (event class to which only background events belong). Similarresults were also obtained for the two last annual cyclesof DAMA/NaI [36] and for DAMA/LIBRA–phase1 [2–5]. Since the same identical hardware and the same identical software procedures have been used to analyse the two classes of events, the obtained result offers an additional strong support for the presence of a DM particle component in the galactic halo. 1-6 keV Residuals (cpd/kg/keV) Fig. 1.5: Experimental residual rates of DAMA/LIBRA–phase2 single-hit events (filled red on-line circles), class of events to which DM events belong, and for multiple-hit events (filled green on-line triangles), class of events to which DM events do not belong. They have been obtained by considering for eachclass of eventsthedataas collectedinasingle annualcycleandbyusinginboth cases the same identical hardware and the same identical software procedures. The initialtimeofthefigureis takenon August7th. The experimental points present the errors as vertical bars and the associated time bin width as horizontal bars. Analogous results were obtained for DAMA/NaI (two last annual cycles) and DAMA/LIBRA–phase1 [2–5, 36]. In conclusion, no background process able to mimic the DM annual modulation signature (that is, able to simultaneously satisfy all the peculiarities of the signature and to account for the measured modulation amplitude) has been found or suggested by anyone throughout some decades thus far (see also discussions e.g. in Ref. [1–5,7,8,19–21,23,34–36]). 1.3 The analysis in frequency In order to perform the Fourier analysis of the data of DAMA/LIBRA–phase1 and of thepresent8annual cyclesof phase2ina widerregionof consideredfrequency, the single-hit eventshavebeengroupedin1day bins.Duetothelow statisticsin 1 New andrecentresults, and perspectivesfrom DAMA/LIBRA–phase2 Normalized Power020406000.10.20.30.40.5 Normalized Power051000.10.20.30.40.5 (2-6) 6-14) Normalized Power020406000.0020.0040.0060.0080.010.0120.014 Fig. 1.6: Power spectra of the time sequence of the measured single-hit events for DAMA/LIBRA–phase1and DAMA/LIBRA–phase2groupedin1day bins.From top to bottom: spectra up to the Nyquist frequency for (2–6) keV and (6–14) keV energy intervals and their zoom around the1y-1 peak, for (2–6) keV (solid line) and (6–14) keV (dotted line) energy intervals. The main mode present at the lowest energy interval correspondstoafrequencyof 2:74 × 10-3 d-1 (vertical line, purple on-line).It correspondstoa periodof ' 1year.Asimilarpeakisnotpresentinthe (6–14) keV energy interval. The shaded (green on-line) area in the bottom figure – calculated by Monte Carlo procedure – represents the 90% C.L. region where all thepeaksare expectedtofallforthe(2–6)keVenergy interval.Inthefrequency range far from the signal for the (2–6) keV energy region and for the whole (6–14) keVspectrum,theupperlimitofthe shadedregion(90%C.L.)canbe calculatedto be 10.8 (continuous lines, green on-line). Authors Suppressed Due to Excessive Length eachtimebin,aprocedure detailedinRef.[64]hasbeenapplied.Fig.1.6showsthe whole power spectra up to the Nyquist frequency and the zoomed ones: a clear peak correspondingtoaperiodof1yearis evidentforthe lowest energy interval, whilethe same analysisinthe(6–14)keV energyregionshowsonlyaliasingpeaks, instead. Neither other structure at different frequencies has been observed.To derive the significance of the peaks present in the periodogram, one can remind that the periodogram ordinate, z, at each frequency follows a simple exponential distribution e-z in case of null hypothesis or white noise [65]. (1-6) Normalized Power020406000.0020.0040.0060.0080.010.0120.014 Fig. 1.7: Power spectrum of the time sequence of the measured single-hit events in the (1–6) keV energy interval for DAMA/LIBRA–phase2groupedin1day bin. The main mode present at the lowest energy interval corresponds to a frequency of 2:77 × 10-3 d-1 (vertical line, purple on-line). It corresponds to a period of ' 1year. The shaded (green on-line) area – calculated by Monte Carlo procedure– represents the 90% C.L. region where all the peaks are expected to fall for the (1–6) keV energy interval. Thus, if M independent frequencies are scanned, the probability to obtain values larger than z is: P(>z)= 1 -(1 - e-z)M. In general M depends on the number of sampled frequencies, the number of data points N, and their detailed spacing. It turns out that M ' N when the data points are approximately equally spaced and whenthe sampledfrequencies coverthefrequency rangefrom0tothe Nyquist one [66, 67]. In the present case, the number of data points used to obtain the spectra in Fig. 1.6 is N = 5047 (days measured over the 5479 days of the 15 DAMA/LIBRA–phase1 and phase2 annual cycles) and the full frequencies region up to Nyquist one has been scanned. Thus, assuming M = N, the significance levels P = 0.10, 0.05 and 0.01, correspond to peaks with heights larger than z = 10.8, 11.5 and 13.1,respectively,in the spectraofFig 1.6.In the case below6keV, a signal is present; thus, to properly evaluate the C.L. the signal must be included. This has been doneby a dedicated Monte Carloprocedure wherea large number of similar experiments has been simulated. The 90% C.L. region (shaded, green on-line) where all the peaks are expected to fall for the (2–6) keV energy interval is 1 New andrecentresults, and perspectivesfrom DAMA/LIBRA–phase2 reported in Fig 1.6. Several peaks, satellite of the one year period frequency, are present. Moreover, for each annual cycle of DAMA/LIBRA–phase1 and phase2, the annual baseline counting rates have been calculated for the (2–6) keV energy interval. Theirpower spectruminthefrequency range 0:00013 - 0:0019 d-1 (corresponding to a period range 1.4–21.1 year) has been calculated according to Ref. [5]. No statistically-significantpeakispresentatfrequencies lowerthan1y-1. This implies that no evidence for a long term modulation in the counting rate is present. Finally, the case of the (1–6) keV energy interval of the DAMA/LIBRA–phase2 data is reported in Fig. 1.7. As previously the only significant peak is the one corresponding to one year period. No other peak is statistically significant being below the shaded (green on-line) area obtained by Monte Carlo procedure. In conclusion,apartfromthepeak correspondingtoa1yearperiod,nootherpeak is statistically significant either in the low and high energy regions. 1.4 The modulation amplitudes by the maximum likelihood approach Theannual modulationpresentatlowenergycanalsobepointedoutbydepicting the energy dependence of the modulation amplitude, Sm(E), obtained by maximum likelihood method considering fixed period and phase: T =1yr andt0 = 152.5 day. For this purpose the likelihood function of the single-hit experimental Nijk µ -ijk ijk data in the k-th energy bin is defined as: Lk = ije , where Nijk is the Nijk! number of events collected in the i-thtime interval(hereafter1day),bythe j-th detector and in the k-th energy bin. Nijk follows a Poisson’s distribution with expectation value ijk =[bjk + Si(Ek)] MjtiEjk. The bjk are the background contributions, Mj is the mass of the j-th detector, ti is the detector running time during the i-th time interval, E is the chosen energybin, jk is the overall efficiency. The signal can be written as: Si(E)= S0(E)+ Sm(E) · cos !(ti - t0), where S0(E) is the constant part of the signal and Sm(E) is the modulation amplitude. The usual procedure is to minimize the function yk =-2ln(Lk)- const for each energy bin; the free parameters of the fit are the (bjk + S0) contributions and the Sm parameter. The modulation amplitudes for the whole data sets: DAMA/NaI, DAMA /LIBRA– phase1 and DAMA/LIBRA–phase2 (total exposure 2.86 tonyr) are plotted in Fig.1.8;thedata below2keVreferonlytothe DAMA/LIBRA–phase2 exposure (1.53 tonyr). It can be inferred that positive signal is present in the (1–6) keV energy interval, while Sm values compatible with zero are present just above. All this confirms the previous analyses. The test of the hypothesis that the Sm values in the (6–14) keV energy interval have random fluctuations around zero yields 2=d:o:f. equal to 20.3/16 (P-value = 21%). For the case of (6–20) keV energy interval 2=d:o:f. = 42.2/28 (P-value = 4%). The obtained 2 value is rather large due mainly to two data points, whose centroids Authors Suppressed Due to Excessive Length Energy Sm (cpd/kg/keV) -0.05-0.02500.0250.0502468101214161820 Fig. 1.8: Modulation amplitudes, Sm, for the whole data sets: DAMA/NaI, DAMA/LIBRA–phase1 and DAMA/LIBRA–phase2 (total exposure 2.86 tonyr) above2keV;below2keV only the DAMA/LIBRA–phase2 exposure (1.53 ton × yr) is available and used. The energy bin E is 0.5 keV.Aclear modulationis present in the lowest energy region, while Sm values compatible with zero are present just above. In fact, the Sm values in the (6–20) keV energy interval have random fluctuations around zero with 2=d:o:f. equal to 42.2/28 (P-value is 4%). are at 16.75 and 18.25 keV, far away from the (1–6) keV energy interval. The P- values obtained by excluding only the first and either the points are 14% and 23%. This method also allows the extraction of the Sm values for each detector. In particular, the modulation amplitudes Sm integrated in the range (2–6) keV for each of the 25 detectors for the DAMA/LIBRA–phase1 and DAMA/LIBRA–phase2 periods can be produced. They have random fluctuations around the weighted averaged value confirmed by the 2 analysis. Thus, the hypothesis that the signal is well distributed over all the 25 detectors is accepted. Aspreviously done for the other datareleases[2–5,19–21,23], the Sm values for each detector for each annual cycle and for each energy bin have been obtained. The Sm are expected to follow a normal distribution in absence of any systematic Sm-hSmi effects. Therefore, the variable x = has been considered to verify that the . Sm are statistically well distributedin the16 energybins(E = 0:25 keV) in the (2–6) keV energy interval of the seven DAMA/LIBRA–phase1 annual cycles and in the 20 energy bins in the (1–6) keV energy interval of the eight DAMA/LIBRA– phase2 annual cycles and in each detector. Here, . are the errors associated to Sm and hSmi are the mean values of the Sm averaged over the detectors and the annual cycles for each considered energy bin. Defining 2 = x2, where the sum is extended over all the 272 (192 for the 16th detector [4]) x values, 2=d:o:f. values ranging from 0.8 to 2.0 are obtained, depending on the detector. The mean value of the 25 2=d:o:f. is 1.092, slightly larger than 1. Although this can be still ascribed to statistical fluctuations, let us ascribe it to a possible systematics. In this case, one would derive an additional error to the modulation amplitude measured below6keV: . 2:4 × 10-4 cpd/kg/keV, if combining quadratically the 1 New andrecentresults, and perspectivesfrom DAMA/LIBRA–phase2 errors, or . 3:6 × 10-5 cpd/kg/keV, if linearly combining them. This possible additional error: . 2:4%or . 0:4%, respectively, on the DAMA/LIBRA–phase1 and DAMA /LIBRA–phase2 modulation amplitudes is an upper limit of possible systematic effects coming from the detector to detector differences. Among further additional tests, the analysis of the modulation amplitudes as a function of the energy separately for the nine inner detectors and the remaining external ones has been carried out for DAMA/LIBRA–phase1 and DAMA/LIBRA– phase2, as already done for the other data sets[2–5,19–21,23]. The obtained values are fully in agreement; in fact, the hypothesis that the two sets of modulation amplitudes belong to same distribution has been verified by 2 test, obtaining e.g.: 2=d:o:f. = 1.9/6 and 36.1/38 for the energy intervals (1–4) and (1–20) keV, respectively(E = 0.5 keV). This shows that the effect is also well shared between inner and outer detectors. Moreover, to test the hypothesis that the amplitudes, singularly calculated for each annual cycle of DAMA/LIBRA–phase1 and DAMA/LIBRA–phase2, are compatible and normally fluctuating around their mean values, the 2 test has been performed together with another independent statistical test: the run test (see e.g. Ref. [69]), which verifies the hypothesis that the positive (above the mean value) and negative (under the mean value) data points are randomly distributed. Both tests accept at 95% C.L. the hypothesis that the modulation amplitudes are normally fluctuating around the best fit values. 1.5 Investigation of the annual modulation phase Finally, let us release the assumption of the phase value at t0 = 152:5 day in the procedure to evaluate the modulation amplitudes, writing the signal as: Si(E)= S0(E)+ Sm(E) cos !(ti - t0)+ Zm(E) sin !(ti - t0) (1.1) = S0(E)+ Ym(E) cos !(ti - t * ). For signals induced by DM particles one should expect: i) Zm ~ 0 (because of the orthogonality between the cosine and the sine functions); ii) Sm ' Ym;iii) * t ' t0 = 152:5 day. In fact, these conditions hold for most of the dark halo models; however, as mentioned above, slight differences can be expected in case of possible contributions from non-thermalized DM components (see e.g. Refs. [38,58,59,61–63]). Considering cumulativelythe data of DAMA/NaI, DAMA/LIBRA–phase1 and DAMA/LIBRA–phase2 the obtained 2. contours in the plane (Sm;Zm) for the (2–6) keV and (6–14) keV energy intervals are shown in Fig. 1.9–left while the obtained 2. contours in the plane (Ym;t ) are depicted in Fig. 1.9–right. Moreover, Fig. 1.9 also shows only for DAMA/LIBRA–phase2 the 2. contours in the (1–6) keV energy interval. The best fit valuesin the considered cases(1. errors) forSm versusZm and Ym versus t * arereportedinTable 1.4. 16 Authors Suppressed Due to Excessive Length Sm (cpd/kg/keV) Zm (cpd/kg/keV) 2-6 keV1-6 keV6-14 keV2s contours-0.0100.01-0.0100.01 Ym (cpd/kg/keV) t* (day) 2-6 keV1-6 keV6-14 keV2s contours80100120140160180200220240-0.04-0.03-0.02-0.0100.010.020.030.04 Sm (cpd/kg/keV) Zm (cpd/kg/keV) 2-6 keV1-6 keV6-14 keV2s contours-0.0100.01-0.0100.01 Ym (cpd/kg/keV) t* (day) 2-6 keV1-6 keV6-14 keV2s contours80100120140160180200220240-0.04-0.03-0.02-0.0100.010.020.030.04 Fig. 1.9: 2. contours in the plane (Sm;Zm) (left)and in the plane(Ym;t ) (right) for: i) DAMA/NaI, DAMA/LIBRA–phase1 and DAMA/LIBRA–phase2 in the (2–6) keV and (6–14) keV energy intervals (light areas, green on-line); ii) only DAMA/LIBRA–phase2 in the (1–6) keV energy interval (dark areas, blue on- line). The contours have been obtainedby the maximum likelihood method.A modulation amplitude is present in the lower energy intervals and the phase agrees with that expected for DM induced signals. Table 1.4: Best fit values(1. errors) forSm versusZm and Ym versus t ,considering: i) DAMA/NaI, DAMA/LIBRA–phase1 and DAMA/LIBRA–phase2 in the (2–6) keV and (6–14) keV energy intervals; ii) only DAMA/LIBRA–phase2 in the (1–6) keV energy interval. See also Fig. 1.9. E(keV) Sm Zm Ym t * (day) (cpd/kg/keV) (cpd/kg/keV) (cpd/kg/keV) DAMA/NaI+DAMA/LIBRA–phase1+DAMA/LIBRA–phase2: 2–6 (0.0097 ± 0.0007) -(0.0003 ± 0.0007) (0.0097 ± 0.0007) (150.5 ± 4.0) 6–14 (0.0003 ± 0.0005) -(0.0006 ± 0.0005) (0.0007 ± 0.0010) undefined DAMA/LIBRA–phase2: 1–6 (0.0104 ± 0.0007) (0.0002 ± 0.0007) (0.0104 ± 0.0007) (153.5 ± 4.0) Finally,the Zm values as functionof the energy have also been determinedbyusing the same procedure and setting Sm in eq. (1.1) to zero. The Zm values as a function of the energy for DAMA/NaI, DAMA/LIBRA–phase1, and DAMA/LIBRA– phase2 data sets are expected to be zero. The 2 test applied to the data supports the hypothesis that the Zm values are simply fluctuating around zero; in fact, in the (1–20) keV energyregion the 2=d:o:f. is equal to 40.6/38 corresponding toa P-value = 36%. The energy behaviors of Ym and of phase t * are also produced for the cumulative exposure of DAMA/NaI, DAMA/LIBRA–phase1, and DAMA/LIBRA–phase2; as in the previous analyses, an annual modulation effect is present in the lower energyintervals and the phase agrees with that expected forDM induced signals. No modulationispresent above6keV and the phaseis undetermined. 1 New andrecentresults, and perspectivesfrom DAMA/LIBRA–phase2 1.6 Perspectives To further increase the experimental sensitivity of DAMA/LIBRA and to disentangle some of the many possible astrophysical, nuclear and particle physics scenarios in the investigation on the DM candidate particle(s), an increase of the exposure (M × trunning, i.e. trunning in our case at fixed M)in the lowest energy bin anda furtherdecreasingofthe softwareenergythresholdare needed.Thisispursuedby running DAMA/LIBRA–phase2 and upgrading the experimental set-up to lower the software energy threshold below1keV with high acceptanceefficiency. Firstly, particular efforts for lowering the software energy threshold have been done in the already-acquired data of DAMA/LIBRA–phase2 by using the same technique as before with dedicated studies on the efficiency. As consequence, a new data point has been added in the modulation amplitude as function of energy downto0.75keV,seeFig.1.10.Amodulationisalsopresentbelow1keV,from0.75 keV. This preliminary result confirms the necessity to lower the software energy threshold by a hardware upgrade and an improved statistics in the first energy bin. Energy Sm (cpd/kg/keV) -0.05-0.02500.0250.0502468101214161820 Fig. 1.10:As Fig. 1.8; the new data point below1keV, with software energy thresholdat0.75keV, showsthatan annual modulationisalsopresent below1keV.This preliminary result confirms the necessity to lower the software energy threshold bya hardware upgrade and to improve the experimental error on the first energy bin. This dedicatedhardware upgradeof DAMA/LIBRA–phase2is underway.It consists in equipping all the PMTs with miniaturized low background new concept preamplifiers andHV dividers mounted on the same socket, andrelated improvements of the electronic chain, mainly the use of higher vertical resolution 14-bit digitizers. 1.7 Conclusions DAMA/LIBRA–phase2 confirms a peculiar annual modulation of the single-hit scintillation eventsin the (1–6) keV energyregion satisfying all the manyrequirements of the DM annual modulation signature; the cumulative exposure by the Authors Suppressed Due to Excessive Length former DAMA/NaI, DAMA/LIBRA–phase1 and DAMA/LIBRA–phase2 is 2.86 ton × yr. As required by the exploited DM annual modulation signature: 1) the single-hit events show a clear cosine-like modulation as expected for the DM signal; 2) the measuredperiodiswell compatiblewiththe1yrperiodas expectedfortheDM signal;3)the measuredphaseis compatiblewiththeroughly ' 152.5 days expected for the DM signal; 4) the modulation is present only in the low energy (1–6) keV interval and not in other higher energy regions, consistently with expectation for theDM signal;5)the modulationispresentonlyinthe single-hit events, while it is absent in the multiple-hit ones as expected for the DM signal; 6) the measured modulation amplitude in NaI(Tl) target of the single-hit scintillation events in the (2–6) keV energy interval, for which data are also available by DAMA/NaI and DAMA/LIBRA–phase1, is: (0:01014 ± 0:00074) cpd/kg/keV (13.7 . C.L.). No systematic or side processes able to mimic the signature, i.e. able to simultaneously satisfy all the many peculiarities of the signature and to account for the whole measured modulation amplitude, has been found or suggested by anyone throughout some decades thus far (for details see e.g. Ref.[1–5,7,8,19–23,35,36]). In particular, arguments related to any possible role of some natural periodical phenomena have been discussed and quantitatively demonstrated to be unable to mimic the signature (see references; e.g. Refs. [7, 8]). Thus, on the basis of the exploited signature, the model independent DAMA results give evidence at 13.7. C.L. (over 22 independent annual cycles and in various experimental configurations) for the presence of DM particles in the galactic halo. The DAMA model independent evidence is compatible with a wide set of astrophysical, nuclear and particle physics scenarios for high and low mass candidates inducingnuclearrecoiland/orelectromagnetic radiation,asalsoshowninvarious literature. Moreover, both the negative results and all the possible positive hints, achieved so-far in the field, can be compatible with the DAMA model independent DM annual modulation results in many scenarios considering also the existing experimental and theoretical uncertainties; the same holds for indirect approaches. Fora discussion see e.g. Ref. [5] andreferences therein. Thepresent new datareleased determine the modulation parameters with increasing precision and will allow us to disentangle with larger C.L. among different DM candidates, DM models and astrophysical, nuclear and particle physics scenarios. 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In the framework of hypercolor scenario of multicomponent Dark Matter, in- elasticinteractionofhigh energyphotonswiththeDark Matter candidatesis considered. This reaction results in production of energetic leptons and neutrinos, and the Dark Matter particles also canbe boosted.Total and differential cross sections have been calculated. The search of correlations between detected signals of high-energy photons and neutrinos can give an information on the Dark Matter scenario and dynamics. Povzetek:Avtor obravnava neelastiˇ cno sipanje visokoenergijskih fotonov na temni snovi, ki temno snov pospeˇcijo pa tudi nastanek visokoenergijskih leptonov. Za temno sijo, povzroˇ snov uporabi hiperbarvni model, ki predvidi, da ima temna snov veˇ c komponent. Za to reakcijo izraˇcuna totalni in parcialne sipalne preseke. Keywords: hypercolor scenario, Dark Matter, high-enery photons, neutrino pro- duction.PACS: 12.60 -i, 96.50.S-,95.35.+d. 2.1 Introduction Dark matter, which occupies such an important place in the observable Universe, both in its role in the formation of gravitating structures and in its contribution to the overall density of matter,still eludes the ”hunters” -ground-based accelerators, measuring complexes and underground laboratories do not find obvious signals of birth, decay or interactions of these objects of unknown nature. Extending the region of the ”hunting”, physicists arestudying in more detail the indirect possible manifestationsof Dark Matterin various astrophysical phenomena[1–5,7,8,8], actively analyzing the detected signals of cosmic rays and individual particles using space telescopes and interpreting observational data in the framework of various scenarios. The so-called indirect methods of searching for traces of dark matter among numerous astrophysical phenomena simultaneously provide a testing ground for highlighting the most viable options for extending the StandardModel. Having no clues from Nature, we have to sort through the options for the DM construction from the main bricks of matter known to us; fermions, scalars, bosons, compound new hadrons or atoms, neutralinos from supersymmetry, axions, representatives 22 V. Beylin ofthe DarkWorld interacting with objectsof our worldby exchanging special mediators -dark bosons. In all cases, we hope to find among the many studied reactions occurring in acts of interaction involving dark matter objects and parts of ordinary matter, unique events that carry information about the dynamics and origin of dark matter. As very important addition we consider events with high-energy neutrino and/or photons that aredetected mostlybyIceCube and LHAASO (and, certainly,byother ground observatories -because of unpredicted ways of cosmic strangers) [9–19]. Note, some interesting data can result from analysis of correlations between observed events with photons and neutrinos from close directions. Aspecialrolehasrecentlybeenplayedbythe analysisof scatteringbydark matter objectsof particles generatedbyprocessesthattakeplaceat enormous densities and energies -in particular, in jets from powerful quasars (or blasars), in the vicinity of black holes. Depending on the DM scenario, possible, in principle, various observed signals: monochromatic photons from DM annihilation, high- energy particle fluxes during the decays of a hypothetical supermassive DM, continuous radiation in a certain energy range due to transitions between DM components, and so on. In this paper, we consider the inelastic interaction of high-energy photons with DM particles. For definiteness, we work within the framework of the SM extension with additional heavy hyperquarks, this is the so-called hypercolor model in its minimal version with two doublets of new fermions and SU(4) symmetry. In this case,itis possibleto constructa vector interactionof hyperquark currents with the gauge bosons, providing the necessary smallness of the Pekin-Tackeuchi parameters. Further,in the frameworkof thellinear sigma model,by analogy with low-energy hadron physics, bound states of hyperfermions are introduced, i.e. new unstable hyperhadrons. On this path, a set of pseudo-Nambu-Goldstone states arises, of which several -the lightest neutral state of a hyperpion triplet and a hyper-diquark with a non-zero conserved hyperdron number -turn out to be stable. These states are interpreted as TM candidates. Some necessary technical details for the description of the process interested will be presented in Section 2. Section3containsresultsof calculations; discussionofresultsis placedin section Conclusions. 2.2 Dark Matter candidates in hypercolor scenario Minimal model of the vector hypercolor SM extension contains one doublet of additional heavy H-fermions (H-quarks in confinement) with zero hypercharge. To provide the necessary smallness of Peskin-Takeuchi parameters, initial fields of H-fermions are redefined resulting in Dirac fields that interact vectorially with the gauge fields; extra SU(2)w symmetry ensures this electroweak interaction. An extra singlet scalar, ~- meson, emerges to provide spontaneous symmetry breaking and, consequently, non-zero masses of new fields. At the next stage, using the linear hyper-- model (as it is done in the low-energy hadron physics), a new H-hadrons generated by H-quarks currents arize with some hierarchy in masses. Besides, the global SO(4) breaking generates a set of 2 Neutrinos production by photons scattered on dark matter pseudo-Nambu-Goldstone (pNG) states, including a triplet of pseudoscalar H- pions and neutral H-baryon (H-diquark with the additive conserving quantum number) along with its antiparticle; H-pions possess a multiplicative conserved quantum number [20]. Thus, the neutral states, ~0 and B0 ;B0, are stable in this — scenario, so they can be interpreted as the DM candidates with equal masses at the tree level. Note, the model which is used here to consider high energy photons scattering offthe DM is described in detail in a series of papers [21–26]. That is why we do not repeate here all known elements of the scenario; remind only, the DM candidates masses were estimated from analysis of the DM burnout kinetics, and we get: mDM ~ 1 TeV. The electroweak mass splitting in the H-pions triplet, i.e. between charged and neutral hyperpions, is nearly constant: m. . 0:16 TeV. Mass splitting between H-pions and neutral H-baryon, another component of the DM, can be as large as . (10 - 15) GeV (see [27]). Mass of ~- meson is connected with the H-pion mass and depends on the the mixing angle, , between ~- and Higgs boson. This mixing should be small, sin . . 0:1, so the standardHiggs bosonhasa small admixtureof additional scalar state, ~. Note also that the density of H-baryons dominates over density of ~0 almost for all possible values of model parameters becaue of different origins of the DM components burning out at different rates [26]. The charged components of H-pion triplet decay, and the dominant channel is the following: ~± › ~0ll with . . 3 · 10-15 GeV. Certainly, the DM objects are neutral, so they do not interact with photons directly. However, in this scenario there is a class of tree-level diagrams which describe intermediate stage in the total process, ~0 › W+ ~- › ll0ll0 ~0. Here, we know that charged H-pion decays into lepton and neutrino with Br . 1, and (in fact, intermediate) gauge W- bosons also eventually decay into lepton-neutrino pairs. Correspondingly, we need only in hyperpion triplet properties. From this point of view,this DM component demonstrates possibilities of any WIMP scenario where the DM candidate interact with the standardgauge bosons in some way. Specific detail is: this scenario deal with heavy DM candidates, so to accelerate them the high energy transfer from the projectile is necessary. In other words, H-pion as the DM component is the almost “pure” WIMP having tree level elctroweak links to the SM fields due to charged (unstable) components of H-pion triplet. Indeed, the model contains two types of stable neutral DM candidates, however, scatterings of photons offneutral H-pion are the most simple due to vertices which are shown in the following part of the model Lagrangian: = igWµ ( ~0 ~- - ~0 ~-)-ieA( ~- ~+ - ~- ~+)+eg~0 ~-AW+ +h:c. (2.1) LEW +;;µ ;;µ µ The B0 DM component participates in EW interactions only via fermion, vector and/or scalar (gauge boson, quark, H-quark or H-pion) loops and via (pseudo) scalar (Higgs boson, ~)exchanges, so, such stable H-diquark presents a hadronic type DM. Some comments onphotons scatteringoff B0, component will be done later. 24 V. Beylin 2.3 Neutrino production by photons Here, we consider inelastic interaction of photons with energies ~ (1 - 10) TeV withtheDM objects. Suchprocessof leptonsand neutrinosproductionby photons seems interesting becausethe yieldof these secondary particles dependsboth on DM dynamics and distribution its density in space. On origin of high-energy photons and neutrinos may be associated, in particular, with physics and structure of jets from active galaxies nucleus(AGN), events with these particles of TeV- energies are repeatedly observed and analyzed [28–39]. So, there is a possibility for DM structures in the AGN vicinity to participate in the generation of leptons and neutrinos fluxes. Certainly, an analogous process is possible when high-energy photons are scattered by the DM clumps. Then, in the framework of minimal hypercolor model we study the photons scattering off ~0- component. Such reaction is the most simple, however, calculations of the total cross section with unstable final states werecarried out in the continued mass approach(see [40] andreferences therein). Three tree-level diagrams describing the photon scattering off ~0 are depicted in Fig.1 Note, there is an additional diagram (see Fig.2) which, however, gives a small contribution to the cross section, because we are interested in dominant configuration with small squared invariant massof intermediateW-boson. Then, the sum of corresponding matrix elements is: (2k1 - p2) (2p1 - k2)µ (p1 + k1)ß µ MC + M . ~+ MW = ige e(-g. ++ · Wd . ~dW . ß ß - WW (g qq (g(p2 - 2k2). - g (2p2 - k2)µ + g (p2 + k2))). (2.2) 2 m W Here, propagators are denoted as d ~;dW. The squared matrix elementis cumbersome,so the exact expressionfor the cross section of this sub-process is not shown here. Instead, we show dependencies of differential cross section on squared invariant massof virtualW-boson, various masses of the DM target and energies of incident photon for fixed energies of final H-pion (see Fig.3). Analogous dependencies of differential cross section on squared invariant mass but fixed energiesof virtualW-boson and for fixed energy of incident photon are shown in Fig.4. In fact, these two-dimensional curves are sections of three-dimensional graph for different values of the parameters; so, these curves contain information about the angle betweenproductsofW-boson decay (lepton and neutrino, in particular). Of course,itis importanttoknownotonlydifferentialcross sectionforthesub- process with virtual states (unstablecharged H-pion andW-boson with known branching of decays into leptons and neutrinos) but total cross section of the process. This next step can be done using model of unstable particles with a smeared masses [40].In this approximation the total cross sectionispresentedina factorized form: Z tot(s)= d2tot(s, µ 2)(µ 2), (2.3) 2 Neutrinos production by photons scattered on dark matter here µ 2 is the variable squared invariant mass of intermediate unstable state and p 12..(2) (µ 2)= p. ((2 - M2(2))2 +(2..(2))2)2 is the density of probability; integration goes over kinematically allowed region. Now, the total cross section for the wholeprocess canbe estimated witha good accuracy (note, the approach above was tested in a numerous reactions, and the accuracy wasreliably evaluated as . 5 %). Total cross section for the process of photons scattering offscalar DM candidates, ~0 › W+ ~- › ll0ll0 ~0, in dependence of photons energy are presented in Fig.5. Here we consider not very high energies of photons, up to 10 TeVbecause for higher energies the photon flux is much lower. These results depend on the DM mass weakly, so we use here some referent value, mDM = 1 TeV. Besides, we get the cross section almost stable in the energy region considered. This result indicates some possible correlations between detected high-energyphotons and neutrinos fluxes -namely, neutrinos of high energy can be produced by photons scattered on the DM objects. But thisis not the whole story because there are loop contributionsto neutrino+ leptons generationby the photons scatteringoffanotherDM component, stable H-baryon. However, this DM candidate presents other type of DM objects which do not interact with the gauge boson directly. So, one from possible contributions into such typereactionis shownin Fig.6. However, there shouldbe also additional channels to transform a part of high-energy photons into fluxes of leptons and neutrinos. These are processes like B0 › W+W-B0 B0 tB0 B0 , › t —, › ZB0 with subsequent decays of heavy standardquarks and gauge bosons. These interesting andpromising channelsof inelastic interactionsofphotons with the DM will be analyzed elsewhere. .±.0W±..0.W± Fig. 2.1:Tree-level diagrams for the subprocess ~0 › W± ~± . 0± W ~ll l Fig. 2.2: Additional contribution to the scattering process. 26 V. Beylin a) b) Fig.2.3:Differentialcross sectionin dependenceonW-boson squared invariant mass for the incident photon energy E. = 10 TeVwith the fixed energy of final H-pion: a) E . ~= 2 TeV; b)E . ~= 5 TeV. a) b) Fig.2.4:Differentialcross sectionin dependenceonW-boson squared invariant mass forthe incident photon energy E. = 10 TeVwiththefixed energyofW-boson: a) EW = 2 TeV; b)EW = 5 TeV. a) b) Fig.2.5:Totalcross sectionin dependenceon photon energies energy;of incident photon: a) up to E. = 1 TeV; b) up toE. = 10 TeV. 2 Neutrinos production by photons scattered on dark matter B0B0h, Zq Fig. 2.6: One from possible loop-level diagrams for the subprocess B0 › ZB0 . 2.4 Conclusions and some open questions So, for tree level production of leptons and neutrinos by inelastic scattering of high-energy photons offthe DM we get tot(E ) ~ 100 nb. Asitis seenfrom calculations,a significant partof E. is converted into energies of secondary leptons and neutrinos which are generated by decays of W-boson both in direct channel and from unstable light standard mesons due to hadronic decay channelsofW. Atthetime,aDMtargetisan activeand necessary participantoftheprocess,so,it also can get sufficiently large portion of photon energy, so the DM components can be accelerated up to energies ~ (1 - 10) TeVor even larger depending on E . Intensityof thisreaction strongly depends on theDM density (the macroscopic cross section is ~ DM, the high-enery secondaries yield can be sufficiently enhanced when the scattering occurs in regions of high DM density. Itisan important notification becausethemain sourcesofhigh-energy photonsare quasars(or blasars shining in the direction of the Earth) which emit dense and very fast fluxes of charged particles as jets (see references above); high-energy photons are radiated by these charged particles, so photons and neutrinos generated in quasars and blasars jets can be detected by ground observatories and cosmic telescopes. Inelastic scattering of photons offthe DM most effectively occurs in the vicinityof quasars,i.e. closeto active nucleiof galaxies.Anditisin theseregions of strongest gravity that the DM density is the largest. Products of this collision of photon with the DM are approximately collimated with the initial photon direction for high E . Flux of neutrinos produced with cross section ~ (0:1 - 1) nb and energies ~ (1 - 10) TeVshould correlate with the initial photon flux, consequently, signalsofTeV-photons detectedby cosmic telescopes and ground observatories can be (approximately) synchronized with neutrinos of close energy which come from the nearly the same direction. Possibly, interactions of such type affect on the DM density fluctuations at early stage of evolution when cosmological plasma is hot and contains a lot of the DM (neutral) objects and also the dense fluxes of photon radiaton. So, these processes can also affect on the density of photons and DM carriers throughot the radiation dominated era. In some sense, these processes can somewhat wash out dense DM clumps and fluctuations due to accelerating the DM particles of various masses. It is, of course, a hardtask to find out and separate from any other sources neutrinos and photons signals correlated in time and spatial direction, however, if it were discovered, it would mean obtaining an important information about the structure and dynamics of the DM and blasars jets, as well as about the DM spatial distribution and its ability to affect to dynamics and composition of cosmic rays in space. 28 V. Beylin Inany case,itis worth considering suchprocessesof photons transformation into leptons and neutrinos fluxes accompanying with the DM accelerated particles in all possible DM scenarios and for various characteristics of cosmological evolution stages. The DM plays an important role of an active and necessary catalyst of mutual transitions between various typesof theSM particles.Asit seems, such processes with the obligatory presence of the DM should be taken into acount when the early Universe dynamics and structure are studied. 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Kuksa, N.I.Volchanskiy: Factorizationin the modelof unstable particles with continuous masses. Cent. Eur. J. Phys. 11, 182 (2013). Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 31) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 3 Elusive anomalies L. Bonora International School for Advanced Studies (SISSA), Via Bonomea 265, 34136Trieste, Italy Abstract. Usually,inorderto computeananomaly(beit chiralortrace)withaperturbative method,the lowest significantorderissufficient.Withthehelpof gaugeordiffeomorphism invariance it uniquely identifies the anomaly. This note is a short review of the ambiguities that arise in the calculation of trace anomalies, and is meant, in particular, to signal cases in which the lowest perturbative order is not enoughto unambiguously identify a trace anomaly.Thismayshedlighton somerecent contradictoryresults. Povzetek:Ko ˇceje teorija anomalna, najveˇsˇcunamo zelimo ugotoviti, ˇckrat zadoˇca,da izraˇ najniˇcelni red v teoriji motenj in preverimo njeno umeritveno ali difeomorfno zji neniˇ invarjanco.Vtem prilspevku avtorna kratkopredstavite ˇcenajniˇ zave,ki se pojavijo, ˇzjired v teoriji motenj ne zadostuje za nesporno odloˇ citev ali je teorija anomalna. Prispevek bo morda pomagal osvetliti nekatereprotislovne nedavno objavljenerezultate. arXiv:2207.03279 [hepth] 3.1 Introduction The first manifestationof anomaliesinQFT(the Adler-Bell-Jackiw anomaly) originated from an apparently technical problem: the constant shift of an integration variable in a fermion loop integral leads to a vanishing result (which, in turn, implies a conserved chiral current), except for the fact that this integral is UV divergent, so that the shift is illegal; on the contrary, a proper treatment of this problem leads to a non-vanishing result, which, in turn, implies an anomalous conservation law. The way it came up the first time might have seemed to be due to a technicality, but in fact it turned out to be the tip of an iceberg. On the one handitwasthefirstofa seriesof similarresultsthatleadtothe discoveryofmany anomalies: many currents which are classically conserved are not anymore so after quantization. On the other hand these anomalies were derived in a number of ways, both perturbative and nonperturbative, and it was discovered that they are far from wild, random violations of the conservation laws, but, on the contrary, they satisfy group theory motivated consistency conditions. Finally, the illuminating connection was found withthe family’s index theorem, illuminating, because it revealed that (consistent) anomalies represent obstructions to the existence of the inverse of the Dirac (or Dirac-Weyl) operator, i.e. to the very existence of the fermionpropagator. The latterisa fundamental ingredientofa quantum theory, therefore consistent anomalies are a spy of its bad health. L. Bonora In quitea similar way, after the ABJ anomalies, also anomaliesin the traceof the energy-momentum tensor were found in theories where, classically, conformal invariance requires a traceless e.m. tensor, [1]. Strangely enough trace anomalies have lived a separate life from chiral anomalies, and any attempt to unify them has failed. Nevertheless it is true that both kind of anomalies are strictly linked to the existence of the inverse kinetic operator: to see it is enough, for instance, to consider that both the derivation of a current conservation law and the trace- lessness condition within the path integral approach requires the existence of the inverse kinetic operator. Leaving aside, for the time being, this link between the two types of anomaly (chiral and trace) let us focus now on their differences. Itis nota mistery that the calculationof trace anomalies has ledto some controversial results. The reason is, on the one hand, the ambiguity in the definition of trace anomaly and, on the other hand on the ambiguities intrinsic to their derivation. It was pointed out above that ABJ chiral anomaly arose from resolving an ambiguity in the definition of a loop integral. These types of ambiguities are the basic ones, and, of course, are present also in the case of trace anomalies. But they are not the only ones. The very definitionof thetrace anomalyin termsof perturbative amplitudes poses a problem. Let us denote by hhT(x)ii the fullone-loop one-point e.m. tensor, i.e. 1Z n X in Y p hhT(x)ii = d4 xig(xi)hii (xi)h0jT T(x)T11 (x1) :::Tnn (xn)j0i 2nn! n=0i=1 (3.1) where h0jT T(x)T11 (x1) :::Tn(xn)j0i are e.m.tensor correlators calculated n with Feynman diagrams. Then we may proceed in two ways to compute the trace of this object. The first is to evaluate g(x)hhT(x)ii, i.e. to take the trace of the one-loop one-point e.m. tensor. The secondis to evaluate hhT µ (x)ii, i.e. the one-loop one-point function of the e.m. trace (which is non-vanishing off-shell). In many examples these two quantities are different, therefore we face the problem of definingwhatwe meanby traceanomaly.It turnsoutthattheright definitionis the difference between the two T(x)= g(x)hhT(x)ii - hhTµ (x)ii (3.2) µ proposedbyM.J.Duff,[2,3]. Accordingtothe physicalinterpretationin[4],itis entirely due to the violation of the equation of motion of the theory (remember that the trace of the e.m. tensor classically incorporates the equation of motion). Butin the Feynman diagram approachthere are other ambiguities. Whenregularizing the loop integral we are faced with more than one possibility, no matter what regularizing prescription we use, dimensional or Pauli-Villars, to name the most frequently used. These possibilities may lead to final results differing by local terms (this may happen also for chiral anomalies). Now, to proceed further, we have to introduce another ingredient that can be, and usually is, disregarded in the caseof chiral anomalies: diffeomorphism invariance.The trace anomalyis theresponseof the functional integral underarescalingof the metric. Therefore the properties of the metric are inevitably brought into the game, and one has to 3 Elusive anomalies checkin particular thatdiffeomorphismsareconserved. Thisrequiresarecourseto cohomology.Aswe shall seeinan example below,dependingontheregularization prescription, the divergence of the e.m. tensor may be vanishing or non-vanishing, giving rise in the latter case to a cocycle of the diffeomorphisms. Such cocycle may be trivial, that is a counterterm can be added to the effective action which cancels this anomaly and, simultaneously, modify the trace anomaly.We shall refer to this as the stabilizing or repairing role of diffeomorphism conservation. In most situations this is what happens: a unique expression for the trace anomaly is identified, accompanied by conserved diffeomorphisms. In other words, as it shouldbe,the finalresultdoesnotdependontheregularizationprescription.Said another way, a regularization prescription determines the anomaly up to trivial cocycles. This is not the end ofthe story. There is another possible ambiguity which we wouldliketo illustrateinthis note.Itis ratherrarebutplaysacrucialrolein specific casesandrendersthe lowestorder calculationofthe trace anomaly unusable. Such an ambiguity is triggered if the three-point function of the energy-momentum tensor (the lowest order as far as the calculation of the trace anomaly in 4d is concerned) vanishes identically. This may not happen for the full e.m. tensor, but it may happen for its odd-parity part. Since even-parity and odd-parity correlators split neatly we can treat them in the anomaly calculations as separate entities. When such vanishing occurs, the first term in (3.2) vanishes, but the second need not vanish. On the other hand the (odd-parity) divergence of the e.m. tensor at the (three-point level) vanishes identically and there is no possibility to adjust the effective action by adding counterterms in such a way as to unambiguously determine the trace anomaly. Now, in most cases the lowest order perturbative calculationis enoughto determine traceor gauge anomalies completely,relyingon gauge or diffeomorphisminvariance,respectively. Butin this caseitis impossible, theproblemis logically undecidableatthelowest perturbativeorder.Thewayout istoresorttohigherorderapproximationsortoa non-perturbativeapproach. In this short noteIwould like to present an example of this pathological phenomenon. But, before, in order to appreciate it, it is necessary to understand the repairing mechanismof diffeomorphism conservation. For thisreasonIpresentin the next section a simple 2d example in which this mechanism works, and devote the thirdsection to the non-working example. Throughoutthe paper the reference action willbe thatofa right-handedWeyl fermion Z  . 1 S = dd x gi R µ @µ + !µ R (3.3) 2 µ where g = det(g), µ = ea a,(, , ::. are world indices, a, b, ::. are flat indices) and !µ is the spin connection: = !ab !µ ab 11+ * where ab =[ a; b] are the Lorentz generators; R = PR , where PR = , 42 and . * is the appropriate chirality matrix. Thereference classical e.m. tensor will be i - T. = R µ r R + fµ - } (3.4) 4 L. Bonora The theory (3.3) is invariant under diffeomorphisms g. = r. + rµ (r is the gravitational covariant derivative and µ = g)andWeyl transformations !g. = 2!g, where (x) and !(x) are the relevant local parameters. As a consequence, classically,  rT(x)= 0, Tµ (x)= 0 (3.5) µ on shell. 3.2 Asimple (working) example In two dimensions, due to the anticommutativity of spinors, the spin connection drops out of the action (3.3). For R the action becomes Z S = id2 x . g R @ R (3.6) Although this is not strictly necessary, we will further simplify it by absorbing the . 1 g into aredefinition of :. › . e= g 4 . ZZ ed2 x eed2 x eµ e S = i R @µ R = i R a e@µ R (3.7) a µ Now let us write e. µ - µ and make the identification 2µ = hµ , where h. aaa aa is the gravitational fluctuation field: g. = . + h. The fermion propagator is = i (3.8) p + i and there is only one graviton-fermion-fermion(Vffg)vertex givenby hi i1 + * (p + p 0) . +(p + p 0) µ . (3.9) . 82 where p and p0 are the two graviton momenta, one entering the other exiting. Other vertices will not be relevant. Our purpose is to compute the two terms in eq.(3.2). In 2d the lowest order contribution is given by the two-point amplitudes h0jT T(x)T(y)j0i and h0jT T µ (x)T(y)j0i, (3.10) respectively. Their Fouriertransformsisprovidedbya Feynman (bubble) diagram with a fermion loop with momentum p and two external gravitons (one entering, one exiting) with momentum k. More in detail, considering the first term in (3.10) we have Z d2k -ik(x-y)e hT(x)T(y)i = 4e T(k) (3.11) (2)2 with 3 Elusive anomalies 35 Z  1d2p1 1 + * 11 + * µ - . e T(k)=- tr (2p - k) . (2p - k) . + 64 (2)2 p=2p=- k=2. - . (3.12) The last bracket means that we have to add three more terms like the first so as torealizea symmetry under the exchanges µ - , . - . Moreover, we have to symmetrize with respect to the exchange (, ) - (, ) (bosonic symmetry). The integralin (3.2)isUVdivergent.Weproceedtoregularizeitwith dimensional regularization.To this end, as usual, we introduce extra space components ofthe momentum running around the loop, pµ › pµ + ` µ —(` µ —= ` 2;:::, ` +1). So (3.2) becomes  (reg) 1 Z d2pd`1 1 + * 11 + * e T(k)=- tr (2p-k)(2p-k) . . . 64 (2)2+. p=+ =` . 2 p=+ =` - k=2 (3.13) 2 22 , == Now let us recall that p== p;=` =-`2 p=` + =`p = 0 and [ , =`]= 0, while . f , p=} = 0. Moreover tr(   )=-21+ 2 .Working out the -matrix algebra and performingaWickrotation k0 › ik0 one can compute the loop integral. Here, for simplicity, we report only the even parity part of the trace and the divergence of the e.m. tensor: hi i e TEµ (k)= kk. + k2(3.14) 192. . and kTeE (k)=- i h 1k2 i (3.15) kkk. + k. + k. 384. 2 where kµ denotes the Euclidean momentum (in particular k2 =-k2). Next one anti-Fourier-transforms these amplitudes and, after returning to the Minkowski background, inserts them into (3.1).To obtain the corresponding integrated cocycles, one multiplies the first by . and saturates the second with . and integrate over space-time. The result are the two cocycles ZZ . = 1d2x!(x) d2yh(y)h0jT Tµ (x)T(y)j0ic = (3.16) µ 2 ZZ Z d2k -ik(x-y)e =2d2x!(x) d2yh(y) eTµ (k)= (2)2  Z  = 1d2x!(x) @@h(x)- h. (x)  96. and ZZ 1 -ik(x-y)e . =- d2x(x) d2yh(y)(-ik)eT(k) 2 Z hi = 1d2x(x) @@@h(x)- @h. (x) (3.17)  192. 36 L. Bonora Let usrecall that the parameters µ and !, accordingtotheruleofBRST quanti zation, are promoted to anti-commuting fields. The lowest order transformation rules for the metric is . (0) h. = 2!. and h. = @. + @. Using this it is easy to prove that (0)(0) (0) (0) = 0, = 0, . + = 0. (3.18) . !. . !. . Both trace and diffeomorphism cocycles are non-vanishing. However the diffeo- morphism one is trivial. For let us consider the local counterterm Z 1 .. C(even) = d2 xh. (x)@@h(x)- h(x)h(x) (3.19)  384. It is easy to prove that 0(even)(even)(0) C(even) . . + = 0 (3.20) . . Therefore diffeomorphisms are conserved. On the other hand the overall even trace cocycle becomes Z 01  (even) . (even) + (0) C(even) d2@@h. - h. . = x. (3.21) !!. . 48. Sofarwehave computedthefirsttermof (3.2).Wehaveto computealsothe second,i.e.the secondonein (3.10).The corresponding amplitude, onceregulated, takes the form ! Z 1d2pd` p=+ =` .. p=- k=1 + * e Tbµ (k)=- tr 2p=+ 2=` - k=(2p - k) . 64 (2)2+. p2 - `2 (p - k)2 - `2 2 (3.22) Adirect calculation shows that it vanishes. Therefore the trace anomaly is deter- minedby (3.21), whichis the first order approximationof Z 1 . A(even) = d2 x g!R (3.23) . 48. 3.2.1 Another prescription Theregularizingprescription (3.2)isnottheonly possibility.Wecouldhave started from Z  1p1 11 + * T e0 (k)=- d2 tr (2p - k) . (2p - k) . :(3.24) 64 (2)2 p=p=- k=2 and regularize it as follows  (reg)0 1 Z d2pd`1 11 + * e T. (k)=- tr (2p - k) . (2p - k) . . 32 (2)2+. p=+ =` p=+ =` - k=2 3 Elusive anomalies 37 We shall refer to it is the rightmost * prescription. The result is now hi 0i e T Eµ (k)= kk. + k2. (3.25)  96. and kTeE (k)= 0 (3.26)  Contrary to the previous prescription this one yields diffeomorphism invariance and the same trace cocycle (3.21). It remains for us to evaluate the second term of (3.2). The corresponding regulated amplitude is ! Z ereg 1d2pd` p=+ =` .. p=+ =` - k=1 + * T b0(k)=- tr 2p=+ 2=` - k=(2p - k) . 32 (2)2+. p2 - `2 (p - k)2 - `2 2 (3.27) which, again, vanishes. Therefore the two prescriptions lead, as expected, to the sameresult, the trace anomaly (3.23), while diffeomorphisms are conserved(as far as the even parity part is concerned.). In this section we have illustrated an example (probably the simplest one) of a perfectly working cohomological mechanism: the first prescription leads both to a trace and a diffeomorphism anomaly; however the latter is trivial and can be eliminated with a counterterm, which in turn modifies the trace anomaly giving it the final (minimal) form. The second prescription preserves diffeomorphism invariance and yields the previous final form of the trace anomaly. In the next section we shall see an example in which this mechanism cannot work. 3.3 The problematic example Theexamplewe considerinthesequelisthatofa right-handedWeyl fermion coupled to a non-trivial metric. The action is (3.3) with d = 4, but in this case the spin connection does not drop out. One can write the action as follows Z  p i - 1 abc S = d4 x jg| R µ @ R - !ab R c 5 R (3.28) 24 where it is understood that the derivative applies to R and R only.We have a used the relation f a;bc} = iabcd d 5.We expand g. and eas before, and, µ accordingly !ab abc =-abc @a. . (3.29) b + ::. Proceeding as in the previous 2d example, after some algebra the action takes the form Z  i - 1 abc S . d4 x (µ - µ ) L a @ L + @a. b . L c 5 L aa 24 L. Bonora from which we can extract the Feynman rules. The fermion propagator and fermion-fermion-graviton vertex(Vffg)are the same as before. In addition we havea two-fermion-two-graviton vertex(Vffgg) 11 + 5 t00(k - k0) . (3.30) 64 2 where t00. = 0 0. + 0 0. + 0 0. + 0 0. (3.31) and the graviton momenta k, k0 are incoming. Other vertices are irrelevant for the sequel. This model has no even-nor odd-parity diffeomorphism anomalies, while it has both even and, as we shall see, odd-parity trace anomalies. The even part works muchin the same way asin theprevious2d example.Our interestin this section is focused on the odd parity part. It is well-known that in 4d there cannot be odd-parity consistent diffeomorphism anomalies, but a priori we cannot exclude other (trivial) anomaliesrelated to the trace anomalies (much like the (3.17) above). Therefore we have to verify that odd-parity divergence of the e.m. tensor vanishes. Therelevant lowestorder contribution (whichwe denote symbolicallyby h@TTTi) may comefroma triangle anda bubble diagram. The triangle diagram contains three fermion propagators and three Vffg vertices.Taughtbythe2d examplewe use the rightmost * . 5 prescription. The corresponding Fourier-transformed contribution after regularization is " Z (odd) 1d4pd` p=+ =` p=- k=1 + =` e qT(k1;k2)=- tr (2p - k1) .  512 (2)4+. p2 - `2 (p - k1)2 - `2 !# p=- q=+ =` .. (2p - 2k1 - k2) ß (2p - q) · q . -(2p - q). q= 5 (p - q)2 - `2 It is not difficult to show that it vanishes identically. Also the contribution from the bubble diagram, constructed with one Vffg, one Vffgg and two propagators, vanishes. Therefore we conclude that with this prescription diffeomorphisms are exactly preserved. We next compute the odd parity contribution of the triangle and bubble diagram to the trace anomaly. At first sight this calculation does not seem to make sense, because a well known result of CFT claims that a conformal odd-parity three- point function h0jT T(x)T00 (y)T (z)j0i(odd) vanishes identically for algebraic reasons. Thisis confirmedby a direct calculation.In fact one canprove that, with both prescriptions above, h0jT T(x)T00 (y)T (z)j0i(odd) vanishes. So, at the lowest significant perturbative order, we can write: hhT(x)ii(odd) = 0 (3.32) However according to the definition (3.2) we must compute also the second term with one insertion of a trace of the e.m. tensor (which we denote by htTTi). The 3 Elusive anomalies 39 triangle diagram gives e T00 (k1;k2)= (3.33) ZZ 1d4pd` p=+ =` (p=+ =` - k=1) =- Tr [(2p - k1) . +(µ - )] 256 (2)4 (2). p2 - `2 (p - k1)2 - `2  (p=+ =` - k=1 - k=2) 1 + 5 [(2p - 2k1 - k2)0 0 +(µ 0 - 0)] (2p=+ 2=` - k=1 - k=2) . (p - k1 - k2)2 - `2 2 (3.34) which, with the addition of the cross diagram, leads to the following result ..  1 ß  (21) e k. k1 2 + k2 , T00 (k1;k2)= 1 k22 + k1 k2 t00 ß - t00 ß 61442 (3.35) where (21) t00. = k2k10 0. + k2k10 0. + k2k10 0. + k2k10 0. (3.36) The contribution from the bubble diagram vanishes. The conclusion of this computation is that the contribution to the odd-parity trace anomaly according to formula (3.2) comes solely from (3.35). To simplify the derivation we set the external lines on-shell,k2 = k2 = 0. This 12 requires a comment. On shell conditions Putting the external lines on shell means that the corresponding fields have to satisfy the EOM of Einstein-Hilbert gravity R. = 0. In the linearized form this means . = @@ . + @@. - @@. = 0 (3.37) . We also choose the De Donder gauge:... . = 0, which at the linearized level g becomes 2@µ - @µ = 0. In this gauge (3.37) becomes µ . . = 0 (3.38) In momentum space this implies that k2 = k2 = 0.We remark that this does 12 not trivially disrupt the cohomology, but define a restricted cohomology of the diffeomorphismsandtheWeyl transformations:the latteris definedupto terms h. and . This is a well defined cohomology, under which we have, in particular, .. 2@µ - @µ = 2 . . 0 (3.39) µ i.e. in this restricted cohomology the De Donder gauge fixing is irrelevant. Simß ilarly, the term corresponding in momentum space to k 1 k2 (k2 1 + k22)t00 ß L. Bonora remains null afterarestricted diffeomorphism transformation. Therestricted co- homology has the same odd class (the Pontryagin one) as the unrestricted one, i.e. it completely determines it (this is not true for the even classes). Since we know that the final result must be covariant and that there is no covariant extension to ß all order of the term k. k(k2 + k2 )t00 , the simplification of considering it 121 2 null does not jeopardize it. This means that this term must be trivial in some way. We will comment on this below. Overall contribution The overall one-loop contribution to the trace anomaly in momentum space, as far as the parity violating part is concerned, is given by (3.35). After returning to the Minkowski metric and Fourier-antitransforming it, we can extract the local expressionof the trace anomaly,by replacing theresults found so far in (3.1). The result, to lowest order, is i .. (x)ii(odd) . . hhT µ @@h . @@h. - @@h . @@h. (3.40) . 7682 Since .. = . @a . @a R. R. @. @@a. - @. @@a. + ::. (3.41) we obtain (x)ii(odd) i 1 hhT µ = R. R. (3.42) µ 7682 2 Now applying the definion (3.2) and recalling (3.32), we obtain the covariant expressionof the parity violating partof the trace anomaly foraWeyl fermion  T[g](x)= i1 R. R. (3.43) 7682 2 3.3.1 The missing mechanism The trace anomaly (3.43) coincides (up to a coefficient) with the KDS (KimuraDelbourgo- Salam) anomalyof the chiral currentina theoryof Dirac fermions immersed in a non-trivial metric background. In [6] this coincidence has been ex ß plained. Therefore, is everything ok? No, because the term k. k(k2 +k2 )t00 ß 1212 we have disregarded has not been explained yet, and the attempt to explain it reveals an unexpected obstacle. It corresponds to an integrated anomalous term R ~ d4x!@h. @h . There is no covariant expression that to the low  est order has this form. Therefore it must be a trivial term. Such a lowest order R @ cocycle canbe canceledbya counterterm ~ d4xhµ h. @h . But this   counterterm term destroys diffeomorphism invariance. There seems to be no way out. Before surrendering, one may tryto change the regularization prescription. For instancewemightusethefirstprescriptionoftheprevious section.Itiseasyto see that with this new prescription diffeomorphisms are still conserved, as one can directly verify (and as it should be, because of the general theorem in [11]). 3 Elusive anomalies But the trace anomaly changes, both in its form and in its overall coefficient (even the bubble diagram givesa nontrivial contribution). Thisisa puzzle.We have seen above an example, but many others can be envisaged, where, after some calculations, nonzero trace anomalies and nonzero diffeomorphism anomalies appear in couples, and (unless the the diffeomorphism anomaly is non-trivial, which is not possible in 4d) by subtracting a counterterm we can recover diffeomorphism invariance and modify the trace anomaly to a minimal form. In other words diffeomorphism invariance playsa‘repairing’ or ‘stabilizing’rolein cohomology. I.e. the diffeomorphism cohomology accompanies the regularization scheme in such a way that the latter preserves the cohomology class. However a necessary conditionforthisroletobeeffectiveisthattherelevant amplitudebe non-vanishing. Which is not what happens in our puzzling case. In fact, the true originof the puzzleis not theregularization scheme, but the accidental vanishingof the odd three-point function of the e.m. tensor. The next question is: does that mean that the perturbative calculation of the trace anomaly is impossible? The answer is: no, it is only moredifficult. In our particular case the problem arises from the vanishing of the odd three-point function of the e.m.tensor. However the three-point function corresponds to the lowest possible order yieldingasignificant contributiontothe calculationofthe trace anomaly.But of course one should consider also the four-point function, the five-point function, and so on. In general there is no such accidental vanishing for the higher order functions. Therefore we should calculate, for instance, the odd four-point function of the energy-momentum tensor and compute both the trace and the divergence in the same way we have done for the three-point function. In this way the stabilizing effect of diffeomorphisms (together with the possible contribution of other graphs, suchasthe bubbleone) wouldunfold undisturbed.Thetroublehereisthe technical complexity. There is an easier way: a non-perturbative approach. Appropriate non-perturbative methods exist, they are the Seeley-Schwinger-DeWitt or heat kernel methods: the diffeomorphism invariance is inbuilt in them and, being non-perturbative, they encompass all the relevant higher order amplitudes. The relevant calculations have been carried out in [8] and more recently in [12] using a method ` a la Fujikawa. The two calculations lead to the same result, eq.(3.43), which is also in accordwith the general formulas of [9]. Remark It is worth pointing out that a missing contribution from the perturbative calculation of an anomaly, such as the one we have come across above, is not rare. Let us consider, for instance, the (multiplet) non-Abelian covariant anomaly ~ tr(TaFF), which appears in the divergence of the chiral current in a theory of Dirac fermions coupled to a vector potential Vµ = Va Ta (with curvature µ F). This anomaly contains a quartic term in the potential Vµ = Va T a, which can µ come only from a pentagon diagram. This diagram however is UV convergent. Therefore the quartic term cannot be produced through a perturbative calculation. It is nevertheless required by the conservation of the vector current in order to guarantee the invariance of the vector gauge symmetry (which plays a role analogous to the diffeomorphisms in solving the above puzzle). L. Bonora 3.4 Conclusions The calculations for the odd-parity trace anomaly have led to controversial results, both with pertubative and non-perturbative methods. But while the non- perturbative approaches, if correctly employed, lead to unambiguous results, [8, 12], and some disagreements can be attributed to inappropriate methods of calculation (see [10] fora discussion), the perturbative ones are intrinsically ambiguous for the reason explained in the previous section. These encompass the perturbative derivationsin[5–7,12,13].As explained before the derivationofa trace anomaly is more complicated than the derivation of the more familiar chiral anomalies and involves the resolution of several ambiguities. The first ambiguity is related to the divergent integrals in the Feynman diagrams: it is resolved by a choice of regularization scheme. The second ambiguity lies in the very definition of the trace anomaly and is resolved by formula (3.2): as explained in [4] this formula selects the very violation of the equation of motion while excluding other irrelevant contributions.Thethirdambiguityisrelatedto cohomology:thereisno suchthingasatraceanomalyunrelatedtodiffeomorphismsandother symmetries of the theory.A trace anomalyisa cocycleof the full symmetryof the theory. When we compute a trace anomaly we must make sure that no other symmetry is violated. Any mis-resolution of these ambiguities may lead to wrong results. For instance, if we compute only the first term in the definition (3.2) the odd parity trace anomaly disappears. Another example: it is always possible to find a counterterm that completely cancels the lowest order odd-parity trace anomaly, but it inevitably breaks diffeomorphism invariance. The calculations in [5–7,12] were made by resolving such ambiguities. But, as far as the odd-parity trace anomaly is concerned, there is a fourth ambiguity generated by the vanishing of the odd three-point amplitude of the e.m. tensor. Asshownabovethisimpliesa dependenceoftheendresultontheregularization scheme. As we have pointed out above, this ambiguity cannot be resolved within the lowest perturbative order, so that this problem is undecidable without going to higher ordersof approximation orresortingto non-perturbative methods.Why the perturbative derivationsof[5–7,12]leadanyhowtothe correctresultis stillto be explained. At this point better avoid any misunderstanding. The true trace anomaly is given by (3.43). The aim of this note is to point out only the ambiguity of its lowest order perturbative derivation. Another example of the type considered before is relatedtotheoddparity traceanomaly inducedbyagaugefield,a caserecently re-examined in [14]. This is because the odd part of correlators < TJJ > vanishes identically, just like the odd part of < TTT >.In this case thereisno needtorestrict the cohomology and, anyhow, an appropriate, and quite simple, non-perturbative approach unambiguously leads toa non-vanishing gauge induced trace anomaly, see [4]. But since in the literature on odd-parity trace anomalies is not univocal, it is worth pointing out that beside the explicit calculations there are also qualitative arguments.Toemd this notewe would liketo brieflyrecall them.The firstis based on the family’s index theorem, [17]. This theorem can be thought of as expressing the obstructions to the existence of the inverse of the kinetic Dirac 3 Elusive anomalies Weyl operator. Any variation of the path integral of the theory defined by the action(3.3),for instanceinorderto seeitsresponse underaWeyl transformation, inevitably involvessuchan inverse,i.e.the(full) fermionpropagator.Therelevant obstructions are expressed in terms of cohomological classes of the classifying space. Among them there are classes that give rise to the well-known chiral consistent anomalies, but in 4d there are also the Pontryagin and Chern classes. The latter are naturally associated to the trace anomaly, generated by the coupling to a background metric or a background gauge field, respectively. The second argument is more ‘phenomenological’: the Pontryagin class or the Chern class densities have the right properties and quantum numbers to couple to trace of the e.m. of a system such as (3.3), which violates parity. The experience teaches us that in such cases quantization usually excites such terms (with non vanishing coefficients). The only exceptions may come from (vector) gauge invariance and diffeomorphism invariance. But in this case the latter is satisfied with a non- vanishing Pontryagin term. So the pertinent question would rather be: why should it vanish? References 1. D. M. Capper and M. J. Duff, Trace Anomalies in Dimensional Regularization, Nuovo Cim. 23A (1974) 173. Conformal anomalies and the renormalizability problem in quantum gravity, Phys. Lett. 53A (1975) 361. 2. M.J.Duff, Twenty years of theWeyl anomaly, Class. Quant. Grav. 11 (1994) 1387 [hepth, 9308075]. 3. M.J.Duff, Weyl, Pontryagin, Euler, Eguchi and Freund, Jour. Phys. A: Mathematical and Theoretical, 53 (2020) 301001 [arXiv:2006.03574]. 4. L.Bonora, Perturbative and Non-PertrubativeTrace Anomalies, Symmetry 13 (2021) 7, 1292. e-Print: 2107.07918 [hep-th] 5. L.Bonora, S.Giaccari and B.Lima de Souza, Trace anomalies in chiral theories revisited, JHEP 1407 (2014) 117 [arXiv:1403.2606 [hep-th]]. 6.L.Bonora,A.D.PereiraandB.L.de Souza, Regularization of energy-momentum tensor correlators and parity-odd terms, JHEP 1506, 024 (2015) [arXiv:1503.03326 [hep-th]]. 7.L.Bonora,M.Cvitan,P.DominisPrester,A.DuartePereira,S. GiaccariandT. ˇ Stemberga, Axial gravity, massless fermions and trace anomalies, Eur. Phys. J. C 77 (2017) 511 [arXiv:1703.10473 [hep-th]]. 8. L.Bonora,M. Cvitan,P. DominisPrester,S. Giaccari,M. Paulisic andT. Stemberga, Axial gravity: a non-perturbative approach to split anomalies, Eur. Phys.J.C 78 (2018) 652 [arXiv:1807.01249]. 9. M.J. DuffandP. van Nieuwenhuizen, Quantum inequivalence of different field representations, Phys.Lett. 94B (1980) 179. 10. L.Bonora and R. Soldati, On the trace anomaly for Weyl fermions, [arXiv:arXiv:1909.11991[hep-th]], and references therein. 11. A. Zhiboedov, Anote on three-point functions of conserved currents,arXiv:1206.6370 [hep- th]. S.Jain,R.R. John,A. Mehta,A.A. NizamiandA. Suresh, Momentum space parity-odd CFT 3-point functions, JHEP 08 (2021) 089; e-Print: 2101.11635 [hep-th] 12. Chang-Yong Liu, Investigationof Pontryagin trace anomaly using Pauli-Villarsregularization, arXiv:2202.13893 [hep-th] L. Bonora 13. S. Abdallah, S. A. Franchino-Vin~as and M. B. Fro¨b, Trace anomaly forWeyl fermions using the Breitenlohner-Maison scheme for . , JHEP 03 (2021) 271 (2021) [arXiv:2101.11382 [hep-th]]. 14. F. Bastianelli and L. Chiese, Chiral fermions, dimensional regularization, and the trace anomaly , arXiv:2203.11668 [hep-th]. 15. R.Delbourgo andA.Salam PCAC anomalies and Gravitation preprint IC/72/86. 16. T. Kimura,Divergence of Axial-Vector Current in the Gravitational Field, Prog. Theor. Phys. 42 (1969) 1191. R.Delbourgo and A.Salam The gravitational correctioin to PCAC, Phys.Lett. 40B (1972) 381. 17. M.F. Atiyah and I.M. Singer, Dirac operators coupled to vector potentials, Proc. Nat. Acad. Sci. 81 (1984), 2597. Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 45) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 4 Maximally PreciseTestsof the Standard Model: Elimination of Perturbative QCD Renormalization Scale and Scheme Ambiguities S. J. Brodsky e-mail: sjbth@slac.stanford.edu SLAC National Accelerator Laboratory, StanfordUniversity Abstract. The Principle of Maximum Conformality (PMC) systematically and rigorously eliminatesorderbyordertherenormalization scaleand scheme ambiguitiesof perturbative QCDpredictions,atopic centralandcrucialfortestingthe StandardModeltohighprecision. The QCD running coupling s(q 2) is defined to sum all ß terms of a pQCD series as required by renormalization group invariance. The PMC thus generalizes the standard Gell-Mann Low scale-setting procedure for high precision tests of QED, where all vacuum polarization contributions are summed into the QED running coupling. The resulting series for pQCD matches the corresponding conformal theory, thus eliminating the non- convergent n-factorialrenormalongrowthofpQCD.ThePMCpredictionsagreewithQED scale-setting in the Abelian limit: PMC scale setting satisfies the property that calculations in QCD with NC colors must analytically match those of Abelian QED theory in the NC › 0 limit. The PMCis alsothe theoretical principle underlying commensurate scalerelations between observables which areindependentof the choiceofrenormalization scheme. The number of active flavors nf in the QCD ß function is also correctly determined. It also satisfiestherequirementthatonemustusethe same scale-settingprocedureinall sectorsof a Grand-UnifiedTheoryofQED,electroweak,andQCD interactions.Iwillalsoreviewa number of successful PMC predictions. Keywords: renormalization scale setting, principle of maximum conformality, light-front holography, color confinement, QCD running coupling at all scales, Abelian limit. 4.1 Renormalization Scale Setting It has become conventional to simply guess therenormalization scale and choose an arbitrary range of uncertainty when making perturbative QCD (pQCD) predictions. However, this ad hoc assignment of the renormalization scale and the estimateofthesizeoftheresulting uncertaintyleadsto anomalousrenormalization scheme-and-scale dependences.Avalid perturbativepredictionforanyphysi- cal observable must be independent of the initial choices of the renormalization scaleandtherenormalization scheme; thisisthe centralpropertyof renormalization group invariance (RGI)[1–5].Infact,relations between physical observables mustbe S. J. Brodsky independent of the theorist’s choice of the renormalization scheme and the renormalization scale in any given scheme at any given order of pQCD. The Principle of Maximum Conformality (PMC) [6–8], which generalizes the conventional Gell- Mann-Low method [9] for scale-setting in perturbative QED to non-Abelian QCD, provides a rigorous method for achieving unambiguous scheme-independent, fixed-order predictions for observables consistent with the principles of the renormalization group. The resulting renormalization scale of the running coupling reflects the physics of the underlying quark and gluon subprocess. A key problem in making precise perturbative QCD predictions has been the uncertainty in determining the renormalization scale µ of therunning coupling s(2). The purpose of the running coupling in any gauge theory is to sum all termsinvolving the ß function; in fact, when the renormalization scale is set properly, all non-conformal ß 6 = 0 terms in a perturbative expansion arising from renormalization are summed into the running coupling. The remaining terms in the perturbative series are then identical to that of a conformal theory; i.e., the corresponding theory with ß = 0. The above discussion was the motivation for the BLM (Brodsky-Lepage-Mackenzie) [10] procedurefor QCD scale-setting at lowest order. The BLM procedureis generalized toallordersbyusingthePMC(the Principleof Maximum Conformality[6–8].The PMC scale-setting procedure sets the renormalization scale s(Q2 ) at every PMC orderbyabsorbing the ß terms appearinginthepQCD series.TheresultingpQCD series thus the matches the corresponding conformal series with all ß terms set to 0. The problematic n!“renormalon” divergenceofapQCD series associated with the nonconformal terms does not appear in the conformal series, and the conformal seriesis independentofthe theorist’s choiceofrenormalizationscheme. This also means that relations between any two perturbatively calculable observables are scheme-independent. These relations are called “commensurate scale relations” [11]. There are no renormalization scale-setting ambiguities for precision tests of quantum electrodynamics. The scale of the QED running coupling at each order of the perturbative QED series is set to absorb all vacuum polarization diagrams; i.e. the ß terms. The coefficients in the pQED series then matches the conformal theory; i.e. the corresponding perturbative series with ß = 0.This defines the standardGell-Mann-Low scale-setting procedure for high precision tests of QED, where all vacuum polarization contributions are summed into the QED running coupling. (For a review, see ref [12]). The same scale-setting procedure applies to the SU(2)- U(1) theory of the electroweak interactions. [14] An important analytic property of non-Abelian QCD with NC colors is that it must agree analytically with Abelian QED in the NC › 0 limit, at fixed ^s = CF s and -1 C fixed n^f = T nf with CF = N2 and T - 1=2. This is the “Abelian correspondence CF 2NC principle.” [13] Thus the setting of the renormalization scale in QCD must agree with Gell-Mann-Low scale setting for QED in the NC › 0 limit. This analytic requirementis satisfiedby the PMC. ThePMC also satisfies therequirement that one must use the same scale-setting procedure in all sectors of a Grand-Unified Theory of QED, the electroweak interactions, and QCD [15]. As in QED, the renormalization scale in the PMC is fixed such that all ß non- conformal terms are systematically eliminated from the perturbative series and Title Suppressed Due to Excessive Length areresummed into therunning coupling; thisprocedureresultsina convergent, scheme-independent conformal series without factorialrenormalon divergences. The resulting scale-fixed predictions relating physical observables using the PMC are thus independent of the choice of renormalization scheme – a key requirement of renormalization group invariance. The PMC predictions are also independent of the choice of the initial renormalization scale 0. Since the PMC sums all of the non-conformal terms associated with the QCD ß function, it provides a rigorous method for eliminating renormalization scale ambiguities in quantum field theory. Predictions based on PMC scale setting also satisfy the self-consistency conditions oftherenormalizationgroup, includingreflectivity, symmetryand transitivity[21]. Theresulting PMCpredictions thus satisfy allof the basicrequirementsof RGI. Fig. 4.1: The thrust mean value h(1 - T)i for three-jet events versus the center-of . mass energy s using the conventional (Conv.) and PMC scale settings [42]. The dot-dashed, dashed and dotted lines are the conventional results at LO, NLO and NNLO,respectively. The solidlineis the PMCresult. The PMCprediction eliminates therenormalization scale µ uncertainty. The bands for theoretical predictions p. are obtained by varying µ . [ s=2;2 s]. The experimental data points are taken from the ALEPH, DELPH, OPAL, L3, JADE,TASSO, MARKII, HRS and AMY experiments . The transition scale between the perturbative and nonperturbative domains of QCD can also be determined by using the PMC [17, 22–24], thus providing a procedure for setting the “factorization” scale for pQCD evolution. The running S. J. Brodsky couplingresums allof the f ig-termsby using the PMC, which naturally leads to a more convergent and renormalon-free pQCD series. In more detail: the PMC scales are determined by applying the RGE of the QCD running coupling. By recursively applying the RGE one establishes a perturbative -pattern at each order in a pQCD expansion. For example, the usual scale-displacement relation for the running couplings at two different scales Q1 and Q2 can be deduced from the RGE, which reads   Q2 Q2 Q2 22 223 aQ2 = aQ1 - 0 ln a + 2 ln2 - 1 ln a Q1 0Q1 Q2 Q2 Q2 1 11      Q2 5Q2 Q2 Q2 2 224 2 +- 3 ln3 + 0 1 ln2 - 2 ln a + 4 ln4 0Q1 0 Q2 2Q2 Q2 Q2 111 1    13 Q2 3Q2 Q2 Q2 2 2 225 - 2 1 ln3 + 2 ln2 + 3 2 0 ln2 - 3 ln a + · , 01 Q1 3Q2 2Q2 Q2 Q2 11 11 (4.1) where aQi = s(Qi)=, the functions 0; 1, · are generally scheme dependent, which correspond to the one-loop, two-loop, · , contributions to the RGE, respectively. The PMC utilizes this perturbative -pattern to systematically set the scale oftherunning couplingateachorderinapQCD expansion. The coefficients of the f ig-terms in the -pattern can be identified by reconstructing the “degeneracy relations” [7,8] among different orders. The degeneracy relations, which underly the conformal features of the resultant pQCD series by applying the PMC, are general properties of a non-Abelian gauge theory [25]. The PMCprediction achievedinthiswayresemblesa skeleton-like expansion[34,35]. TheresultingPMCscalesreflectthe virtualityofthe amplitudesrelevanttoeach order, which are physical in the sense that they reflect the virtuality of the gluon propagatorsatagivenorder, aswellassettingtheeffective number(nf)of active quark flavors. The momentum flow for the process involving three-gluon vertex canbe determinedbyproperly dividing the total amplitude into gauge-invariant amplitudes [19]. Specific values for the PMC scales are computed asa perturbative expansion, so they have small uncertainties which can vary order-by-order. The PMC scales and the resulting fixed-order PMC predictions are to high accuracy independent of the initial choice of renormalization scale, e.g. the residual uncertainties due to unknown higher-order terms are negligibly small because of the combined suppression effect from both the exponential suppression and the s-suppression [7,8]. When one applies the standardPMC procedures, different scales generally appear at each order; this is called the PMC multi-scale approach which often requires considerabletheoretical analysis.To make the PMC scale-settingprocedure simpler and more easily to be automatized, a single-scale approach (PMC-s), which achieves many of the same PMC goals, has been suggested in Ref. [16]. This method effectively replaces the individual PMC scale at each order by a single (effective) scale in the sense of a mean value theorem; e.g., it can be regarded as a weighted average of the PMC scales at each order derived under PMC multi-scale approach. The PMC-s inherits the main features of the multi-scale approach; for Title Suppressed Due to Excessive Length example, its predictions are scheme independent, and the pQCD convergence is greatly improved due to the elimination of divergent renormalon terms.The single “PMC-s” scale shows stability and convergence with increasing order in + +- pQCD, as observed by the ee- annihilation cross-section ratio Reeand the Higgs decay-width ..(H › bb— ), up to four-loop level. Moreover, its predictions are again explicitly independent of the choice of the initial renormalization scale. Thus the PMC-s approach, which involvesa simpler analysis, canbe adopted asa reliable substitute for the PMC multi-scale approach, especially when one does not need detailed information at each order.We have givena detailed comparison of these two PMC approaches by comparing their predictions for three important quantities Re+e, R. and ..H!b — upto four-loop pQCD corrections [6]. The numeri- b cal results show that the single-scale PMCs method, which involves a somewhat simpler analysis, can serve as a reliable substitute for the full multi-scale PMCm method, and that it leads to more precise pQCD predictions with less residual scale dependence. There are also cases in which additional momentum flows occur, whose scale uncertainties can also be eliminated by applying the PMC. For example, there are A two types of log terms, ln(=MZ) and ln(=Mt) [26–30], for the axial singlet r S of the hadronic Z decays. By applying the PMC, one finds the optimal scale is A QAS ' 100 GeV [32], indicating that thetypical momentum flow for ris closer to S MZ than Mt. The PMC can also be systematically applied to multi-scale problems. The typical momentum flow can be distinct; thus, one should apply the PMC separatelyineachregion.For example,twooptimal scalesariseattheN2LO level for theproductionof massive quark-anti-quark pairs(QQ—)close to threshold [33], p. with one being proportional to s ^and the other to vs^, where v is the Q and — Q relative velocity. The PMC thus greatly improves the reliability and precision of QCD predictions at the LHC and other colliders [6] and greatly increases the sensitivity of experiments at the LHC to new physics beyond the StandardModel. 4.1.1 An overview of PMC renormalization-scale setting The PMC procedure follows these steps -First, we perform a pQCD calculation of an observable by using general regularization and renormalization procedures at an arbitrary initial renormalization scale µ and by taking any renormalization scheme. The initial renormalization scale can be arbitrarily chosen, which only needs to be large enough(µ >> QCD)to ensurethereliabilityofthe perturbative calculation. One may choose the renormalization scheme to be the usually adopted MS- scheme; after applying the PMC, the final pQCD prediction will be shown to be independent to this choice, since the PMC is consistent with RGI. -Second, we identify the non-conformal f ig-terms in the pQCD series. This can be achieved with the help of the degeneracy relations among different orders [7,8], which identify which terms in the pQCD series are associated with the RGE and which terms are not. By using the displacement relation for the running coupling at any two scales, e.g. Eq.(4.1), one can obtain thegeneral patternof the f ig-terms at each order, S. J. Brodsky which naturally implies the wanted degeneracy relations among different 2 , iai+2 terms; e.g., the coefficients for 0a, 1a3 ,· arethe same. It has been µ demonstrated that the degeneracy relations hold using any renormalization scheme [25]. The dimensional-like R-scheme provides a natural explanation of the degeneracy relations which are general properties of the non-Abelian gaugetheoryandunderlytheresulting conformal featuresofthepQCD series. Alternatively, one can use the . dependence of the series to identify the f igterms [8]. One can alsorearrange all the perturbative coefficients, which are usually expressed as an nf-power series, into f ig-terms or non-f ig-terms. One needs to be careful using this method to ensure that the UV-free light- quark loops are not related to the f ig-terms; they should be identified as conformal terms and should be kept unchanged when doing the nf › f i} transformation. The separation of UV-divergent and UV-free terms is very important. This fact has already been shown in QED case, in which electron- loop light-by-light contribution to the sixth-order muon anomalous moment is sizable but UV-free and should be treated as conformal terms [41]. There are many examples for the QCD case. For example,by carefully dealing with the UV-free light-by-light diagrams at theN2LO level, the resulting PMC prediction agrees with the BaBar measurements within errors, thus provides a solution for the . * › c form factor puzzle [39]. In practice, one can also apply the PMCby directly dealing with the nf-power series without transforming them into the f ig-terms [40]. This procedure is based on the observation that one can rearrange all the Feynman diagrams of a process in form of a cascade; i.e., the “new” terms emerging at each order can be equivalently regarded as a one-loop correction to all the “old” lower-order terms. Allof the nf-terms can then be absorbed into the running coupling following the basic -pattern in the scale-displacement formula, i.e. Eq.(4.1). More explicitly, in this treatment, the PMC scales can be derived in the following way: The LO PMC scale Q1 is obtained by eliminating all the nf-terms with the highest power at each order, and at this step, the coefficients of the lower-power nf-terms are changed simultaneously to ensure that the correct LO s-running is obtained; the NLO PMC scale Q2 is obtained by eliminating the nf-termsof one less powerin the new series obtaininga third series with less nf-terms; and so on until all nf-terms are eliminated. If the nf-terms are treated correctly, the results for both treatments shall be equivalent since they lead to the same resummed “conformal” series up to all orders. Those two PMC approaches differ, however, at the non-conformal level,bypredicting slightly different PMC scalesof therunning coupling. This difference arisesduetodifferentwaysofresummingthe non-conformal f igterms, but this difference decreases rapidly when additional loop corrections are included [25]. -Third, we absorb different types of f ig-terms into therunning coupling via an order-by-order manner with the help of degeneracy relations. Different types of f ig-terms as determined from the RGE lead to different running behaviors of the running coupling at different orders, and hence, determine the distinct scales at each order. As a result, the PMC scales themselves are perturbative Title Suppressed Due to Excessive Length expansion series in the running coupling. Since a different scale generally appears at each order, we call this approach as the PMC multi-scale approach. -Finally, since all the non-conformal f ig-terms have beenresummed into the running coupling, theremaining termsin the perturbative series willbe identical to those of the corresponding conformal theory, thus leading to a generally scheme-independent prediction. Because of the uncalculated high-order terms, there is residual scale dependence for the PMC prediction. However such residualrenormalization scale dependenceis generally small eitherduetothe perturbative nature of the PMC scales or due to the fast convergence of the conformal pQCD series 1. This explains why one refers to the PMC method as “principle of maximum conformality”. The scheme independence of the PMC predictionisa generalresult, satisfying the centralpropertyof RGI. 4.2 Some Recent Applications of PMC scale setting Inthis section, somerecent PMC applications, which show essential featuresof PMC and the importance of proper renormalization scale-setting are reviewed. Further details can be found in refs. [31, 36–38] The hadroproduction of the Higgs boson The total cross section for the production of Higgs boson at hadron colliders can be treated as the convolution of the hard-scattering partonic cross section ^ij with the corresponding parton luminosity Lij, i.e. Z S X H1H2!HX = ds Lij(s, S, f)^ij(s, L, R), (4.2) i;j 2 m H where the parton luminosity Z S 1d^s Lij = fi=H1 (x1;f) fj=H2 (x2;f) . (4.3) S ^s s Here the indices i, j run over all possible parton flavors in proton H1 or H2, x1 =^s=S and x2 = s=s^. S denotes the hadronic center-of-mass energy squared, and s = x1x2S is the subprocess center-of-mass energy squared. The subprocess cross section ^ij depends on both therenormalization scale r and the factorization scale 2 f,and the parton luminosity depends onf.Wedefine two ratios L = =m2 and fH 2 R = =2 , where mH is the Higgs boson mass. The parton distribution functions rf (PDF) underlying the parton luminosity fi=H. (x ;f) (. = 1 or 2)describes the probability of finding a parton of type i with light-front momentum fraction between x. and x. + dx. in the proton H . The two-dimensional integration over 1 By choosing a proper scale for the highest-order terms, whose value cannot be fixed, one can achievea scheme-independentpredictiondueto commensurate scalerelations among the predictions under different schemes [11]. 52 S. J. Brodsky s and ^s can be performed numerically by using the VEGAS program [44]. For 2 this purpose, one can set s = m(S=m2 )y1 and s ^= s(S=s)y2 , and transform the HH two-dimensional integration into an integration over two variables y1;2 . [0, 1]. Analytic expressions using the MS-scheme for the partonic cross section ^ij up to N2LO level can be found in Refs. [46, 47], which can be used for the PMC analysis. There are two types of large logarithmic terms ln(r=mH) and ln(r=mt) in ^ij. Thus a single guessed scale, using conventional scale-setting, such as r = mH, cannot eliminate all of the large logarithmic terms. This explains why there are large K factors for the high-order terms, confirming the importance of achieving exact values for each order. The PMC uses the RGE to determine the optimal running behavior of s at each order, and the large scale uncertainty for each order using conventional scale setting canbe eliminated.Tobe specific, the PMC introduces multiple scales for physical applications which depend on multiple kinematic variables, which is caused by the fact that different typical momentum flows could exist in different kinematic regions. Similar conditions have been observed in the hadronic Z decays [32] and the heavy-quark pair production via qq —fusion [33]. For example, the process qq —› QQ —near the heavy quark(Q) threshold involves not only the invariant variable ^s ~ 4M2 , but also the variable Q 2 vs with vrel being the relative velocity, which enters the Sudakov final-state rel ^ corrections. Tevatron LHC . S 1.96TeV 7TeV 8TeV 13TeV 14TeV Conv. 0:63+0:13 13:92+2:25 18:12+2:87 44:26+6:61 50:33+7:47 -0:11 -2:06 -2:66 -6:43 -7:31 PMC 0:86+0:13 18:04+1:36 23:37+1:65 56:34+3:45 63:94+3:88 -0:12 -1:32 -1:59 -3:00 -3:30 Table 4.1: The total hadronic cross section sum (in unit: pb) using the conventional (Conv.) and PMC scale-settings [48], where the uncertainties are for r . [mH=2, 2mH] and f . [mH=2, 2mH]. We usesum to stand for the sum of the total hadronic production cross sections (ij) with (ij)=(gg), (qq—), (gq), (gq—) and (qq0), respectively. Numerical results for sum at the Tevatron and LHC are presented in Table 4.1 [48], where the uncertainties are for r . [mH=2, 2mH] and f . [mH=2, 2mH]. As a comparison, the results using conventional scale-setting are also presented. After applying the PMC, sum is increased by ~ 37%at theTevatron, andby~ 30%at the LHC for . S =7,8,13 and14TeV, respectively. To compare with the LHC measurements for Higgs boson production cross- section [49–51], one needs to include the contributions from other known production modes, such as the vector-boson fusion production process, the WH=ZH Higgs associated production process, the Higgs production associated with heavy quarks, etc.Weuse xH to stand for the sum of those production cross sections from the channels via Z, W, tt— , bb —and · , and use EW to stand for those production cross sections from the channels with electroweak corrections. The values of xH and EW are small in comparison to the dominant gluon-fusion ggH contribution. Title Suppressed Due to Excessive Length Decay channel Incl 7TeV 8TeV 13TeV H › . [49–51] 35+13 30:5+7:5 47:9+9:1 -12 -7:4 -8:6 H › ZZ * › 4l [49–51] 33+21 37+9 68:0+11:4 -16 -8 -10:4 LHC-XS [56] 19:2 ± 0:9 24:5 ± 1:1 55:6+2:4 -3:4 PMC 21:21+1:36 27:37+1:65 65:72+3:46 -1:32 -1:59 -3:01 Table4.2:Total inclusivecross sections(in unit:pb)forHiggsproductionatthe . LHC for the CM collision energies S = 7,8and13TeV,respectively [48].The inclusive cross section is Incl = sum + xH + EW. 10203040506070.Incl(pb) PMCConv. LHC-XSNNLO+NNLLNNNLOH›..H›ZZ*›.S=8TeV Fig.4.2: ComparisonoftheN2LO conventional versus PMC predictions for the total inclusive cross section Incl [48] with the latestATLAS measurements at8 TeV [49]. The LHC-XSpredictions [52], theN2LO+NNLL prediction [57], and the N3LO prediction [58] are presented as a comparison. The solid lines are central values. . Taking S = 8 TeV andmH = 125 GeV, one predicts xH = 3:08 + 0:10 pb [49, 52]; the electro-weak correctionupto two-loop level leadstoa +5:1%shift with respect to theN2LO-levelQCDcross sections[54,55].Taking those contributions into consideration, the PMC predictions for the total inclusive cross section Incl at the LHC for several center-of-mass (CM) collision energies are presented in Table 4.2; the LHCATLAS predictions viaH › . and H › ZZ * › 4l decay channels [49,50] are also given. The PMCresults are larger than the central values . of the LHC-XS prediction [56] by about 10%, 12%and18%for S = 7,8and 13 TeV, respectively, which shows a better agreement with the data. This is clearly S. J. Brodsky shownby Figure 4.2,in whicha comparisonof ourpresentN2LO conventional and PMC predictions for Incl withtheATLAS measurementsat8TeVispresented. Becauseofthelarge uncertaintyfortheATLASdata,moredatais neededtodraw definite conclusions on the SM predictions. More accurate measurements with . high integrated luminosity for S=13TeV will be helpful to test the PMC and conventional predictions. fid(pp › H › ) 7TeV 8TeV 13TeV ATLAS data [59] 49 ± 18 42:5+10:3 52+40 -10:2 -37 CMS data [60] --84+13 -12 ATLAS data [61] --60:4 ± 8:6 LHC-XS [56] 24:7 ± 2:6 31:0 ± 3:2 66:1+6:8 -6:6 PMC prediction 30:1+2:3 38:3+2:9 85:8+5:7 -2:2 -2:8 -5:3 Table 4.3: The fiducial cross sectionfid(pp › H › ) (in unit: fb) at the LHC for . CM collision energies S =7,8and13TeV,respectively [48]. It has been suggested that the fiducial cross section fid can also be used to test the theoretical predictions, which is defined as fid(pp › H › )= InclBH! A. (4.4) The A is the acceptance factor, whose values for three typical proton-proton CM collision energies are [59], Aj7TeV = 0:620 ± 0:007, Aj8TeV = 0:611 ± 0:012 and Aj13TeV = 0:570 ± 0:006. The BH! . is the branching ratio of H › . By using the ..(H › ) with conventional scale-setting, the LHC-XS group predicts BH! . = 0:00228 ± 0:00011 [52].A PMC analysis for . (H › ) up to three- loop or five-loop level has been given in Refs. [53, 133]. Using the formulae given there, we obtain ..(H › )jPMC = 9:34 × 10-3 MeV for mH = 125 GeV. Using this value, together with Higgs total decay width ..Total =(4:07 ± 0:16) × 10-3 GeV [52], we obtain BH! jPMC = 0:00229 ± 0:00009. The PMC predictions for fid(pp › H › ) attheLHCaregiveninTable4.3, wheretheATLASandCMS measurements [59–61] and the LHC-XS predictions [56] are also presented. The PMC fiducial cross sections are larger than the LHC-XS onesby ~ 22%, ~ 24%and . ~ 30%for S =7TeV,8TeVand13TeV,respectively.Table4.3 showsno significant differences between the measured fiducial cross sections and the SM predictions, . and the PMC predictions show better agreement with the measurements at S = 7 TeV and8TeV. Top-quark pair production at hadron colliders and the top-quark pole mass As in the case of the hadronic production of the Higgs boson, the total cross section for the top-quark pair production at the hadronic colliders can also be written as the convolution of the factorized partonic cross section ^ij with the parton Title Suppressed Due to Excessive Length luminosities Lij: Z S X H1H2!t —= ds Lij(s, S, f)^ij(s, s(r);r;f), (4.5) tX i;j 4m2 t where the parton luminosities Lij has been defined in Eq.(4.3), and the partonic cross section ^ij has been computeduptoN2LO level,  1 f0 ^= (, r;f) 2 (r)+ f1 (, r;f) 3 (r)+ f2 (, r;f) 4 (r)+ O( 5 ) ij m2 ijsijsijss t (4.6) where . = 4m2 =s, (ij)= f(qq—), (gg), (gq), (gq—)} stands for the four production t channels,respectively.Inthe literature,the perturbativecoefficientsuptoN2LO level have been calculated by various groups, e.g. Refs. [62–72]. More explicitly, the LO, NLO andN2LO coefficients f0 ij and f2 in an nf-power series can be ij, f1 ij explicitlyreadfrom theHATHORprogram [73] and theTop++program [74]. By identifying the nf-terms associated with the f ig-terms in the coefficients f0 ij, f1 and fij 2 , and by using the degeneracy relations of -pattern at different ij orders, one can determine the correct arguments of the strong couplings at each order and hence the PMC scales at each order by using the RGE via a recursive way [97,99]. The Coulomb-type corrections near the thresholdregion shouldbe treated separately, since their contributions are enhanced by factors of . and are . sizable (e.g. those terms areproportional to (=v) or (=v)2 [33], where v = 1 - , the heavy quark velocity). For this purpose, the Sommerfeld re-scattering formula is useful for a reliable prediction [75, 76]. Conventional scale-setting PMC scale-setting LO NLON2LO Total LO NLON2LO Total (qq—) channel 4.87 0.96 0.48 6.32 4.73 1.73 -0:063 6.35 (gg) channel 0.48 0.41 0.15 1.04 0.48 0.48 0.15 1.14 (gq) channel 0.00 -0:036 0.0046 -0:032 0.00 -0:036 0.0046 -0:032 (gq—) channel 0.00 -0:036 0.0047 -0:032 0.00 -0:036 0.0047 -0:032 sum 5.35 1.30 0.64 7.29 5.21 2.14 0.096 7.43 Table 4.4: The top-quark pair production cross sections (in unit: pb) before and . after PMC scale-setting at theTevatron with S = 1:96 TeV.r = f = mt. Numerical results for the total top-quark pair production cross sections at the hadronic collidersTevatron and LHC for both conventional and PMC scale settings arepresentedinTables 12.2, 4.5, 4.6, and 4.7,respectively.We have updatedprevious predictions by using mt = 173:3 GeV [77] and the CTEQ version CT14 [78] as the PDF. The cross sections for the individual production channels, i.e. (qq—), (gq), (gq—) and (gg) channels are presented. In these tables, the initial choice of renormalization scale and factorization scale is fixed to be r = f = mt. Wepresent theN2LO top-quarkpairproductioncross sectionsattheTevatronand LHC for both conventional and PMC scale settingsinTable 4.8, where fourCM S. J. Brodsky Conventional scale setting PMC scale setting LO NLON2LO Total LO NLON2LO Total (qq—) channel 23.37 3.42 1.86 28.69 22.32 7.23 -0:78 28.62 (gg) channel 80.40 46.87 10.87 138.15 80.10 54.70 8.77 145.54 (gq) channel 0.00 -0:43 1.41 1.03 0.00 -0:43 1.41 1.03 (gq—) channel 0.00 -0:44 0.24 -0:20 0.00 -0:44 0.24 -0:20 sum 103.77 49.42 14.38 167.67 102.42 61.06 9.64 174.98 Table 4.5: The top-quark pair production cross sections (in unit: pb) before and . after PMC scale-setting at the LHC with S = 7 TeV.r = f = mt. Conventional scale setting PMC scale setting LO NLON2LO Total LO NLON2LO Total (qq—) channel 29.88 4.20 2.31 36.43 28.46 9.09 -1:06 36.29 (gg) channel 118.10 67.43 15.01 200.57 117.66 78.53 11.92 210.86 (gq) channel 0.00 0.18 2.02 2.18 0.00 0.18 2.02 2.18 (gq—) channel 0.00 -0:53 0.37 -0:15 0.00 -0:53 0.37 -0:15 sum 147.98 71.28 19.71 239.03 146.12 87.27 13.25 249.18 Table 4.6: The top-quark pair production cross sections (in unit: pb) before and . after PMC scale-setting at the LHC with S = 8 TeV.r = f = mt. Conventional scale setting PMC scale setting LO NLON2LO Total LO NLON2LO Total (qq—) channel 66.47 8.30 4.73 79.58 62.86 19.38 -2:74 79.08 (gg) channel 415.06 224.43 43.36 682.98 413.52 259.35 32.56 713.60 (gq) channel 0.00 7.09 6.52 13.82 0.00 7.09 6.52 13.82 (gq—) channel 0.00 -0:25 1.59 1.33 0.00 -0:25 1.59 1.33 sum 481.53 239.57 56.20 777.72 476.38 285.57 37.93 807.83 Table 4.7: The top-quark pair production cross sections (in unit: pb) before and . after PMC scale-setting at the LHC with S = 13 TeV.r = f = mt. . collision energies S = 1:96 TeV,7 TeV,8 TeV,and13 TeV,and three typical choices of initial renormalization scale r = mt=2, mt, and 2mt have been assumed. Table 4.8 shows the PMC predictions for the top-pair total cross section:1:96TeV = Tevatron 7:43+0:14 pb at theTevatron, 7TeV = 175:0+3:5 pb, 8TeV = 249:2+5:0 pb, and -0:13 LHC -3:5 LHC -4:9 13TeV = 807:8+16:0 pbattheLHC. ThesepredictionsagreewiththeTevatronand LHC -15:8 LHC measurements withinerrors[79–95].Table4.8showsthatusing conventional scale setting,therenormalization scale dependenceoftheN2LO-level cross section is about 6%- 7%forr . [mt=2, 2mt]. Thus achieving the exact value for each order is important for a precise lower-order pQCD prediction, especially for those observables that are heavily dependent on the value at a particular order. By analyzing theN2LO pQCD seriesin detail,itis found that therenormalization scale dependence of each perturbative term is rather large using conventional scale setting [43]. On the other hand, by using the PMC, the cross sections at each order Title Suppressed Due to Excessive Length Conventional PMC r mt=2 mt 2mt mt=2 mt 2mt 1:96TeV Tevatron 7.54 7.29 7.01 7.43 7.43 7.43 7TeV LHC 172.07 167.67 160.46 174.97 174.98 174.99 8TeV LHC 244.87 239.03 228.94 249.16 249.18 249.19 13TeV LHC 792.36 777.72 746.92 807.80 807.83 807.86 Table 4.8: The N2LO top-pair production cross sections for the Tevatron and LHC (in unit of pb), comparing conventional versus PMC scale settings. Here all production channels have been summed. Three typical choices for the initial renormalization scales r = mt=2, mt and 2mt have been adopted. are almost unchanged, indicating a nearly scale-independent prediction can be achieved even at lower orders. If one sets r = mt=2 for conventional scale setting, the total cross section is close to the PMC prediction, whose pQCD convergence is also betterthan the cases of r = mt and r = 2mt as has been observed in Ref. [96]. Thus, the PMC provides support for “guessing” the optimal choice of r ~ mt=2 using conventional scale setting [45]. After applying the PMC, we obtain the optimal scale of the top-quark pair productionateach perturbativeorderinpQCD,andtheresultingtheoreticalpredictions are essentially free of the initial choice of renormalization scale. Thus a more accurate top-quark pole mass and a reasonable explanation of top-quark pair forward- backwardasymmetryatthehadronic colliderscanbe achieved[43,45,98–100]. First, to fix the top-quark mass, one can compare the pQCD prediction on the top- quark pair production cross-section with the experimental data. For this purpose, one can definea likelihood function [101] Z +. f(mt)= fth(jmt) · fexp(jmt) d. (4.7) -. Here fth(jmt) is the normalized Gaussian distribution determined theoretically, "# 2 (. - th(mt)) fth(jmt)= . 1 exp - . (4.8) 2th(mt) 22 th(mt) The top-quark pair production cross-section is a function of the top-quark pole mass mt;its decrease with increasingmt can be parameterized as [69] 4  172:5 mt mt th(mt)= c0 + c1(- 172:5)+ c2(- 172:5)2 + (4.9) mt=GeV GeV GeV mt + c3(- 172:5)3 , GeV (4.11) where all masses are given in units of GeV. th(mt) stands for the maximum error of the cross-section for a fixed mt. One can estimate its value by using S. J. Brodsky the CT14 error PDF sets [78] with range of s(MZ) . [0:117, 0:119]. The values for the coefficients c0;1;2;3 can be determined by using a wide range of the top- quark pole mass, mt . [160 GeV, 190 GeV]. Here th(mt) is defined as the cross- section at a fixed mt, where all input parameters are set to be their central values, [th(mt)++ th(mt)] is the maximum cross-section within the allowable parameter range, and [th(mt)- - th(mt)] is the minimum value. The function fexp(jmt) is the normalized Gaussian distribution determined experimentally, "# ..2 1. - exp(mt) fexp(jmt)= . exp - , (4.12) 2exp(mt) 2exp2 (mt) where exp(mt) is the measured cross-section, and exp(mt) is the uncertainty for exp(mt). By evaluating the likelihood function, we obtain mt = 174:6+3:1 -3:2 GeV [98], where the central value is extracted from the maximum of thelikeli- hood function, and the error ranges are obtained from the 68%area around the maximum. Because the PMC predictions have less uncertainty compared to the predictions by using conventional scale-setting, the precision of top-quark pole mass is dominated by the experimental errors. For example, the PMC determination for the pole mass via the combined dilepton and the lepton+jets channels data is about 1:8%, which is almost the same as that of the recent determination by the D0 collaboration, 172:8+3:4 GeV [102], whose error is ~ 1:9%. -3:2 Asummaryofthe top-quarkpole masses determinedatboththeTevatronand LHC is presented in Figure 4.3, where the PMC predictions and previous predictionsfrom other collaborations [87,88,101–107] arepresented. Second, it has been found that by applying the PMC, the SM predictions for the top-quark forward-backwardasymmetry at theTevatron are only 1. deviation from the CDF andD0 measurements [43,99,100].In fact, the PMC givesa scale- independent precise top-quark pair forward-backwardasymmetry, APMC = 9:2% FB and AFB(Mtt — > 450 GeV)= 29:9%, in agreement with the corresponding CDF and D0 measurements [108–114]. The large discrepancies of the top-quark forward- backwardasymmetry between theSM estimate and theTevatron data are thus greatly reduced. Moreover, the PMC prediction for AFB(Mtt — >Mcut) displays an “increasing-decreasing” behavior as Mcut is increased, consistent within errors with the measurements recently reported by D0 experiment [113]. The top-quark charge asymmetry at the LHC for the pp › t — tX process is defined as N(jy| >0)- N(jy| <0) AC = , (4.13) N(jy| >0)+ N(jy| <0) where jy| = jyt| - jyt— | is the difference between the absolute rapidities of the top and anti-top quarks, and N is the number of events. Measurements of the top- quark charge asymmetry at the LHC have been reported in Refs. [115–120]. Figure 4.4 gives a summary of the LHC measurements, together with the theoretical predictions. In contrast to the Tevatron pp —› t — tX processes, the asymmetric channel qq —› tt — gives a small pQCD contribution to the top-pair production at the LHC, and the symmetric channel gg › tt — provides the dominant contribution. Thus, the predicted charge asymmetry at the LHC is smaller than the one at the Title Suppressed Due to Excessive Length 135140145150155160165170175180185190mt(GeV) TevatronLHCCMS: JHEP 1608,029 (2016) CMS: Phys.Lett.B 728,496 (2013) ATLAS: JHEP 1510,121 (2015) Direct measurement LHC+TevatronD0: Phys.Rev.D 80,071102 (2009) D0: Phys.Lett.B 703,422 (2011) this work, PMC prediction at 7TeVthis work, PMC prediction at 8TeVD0: Phys.Rev.D 94,092004 (2016) ATLAS: ATLAS-CONF2011-054ATLAS: Eur.Phys.J.C 74,3109 (2014) this work, PMC prediction Fig.4.3:Asummaryofthe top-quarkpole mass determined indirectlyfromthe top-quarkpairproduction channelsattheTevatronandLHC[98].Forreference, the combinationofTevatron and LHC direct measurementsof the top-quark mass ispresented asa shaded band, which gives mt = 173:34 ± 0:76 GeV [107]. Tevatron.Two typicalSMpredictionsforthecharge asymmetryattheLHCare: ACj7TeV =(1:15 ± 0:06)%andACj8TeV =(1:02 ± 0:05)%[121];ACj7TeV =(1:23 ± 0:05)%andACj8TeV =(1:11 ± 0:04)%[122]. The uncertainties of the theoretical prediction are dominated by the choice of scale. The scale errors for conventional scale setting are obtainedby varying r . [mt=2, 2mt], and fixing the factorization scale f . r. As a representation, Figure 4.4 shows the prediction of Ref. [122]. On the other hand, the PMC prediction is almost scale independent and a more precise comparison with the data can be achieved. The . * › c transition form factor The simplest exclusive charmonium production process, . * . › c, measured in two-photon collisions, provides another example of the importance of a proper scale-setting approach for fixed-order predictions. This is also helpful for testing Nonrelativistic QCD (NRQCD) theory [123]. One can definea transition form factor (TFF) F(Q2) via the following way [124]:  EMj (k, ")i = ie2(4.14) hc(p)jJµ "qkF(Q2), where Jµ EM is the electromagnetic current evaluated at the time-like momentum transfer squared, Q2 =-q2 = -(p - k)2 >0. The BaBar collaboration has measured its value and parameterized it as jF(Q2)=F(0)| = 1=(1 + Q2=) [125], where . = 8:5 ± 0:6 ± 0:7 GeV2. In the case of conventional scale setting, the renormal- p 2 ization scale is simply set as the typical momentum flow Q = Q2 + m;the c S. J. Brodsky -20246810AC(%) ATLAS, ATLAS-CONF-2012-057 (2012) Conv., BS programCMS, Phys.Lett.B 717,129 (2012) CMS, CMS PAS TOP-12-010 (2012) ATLAS, JHEP 1402,107 (2014) CMS, JHEP 1404,191 (2014) PMC, this workF. Derue, arXiv:1408.6135 (2014) weighted average Fig. 4.4: The top-quark charge asymmetry AC assuming conventional scale setting . (Conv.) and PMC scale setting for S = 7 TeV [45]; the error bars are forr . [mt=2, 2mt] and f . [mt=2, 2mt].Asacomparison,the experimentalresults [115– 120] and the prediction of Ref. [122] are also presented. N2LO NRQCD prediction cannot explain the BaBar measurements over a wide Q2 range [126]. Here mc is the c-quark mass and we set its value as 1:68 GeV. This disagreement cannotbe solvedby taking higher Fock states into consideration [127, 128]. Numerically, the choice of renormalization scale r = Q leads to a substantially negativeN2LO contribution and hencea large jF(Q2)=F(0)j, in disagreement with the data. Following the standardPMC scale-setting procedures, one can determine PMC the PMC scale of the process by carefully dealing with the light-by-light r diagrams at theN2LO level. The determined PMC scale varies with momentum transfer squared Q2 at which the TFF is measured, and it is independent of the initial choice of r (thus the conventional scale uncertainty is eliminated).We PMC present the PMC scale µ versus Q2 in Figure 4.5, which is larger than the r “guessed” value Q in the small and large Q2-regions. In the intermediate Q2 PMC region, e.g. Q2 ~ [20, 60] GeV2, the discrepancy between and Q is small; r and the largest difference occurs at Q2 = 0. Acomparisonoftherenormalization scale dependenceforthe ratiojF(Q2)=F(0)| is giveninFigure4.6, whichis obtainedbyusingthe sameinput parametersas those of Refs. [39, 126]. It shows that the PMC prediction is independent of the initial choice of scale r, whereas the conventional scale uncertainty is large, especially in low Q2-region. The PMC prediction is close to the BaBar measurement. Thus Title Suppressed Due to Excessive Length Fig. 4.5: The PMC scale of the transition form factor F(Q2) [39], defined in Eq.(4.14), versus Q2. The conventional choice of scale r = Q is presented as a comparison. the application of PMC supports the applicability of NRQCD to hardexclusive processes involving heavy quarkonium. The determination of the factorization scale is a separate issue from renormalization scale setting, sinceitispresent even fora conformal theory. The factorization scale can be determined by matching nonperturbative bound-state dynamics with perturbative DGLAP evolution [157–159]. Recently, by using light-front holography [160,161],it has been shown that the matchingof high-and-low scaleregimes of s can determine the scale which sets the interface between perturbative and nonperturbative hadron dynamics [17,22–24]. Figure 4.6 also shows the factorization scale dependence for the ratio jF(Q2)=F(0)j. In the case of conventional scale-setting, thereis large factorization scale dependence. Choosinga smaller factorization scale could lower theN2LO-level ratio jF(Q2)=F(0)| to a certain degree, but it cannot eliminate the large discrepancy withthe data. In contrast, after applying the PMC, theprediction showsa small factorization scale dependence. This in some sense also shows the importance of a proper scale-setting approach. More explicitly, in the case Q2 = 0, a large factorization scale uncertainty is observed using conventional scale-setting; i.e., FConv(0)jr =mc = 0:43c(0) , 0:22c(0) , -0:06c(0) (4.15) S. J. Brodsky Conv.( .= mc) Conv.( .= 1GeV) PMC( .= 1GeV) PMC( .= mc) BABAR data0204060800.00.51.01.52.0Q2(GeV2)F(Q2)/F(0) Fig. 4.6: The ratio jF(Q2)=F(0)| up toN2LO-level versus Q2 using conventional (Conv.) and PMC scale-settings [39], where the BaBar data are presented as a comparison [125].Two typical factorization scales, . = 1 GeV and mc are adopted. p 22 2 The error bars are for µ =[µ =2, 22 ] with Q = Q2 + m. rQQc for factorization scale . = 1 GeV, mc and2mc,respectively. Here the LO coefficient c(0) is 4e2 hcj y()j0i (0) c c= . , (4.16) (Q2 + 4m2 ) mc c where ec =+2=3 is the c-quark electric charge, and hcj y()j0i represents the nonperturbative matrix-element which characterizes the probability of the (cc—)-pair to form a c bound state. The magnitude of the negativeN2LO term increases with increasing , and the FConv (0) is even negative for . = 2mc. On the other hand, by applying the PMC, we obtain a reasonable small factorization scale dependence FPMC(0)= 0:61c(0), 0:50c(0), 0:34c(0). (4.17) again for . = 1 GeV, mc and2mc,respectively. The conventional renormalization scheme-and-scale ambiguities for fixed-order pQCD predictions are caused by the mismatch of the perturbative coefficients and the QCD running coupling at any perturbative order. The elimination of such ambiguities relies heavily on how well we know the precise value and analytic properties of the strong coupling s. An extended RGE has been suggested to determine the s scheme-and-scale running behaviors simultaneously based on the conventional RGE,. However, those dependences are usually entangled with Title Suppressed Due to Excessive Length each other and can only be solved perturbatively or numerically. More recently, a C-scheme coupling ^s has been suggested, whose scheme-and-scale running behavior is exactly separated; it satisfies a RGE free of scheme-dependent f i2gterms. The C-scheme coupling canbe matchedtoa conventional coupling s viaa proper choice of the parameter C.We have demonstrated that the C-dependence of the PMC predictions can be eliminated up to any fixed order; since the value of C is arbitrary, it means the PMC prediction is independent of any renormalization scheme.We have illustrated these features for three physical observables which are known up to the four-loop level. -------------------------- 468101214160.140.160.180.200.220.240.260.28QHGeVL.sHQL Fig. 4.7: The extracted s(Q) in the MS-scheme from the comparison of PMC predictions with ALEPH data [129]. The error bars are from the experimental data. The three lines are the world average evaluated from s(MZ)= 0:1181 ± 0:0011 [130]. The renormalization scale depends on kinematics such as thrust (1 - T) for three + jet production via ee- annihilation.Adefinitive advantageof using the PMC is that since the PMC scale varies with (1 - T), we can extract directly the strong coupling s at a wide range of scales using the experimental data at single center . of-mass-energy, s = MZ. In the case of conventional scale setting, the predictions are scheme-and-scale dependent and do not agree with the precise experimental results; the extracted coupling constants in general deviate from the world average. In contrast, after applying the PMC, we obtainacomprehensive and self-consistent analysis for the thrust variable results including both the differential distributions and the mean values [42]. Using the ALEPH data [129], the extracted s are S. J. Brodsky presented in Figure 4.7. It shows that in the scale range of 3:5 GeV < Q < 16 GeV (corresponding(1 - T)range is0:05 < (1 - T) < 0:29), the extracted s are in excellent agreement with the world average evaluated from s(MZ). The PMC provides first-principle predictions for QCD; it satisfies renormalization group invariance and eliminates the conventionalrenormalization scheme-and- scale ambiguities, greatly improving the precision of tests of the StandardModel and the sensitivity of collider experiments to new physics. Since the perturbative coefficients obtained using the PMC are identical to those of a conformal theory, one can derive all-orders commensurate scale relations between physical observables evaluated at specific relative scales. Because the divergentrenormalon series does not appearin the conformal perturbative series generatedbythePMC,thereisan opportunitytouseresummation procedures such as thePA approach to predict the values of the uncalculated higher-order terms and thus to increase the precision and reliability of pQCD predictions.WehaveshownthatifthePMCpredictionforthe conformal seriesfor an observable has been determined at order n , then the [N=M]=[0=n - 1]-type s PAseries provides an important estimate for the higher-order terms. An essential property of renormalizable SU(N)]/U(1) gauge theories, is “Intrinsic Conformality,” [131]. It underlies the scale invariance of physical observables and canbeusedtoresolvethe conventionalrenormalization scale ambiguity at every order in pQCD. This reflects the underlying conformal properties displayed by pQCD at NNLO, eliminates the scheme dependence of pQCD predictions and is consistent with the generalpropertiesof the PMC.We have also introduceda new method [131] to identify the conformal and ß terms which can be applied either to numerical or to theoretical calculations and in some cases allows infinite resummation of the pQCD series, The implementation of the PMC. can significantly improve the precision of pQCD predictions; its implementation in multi-loop analysis also simplifies the calculation of higher orders corrections in a generalrenormalizable gauge theory. This method has also been used to improve the NLO pQCD prediction for tt — pair production and other processes at the LHC, where subtle aspects of the renormalization scale of the three-gluon vertex and multi gluon amplitudes, as well as large radiative corrections to heavy quarks at thresholdplayacrucialrole.Thelarge discrepancyofpQCDpredictionswiththe forward-backwardasymmetry measuredattheTevatronis significantlyreduced from3 . to approximately1 . The PMC has also been used to precisely determine the QCD running coupling constant s(Q2) over a wide range of Q2 from event . shapes for electron-positron annihilation measured at a single energy s [132]. ThePMC methodhasalsobeenappliedtoaspectrumofLHCprocesses including Higgsproduction,jetshapevariables,and final states containingahigh pT photon plus heavy quark jets, all of which, sharpen the precision of the StandardModel predictions. Title Suppressed Due to Excessive Length 4.3 Extending the Predictive Power of Perturbative QCD Using the Principle of Maximum Conformality and Bayesian Analysis In addition to the evaluation of high-order loop contributions, the precision and predictive power of perturbative QCD (pQCD) predictions depends on two important issues: (1) how to achieve a reliable, convergent fixed-order series, and (2) how to reliably estimate the contributions of unknown higher-order terms. Therecursive useofrenormalizationgroup equation, together with the Principle of Maximum Conformality (PMC), eliminates the renormalization scheme-and- scale ambiguities of the conventional pQCD series. The result is a conformal, scale-invariant series of finite order which also satisfies all of the principles of the renormalization group. In a recent paper [18] a novel Bayesian-based approach is proposed to estimate the size of the unknown higher order contributions based on an optimized analysis of probability distributions. One finds that by using the PMC conformal series, in combination with the Bayesian analysis, one can consistently achieve high degree of reliability estimates for the unknown high order terms. Thus the predictive power of pQCD can be greatly improved. This procedure has been applied to three pQCD observables: Re+e- R. and . (H › bb— ), which are each known up to four loops in pQCD. Numerical analyses confirm thatby using the convergent and scale-independent PMC conformal series, one can achieve reliable Bayesian probability estimates for the unknown higher-order contributions. For further details, see Ref. [18] 4.4 Color Confinement, Light-Front Holography, and the QCD Coupling at all Scales A key problem in hadron physics is to obtain a first approximation to QCD which can accurately predict not only the spectroscopy of hadrons, but also the light-front wave functions which underly their properties and dynamics. Guy de T´ eramond, Guenter Dosch, and I [15] have shown that a mass gap and a fundamental color confinement scale can be derived from light-front holography – the duality between five-dimensional anti-de Sitter (AdS) space physical 3+1 spacetime using light-front time. The combination of superconformal quantum mechanics [13, 134], light-front quantization [2] and the holographic embedding on a higher dimensional gravity theory [5] (gauge/gravity correspondence) has led to new analytic insights into the structure of hadrons and their dynamics [7, 9,15–17,148]. This newapproachto nonperturbativeQCD dynamics, holographic light-front QCD,hasledtoeffective semi-classicalrelativistic bound-state equations for arbitrary spin [8], and it incorporates fundamental properties which are not apparent from the QCD Lagrangian, such as the emergence of a universal hadron mass scale, thepredictionofa massless pionin the chiral limit, andremarkable connections between the spectroscopy of mesons, baryons and tetraquarks across the full hadron spectrum [39–41, 151]. Light-Front Hamiltonian theory provides a causal, frame-independent, ghost-free nonperturbative formalism for analyzing gauge theories such as QCD. Remarkably, 66 S. J. Brodsky LFtheoryin3+1 physical space-timeis holographicallydualto five-dimensional AdS space, if one identifies the LF radial variable . with the fifth coordinate z of AdS5. If the metric of the conformal AdS5 theory is modified by a dila- +2 z ton of the form e 2 , one obtains an analytically-solvable Lorentz-invariant color-confiningLF Schr ¨ odinger equations for hadron physics. The parameter . of the dilaton becomes the fundamental mass scale of QCD, underlying the color- confining potential of the LF Hamiltonian and the running coupling s(Q2) in the nonperturbative domain. When one introduces super-conformal algebra, theresult is “HolographicLFQCD”,whichnotonlypredictsa unified Regge-spectroscopy of mesons, baryons, and tetraquarks, arranged as supersymmetric 4-plets, but also the hadronic LF wavefunctions which underly form factors, structure functions, and other dynamical phenomena. In each case, the quarks and antiquarks cluster in hadrons as 3C diquarks, so that mesons, baryons and tetraquarks all obey a two-body 3C - 3— C LF bound-state equation. Thus tetraquarks arecompact hadrons, as fundamental as mesons and baryons. Holographic LF QCD also leads to novel phenomena such as the color transparency of hadrons produced in hard-exclusive reactions traversing a nuclear medium and asymmetric intrinsic heavy-quark distributions Q(x) 6Q(x), appearing at high — states of hadrons. Phenomenological extensions of the holographic QCD approach have also led to nontrivial connections between the dynamics of form factors and polarized and unpolarized quark distributions with pre-QCD nonperturbative approaches such as Regge theory and theVeneziano model [18,19,136].As discussedin the next section, it also predicts the analytic behavior of the QCD coupling s(Q2) in the nonperturbative domain [17, 139]. = x in the non-valence higher Fock 4.4.1 The QCD Coupling at All Scales The QCD running coupling can be defined [154] at all momentum scales from any perturbatively calculable observable, such as the coupling s (Q2) which is g1 defined from measurements of the Bjorken sum rule. At high momentum transfer, such “effective charges” satisfy asymptotic freedom, obey the usual pQCD renormalization group equations, and can be related to each other without scale +2 z ambiguity by commensurate scale relations [11]. The dilaton e 2 soft-wall modification [156] of the AdS5 metric, together with LF holography, predicts the functional behavior in the small Q2 domain [139]: s (Q2)= e-Q2=42 . Mea g1 surements of s (Q2) are remarkably consistent with this predicted Gaussian g1 form. The predicted coupling is thus finite at Q2 = 0. We have also shown how the parameter, which determines the mass scale of hadrons in the chiral limit, can be connected to the mass scale s controlling the evolution of the perturbative QCD coupling [17, 139, 140]. This connection can be done for any choice of renormalization scheme, including the MS scheme, as seen in Fig. 4.8. The relation between scales is obtained by matching at a scale Q2 the nonperturbative behavior of the effective QCD coupling, as determined 0 from light-front holography, to the perturbative QCD coupling with asymptotic Title Suppressed Due to Excessive Length freedom. The result of this perturbative/nonperturbative matching at the analytic inflection point is an effective QCD coupling which is defined at all momenta. Let us assume that the QED, electroweak, and QCD gauge theories satisfy grand unification. One can then argue that each of their gauge couplings at Q2 = 0 in the unified theory. is independent of the choice of renormalization scheme. The nonperturbative behavior of s(Q2) is driven by the color confining potential U(2)= 42, where 2, like the hadron masses, is scheme independent. The couplinghas noUV divergencesin the nonperturbative domain. Thus QCD(Q2) is analytically universal and scheme independent for Q2 belowthe transition scale, the infection point, and It becomes scheme dependent only above that scale. Knowing the QCD coupling in the nonperturbative and timelike domains also can have important implications for QCD predictions, For example, PMC scale- setting can require knowledge of the coupling outside of its usual spacelike and perturbative domains. The analytic determination of s(Q2) over all domains will clearly greatly increase the precision and reliability of QCD predictions. 4.5 Summary It has become conventional to simply guess therenormalization scale and choose an arbitrary range of uncertainty when making perturbative QCD (pQCD) predictions. However, this ad hoc assignment of the renormalization scale and the estimateofthesizeoftheresulting uncertaintyleadsto anomalousrenormalization scheme-and-scale dependences. In fact, relations between physical observables mustbe independentof the theorist’s choiceof therenormalization scheme, and the renormalization scale in any given scheme at any given order of pQCD is not ambiguous. The Principle of Maximum Conformality (PMC), which generalizes the conventionalGell-Mann-Low methodfor scale-settingin perturbativeQED to non-Abelian QCD, provides a rigorous method for achieving unambiguous scheme-independent, fixed-order predictions for observables consistent withthe principles of the renormalization group. The renormalization scale of the running coupling depends dynamically on the virtuality of the underlying quark and gluon subprocess and thus the specific kinematics of each event. Therenormalization scalein the PMCis fixed such that all ß nonconformal terms are eliminated from the perturbative series and are resummed into the running coupling; this procedure results in a convergent, scheme-independent conformal series without factorial renormalon divergences. The resulting scale-fixed predictions for physical observables using the PMC are also independent of the choice of renormalization scheme –akeyrequirementofrenormalizationgroup invariance. The PMC predictions are also independent of the choice of the initial renormalization scale 0. Other important properties of the PMC are that the resulting series are free of renormalon resummation problems, and the predictions agree with QED scale-setting in the Abelian limit. The PMC is also the theoretical principle underlying the BLM procedure, commensurate scale relations between observables, and the scale-setting methodused in lattice gauge theory. The number of active flavors nf in the QCD ß function is also correctly determined.We have also showed that a single global PMC scale, valid at leading order, can be derived S. J. Brodsky 1110Q .g1(Q)/.Transition .sg1(Q2) . Nonperturbative (Quark All-Scale QCD CouplingQ20=1.08±Q24.2Deur, sjbm.=p2. mp=2. !..2.!s(Q) . defined at ..s(Q)/....g1/.. (pQCD) ..g1/.. world data ..../.. OPALAdSModified AdSLattice QCD (2004)(..g1/.. Hall A/CLAS ..g1/.. JLab CLAS ..F3/GDH limit00.20.40.60.8110-1110Sublimated gluons GeVAdS/QCD dilaton captures the higher twist e+.2z2 Fig. 4.8: (A). Prediction from LF Holography for the QCD running coupling s (Q2). The magnitude and derivative of the perturbative and nonperturba g1 tive coupling are matched at the scale Q0. This matching connects the perturbative scale to the nonperturbative scale . which underlies the hadron mass scale. MS (B). Comparison of the predicted nonperturbative coupling with measurements of the effective charge s (Q2) defined from the Bjorken sum rule. See Ref. [140]. g1 Title Suppressed Due to Excessive Length from basicpropertiesofthe perturbativeQCDcross section.Wehavegivena detailed comparison of these PMC approaches by comparing their predictions for three important quantities Re+e, R. and ..H!b —up to four-loop pQCD cor- b rections. The numerical results show that the single-scale PMCs method, which involvesa somewhat simpler analysis, can serve asareliable substitute for the full multi-scale PMCm method, and that it leads to more precise pQCD predictions withlessresidualscale dependence.ThePMCthusgreatlyimprovesthereliability and precision of QCD predictions at the LHC and other colliders. As we have demonstrated, the PMC also has the potential to greatly increase the sensitivity of experiments at the LHC to new physics beyond the StandardModel. 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Frampton paul.h.frampton@gmail.com Dipartimento di Matematica e Fisica ”Ennio De Giorgi”, Universit`a del Salento,Via Arnesano, 73100 Lecce, Italy Abstract. We discuss the predictions of the bilepton model which is an extension of the standard model in which the group SU(2) × U(1) is changed to SU(3) × U(1) and the fermion families are treated non-sequentially with the thirdassigned differently from the first two. Cancellation of triangle anomalies and asymptotic freedom require three families. The predicted new physics includes bileptons and three heavy quarks D. S and T . QCD will bind the heavy quarks to light quarks andto each other to form baryons and mesons which, unlike bileptons, are beyond the reach of the LHC but accessible in a hypothetical 100TeVproton-proton collider. Povzetek:Snov, ki jo gradijo kvarki in leptoni ter ter njihova umeritvena polja, prispeva komaj dvajsetino entropije.Avtoriˇsˇce odgovorspredlogom,da prispevakentropiji vesolja poleg temne snovi, ki v prete ˇca gibanje snovi v zni meri doloˇ galaksijah in v intergalaktiˇcnem prostoru, ipredvsem zjemno masivna temna snov iz prvobitnihˇ crnih lukenj. Keywords:Triangle anomaly cancellation; three families;TeV quarks; additional baryons; additional mesons. arXiv:2209.05349 5.1 Introduction In this talk we shall discuss what now seems likely to be the first new particle beyondthe standardmodelandwhichisnowbeing actively searchedforatthe LHC. The bilepton model, a better name than the 331-Model, was invented as an example of what then was expected to be a new class of models which require the existence three families. That invention was in 1992 [1] but it required a couple moreyearstorealisethatthe expectednewclassof modelshasonlyone member. We remain optimistic that LHC can find a discovery signal for the bilepton gauge boson in the remainder of 2022. What we can say generally is that to invent a model which is beyond the standardmodel, one generally aims to both (i)address and solve a question unanswered within the standardmodel, and to (ii) provide explicit predictions which are testable. The bilepton model beautifully fulfils both of these criteria. Paul H. Frampton Although not a sub-theory, the model originated from studying an interesting SU(15) model [2] in which the 224 gauge bosons couple to all possible pairs of the 15 states (u, d) ;( u—) ;( d— ) ;(e:e); (e—). (5.1) Every gauge boson therefore hasa well-definedBandL so there canbe noproton decay by tree-level gauge boson exchange. In the SU(15) model there is one unaesthetic feature that anomalies are cancelled by adding mirror fermions as in 15 + 15. But persisting further, we considered the — subgroups in SU(15) › SU(12)q SU(3)l,especially theSU(3)l which contains an antitriplet (e+;e;e-) where the jL| = jQ| = 2 bilepton can first be seen, coupling electron to positron. The question then was: can a chiral model contain bileptons? After hundreds of trials and errors we found only one solution of the anomaly cancellation equations. Thisrequired non-sequential families where the thirdis assigned differentlyfrom the first two and explains why there must be three families. This is the bilepton model. It provides an answer to Rabi’s famous question when the muon was discovered in 1936: ”Who ordered that?” The non-sequentiality of families offers one explanation for the failure of the SU(5) model studied firstin 1974[3,4] thenin hundredsof other papers. SU(5) assumed sequentiality of families of the form 3(10 + 5— ). In 1977Weinberg[5] and in 1984 Glashow [6] both considered upgrading the electroweak SU(2) of the standardmodel to SU(3) but overlooked the assignments which explain three families. 5.2 Bilepton model The gauge group is: SU(3)C × SU(2)L × U(1)X (5.2) The simplest choice for the electric charge is Q = 1 3 L + 2 . ! 3 8 L + X 2 . ! 3 . 9 2 (5.3) where Tr(a b ) = 2ab LL(5.4) and 9 . . ! 2 . diag(1, 1, 1) 3 (5.5) Thus a triplet has charges (X + 1, X, X - 1). 5 Predictions of Additional Baryons and Mesons 77 Leptons are treated democratically in each of the three families. They are colour singlets in antitriplets of SU(3)L : (e+;e;e-)L (µ +;;µ -)L (+;;-)L All have X = 0. Quarksinthefirstfamilyare assignedtoa left-handedtripletplusthree singlets of SU(3)L. (u ;d ;D )L ( u— )L, ( d— )L, ( D— )L Similarly for the second family (c;s ;S )L (c— )L, (s— )L, ( S— )L TheXvaluesareforthe tripletsare X =-1=3 and for the singlets X =-2=3, +1=3, +4=3 respectively. The electric charge of the new quarks D, S is -4=3. The quarks of the third family are treated differently. They are assigned to a left-handed antitriplet and three singlets under SU(3)L (b ;t ;T )L ( b— )L, (t— )L, ( T— )L The antitriplet has X =+2=3 and the singlets carry X =+1=3, -2=3, -5=3 respectively. The new quark T has Q = 5=3. SomeoftherelevantLHC phenomenologyis discussedin[7].A refined mass estimate [8] for the bilepton is is M(Y)=(1:29 ± 0:06) TeV wherefaute de mieux it was assumed that the symmetry breaking of SU(3)L is closely similar to that of SU(2)L. It will be pleasing if the physical mass is consistent with this. 5.3 New Quarks Because the quarks are in triplets and anti-triplets of SU(3)L, rather than only in doublets of SU(2)L as in the standardmodel, there is necessarily an additional quark in each family. In the first and second families they are the D and S respectively, both with charge Q =-4=3 and lepton number L =+2. In the thirdfamily is the T with charge Q =+5=3 and lepton number L =-2. All the threeTeV scale 1 quarks are colour triplets with spin-1 and baryon number B = . Their masses are 23 yet to be measured but may be expected to be below the ceiling of 4:1TeV which Paul H. Frampton is the upper limit for symmetry breaking of SU(3)L and probably above 1TeV. By analogy with the known quarks, one might expect M(T ) >M(S) >M(D), although without experimental data this is conjecture. The heavy quarks and antiquarks will be bound to light quarks and antiquarks, and to each other, to form an interesting spectroscopy of mesons and baryons. Let us first display,inTables1,2theTeV mesons, theninTables 3,4,5 theTeV baryons. The charge conjugate states are equally expected, and will reverse the signs of Q and L. 5.4 Additional Baryons and Mesons . Table 5.1:TeV mesonsQ q — Q —q Q L D=S D=S —u etc. — d etc. -2 -1 +2 +2 T T —u etc. — d etc. +1 +2 -2 -2 Although the Q massesare unknown,itmaybereasonable firsttomakeaprelimi- nary discussion of these states by assuming that M(T ) >M(S)+ 2Mt >M(D)+ 4Mt (5.6) where Mt isthetopquarkmasssothatthelightestoftheTeVbaryonsand mesons are those containing just one D quark or one D antiquark. The next lightest are the — TeV baryons and mesons containing just oneS quark or one S antiquark. — We begin by discussing the decay modes of theD q —mesonsinTable1, focusing on final states from the first family. The decays of D include, taking care of L conservation, 5 Predictions of Additional Baryons and Mesons Table 5.2:TeV mesonsQQ — Q —Q Q L D=S —D —S 0 0 D=S —T -3 +4 T —T 0 0 Table 5.3:TeV baryonsQqq Q qq Q L D=S dd etc. -2 +2 D=S ud etc. -1 +2 D=S uu etc. 0 +2 T dd etc. +1 -2 T ud etc. +2 -2 T uu etc, +3 -2 Table 5.4:TeV baryonsQQq QQ q Q L (D=S)(D=S) detc. -3 +4 (D=S)(D=S) u etc. -2 +4 (D=S)T detc. 0 0 (D=S)T u etc. +1 0 T T detc. +3 -4 T T u etc. +4 -4 D › d + Y- - › d +(e + e) - › d +(µ + ) › d +(- + ) (5.7) Paul H. Frampton Table 5.5:TeV baryonsQQQ QQQ Q L (D=S)(D=S)(D=S) -4 +6 (D=S)(D=S)T -1 +2 (D=S)T T +2 -2 T T T +5 -6 which implies that decays of the (D u—) meson include - (D u—) › - +(e + e) - › - +(µ + ) › - +(- + ) (5.8) and variants thereof where - is replaced by any other non-strange negatively charged meson. The d in Eq.(5.7) can be replaced by s or b which subsequently decay. An alternative to Eq.(5.7) is D › u + Y-- - › u +(e + e -) - › u +(µ + µ -) › u +(- + -) (5.9) which implies additional decay modes of the (D u—) meson which include - (D u—) › 0 +(e + e -) - › 0 +(µ + µ -) › 0 +(- + -) (5.10) and variants obtained by flavour replacements. Eqs.(5.8) and (5.10), and their generalisations to other flavours, suffice to illustrate the richness of (D u—) decays. Turning to the mesonD d— , we can use Eq.(5.7) to identify amongst its possible decays 5 Predictions of Additional Baryons and Mesons - (D d— ) › 0 +(e + e) - › 0 +(µ + ) › 0 +(- + ) (5.11) and variants thereof where 0 isreplacedby any other non-strange neutral meson. When u in Eq.(5.7) is replaced by c or t which subsequently decay, we arrive at many other decay channels additional to Eq.(5.11). Employing instead the D decays in Eq.(5.9) implies additional decay modes of (D d— ) meson that include - (D d— ) › + +(e + e -) - › + +(µ + µ -) › + +(- + -) (5.12) and variants obtainedby flavourreplacement. Eqs.(5.11) and (5.12), merely illustrate a few of the simplest (D d— ) decays. There are many more. Next we consider the lightestTeV baryons inTable3with Q = D. Using the D decays from Eq.(5.7) we find for (Duu) decay (Duu) › p +(l- + i). i (5.13) togetherwith flavourrearrangements.Here,asin subsequent equations, i = e, , . Alternatively, the D decays from Eq.(5.9) lead to (Duu) › N ++ + Y-- . › p + + +(l- + l- ):. ii (5.14) Looking at theTeV baryon (Dud) the respective sets of decays corresponding to Eq.(5.7) are (Dud) › n +(l- + i) i (5.15) where only the simplest light baryon is exhibited. Corresponding to D decays in Eq.(5.9) there are also (Dud) › p +(l- + l- ) ii (5.16) Paul H. Frampton in the simplest cases. Finally, of the (Dqq) TeV baryons, we write out the decays for(Ddd), first for the D decays in Eq.(5.7) (Ddd) › N - + Y- › n + - +(l- + i). i (5.17) within flavour variations. With the Eq.(5.9) decays ofD there are also decays (Ddd) › n +(l- + l- ) ii (5.18) again with more possibilities by choosing alternative flavours. We nowreplace theTeV quarkD by the next heavierTeV quark S and repeat our study of decays whereupon we shall encounter the first example of decay not only to the known quarks but also toaTeV quark. TheTeV quark S has possible decay channels S › d + Y- - › d +(e + e) - › d +(µ + ) › d +(- + ) › D + Z0 -+ › d +(e + e)+(e + e -) -+ › d +(e + e)+(µ + µ -) - › d +(e + e)+(+ + -) -+ › d +(µ + )+(e + e -) -+ › d +(µ + )+(µ + µ -) - › d +(µ + )+(+ + -) + › d +(- + )+(e + e -) + › d +(- + )+(µ + µ -) › d +(- + )+(+ + -) (5.19) where we note the opening up of channels due to S › D decay. 5 Predictions of Additional Baryons and Mesons With Eq.(5.19)in mind, the decaysof theTeV meson(S u—) include (S u—) › - +(l- + i) i + l- › - +(l- i + i)+(lj + j ) (5.20) where the second line involvesa D intermediary. An alternative to Eq.(5.19) is S › u + Y-- - › u +(e + e -) - › u +(µ + µ -) › u +(- + -) (5.21) which implies additional decay modes of (S u—) (S u—) › 0 +(l- + l- ) ii (5.22) and variants whichreplace 0 by another neutral non-strange meson. Eqs.(5.20) and (5.22), illustrate sufficiently (S u—) decays. Turning to the meson(S d— ), we can use Eq.(5.19) to identify its possible decays (S d— ) › 0 +(l- + i) i (5.23) When u in Eq.(5.19) is replaced by c or t which subsequently decay, we arrive at many other decay channels additional to Eq.(5.23). Employing instead the S decays in Eq.(5.21) implies additional decay modes of (S d— ) that include (S d— ) › + +(l- + l- ) ii (5.24) and variants obtained by flavour replacement. Eqs.(5.23) and (5.24), illustrate only a few of the simplest (S d— ) decays. There are many more. Next we consider the lightestTeV baryonsinTable3with one Q = S. Usingthe S decaysfrom Eq.(5.19) we find for(Suu)decay (Suu) › p +(l- + i). i (5.25) Paul H. Frampton together with flavour rearrangements. Alternatively, the S decays from Eq.(5.21) lead to (Suu) › N ++ +(l- + l- › p + + +(l- + l- i ii i ):. ). (5.26) Looking at theTeV baryon (Sud) the respective sets of decays corresponding to Eq.(5.19) are (Sud) › n +(l- i + i) (5.27) i i where only the simplest version is exhibited. Corresponding to the S decays in Eq.(5.21) there are the decays (Sud) › p +(l- + l- ) (5.28) i For baryon (Sdd), firstly from the S decays in Eq.(5.19) we have (Sdd) › N - + Y- › n + - +(l- + i). (5.29) i i within flavour variations. Secondly, from the Eq.(5.21) decays of S there are baryon decays of the type (Sdd) › n +(l- + l- ) (5.30) with more possibilities by choosing alternative flavours. 5.5 Discussion We could continue further to study decays of all the baryons and mesons in our Tables. However, it seems premature to do so, until we know from experimental data the masses and mixings of D, S, T .We remark only that the type of lepton cascade which we have exhibited in Eq.(5.19) becomes ever more prevalent as the lepton number of the decaying hadron increases. 5 Predictions of Additional Baryons and Mesons We may expect, by analogy with the top quark mass being close to the weak scale that the mass of the T quark, although probably below 4:1 TeV for the symmetry- breaking reason discussed ut supra, might be not much below. For example it might exceed 3 TeV whereupon the massofa(TTT )baryon could exceed9 TeV. Since this baryon has high lepton number, it must be pair produced and such production is far beyond the reach of the 14 TeV LHC. Its study would require a 100 TeV colliderof the typepresently underpreliminary discussion.Asa foretaste of the physics accessible to such a hypothetical collider, the simplest decay of the (TTT ) baryon we can find is p + 4(e +)+ 2( —e). which would be very exciting to confirm. At the timeof writing, the particles exhibitedin ourTables are conjectural. After the bilepton is discovered the existence of all the additional baryons and mesons in our fiveTables would become sharppredictions.  The bileptonresonancein ± hasbeenthesubjectof searchesbytheATLASand CMS CollaborationsattheLHC, startinginMarch 2021.InMarch2022,ATLAS publishedan inconclusiveresult[9] aboutthe existenceoftheresonance, putting only a lower mass limit MY > 1:08 TeV. CMS has better momentum resolution and, whatis the same thing, charge identification thanATLAS and shouldbe able to investigatethe bileptonresonanceproper.Thehigh sensitivityofCMSisaresult of serendipity because it was designed in 1993 not for the bilepton but to search forheavyZ-primes[11].Asecondserendipitywasan accidental2015meetingin London between us and SirTejinderVirdee who helped design the CMS detector. Ourstrongbeliefinthe existenceofthebileptonliespartlyinthecloserelationship between the 1961 paper [10] which solved the parity puzzle and our 1992 paper [1] which solved the family puzzle.Weregardthese two papers which span three decades as well-matched bookends, Accordingtoour calculations[7],theRun2datawith139/fb collectedby2018 are sufficient for a CMS discovery of the bilepton. If not, future LHC runs up to their target integrated luminosity of 4/ab can provide 28 times as many events and bilepton discovery wouldbe merely postponed.Wedo hope, however, thata great discoverywillbemadebytheLHC withinsix monthsfromtoday(July25, 2022). Note added: We answer here one interesting question received after our talk: Why are these heavy states not as unstable as the top quark which lives for less than a trillion trillionth of a second? The answer is that they decay via bilepton exchange. This fact renders their lifetimes a trillion times longer than the top quark lifetime. Paul H. Frampton Acknowlegement We thank the organisers Norma Borstnik, Maxim Khlopov and Holger Nielsen for their invitation to present this talk. References 1. P.H. Frampton, Phys. Rev. Lett. 69, 2889 (1992). 2. P.H. Frampton and B.-H. Lee, Phys. Rev Lett. 64, 619 (1990). 3. H. Georgi and S.L. Glashow, Phys. Rev. Lett. 32, 438 (1974). 4. H. Georgi, H.R. Quinn andS.Weinberg, Phys. Rev. Lett. 33, 451 (1974). 5. B.W.Lee and S.Weinberg, Phys.Rev. Lett. 38, 1237 (1977). 6. S.L. Glashow, FifthWorkshop on Grand Unification. World Scientific Publishing Company (1984). 7. G. Corcella,C, Coriano,A, Costantini andP.H. Frampton, Phys. Lett. B773, 544 (2017); ibid. B785, 73 (2018); ibid. B826, 136904 (2022). 8. C. Coriano andP.H. Frampton, Mod. Phys. Lett. A36, 2150118 (2021). 9. ATLAS Collaboration, ATLAS-CONF-2022-010 (March 2022). 10. S.L. Glashow, Nucl. Phys. 22, 579 (1961). 11. T.Virdee, Several private communications beginning in 2015. Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 87) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 6 Possibility of Additional Intergalactic and Cosmological Dark Matter Paul H. Frampton paul.h.frampton@gmail.com Dipartimento di Matematica e Fisica ”Ennio De Giorgi”, Universit`a del Salento,Via Arnesano, 73100 Lecce, Italy Abstract. The entropiesofthe known entitiesinthe universeaddtoa total whichis some twenty ordersof magnitude belowthe holographic limit. Based on an assumption that the entropies should saturate the limit, we suggest that there exists dark matter, in the form of extremely massive primordial black holes, in addition to the dark matter known to exist inside galaxies and clusters of galaxies. Povzetek:Avtor ponudi napovedi bileptonskega modela, ki je razˇ siritev standardnega modela, v katerem zamenja grupo SU(2) × U(1) z grupo SU(3) × U(1).Spodobno spremembo grupe pose ˇstevilo dru ˇcemer meni, da zahteva ze tudiv ˇzin kvarkovin leptonov,priˇ odprava trikotniˇzine kvarkov in lep ske anomalije in asimptotska svoboda teorije tri druˇ tonov. Napoveduje tri nove te ˇ zke kvarke, D. S in T , in nova vezana stanja dileptonov. Dileptone bodo izmerili na LHC, novi barioni in mezoni pa potrebujejo protonski trkalnik, ki bi dosegel 100Tev. Keywords: Dark matter; Entropy of the universe; Black holes. 6.1 Introduction In particle theory, the concept of entropy is usually not regarded as fundamental. Particle theorists rarely even use the wordentropy. For one elementary particle, entropy is neither defined nor useful. In general relativity and cosmology, the situation is different. For black holes, entropyisa central and useful concept.We shallin this talk argue that the origin and nature of cosmological dark matter can be bestunderstood by consideration of the entropy of the universe. Paul H. Frampton We have made such an argument some years ago but that discussion wasperhaps too dilutedby considering simultaneously dark matter being madefrom elementary particles such as WIMPs and axions. In this talk, we dispose of microscopic candidates in one paragraph. The standardmodel of particle theory (SM) has two examples of lack of naturalness, the Higgs boson and the strong CP problem. Our position is that to understand these we still need to understand better the SM itself. Regarding the strong CP problem, it is too ad hoc to posit a spontaneously broken global symmetry and consequences which include an axion. Concerning the WIMP, the idea that dark matter experiences weak interactions arose from assumingTeV-scale supersymmetry whichis now disfavouredby LHC data.To identify the dark matter, we therefore instead look up at the night sky. Assuming dark matter is astrophysical, and that the reason for its existence lies in the SecondLawof Thermodynamics,we shallbeled uniquelytothedark matter constituent as the Primordial Black Hole (PBH).We must admit that thereis no observational evidence for any PBH, but according to our discussion PBHs must exist. In the ensuing discussion, we shall speculate that they exist in abundance in three tiers of mass up to and including extremely high masses which are far greaterthanthe massesofgalaxy clustersandapproachcloselytothemassofthe visible universe. Because PBH entropy goes like mass squared, we are mainly interested in masses satisfying MPBH > 100M .WithintheMilkyWay,we usethe acronym PIMBHfor intermediate mass PBHs in the mass range 102M 15, a population of PBHs would be expected to accrete matter and emit in X-ray and UV radiation which will be redshifted into the CIB tobeprobed for the first timeby the JamesWebb SpaceTelescope which could therefore provide support for PBH formation. Analysis of a specific PBH formation model supports this idea that the JWST observations in the infrared could provide relevant information about whether PBHs really are formed in the early universe. This is important because although we have plenty of evidence for the existence of black holes, whether any of them is primordial is not known. The gravitational wave detectors LIGO, VIRGO and KAGRA have discovered mergers in black hole binaries with initial black holes in the mass range 3-85M .Wesuspect that all or most of these arenot primordial but that is only conjecture. The supermassive black holes at galactic centres, including SgrA*atthe centreoftheMilkyWay, arewell establishedandareprimordialin our toy model. Whether that is the case in Nature is unknown. Because of the no-hair theorem that black holes are completely characterised by their mass, spin and electric charge (usually taken to be zero), there is no way to tell directly whethera given black holeis primordial or theresultof gravitational collapseofa star.The distinction betweena primordialanda non-primordialblack hole can be made only from knowledge of its history. For example, if it existed before star formation, it must be primordial. The infra-red data from JWST will able to provide insight into the central question of PBH formation. Asecond deep insight likely to be provided by the JWST is whether or not Population III stars existed at high red shifts. Their existence looks inevitable from metallicity arguments. Our Sun and other typical stars havea surprisingly high metallicity close to 2%. Such stars cannot be formed directly from the primordial gases which have vanishing metallicity so there must be, and is, an earlier generation of Poplulation II stars with metallicities orders of magnitude below that of the Sun. Even this is insufficient to account for the existence of the Sun and therefore Population III stars are expected to have existed at Z > 15. These extremely low metallicity stars would have lifetimes of only about ten million years and have longago disappeared. Evidencefromthe infra-red observationsbytheJWST could find evidenceof PopulationIII stars,iftheyreally existed. Itis familiartostudya mass-energy pie-chartofthe universewithapproximately 5% baryonic normal matter, 25% dark matter and 70% dark energy. The entropy Paul H. Frampton pie-chart is very different if the toy model considered in this papers resembles Nature. The slices corresponding to normal matter and dark energy are extremely thin and the pie is essentially all dark matter. In this talk we have attempted to justify better that entropy and the second law appliedtotheearly universeprovidea raison d’^etre forthedarkmatter.Wepropose that the dark matter constituents are PBHs with a very wide range of masses from 102M to 1022M . Since it has never been observed except by its gravity, it does seem most likely that dark matter has no direct or even indirect connection to the standardmodel of strong and electroweak interactions in particle theory. The three clues we have mentioned: the dominance of black holes in the entropy inventory, the CMB spectrum and the holographic entropy maximum all hint towardPBHs as the dark matter constituent. Assuming that the maximum entropy limit suggested by holography is saturated the mass function for the PBHs must extend to maximally high mass values. 6.6 Testability So far, our discussion has been highly speculative and has populated the visible universe with objects which may well be the most massive ever contemplated. The nearest may be [7] which considered almost as massive black holes. From the point of view of entropy, all these very massive black holes are a natural extension of the dark matter expected inside galaxies and clusters. Thus, dark matter in this generalised sense permeates all of space not as condensed clumps of mass but spread out on all scales up to cosmological ones. This occurrence of such extremely massive black holes seems inevitable, if we adopt the hypothesis that the bulk contents of the universe possess an entropy which saturates the holographic limit. An obvious question is how to test this novel view of the universe. Additional great attractors, along the lines of [3], if they exist, require better technology to observe galaxy distributions at larger distances. As for the most extreme black holes comparable to the size of the universe itself, we are unaware of any good and practicable observational test. Thereisthe importantquestionof whetherandhowPBHswereformed.According to Eq.(6.3), masses 1018M and 1022M would be formed at, respectively, t = 300ky and 3Gy so there can be natural concern about distorting too much the CMB and of adversely affecting the formation of large-scale structure. OnepossibilitywouldbetotestthePBHtheoryby numericaldarkmattersimulations, similar to those pioneered in [7], but this seems very challenging because 6 Possibility of Additional Intergalactic and Cosmological Dark Matter they give a qualitatively acceptable result for the Large Scale Structure independent of the mass of the dark matter constituents. It is conceivable that more powerful computers than presently available will be able to discriminate between the predicted LSS estimated both with and without such large PBHs. In a similar vein, more advanced technology in telescope construction, both terrestrial and in space, is necessary to make astronomical observations sensitive enoughto detect the existence of more examples similar to the Great Attractor. Discussion of the central assumption of this article, that the holographic entropy maximum is reached by summing the entropies of all the objects within the universe might prove fruitless but hopefully not. What prompted us to publish this discussion was partly the response ”pure cowardice” by Dirac when asked why he did not predict the positron in his 1928 paper which announced the discovery of his eponymous equation. Acknowlegement We thank the organisers Norma Borstnik, Maxim Khlopov and Holger Nielsen for their invitation to present this talk. References 1. P.H. Frampton, arXiv:2202.04432[astro-ph.GA]. 2. N. Secrest, et al., Astrophys. J. Lett. 908, L51 (2021). arXiv:2009.14826[astro-ph.CO]. 3. A. Dressler, Nature 350, 391 (1991). 4. P.H. Frampton, Mod.Phys. Lett. A31, 1650093 (2016). arXiv:1510.00400[hep-ph]. 5. G. ’t Hooft, in SalamFestschrift. Editors: A. Ali, D. Amati and J. Ellis. (1993). arXiv:gr-qc/9310026. 6. P.H. Frampton,M. Kawasaki,F.Takahashi andT.Yanagida. JCAP 04:023 (2010). arXiv:1001.2308 [hep-ph]. 7. B. Carr,F. Kuhnel andL.Visinelli, MNRAS 501, 2029 (2021). arXiv:2008.08077[astro-ph.CO]. 8. J.F. Navarro, C.S. Frenk and S.D.M. White, Astrophys. J. 462, 563 (1996). arXiv:astro-ph/9508025. ibid. 490, 493 (1997). arXiv:astro-ph/9611107. Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 96) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 7 Near-inflection point inflation and production of dark matter during reheating Anish Ghoshal1, Gaetano Lambiase2, Supratik Pal3, Arnab Paul4, Shiladitya Porey5 1Instituteof Theoretical Physics, Facultyof Physics, UniversityofWarsaw,ul. Pasteura5, 02-093Warsaw, Poland, email: anish.ghoshal@fuw.edu.pl 2Dipartimento di Fisica ”E.R. Caianiello”, Universita’ di Salerno, I-84084 Fisciano (Sa), Italy Gruppo Collegato di Salerno, I-84084 Fisciano (Sa), Italy,email:lambiase@sa.infn.it 3Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700108, India Technology Innovation Hub on Data Science, Big Data Analytics and Data Curation, Indian Statistical Institute, Kolkata-700108, India, email:supratik@isical.ac.in, 4Physics and Applied Mathematics Unit, Indian Statistical Institute, Kolkata-700108, India, Indian School of Physical Sciences, Indian Association for the Cultivation of Science, Kolkata-700032, India, email:arnabpaul9292@gmail.com, 5Department of Physics and Astronomy, Novosibirsk State University, email: shiladityamailbox@gmail.com Abstract. We study slow roll single field inflationary scenario and the production of non- thermal fermionic dark matter, together with standardmodel Higgs, duringreheating. For the inflationary scenario, we have considered two models of polynomial potential – one is symmetric aboutthe origin and another oneis not.We fix the coefficientsof the potential fromthe current Cosmic Microwave Background (CMB) datafrom Planck/BICEP. Next, we explorethe allowed parameter space on the coupling (y) with inflaton and mass (m) of dark matter (DM) particles () produced duringreheating and satisfying CMB andseveral other cosmological constraints. http://arxiv.org/abs/2211.15061 7.1 Introduction Cosmic inflation whichis postulated asa fleeting cosmological epoch, occurred at the very early time of the universe. During this primordial epoch, spacetime expanded exponentially resulting in statistical homogeneity and isotropy on large angular scales, the exceedingly flat universe, and providing a proper explanation forthe horizonproblem.In additionto that, inflation can generate quantum fluctuations, which transform into scalar and tensor perturbations. Scalar perturbation acts as the mechanism for the formation of the large-scale structure, while tensor perturbation is responsible for generating gravitational wave. The simplest way to fabricate such an epoch is to assume that the universe was dominated by the energy densityofasingle scalar field, called inflaton, minimally coupledto gravity Title Suppressed Due to Excessive Length andhaving canonicalkineticenergy,slowlyrollingalongtheslopeofthepotential. However, current data from CMB measurements, e.g. Planck [1] and BICEP [2], favour plateau-like potential over the inflaton-potential of the form V() . p with p . 1. One of the other alternatives to get such a potential is to consider inflection-point inflation. On the other hand, CMB measurements suggest that approximately one-quarter of the total mass-energy density of the present universe is in the form of Dark Matter (DM) whose true nature is still not known with certainty. All proposed possible particlesofDM canbe categorized into twogroups -Weakly Interacting Massive Particles (WIMP) and Feebly Interacting Massive Particles (FIMP).Till now, the signature of the presence of WIMP particles has not been detected in particle detector experiments [3]. In that case, FIMP which were never in thermal equilibrium with the relativistic plasma of the universe, seems more favorable as the viable DM candidate [4]. In the paper [5] we studied a single unified model of inflation and the production of non-thermal dark matter particles. For the inflationary part, we have considered two small-field inflection point inflationary scenarios.Wehave also assumed direct coupling between the inflaton and the DM, a vector-like fermionic field . which transforms as gauge singlet under the SM gauge groups. The inflaton either decays to DM or may undergo scattering with the dark sector to produce the observed relic. As we will see, additional irreducible gravitational interaction may also mediate theDMproduction, eitherby 2-to-2 annihilationof the StandardModel (SM) Higgs bosons or of the inflatons during the reheating era. This paper is organized as follows: in Section 7.2, we discuss the condition of getting an inflection point for a single field potential. In Section 7.3, we study the slow roll inflationary scenario for two potentials and findthe location of inflection point and fix the coefficients of the potentials from CMB data. Reheating and production of dark matter have been discussed in Section 7.5. Section 7.6 contains conclusion. 7.2 Inflection-point inflation models Near the location of the inflection point, the potential takes a plateau-like shape. Because of that, inflection point of the inflationary potential is important for the slowroll inflationary scenario.If inflaton startsrollingalongthe potentialfromthe vicinity of the inflection point, the number of e-foldings (described in Section 7.3) increases without significant change in the inflaton value. To determine the stationary inflection point of an inflationary potentialV( ) ofa single scalar field , we need the solution of d2 dVV == 0. (7.1) d. d 2 In the following sections (Section 7.3) we discuss two different slow roll small- field inflationary scenarios, where each of the inflationary potentials possesses an inflection point. 98 A. Ghoshal, G. Lambiase, S. Pal, A. Paul, S. Porey 7.3 Slow roll inflationary scenario The Lagrangian density we are interested in, is given by in —h = c = kB = 1 unit, M2 LI = P R + LKE;INF + UINF + LKE;. - U()+ LKE;H - UH(H)+ Lreh , 2 (7.2) where MP ' 2:4 × 1018 GeV is the reduced Planck mass and R is the Ricci scalar with metric-signature (+, -, -, -). LKE;INF and UINF are respectively the kinetic energy and potential energy term of the single scalar inflaton. Since, those two terms are function of inflaton, they alter when we change the model of inflation. In this work, we use . to symbolize inflaton for ModelIinflation and. for Model II. Accordingly, UINF U. = V0 + a. - b2 + d4 (for ModelI) , (7.3) U. = p'2 - q'4 + w'6 (for Model II) . (7.4) Here V0, a, b, d, p, q, and w are all assumed to be positive, real; and we choose d, w >0. The potential of Eq. (7.3) contains a term of linear order of inflaton. Due to this term U. is not symmetric about the origin. On the contrary, the U. is symmetric about the origin. In Eq. (7.2), LKE;, and LKE;H represent the kinetic energy of the vector-like fermionic DM, , and StandardModel (SM) Higgs field, H, respectively. And the potential term for . and H are given by U()= m. —(7.5) . , ..2 2 UH(H)=-mHyH + H HyH. (7.6) H Furthermore, the last term on the right side of Eq. (7.2), Lreh, takes care of the interactions of . and H with (') duringreheating anditis defined as . - 12HyH - 222HyH (for Model I) , Lreh . Lreh;I =-y. — Lreh;II =-y. —(for Model II) , . - 12'HyH - 22'2HyH (7.7) where 12, 22, andYukawa-like y. are the couplings of SM Higgs and fermionic DM with inflaton. During the slow roll inflationary epoch, contribution from the terms except the first three terms in Eq. (7.2) is negligible. The slow-roll condition is measured in terms of four potential-slow-roll parameters – V ;V ;V , and V . During slow roll inflationary epoch, jV | , jV | , jV | , jV | 1. These four potential-slow-roll Title Suppressed Due to Excessive Length parameters for ModelI are defined as 2 ..2 M2 U0 a - 2b . + 4d 3 P. V . = M2 , (7.8) P 2U. 2 (. (a - b. + d3)+ V0) 2 .. U00 . 2b - 6d 2 V . M2 =-M2 , (7.9) PP U. . (a - b. + d3)+ V0 .. U000 U0 24d. a - 2b . + 4d 3 . V . M4 = M4 , (7.10) PP U2 (. (a - b. + d3)+ V0) 2 . ..2 2 U0000 U0 24d a - 2b . + 4d 3 . V . M6 P U3 = M6 P . (7.11) (. (a - b. + d3)+ V0) 3 . Here, prime denotes derivative with respect to inflaton. For Model II inflation, the potential-slow-roll parameters are ..2 2 p. - 2q'3 + 3w'5 V = MP 2 , (7.12) (p'2 - q'4 + w'6) 2 .. 2p - 6q'2 + 15w'4 V = M2 , (7.13) P p'2 - q'4 + w'6 ....  48'2 -q + 5w'2 p - 2q'2 + 3w'4 V = M4 , (7.14) P (p'2 - q'4 + w'6) 2 .... 2 96 -q + 15w'2 p. - 2q'3 + 3w'5 V = M6 . (7.15) P (p'2 - q'4 + w'6) 3 By the time any one of these slow-roll parameters becomes ~ 1 at . ~ end (for Model I) or at . ~ 'end (for Model II), slow roll inflation terminates. The duration of slow roll inflation is measured in terms of the total number of e- foldings, NCMB, tot as ZCMB('CMB) ZCMB('CMB ) UINF 1 NCMB, tot = M-2 d(')= . d(') , (7.16) P 2V U0 end('end) INF end('end) where CMB('CMB) is the inflaton value at which the length scale, which had previously left the causal horizon during inflation, hasreentered during the period of recombination. Moreover, inflation generates primordial scalar and tensor perturbations. The primordial scalar and tensor power spectrum for ’k’-th Fourier mode are defined as  ns-1+(1=2) s ln(k=k)+(1=6) s(ln(k=k))2 Ps (k)= As k , (7.17) k*  nt+(1=2)dnt=d ln k ln(k=k)+· Ph (k)= At k , (7.18) k* where k* = 0:05Mpc-1;ns and nt are the scalar and tensor spectral index, s is therunningof scalar spectral index, and s is called the ’running of running’. 100 A. Ghoshal, G. Lambiase, S. Pal, A. Paul, S. Porey Moreover, in Eq. (7.17)-(7.18), As and At are the normalizations. The relation between As and inflationary potential is UINF 2UINF As . . (7.19) 242M4 V 32M4 r PP Here, r is the tensor-to-scalar ratio. r, ns, s and s depend on potential-slow-roll parameters as At dlnPs r = . 16V :ns == 1 + 2V - 6V , (7.20) As dlnk dns s . = 16V V - 242 - 2V . (7.21) dlnk V d2 ns s . V + 1922 - 24V V + 2V V + 2V . =-1923 V V - 32V 2 dlnk2V (7.22) The observed values of all these inflation parameters measured at . = CMB (at k* ' 0:05Mpc-1)fromPlanck, WMAP, and other CMB observations are presented inTable 7.1. 1 Table 7.1:CMB constraints on inflationary parameters. ln(1010As) 3:047 ± 0:014 68%, TT,TE,EE+lowE+lensing+BAO [1] ns 0:9647 ± 0:0043 68%, TT,TE,EE+lowE+lensing+BAO [1] dns=dlnk 0:0011 ± 0:0099 68%, TT,TE,EE+lowE+lensing+BAO [1] d2 ns=dlnk2 0:009 ± 0:012 68%, TT,TE,EE+lowE+lensing+BAO [1] r 0:014+0:010 and -0:011 < 0:036 95%, BK18, BICEP3, Keck Array 2020, and WMAP and Planck CMB polarization [1,2,7,8] 7.3.1 Estimating coefficients from CMB data In this subsection, we find the location of inflection points and also, fix the coefficient of the potentials of both inflationary models, mentioned in Eq. (7.3) and Eq. (7.4), from the CMB data. At first, we start the calculation with Model I. SolutionofEq.( 7.1)providesthe locationof inflectionpointfor ModelIpotential 3a 8b3 0 = when d = . (7.23) 4b 27a2 To fix the coefficients of the potential of Eq. (7.3), following [9,10], we can write 0@ CMB 2 4 CMB CMB 1 2CMB 43 CMB 0@ 1A a b 1A = 0@ U(CMB)- V0 U0  (CMB) 1A , (7.24) U00 0 2 122 d (CMB) CMB  1TandEcorrespondsto temperatureand E-mode polarisationofCMB. Title Suppressed Due to Excessive Length 101 where d isknownfromEq. (7.23) and U(CMB);U0 (CMB) and U00 (CMB) can  be derived using Eq. (7.8), (7.9), (7.10), (7.19), (7.20) as U(CMB)= 3Asr2MP 4 , (7.25) 2 r 3r .. U0 r2 M3 (7.26) (CMB)= 2 8AsP ,  3 3r .. U00 (CMB)= + ns - 1Asr2 MP 2 . (7.27) 48 Using these together withTable 7.1, we can find the coefficientsof the potential. However,for cosmological purpose,itis adequateto designthe potentialinaway such that CMB is adjacent to 0 [11]. In order to implement this, let us modify the potential (Eq. (7.3)) as U()= V0 + A. - B2 + d4 , (7.28) with A = a(1- I );B = b(1- I ) (where I ; I aredimensionless) and in the limit 1212 I ; I › 0, the slope of the potential vanishes at 0. Using this modification, we 12 have found the benchmark value for this potential whichis exhibitedinTable 7.2, and using this value, the evolution of the potential and slow roll parameters with . are illustrated in Fig. 7.1. From this Fig. 7.1 it is clear that V ;V ;V < jV j. Besides, at . = CMB, V , jV | ;V ;V << 1, and at . = end, jV | ' 1. This last condition leads to the ending of slow roll phase. Table 7.2:Benchmark value for linear term potential(ModelI)(min is the minimum of potential in Eq. (7.28)) V0=M4 P a=M3 P b=M2 P d I 1 I 2 2:788 × 10-19 9:29 × 10-19 6:966 × 10-18 1:16 × 10-16 6 × 10-7 6 × 10-7 CMB=MP end=MP min=MP 0=MP 0:1 0:098889 -0:200045 0:100022 r ns As e-folding s s 9:87606 × 10-12 0:960249 2:10521 × 10-9 53:75 -1:97 × 10-3 -3:92 × 10-5 Next, we follow similar steps for the inflationary potential of Model II. The potential of Eq. (7.4) has an inflection point at . 2 '0 = . q for p = q . (7.29) 3 w 3 w Likewise, we can also redefine the potential Model II as U'(')= p'2 - Q'4 + W'6 , (7.30) 102 A. Ghoshal, G. Lambiase, S. Pal, A. Paul, S. Porey Fig. 7.1: In the top-left panel: normalised inflaton-potentialof ModelIinflation asa function of ’'=MP’ for benchmark value showninTable 7.2. The evolutionof inflationary slow-roll parameters(V , -V ;V ;V )as a function of =MP is presented in the top-right panel; second row -left panel: V , and second row – right panel: V of ModelI slow roll inflation against =MP are shown individually for benchmark values listed in Table 7.2. The dashed line is for1. Whenever jV | becomes ~ 1, the slowroll inflation ends. From these figures, it is clearly visible that jV | < jV | < jV | < jV | during the slow-roll regime. q(1 - II w(1 - II 1 , II such that Q =) and W =) and II have zero mass 1 22 dimension. Then, we can estimate p, q and w, and the values are mentioned in Table 7.3. For this value, the variation ofU'(') of Eq. (7.30) and V , jV | ;V ;V as a function of . is shown in Fig. 7.2. The slow roll inflationary phase ends at 'end when jV | ' 1 (because for Model II V < jV j). Title Suppressed Due to Excessive Length 103 Fig. 7.2: Top-left panel: evolution of normalised inflaton-potential ofModel II for benchmark valuefromTable 7.3.Top-right panel: absolute valuesof four slowroll parameters (V , -V ;V ;V )are plotted against '=MP. Left and right panel of the second row displays V and V ,respectively,against'=MP for benchmark values mentionedinTable 7.3. The dashed line indicates 1. These graphs demonstrate that jV | < jV | < jV | < jV | <1 during the slow-roll inflation, similar to what we have found in Model I. Table 7.3: Benchmark values for sextic potential('min is the minimum of potential Eq. (7.30)) p=M2PqwM2P II1 II21:4510-181:6210-175:9810-171:5310-81:5310-8'CMB=MP'end=MP'min=MP'0=MP0:30:29944400:300011rnsAse-folding s s1:410-120:960012:1052110-960:247-1:48710-3-2:97210-5 7.4 Stability analysis In this section, we attempt to determine the upper bound of y. and 12 so that Lreh;I and Lreh;II do not affect the inflationary scenario set forth in Section 7.3. 104 A. Ghoshal, G. Lambiase, S. Pal, A. Paul, S. Porey The Coleman–Weinberg(CW) radiative correction at 1-loop order to the inflaton- potential is given by [6] " !# X m2 nj4 ej (-1)2sj e VCW = m j ln - cj . (7.31) 2 642  j Here, j . H, . and inflaton; nH;. = 4, nj for inflaton is 1. Furthermore, sH = 0, s. = 1=2, and = 0. e s(') mj is inflaton dependent mass of the component j and µ istherenormalization scale, whichis taken ~ 0 (for ModelI) or '0 (for Model II). 3 Besides, cj = . Now, the second derivative of the CW term w.r.t. inflaton is 2 !  X.. 2 e2 .. .. m V00 nj2 0 22 00 j 22 00 CW = (-1)2sj ej + ej ej ln 2 m j m j m mm - ee. 322  j (7.32) In the next two subsections, we investigate the stability relative to the couplings y. and 12 for the two inflation-potentials (Eq. (7.28))and Eq.(7.30))we have considered. 7.4.1 Stability analysis for linear term inflation From Eq. (7.7), the field-depended mass of the . and H are respectively 2 2 e()=(m. + y)(7.33) m, 22 e()= m(7.34) mHH + 12. 2 For the stability of the inflation-potential, the terms of the order of 2 and y. on 12 the right-hand side in Eq. (7.32) should be less than corresponding tree level terms from Eq. (7.28) 32b32 V00 0 tree(0) . U00 (0)= 9a2 - 2b(1 - ) , (7.35)  where I = I = I (as we have chosen the benchmark value I = I ). The 12 12 second derivative (Eq. (7.32)) of CW term for Higgs field is  jV00 12 2 12. | = ln . (7.36) CW;H82 2 0 The upper bound of the value of 12 at . ~ 0 can be deduced from jV00 | < CW;H V00 tree(0), and it is depicted on the right panel of Fig. 7.3. Thus, allowed value of 12=MP is < 5:283 × 10-12 . Similarly, for y, !! 2 12y V00 4. 4 CW;. = 62 y . ln - 22 y . . (7.37) 82 2 0 V00 The upper bound on y. around . ~ 0 can be obtainedfrom ..(')!. such that total decay width of inflaton . = ..(')!. + ..(')!hh ' ..(')!hh. Hence, 12 6:15 × 1062 (for ModelI) , MP (7.45) . = 12 2:33 × 1072 (for Model II) . MP . >. Now, the branching ratio for the production of . is 2 ..(')!. ..(')!. y. 2 Br = ' = m (') (7.46) ..(')!. + ..(')!hh ..(')!hh 12 2 y. 4:18 × 10-17 M2 (for ModelI) , P  12 2 (7.47) = >. y. M2 2:91 × 10-18 (for Model II) . P 12 These produced particles cause the development of the local-thermal relativistic fluid of the universe and consequently, raise the temperature of the universe. Title Suppressed Due to Excessive Length 107 At the beginning of reheating, due to the small value of couplings to inflaton, ..< H(ß), where HH(ß) is the Hubble parameter andßis the cosmological scale factor. Meanwhile, H continues to decrease. At the moment when H becomes ~ .., the temperatureof the universeis called asreheating temperature, Trh, and it is can be computed as [12] r  1=4 p. 2 10 1095:07 12 Trh = MP . = . g? 2132:0912 (for Model I) , (for Model II) . (7.48) We have assumedg? = 106:75. At temperature below Trh, the universe behaves as if it is dominated by relativistic particles [13]. Additionally, we have assumed here that the process of particle production from inflaton is instantaneous [14]. In general, reheating is not an instantaneous process. The maximum temperature of the universe during the whole process of reheating may be many orders greater than Trh and it can be estimated as [14] 1=4 2=5 = ..1=4 60 3 1=4 1=2 Tmax HM, (7.49) IP g?2 8 where HI isthevalueofthe Hubbleparameteratthe beginningofreheatingwhen no particle, including the DM, is produced. This can be taken as . q U(0) . = 3:23 × 10-10MP (for Model I) , P HI ' q3M2 (7.50) U'('0) := 1:206 × 10-10MP (for Model II) . 3M2 P The Eq. (7.48)with Trh & 4MeV puts down the lower limit on 12 1:52 × 10-24 (for ModelI) , 12 & (7.51) MP 7:82 × 10-25 (for Model II) . From Eq. (7.49), we can write 2=5 1=4 Tmax 3 HI = , (7.52) Trh 8 H(Trh) where r g? H(Trh)= T2 (7.53) rh . 3MP 10 The allowed ranges for Tmax=Trh for two inflationary models areshown in Fig. 7.5. The upper limit for the allowed region comes from Eq. (7.52) and the lower limit from the fact that Trh & 4MeV which is needed for successful Big Bang nucleosyn- thesis (BBN) [15]. 7.5.1 Dark Matter Production and Relic Density In this subsection, we estimate, following Ref. [12], the amount of DM produced during reheating and compared it with DM relic density of the present-day universe. The Boltzmann equation for the evolution of DM number density, n, of 108 A. Ghoshal, G. Lambiase, S. Pal, A. Paul, S. Porey Fig. 7.5: Allowed range (colored region) for Tmax=Trh:left panelis for ModelIinflation, where right panel is for the Model II.The green color line points to Tmax=Trh when Tmax = 4MeV. The gray colored area indicates the lower(Trh 4MeV) and upper bound on Trh obtained from the stability analysis (see Eq. (7.36) and Eq. (7.41)). DM particles is d n. + 3H n. = , (7.54) dt where t is the physical time, . is the rate of production of DM per unit volume. Then the evolution equation of comoving number density, N. = nß3 (ß(t) is the cosmological scale factor, as mentioned earlier), of DM particles dN. = ß3 . (7.55) dt While the temperature, T of the universe is Tmax >T>Trh, the energy density of the universeis dominatedby inflatonandthe first Friedman equation leadsto[12] r g? T4 H = . (7.56) 3 10MP T2 rh Therefore, energy density of inflaton g? (') = 2TT4 8 . (7.57) 30 rh Since, duringreheating, . behaves as a non-relativistic fluids, (') . ß-3, the scale factor behaves as ß. T -8=3 . (7.58) Using Eq. (7.56) and (7.58) in Eq. (7.55) we obtain 1=2 T10 dN. 8MP 10 rh =- T13 ß3(Trh) . (7.59) dT g? DMYield, Y. is defined as the ratio of the number density of DM to the entropy n(T) density of photons, i.e., Y. = , where entropy density s(T)= 22 g?;sT3 and s(T) 45 Title Suppressed Due to Excessive Length 109 g?;s is the effective number of degrees of freedom of the constituents of the relativistic fluid. If we assume that there is no entropy generation in any cosmological process, afterreheating epoch, then the evolutionof Y. can be expressed as s dY. dT 135 = - 23 g?;s 10 g? MP T6 . . (7.60) We are assuming that the DM particles, produced during reheating, were never in thermal equilibrium with the relativistic fluid of the universe. Those DM particles contribute to the cold dark matter (CDM) density of the present universe. Thus, followingTable 7.4,present-day CDM yield[12]is 4:3. × 10-10 YCDM;0 = , (7.61) m. where m. isexpressedin GeV.Now,the amountofDMproducedduringreheating throughdecayorviascatteringinboth ModelIandModelII,hasbeen estimated and compared with YCDM;0 in the following part of this subsection. Table 7.4:Data about CDM(hCMB . 0:674) CDM 0:120 h-2 CMB [16] c -3 1:878 × 10-29 h2 CMB gcm s0 -3 2891:2 (T=2:7255K)3 cm Inflaton decay If DM particles are generated from the inflaton decay (') . = 2Br ... (7.62) m(') Substituting this in Eq. (7.60), the DM yield from the decay of inflaton, ss 3g? 10 MP . 3g? 10 MP (y)2 Y;0 ' Br = (7.63) g?;s g? m(') Trh g?;s g? Trh 8. 2 . = 1:163 × 10-2MP y . (7.64) Trh Here, we assume g?;s = g?. Equating Eq. (7.64) with Eq. (7.61), we get the condition to generate the complete CDM energy density 2 Trh ' 6:49 × 1025 y m. . (7.65) Fig. 7.6 depicts the allowed range of the coupling y. from Eq. (7.65),to generate the complete CDM density of the contemporary universe only via the decay channel of inflaton. From this figure, we can deduce that the allowed range for y. and m. to construct the CDM density of the universe is 10-10 & y. & 10-15 (for 2:5 × 103 GeV . m. . 8:1 × 109 GeV in Model I) and 10-11 & y. & 10-15 (for 8:4 × 103 GeV . m. . 2 × 109 GeV in Model II). 110 A. Ghoshal, G. Lambiase, S. Pal, A. Paul, S. Porey Fig. 7.6: The allowedregion (unshaded) for theYukawa-like coupling y. to produce the complete CDMof thepresent universe: left panelis for ModelIinflation and right for Model II inflation. The constraints (colored regions) are from (a) BBN (light green colored region): Trh >4MeV, (b) from stability analysis (blue colored region): Trh ' 1:388 × 1010GeV (for ModelI) or Trh ' 1:83 × 109GeV (for Model II) from the upper bound of 12 from Eq. (7.36) or Eq. (7.41),(c) stability(red-coloredregion):from the upper bound of y. from Eq. (7.37) or Eq. (7.42),(d) (deepgreenregion): m. must be T. (7.68) 48 g?;s g? m4 (') Title Suppressed Due to Excessive Length 111 Fig. 7.7: myield of DM generated from the 2-to-2 scattering with graviton as mediator for different values of m. The left panel shows theresult for ModelIand the right panel for Model II inflation. YSMi;0 ~ 10-60 (~ 10-62)forTrh ~ 105GeV ' 10-5m. (m')forg? = g?;s = 106:75, 12 ~ 10-12 (10-13)andy. ~ 10-6 (10-7). Therefore, the DM produced from 2-to-2 scattering during reheating is insignificant in comparison to total CDM density of the universe. 7.6 Conclusions and Discussion Weinvestigatedasimple possibilityofascalar inflaton andanon-thermal fermionic particle that originated during thereheating epoch and acted asthe CDM. Satisfyingthe correctrelic densityofDMand otherCMB bounds,we discoveredthe following features of our analysis: • We investigated two polynomial potential models for slow roll single field cosmic inflation. Each of these models features an inflection point. Moreover, duetothepresenceofaterm correspondingtothe linear powerof inflaton(see Eq.(7.3)),the potentialofModelIisnot symmetricabouttheorigin.In contrast, the potential of Model II (Eq. (7.4))is symmetric under the transformation of . › -'. • We computed the coefficients of the potentials of both models satisfying the current CMB bounds and under the assumption of near-inflection point inflationary scenario.We also found ns ~ 0:96, r ~ 10-12; s ~ 10-3, and s ~ 10-8 (seeTable 7.2 andTable 7.3). • We assumed that inflaton decays to SM Higgs(H)together with DM(). From stability analysis of the inflation-potential in Fig. 7.3 and Fig. 7.4, we deduced that the upper bounds of the couplings for two decay channels are 12=MP . O(10-12) and y. . O(10-6). The former upper bound defines the highest permissible value of Trh. • We studied the formation of non-thermal vector-like fermionic DM particles, during reheating from the inflaton decay. The rate of DM creation through this decay is temperature dependent; when the temperature of the universe’s relativistic fluid increases duringreheating, the rateofDM generationreduces 112 A. Ghoshal, G. Lambiase, S. Pal, A. Paul, S. Porey (Eq. (7.60)). Fig. 7.5 depicts the permissible range for the ratio of the highest temperature Tmax to the reheating temperature Trh during that period, Tmax=Trh. For Trh = 4MeV,the ratio mightreachO(107). Thepermitted range of Tmax=Trh is determinedbythe inflectionpoint(seeEq. (7.49)andEq. (7.50)). Because we chose the CMB scale around the inflection point, the inflection point determines the CMB observables, such as ns and r on one hand, and controls theproductionregimes (viaTmax)of DM and consequently DM relic on the other hand. • Fig.7.6 depictsthe allowedregionin Trh-m. space for two models of potential we have considered and the constraints on that space are coming from bound on Trh from BBN, radiative stability analysis of the potential for slow roll inflation,Ly-. bound, and the maximum possible value of m. for the effective mass of the inflaton. From this figure we can conclude that . produced only through the decay of inflaton may explain the total density of CDM of the current universe if10-10 & y. & 10-15 (for 2:5 103 GeV . m. . 8:1 109 GeV in ModelI) and 10-11 & y. & 10-15 (for 8:4 × 103 GeV . m. . 2 × 109 GeV in Model II). • . can also be produced from 2-to-2 scattering of either SM particles or infla- tons. Among all those scattering processes, the promising one is – from the scattering of inflaton with graviton as the mediator. In Fig. 7.7 we showed that Y. produced through 2-to-2 scattering of inflaton with graviton as mediator, is more than the DM production via other scattering channels, and it is YIS;0 ~ O(10-36)forTrh = 108 GeV;m. = 103 GeV. But, YIS;0 produced through this channel is much less than YCDM;0 and thus . produced through 2-to-2 scattering channels can contribute only a negligible fraction of YCDM;0. In conclusion,we considertwo membersofthebeyondthe standardmodelphysics -inflaton and the non-thermal DM, to connect the CMB data and the DM mystery. This work can be further extended to study the formation of Primordial Black Holes for inflection point inflationary scenario, non-Gaussianitiesin the CMB spectrum, and generation of GravitationalWaves which can be tested from future CMB experiments. Acknowledgement Shiladitya Porey wants to thank Professor Norma Susana Manko ˇstnik, Pro- c Borˇ fessor Maxim Khlopov, Professor Astri Kleppe, and the organizers of the Bled 25thWorkshop.Workof ShiladityaPoreyis fundedbyRSF Grant 19-42-02004. Supratik Pal thanks Department of Science andTechnology, Govt. of India for partial support through Grant No. NMICPS/006/MD/2020-21. References 1. N. Aghanim et al. [Planck], Astron. Astrophys. 641, A6 (2020) [erratum: Astron. As- trophys. 652, C4 (2021)] doi:10.1051/0004-6361/201833910 [arXiv:1807.06209 [astro- ph.CO]]. 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(p. 114) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 Quark masses and mixing from a SU(3) gauge family symmetry A. Hernandez-Galeana DepartamentodeF´isica, ESFM -Instituto Polit´ecnico Nacional. U.P. ”AdolfoL´opez Mateos”.C.P. 07738, CiudaddeM´exico,M´exico. e-mail: ahernandez@ipn.mx Abstract. Withinabroken local vector-likeSU(3) family symmetry,we address the problem of the hierarchical spectrumof quark masses and mixing.Inthis scenario heavy fermions, top and bottom quarks and tau lepton become massive at tree level from Dirac See-saw mechanisms implementedby the introductionofa new setof SU(2)L weak singlets vector- like fermions U, D, E, N,withN asterile neutrino. Light fermions, quarks and leptons obtain massesfrom loop radiative corrections mediatedby the massive SU(3) gauge bosons. We provide a parameter space region where this framework can account for the known hierarchical spectrum of quark masses and mixing, and simultaneously suppress properly —— the current experimental constraints on Ko - Ko and Do - Do meson mixing. In addition, we find out that the mass of the SU(2)L weak singlet vector-likeDquark introducedin this scenariomaylie withinafewTeV’sregion,and hence within currentLHC possibilities. Povzetek:Avtor predstavi svoj predlog modela z umeritveno druˇ zinsko simetrijo SU(3), ki poskrbi za mase kvarkov in leptonov in za meˇsibke singlete fermionov salni matriki. Uvede ˇ SU(2)L (U, D, E, N, N je nevtralni nevtrino),ki prinesejo kvarkomatinbter leptonu tau maso ˇ ze na drevesnem nivoju preko Diracovega mehanizma See-saw. Za maso ostalih kvarkov in leptonov poskrbijo massivna umeritvena polja dru ˇ zinske simetrije SU(3) s popravki v naslednjem redu. Avtor predstavi obmoˇ cje parametrov, znotraj katerega so dobljenirezulati skladniz eksperimenti Ko - Ko in Do - Do. Masa kvarkaDjenekajTeV,torejvdosegutrenutnih zmogljivosti —— LHC. Keywords: Quark masses and mixing, Flavor symmetry, Dirac See-saw mechanism. 8.1 Introduction In this report we study the quark masses and mixing within the framework of a broken SU(3) gauged family symmetry model [1,2]. This framework introducea hierarchical mass generation mechanism in which light fermions become massive from radiative corrections, mediated by the massive gauge bosons associated to the SU(3) family symmetry that is spontaneouslybroken, while the masses of the 8 Quark masses and mixing from a SU(3) gauge family symmetry 115 top and bottom quarks as well as for the tau lepton, are generated at tree level from ”Dirac See-saw”mechanisms implementedby the introductionofa new set of SU(2)L weak singlets U, D, E and N vector-like fermions. Flavor physics and rare processes play an important role to test any Beyond StandardModel(BSM) physics proposal, and hence, it is crucial to compute the the F = 2 processes [3]-[6] in neutral mesons at tree level exchange diagrams mediated by the horizontal gauge bosons. Previous theories addressing theproblemof quark and lepton masses and mixing with spontaneously broken SU(3) gauge symmetry of generations include the ones withchiral SU(3) family symmetry [8]-[12], as well as other SU(3) family symmetry proposals [7], [13]-[16]. 8.2 SU(3) family symmetry model The model is based on the gauge symmetry G . SU(3) . SU(3)C . SU(2)L . U(1)Y (8.1) where SU(3) is a completely vector-like and universal gauged family symmetry. That is, the corresponding gauge bosons couple equally to Left and Right Handed ordinary Quarks and Leptons, with gH, gs, g and g0 the corresponding coupling constants. The content of fermions assumes the standardmodel quarks and leptons: o =(3;3;2, 1 )L ; o =(3, 1, 2, -1)L (8.2) ql 3 o =(3;3;1, 4 )R ; o (3, 3, 1, - 2 )R ; o =(3, 1, 1, -2)R (8.3) u de 33 where the last entry is the hypercharge Y, with the electric charge defined by 1 Q = T3L + 2 Y. The model includes two types of extra fermions: Right Handed Neutrinos: o = R (3, 1, 1, 0)R, introduced to cancel anomalies [7], and a new family of SU(2)L weak singlet vector-like fermions:Vector like quarks Uo ;Uo =(1, 3, 1, 4 ) and LR 3 Do ;Do =(1, 3, 1, - 2 ),Vector Like electrons: Eo ;Eo =(1, 1, 1, -2), and New Ster LR3 LR ile Neutrinos: No ;No =(1, 1, 1, 0). LR The particle content and gauge symmetry assignments are summarized inTable 8.1. Notice that all SU(3) non-singlet fields transform as the fundamental representation under the SU(3) symmetry. 116 A. Hernandez-Galeana SU(3) SU(3)C SU(2)L U(1)Y o q 3 3 2 1 3 o uR 3 3 1 4 3 o dR 3 3 1 2 -3 o l 3 1 2 -1 o eR 3 1 1 -2 o R 3 1 1 0 Uo L;R 1 3 1 4 3 Do L;R 1 3 1 2 -3 Eo L;R 1 1 1 -2 No L;R 1 1 1 0 u 3 1 2 -1 d 3 1 2 +1 1 , 2 3 1 1 0 Table 8.1: Particle content and charges under the gauge symmetry 8.3 SU(3) family symmetry breaking SU(3) family symmetry is broken spontaneously by heavy SM singlet scalars 1 =(3, 1, 1, 0) and 2 =(3, 1, 1, 0) in the fundamental representation of SU(3), with the ”Vacuum ExpectationValues” (VEV’s): h1iT =(1, 0, 0) , h2iT =(0, 2;0) . (8.4) It is worth to mention that these two scalars in the fundamental representation is the minimal set of scalars to break down completely the SU(3) family symmetry. The interaction of the SU(3) gauge bosons to the SM massless fermions is 0 BBBB@ 1 CCCCA +µ +µ  1  2 . Z Z Y Y 0 B@ 1 CA (8.5) 2. 1. + 2 o 1 2 2 3 2 f -µ +µ Y 2Z . 3. 1. o3o2o1 = gH (f —f —f — Y where gH is the SU(3) coupling constant, Z1, Z2 and Y± = j Y iLint;SU(3) ) µ o 2 f - , , 2 3 2 o 3 f -µ -µ  1  2 . Y Y Z Z 3. 2. - + 2 2 2 3 2 1j . the eight gauge bosons. Thus, the contribution to the horizontal gauge boson masses from the VEV’s of Eq.(8.4) read 2 j iY ;j = 1, 2, 3 are 2 h1i : g 2H  2 1 (Y+ 1 Y- + Y+ Y- 1 22 )+ g 2H  2 1 (Z2 1 + + 2Z1 pZ 2 ) 2 2 Z 3 3 2 4 2H  2 2 2 (Y+ 1 Y- + Y+ Y- 1 33 )+ g 2 2 Z 3 h2i : g 2 2 2 H The ”Spontaneous Symmetry Breaking” (SSB) of SU(3) occurs in two stages 8 Quark masses and mixing from a SU(3) gauge family symmetry 117 SU(3)F × GSM › h2i SU(2)F × GSM › h1i GSM Z1;Y± 2 Notice that the hierarchy of scales 2 >1 yield an ”approximate SU(2) global symmetry” in the spectrum of SU(2) gauge boson masses of order gH 1. Therefore, neglecting tiny contributions from electroweak symmetry breaking, the gauge boson massesread (M2 1 + M2 ) Y+ Y- + M2 1 Y+ Y- + M2 2 Y+ Y- 211 22 33 1 1M2 1 + 4M2 2 12 + M2 1 Z1 2 + Z2 2 +(M1 2 ) . Z1 Z2 (8.6) 2 23 23 22 g2 g2 H1 H2 M2 = ;M2 = (8.7) 12 22 Z1 Z2 2 1 M M2 1 . Z1 3 2 2 21 2 1 MM+4M . 3 3 Z2 Table 8.2:Z1 - Z2 mixing mass matrix 8.4 Electroweak symmetry breaking The ”Electroweak Symmetry Breaking” (EWSB) is achieved by the Higgs fields u and d , which transform simultaneously as triplets under SU(3) and as Higgs ii doublets with hypercharges -1 and +1 under the SM,respectively, explicitly: 1 CCCCCCCCCCCA , d = 0 BBBBBBBBBBBB@ 1 CCCCCCCCCCCCA  d u = 0 BBBBBBBBBBB@  u 1 u + o - o 1  d o - o + o 2 u 2   d + - o 3 3 with the VEV’s 118 A. Hernandez-Galeana     0 vu1 2 0 1. 1 2 vd1 . hui = 0 BBBBBBBBBB@ 1 CCCCCCCCCCA , hdi = 0 BBBBBBBBBB@ 1 CCCCCCCCCCA     0 vu2 2 0 1. 1 2 vd2 .     0 vu3 1122 0 vd3 . The contributions from hui and hdi generate the W and Zo SM gauge boson masses 22 02 g(g+ g) 22 22 (v + v ) W+W- +(v + v ) Z2 (8.8) ududo 48 + tiny contribution to the SU(3) gauge boson masses and mixing with Zo, 22222222 1 v= v+ v+ vv= v+ v+ v. So, if MW . gv, we may write 3u , . 1u 2u d 1d 2d 3d 2 u q v2 + v2 . 246 GeV. v = d u 8.5 Fermion masses 8.5.1 Dirac See-saw mechanisms SM quarks and leptons get tree level mass contribution after EWSB from the generic diagramin Fig.1 The gauge symmetry G . SU(3) × GSM, the fermion content, and the transformation of the scalar fields, all together, avoidYukawa couplings between SM fermions. The allowedYukawa couplings involve terms between theSM fermions andthe corresponding vector-like fermionsU,D,EandN.The scalarsand fermion content allow the gauge invariantYukawa couplings o u Uo 1 Uo 2 Uo + MU Uo Uo hu + h1u o + h2u o q R uRL uRL LR d Do 1 Do 2 Do Do hd o + h1d o + h2d o + MD Do q R dRL dRL LR h. o u No 1 No 2 No + mD No No lR + h1. o L + h2. o L LR R R o d Eo 1 Eo 2 Eo + ME Eo Eo heR + h1e o L + h2e o L R +h:c l eR eRL 8 Quark masses and mixing from a SU(3) gauge family symmetry 119 Neutrinos may also obtain left-handed and right-handed Majorana masses both from tree level and radiative corrections. hL o u (No )c + mL No (No )c lLLL h1R o 1 (No )c + h2R o 2 (No )c + mR No (No )c + h:c RRRRRR T When the involved scalar fields acquire VEV’s, we get in the gauge basis o = L;R o (e;o;o;Eo)L;R, the mass terms — Mo o + h:c, where LR Mo = 0 BB@ 0 0 0 h v1 0 0 0 h v2 0 0 0 h v3 1 CCA . 0 BB@ 0 00a1 0 00a2 0 00a3 1 CCA . (8.9) h11 h22 0 M b1b2 0M Mo is diagonalizedby applyinga biunitary transformation o = Vo L;R L;R L;R. T Vo Mo Vo = Diag(0, 0, -3;4) (8.10) LR TT Vo MoMoT Vo = Vo MoT Mo Vo = Diag(0, 0, 2 ;2 ) , (8.11) LLRR 34 where 3 and 4 are the nonzero eigenvalues, 4 being the fourth heavy fermion mass, and 3 oftheorderofthetop, bottomandtau massforu,dande fermions, respectively.Weseefrom Eqs.(8.10,8.11)thatfromtreelevelthereexisttwo massless eigenvalues associated to the light fermions: 8.6 One loop contribution to fermion masses The one loop diagram of Fig.8.1 gives the generic contribution to the mass term oo ee mij —iLjR, Fig. 8.1: Generic one loop diagram mass contribution 120 A. Hernandez-Galeana X 2 o oH mij = cY . H m k (VL o )ik(VR o )jkf(MY;m k) ; H . g4. , (8.12) k=3;4 o MY being the mass of the gauge boson, cY isa factor coupling constant, m=-3 3 22 ox and m = 4, and f(x, y)= ln x 2 , 4x2-y2 y X ai bj M oo m k (VL o )ik(VR o )jkf(MY;m k)= F(MY) , (8.13) 2 - 2 43 M2 M2 k=3;4 YYYY i = 1, 2, 3 , j = 1, 2, and F(MY) . ln M2 - ln M2 . Adding up all M2 -2 2 M2 -2 2 Y44 Y33 possible one loop diagramss, we get the contribution —o Mo o + h:c:, L1R Mo = 1 0 BB@ D11 D12 00 D21 D22 00 D31 D32 D33 0 1 CCA H . , (8.14) 0 0 00 FZ1 FZ2 FZ2 D11 = 11(+ + Fm)+ 1 22F1 ; D12 = 12(- - Fm) 4122 6 FZ2 11 D21 = 21(- - Fm); D22 = 11F1 + 22FZ2 6 23 FZ1 FZ2 FZ2 D31 = 31(- 4 + ); D32 = 32(- + Fm) 12 6 1 D33 =(11F2 + 22F3) 2 F1 . F(MY1 ) ;F2 . F(MY2 ) ;F3 . F(MY3 ) (8.15) 2 FZ1 = cosF(M-)+ sin2 F(M+) (8.16) 2 FZ2 = sin2 F(M-)+ cosF(M+) (8.17) cos . sin . Fm = . [ F(M+)- F(M-)] . (8.18) 23 FZ1 ;FZ2 are the contributions from the diagrams mediated by the Z1 ;Z2 gauge bosons, Fm comes from the Z1 - Z2 mixing diagrams, with M1;M2 , M-;M+ the horizontal boson mass eigenvalues, Eqs.(7-11), ai bj Mai bj ij == 3 c. cß , (8.19) 2 4 - 2 3 ab c. = cos , cß = cos , s. = sin , sß = sin ß are mixing angles from the diagonalization of Mo. Therefore, up to one loop corrections the fermion masses are 8 Quark masses and mixing from a SU(3) gauge family symmetry 121 — Mo o —o Mo o = (8.20) R + — L M R , LL1R i h T where o = Vo L;R, and M. Diag(0, 0, -3;4)+ Vo Mo Vo , namely: L;R L;R L1R M = 0 BBBBBBBB@ m11 m12 cß m13 sß m13 m21 m22 cß m23 sß m23 c. m31 c. m32 (-3 + c cß m33) c sß m33 s. m31 s. m32 s cß m33 (4 + s sß m33) 1 CCCCCCCCA , (8.21) The diagonalization of M,Eq.(8.21) gives the physical masses foru anddquarks, e charged leptons and . Dirac neutrino masses. Using a new biunitary trans- T (1)(1)(1) T — formation L;R = V L;R; —L M R = L VM V R, with L;R = L;R LR (f1;f2;f3;F)L;R the mass eigenfields, that is TT (1)(1)(1)(1) 222 VMMT V= VMT M V= Diag(m 1;m 2;m 3;M2 ) , (8.22) LLRR F 2 22 22 2 m = m , m = m , m = m and M2 = M2 for charged leptons. So, the 1e23. F E rotations from massless to mass fermion eigenfields in this scenario reads (1)(1) o = Vo V L and o = Vo V R (8.23) LLL RRR 8.6.1 Quark Mixing Matrix VCKM We recall that vector like quarks are SU(2)L weak singlets, and hence the in- TooT teraction of L-handed up and down quarks; fo =(u;c;to)L and fo = uL dL (do;so;bo)L, to the W charged gauge boson is . g f —o fo . g— uL dLW+µ = uL (VCKM)44  dL W+µ , (8.24) 22 where the non-unitary quark mixing matrix VCKM of dimension 4 × 4 is (1)(1) (VCKM)44 = [(Vo V)34]T (Vo V)34 (8.25) uLuL dLdL 8.7 Numerical results for quark masses and mixing As an example of the possible spectrum of quark masses and mixing from this scenario, we show up the following fit of parameters at the MZ scale [17] Using theinput values for thehorizontal boson masses, Eq.(8), and the coupling constant of the SU(3) family symmetry: 122 A. Hernandez-Galeana M1 = 2800 TeV ;M2 = 103 M1 ; H = 0:05 , (8.26) . we write the tree level Mo , and up to one loop corrections Mq quark mass q matrices, as well as the corresponding mass eigenvalues and mixing: d-quarks: Tree level see-saw mass matrix: Mo = d 0 BB@ 0 0 0 817:977 0 0 0 9224:67 0 0 0 4139:08 3:072 × 106 -132120. 0 9:1 × 106 1 CCA MeV , (8.27) the mass matrix up to one loop corrections: Md = 0 BB@ 0. -2:31807 -48:2688 -16:3033 46:5611 20:7889 -0:89430 -0:30206 -20:8102 46:5138 -2859:86 130:424 -0:02081 0:04651 0:38614 9:61 × 106 1 CCA MeV , (8.28) the d-quark mass eigenvalues ( md;ms;mb;MD)=( 2:97 , 51 , 2860:72 , 9:61 × 106 ) MeV , (8.29) and the product of mixing matrices: 0 BB@ -0:99508 -0:01754 -0:09742 8:0 × 10-5 0:09608 -0:40801 -0:90790 9:218 × 10-4 0:02382 0:91280 -0:40769 4:136 × 10-4 1 CCA (1) = Vo VdL dL VdL = -1:87 × 10-5 7:19 × 10-10 1:34 × 10-5 0:99999 (8.30) 0 BB@ 0:02236 -0:01754 -0:94709 0:31970 0:91254 -0:40802 0:02446 -0:01374 0:40833 0:91280 -0:00726 -2:19 × 10-9 1 CCA (8.31) = Vo V VdR dR (1) dR = 0:00569 -6:24 × 10-8 0:31994 0:94741 u-quarks: 0 BB@ 0 0 0 23924:3 0 0 0 216134. 1 CCA Mo = u 0 0 0 104083. 8:65 × 1010 -6:91 × 109 0 6:96 × 1010 MeV , (8.32) 8 Quark masses and mixing from a SU(3) gauge family symmetry 123 0 BB@ 0 -150:079 -678:614 -845:855 4:02292 586:756 2635:9 3285:51 1 CCA MeV , (8.33) Mu = -1:92554 2086:27 -172961. 18797:7 -2:61 × 10-6 0:00282 0:02045 1:11 × 1011 the u-quark mass eigenvalues (mu;mc;mt;MU)=(1:38 , 638:36 , 172995 , 1:11 × 1011) MeV (8.34) and the product of mixing matrices: = Vo VuL uL V (1) uL = 0 BB@ 1 CCA 0:97438 0:20022 -0:10239 1:45 × 10-7 0:00044 -0:45700 -0:88946 1:38 × 10-6 -0:224887 0:866634 -0:445389 6:31 × 10-7 (8.35) = 0 BB@ 0 5:44 × 10-8 1:52 × 10-6 1. -0:00007 0:07222 -0:6247 0:77751 -0:00087 0:99734 0:03791 -0:06217 1. 0:00087 -0:00001 0 7:09 × 10-7 0:00936 0:77994 0:62578 1 CCA = Vo VuR uR V (1) uR (8.36) and the quark mixing matrix: VCKM = 0 BB@ -0:97491 -0:22255 -0:00364 -0:000014 -0:22250 0:97402 0:04208 -0:000046 0:00581 -0:04184 0:99910 -0:001012 3:24 × 10-9 1:07 × 10-8 -1:52 × 10-6 1:54 × 10-9 1 CCA (8.37) 8.8 F = 2 Processes in Neutral Mesons The SU(3) family gauge bosons contribute to new FCNC’s, in particular they mediate Ko - K—o, Do - D—o mixing via single exchange from the depicted diagram in Fig.8.2 2 2 12 YiY . the SU(3) symmetry breaking, and have flavor changing couplings in both left- and right-handed fermions, and then contribute the F = 2 effective operators OLL =( d— L sL)( d— L sL) , ORR =( d— R sR)( d— R sR) (8.38) OLR =( d— L sL)( d— R sR) (8.39) The SU(3) couplings to fermions when written in the mass basis yield the effective couplings The Z1 ;Y± (Y± 22 )gauge bosons become massive at the second stage of = 2 124 A. Hernandez-Galeana — gauge bosons. Fig. 8.2: Generic tree level exchange contribution to Ko - Ko from the SU(3) family 2 g H HSU(2) = 2 L OLL + 2 R ORR + 2LR OLR (8.40) 4M2 1 The suppression of the generic meson mixing couplings zij ( q—iL PL qj)2 come 2 out as follows — 8.8.1 Ko - Ko meson mixing L = 0:0392053 , M1 = 101667. TeV’s gH 2 jL| R = 0:372337 , M1 = 10705. TeV’s g2 H jR| (8.41) p M1 jLR| = 0:170869 , gH . = 23327. TeV’s 2 jLR| — 8.8.2 Do - Do meson mixing L = 0:000201739 , M1 = 1:97 × 107 TeV’s gH 2 jL| R = 0:000872865 , M1 = 4:56 × 106 TeV’s g2 H jR| (8.42) p M1 jLR| = 0:49322 , gH . = 8081:33 TeV’s 2 jLR| These values are within the suppression required for BSM contributions reported for instance in the review ”CKM Quark -Mixing Matrix” in PDG2022 [18]. 8 Quark masses and mixing from a SU(3) gauge family symmetry 125 8.9 Conclusions We have updated the analysis of quark masses and mixing within the context of a broken local vector-like SU(3) family symmetry, which combines tree level ”Dirac See-saw” mechanisms and radiative corrections to implement a successful hierarchical spectrum for fermion masses and mixing. We provided a parameter space region where this scenario can accommodate the known hierarchy spectrum of quark masses and mixing, and simultaneously suppress properlythe S = 2 and C = 2 processes. 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Buras, arXiv:1611.06206[hep-ph]; arXiv:1609.05711[hep-ph]; 7.T.Yanagida, Phys. Rev.D 20, 2986 (1979). 8. Z.G. Berezhiani: The weak mixing angles in gauge models with horizontal symmetry: Anew approach to quark and lepton masses, Phys. Lett.B 129, 99 (1983). 9. Z. Berezhiani and M. Yu.Khlopov: Theory of broken gauge symmetry of families, Sov.J.Nucl.Phys. 51, 739 (1990). 10. Z. Berezhiani andM.Yu.Khlopov: Physical and astrophysical consequencesof family symmetry breaking, Sov.J.Nucl.Phys. 51, 935 (1990). 11. A.S. Sakharov andM.Yu. Khlopov: Horizontal unification as the phenomenologyof the theory of “everything”, Phys.Atom.Nucl. 57, 651 (1994). 12. Z. Berezhiani,M.Yu.Khlopov andR.R. Khomeriki:On the possible testof quantum flavor dynamics in the searches for rare decays of heavy particles, Sov.J.Nucl.Phys. 52, 344 (1990). 13. J.L. Chkareuli, C.D.Froggatt, and H.B. Nielsen, Nucl. Phys.B 626, 307 (2002). 126 A. Hernandez-Galeana 14.T. Appelquist,Y. Bai andM.Piai: SU(3) Family Gauge Symmetry and the Axion, Phys. Rev.D 75, 073005 (2007). 15.T. Appelquist,Y. Bai andM. Piai: Neutrinos and SU(3) family gauge symmetry, Phys. Rev.D 74, 076001 (2006). 16.T. Appelquist,Y. Bai andM. Piai: Quark mass ratios and mixing anglesfrom SU(3) family gauge symmetry, Phys. Lett.B 637, 245 (2006). 17. Zhi-zhongXing,He Zhang and Shun Zhou, Phys. Rev.D 86, 013013 (2012). 18. R.L.Workmanetal. (ParticleDataGroup),Prog.Theor.Exp.Phys. 2022, 083C01 (2022) 8 Quark masses and mixing from a SU(3) gauge family symmetry 127 8.11 Appendix 8.11.1 Diagonalization of the generic Dirac See-saw mass matrix Mo = 0 BB@ 0 00a1 0 00a2 0 00a3 b1 b2 0c 1 CCA (8.43) The tree level Mo 4 × 4 See-saw mass matrix is diagonalized by a biunitary transformation o = Vo L and o = Vo R. The diagonalization of MoMoT LL RR (MoT Mo)yield the nonzero eigenvalues and rotation mixing angles  p   p 1 1 2 3 = , 2 4 = (8.44) B2 - 4D B2 - 4D B - B + 2 2 s s 2 4 - a2 a2 - 2 3 cos . = , sin . = , 2 - 2 2 - 2 43 43 (8.45) s s 2 - b2 b2 - 2 43 cos ß = , sin ß = . 2 - 2 2 - 2 43 43 22 B = a + b2 + c = 2 3 + 2 ;D = a 2b2 = 2 2 (8.46) 4 34 , 2 222 a = a 1 + a 2 + a ;b2 = b2 1 + b2 (8.47) 32 The rotation matrices Vo ;Vo admit several parametrizations related to the two LR zero mass eigenstates, for instance 0 BB@ c1 s1 c2 s1 s2 c. s1 s2 s. -s1 c1 c2 c1 s2 c. c1 s2 s. 0 -s2 c2 c. c2 s. 1 CCA , Vo R = 0 BB@ 0 cr sr cß sr sß 0 -sr cr cß cr sß 100 0 1 CCA Vo L = 00 -s. c. 00 -sß cß = q a , a = q ap 2 + a 2 3 , b = q b2 + b2 12 22 + a ap 1 2 a1 a2 ap a3 b1 b2 s1 = ;c1 = ;s2 = ;c2 = ;sr = ;cr = ap ap aabb a1 = s1 s2 a, a2 = c1 s2 a, a3 = c2 a, b1 = sr b, b2 = cr b Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 128) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 9 Evolution and Possible Forms of Primordial Antimatter and Dark Matter celestial objects MaximYu. Khlopov, O.M. Lecian email: khlopov@apc.in2p3.f email:orchideamaria.lecian@uniroma1.it Institute of Physics, Southern Federal University,Rostov on Don, Russia Virtual Institute of Astropartcie physics, Paris, France and National Research Nuclear University ”MEPHI”, Moscow, Russia; Sapienza University of Rome, Faculty of Medicine and Pharmacy, Viale Regina Elena, 324 -00185 Rome, Italy; Sapienza University of Rome, Faculty of Medicine and Dentistry, Piazzale Aldo Moro,5 -00185 Rome, Italy; Kursk State University, Faculty of Physics, Mathematics and Information Sciences, Chair of Algebra, Geometry and Didactics of Mathematics Theory, ul. Radis’c’eva, 33, aud. 201, Kursk, Russia Abstract. The structure and evolution of Primordial Antimatter domains and Dark matter objects are analysed. Relativistic low-density antimatter domains are described. The Relativistic FRWperfect-fluid solution is found for the characterization of i) ultra-high density antimatter domains, ii) high-density antimatter domains, and iii) dense anti-matter domains. The possible sub-domains structures is analyzed. The structures evolved to the time of galaxy formation are outlined. Comparison is given with other primordial celestial objects. The features of antistars are outlined. In the case of WIMP dark matter clumps, the mechanisms of their survival to the present time are discussed. The cosmological features of neutrino clumping due to fifth force are examined. Povzetek:ˇ Clanek obravnava strukturo in dinamiko domen anti-snovi majhne gostote in temne snovi v zgodnjem vesolju. Avtorja opiˇcno seta domene antisnovi z relativistiˇ idealno teko cino in poiˇsˇˇsitve za majhne, srednje velike in velike gostote tekoˇ cetareˇcine. Di- namiko domen antisnovi spremljata do nastanka galaksij in obravnavata njihovo pre ˇ zivetje do danes. Studirata tudi kozmoloˇˇske posledice zdruˇ zevanja nevtrinov zaradi pete sile. Keywords: perfect-fluid plasma solution; nonhomogeneus baryosynthesis, antimatter; cosmology, celestial bodies. 9.1 Introduction The formationof antimatterregionsand antimatter domainsinamatter/antimatter asymmetric Universe has long been studied according to the properties of the Title Suppressed Due to Excessive Length 129 pertinent celestial objects, as well as to the observational signatures expected, i.e. the energetic gamma rays descending from the matter-antimatter interaction at the boundaries of the antimatter domains. Several scenarios can be envisaged, i.e. also ones in which strong antimatter inhomogeneities interact with the surrounding medium(see[1,3,4,4,5] forreview andreferences). The mechanisms of survival for the antimatter domains can be analyzed. Comparison with other celestial bodies enables one to extrapolate the properties of both the formation and evolution of such celestial bodies, as well as the interactions under which the celestial bodies are formed. In thepresent paper, low-density antimatter domains willberevisedin the non- relativistic description, under the suitable hypotheses. The Relativistic diffusion equation of low-density antimatter domains will be solved; the Relativistic radius and the Relativistic spherical shell interaction width will be calculated. Dense antimatter domains will be introduced and classified according to the density, i.e. ultra-high density antimatter domains, very-highdensity ones and high-density ones. The Relativistic FRWdiffusion equation of dense antimatter domains will be solved in the perfect-fluid FRWplasma solution. The Relativistic radius of the dense antimatter domains in the FRWsymmetry and the Relativistic spherical shell interaction width in the FRW symmetry will be calculated; the calculated expressions will be shown to depend on the Relativistic quantities in a non-trivial manner. Baryon subdomains inside the antimatter domains will be investigated; in particular, the analysis will be conduced in the cases pertinent to the epoch before the second phase transition and that after the second phasetransition. This way, the formation of non-trivial structures will be assessed; more in detail, ’Swiss-cheese’ structures and ’Chinese-boxes’ structures will be reconducted to the analytical quantification. Antimatter-excess regions will be explored wrt the diffusion process taking place at the boundary regions. The density of antimatter domains at the time of galaxy formation will be written down. Experimental-verification methodswillberecapitulated for antimatter domains in a matter/antimatter asymmetric Universe within the framework of inhomogeneous baryosynthesis. The investigation methods for these purposes will be specialized to the study of the -ray background and of the expected anti-Helium. Further experimental purposes will be recalled. Comparison with other celestial bodies will be brought. The features of antimatter celestial bodies in the Galaxy, WIMP dark-matter clumps and Fifth-Force neutrino lumps will be revised for the sake of the study of the formation mechanisms, the Universe-evolution survival models, and of the interaction ruling the structure of the celestial bodies. 9.2 Low-density antimatter domain: diffusion equation The Relativistic diffusion equation of 130 MaximYu. Khlopov, O.M. Lecian n — the antibaryon number density as a function of nb the baryon number density, b and n. the photon number density reads [6] dn — 3d b =- < v > n b— nb - n — (9.1) b dt R being R the antimatter domain non-Relativistic radius, and d the antimatter domain spherical shell boundary interaction width; furthermore, < v > antibaryon- baryon annihilation cross-section within the interaction region is defined, and ß the FRWRelativistic factor is introduced. Being —r the antibaryon-to-photon ratio and r baryon-to-photon ratio the RelativisticFRWdiffusion equationof low-density domainsrewrites r —=- 3d <v>rn. r —- r —(9.2) R 9.2.1 Low-density antimatter domains: non-Relativistic approximation The non-Relativistic diffusion equation of low-density antimatter domains can be approximated after neglecting r—, and posing < v > rn t ~ 1, and solved as Z t. r—. 3d = exp[- < v > rn dt] (9.3) r—0 R t0 9.2.2 Low-density antimatter domains: Relativistic solution The Relativistic diffusion equation of low-density antimatter domains under the assumption — r << 1 can be solved as 1=3 dr. 14. (t) (ln[- ~ (t. - t0)]) = - (9.4) a(t)1=3 dt. r0 33 with ß ~=-beta. The Relativistic quantities d › (t) Relativistic spherical-shell width interaction 4R(t)3 region, and a(t)= FRWvolume have been upgraded. 3 9.3 General implemetation-Relativistic After hypothesizing - n — . ß ~small but not negligible, and ß ~' const, the b following solution is found r—f . - ~ ln[ t] ' - t (9.5) r— 0 3a in the case of perturbed Minkowski space-time. In the case of an FRWsymmetry, the following solution is written dr—f (t) - ~ dt. r— 0 3a(t) ln[ t] ' - (9.6) Title Suppressed Due to Excessive Length 131 9.4 Perfect-fluid Relativistic FRWequation of dense antimatter domains The prefect-fluid FRWdiffusion equation of the antibaryon number density writes dn — 3d n — bb b + Q(r, p, t)- ~ =- < v >ext n b— nb - n — + dt R td X 2 Fi(p, p˙; :::)- f(E, (9.7) p; Rd;ld; i Here, < v >ext is cross-section of the antibaryon-baryon annihilation process ~ ~ vT ;vf; i~ )- r + n — b attheboundaryoftheantimatterdomain, isasourceterm(canbe () Qr;p;t~ neglected),˙arefurthertermsdependingonthemomentum(canbene-() F; p;p:::i glected), f(E, vT ;vf; i~ ) is plasma characterization in terms of the viscosity p; Rd;ld; b properties and of the turbulent velocity (can be neglected), n — '< ~v>int n — ~bnb td is the decay rate inside the interior of the domain, td = const? is the time scale of annihilation, < v >int is cross-section of the antibaryon-baryon annihilation process in the interior of the antimatter domain, n — accounts for the FRWhomo- b geneous Relativistic expansion of the universe, and µ is the chemical potential, i.e. 2 n ~b — =-rn b — for the self-similarity properties of the equation. ~ 9.5 Dense antimatter domains ~ By construction, both the antibaryon density and the baryon one are much higher than average baryon density in all the Universe; several cases can be distinguished: i)ultra-high densities the antibaryon excess and baryon ones start to exceed the contribution of thermal quark-antiquark pairs before QCD phase transition ii)very-high densities the antibaryon density and the baryon ones exceed the contribution of plasma and radiation after the QCD phase transition iii)high densities the antibaryon densities and the baryon ones exceed the DM density 9.5.1 i) ultra-high density antimatter domains Let n b — be the number density of antibaryons. The following diffusion equation is outlined dn —3d n — b 2b =- < v >ext n b— nb - n b — - r n b — . -- n b — + ß ~+ µ ~(9.8) dtR ts and solved as 1=3 Z t. r—. 14. (t) ln[ -( ß ~+ ~)(t. - t0)] = - < v >ext rn. dt (9.9) (a(t))1=3 r—0 33 ti 132 MaximYu. Khlopov, O.M. Lecian with a(t)= 4R(t)3=3 the Relativistic FRWvolume, and d › (t) the Relativistic interaction spherical-shell width, which simplifies as 1=3 dr—. 14. (t) (ln[ -( ß ~+ ~)(t. - t0)]) = - < v >ext rn. (9.10) (a(t))1=3 dt. r—0 33 Relativistic expression for the radius of the antimatter domain Relativistic expression for the radius of the antimatter domain reads  3 1=3 < v >ext rn (t)[ r— — . -( ß ~+ ~)(t. - t0)] 1=3 r0 (a(t))=- dr. (9.11) 4. 3 [ —-( ß ~+ ~)(t. - t0)] dt. r—0 Relativistic expression for the interaction width Relativistic expression for the interaction width is obtained as 1=3 dr. 33(a(t))1=3 [ —-( ß ~+ ~)(t. - t0)] dt. r—0 (t) ~ - (9.12) 4. <v>ext rn. [ r. -( ß ~+ ~)(t. - t0)] r0 The relativistic expression of the interaction width of the antimatter domain depends therefore also on the Relativistic radius in a non-trivial manner, i.e. as a prefactor. 9.5.2 ii) very-high-density antimatter domains In the case of very-high-density antimatter domains, the diffusion equation of the baryon number density becomes dn —3d n —n —n — b b2b b2 =- < v >ext n b— nb - n b —-- r n b —. -- n b —-- r n b —(9.13) dtR td ts td solved as 1=3 Z r—. 1 4. t. (t) ln[+ < ~>int r -( ß ~+ ~)(t. - t0)] = - dt (9.14) . ~j rn. — r—0 33 ti (a(t))1=3 with a(t)= 4R(t)3=3 Relativistic FRWvolume, and d › (t) Relativistic interaction spherical-shell width  dr—. 1 4. 1=3 (t) (ln[+ < ~>int r(t. - t0)-( ~)(t. - t0)]) = - . ~j rn. —ß + ~(9.15) (a(t))1=3 dt. r—0 33 Relativistic expression of the radius of the antimatter domain The Relativistic expression of the radius of very-high-density antimatter domains is obtained as 1=3 3 <v>ext rn (t) (a(t))1=3 =- · 4. 3 r. [ —+ <. ~~j rn. —ß + ~)(t. - t0)] >int r(t. - t0)-( ~ · r—0 (9.16) dr. [ —-+ <. ~~j rn. —ß + ~)(t. - t0)] >int r(t. - t0)-( ~ dt. r—0 Title Suppressed Due to Excessive Length 133 Relativistic expression of the spherical shell interaction width The Relativistic expression of the spherical shell interaction width of very-high density antimatter domains is !dr—. 1=3 [+ <. ~~r(t. - t0)-( ß ~+ ~)(t. - t0)] 33(a(t))1=3 dt. —µ >int j rn. — r0 (t) ~ - (9.17) r—. r—0 µ >int j rn. — 4. < v >ext rn. [+ <. ~~r(t. - t0)-( ß ~+ ~)(t. - t0)] The Relativistic expression of the interaction width of the antimatter domain dependsthereforealsoonthe Relativistic radiusina non-trivial manner,i.e.asa prefactor. 9.5.3 iii) high-density antimatter domains The diffusion equation of the antibaryon number density of high-density antimatter domains is characterized as dn — 3d n — b b2 =- < v >ext n b— nb -- r n b — . dtR td 3d n — n — bb 2 . - < v >ext n b— nb --- r n — (9.18) Rts td b and solved as 1=3 Z t. r—. 14. (t) ln[ -( ~)(t. - t0)] = - < v >ext rn. dt (9.19) (a(t))1=3 r—0 33 ti Here, a(t)= 4R(t)3=3 is the Relativistic FRW volume, and d › (t) is the Relativistic interaction spherical-shell width. Eq. (9.19)rewrites 1=3 dr—. 14. (t) (ln[ -( ~)(t. - t0)]) = - < v >ext rn. (9.20) (a(t))1=3 dt. r—0 33 Relativistic expression for the radius of the antimatter domain In the case of high-density antimatter domains, the Relativistic expression for the radius of the antimatter domain is expressed as 1=3 [ r—.  3 <v>ext rn (t) r—0 - ~(t. - t0)] (a(t))1=3 =- (9.21) d [ r—. 4. 3 - ~(t. - t0)] dt. r—0 Relativistic expression for the spherical shell interaction width The Relativistic expression for the spherical shell interaction width of high-density antimatter domains is solved as 1=3 dr. 33(a(t))1=3 [ —- ~(t. - t0)] dt. r—0 (t) ~ - (9.22) r. 4. <v>ext rn. [ r — —0 - ~(t. - t0)] The Relativistic expression of the spherical-shell interaction width of the antimatter domain depends therefore also on the Relativistic radius in a non-trivial manner, i.e. as a prefactor. 134 MaximYu. Khlopov, O.M. Lecian 9.6 Conditions and evolution of different types of of strong primordial inhomogeneities in non-homogeneous baryosynthesis In the case of non-homogeneous primordial baryosynthesis, various types of scenarios can accomplish: antimatter consisting of axion-like particles; closed walls for baryogenesis with excess of antibaryons, and phase fluctuations such that a baryon excessis created everywhere and with non-homogeneous distribution.To avoid large-scale fluctuations, the fluctuations have to be imposed to be small. In the latter cases of small fluctuations, the following inequality holds 3B >>B (9.23) 4R(t)3 The diffusion process of the model is described as follows. Three Regions can be outlined: 1)thedense antimatter domain of radius R . R1 of antibaryon number density nb1, and of chemical potential 1, 2)theouter spherical shell region of radius R1 . R . R2 of antibaryon number density nb2, and of chemical potential 2, where the diffusion process happens, and 3)the outmost region of radiusR3 . R2 of antibaryon number density nb3 of low antimatter density. The chemical potentials of related to the three regions are assumed to be small but not negligible, i.e. | ~1 j<< 1, | ~2 j<< 1, and | ~3 j<< 1. The differential equation of the antibaryon number density in Region 1)is nb1 2 =-1r nb1 ~ ~1nb1; (9.24) dt The differential equation of the antibaryon number density in Region 2)is nb2 2 =-2r nb2 ~ ~2nb2 (9.25) dt The differential equation of the antibaryon number density in Region 3) nb3 2 =-3r nb3 ~ ~3nb3 (9.26) dt The solutions of Eq. (9.24), Eq. (9.6) and (9.25) must satisfy the continuity conditions nb1(t, R1)= nb2(t, R1) (9.27) on the boundary of Region 1), and nb2(t, R2)= nb3(t, R2) (9.28) on the boundary of Region 2). Title Suppressed Due to Excessive Length 135 9.7 Dense Baryon subdomains It is possible to hypothesize the presence of antibaryons inside the baryon subdomains, which exceed the survival size of volume Vj = 4R3 j =3;such a possibilty is dependent on the second phase transition. For axion-like particles, it is dependent on the QCD phase transition. Two possibilities are outlined, i.e. according to whether the description is taken before the QCD phase transition, or after it. I) In the case <QCD, the baryon number density nb in a baryon subdomain filled with (grazing) antibaryons obey the following plasma characterization dnb2 =- < v >j ext nbn b— - < v >j int nbn b — - r nb (9.29) dt The following perfect-fluid Relativistic FRWsolution is found . ~ (t) - 43 rint r—intn. < v >j ext = 3a~(t) = d ln[rint r—intn. int < v >j ext +- ~(t. - t0)+] (9.30) dt. In the case II) >QCD the baryon number density nb in a baryon subdomain without free antibaryons inside is described by the following plasma characterization dnb2 =- < v >j ext nbn b — - r nb (9.31) dt The following perfect-fluid Relativistic FRWsolution is found . (t~ ) d - 43 rint r—intn. < v >j ext = ln[+ ~(t. - t0)] (9.32) 3a~(t) dt. 9.8 Further structures Further structures can be analysed, according to the presence of baryon subdo- main(s) inside the antibaryon domain. 9.8.1 ’Swiss-cheese’ structures Adescription of ’Swiss-cheese’ structures can be hypothesized as an antimatter domain containing one matter domain, in the simplest instance, and more complicated ’Swiss-cheese’ structures, such as an antimatter domain containing several matter subdomains. 136 MaximYu. Khlopov, O.M. Lecian Antibaryon domain containing one baryon subdomain In the case of an an- tibaryon domain containing one baryon subdomain, the antibaryon number density obeys the differential equation dn — 3d n — bb =- < ~v>ext n — b + Q(r, p, t)- p; :::)+ ~bnb - n — ~++ Fi(p, ˙ dtR td 2 3di2 -r n b — - <^i ^vi >n b— nbi - ir n — (9.33) Ri b Swiss-cheese structure: baryon domain containing several baryon subdomains dn — 3d n — bb =- <~~bnb - n b — + Q(~r, p, t)- + v>ext n — dtR td i=I X 2 3di2 + Fi(p, p˙; :::)- r n b — - Ri <^i ^vi >n b— nbi - ir n b — (9.34) i=1 Chinese-boxes structures ’Chinese-boxes’ structures are described as dn — 3d n — b b2 =- < v >ext n b— nb - n b — + Q(~r, p, t)- - r n — dtR td b i=I j=J XX 2 3di - Fi(p, p˙; :::)- r n — - <v >i ext n b— nbi +[Fj(p, p˙; :::) bi Ri i=1j=1 3dj 2 -r n b— j - <v >j ext n b— nbj ] (9.35) Rj 9.9 Galaxy formation: Relativistic density of the surviving domains The present section is aimed at studying the density of the antimatter domains at the time of galaxy formation. 9.9.1 Plasma characterization The plasma characterization of the antimatter domains at the time of galaxy formation is given after the condition < v >int r = 0, (9.36) i.e. after the antibaryon/baryon interactions in the interior of the antimatter domains have exhausted. The following conditions are taken into account: ~(t. - t0)n. << 1, (9.37) Title Suppressed Due to Excessive Length 137 and ~ (t. - t0)n. << 1, (9.38) with ( µ ~+ ~ )(t. - t0)n. << 1, (9.39) i.e. that the chemical-potential etrms and the Relativistic FWR terms be small but not negligible. i) ultra-high-density antimatter domains In the case of ultra-high-density antimatter domains, the antimatter-domain density at the time of galaxy formation reads r—n. 1n. = · 1 a(t) a(t) r—0 +( ß ~+ ~)(t. - t0)n. " 1=3 Zt. # 1 4. t exp <v>ext r0n. dt (9.40) 33 t0 a(t) ii) very-high-density antimatter domains In the case of very-high density antimatter domains, the antimatter-domain density at the time of galaxy formation is r—n. 1n. = · 1 a() a(t) +( ß ~+ ~)(t. - t0)n. r—0n. Z 1=3 t. 1 4. t exp 3 () < v >ext r0n. dt (9.41) 3 t0 a(t) iii) high-density antimatter domains In the case of high-density antimatter domains, the antimatter-domain density at the time of galaxy formation becomes Z 1=3 t. r—n. 1n. 1 4. t = n. e 3 () < v >ext r0n. dt (9.42) a() a(t)+ ~(t. - t0)n. 3a(t) r—0 t0 9.10 Experimental verification The signatures of the experimental verification of the existence of antimatter domains have to be analysed. In particular, the -ray background is expected to be modified after the baryon/antibaryon interaction within the boundary interaction region of the antimatter domains. Furthermore, the detection of anti-Helium flux after the AMS2 experiment is awaited. The properties of pp atoms have been studied in [7] In [8], the -ray spectrum originated after the pp —annihilationin liquidHydrogenis analysedby meansof two spectrometers. As a result, no exotic narrow peacks are evidentiated, and the upper limit is calculated. The -ray signal due to matter-antimatter annihilation on the boundary of an antimatter domaincanthereforebe analyzed[9];the hypothesesofamatter/antimatter 138 MaximYu. Khlopov, O.M. Lecian symmetric Universe and of a matter/antimatter asymmetric Universe can be scrutinized and compared. In the case of a matter/antimatter-symmetric Universe: more -rays than the observed quantities are predicted; therefore, a matter/antimatter-symmetric Universe is possible iffthe present Universe is one consisting of the matter quantity. The pp —interaction process is studied as resolving in photons after the 0 decay. Be g —the mean photon multiplicity; each pp —annihilation process is estimated to produce g —' 3:8 electrons and positrons, and an approximate similar number of photons. The annihilation electronsare describedataredshift y s.t. 20 < y < 1100. The mechanisms that control the electrons motion must therefore be studied. Such mechanisms are evaluated to be the cosmological redshift, the collision with CBR photons, and the collision with ambient plasma electrons. At the considered values of the redshift, the collisions with CBR photons are considered the most important control mechanism of the electron trajectory. For initially-Relativistic electrons of energy E0 = 0me, the dependece of the width of the reheated zone where the electrons produced after the annihilations directly deposit energy into the fluid, i.e. the electron range, on 0 is negligible. The inclusive photon spectrum in the pp —process is normalized to g—;the average number of photons made per unit volume is calculated: the transport equation of the photons scatter and redshift, (which lead to a spectral flux of annihilation photons),is therefore assessed.Aconservative lower limit for the -ray signal can this way estimated. The -rays energy is expected to be of order 100MeV - 10Mev at modern times;a different value can be expected for the opaque universe at the early stages). The results are awaited after the experiment AMS2 as far as the presence of the anti-Helium flux is concerned. 9.11 Further experimental verifications Further experimental verifications of a matter/antimatter Universe can be expected . As an example [10], annihilation and transformation of annihilating matter’s rest mass into energy particles and radiation with 100%efficiency can be looked for at different length scales. Asubstantial lack of antimatter on the Earth is evidentiated within the due limits. Alack of antimatter in the vicinity of the Earth is found. Matter asymmetry in the Solar System can be revealed within the study of the: Solar wind, i.e. the continuous outflow of particles from the Sun. For antiplanets of radius r and distance d from the Earth intercepting the Solar wind, the expected annihilation flux is -2 -1 F(. = 100MeV) ~ 108(r=d)2 photons cm s . Title Suppressed Due to Excessive Length 139 At scales larger than the Solar System, i.e. at the Galaxy scales, the -ray analysis must be investigated( -ray detectors have better detection capabilities and localization ones at E ~ 100 MeV than neutrino detectors). Antimatter mixed in with matter inside our Galaxy’s gas at E ~ 100 MeV is expected to be present in a matter/antimatter-symmetric Universe. Models can be postulated [11], such that SUSY-condensate baryogenesis models motivate the possibilities of antimatter domains in the Universe. In this case, vast antimatter structures in Early-Universe evolution possible after initial space distribution at the inflationary stage of the quantum fluctuation field (r, t0), unharmonic potential of the field carrying the baryon charge, and inflationary expansion of the initially microscopic baryon distribution. The vast antimatter regions are calculated to be separated at distances larger than 10 Mpc from the Earth, and separatedfrom the matter onesby baryonically-empty voids. Such models are not ruled out after cosmic rays data, -rays ones, and CMB anisotropy ones. Antimatterina matter/antimatter-symmetric Universe canbe further verified[12] after the presence of antimatter at the Galactic scale and above. As far as hydrogen in ”clouds” is concerned, the experimental verification is based on the observation of -rays from their directions, compatible with 0 decay, and non-observation of a . excess. In this case, form the observational data, the an- tibaryon presence in the media is calculated not to exceed one part in 1015 . The instance of galaxy-antigalaxy collisions can be studied. Such events have not been verified after devices s.t. Antennae pair NGC4038(9). Clusters of certain galaxies, dense enough and active in order to allow for intergalactic hot plasma in the central parts (at temperatures of order ~ 10 keV:it is therefore possible to verify the presence of antimatter as a few parts per million from the observation of absence of enough -ray excess on the thermal spectral tail. Large antimatter regions with sizes larger than the critical surviving size can be verified in different observational proofs [13]. The absence of anti-Helium in the cosmic rays and annihilation signals can be consistent as an indicator: their fraction in the Galaxy is smaller than 10-4. The antimatter islands must be separated from a space filled with matter at least by the distance of about 1Mpc. For this, the possibility for antimatter islands (antistars) in the Galaxy still allowed [14]. Large antimatterregionswithhigh antimatterdensityevolvetosinglegalaxies[15]. They are detected after particular content of anti-Helium and anti-deuterium. Further Cosmic antimatter searches can be pursued [16]. The presence of antistars in our Galaxy can be verified after the possibility to detect antinuclei with Z . 2. Domain sizes of the scale of galaxies or scales of galaxy clusters can be testified after antimatter cosmic rays (CR) originating from the nearest domain for uniform domains, non-uniform domains, and condensed antimatter bodies (i.e. antistars, antiplanetoids). The upper limitof antistarsin the MilkyWay has been estimated as 107 (i.e. 10-4 of thetotal numberof stars). Antistars canbe described as con 140 MaximYu. Khlopov, O.M. Lecian fined into compact structures separated from the matter environment and able to survive for a longer period rather than in gas clouds. Antistars are not expected to be strong -ray emitters, unless they at least cross a galactic cloud or impact on other condensed bodies. The lower limit on the distance of the nearest antistar [17] has been set as ~ 30 pc. The upper limit on the fraction of antistars in the Andromeda Galaxy has been estimated as ~ 10-3 . Experimental verification of presence of matter regions and antimatter ones in a matter/antimatter-symmetric universe should be studied after the pre-recombinational signals and the post-recombinational ones. The prerecombination signal [18] allows one for the verification of the presence of domainsof larger size. The assumptionthat matter domains and antimatter ones were in contact before the last scattering exhibits such effects after which contact and annihilation significantly distort the radiation from the last-scattering surface: asingle domain boundary,ora fraction,canbe detectable;differently,the absence of such signatures rules out a matter/antimatter-symmetric universe. The postrecombination signal [18] would consists of the observable unobserved -ray flux, due to nuclear annihilation rate of matter/antimatter near domain boundaries; the a resulting relic diffuse -ray flux exceeds the observed cosmic diffuse . spectrum, so that a matter/antimatter-symmetric Universe is ruled out unless the matter region consists of almost the entire Universe 9.12 Antistars The analysis of the mean free path of the cross section of the matter/antimatter annihilation products in the interaction spherical shell boundary of the antistars is conisistent for the comparison with the -ray-background constraints [19]. After the compilation of the 10-years Fermi Large AreaTelescope (LAT) -ray- sources catalog, constraints on the abundance of antistars around the Sun are obtained: 14 antistar candidates arepresent around the Sun. In particular,they have been chosen as they are not associated with any objects belonging to established -ray source classes, and exhibit a spectrum compatible with baryon-antibaryon annihilation [20]. 9.13 Antimatter celestial objects in the Galaxy The exist observational evidences of the existence of antimatter celestial objects in the Milky Way; more in detail [21], they are point-like sources of gamma radiation, and diffuse galactic -ray background, where the latter possible antimatter sources are to be verified after an anomalous abundance of chemical (anti-)elementsaroundit possibly measuredby spectroscopy, anti-nucleiin cosmic Title Suppressed Due to Excessive Length 141 rays, and more exotic events, where large amounts of matter and antimatter interact. In the latter case, star-antistar annihilation can be considered: huge energy produced, even though their total destruction is prevented by the radiation pressure produced in the collision; and collision of a star and an anti-star with similar massesis calculatedtoprovokea peculiarresult. 9.13.1 Antistars The creation of stellar-like objects in the very early universe [22], from the QCD phase transition until the BBN and later, can be witnessed as the presence of some of the celestial objects created which can consist ofantimatter. The . cosmological NB baryon asymmetry . = can be close to unity, i.e. much larger than the ob N. served value . ' 6 · 10-10. The ratio . can also be negative: this way, the amount of antimatter constituting compact objects in the Galaxy is expected. 9.14 WIMP’s clumps 9.14.1 Neutralino clumps Within the standardcosmological scenario (FRWwith its thermal history,inflationary- produced primordial fluctuation spectrum and with a hierarchical clustering), the neutralino clumps [23] undergo tidal destruction in the hierarchical clustering (i.e. the smaller clumps are captured by the larger clumps) at earlystages of the structure-formationprocess, startingfroma timeof clump detachmentfrom the Universe expansion. In the case of small-scale dark-matter clumps, a mass function can be calculated for the survived clumps: the tidal destruction of clumps by the Galactic disk, the life-time of clumps in the central stellar bulge, and the life-time of clumps in the stellar halo spheroid can be calculated; as a result, the minimal mass is the evaluated as the Moon-scale mass. 9.14.2 Neutralino annihilation in the Galaxy Within the standardcosmological scenario, neutralino annihilation of small-scale neutralino clumps [24] would produce a signal from the galactic halo: the clump destruction is due to larger-scale clumps, gravitational field of the galactic disk, stars in the galactic bulge, and stars in the galactic halo. The mutual tidal clumps interactions would become important at early stages of hierarchical clustering, and for the galactic halo formation. The hierarchical clustering implies clumps surviving the hierarchical clustering to be continuously destroyed by interactions with the galactic disk and stars. This way, 142 MaximYu. Khlopov, O.M. Lecian 20%of neutralino clumps surviving the hierarchical clustering between the Earth and the Moon can ’survive the Sun position’ because of tidal destruction due to Galactic disk. Furthermore, the diminishing of the expected DM annihilation signal from the galactic halo would be awaited. 9.14.3 Small-scale DM clumps The clumps scenarios comprehend spherical models, non-spherical models, and clumps around topological defects [25]. The possible observational verifications are established DM-particles direct detection, recordof clumpsin gravitational-wave detectors, neutralino stars, baryonsin clumps, and clump motion in the Sky sphere. 9.15 Fifth-Force neutrino lumps 9.15.1 Fifth-Force codifications The Fifth Force potential can be codified as [26], [27]    ~. Gm1 -r=. m1m2 -ß m1m2 G r V(r)= 1 + e= G1 + e = G + . (9.43) r r rG Sucha codification allows for the descriptionof dark-matter gravitational clustering. 9.15.2 Modellizations for neutrino cosmology The parameter ß is intended as the Fifth-Force parameter, and . - . coupling is postulated. The fifth force is requested to be subdominant with respect to the gravitational force [28], [29].As an example, therequest canbe expressed as d ln m. ß =- . (9.44) d. Is is also possibe to set a . comoving length scale larger than the typical lump sizes, but smaller than their typical distances, such that the mean distance between neighboring lumps be of order 100h-1Mpc. l lumps of masses Ml are expressedvia smoothed fields ^ . The effective coupling d ln Ml l = - d^. (9.45) is worked out. Title Suppressed Due to Excessive Length 143 9.15.3 Applications for fluids of composite objects Neutrino lumps are described within a hydrodynamic framework, i.e. endowed witha balance equation [29], anda stability equation [30] based on theTolmanOppenheimer- Volkoffequation. Within the framework of the. - . coupled fluid, neutrino fluctuations are hypothesized to grow under the effect of the Fifth-Force [31]. Non-Relativistic neutrino clusters under the effect of the fifth-force are hypothesized at scales estimated to be around a few 10–100Mpc.Astatistical distribution of neutrino lumps is expressed as a function of the mass at different redshifts z . 1. The oscillating structure formation is described as at the time a large number of neutrinos were staying in gravitationally-bounded lumps at z = 1:3. 9.15.4 Formationof large-scale neutrino lumpsina recent cosmological epoch Within the framework of a. - . interaction, the non-linear features of the Fifth- Force can be outlined [32]. The averaged interaction strength < > ofthe neutrinosina neutrinolumpreads d ln < >=-M. (9.46) d. The effective suppression of the -mediated attractive force between neutrino lumps is proportional to 2 . In particular, the attraction between two equal lumps 2 < > isreducedbya factor . Furthermore, the characteristic time scale for the ß ß infall increased by a factor compared to the consideration excluding non < > linear effects and thus results in a slow down of the infall: the time scale for the 2 ß clumpingoflumpstolargerlumps enhancedby a factor . < > In the interior of the lump, the possibility of a time variation of fundamental constants results much smaller than the cosmological evolution; therefore, it is possibletoreconcilethe cosmological variationsofthefinestructure constantwith geophysical bounds. 9.15.5 CMB verification for neutrino lumps Within the framework of a. - . interaction, theintegrated Sachs-Wolfeeffectof CMB [33] can be considered. The size of the gravitational potential induced by the neutrino lumps, and the time evolution of the gravitational potential induced by the neutrino lumps have to be analyzed. asaresult,aproportionality betweenthe scalar potentialandthe neutrino-induced gravitational potential is found as . = 2 2. (9.47) 144 MaximYu. Khlopov, O.M. Lecian for the local potential and the cosmological-averaged potential. The population of lumps of size . 100Mpc can lead to observable effects from the CMB anisotropies for low angular momenta. 9.16 Outlook and perspectives Evolution of antimatter domains have been studied: an analysis of low-density antimatter domains and dense antimatter domains has been performed. More in detail, ultra-high density antimatter domains, very-high density antimatter domains, and high-density antimatter domains. Experimental verificationoftheir signatures consistsofthe searchfor confirmation in the observed -ray background, and for the expected anti-Helium flux in AMS02 experiment. Comparison with other celestial objects has been accomplished: studyof formation mechanisms, Universe-evolution survival models, and comparison of interactions characterizing the structure of the celestial bodies has been performed. Acknowledgements OML acknowledges the Programme Education in Russian Federation for Foreign Nationals of the Ministry of Science and Higher Education of the Russian Federa- tion.TheworkbyMKwas performedinMEPHIinthe frameworkof cosmological studies of Prioritet2030 programme. References 1. M.Yu.Khlopov: What comes after the Standard model? Prog.Part. Nucl. Phys. 116, 103824 (2021). 2. Khlopov, M. Cosmological Reflection of Particle Symmetry. Symmetry 8, 81 (2016). 3. V.M. Chechetkin, M.Y. Khlopov, M.G. Sapozhnikov, Y.B. 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Casadei: Searches for Cosmic Antimatter, Frontiers in Cosmic Ray Research, Nova Science Publishers, Hauppauge, NewYork, USA, 2007. 17. A. Dudarewicz, A.W.Wolfendale: Anti-matter in the Universe on very large scales, Mon. N. R. Astr. Soc.268, 609 (1994). 18. A.G. Cohen,A.DeR ´ujula, S.L. Glashow:AMatter-Antimatter Universe?,Astrophys.J. 495, 539 (1998). 19. A.D. Dolgov: Antimatterin the MilkyWay,Preprint arXiv:2112.15255 [astro-ph.GA]. 20. S. Dupourqu´e,L.Tibaldo,P. von Ballmoos: Constraints on the antistar fractionin the Solar System neighborhoodfrom the 10-year Fermi LargeAreaTelescope -ray source catalog, Phys. Rev.D 103 083016 (2021). 21. C. Bambi,A.D. Dolgov:Antimatterin the MilkyWay, Nucl. Phys.B 784, 132 (2007). 22. A. D. Dolgov, S. I. Blinnikov: Stars and Black Holes from the very Early Universe, Phys. Rev.D 89, 021301 (2014). 23.V. Berezinsky,V. Dokuchaev,Y.Eroshenko: Destructionof small-scale dark matter clumpsin the hierarchical structures and galaxies, Phys. Rev.D 77, 083519 (2008). 24.V.Berezinsky,V. Dokuchaev,Y.Eroshenko: Dark Matter Annihilationinthe Galaxy, Phys. Atom. Nucl. 69, 2068 (2006). 25.V. Berezinsky,V. Dokuchaev,Y.Eroshenko: Small-scale clumpsof dark matter,Usp. Fiz. Nauk 184,3(2014). 26. A. Nusser,S.S. Gubser, andP.J. Peebles: Structure formation witha long-range scalar dark matter interaction, Phys. Rev.D 71, 083505 (2005). 27. W. A. Hellwing and R. Juszkiewicz: Dark matter gravitational clustering with a long- range scalar interaction, Phys. Rev.D 80, 083522 (2009). 28. S. Casas,V. Pettorino,C.Wetterich: Dynamicsof neutrino lumpsingrowing neutrino quintessence, Phys. Rev.D 94, 103518 (2016). 29. Y. Ayaita, M. Weber, C. Wetterich: Neutrino Lump Fluid in Growing Neutrino Quintessence, Phys. Rev.D 87, 043519 (2013). 30. A.E. Bernardini andO. Bertolami: Stabilityof mass varying particle lumps Phys.Rev.D 80, 123011 (2009), arXiv:0909.1541 [gr-qc] 31. M. Baldi,V. Pettorino,L. Amendola andC.Wetterich: Oscillating non-linear large-scale structures in growing neutrino quintessence, Mon. Not. R. Astron. Soc. 418, 214 (2011). 32. N.J. Nunes,L. Schrempp,C.Wetterich: Massfreezingingrowing neutrino quintessence, Phys. Rev. D83 083523 2011. 33. V. Pettorino, N. Wintergerst, L. Amendola, C. Wetterich: Neutrino lumps and the Cosmic Microwave Background, Phys. Rev.D 82, 123001 (2010). Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 146) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 10 The Problem of Particle-Antiparticle in Particle Theory FelixMLev Artwork Conversion Software Inc. 509 N. Sepulveda Blvd Manhattan Beach CA 90266 USA Email: felixlev314@gmail.com Abstract. The title of this workshop is: ”What comes beyond standardmodels?”. Standard models are based on standardPoincare invariant quantum theory (SQT). Here irreducible representations (IRs) of the Poincare algebra are such that in each IR, the energies are either . 0 or . 0. In the first case, IRs are associated with particles and in the second case — with antiparticles, while particles for which all additive quantum numbers (electric charge, baryon and lepton quantum numbers) equal zero are called neutral. However, SQT is a special degenerate case of finite quantum theory(FQT) in the formal limit p !. where p is a characteristic of a ring in FQT. In FQT, one IR of the symmetry algebra describes a particle and its antiparticle simultaneously, and there are no conservation laws of additive quantum numbers. One IR in FQT splits into two standardIRs with positive and negative energies asaresultof symmetrybreakingin the formal limit p !1. The construction of FQT is one of the most fundamental (if not the most fundamental) problems of particle theory. Povzetek: Standardni modeli temeljijo na standardni Poincarejevi invariantni kvantni teoriji (SQT). Nerazcepne upodobitve (IR) Poincarejeve algebre privzamejo, da imajo delci (fermioni) pozitivno energijo, antifermioni (antidelci) pa negativno energijo. Nevtralne imenujemo fermione, ki ne nosijo nabojev in je njihovo barionsko ali leptonsko ˇ stevilo enako niˇcne kvantne teorije (FQT)v limiti p !1, kjerjep c. SQT je poseben primer konˇ radij ustrezne sfere. FQT opiˇ se hkrati delce in antidelce in ne ohranja aditivnih kvantnih ˇce ne kar osnovni problem stevil.Avtor meni,daje konstruiranjeFQT eno najbolj nujnih, ˇ fizike osnovnih delcev. Keywords: irreducible representations, particle-antiparticle, de Sitter symmetry PACS numbers: 02.20.Sv, 03.65.Ta, 11.30-j, 11.30.Cp, 11.30.Ly 10.1 Introduction: problems with the physical interpretation of the Dirac equation Modern fundamental particle theories (QED, QCD and electroweak theory) are based on the concept of particle-antiparticle. Historically, this concept has arisen as a consequence of the fact that the Dirac equation has solutions with positive and negative energies. The solutions with positive energies are associated with particles, and the solutions with negative energies -with corresponding antiparticles. Andwhenthepositronwasfound,itwastreatedasagreat successofthe 10 The Problem of Particle-Antiparticle in Particle Theory 147 Dirac equation. Another great success is that in the approximation (v=c)2 the Dirac equation reproduces the fine structure of the hydrogen atom with a very high accuracy. However, now we know that there are problems with the physical interpretation of the Dirac equation. For example, in higher order approximations, the probabilistic interpretation of non-quantized Dirac spinors is lost because the coordinate description impliesthattheyare describedbyrepresentations inducedfrom non- unitary representations of the Lorenz algebra. Moreover, this problem exists not only for the Dirac spinors but for any functions described by relativistic covariant equations (Klein-Gordon, Dirac, Rarita-Schwinger and others).As shownby Pauli [1] in the case of fields with an integer spin there is no invariant subspace where the spectrum of the charge operator has a definite sign while in the case of fields witha half-integer spin thereis no invariant subspace where the spectrum of the energy operator has a definite sign. It is also known that the description of the electron in the external field by the Dirac spinor is not accurate (e.g., it does not take into account the Lamb shift). Another fundamental problem in the interpretation of the Dirac equation is as follows.Oneofthekey principlesof quantumtheoryisthe principleof superposition. This principle states that if 1 and 2 are possible statesofa physical system then c1 1 + c2 2, when c1 and c2 are complex coefficients, also is a possible state. The Dirac equation is the linear equation, and, if 1(x) and 2(x) are solutions of the equation, then c1 1(x)+ c2 2(x) alsoisa solution,in agreement with the principle of superposition. In the spirit of the Dirac equation, there should be no separate particles the electron and the positron. It should be only one particle which can be called electron-positron such that electron states are the states of this particle with positive energies, positron states are the states of this particle with negative energies and the superposition of electron and positron states should not be prohibited. However, in view of charge conservation, baryon number conservation, and lepton numbers conservations, the superposition of a particle and its antiparticle is prohibited. Modern particle theories are based on Poincare (relativistic) symmetry. In these theories, elementary particles are described by irreducible representations (IRs) of the Poincare algebra. Such IRs have a property that energies in them can be either strictly positive or strictly negative but there are no IRs where energies have different signs. The objects described by positive-energy IRs are called particles, and objects described by negative-energy IRs are called antiparticles, and energies of both, particles and antiparticles become positive after second quantization. In this situation, there are no elementary particles which are superpositions of a particle and its antiparticle, and as explained above, this is not in the spirit of the Dirac equation. In particle theories, only quantized Dirac spinors (x) are used. Here, by analogy with non-quantized spinors, x is treated as a point in Minkowski space. However, (x) is an operator in the Fock space for an infinite number of particles. Each particle in the Fock space can be described by its own coordinates (in the approximation when the position operator exists — see e.g., [3]). In view of this fact, the following natural question arises: why do we need an extra coordinate x which 148 FelixMLev does not have any physical meaning because it does not belong to any particle and so is not measurable? Moreover,I can ask the following seditious question: in quantum theory, do we need Minkowski space at all? When there are many bodies, the impression may arise that they are in some space but this is only an impression. In fact a background space-time (e.g., Minkowski space)is onlya mathematical concept neededin classical theory. For illustration, consider quantum electromagnetic theory. Here we deal with electrons, positrons and photons. In the approximation when the position operator exists, each particle canbe describedby its own coordinates. The coordinatesofthe background Minkowskispacedonothavea physicalmeaning becausetheydonotrefertoany particle and therefore are not measurable. However, in classical electrodynamics we do not consider electrons, positrons and photons. Here the concepts of the electric and magnetic fields (E(x);B(x)) have the meaning of the average contribution of all particles in the point x of Minkowski space. This situation is analogous to that in statistical physics. Here we do not consider each particle separately but describe the average contributionof all particlesby temperature, pressure etc. Those quantities have a physical meaning not for each separate particle but for ensembles of many particles. A justification of the presence of x in quantized Dirac spinors (x) is that in quantum field theories (QFT) the Lagrangian density depends on the four-vector x, but this is only the integration parameter which is used in the intermediate stage. The goal of the theory is to construct the S-matrix, and when the theory is already constructed one can forget about Minkowski space because no physical quantity depends on x. This is in the spirit of the HeisenbergS-matrix program according to whichinrelativistic quantum theoryitis possibleto describeonly transitionsof states from the infinite past when t › -. to the distant future when t !1. The fact that the theory gives the S-matrix in the momentum representation does not mean that the coordinate description is excluded. In typical situations, the position operatorin momentumrepresentation existsnotonlyinthe nonrelativistic case butin therelativistic case as well.In the latter case,itis known, forexample, as the Newton-Wigner position operator [3] or its modifications. However, as pointed out even in textbooks on quantum theory, the coordinate description of elementary particles can work only in some approximations. In particular, even in most favorable scenarios, for a massive particle with the mass m its coordinate cannot be measured with the accuracy better than the particle Compton wave length —h=mc. 10.2 Is Poincare symmetrythe most general symmetry in particle theory? The above discussion of the problems with Dirac spinors was based on the assumption that Poincare (relativistic) symmetry is the most general symmetry in particle theory, and StandardModel is based on this assumption. But suppose thatIaskaquestion:whynotto considerparticletheorybasedonGalilei(non- relativistic) symmetry? Probably, most physicists will immediately say that this question is silly because everybody knows that Poincaresymmetry is moregeneral 10 The Problem of Particle-Antiparticle in Particle Theory 149 (fundamental) than Galilei one and many facts in particle physics show that Galilei symmetrydoesnotworkhere.But supposethatI amnota physicist,Idonot know experimental data andIask whether the fact that Poincare symmetry is more general than Galilei one follows only from mathematics? Is this question legitimate? In his famous paper ”Missed Opportunities” [5] Dyson explains that the fact that Poincare symmetry is more general than Galilei one follows from pure mathematical considerations. The Poincare group is more symmetric that the Galilei one: the former contains a formal parameter c (I even do not discuss its physical meaning), and the latter can be obtained from the former by a procedure called contraction when formally c !1. In viewof this observation,I can ask whether Poincare symmetryis most general, maybe there are groups more symmetric that Poincare one such that the Poincare group can be obtained from these more symmetric groups by contraction? In his paper Dyson explains that indeed the de Sitter (dS) and anti-de Sitter (AdS) groups are more symmetric than Poincare one and the transition from the former to the latteris describedby contraction whena parameter R (see below) goes to infinity. At the same time, since dS and AdS groups are semisimple, they have a maximum possible symmetry and cannot be obtained from more symmetric groups by contraction. The paper[5] appearedin 1972, i.e.,50 years ago, and,in viewof Dyson’sresults, a question arises why the fundamental particle theories are still based on Poincare symmetry and not dS or AdS ones. The parameter R arises from particle theory but in the literature it is often interpreted as the radius of the universe. Probably, physicists believe that, since R is even much greater than sizes of stars, the dS and AdS symmetriescanplayan importantroleonlyin cosmologyandthereisnoneed tousethemfor describing elementary particles.Ibelievethatthisargumentisnot consistent because usually more general theories shed a new light on standard concepts, andmy talkisa good illustrationof this point. InSec.10.3Idescribethe conceptof symmetryon quantum level.InSecs.10.7and 10.8Iconsider the conceptof particle-antiparticle fordS and AdS symmetriesin standardquantum theoryandina quantum theorybased on finite mathematics (FQT). Here I give a popular explanation why standard concepts of particle- antiparticle, electric charge and baryon number have onlya limited meaning when the symmetry in FQT is broken to Poincare or standardAdS symmetries. Finally, Sec. 19.3is discussion.Idescribe all physical quantitiesin units c = h —= 1. 10.3 Symmetry on quantum level In the usual treatment of relativistic quantum theory, the approach to symmetry on quantum level follows. Since the Poincare group is the group of motions of Minkowski space, quantum states shouldbe describedby representationsof this group. This implies that therepresentation generators commute according to the 150 FelixMLev commutationrelationsof the Poincaregroup Lie algebra: P. -  [P;P]= 0, [P;M]=-i(P), [MM. + M. - M. - M) ;M]=-i((10.1) where , . = 0, 1, 2, 3, Pµ are the operators of the four-momentum, M. are the operators of Lorentz angular momenta, and . is such that 00 =-11 =-22 = -33 = 1 and . = 0 if µ = 6. This approach is in the spirit of Klein’s Erlangen program in mathematics. However,as noted in Sec. 18.2 and discussed in detail in [3], in quantum theory,the concept of space-time background does not have a physical meaning. As argued in[3,5],theapproach shouldbethe opposite.Each systemis describedby aset of linearly independent operators. By definition, the rules how they commute with each other define the symmetry algebra. In particular, by definition, Poincare symmetry on quantum level means that the operators commute according to Eq. (18.1). This definition does not involve Minkowski space at all. In particular, the fact that . coincides with the metric tensor in Minkowski space, does not imply thatthisspaceis involved.Iamvery gratefulto LeonidAvksent’evich Kondratyuk for explaining me this definition during our collaboration. By analogy with the definition of Poincare symmetry on quantum level, the definition of dS symmetry on quantum level should not involve the fact that the dS group is the group of motions of dS space. Instead, the definition is that the operators Mab (a, b = 0, 1, 2, 3, 4, Mab =-Mba)describing the system under consideration satisfy the commutation relations of the dS Lie algebra, i.e., [MabMbd + bdMac - adMbc - bcMad) ;Mcd]=-i(ac(10.2) -11 -22 -33 -44 where ab is such that 00 ===== 1 and ab = 0 if a 6b. The definition of AdS symmetry on quantum level is given by the same = equations but 44 = 1. The procedure of contraction from dS and AdS symmetries to Poincare one is defined as follows. If we define the operators P. as P. = M4=R where R is a parameter with the dimension length then in the formal limit when R !1, M4 !. but the quantities P. are finite, Eqs. (18.2) become Eqs. (18.1). This procedure is the same for the dS and AdS symmetries. The above contraction is analogous to the contraction from Poincare symmetry to Galilei one, where the parameter of contraction is c. On quantum level, R and c are onlythe parameters describing therelations between Lie algebrasof higher and lower symmetries. On classical level, the physical meaning of c is well-known, while R is the radius of the dS or AdS space.Adetailed discussion of the both contractions is described in a vast literature, in particular, in [3] where it has been proposed the following Definition: Let theoryAcontaina finite nonzero parameterand theoryBbe obtained fromtheoryAintheformallimitwhenthe parametergoestozeroorinfinity.Supposethat, withany desired accuracy, theoryA canreproduceanyresultof theoryBby choosinga valueof the parameter.On the contrary, when the limitis already taken, one cannotreturn backto theoryA,and theoryBcannotreproduceallresultsof theoryA. Then theoryA 10 The Problem of Particle-Antiparticle in Particle Theory 151 is more general (fundamental) than theoryBand theoryBisa special degenerate caseof theoryA. As proved in [3], dS and AdS symmetries are more general (fundamental) than Poincare symmetry. The latter is a special degenerate case of the former in the formal limit R !1. As noted above, in contrast to Dyson’s approach based on Lie groups, our approach is based on Lie algebras. Then, as proved in [3], classical theory is a special degenerate case of quantum one in the formal limit h —› 0, and nonrelativistictheory(NT)isaspecial degenerate caseofrelativisticone(RT)inthe formal limit c !1. In the literature the above facts are explained from physical considerations but, as shown in [3] they can be proved mathematically by using properties of Lie algebras. In particular, since, from mathematical point of view, de Sitter symmetry is more general (fundamental) than Poincare one, there should exist physical phenomena which can be explained by de Sitter symmetries but cannotbe explainedby Poincare symmetry. BelowIwill discuss such phenomena. 10.4 Problems with describing natureby classical mathematics Standardquantumtheory(SQT)isbasedon classical mathematics involvinglimits, infinitesimals, continuity etc. Mathematical education at physics departments develops a belief that classical mathematics is the most fundamental mathematics, while, for example, discrete and finite mathematics is something inferior what is used only in special applications. And many mathematicians have a similar belief. Historically it happened so because more than 300 years ago Newton and Leibniz proposed the calculus of infinitesimals, and, since that time, a titanic work has been done on foundation of classical mathematics. This problem has not been solved till the present time, but for most physicists and many mathematicians the most important thing is not whether a rigorous foundation exists but that in many cases standardmathematics works with a very high accuracy. The ideaof infinitesimals wasin the spiritof existed experience that any macroscopic object can be divided into arbitrarily large number of arbitrarily small parts, and, even in the 19th century, people did not know about atoms and elementary particles.Butnowweknowthatwhenwereachthelevelof atomsand elementary particles, standarddivision loses its usual meaning and in nature there are no arbitrarily small parts and no continuity. For example, typical energies of electrons in modern accelerators are millions of times greater than the electron rest energy, and such electrons experience many collisions with different particles. If it werepossible to break the electron into parts, then it would have been noticed long ago. Another exampleisthatifwedrawalineona sheetofpaperandlookatthisline by a microscope then we will see that the line is strongly discontinuous because it consists of atoms. That is why standardgeometry (the concepts of continuous lines and surfaces) can work well onlyinthe approximation when sizesof atoms are neglected, standardmacroscopictheory can work wellonlyin this approximation etc. Of course, when we consider water in the ocean and describe it by differential equations of hydrodynamics, this works well but this is only an approximation 152 FelixMLev since water consists of atoms. However, it seems unnatural that even quantum theory is based on continuous mathematics. Even the name ”quantum theory” reflects a belief that nature is quantized, i.e., discrete, and this name has arisen because in quantum theory some quantities have discrete spectrum (i.e., the spectrum of the angular momentum operator,the energy spectrum of the hydrogen atom etc.). But this discrete spectrum has appeared in the framework of classical mathematics. Iasked physicists and mathematicians whether, in their opinion, the indivisibility of the electron shows that in nature there are no infinitesimals, and standard division does not work always. Some mathematicians say that sooner or later the electron will be divided. On the other hand, as a rule, physicists agree that the electron is indivisible and in nature there are no infinitesimals. They say that, for example, dx=dt should be understood as x=t where x and t are small but not infinitesimal.I ask them: but you work with dx=dt, not x=t. They reply that since mathematics with derivatives works well then there is no need to philosophize and develop something else (and they are not familiar with finite mathematics). One of the key problems of modern quantum theory is the problem of infinities: the theory gives divergent expressions for the S-matrix in perturbation theory. Inrenormalized theories, the divergencies are eliminatedby therenormalization procedure where finite observable quantities are formally expressed as products of singularities. Although this procedure is not well substantiated mathematically, in some cases it results in excellent agreement with experiment. Probably the most famous case is that the results for the electron and muon magnetic moments obtained at the end of the 40th agree with experiment at least with the accuracy of eight decimal digits (see, however, a discussion in [6]). In view of this and other successes of quantum theory, most physicists believe that agreement with the data is much more important than the rigorous mathematical substantiation. At the same time, in nonrenormalized theories, infinities cannot be eliminated by the renormalization procedure, and this a great obstacle for constructing quantum gravity based on quantum field theory (QFT). As the famous physicist and the Nobel Prize laureate StevenWeinbergwritesin his book [7]: ”Disappointingly this problem appeared with even greater severity in the early days of quantum theory, and althoughgreatly amelioratedby subsequent improvementsin the theory,itremains with us to the present day”. The titleofWeinberg’s paper[8]is ”Living with infinities”. In viewofeffortsto describe discrete naturebycontinuous mathematics,my friend toldmethe followingjoke:”Agroupof monkeysisorderedtoreachthe Moon. For solving thisproblem each monkey climbsatree.The monkeywhohasreached the highest point believes that he has made the greatest progress and is closer to the goal than the other monkeys”. Is it reasonable to treat this joke as a hint on some aspects of the modern science? Indeed, people invented continuity and infinitesimals whichdonot existin nature,createdproblemsfor themselvesand now apply titanic efforts for solving those problems. The founders of quantum theory and scientists who essentially contributed to it were highly educated. But they used only classical mathematics, and even now finite mathematics is not a part of standardeducation for physicists. The 10 The Problem of Particle-Antiparticle in Particle Theory 153 development of quantum theory has shown that the theory contains anomalies and divergences. Most physicists considering those problems, worked in the framework of classical mathematics and did not acknowledge that they arise just because this mathematics was used. 10.5 Quantum theory based on finite mathematics Several well-known physicists, including the Nobel Prize laureates Gross, Nambu and Schwinger, discussed approaches when quantum theory involves finite mathematics. While classical mathematics starts from the ring of integers Z = (-1, :::- 1, 0, 1, :::1), finite mathematics rejects infinities from the beginning. It starts from the ring Rp =(0, 1, 2, :::p - 1) where addition, subtraction and multiplication are performed as usual but modulo p, and p is called the characteristic of the ring. In number theory, p is the usual notation forthe characteristic and this has nothing to do with the fact that in particle theory the notation p is used for denoting a particle four-momentum. Since the operations in Rp are modulo p, then, if p is odd, one can say that Rp contains the numbers (-(p - 1)=2, ::. - 1, 0, 1, :::(p - 1)=2). Then, if elements of Z are depicted as integer points on the x axis of the xy plane, the elements of Rp can be depictedas pointsofthecircleinFigure1and analogouslyif p is even. Fig. 10.1: Relation between Rp and Z The analogy between Rp and the circle follows from the following observations. If we take an element of Rp and successively add1 to it, then after p steps we will return to the original element because addition in Rp is modulo p. This is analogous to the fact that if we are moving along the circle in the same direction then, sooner or later, we will arrive to the initial point. Figure1is naturalfrom the following historical analogy. For many years people believed that the Earth was flat and infinite, and only aftera long periodof time they realized that it was finite and curved. It is difficult to notice the curvature 154 FelixMLev when we deal only with distances much less than the radius of the curvature. Analogously, when we deal with numbers the modulus of which is much less than p,theresultsarethe samein Z and Rp, i.e., we do not notice the ”curvature” of Rp. This ”curvature” is manifested only when we deal with numbers the modulus of which is comparable to p. As proved in my book [3], as follows from Definition, classical mathematics (involvingtheconceptsoflimits, infinitesimals,continuityetc.)isaspecialdegenerate caseof finite mathematicsinthe formal limit when the characteristic p of the ring orfieldinthelattergoestoinfinity.Therefore standarddSandAdSsymmetries overthe fieldof complex numberscanbe generalizedtodSandAdS symmetries over a finite ring or field of characteristic p. We use the abbreviation FQT (finite quantum theory) to denote quantum theory over the ring or field of characteristic p. Since mathematically FQT is more general (fundamental) than SQT, there are physical phenomena which can be explain only by FQT but cannot be explained by SQT. An example of such a phenomenon is discussed in Sec. 10.8, for other examples —see [3]. 10.6 Particles and antiparticles in Poincare invariant theories As noted in Sec. 18.2, solutions of the Dirac equation with positive energies are associated with particles and solution with negative energies — with antiparticles. It has been noted that there are problems with the interpretation of the non- quantized Dirac spinor (x) and for the quantized Dirac spinor the problem is that the quantity x does not have the physical meaning. Elementary particles in Poincare invariant theory are described by IRs of the Poincare algebra by selfadjoint operators. Thereforeaproblem arises whether the conceptof particle- antiparticle can be defined proceeding only from such IRs without mentioning the nonphysical parameter x. Let p. be the four-momentum of a particle in Poincare invariant theory. Define 2 p= pp,whereasum overrepeated indicesis assumed.Thenforusual particles p2 . 0 while for tachyons p2 <0. The existence of tachyons is a problem, and we will consider only usual particles. Then the mass of the particle can be defined as 22 a nonnegative number m such that m= p. 22)1=2 The energy E ofaparticle with the momentum p and mass m equals (m+p. The choiceof the signof the squarerootis only the matterof convention but not the matter of principle. Depending on this sign, there are IRs where energies can be only either positive or negative while the probability to have zero energy is zero. Whenweconsiderasystemconsistingofparticlesand antiparticlesthentheenergy sign of both, particles and antiparticles should be the same. Indeed, consider, for example a system of two particles with the same mass m and let the momenta p1 and p2 be such that the total momentum p1 + p2 equals zero. Then, if the energyof particle1is positive, and the energyof particle2is negative then the total four-momentum of the system would be zero what contradicts experimental data. By convention, the energy sign of all particles and antiparticles in question is chosen to be positive. For this purpose, the procedure of second quantization 10 The Problem of Particle-Antiparticle in Particle Theory 155 is defined such that after the second quantization the energies of antiparticles become positive. Then the massof any particleisthe minimum valueof its energy in the case when the momentum equals zero. Suppose now that we have two particles such that particle1has the mass m1, spin s1 andis characterizedby some additional quantum numbers (e.g., electric charge, baryon quantum number etc.), and particle2has the mass m2, spin s2 = s1 and all additional quantum numbers characterizing particle2equal the corresponding additional quantum numbers for particle1with the opposite sign.Aquestion ariseswhen particle2canbetreatedasan antiparticlefor particle1.Isit necessary that m1 should be exactly equal m2 or they can slightly differ each other? In particular, can we guarantee that the mass of the positron exactly equals the mass of the electron, the mass of the proton exactly equals the mass of the antiproton etc.? If particle2(for somereasons)istreatedasan antiparticlefor particle1,andthe particles are considered only on the level of IRs, then the relation between m1 and m2 is fully arbitrary. However, in QFT, m1 = m2 because IRs for a particle and its antiparticle are combined together in the framework of a local field. For example, the Dirac spinor combines together two IRs for the electron and positron. However, as noted in Sec. 18.2, this procedure encounters the following problems: • The quantity x in quantized fields (x) does not have a physical meaning. • There is no probabilistic interpretation of (x) because it is described by a non-unitary representation of the Poincare algebra. • Although (x) satisfies a linear equation, a superposition of solutions with positive and negative energies is prohibited. Ausual statement in the literature is that in QFT the fact thatm1 = m2 follows from the CPT theorem which is a consequence of locality since we construct local covariant fields from a particle and its antiparticle with equal masses. However, as noted in Sec. 18.2, since on quantum level there are problems with the physical interpretation of covariant fields and the quantity x, the very meaning of locality on quantum level is problematic. Also, a question arises what happens if locality is only an approximation: in that case the equality of masses is exact or approximate? Consider a simple model when electromagnetic and weak interactions are absent. Then the fact that the proton and the neutron have equal masses has nothing to do with locality; it is only a consequence of the fact that the proton and the neutron belong to the same isotopic multiplet. In other words, they are simplydifferent states of the same object—the nucleon. Since the concept of locality is not formulated in terms of selfadjoint operators, this concept does not havea clear physical meaning, and this fact has been pointed out eveninknown textbooks(seee.g.[9]).Therefore,QFTdoesnotgivea physical proof that m1 = m2. Note also that in Poincare invariant quantum theories there can exist elementary particles for which all additional quantum numbers are zero. Such particles are called neutral because they coincide with their antiparticles. InSecs.10.7and10.8Iconsiderhowthe conceptof particle-antiparticleintreated for dS and AdS invariant theories, respectively. 156 FelixMLev 10.7 Particles and antiparticles in dS invariant theories The descriptions of elementary particles in the dS and AdS cases are considerably different. In the former case all the operators M4 (. = 0, 1, 2, 3)are on equal footing. Therefore, M04 can be treated as the Poincare analog of the energy only in the approximation when R is rather large. In the general case, the sign of M04 cannot be used for the classification of IRs. In his book[7] Mensky describes the implementationofdS IRs when therepresentation spaceisthe three-dimensional unit spherein the four-dimensional space. In this implementation, there exist one-to-one relations between the northern hemisphere and the upper Lorentz hyperboloid with positive Poincare energies and between the southern hemisphere and the lower Lorentz hyperboloid with negative Poincare energies, while points on the equator correspond to infinite Poincare energies. However, the operators of IRs are not singular in the vicinity of the equator and, since the equator has measure zero, the properties of wave functions on the equator are not important. Since the number of states in dS IRs is twice as big as the number of states in IRs of the Poincare algebras, one might think that each IR of the dS algebra describes a particle and its antiparticle simultaneously. However, a detailed analysis in [3] shows that states described by dS IRs cannot be characterized as particles or antiparticles in the usual meaning. For example, let us call states with the support of their wave functions on the northern hemisphereas particles and states with the support on the southern hemisphereas their antiparticles.Then states whichare superpositionsofa particleand its antiparticle obviously belong to the representation space under consideration, i.e., they are not prohibited. As noted in Sec. 18.2, in the spirit of the Dirac equation, thereshould be no separate particles the electron and the positron. It should be only one particle which can be called electron-positron such that electron states are the states of this particle withpositive energies,positron statesarethe statesofthisparticlewith negative energies and, as follows from the principle of superposition in quantum theory,the superposition of electron and positron states should not be prohibited. However, since in standard particle theory, charge conservation is treated as more fundamental than the principleof superposition, the superpositionofa particle and its antiparticleis prohibited. However, we see that in the dS case the situation is in the spirit of the Dirac equation: there are no independent particles and antiparticles, there are only objects describedbyIRsofthedS algebra,and,if statesofeachobjectwith positive energies are called particle states and states with negative energies — antiparticle states, superpositions of such states are not prohibited. Therefore, in the dS case, the principle of superposition is stronger than the electric charge conservation. Note that the law of electric charge conservation comes from classical physics. The existing experimental data confirms that this law takes place. However, a problem arises whether those data describe all possible situations.We discuss thisproblem below. As noted in Sec. 10.3, dS symmetry is more general than Poincare one, and the latter canbetreatedasa special degenerate caseofthe formerinthe formal limit 10 The Problem of Particle-Antiparticle in Particle Theory 157 R !1. This means that, with any desired accuracy, any phenomenon described in the framework of Poincare symmetry can be also described in the framework of dS symmetry if R is chosentobesufficientlylarge,buttherealso existphenomena for explanation of which it is important that R is finite and not infinitely large (see [3]). As shown in [3, 9], dS symmetry is broken in the formal limit R !. because one IR of the dS algebra splits into two IRs of the Poincare algebra with positive and negative energies and with equal masses. Therefore, the fact that the masses of particles and their corresponding antiparticles are equal to each other, can be explained as a consequence of the fact that observable properties of elementary particles can be described not by exact Poincare symmetry but by dS symmetry withaverylarge(but finite) valueof R.In contrasttoQFT,for combiningaparticle and its antiparticle into one object, there in no need to assume locality and involve local field functions because a particle and its antiparticle already belong to the same IR of the dS algebra (compare with the above remark about the isotopic symmetry in the proton-neutron system). The fact thatdS symmetryis higher than Poincare oneis clear evenfrom the fact that, in the framework of the latter symmetry, it is not possible to describe states which are superpositions of states on the upper and lower hemispheres. Therefore, breaking the IR into two independent IRs defined on the northern and southern hemispheres obviously breaks the initial symmetry of the problem. This fact is in agreement with the Dyson observation (mentioned above) that dS group is more symmetric than Poincare one. When R !1, standard concepts of particle-antiparticle, electric charge and baryonand leptonquantum numbersarerestored, i.e.,in this limit, superpositions of particle and antiparticle become prohibited because now a particle and its antiparticle belong to different IRs. Therefore, those concepts arise as a result of symmetry breaking, i.e., they are not universal. 10.8 Particles and antiparticles in AdS invariant theories In theories where the symmetry algebra is the AdS algebra, the structure of IRs is known(see e.g.,[3,12]).The operator M04 P is the AdS analog of the energy operator. 1 Let W be the Casimir operator W = MabMab where a sum over repeated 2 indices is assumed. As follows from the Schur lemma, the operator W has only one eigenvalueineveryIR.Byanalogywith Poincareinvarianttheory,wewillnot consider AdS tachyons and then one can define the AdS mass µ such that µ . 0 and 2 is the eigenvalue of the operator W. As noted in Sec. 10.3, the procedure of contraction from the AdS algebra to the Poincare one involves the definition of P. such that M4 = RP. This relation has a physical meaning only if R is rather large. In that case the AdS mass µ and the Poincare mass m are related as µ = Rm, and the relation between the AdS and Poincare energies is analogous. Since AdS symmetry is more general (fundamental) then Poincare one then µ is more general (fundamental) than m. In contrast to the Poincare masses and energies, the AdS masses and energies are dimensionless. From cosmological considerations (see e.g., [3]), the value of R is 158 FelixMLev usually accepted to be of the order of 1026 m. Then the AdS masses of the electron, the Earth and the Sun are of the order of 1039 , 1093 and 1099 , respectively. The fact that even the AdS mass of the electron is so large might be an indication that the electron is not a true elementary particle. In addition, the present accepted upper level for the photon mass is 10-17ev. This value seems to be an extremely tiny quantity. However, the corresponding AdS mass is of the order of 1016, and so, even the mass which is treated as extremely small in Poincare invariant theory might be very large in AdS invariant theory. In the AdS case there are IRs with positive and negative energies, and they belong to the discrete series [3,12]. Therefore, by analogy with standardparticle theory, one can define particles and antiparticles. Consider first the construction of positive energy IRs.We startfrom ”therest state” where the AdS energy equals the AdS mass 1. Then we obtain the states with the AdS energies 1;1 + 1, 1 + 2, :::. (see Figure 2). Analogously, if 2 is the AdS mass of the antiparticle, we start from the state where the energy equals -2 and obtain the states with the AdS energies -2, -2 -1, -2 -2, :::-1.(seeFigure2)Therefore,the situationisprettymuch analogous to that in Poincare invariant theories, and there is no way to conclude whether the massofa particle equals the massof the corresponding antiparticle. In view of the results in this and preceding sections, we conclude that the descriptions of elementary particles in the cases of dS and AdS symmetries are considerably different. In the dS case, one IR describes particle and antiparticle states simultaneously and their superpositions are not prohibited, i.e. the principle of superposition is more fundamental than the conservation of electric charge and other additive quantum numbers. On the other hand, in the AdS case, the situation is analogous to that in Poincare invariant theories; in particular the electric charge conservation is more fundamental than the principle of superposition. So,a question arises whichof those possibilitiesinSQTis more physical. However, as discussed in [3], FQT is more general (fundamental) than SQT, in FQT it is also possible to define the concepts of dS and AdS symmetries and herethe dS and AdS cases are physically equivalent. Below we will consider a direct generalization of the AdS symmetry from SQT to FQT. The description of the energy spectrum in standardIRs of the AdS algebra has beengiven above.WewillnowexplainwhyinFQTthespectrumisdifferent,and in FQT the situation is similar to that in standarddS case but not standardAdS one because IRs in FQT contain both, positive and negative energies. Letus note first that, while in SQT the quantity µ can be an arbitrary real number, in FQT µ is an element of Rp. As noted above, if p is odd then Rp contains the elements -(p - 1)=2, ::. - 1, 0, 1, :::(p - 1)=2 (see Figure 1) and the case when p is even is analogous. For definiteness, we consider the case when p is odd. Byanalogy with the construction of positive energy IRs in SQT,in FQT we start the construction from ”the rest state”, where the AdS energy is positive and equals . Then we act on this stateby raising operators and gradually get states with higher and higher energies, i.e., µ + 1, µ + 2, :::. However, now we are moving not along the straightlinebutalongthecircleinFigure1and,in contrasttothe situationin SQT, we cannot obtain infinitely large numbers. When we reach the state with the energy (p - 1)=2, the next state has the energy (p - 1)=2 + 1 =(p + 1)=2 and, since 10 The Problem of Particle-Antiparticle in Particle Theory 159 Fig. 10.2: Spectrum of Energies of Elementary Particle the operations are modulo p, this value also can be denoted as -(p - 1)=2 i.e., it may be called negative. When this procedure is continued, one gets the energies -(p - 1)=2 + 1 = -(p - 3)=2, -(p - 3)=2 + 1 = -(p - 5)=2, ::. and, as shown in [3], the procedure finishes when the energy -µ isreached (see Figure 2). Therefore, in contrast to the situation in SQT, in FQT IRs are finite-dimensional (and even finite since the ring Rp and its complex extension Rp + iRp are finite). By analogy with the dS case in SQT, one can say that the states with the energies , +1, +2, ::. refertoaparticleand stateswiththe energies:::--2, --1, -µ —toan antiparticle.Therefore,inFQTthe massofa particle automaticallyequals the mass of the corresponding antiparticle. This is an example when FQT can solve a problem which standardquantum AdS theory cannot. By analogy with the situation in the dS case, for combining a particle and its antiparticle together, there is no need to involve additional coordinate fields because a particle and its antiparticle are already combined in the same IR. Then, since states which are superpositions of particles and antiparticles belong to therepresentationspace,we concludebyanalogywiththe situationinSec.10.7, that in FQT there are no superselection rules which prohibit superpositions of states with opposite electric charges, baryon quantum numbers etc. Moreover, the representation operators of the enveloping algebra can perform transformations particle - antiparticle. As shown in Ref. [3], in the formal limit p !1, one IR in FQT splits into two standardIRs of the AdS algebra with positive and negative energies. This result seems naturalfrom Figure2 sincethe spectrumof positive energies becomes , µ + 1, µ + 2, :::. and the spectrum of negative energies becomes -1, ::. - µ - 160 FelixMLev 2, -µ - 1, -µ by analogy with the spectrum in SQT (see Figure 2). Therefore, in this limit the conceptof particle-antiparticle and the superselectionruleshave the usual meaning. In turn, in situations when one can define the quantity R such that the contraction to the Poincare algebra works witha high accuracy, one can describe particles and antiparticles in the framework of Poincare symmetry. Even from the fact that in standardquantum theory, there are no superpositions of statesbelongingtoaparticleandits antiparticle,itisclearthatsymmetry described by one IR in FQT is higher than symmetry described by two IRs obtained from one IR in FQT in the formal limit p !1. Therefore standardconcepts of particle- antiparticleand superselectionrules ariseasaresultof symmetrybreaking,i.e., they are not universal. 10.9 Discussion As explained in Sec. 10.6, in quantum theory based on Poincare symmetry, the concept of particle-antiparticle arises because IRs have the property that energies in them can be either positive or negative, and thereareno IRs whereenergies have different signs. Then IRs with positive energies are associated with particles and IRs with negative energies — with antiparticles, and superpositions of particles and antiparticles are prohibited because they belong to different IRs. As shown in Sec. 10.8, in SQT based on AdS symmetry, the situation is analogous. On the other hand, as shown in Secs. 10.7 and 10.8, in SQT based on dS symmetry and in FQT, IRs contain states with both, positive and negative energies. If states with positive energies are called particle states and states with negative energies — antiparticle states then their superpositions are not prohibited because they belong to the same IR. The principle of superposition is a fundamental principle of quantum theory but in SQT based on Poincare and AdS symmetries, superpositions of particles and antiparticles are prohibited because they contradict the electric charge conservation, baryon number conservation etc. Therefore, in those cases, e.g., the electric charge conservation is treated as more fundamental than the principle of superposition but in SQT based on dS symmetry and in FQT the situation is the opposite. One might think that for this reason the latter theories are not physical but in fact they are more physical than the former theories. The matter is that, as explained in Secs. 10.7 and 10.8: • StandardPoincare invariant theory arises as a result of symmetry breaking at R !. in dS invariant quantum theory because in this limit one IR in the latter splits into two IRs in the former. • StandardPoincare and AdS invariant theories arise as a result of symmetry breaking at p !. in FQT because in this limit one IR in the latter splits into two IRs in the former. Then experimentally the electric charge conservation, baryon number conservation etc. are observed with a very high accuracy as a consequence of the fact that at the present stage of the universe the quantities R and p are extremely high and then standardquantum theory based on Poincare symmetry works with a very 10 The Problem of Particle-Antiparticle in Particle Theory 161 high accuracy. However, there are reasons to think [3] that at early stages of the universe those quantities were much less than now . That is why at those stages the conservation of the electric charge and baryon quantum number did not take place.Asarguedin[13],thisisthereasonofthebaryon asymmetryofthe universe. The present fundamental particle theories are based on Poincare invariant QFT, and,asnotedinSec.10.6,forsolvingtheproblemwhyaparticleandits antiparticle have equal masses, those theories involve local quantized field (x) where x does not belong to any particle and is simply a parameter arising from the second quantization of a non-quantized field. So, the physical meaning of x is not clear. Although QFT has many successes, it also has problems because, as noted, for example, in the textbook [9], (x) is an operatorial distribution, and the product of distributions at the same point is not a well defined mathematical operation. As explained in Secs. 10.7 and 10.8, in quantum theories based on dS symmetry and FQT, the masses of a particle and the corresponding antiparticle are automatically equal, and thisis achieved without introducing local quantized fields. However, as noted above, in those theories the concepts of particle-antiparticle and additive quantum numbers differ from standardones because one IR combines together a particle and its antiparticle. The construction of such theories is one of the most fundamental (if not the most fundamental) problems of particle theory. References 1. W. Pauli: The connection between spin and statistics, Phys. Rev. 58, 716–722 (1940). 2. F. Lev: Finite mathematics as the foundation of classical mathematics and quantum theory.With applicationtogravityandparticletheory.ISBN 978-3-030-61101-9.Springer, https://www.springer.com/us/book/9783030611002 (2020). 3.T.D. NewtonandE.P.Wigner: Localized Statesfor Elementary Systems,Rev.Mod.Phys. 21, 400–405 (1949). 4. F.G. Dyson: Missed Opportunities. Bull. Amer. Math. Soc., 78, 635–652 (1972). 5.F.M.Lev:de SitterSymmetryand QuantumTheory,Phys.Rev.D85, 065003 (2012). 6. O. Consa: Something is wrong in the state of QED, arXiv preprint (2021), https://arxiv.org/abs/2110.02078. 7. S.Weinberg: The Quantum Theoryof Fields,Vol.I, Cambridge UniversityPress, Cambridge, UK, 1999. 8. S. Weinberg: Living with Infinities, arXiv preprint (2009), https://arxiv.org/abs/0903.0568. 9. N.N. Bogolubov, A.A. Logunov, A.I. Oksak and I.T.Todorov: General Principles of Quantum Field Theory, Nauka: Moscow (1987). 10. F.M. Lev: Could Only Fermions Be Elementary? J. Phys. A: Mathematical and Theoretical, A37, 3285-3304 (2004). 11. M.B. Mensky: Method of Induced Representations. Space-time and Concept of Particles. Moscow: Nauka (1976) 12. N.T. Evans: Discrete SeriesfortheUniversal CoveringGroupofthe3+2de SitterGroup, J. Math. Phys. 8, 170-184 (1967). 13. F.M. Lev: The Concept of Particle-Antiparticle and the Baryon Asymmetry of the Universe, Physics of Particles and Nuclei Letters, 18, 729-737 (2021). Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 162) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 11 Clifford odd and even objects, offering description of internal space of fermion and boson fields, respectively, open new insight into next step beyond standard model N. S. Mankoˇc Borˇstnik Department of Physics, University of Ljubljana SI-1000 Ljubljana, Slovenia norma.mankoc@fmf.uni-lj.si Abstract. In a long series of works the author demonstrated, together with collaborators, that the model named the spin-charge-family theory offers the explanation for all in the standard model assumed properties of fermion and boson fields, with the families of fermions and the Higgs’s scalars included. The theory starts with a simple action in . (13 + 1)dimensional space-time with massless fermions which interact with massless gravitational fields only (vielbeins and the two kinds of spin connection fields). The internal spaces of fermion and boson fields are describedby the Cliffordodd and even objects,respectively. The corresponding odd and even ”basis vectors” in a tensor product with the basis in ordinary momentum or coordinate space define the creation and annihilation operators, which explain the second quantization postulates for fermion and boson fields. The break of the starting symmetry leads at low energies to the action for families of quarks and leptons and the corresponding gauge fields, with Higgs’s fields included, offering several predictions and several explanations of the observed cosmological phenomena. The properties of the odd dimensional spaces are also discussed. Povzetek:V dolgem nizu ˇ clankov je avtorica, skupaj s sodelavci, pokazala, da ponuja model, ki ga avtorica poimenuje teorija spinov-nabojev-druˇzin, razlago za vse v standardnem modelu privzete lastnosti fermionskih in bozonskih polj, vklju ˇzinami fermionov in cnoz dru ˇ Higgsovimi skalarji.Teorijapredpostavipreprosto akcijov . (13 + 1)-razseˇ znem prostoru- ˇ casu, v kateri fermioni nimajo mase, interagirajo pa samo z brezmasnim gravitacijskim poljem (tetradani, ki doloˇcajnem prostoru in dvema vrstama cajo gravitacijsko poljev obiˇ spinskih povezav,ki so umeritvena polja Lorentzovih transformacijv notranjemprostoru fermionov). Notranji prostor fermionov opiˇcnimi vektorji”, ki so lihi se avtorica z ”baziˇ objekti Clifordove algebre, notranjiprostor bozonovpas Cliffordovo sodimi objekti. Us- trezni lihi in sodi ”baziˇ cni vektorji” v tenzorskem produktu z bazo v prostoru gibalnih koliˇcih fermionskih polj cin definirajo kreacijske in anihilacijske operatorje antikomutirajoˇ in komutirajoˇ cih bozonskih polj, kar pojasni postulate za drugo kvantizacijo za fermionska in bozonska polja. Zlomitev zaˇ cetne simetrije akcije vodi pri nizkih energijah do akcije kot jo predpostavi standardni model— za druˇ zine kvarkov in leptonov in za ustrezna umeritvena polja ter za Higgsove skalarje.Teorija ponuja ˇ stevine napovedi in pojasni vzroke za kozmoloˇzenja. Predstavi tudi lastnosti Cliffordovih objektov v prostorih z lihim ska opaˇ ˇ stevilom dimenzij. Title Suppressed Due to Excessive Length 163 Keywords: Second quantization of fermion and boson fields in Cliffordspace; beyond the standard model; Kaluza-Klein-like theories in higher dimensional space, explanation of appearance of families of fermions, scalar fields, fourth family, dark matter. 11.1 Introduction The standard model (with massive neutrinos added) has been experimentally confirmed without raising any serious doubts so far on its assumptions, whichremain unexplained 1. The assumptions of the standard model has in the literature several explanations, mostly with many new not explained assumptions. The most popular seem to be the grand unifying theories([1–6]. Among the questions for which the answers are needed are: i. Where do fermions, quarks and leptons, originate? ii. Why do family members, quarks and leptons, manifest so different masses if they all start as massless? iii. Why are charges of quarks and leptons so different and why have the left handed family members so different charges from the right handed ones? iv. Where do antiquarks and antileptons originate? v. Where do families of quarks and leptons originate and how many families do exist? vi. What is the origin of boson fields, of vector fields which are the gauge fields of fermions? vii. Whatis the originof the Higgs’s scalars and theYukawa couplings? viii. How are scalar fields connected with the origin of families and how many scalar fields determine properties of the so far (and others possibly be) observed fermions and of weak bosons? ix. Why have the scalar fields half integer weak and hyper charge? Do possibly exist also scalar fields with the colour chargesin the fundamentalrepresentation? ix. Could all boson fields, with the scalar fields included, havea common origin? x. Where does the dark matter originate? Does the dark matter consist of fermions? xi. Where does the ”ordinary” matter-antimatter asymmetry originate? xii. Where does the dark energy originate? xiii. How can we understand the postulates of the second quantized fermion and boson fields? xiv. What is the dimension of space? (3 + 1)?, ((d - 1)+ 1)?, 1? xv. Are all the fields indeed second quantized with the gravity included? And consequently are all the systems second quantized (although we can treat them in simplified versions, like it is the first quantization and even the classical treatment), with the black holes included? xvi. And many others. 1 This introduction is similar to the one appearing in the arxiv:2210.07004. Also most of sections and subsections are similar. There are, however, some new parts added. 164 N. S. Mankoˇc Borˇstnik Ina long seriesof works([1–3,5,23,25,27–29,31,32] and thereferences therein), the author has succeeded, together with collaborators, to find the answer to many of the above, and also to other open questions of the standard model, as well as to several open cosmological questions, with the model named the spin-charge-family theory. The more work is put into the theory the more answers the theory offers. The theory assumes that the space has more than (3 + 1) dimensions, it must have d . (13 + 1), so that the subgroups of the SO(13, 1) group, describing the internal space of fermions by the superposition of odd products of the Cliffordobjects a’s, manifest from the point of view of d =(3 + 1)-dimensional space the spins, handedness and charges assumed for massless fermions in the standard model. Correspondingly each irreducible representation of the SO(13, 1) group carrying the quantum numbers of quarks and leptons and antiquarks and antileptons, represents one of families of fermions, the quantum numbers of which aredetermined by the second kind of the Cliffordobjects, by ~ a (by S~ ab (= i { ~a ; ~bg-). 4 Fermions interact in d =(13 + 1) with gravity only,with vielbeins (the gauge fields of momenta) and the two kinds of the spin connection fields, the gauge fields of the two kinds of the Lorentz transformations in the internal space of fermions, of Sab(= i f a; bg-) and of S~ ab (= i { ~a ; ~bg-). 44 The theory assumesa simple starting action([5]andthereferences therein)forthe second quantized massless fermion and antifermion fields, and the corresponding massless boson fields in d = 2(2n + 1)-dimensional space Z 1 dd( — 2 A = xE a p0a )+ h:c. + Z ddxE ( R + . ~R~ ) , p0a = f ap0. + 1 fp , Ef ag- , 2E 11 SabS~ ab ~ p0. = p. - !ab. - !ab. , 22 R = 1 ff [af b] (!ab ;ß - !ca. !cb )} + h:c. , 2 ~ff [af b] ( ~!c R = 1!ab ;ß - !~ca. ~b )} + h:c. . (11.1) 2 = f af b - f bf a Here 2 f [af b] . fa , and the two kinds of the spin connection fields, !ab. (the gauge fields of Sab)and !~ab. (the gauge fields of S~ ab), manifest in d =(3 + 1) as the known vector gauge fields and the scalar gauge fields taking 2 a aaß f a are inverted vielbeins to e . with the properties e f b = ab;e f a =  , E = det(ea ). Latin indices a, b, ::, m, n, ::, s, t, :. denote a tangent space (a flat index), while Greek indices , , ::, , , ::, , :. denoteanEinstein index(a curved index). Lettersfrom the beginningof boththe alphabets indicatea general index(a, b, c, :. and , , , :. ), from the middle of both the alphabets the observed dimensions 0, 1, 2, 3 (m, n, :. and , , ::),indexesfrom the bottomof the alphabets indicate the compactified dimensions (s, t, :. and , , ::).We assume the signature ab = diagf1, -1, -1, · , -1g. Title Suppressed Due to Excessive Length 165 careof massesofquarksandleptonsand antiquarksandantileptonsandtheweak boson fields [27] 3 While in any even dimensional space the superposition of odd products of a’s, forming the Cliffordodd ”basis vectors”, offer the description of the internal space of fermions with the half integer spins, (manifesting in d =(3 + 1) properties of quarks and leptons and antiquarks and antileptons, with the families included if d =(13 + 1), the superposition of even products of a’s, forming the Clifford even ”basis vectors”, offer the description of theinternal space of boson fields with integer spins, manifesting as gauge fields of the corresponding Cliffordodd ”basis vectors”. From the point of view of d =(3 + 1) one family of the Cliffordodd ”basis vectors” d=14 -1 members manifest spins, handedness and charges of quarks and with 2 2 leptons and antiquarks and antileptons appearing in 2 d=14 2 -1 families, while their d 2 d -1 members in 2 -1 Hermitian conjugated partners appear in another group of 2 4 families . 2 d 2 -1 × 2 d -1 The Clifford even ”basis vectors” appear in two groups, each with 2 2 members, with the Hermitian conjugated partners within the same group and have correspondingly no families. The Cliffordeven ”basis vectors” manifest from the point of view of d =(3 + 1) all the properties of the vector gauge fields before the electroweak break and for the scalar fields causing the electroweak break (as assumed by the standard model). Tensor products of the Cliffordodd and Cliffordeven ”basis vectors” (describing the internal space of fermions and bosons, respectively) with the basis in ordinary space form the creation operators to which the ”basis vectors” transfer either anticommutativity or commutativity. The Cliffordodd ”basis vectors” transfer their anticommutativity to creation operators and to their Hermitian conjugated partners annihilation operators for fermions. The Clifford even ”basis vectors” transfer their commutativity to creation operators and annihilation operators for bosons. Correspondingly the anticommutation properties of creation and annihilation operators of fermions explain the second quantization postulates of Dirac for fermion fields, while the commutation properties of creation and annihilation operatorsfor bosons explain the corresponding second quantization postulates for boson fields 5. In Sect. 11.2 the Grassmann and the Cliffordalgebra are explained and creation and annihilation operators described as a tensor products of the ”basis vectors” 3 Since the multiplication with either a’s or ~a’s changes the Cliffordodd ”basis vectors” into the Clifford even objects, and even ”basis vectors” commute, the action for fermions can not include an odd numbers of a’s or ~a’s, what the simple starting action of Eq. (19.1) does not. In the starting action a’s and ~a’s appear as 0 a p^a or as 0 c Sab c S~ ab ~ !abc and as 0!abc. 4 The appearance of the condensate of two right handed neutrinos causes that the number ofthe observedfamiliesreducestotwoatlow energies decoupledgroupsoffourgroups. 5 The creation and annihilation operators for either fermion or boson fields with the momenta zero, have no dynamics, and consequently no influence on clusters of fermion and boson fields. 166 N. S. Mankoˇc Borˇstnik offering explanation of the internal spaces of fermion (by the Cliffordodd algebra) and boson (by the Clifford even algebra) fields and the basis in ordinary space. In Subsect. 11.2.1 the ”basis vectors” are introduced and their properties presented. In Subsect. 11.2.2 the properties of the Cliffordodd and even ”basis vectors” are demonstrated in the toy model in d =(5 + 1). The simplest cases with d =(1 + 1) and d =(3 + 1) are also added. In Subsect. 11.2.3 the properties of the creation and annihilation operators for the second quantized fields are described. In Sect. 11.3 a short overview of the achievements and predictions so far of the spin-charge-family theory is presented, Sect. 11.4presents what thereadercould learnfrom the main contributionof this talk. In Sect. 11.5 the properties of Cliffordodd and Cliffordeven ”basis vectors” in odd dimensional spaces are presented, demonstrating how much properties of ”basis vectors” in odd dimensional spaces differ from the properties in even dimensional spaces. 11.2 Creation and annihilation operators for fermions and bosons The second quantization postulates for fermions [16–18]require that the creation operators and their Hermitian conjugated partners annihilation operators, depending on a finite dimensional basis in internal space, that is on the space of half integer spins and on charges described by the fundamental representations of the appropriate groups, and on continuously infinite number of momenta (or coordinates)([5], Subsect. 3.3.1), fulfil anticommutationrelations. The second quantization postulates for bosons [16–18] require that the creation and annihilation operators, depending on finite dimensional basis in internal space, that is on the space of integer spins and on charges described by the adjoint representations of the same groups, and on continuously infinite number of momenta(or coordinates)([5], Subsect. 3.3.1), fulfil commutationrelation. Idemonstrate in this talk that using the Cliffordalgebra to describe the internal space of fermions and bosons, the creation and annihilation operators which are tensor products of the internal basis and the momentum/coordinate basis, not only fulfil the appropriate anticommutation relations (for fermions) or commutation relations(for bosons)butalsohavetherequiredpropertiesfor either fermion fields (if the internal space is describedwith the Cliffordodd products of a’s) or for boson fields (if the internal space is described with the Clifford even products of a’s). The Cliffordodd and Clifford even ”basic vectors” correspondingly offer the explanation for the second quantization postulates for fermions and bosons, respectively. Therearetwo Cliffordsubalgebras which can be used to describe the internal space of fermions and of bosons, each with 2d members. In each of the two subalgebras there are2 × 2 -1× 2 -1 Cliffordodd and2 × 2 -1× 2 -1 Clifford even ”basic vectors” which can be used to describe the internal space of fermion fields, the d d d d 2 2 2 Title Suppressed Due to Excessive Length 167 Cliffordodd ”basic vectors”, and of boson fields, the Cliffordeven ”basic vectors” in any even d. d =(13 + 1) offers the explanation for all the properties of fermion fields, with families included, and of boson fields which are the gauge fields of fermion fields. In any even d, d = 2(2n + 1) or d = 4n, any of the two Clifford subalgebras offers twice 2 d 2 -1 irreducible representations, each with 2 d 2 -1 members, which can represent ”basis vectors” and their Hermitian conjugated partners. Each irreducible representation offers in d =(13 + 1) the description of the quarks and the antiquarks and the leptons and the antileptons (with the right handed neutrinos and left handed antineutrinos included in addition to what is) assumed by the standard model. There are obviously only one kind of fermion fields and correspondingly also of their gauge fields observed. There is correspondingly no need for two Clifford subalgebras. The reduction of the two subalgebras to only one with the postulate in Eq. (19.6), (Ref. [5], Eq. (38)) solves this problem. At the same time the reduction offers the quantum numbers for each of the irreducible representations of the Clifford subalgbebra left, a’s, when fermions are concerned([5] Subsect. 3.2). Boson fields have no families as it will be demonstrated. Grassmann and Clifford algebras The internal space of anticommuting or commuting second quantized fields can be describedby using either the Grassmann or the Cliffordalgebras[1–3,31]. What follows is a short overview of Subsect.3.2 of Ref. [5] and of references cited in [5]. In Grassmann d-dimensional space there are d anticommuting (operators) a, and . d anticommuting operators which are derivatives with respect to a ,, @a fa;bg+ = 0, { @ , . g+ = 0, @a @b fa, . g+ @b = ab , (a, b) = (0, 1, 2, 3, 5, · · · , d) . (11.2) Defining [32] (a)† . = aa @a , leads to . ( )† @a = aaa , (11.3) with ab = diagf1, -1, -1, · · · , -1g. . a and are, up to the sign, Hermitian conjugated to each other. The identity @a is the self adjoint member of the algebra. The choice for the following complex . properties of a and correspondingly of are made @a fa} * =(0;1 , -2;3 , -5;6 , :::, -d-1;d) , . @@@@@. @. f} * =( ;, - ;, - , , :::, - , ) . (11.4) @a @0 @1 @2 @3 @5 @6 @d-1 @d The are 2d superposition of products of a, the Hermitian conjugated partners of . which are the corresponding superposition of products of . @a 168 N. S. Mankoˇc Borˇstnik There exist two kinds of the Clifford algebra elements (operators), a and ~a , a . expressible with a’s and their conjugate momenta p= i [2], Eqs. (11.2, @a 11.3), @. a =(a +) ; ~a = i (a -) , @a @a 1 @1 a =( a - i ~a) , =( a + i ~a) , 2 @a 2 (11.5) offering together 2 · 2d operators: 2d are superposition of products of a and 2d of ~a. It is easy to prove, if taking into account Eqs. (11.3, 11.5), that they form two anticommuting Cliffordsubalgebras, f a ; ~bg+ = 0, Refs.([5]andreferences therein) f a; bg+ = 2ab = { ~a ; ~bg+ , f a ; ~bg+ = 0, (a, b)=(0, 1, 2, 3, 5, · ;d) , ( a)† = aa a a)† = aa ~a , ( ~. (11.6) While the Grassmann algebra offers the description of the ”anticommuting integer spin second quantized fields” and of the ”commuting integer spin second quantized fields” [5,35], the Cliffordalgebras which are superposition of odd products of either a’s or ~a’s offer the description of the second quantized half integer spin fermion fields, which from the point of the subgroups of the SO(d - 1, 1) group manifest spins and charges of fermions and antifermions in the fundamental representations of the group and subgroups. The superposition of even products of either a’s or ~a’s offer the description of the commuting second quantized boson fields with integer spins (as we can see in [9] and shall see in this contribution) which from the point of the subgroups of the SO(d - 1, 1) group manifest spins and chargesin the adjointrepresentations of the group and subgroups. The following postulate, which determines how does ~a’s operate on a’s, reduces the two Cliffordsubalgebras, a’s and ~a’s,to one,tothe one describedby a’s [2, 14,29,31,32] { ~aB = (-)B i B agj oc >, (11.7) with (-)B =-1, if B is(a functionof) an oddproductsof a’s, otherwise (-)B = 1 [14], j oc > is defined in Eq. (19.8) of Subsect. 11.2.1. After the postulate of Eq. (19.6) it follows: a. The Cliffordsubalgebra describedby ~a’s looses its meaning for the description of the internal space of quantum fields. b. The ”basis vectors” which are superposition of an odd or an even products of a’s obeythe postulates for the second quantization fields for fermions or bosons, respectively, Sect.11.2.1. c. It can be proven that the relations presented in Eq. (19.3) remain valid also after Title Suppressed Due to Excessive Length 169 the postulate of Eq. (19.6). The proof is presented in Ref.([5], App. I, Statement 3a. d. Each irreduciblerepresentationof the Cliffordodd ”basis vectors” describedby a’s are equippedby the quantum numbersof the Cartan subalgebra membersof S~ ab, chosen in Eq. (19.4), as follows S03 , S12 , S56 · , Sd-1d , · , S03· ;Sd-1d ;S12;S56 , · , S~ 03 S~ 12 S~ 56 S~ d-1d ;;, · ;, @. Sab = Sab Sab + ~= i (a - b ) . (11.8) @b @a After the postulate of Eq. (19.6) no vector space of ~a’s needs to be taken into account for the description of the internal space of either fermions or bosons, in agreement with the observed properties of fermions and bosons. Also the GrassmannalgebraisreducedtoonlyoneoftheCliffordsubalgebras. The operators ~a’s describe from now on properties of fermion and boson ”basis vectors” determined by superposition of products of odd or even numbers of a’s,respectively. ~a’s equip each irreduciblerepresentationof the Lorentzgroup (with the infinites- i imal generators Sab = f a; bg-)when applying on the Clifford odd ”basis 4 vectors” (which are superposition of odd products of a 0 s)with the family quan- Sab i a tum numbers (determinedby ~= { ~; ~bg-). 4 Correspondingly the Cliffordodd ”basis vectors” (they are superposition of an odd products of a’s) form 2 2 d -1 families, with the quantum number f, each family have 2 2 d -1 members, m. They offer the description of the second quantized fermion fields. The Clifford even ”basis vectors” (they are superposition of an even products of a’s) have no families as we shall see in what follows, but they do carry both quantum numbers, f and m. They offer the description of the second quantized boson fields as the gauge fields of the second quantized fermion fields. The generators of the Lorentz transformations in the internal space of the Clifford even ”basis = Sab Sab vectors” are Sab + ~. Properties of the Cliffordodd and the Clifford even ”basis vectors” are discussed in the next subsection. 11.2.1 ”Basis vectors” of fermions and bosons After the reduction of the two Cliffordsubalgebras to only one, Eq. (19.6), we only need to define ”basis vectors” for the case that the internal space of second quantized fields is described by superposition of odd or even products a’s 6. Let us use the technique which makes ”basis vectors” products of nilpotents and projectors[2,3,13,14] whichare eigenvectorsofthe (chosen) Cartan subalgebra 6InRef.[5]thereadercanfindin Subsects.(3.2.1and3.2.2) definitionsforthe ”basis vectors” forthe GrassmannandthetwoCliffordsubalgebras,whichareproductsof nilpotentsand projectors chosen to be eigenvactors of the corresponding Cartan subalgebra members of the Lorentz algebras presented in Eq. (19.4). 170 N. S. Mankoˇc Borˇstnik members, Eq. (19.4), of the Lorentz algebra in the space of a’s, either in the case of the Cliffordodd or in the case of the Cliffordeven products of a’s . d There are members of the Cartan subalgebra, Eq. (19.4), in even dimensional 2 spaces. One finds for any of the d Cartan subalgebra member, Sab or S~ ab, both applying 2 ab ab on a nilpotent (k) or on projector [k] ab aa ab (k):= 1 ( a + b) , ((k))2 = 0, 2 ik ab abab [k]:= 1 (1 + i a b) , ([k])2 =[k] 2k the relations abab abab Sab kS~ ab k (k)= (k) , (k)= (k) , 22 abab ab ab kk Sab Sab [k]= [k] , ~[k]=- [k] , (11.9) 22 with k2 = aabb, demonstrating that the eigenvalues of Sab on nilpotents and projectors expressed with a’s differ from the eigenvalues of S~ ab on nilpotents and projectors expressed with a’s, so that S~ ab can be used to equip each irreducible representation of Sab with the ”family” quantum number. 7 We define in evend the ”basis vectors” as algebraic, * A, products of nilpotents and projectors so that each product is eigenvector of all d Cartan subalgebra members. 2 We recognize in advance that the superposition of an odd products of a’s,that is the Cliffordodd ”basis vectors”, must include an odd number of nilpotents, at least one, while the superposition of an even products of a”s, that is Cliffordeven ”basis vectors”, must include an even number of nilpotents or only projectors. To define the Cliffordodd ”basis vectors”, we shall see that they have properties appropriate to describe the internal space of the second quantized fermion fields, and the Clifford even ”basis vectors”, we shall see that they have properties appropriate to describe the internal space of the second quantized boson fields, we need to know the relations for nilpotents and projectors ab aa ab 1 1i (k):= ( a + b) , [k]:= (1 + a b) , 2 ik 2k ab aa ab 1 1i (k~ ):= ( ~a + ~b) , [k~ ]: (1 + ~a ~b) , (11.10) 2 ik 2k 7 The reader can find the proof of Eq. (19.7) in Ref. [5], App. (I). Title Suppressed Due to Excessive Length 171 which can be derived after taking into account Eq. (19.3) ab ab abababab ab ab ab ab (k)= aa [-k]; (k)= -ik [-k], [k]=(-k); [k]= -ikaa (-k) , ab ab abab abab ab ab ~a (k)=-iaa [k]; ~b (k)= -k [k]; ~a [k]= i (k); ~b [k]= -kaa (k) , † ab ab ab abab ab = aa 2 (k) (-k) , ((k))= 0, (k)(-k)= aa [k] , † ab ab ab ab abab [k] =[k] , ([k])2 =[k] , [k][-k]= 0, abab abab ab ab ab ab ab ab (k)[k]= 0, [k](k)=(k) , (k)[-k]=(k) , [k](-k)= 0, † ab ab ab abab ab = aa 2 (k~ ) (-~k) , ((k~ ))= 0, (k~ )(-~k)= aa [k~ ] , † ab ab ab ab abab 2 [k~ ] =[k~ ] , ([k~ ])=[k~ ] , [k~ ][-~k]= 0, abab abab ab ab ab ab ab ab (k~ )[k~ ]= 0, [k~ ](k~ )=(k~ ) , (k~ )[-~k]=(k~ ) , [k~ ](-~k)= 0. (11.11) Looking at relations in Eq. (19.9) it is obvious that the properties of the ”basis vectors” which include odd number of nilpotents differ essentially from the ”basis vectors” which include even number of nilpotents. One namelyrecognizes: † ab ab i. Since the Hermitian conjugated partner of a nilpotent (k) is aa (-k) and since neither Sab nor S~ ab nor both can transform odd products of nilpotents to belong to one of the 2 d 2 -1 members of one of 2 d 2 -1 irreducible representations (families), the Hermitian conjugated partners of the Cliffordodd ”basis vectors” must belong toa differentgroupof 2 d 2 -1 members of 2 d 2 -1 families. abcd ab cd Sab Since Sac transforms (k) * A (k0) into [-k] * A [-k0], while ~transforms abcd ab cd [-k] * A [-k0] into (-k) * A (-k0) it is obvious that the Hermitian conjugated partners of the Cliffordodd ”basis vectors” must belong to the same group of d -1 × 2 d 2 -1 members. Projectors are self adjoint. 2 2 ii. Since an odd products of a’s anticommute with another group of an odd product of a, the Cliffordodd ”basis vectors” anticommute, manifesting in a tensor product with the basis in ordinary space together with the corresponding Hermitian conjugated partners properties of the anticommutation relations postulated by Dirac for the second quantized fermion fields. The Cliffordeven ”basis vectors” correspondingly fulfil the commutationrelations for the second quantized boson fields. iii. The Cliffordodd ”basis vectors” have all the eigenvalues of the Cartan subalgebra members equal to either 1 or to ± i . 22 The Clifford even ”basis vectors” have all the eigenvaluesofthe Cartan subalge- bra members Sab equal to either 1 and zero or to i and zero. Let us define odd an even ”basis vectors” as products of nilpotents and projectors in even dimensional spaces. 172 N. S. Mankoˇc Borˇstnik a. Clifford odd ”basis vectors” The Cliffordodd ”basis vectors” must be products of an odd number of nilpotents and therest,upto d ,of projectors, each nilpotent and projector must be the ”eigen 2 state” of one of the members of the Cartan subalgebra, Eq. (19.4), correspondingly are the ”basis vectors” eigenstates of all the members of the Lorentz algebras: Sab’s determine 2 2 d -1 members of one family, S~ ab’s transform each member of one family to the same member of the rest of 2 2 d -1 families. bm† Let us name the Cliffordodd ”basis vectors” ^, where m determines member- f ship of ’basis vectors” in any family and f determines a particular family. The Hermitian conjugated partner of b^m† is named by b^ =(b^m† )† . f ff 1† ^ Let us start in d = 2(2n + 1) with the ”basis vector” bwhich is the product 1 of only nilpotents, all the rest members belonging to the f = 1 family follow , S03 by the application of S01 , :::;S0d;S15 , :::;S1d;S5d :::;Sd-2d. The algebraic product mark * A is skipped. d = 2(2n + 1) , 03 1256 d-1d ^ b11† =(+i)(+)(+) · (+) , 03 1256 d-1d ^ b2 1 † =[-i][-](+) · (+) , · 2 d -103 1256 d-3d-2d-1d † b^ 2 1 [-i][-](+) ::. [-] [-] = , · . (11.12) bm† The Hermitian conjugated partnersof the Cliffordodd ”basis vector” ^, pre 1 sented in Eq. (11.12), are d = 2(2n + 1) , 0312 d-1d ^ b1 =(-i)(-) · (-) , 1 03 1256 d-1d ^ b1 2 =[-i][-](-) · (-) , · 2 d -103 125678 d-3d-2d-1d † b^ 2 1 [-i][-](-)[-] ::. [-] [-] = , · . (11.13) In d = 4n the choice of the starting ”basis vector ”with maximal number of nilpo- tents must have one projector d = 4n , 0312 d-1d ^ b11† =(+i)(+) · [+] , 03 1256 d-1d ^ b12† =[-i][-](+) · [+] , · b^ 2 1 2 d 03 1256 d-3d-2d-1d -1 † = [-i][-](+) ::. [-] [+] , ::. . (11.14) Title Suppressed Due to Excessive Length 173 bm† The Hermitian conjugated partnersof the Cliffordodd ”basis vector” ^, pre 1 ab ab sented in Eq. (11.14), follow if all nilpotents (k) are transformed into aa (-k). For either d = 2(2n + 1) or for d = 4n all the 2 d -1 families follow by applying 2 S~ ab’s on all the members of the starting family. (Or one can find the starting b^ for f all families f and then generate all the members b^ from b^ by the application of ff S~ ab on the starting member.) It is not difficult to see that all the ”basis vectors” within any family as well as the ”basis vectors” among families are orthogonal, that is their algebraic product is zero, and the same is true for the Hermitian conjugated partners, what can be provedby the algebraic multiplication using Eq.(19.9). bm† bm† bm‘ 0 ^^ ^* A ^= 0, bm * A = 0, 8m;m ;f;f‘ . (11.15) ff‘ ff‘ If we require that each family of ”basis vectors”, determined by nilpotents and projectors described by a’s, carries the family quantum number determinedby S~ ab and define the vacuum state on which ”basis vectors” apply as d -1 2 2 X b^ m b^m† j oc >= f * A f | 1 >, (11.16) f=1 it follows that the Cliffordodd ”basis vectors” obey the relations b^m f * A j oc > = 0. j oc >, b^m f y * A j oc > = j m f >, bm b^ m 0 f^ f , f‘ g* A+j oc > = 0. j oc >, fb^m f † ;b^mf‘ 0† g* A+j oc > = 0. j oc >, > = mm fb^ m ;b^ m 0† g* A+j oc 0 ffj oc >, (11.17) ff 0† bm† >= mm 0 while the normalization < ocjb^mf0 * A ^ f * A j oc ff0 is used and the 0† 0† 0† anticommutationrelation mean fb^m† ;b^ m = b^m† * A b^ m + b^ m * A b^m† . ff‘ g* A+ ff‘ f‘ f If we write the creation and annihilation operators as the tensor, * T , products of ”basis vectors” and the basis in ordinary space, the creation and annihilation operators fulfil the Dirac’s anticommutation postulates since the ”basis vectors” transfer their anticommutativity to creation and annihilation operators. It turns out that not only the Clifford odd ”basis vectors” offer the description of the internal space of fermions, they offer the explanation for the second quantization postulates for fermions as well. Table 11.1, presented in Subsect. 11.2.2, illustrates the properties of the Clifford odd ”basis vectors” on the case of d =(5 + 1). b. Clifford even ”basis vectors” The Cliffordeven ”basis vectors” must be products of an even number of nilpotents and the rest, up to d , of projectors, each nilpotent and projector in a product must 2 be the ”eigenstate” of one of the members of the Cartan subalgebra, Eq. (19.4), 174 N. S. Mankoˇc Borˇstnik correspondingly are the ”basis vectors” eigenstates of all the members of the Lorentz algebra: Sab’s and S~ ab’s generate from the starting ”basis vector” all the 2 d 2 -1× 2 d 2 -1 members of one group which includes as well the Hermitian conjugated partners of any member. 2 projectors only. They are self adjoint. 2 d -1 members of the group are products of d 2 -12 d 2 -1 members There are two groups of Cliffordeven ”basis vectors”with 2 each. The members of one group are not connected with the members of another group by either by Sab’s or S~ ab’s or both. Let us name the Clifford even ”basis vectors” i A^m† , where i =(I, II) denotes that f there are two groups of Clifford even ”basis vectors”, while m and f determine membership of ’basis vectors” in any of the two groups, I or II. Let merepeat that the Hermitian conjugated partnerof any ”basis vector” appears eitherin the case of I A^m† or in the case of II A^m† within the same group. ff LetuswritedowntheCliffordeven”basis vectors”asaproductofanevennumber of nilpotents and the rest of projectors, so that the Clifford even ”basis vectors” are eigenvectors of all the Cartan subalgebra members, and let us name them as follows d = 2(2n + 1) 0312 d-1d 0312 d-1d I A1† II A^1† ^ =(+i)(+) · [+] , =(-i)(+) · [+] , 11 031256 d-1d 031256 d-1d I II A^2† =[-i][-](+) · [+] , A^2† =[+i][-](+) · [+] , 11 03 1256 d-3d-2d-1d 031256 d-3d-2d-1d I A3† II A^3† ^ =(+i)(+)(+) · [-] (-) , =(-i)(+)(+) · [-] (-) , 11 ::. ::. d = 4n 0312 d-1d 0312 d-1d I A1† II A1† ^^ =(+i)(+) · (+) , =(-i)(+) · (+) , 11 031256 d-1d 031256 d-1d I II A^2† =[-i][-i](+) · (+) , A^2† =[+i][-i](+) · (+) , 11 03 1256 d-3d-2d-1d 031256 d-3d-2d-1d I A3† II A3† ^^ =(+i)(+)(+) · [-] [-] , =(-i)(+)(+) · [-] [-] 11 ::. ::. (11.18) d 2 d 2 -1 Clifford even ”basis vectors” of the kind I A^m† f and there There are 2 -1 × 2 d d 2 -1 Clifford even ”basis vectors” of the kind II A^m† f . -1 2 are 2 2 Table 11.1, presented in Subsect. 11.2.2, illustrates properties of the Cliffordodd and Clifford even ”basis vectors” on the case of d =(5 + 1). Looking at this particular case it is easy to evaluate properties of either even or odd ”basis vectors”. Ishall present here the general results which follow after careful inspection of properties of both kinds of ”basis vectors”. The Cliffordeven ”basis vectors” belonging to two different groups are orthogonal due to the fact that they differ in the sign of one nilpotent or one projectors, or the algebraic products of members of one group give zero according to Eq. (19.9). I Am† II Am† II Am† I Am† ^^^^ * A = 0 = * A . (11.19) ff ff Title Suppressed Due to Excessive Length 175 The members of each of this two groups have the property Am† I;II Am† f ^ * A I;II Am f‘ ^ 0† › I;II f‘ ^ , only one for 8f‘ , (11.20) or zero . 0† , the algebraic product, * A, of which gives Two ”basis vectors” I Am† f ^ and I Am f0 ^ Am† f‘ ^ . The same is true also for nonzero contribution, ”scatter” into the third one I the ”basis vectors” II Am† . f ^ Let us write the commutation relations for Clifford even ”basis vectors” taking into account Eq. (11.20). 0† 0† Am† f ^ Am f‘ ^ › I Am† f‘ ^ ^ Am f‘ ^ Am† f In the case that I * A I and I I = 0 it follows i. * A 0† Am† f‘ ^ ^ Am† f ^ Am f‘ ^ Am† f‘ I I I › I (if * A 0† , fI I Am† , Am ff‘ ^ ^ (11.21) g* A - › 0† ^ Am f‘ ^ Am† f I I and * A = 0) , 0† 0† ^ Am f 0† it Am† f ^ ^ Am f‘ ^ Am† f‘ ^ Am f‘ ^ Am† f In the case that I I › I and I I › I ii. * A * A follows 0† 0† ^ Am† f‘ ^ Am f ^ Am† f ^ Am f‘ ^ Am† f‘ (11.22) ) , I - I I I › I (if * A 0† , fI Am† f ^ I , Am f‘ ^ g* › - A 0† 0† ^ Am f‘ ^ Am† f › I ^ Am f I * A I and iii. In all other cases we have 0† fI I ^ ^ Am† , Am ff‘ g* A - = 0. (11.23) 0† 0† 0† ^ Am† f I ^ , Am f‘ ^ Am† f * A I ^ Am f‘ - I ^ Am f‘ * A I ^ Am† f . fI g* A - means I ^ It remains to evaluate the algebraic application, * A, of the Clifford even ”basis 0† ^ Am† f vectors” I on the Cliffordodd ”basis vectors” . One finds bm f‘ ^ or zero . 0† , I ^ Am† f‘ ^ -1 × 2 (11.24) bm† f * A › bm f d 2 d 2 ^ Am† f -1 members of the Cliffordodd ”basis For each I there are among 2 0† d 2 ^ -1 - 1), give zero -1 members, vectors” (describing the internal space of fermion fields) 2 bm , f‘ fulfillingtherelationofEq. (11.24).Alltherest(2 contributions. d 2 -1 × (2 d 2 Eq. (11.24) clearly demonstrates that I ^ Am† f transforms the Clifford odd ”basis vector” in general into another Clifford odd ”basis vector”, transfering to the Cliffordodd ”basis vector” an integer spin. We can obviously conclude that the Cliffordeven ”basis vectors” offer the description of the gauge fields to the corresponding fermion fields. 176 N. S. Mankoˇc Borˇstnik While the Cliffordodd ”basis vectors” offer the description of the internal space of the second quantized anticommuting fermion fields, appearing in families, the Clifford even ”basis vectors” offer the description of the internal space of the second quantized commuting boson fields, having no families and manifesting as the gauge fields of the corresponding fermion fields. 11.2.2 Example demonstrating properties of Clifford odd and even ”basis vectors” for d =(1 + 1), d =(3 + 1), d =(5 + 1) ‘ Subsect. 11.2.2 demonstrates properties of the Cliffordodd and even ”basis vectors” in special cases when d =(1 + 1), d =(3 + 1), and d =(5 + 1). Let us start with the simplest case: d=(1+1) There are 4 (2d=2) ”eigenvectors” of the Cartan subalgebra members S01 and S01 of the Lorentz algebra Sab and Sab , Eq. (19.4), representing one Cliffordodd 01 1† ”basis vector” b^=(+i) (m=1), appearing in one family (f=1) and correspondingly 1 01 one Hermitian conjugated partner b^ 1 = (-i) 8 and two Cliffordeven ”basis vector” 1 01 01 IA1† =[+i] and IIA1† =[-i], each of them is self adjoint. 11 Correspondingly we have two Cliffordodd 01 01 ^1† b^ 1 b1 =(+i) , 1 =(-i) and two Clifford even 01 01 IA1† IIA1† =[+i] , =[-i] 11 ”basis vectors”. The first two Cliffordodd ”basis vectors” are Hermitian conjugated to each other. 1† ^ Imake a choice that bis the ”basis vector”, the second Cliffordodd object is 1 its Hermitian conjugated partner. Defining the handedness as ..(1+1) = 0 1 it 1† 1† 1† ^ follows, usingEq. (19.5),that ..(1+1) b^ = b^ , which means that bis the right 11 1 handed ”basis vector”. 01 1† ^ We could makea choiceoflefthanded ”basis vector”if choosingb=(-i), but 1 the choice of handedness would remain only one. )† I;IIA1† Each of the two Clifford even ”basis vectors” is self adjoint((I;IIA1† = ). 11 01 01 8 It is our choice which one, (+i) or (-i), we chose as the ”basis vector” b^1 1 † and which oneis its Hermitian conjugated partner. The choiceofthe ”basis vector” determines the 01 01 ^ vacuum state j oc >, Eq. (19.8). For b1 1 † =(+i), the vacuum state is j oc >=[-i] (due to therequirement that b^11† j oc > is nonzero) which is the Clifford even object. Title Suppressed Due to Excessive Length 177 Let us notice, taking into account Eqs. (19.5, 19.9), that 0101 01 1† IIA1† fb^ (. (-i)) * A b^(. (+i))gj oc >=(. [-i])j oc >, 11 1 >= j oc 01 01 1† b^ fb^(. (+i)) * A (. (-i))gj oc >= 0, 11 0101 01 IA1† (. [+i]) * A b^ (. (+i))j oc >= b^ (. (+i))j oc >, 111 01 01 IA1† b1 (. [+i])^(. (-i))j oc >= 0. 11 We find that IA1† IIA1† IIA1† IA1† * A = 0 = * A . 11 11 From the case d =(3 + 1) we can learn a little more: d=(3+1) There are 16 (2d=4) ”eigenvectors”of the Cartan subalgebra members(S03;S12) and(S03 , S12)of the Lorentz algebrasSab and Sab , Eq. (19.4), in d =(3 + 1). There are two families(2 4 2 -1, f=(1,2)) with two(2 4 2 -1, m=(1,2)) members each of bm† the Cliffordodd ”basis vectors” ^ f , with 2 4 2 -1 × 2 4 2 -1 Hermitian conjugated partners b^ m in a separate group (not reachable by Sab). f 4 -1 2 4 -1IAm† members of the group of f There are 2 , which are Hermitian × 2 2 conjugatedtoeach otherorareself adjoint,allreachableby Sab from any starting ”basis vector IA1† . 1 And there is another group of 2 4 2 -1 2 4 2 -1 members of IIAm† f , again either Hermi tian conjugatedtoeachotherorareself adjoint.Allarereachablefromthe starting vector IIA1† by the application of Sab . 1 Again we can make a choice of either right or left handed Cliffordodd ”basis vectors”, but notof both handedness. Makinga choiceof the right handed ”basis vectors” f = 1f = 2 S03 i S12 1 S03 S12 ~= ;S~ 12 =- 1 ;S~ 03 =- i , ~= ;, 22 22 0312 0312 1† 1† i1 ^^ b=(+i)[+] b=[+i](+) 1 222 0312 0312 2† 2† b^=[-i](-) b^=(-i)[-] - i - 1 , 1 222 we find for the Hermitian conjugated partners of the above ”basis vectors” 1 S03 i S~ 03 S~ 12 S03 =- i ;S12 = , = ;S12 =- 1 ;, 222 2 0312 0312 b^ 1 b^ 1 - i - 1 =(-i)[+] =[+i](-) 1 222 0312 0312 b^ 2 b^ 2 i1 =[-i](+) =(+i)[-] . 1 222 Let us notice that if we look at the subspace SO(1, 1) with the Cliffordodd ”basis vectors” with the Cartan subalgebra member S03 of the space SO(3, 1),and neglect 178 N. S. Mankoˇc Borˇstnik 03 03 1† 2† the values of S12, we do have b^=(+i) and b^=(-i), which have opposite 12 handedness ..(1;1) in d =(1+1),but they have different”charges”S12 in d =(3+1). In the whole internal space all the Cliffordodd ”basis vectors” have only one handedness. 0312 0312 1 We further find thatj oc >= . ([-i][+] + [+i][+]). All the Cliffordodd ”basis 2 bm† 0† vectors” are orthogonal: ^* A b^ 0. ff0 For the Clifford even ”basis vectors” we find two groups of either self adjoint members or with the Hermitian conjugated partners within the same group. The two groups are not reachable by S03.We have for IAm† ;m =(1, 2);f =(1, 2) f S03 S12 S03 S12 0312 03 12 IA1† =[+i][+] 0 0, IA1† =(+i)(+) i1 12 03 12 0312 IA2† IA2† =(-i)(-) -i -1, =[-i][-] 0 0, 12 and for IIAm† ;m =(1, 2);f =(1, 2) f S03 S12 S03 S12 0312 03 12 IIA1† =[+i][-] 0 0, IIA1† =(+i)(-) i1 12 03 12 0312 IIA2† IIA2† =(-i)(+) -i 1, =[-i][+] 0 0. 12 The Clifford even ”basis vectors” have no families. IAm† * AIAm 0† = 0, for any ff‘ (m, m’,f,f‘). d =(5 + 1) InTable 11.1 the 64 (= 2d=6) ”eigenvectors” of the Cartan subalgebra members of the Lorentz algebra Sab and Sab, Eq. (19.4), are presented. The Cliffordodd ”basis vectors”, they appear in 4 (= 2 d=6 -1) families, each family has 4 members, 2 are products of an odd number of nilpotents, that is either of three nilpotents or bm† ofone nilpotent.TheyappearinTable11.1inthegroupnamed odd I ^. Their f Hermitian conjugated partners appear in the second group named odd II b^ m . f Within each of these two groups, the members are orthogonal, Eq. (11.15), which means that the algebraic product of b^m† * A b^ m 0† = 0 for all (m, m0, f, f). This can ff‘ be checkedbyusingrelationsinEq. (19.9). Equivalently,the algebraicproductsof their Hermitian conjugated partners are also orthogonal among themselves. The ”basis vectors” and their Hermitian conjugated partners are normalized as follows bm 0† >= mm 0 ^ < ocjb^ m * A j oc ff‘ , (11.25) ff‘ 03 1256 03 1256 03 1256 1 since the vacuum state j oc >= . d=6 ([-i][-][-] + [-i][+][+] + [+i][-][+] -1 2 2 03 1256 + [+i][+][-]) is normalized to one: < ocj oc >= 1. The longer overview of the properties of the Cliffordodd ”basis vectors” and their Hermitian conjugated partners for the case d =(5 + 1) can be found in Ref. [5]. Title Suppressed Due to Excessive Length 179 The Clifford even ”basis vectors” are products of an even number of nilpo- tents, of either two or none in this case. They are presented in Table 11.1 in d=6d=6 two groups, each with 16 (= 2 2 -1 × 2 2 -1) members, as even I Am† and f even II Am† . One can easily check, using Eq. (19.9), that the algebraic product f IAm† IIAm 0† * A = 0, . (m, m0, f:f), Eq. (11.19). The longer overview of the Clif- ff‘ ford even ”basis vectors” and their Hermitian conjugated partners for the case d =(5 + 1)-can be found in Ref. [9]. While the Cliffordodd ”basis vectors” are (chosen tobe) right handed, ..(5+1) = 1, have their Hermitian conjugated partners opposite handedness 9 While the Cliffordodd ”basis vectors” have half integer eigenvalues of the Cartan subalgebra members, Eq.(19.4), thatis of S03;S12;S56 in this particular case of d =(5 + 1), the Clifford even ”basis vectors” have integer spins, obtained by S03 = S03 S03 , S12 = S12 S12 , S56 = S56 S56 + ~+ ~+ ~. Let us check what does the algebraic application, * A, of I ;m =(1, 2, 3, 4), A^m† f=3 presentedinTable 11.1in the thirdcolumnof even I, do on the Cliffordodd ”basis vectors” b^m=1† , presented as the first odd I ”basis vector”inTable 11.1. This can f=1 easily be evaluated by taking into account Eq. (19.5) for any m. 03 1256 I Am† 1† ^^ 3 * A b1 (. (+i)[+][+]) : 03 1256 03 1256 I1† 1† A^1† (. [+i][+][+]) * A b^(. (+i)[+][+]) › b^ , selfadjoint 31 1 03 1256 03 1256 I1† 2† A^2† (. (-i)(-)[+]) * A b^ › b^(. [-i](-)[+]) , 3 11 03 1256 03 1256 I A3† 1† 3† ^(. (-i)[+](-)) * A b^ › b^(. [-i][+](-)) , 3 11 03 1256 03 1256 I1† 4† A^4† (. [+i](-)(-)) * A b^ › b^(. (+i)(-)(-)) . (11.26) 3 11 A^1† The sign › meansthattherelationis validuptothe constant. I is self adjoint, 3 the Hermitian conjugated partner of I is I , of is I and of I is A^2† A^1† I A^3† A^1† A^4† 3432 3 I A^1† . 1 03 1256 I Am† 1† ^ We can conclude that the algebraic,* A, application of (. (-i)[+](-)) on b^ 31 leads to the same or another family member of the same family f = 1, namely to b^m† , m =(1, 2, 3, 4). 1 Calculating the eigenvalues of the Cartan subalgebra members, Eq. (19.4), before I A^m† and after the algebraic multiplication, * A, one sees that carry the integer 3 = Sab eigenvalues of the Cartan subalgebra members, namely of Sab + S~ ab, since they transferwhen applyingontheCliffordodd ”basis vector”toitthe integer eigenvalues of the Cartan subalgebra members, changing the Cliffordodd ”basis vector” into another Cliffordodd ”basis vector”. I A^m† We therefore find out that the algebraic application of, m = 1, 2, 3, 4, on 3 1† 1† bm† ^^ btransforms binto ^, m =(1, 2, 3, 4). Similarly we find that the algebraic 1 11 2† 2† application of I A^m 4 ;m =(1, 2, 3, 4) on b^ transforms b^ into b^m† ;m =(1, 2, 3, 4). 1 11 9 The handedness . (d), one of the invariants of the group SO(d), with the infinitesiSa1a 2 mal generators of the Lorentz group Sab, is defined as ..(d) = "a1a2:::ad-1ad · Sa3a4 · Sad-1ad , with . chosen so that . (d) = 1. 180 N. S. Mankoˇc Borˇstnik Table 11.1: 2d = 64 ”eigenvectors” of the Cartan subalgebra of the Cliffordodd and even algebras — the superposition of odd and even products of a’s — in d =(5 + 1)-dimensional space are presented, divided into four groups. The first group, odd I, is chosen to represent ”basis vectors”, named b^m† , appearing in f -1 -1 22 2 d = 4 ”families”(f = 1, 2, 3, 4), each ”family” with 2 d = 4 ”family” mem- bers(m = 1, 2, 3, 4). The second group, odd II, contains Hermitian conjugated partners of the first group for each family separately, b^ =(b^m† )y. Either odd I or ff odd II areproductsof an odd numberof nilpotents, therest areprojectors. The b^m† S03 S~ 12 S~ 56), ”family” quantum numbers of , that is the eigenvalues of ( ~;, are f for the first oddI group written above each ”family”, the quantum numbers of the members (S03;S12;S56) are written in the last three columns. For the Hermitian conjugated partners of oddI, presented in the group odd II, the quantum numbers (S03;S12;S56) are presented above each group of the Hermitian conjugated part S03 S~ 12 ~ ners,thelastthree columnstell eigenvaluesof ( ~, ;S56). The two groups with the even number of a’s, even Iand even II, each has their Hermitian conjugated partners within its own group, have the quantum numbers f, that is the eigen S03 S12 values of ( ~, ~;S~ 56), written above column of four members, the quantum numbers of the members, (S03;S12;S56), are written in the last three columns. 00basis vectors 00 ( ~S03 , ~S12 , ~S56) m › f = 1 ( i , - 1 , - 1 ) 2 2 2 f = 2 (- i , - 1 , 1 ) 2 2 2 f = 3 (- i , 1 , - 1 ) 2 2 2 f = 4 ( i , 1 , 1 ) 2 2 2 S03 S12 S56 m† odd I ^b f 1 03 12 56 (+i) [+] [+] 03 12 56 [+i] [+] (+) 03 12 56 [+i] (+) [+] 03 12 56 (+i) (+) (+) i 2 1 2 1 2 2 [-i](-)[+] (-i)(-)(+) (-i)[-][+] [-i][-](+) - i 2 - 1 2 1 2 3 [-i][+](-) (-i)[+][-] (-i)(+)(-) [-i](+)[-] - i 2 1 2 - 1 2 4 (+i)(-)(-) [+i](-)[-] [+i][-](-) (+i)[-][-] i 2 - 1 2 - 1 2 (S03, S12, S56) › (- i , 1 , 1 ) 2 2 2 03 12 56 ( i , 1 , - 1 ) 2 2 2 03 12 56 ( i , - 1 , 1 ) 2 2 2 03 12 56 (- i , - 1 , - 1 ) 2 2 2 03 12 56 ~S03 ~S12 ~S56 odd II ^b m f 1 (-i)[+][+] [+i][+](-) [+i](-)[+] (-i)(-)(-) - i - 1 - 1 2 2 2 2 [-i](+)[+] (+i)(+)(-) (+i)[-][+] [-i][-](-) i 1 - 1 2 2 2 3 [-i][+](+) (+i)[+][-] (+i)(-)(+) [-i](-)[-] i - 1 1 2 2 2 5 -1 -1 4 (-i)(+)(+) [+i](+)[-] [+i][-](+) (-i)[-][-] - i 1 1 2 2 2 ( ~S03 , ~S12 , ~S56) › (- i , 1 , 1 ) 2 2 2 03 12 56 ( i , - 1 , 1 ) 2 2 2 03 12 56 (- i , - 1 , - 1 ) 2 2 2 03 12 56 ( i , 1 , - 1 ) 2 2 2 03 12 56 S03 S12 S56 even I IAm f 1 [+i](+)(+) (+i)[+](+) [+i][+][+] (+i)(+)[+] i 2 1 2 1 2 2 (-i)[-](+) [-i](-)(+) (-i)(-)[+] [-i][-][+] - i 2 - 1 2 1 2 3 (-i)(+)[-] [-i][+][-] (-i)[+](-) [-i](+)(-) - i 2 1 2 - 1 2 4 [+i][-][-] (+i)(-)[-] [+i](-)(-) (+i)[-](-) i 2 - 1 2 - 1 2 ( ~S03 , ~S12 , ~S56) › ( i , 1 , 1 ) 2 2 2 03 12 56 (- i , - 1 , 1 ) 2 2 2 03 12 56 ( i , - 1 , - 1 ) 2 2 2 03 12 56 (- i , 1 , - 1 ) 2 2 2 03 12 56 S03 S12 S56 even II IIAm f 1 [-i](+)(+) (-i)[+](+) [-i][+][+] (-i)(+)[+] - i 2 1 2 1 2 2 (+i)[-](+) [+i](-)(+) (+i)(-)[+] [+i][-][+] i 2 - 1 2 1 2 3 (+i)(+)[-] [+i][+][-] (+i)[+](-) [+i](+)(-) i 2 1 2 - 1 2 4 [-i][-][-] (-i)(-)[-] [-i](-)(-) (-i)[-](-) - i 2 - 1 2 - 1 2 Title Suppressed Due to Excessive Length 181 3† 3† A^m ^^ The algebraic application of I 2 ;m =(1, 2, 3, 4) on btransforms binto 11 bm† 4† ^;m =(1, 2, 3, 4). And the algebraic application of I A^m 1 ;m =(1, 2, 3, 4) on b^ 11 4† transforms b^ into b^m† ;m =(1, 2, 3, 4). 11 The statement of Eq. (11.24) is therefore demonstrated on the case of d =(5 + 1). Itremainsto stress and illustratein the caseof d =(5 + 1) some general properties of the Clifford even ”basis vector” I A^m† when they apply on each other. Let us f denote the self adjoint member in each group of ”basis vectors” of particular f as I A^m0† .We easily see that f I 0 fI A^m f † , A^m f 0† g- = 0, if (m, m 0)= 6m0 or m = m0 = m, . f, I A^m† I Am0† › I A^m† ^ * A , . m, . f. (11.27) ff f I A^m† II A^m† InTable 11.1 we see that in each column of either even f or of evenf there is one self adjoint I;II A^m0† .We also seethattwo ”basis vectors” I A^m† and ff I 0† A^m of the same f and of (m, m 0) 6m0 are orthogonal.We only have to take = f into account Eq. (19.9), which tells that abab abab ab ab ab ab ab ab (k)[k]= 0, [k](k)=(k) , (k)[-k]=(k) , [k](-k)= 0. A^1† I A^2† I A^1† These relations tell us that I * A = , what illustrates Eq. (11.23), while 4 33 I A^2† I A^1† I A^2† I A^1† I A^2† * A = illustrating Eq. (11.22), while * A = 0 illustrates 344 34 Eq. (11.21). Table 11.2presentsthe Cliffordeven ”basis vectors” I A^m† for d =(5 + 1) withthe f ab properties: i.They are products of an even number of nilpotents, (k), with the ab rest up to d of projectors, [k]. ii. Nilpotents and projectors are eigenvectors of 2 = Sab the Cartan subalgebra members Sab + S~ ab, Eq. (19.4), carrying the integer eigenvalues of the Cartan subalgebra members. A^m† iii. They have their Hetmitian conjugated partners within the same group of I f dd 2 with 2 2 -1 × 2 -1 members. iv. They have properties of the boson gauge fields. When applying on the Clifford odd ”basis vectors” (offering the description of the fermion fields) they transform the Cliffordodd ”basis vectors” into another Cliffordodd ”basis vectors”, transferring to the Cliffordodd ”basis vectors” the integer spins with respect to the SO(d - 1, 1) group, while withrespectto subgroupsof the SO(d - 1, 1) group they transfer appropriate superposition of the eigenvalues (manifesting the properties of the adjoint representations of the corresponding groups). To demonstrate that the Cliffordeven ”basis vectors” have properties of the gauge fields of the corresponding Cliffordodd ”basis vectors” we study properties of the SU(3) U(1) subgroups of the Cliffordodd and Clifford even ”basis vectors”. We present in Eqs. (11.28, 11.29) the superposition of members of Cartan subalgebra, Eq. (19.4), for Sab for the Cliffordodd ”basis vectors”, for the subgroups SO(3, 1) × U(1) (N3 SU(3) U(1):(0;8). The same relations can be used ± ;)and for the subgroups;3 182 N. S. Mankoˇc Borˇstnik also for the corresponding operators determining the ”family” quantum numbers( N~ ;~) of the Cliffordodd ”basis vectors’, if Sab’s are replaced by S~ ab’s. For the Clifford even objects Sab(= Sab + S~ ab) must replace Sab . 3 3 120356 N(= N(L;R)) := 1 (S± iS) ;. = S, (11.28) 2 11 3 12 038 03 12 56 := (-S- iS) ;= . (-iS+ S- 2S) , 2 23 0 1 03 12 56 =- (-iS+ S+ S) . (11.29) 3 03 1256 Let us, for example, algebraically apply I A^2 (. (-i) (-) [+]), denoted by on 3 2 Table 11.2, carrying(3 = 0, 8 =- . 1;0 =), represented also on Fig. 11.2 by 3 03 1256 3 1† ^ , on the Cliffordodd ”basis vector” b(. (+i)(+)(+)),presented onTable 11.1, 1 with (3 = 0, 8 = 0, 0 =- 1 ), as we can calculate using Eq. (11.29) and which 2 I 1† is represented on Fig. 11.1 by a square as a singlet. A^2 transforms b^ (by trans 31 1† 21† 1 ferring to b^(3 = 0, 8 =- . 1;0 =))tob^ with (3 = 0, 8 =- . 1;0 =), 132 6 33 belonging on Fig. 11.1 to the triplet, denoted by . The corresponding gauge fields, presented on Fig. 11.2, if belonging to the sextet, would transform the triplet of quarks among themselves. (1/(0,0,-1/(-1/2,1/2.3,1/6) (0,-1/.3,1/6) Fig. 11.1: Representations of the subgroups SU(3) and U(1) of the group SO(5, 1), thepropertiesof which appearinTable 11.1, arepresented.(3;8 and 0)can be calculatedifusing Eqs.(11.28, 11.29). On the abscissa axis, on the ordinate axis and on the third axis the eigenvalues of the superposition of the three Cartan subalgebra members, 3 , 8 , 0 are presented. One notices one triplet, denoted by 1 11 11 with the values 0 = ,(3 =- 1 ;8 = . ;0 =),(3 = ;8 = . ;0 = 62 62 23 23 11 ),(3 = 0, 8 =- . 1;0 = ),respectively,and one singlet denotedbythe square. 66 3 (3 = 0, 8 = 0, 0 =- 1 ). The triplet and the singlet appear in four families. 2 In the case of the group SO(6) (SO(5, 1)indeed), manifesting as SU(3) × U(1) and representing the SU(3) colour group and U(1) the ”fermion” quantum number, embedded into SO(13, 1) the triplet wouldrepresent quarks and the singlet leptons. The corresponding gauge of the fields, presented on Fig. 11.2, if belonging to the sextet, would transform the triplet of quarks among themselves, changing the Title Suppressed Due to Excessive Length 183 1 colour and leaving the ”fermion” quantum number equal to . 6 .(1,0,0)(-1,0,0) (1/2,.3/2,0)(-1/2,.3/2,0) (-1/2,-.3/2,0)(1/2,-.3/2,0) (0,1/.3,-2/3) (-1/2,-1/(2.3),-2/3) (1/2,-1/((1/2,1/(2.3),2/3)(-1/2,1/(2.3),2/3) (0,-1/.3,2/3)..38' I ^ Fig. 11.2: The Clifford even ”basis vectors” Am , in the case that d =(5 + 1), f are presented with respect to the eigenvalues of the commuting operators of 1 the subgroups SU(3) and U(1) of the group SO(5, 1): 3 = (-S12 - iS03), 2 1 (S12 - iS03 8 = . (S12 - iS03 - 2S56), 0 =- 1 + S56). Their properties appear 3 inTable 11.2. The abscissa axis carries the eigenvalues of 3, the ordinate axis of 8 and the third axis the eigenvalues of 0, One notices four singlets with (3 = 0, 8 = 0, 0 = 0),denotedby , representing four self adjoint Clifford even ”basis vectors” I Am , one sextet of three pairs with 0 = 0, Hermitian conjugated 23 ^ f 3 to each other, denoted by 4 (with(0 = 0, 3 =- 1 ;8 =- . )and(0 = 2 23 13 0, 3 = ;8 = . )), respectively, by‡ (with(0 = 0, 3 =-1, 8 = 0)and 2 23 13 (0 = 0, 3 = 1, 8 = 0), respectively, and by . (with(0 = 0, 3 = ;8 =- . ) 2 23 3 and(0 = 0, 3 =- 1 ;8 = . )), respectively, and one triplet, denoted by?? 2 23 211 21 with(0 = ;3 = ;8 = . ), by • with(0 = ;3 =- 1 ;8 = . ), and 32 32 23 23 2 by with(0 = ;3 = 0, 8 =- . 1), as well as one antitriplet, Hermitian 3 3 1 conjugated to the triplet, denotedby ?? with(0 =- 2 ;3 =- 1 ;8 =- . ),by • 32 23 11 1 with(0 =- 2 ;3 = ;8 =- . ), and by with(0 =- 2 ;3 = 0, 8 = . ). 32 3 23 3 I A^m† 1† We can see thatwith (m = 2, 3, 4), if applied on the SU(3) singlet b^ with 31 m=2;3;4)† (0 =- 1 ;3 = 0, 8 = 0), transforms it to b^ , respectively, which are 21 members of the SU(3) triplet. All these Cliffordeven ”basis vectors” have 0 equal 1 to 2 , changing correspondingly 0 =- 1 into 0 = and bringing the needed 3 26 values of 3 and 8 . InTable 11.2 we find (6 + 4) Cliffordeven ”basis vectors” I with = 0. Six of A^m† f them are Hermitian conjugated to each other — the Hermitian conjugated partners are denoted by the same geometric figure on the thirdcolumn. Four of them are self adjoint and correspondingly with(0 = 0, 3 = 0, 8 = 0), denoted in the third columnofTable 11.2by . The rest 6 Clifford even ”basis vectors” belong to one 184 N. S. Mankoˇc Borˇstnik 2 11 triplet with 0 = and (3;8) equal to[(0, - . 1), (- 1 , . ), ( 1 , . )]and one 3 22 3 23 23 1 11 antitriplet with 0 =- 2 and((3;8) equal to[(- 1 , - . ), ( 1 , - . ), (0, . )]. 3 22 23 233 Each triplet has Hermitian conjugated partner in antitriplet and opposite. In Table 11.2 the Hermitian conjugated partners of the triplet and antitriplet are denotedby the same signum:(I A^1† , I A^4† )by??,(I A^1† , I A^3† )by, and(I A^2† , 1323 3 I A^1† 4 )by . The octet and the two triplets are presented in Fig. 11.2. Table 11.2: The Cliffordeven ”basis vectors” I A^m† , each of them is the product f of projectors and an even number of nilpotents, and each is the eigenvector of , S12 all the Cartan subalgebra members, S03 , S56, Eq. (19.4), are presented for dd 2 d =(5 + 1)-dimensional case. Indexes m and f determine 2 2 -1 × 2 -1 different A^m† A^m† members I f . In the thirdcolumn the ”basis vectors” I f which are Hermitian conjugated partners to each other (and can therefore annihilate each other) are pointed out with the same symbol. For example, with ?? are equipped the first member with m = 1 and f = 1 and the last member of f = 3 with m = 4. The sign denotes the Clifford even ”basis vectors” which are self adjoint (I A^m† )† f I = A^m 0† . It is obvious that † has no meaning, since I A^m† are self adjoint or are f‘ f I Hermitian conjugated partner to another A^m 0† . This table represents also the f‘ eigenvalues of the three commuting operators N 3 and S56 of the subgroups L;R SU(2) × SU(2) × U(1) of the group SO(5, 1) and the eigenvalues of the three commuting operators 3;8 and 0 of the subgroups SU(3) × U(1). f m * I ^m† Af S03 S12 S56 N 3 L N3 R 3 8 . 0 03 12 56 I 1 ?? [+i] (+) (+) 03 12 56 0 1 1. 1 2 1 2 - 1 2 - 1 . 2 3 - 2 3 2 4 (-i) [-] (+) 03 12 56 -i 0 1 1 2 - 1 2 - 1 2 - 3 . 2 3 0 3 ‡ (-i) (+) [-] 03 12 56 -i 1 0 1 0 -1 0 0 4 [+i] [-] [-] 0 0 0 0 0 0 0 0 03 12 56 II 1 • (+i) [+] (+) 03 12 56 i 0 1 - 1 2 1 2 1 2 - 1 . 2 3 - 2 3 2 . [-i] (-) (+) 03 12 56 0 -1 1 - 1 2 - 1 2 1 2 - 3 . 2 3 0 3 [-i] [+] [-] 03 12 56 0 0 0 0 0 0 0 0 4 ‡ (+i) (-) [-] i -1 0 -1 0 1 0 0 03 12 56 III 1 [+i] [+] [+] 0 0 0 0 0 0 0 0 2 03 12 56 (-i) (-) [+] -i -1 0 0 -1 0 - 1. 3 2 3 3 • 03 12 56 (-i) [+] (-) -i 0 -1 1 2 - 1 2 - 1 2 1 . 2 3 2 3 03 12 56 4 ?? [+i] (-) (-) 0 -1 -1 - 1 2 - 1 2 1 2 1 . 2 3 2 3 IV 1 03 12 56 (+i) (+) [+] i 1 0 0 1 0 1. 3 - 2 3 03 12 56 2 [-i] [-] [+] 0 0 0 0 0 0 0 0 3 . 03 12 56 [-i] (+) (-) 0 1 -1 1 2 1 2 - 1 2 3 . 2 3 0 03 12 56 4 4 (+i) [-] (-) i 0 -1 - 1 2 1 2 1 2 3 . 2 3 0 Title Suppressed Due to Excessive Length 185 A^m d d -1 -1 members I Fig. 11.2 represents the 2 of the Clifford even ”basis × 2 2 2 f vectors” for the case that d =(5 + 1). The properties of I A^m are presented also in f Table 11.2. There are in this case again16 members. Manifesting the structure of subgroups SU(3)U(1) of the group SO(5, 1) they arerepresented as eigenvectors , S12 of the superposition of the Cartan subalgebra members(S03 , S56),that is with 31 (-S12 - iS03), 81 (S12 - iS03 == . (S12 - iS03 - 2S56), and 0 =- 1 + 23 23 S56). There are four self adjoint Clifford even ”basis vectors” with(3 = 0, 8 = 0, 0 = 0), one sextet of three pairs Hermitian conjugated to each other, one triplet and one antitriplet with the members of the triplet Hermitian conjugated to the corresponding members of the antitriplet and opposite. These 16 members of the Clifford even ”basis vectors” I A^m are the boson ”partners” of the Cliffordodd f b^m† ”basis vectors” , presented in Fig. 11.1 for one of four families, anyone. The f I reader can check that the algebraic application of A^m , belonging to the triplet, f transformstheCliffordoddsinglet, denotedonFig.11.1byasquare,tooneofthe members of the triplet, denoted on Fig. 11.1 by the circle . Looking at the boson fields I A^m† from the point of view of subgroups SU(3)U(1) f of the group SO(5 + 1) we will recognize in the part of fields forming the octet the colour gauge fields of quarks and leptons and antiquarks and antileptons. 11.2.3 Second quantized fermion and boson fields the internal spaces of which are described by the Clifford basis vectors. We learned in the previous subsection that in even dimensional spaces(d = 2(2n + 1) or d = 4n)the Cliffordodd and the Cliffordeven ”basis vectors”, which are the superposition of the Cliffordodd and the Clifford even products of a’s, respectively, offer the description of the internal spaces of fermion and boson fields. The Clifford odd algebra offers 2 d 2 -1 b^m† ”basis vectors” f , appearing in 2 d -1 2 Sab i families (with the family quantum numbers determinedby ~= { ~a ; ~bg-), 2 d d -1 Hermitian conjugated partners b^ m f fulfil -1 which together with their 2 × 2 2 2 the postulates for the second quantized fermion fields, Eq. (11.17) in this paper, Eq.(26) in Ref. [5], explaining the second quantization postulates of Dirac. The Clifford even algebra offers 2 d 2 -1× 2 d 2 -1 ”basis vectors” of I A^m† f (and the same number of II A^m† )with the properties of the second quantized boson fields f manifesting as the gauge fields of fermion fields described by the Cliffordodd ”basis vectors” b^m† . f The Cliffordodd and the Clifford even ”basis vectors” are chosen to be products ab of nilpotents, (k) (with the odd number of nilpotents if describing fermions and ab the even number of nilpotents if describing bosons), and projectors, [k]. Nilpotents and projectors are (chosen to be) eigenvectors of the Cartan subalgebra members of the Lorentz algebra in the internal space of Sab for the Clifford odd ”basis vectors” and of Sab(= Sab + S~ ab)for the Cliffordeven ”basis vectors”. To define the creation operators, either for fermions or for bosons besides the ”basis vectors” defining the internal space of fermions and bosons also the basis in 186 N. S. Mankoˇc Borˇstnik ordinary spacein momentumor coordinaterepresentationis needed.Here Ref.[5], Subsect. 3.3 and App.Jis overviewed. Let us introduce the momentum part of the single particle states. The longer version is presented in Ref. [5] in Subsect. 3.3 and in App. J. † ^ bp | =<0p ~ | b^~ j~p> = | 0p p - ~ >, < p , ~p † 0 0)=<0p jb^~b^ p p jp> = ( b ^† ^ ~~ 0 | 0p < >, p leading to p) , ~ ~ ~ p (11.30) 0 - ~ = ( p~ 0 b~p p ~ 1. While the quantized operators p ^ and ik i ;p^jg- = 0 and fx^;x^lg- = 0, it follows for fp^;x^jg- = iij. One with the normalization < 0p | 0p >= ^ correspondingly finds † y† < ~p j~x> = <0~p | b^~p b^~xj0~x >=(<0~x | b^~x b^~p j0~p >) y† † fb^;b^ = 0, f^^0 g- = 0, f^b^ = 0, ~p ~p 0 g- b~p;b~p b~p, ~p 0 g- y† † ^^^ fb^;b0 g- = 0, fb^~x;b~x 0 g- = 0, fb^~x;b0 g- = 0, ~x~x ~x 11 † i~p~x † -i~p~x ^^ f^b= e p;, f^b= e p, (11.31) b~p, ~xg- b~x, ~pg- (2)d-1 (2)d-1 . The internal space of either fermion or boson fields has the finite number of ”basis -1 22 vectors”, 2 d × 2 d -1, the momentum basis is continuously infinite. The creation operators for either fermions or bosons must be a tensor product, * T , of both contributions, the ”basis vectors” describing the internal space of fermions or bosons and the basis in ordinary, momentum or coordinate, space. ~ x commute fp^ 0 Thecreation operatorsforafree massless fermionofthe energy p ~ = jpj,belonging toa family f andtoa superpositionof family members m applying on the vacuum state j oc > * T j0~> canbe written as([5], Subsect.3.3.2, and thereferences p therein) X b^s† † b^m† * Tf ~ ~ f (p) f(p)^~p m where the vacuum state for fermions j oc , (11.32) sm b = c > * T j0~p > includes both spaces, the ~ internal part, Eq.(19.8), and the momentum part, Eq. (11.30) (in a tensor product for a starting single particle state with zero momentum, from which one obtains † the other single fermion statesofthe same ”basis vector”bythe operator b^ which ~p p 10). pushes the momentum by an amount 10 The creation operators and their Hermitian conjugated partners annihilation operators in bs† 0 the coordinaterepresentationcanbereadin[5]andthereferencestherein: ^ f (~x, x )= PRdd-10 bm† +. p ms † -i(px 0-"~p~ ^ . c (~p) b^ e x) ([5], subsect. 3.3.2., Eqs. (55,57,64) mf -. ( 2)d-1 f ~p and thereferences therein). Title Suppressed Due to Excessive Length 187 The creation operators fulfil the anticommutation relations for the second quantized fermion fields fb^ 0) ;b^s† > = ss 0 0 f‘ ( pf (~p)g+ j oc > j0~p ff 0 (~p - ~p) j oc > j0~p >, fb^f s ‘ 0 ( p~ 0) ;b^f s (~p)g+ j oc > j0~p > = 0. j oc > j0~p >, bs† fb^s f 00† ( p~ 0) , ^ f (~p)g+ j oc > j0~p > = 0. j oc > j0~p >, b^s† f (~p) j oc > j0~p > = j s f (~p) > b^sf(~p) j oc > j0~p > = 0. j oc > j0~p > jp 0| = j~p| . (11.33) (p)) and their Hermitian conjugated partners annihila- ~ bs† The creation operators ^ f tion operators b^ s (p), creating and annihilating the single fermion states, respectively, fulfil when applying on the vacuum state, j oc > * T j0~p >, the anticommutationrelations for the second quantized fermions,postulatedby Dirac (Ref. [5], Subsect. 3.3.1, Sect. 5). 11 To write the creation operators for boson fields we must take into account that boson gauge fields have the space index , describing the . component of the boson fieldin the ordinary space 12.We therefore add the space index . as follows ~ f A^m† f. Cm A^m† † * T f I f I b^ . (11.34) ~ (p) We treat free massless bosons of momentum = ~p 0 ~ ~ ~ p and energy p= jp| and of particular ”basis vectors” I A^m† ’s which are eigenvectors of all the Cartan subalgebra f members 13, Cmf. carry the space index . of the boson field. Creation operators operate on the vacuum state j ocev > * T j0~> with the internal space part justa p constant, j ocev >= | 1>, and fora starting single boson state witha zero momentum from which one obtains the other single boson states with the same ”basis † ^ bp, making vector”by the operators which push the momentum by an amount ~ p also Cmf. depending on ~ ~ ~ p. For the creation operators for boson fields in a coordinate representation we find using Eqs. (11.30, 11.31) Z+. dd-1 00 I A^m† p I A^m† -i(px (x, x (p) e p~ -"~x) 0)= (11.35) . j 0=j~ p| . f. f. p -. ( 2)d-1 11The anticommutationrelationsofEq.(11.33)arevalidalsoifwereplacethe vacuumstate, j oc > j0~p >, by the Hilbert space of Cliffordfermions generated by the tensor product multiplication, * TH , of any number of the Clifford odd fermion states of all possible internal quantum numbers and all possible momenta (that is of any number of b^s f † (~p) of any (s, f, ~p)), Ref.([5], Sect. 5.). 12 In the spin-charge-family theory the Higgs’s scalars origin in the boson gauge fields with the vector index (7, 8), Ref.([5], Sect. 7.4.1, and the references therein). 13 In general the energy eigenstates of bosons are in superposition of I A^m f † . One example, which uses the superposition of the Cartan subalgebra eigenstates manifesting the SU(3) × U(1) subgroups of the group SO(6), is presented in Fig. 11.2. 188 N. S. Mankoˇc Borˇstnik Tounderstand what new does the Cliffordalgebra description of the internalspace of fermion and boson fields, Eqs. (11.34, 11.35, 11.32), bring to our understanding of the second quantized fermion and boson fields and what new can we learn ~ PP I A^m† Cmf from this offer, we need to relate cab!ab. and , recognizing ab mff. A^m† Cmf that I are eigenstates of the Cartan subalgebra members, while !ab. are f. not. The gravity fields, the vielbeins and the two kinds of the spin connection fields, fa , !ab , !~ab ,respectively, are in the spin-charge-family theory (unifying spins, charges and families of fermions and offering not only the explanation for all the assumptions of the standard model but also for the increasing number of phenomena observed so far) the only boson fields in d =(13+1),observed ind =(3+1) besides as gravity also as all the other boson fields with the Higgs’s scalars included [27]. We therefore need to relate XX XX 1 0† 0f0 Sab mf b^m† I A^m Cm mf b^m† { (p) relate to f} (p) , !ab } f f ~ f0 2 abm m0f0 m 8f and . mf , X Scd ab Cmf (c mf !ab ) relate to Scd (I A^m† ) , f. ab . (m, f), . Cartan subalgebra member (11.36) Scd : Letberepeated that I are chosen to be the eigenvectors of the Cartan subal- A^m† f gebra members,Eq. (19.4). Correspondinglywe canrelatea particular I A^m† Cmf f. with sucha superpositionof !ab ’s which is the eigenvector with the same values A^m† Cmf of the Cartan subalgebra members as there is a particular I .We can do f. this in two ways: i. UsingthefirstrelationinEq. (11.36).Onthelefthandsideofthisrelation Sab’s bm† bm† apply on ^part of ^ ff A^m† f ~ (p). On the right hand side I same ”basis vector” f . b^m† ii. Usingthe secondrelation,in which Scd apply on the left hand side on !ab ’s XX Scdab ab c = c (11.37) mf !ab. mf i (!cb ad - !db ac + !ac bd - !ad bc);ab ab ab on each !ab. separately; cmf are constants to be determined from the second relation, where on the right hand side of this relation Scd(= Scd + S~ cd) apply on the ”basis vector” I A^m† of the corresponding gauge field. f Let us conclude this section by pointing out that either the Cliffordodd ”basis b^m† i A^m† vectors” or the Clifford even ”basis vectors” ;i =(I, II) have in any ff apply as well on the d -1 2 d 2 -1 members, while !ab. as well as !~ab. have each for each . even d2 d (d - 1) members. It is needed to find out what new does this difference bring into the -unifying theories of the Kaluza-Klein theories are. Title Suppressed Due to Excessive Length 189 11.3 Short overview and achievements of spin-charge-family theory The spin-chare-family theory[1,2,23,25,27–32]isakindofthe Kaluza-Klein theories [27,38–45] sinceitis built on the assumption that the dimensionof space-time is . (13 + 1) 14, and that the only interaction among fermions is the gravitational one (vielbeins, the gauge fields of momenta, and two kinds of the spin connection fields, the gauge fields of Sab and of S~ ab 15). This theory assumes as well that the internal space of fermion and boson fields are describedby the Cliffordodd and Clifford even algebra,respectively[6,7] 16. The theory is offering the explanation for all the assumptions of the standard model, unifying not only charges, but also spins, charges and families, [36,37,46,48,51] and consequently offering the explanation for the appearance of families of quarks and leptons and antiquarks and antileptons, of vector gauge fields [27], of Higgs’s scalar field and theYukawa couplings [28,30,32,36], for the differencesin masses among quarks and leptons [46, 51], for the matter-antimatter asymmetry in the universe [51], for the dark matter [49], making several predictions. The spin-charge-family theory shares with the Kaluza-Klein like theories their weak points, like: a. Not yet solved the quantization problem of the gravitational field 17. b. The spontaneous symmetry breaking which would at low energies manifest the observed almost massless fermions [30,32,34,39]. The spontaneously break of the starting symmetry of SO(13 + 1) with the condensate of the two right handed neutrinos (with the family quantum numbers of the group of four families, which does not include the observed three families([19],Table III),([5],Table6) bringing masses of the scale . 1016 GeV or higher to all the vector and scalar gauge fields, which interact with the condensate [25] is promising to show the right way [32–34]. The scalar fields (scalar fields arethe spin connection fields with the space index . higher than (0, 1, 2, 3))with the space index(7, 8) offer, after gaining constant non zero vacuum values, the explanation for the Higgs’s scalar and theYukawa couplings. They namely determine the mass matrices of quarks and leptons and antiquarks and antileptons. In Refs. [24,27] it is pointed out that the spin connection 14 d =(13 + 1) is the smallest dimension for which the subgroups of the group SO(13, 1) offer the description of spins and charges of fermions assumed by the standard model and correspondingly also of boson gauge fields. 15 If thereareno fermions present both spin connection fields areexpressible with vielbeins( [5], Eq. (103)). 16 Fermions and bosons internal spaces are assumed to be superposition of odd products of a’s(fermion fields)orof evenproductsof a’s (boson fields) what offers the explanation for the second quantized postulates of Dirac [16]. The ”basis vectors” of the internal spaces namely determine anticommutativity or commutativity of the corresponding creation and annihilation operators. 17 The description of the internal space of fermions and bosons as superposition of odd (for fermion fields) or even (for boson fields) products of the Cliffordobjects a’s seems verypromisinginlookingforanewwayto second quantizationofallfields,withgravity included, as discussed in this talk. 190 N. S. Mankoˇc Borˇstnik gauge fields do manifest in d =(3 + 1) as the ordinary gravity and all the observed vector and scalar gauge fields. The spin-charge-family theory assumesasimple starting action for second quantized massless fermion and the corresponding gauge boson fields in d =(13 + 1)dimensional space, presented in Eq. (19.1). The fermion part of the action, Eq. (19.1), can be rewritten in the way that it manifests in d =(3 + 1) in the low energy regime before the electroweak break by the standard model postulated properties of: i. Quarks and leptons and antiquarks and antileptons with the spins, handedness, charges and family quantum numbers. Their internal space is described by the Cliffordodd ”basis vectors” which are eigenvectors of the Cartan subalgebra of Sab and S~ ab, Eqs. (19.4, 11.29, 11.28). ii. Couplings of fermions to the vector gauge fields, which are the superposition of gauge fields !st , Sect. 6.2 in Ref. [5], with the space index . =(0, 1, 2, 3) and with the charges determinedby the Cartan subalgebraof Sab and S~ ab manifesting the symmetry of space (d - 4),andtothescalar gauge fields[1,2,23,24,26,29, 31,36,37,48–50] with the space index . . 5 and the charges determined by the Cartan subalgebra of Sab and S~ ab (as explained in the case of the vector gauge !ab fields), and which are superposition of either !st . or ~ , X — AiAiAAi = m(pm - g ). + Lfm A;i X — { s p0s } + s=7;8 X — { t p0t } , (11.38) t=5;6;9;:::;14 00 Sab ~Sab ~ where p0s = ps - 1 Sss" !s0s}s - 1 ~!abs, p0t = pt - 1 Stt" !t0t}t - 1 ~!abt, 22 22 . 0 with p0s = es p0 , m . (0, 1, 2, 3), s . (7, 8), (s;s}) . (5, 6, 7, 8), (a, b) (appearing 0 in S~ ab)run within either (0, 1, 2, 3) or (5, 6, 7, 8), t runs . (5, . . . , 14), (t;t}) run either . (5, 6, 7, 8) or . (9, 10, . . . , 14). The spinor function . represents all family -1 2 members of all the 2 7+1 = 8 families. The first line of Eq. (11.38) determines in d =(3+1) the kinematics and dynamics of fermion fields coupledtothe vector gauge fields[23,27,31].The vector gauge fields are the superposition of the spin connection fields !stm, m =(0, 1, 2, 3), (s, t)=(5, 6, · , 13, 14), and are the gauge fields of Sst, Subsect. (6.2.1) of Ref. [5]. Thereader can findin Sect.6of Ref.[5]a quite detailed overviewof theproperties which the masslessfermion and boson fields appearing in the simple starting action, Eq. (19.1), (the later only as gravitational fields) manifest in d =(3 + 1) as all the observed fermions — quarks and leptons and antiquarks and antileptons in each family — appearing in twice four families, with the lower four families including the observed three families of quarks and leptons and antiquarks and antileptons. The higher four families offer the explanation for the dark matter [49]. Table5andEq.(110)ofRef.[5]explainthatthe scalar fieldswiththespaceindex . =(7, 8) carry the weak charge 13 = 1 and the hyper charge Y = 1 , just as 22 assumed by the standard model. Title Suppressed Due to Excessive Length 191 Massesof familiesof quarks and leptons are determinedby the superpositionof the scalar fields, Eq. (108-120) of Ref. [5], appearing in two groups, each of them manifesting the symmetry SU(2)SU(2) U(1) 18. The scalar gauge fields with the space index (7, 8) determine correspondingly the symmetryof mass matricesof quarks and leptons([5],Eq. (111)) which appear in twogroups as the scalar fieldsdo [49,51].InTable5in Ref. [5]) the symmetry SU(2) × SU(2) × U(1) for each of the two groups is presented and explained. Although spontaneous symmetry braking of the starting symmetry has not (yet consistently enough) been studied and the coupling constants of the scalar fields among themselves and with quarks and leptons are not yet known, the known symmetry of mass matrices, presented in Eq. (111) of Ref. [5], enables to determine parameters of mass matrices from the measured data of the 3 × 3 sub mixing matrices and the masses of the measured three families of quarks and leptons. Although the known 3 × 3 submatrix of the unitary 4 × 4 matrix enables to determine 4 × 4 matrix, the measured 3 × 3 mixing sub matrix is even for quarks far accurately enough measured, so that we only can predict the matrix elements of the 4 × 4 mixing matrix for quarks if assuming that masses (times c2)of the fourth family quarks are heavy enough, thatis above oneTeV [46,49]. The new measurements of the matrix elements among the observed 3 families agree better with the predictions obtained by the sspin-charge-family theory than the old measurements. Thereader can findpredictionsin Refs.([50,51]) and the overviewin Ref.([5], Subsect. 7.3.1). The upper groupof four families offers the explanation for the dark matter,to which the quarks and leptons from the (almost) stable of the upper four families mostly contribute.Thereader canfindthereporton thisproposalforthe dark matter origin in Ref. [49] and a short overview in Subsect. 7.3.1 of [5], where the appearance, development and properties of the dark matter are discussed. The upper four families predict nucleons of very heavy quarks with the nuclear force among nucleons which is correspondingly very different from the known one [49, 52]. Besides the scalar fields with the space index . =(7, 8), which manifest in d =(3 + 1) as scalar gauge fields with the weak and hyper charge 1 and 2 1 , respectively, and which gaining at low energies constant values make fam 2 ilies of quarks and leptons and the weak gauge field massive, there are in the starting action, Eqs. (19.1), additional scalar gauge fields with the space index . =(9, 10, 11, 12, 13, 14). They are withrespect to the space index . either triplets or antitriplets causing transitions from antileptons into quarks and from antiquarks into quarks and back. 18 The assumption that the symmetry SO(13, 1) first breaks into SU(3)× U(1) × SO(7, 1) makes that quarks and leptons distinguish only in the part SU(3) × U(1), while the SO(7, 1) part is identical separately for quarks and leptons and separately for antiquarks and antileptons.Table7of Ref. [5],presenting one family, which includes quarks and leptons and antiquarks and antileptons, manifests these properties. The !ab , with the space index (7, 8) carry with respect to the flat index ab only quantum numbers + Y),13 (S56 - S78+ 23(S9 10 + S11 12 Q, Y, 4,(Q (= 13 (= 1 2 ), Y (= 4 ) and 4 =- 1 3 + S13 14 ), the flat index (ab) of !~ab , with the space index (7, 8), includes all(0, 1, . . . , 8) correspondingly forming the symmetry SU(2)SU(2) U(1). 192 N. S. Mankoˇc Borˇstnik Their properties are presented in Ref. [25] and briefly inTable9 and Fig.1 of Ref. [5]. Concerning this second point we proved on the toy model of d =(5 + 1) that the break of symmetry can lead to (almost) massless fermions [34]. In d =(3 + 1)-dimensional space — at low energies — the gauge gravitational fields manifest as the observed vector gauge fields [27], which can be quantized in the usual way. The author is in mean time trying to find out (together with the collaborators) how far can the spin-charge-family theory — starting in d =(13 + 1)-dimensional space with a simple and ”elegant” action, Eq. (19.1) — reproduce in d =(3 + 1) the observedpropertiesof quarks and leptons [23,25,27–32], the observed vector gauge fields, the scalar field and theYukawa couplings, the appearanceof the dark matter and of the matter-antimatter asymmetry,as well as the other open questions, connecting elementary fermion and boson fields and cosmology. The work done so far on the spin-charge-family theory seems promising. 11.4 Conclusions In the spin-charge-family theory[1,2,5,23,25,27–32] the Cliffordodd algebrais used to describe the internal space of fermion fields. The Clifford odd ”basis vectors” — the superposition of odd products of a’s — in a tensor product with the basis in ordinary space form the creation and annihilation operators, in which the anticommutativity of the ”basis vectors” is transferred to the creation and annihilation operators for fermions, offering the explanation for the second quantization postulates for fermion fields. The Cliffordodd ”basis vectors” have all the properties of fermions: Half integer spins with respect to the Cartan subalgebra members of the Lorentz algebra in the internal space of fermions in even dimensional spaces(d = 2(2n + 1) or d = 4n), as discussed in Subsects. (11.2.1, 11.2.3). Withrespecttothe subgroupsoftheSO(d - 1, 1) group the Cliffordodd ”basis vectors” appear in the fundamental representations, as illustrated in Subsects. 11.2.2. In this article it is demonstrated that the Clifford even algebra is offering the description of the internal space of boson fields. The Clifford even ”basis vectors” — the superposition of even products of a’s —ina tensorproduct with the basis in ordinary space form the creation and annihilation operators which manifest the commuting properties of the second quantized boson fields, offering explanation for the second quantization postulates for boson fields[9]. The Cliffordeven ”basis vectors”haveallthepropertiesof bosons: Integerspinswithrespecttothe Cartan subalgebra members of the Lorentz algebra in the internal space of bosons, as discussed in Subsects. (11.2.1, 11.2.3). With respect to the subgroups of theSO(d - 1, 1) group the Clifford even ”basis vectors” manifest the adjoint representations, as illustrated in Subsect. 11.2.2. There are two kinds of anticommuting algebras [2]: The Grassmann algebra, . offering in d-dimensional space 2:2d operators(2d a’s and 2d @a ’s, Hermitian Title Suppressed Due to Excessive Length 193 conjugated to each other, Eq. (11.3)), and the two Cliffordsubalgebras, each with 2d operators named a’s and~ a’s,respectively,[2,13,14],Eqs. (11.2-19.3). The operators in each of the two Cliffordsubalgebras appear in two groups of d d 2 -1× 2 2 -1 of the Cliffordodd operators (the odd products of either a’s in one subalgebra or of ~a’s in the other subalgebra), which are Hermitian conjugated to each other: In each Cliffordodd group of any of the two subalgebras there appear d -1 irreducible representation each with the 2 d 2 -1 members and the group of 2 their Hermitian conjugated partners. There are as well the Clifford even operators (the even products of either a’s in one subalgebra or of ~a’s in another subalgebra) which again appear in two groups of 2 d 2 -1× 2 d 2 -1 members each. In the case of the Clifford even objects the members of each group of 2 d 2 -1× 2 d 2 -1 members have the Hermitian conjugated partners within the samegroup, Subsect. 11.2.1,Table 11.1. The Grassmann algebra operators are expressible with the operators of the two Cliffordsubalgebras and opposite, Eq. (11.5). The two Cliffordsubalgebras are independent of each other, Eq. (19.3), forming two independent spaces. Either the Grassmann algebra [15,20] or the two Cliffordsubalgebras canbe used to describe the internal space of anticommuting objects, if the superposition of odd products of operators(a’s or a’s, or ~a’s) are used to describe the internal space of these objects. The commuting objects must be superposition of even products of operators(a’s or a’s or~ a’s). No integer spin anticommuting objects have been observed so far, and to describe the internal space of the so far observed fermions only one of the two Cliffordodd subalgebras are needed. Theproblemcanbe solvedby reducingthetwoCliffordsub algebrastoonlyone, the one (chosen to be) determined by ab’s. The decision that ~a’s apply on a as follows: { ~aB = (-)B i B agj oc >, Eq. (19.6), (with (-)B =-1, if B is a function of an odd products of a’s, otherwise (-)B = 1)enables that2 2 d -1 irreducible representations of Sab = i 2 f a ; bg- (each with the 2 2 d -1 members) obtain the Sab i a family quantum numbers determined by ~= { ~; ~bg-. 2 The decision to use in the spin-charge-family theory in d = 2(2n + 1), n . 3 (d . (13 + 1) indeed), the superposition of the odd products of the Cliffordalgebra elements a’s to describe the internal space of fermions which interact with the gravityonly(withthevielbeins,thegaugefieldsof momenta,andthetwokindsof the spin connection fields, the gauge fields of Sab and S~ ab,respectively), Eq.(19.1), offers not only the explanation for all the assumed properties of fermions and bosons in the standard model, with the appearance of the families of quarks and leptons and antiquarks and antileptons([5] and the references therein) and of the corresponding vector gauge fields and the Higgs’s scalars included [27], but also for the appearance of the dark matter [49] in the universe, for the explanation of the matter/antimatter asymmetry in the universe [25], and for several other observed phenomena, making severalpredictions [37,47,48,50]. Recognition that the use of the superposition of the even products of the Clifford algebra elements a’s to describe the internal space of boson fields, what appear 194 N. S. Mankoˇc Borˇstnik to manifest all the properties of the observed boson fields, as demonstrated in this articles, makes clear that the Cliffordalgebra offers not only the explanation for the postulates of the second quantized anticommuting fermion fields but also for the postulates of the second quantized boson fields. TherelationsinEq. (11.36) XX XX 1 0† 0f0 Sab mf b^m† I A^m Cm mf b^m† { !ab } (~p) relate to { f0 } (~p) , f f 2 abm m0f0 m 8f and . mf , X Scd ab A^m† Cmf (c mf !ab ) relate to Scd (I ) , f. ab . (m, f), . Cartan subalgebra member Scd , offers the possibility to replace the covariant derivative p0. 1Sab1S~ ab ~ p0. = p. - !ab. - !ab. 2 2 in Eq. (19.1) with X X m† I ^ e A I A^m† ICm f f. I Cem = p. - - p0. f. , f mf mf m† m† e where the relation among I A ^ !~ab , not discussed directly in this article, needs additional study and explanation. Although the properties of the Cliffordodd and even ”basis vectors” and correspondingly of the creation and annihilation operators for fermion and boson fields are, hopefully, clearly demonstrated in this article, yet the proposed way of the second quantization of fields, the fermion and the boson ones, needs further study to find out what new can the description of the internal space of fermions and bosons bring in understanding of the second quantized fields. Let be added that in even dimensional spaces the Cliffordodd ”basis vectors” carry only one handedness, either right or left, depending on the definition of handedness and the choice of the ”basis vectors”. Their Hermitian conjugated partners carry opposite handedness. The ”basis vectors” in the subspace of the whole spacedo have both handedness.In odd dimensional spaces(d =(2n + 1)) the operator of handedness is a superposition of an odd products of a’s. The eigenstates of the operator of handedness must be therefore the superposition of the Cliffordodd and the Clifford even ”basis vectors”. These eigenstates can have either right or left handed.Thepropertiesof ”basis vectors”in odd dimensional spaces are demonstrated in the App. 11.5 of this contribution for d = 1 and d =(2 + 1) spaces. It looks like that this study, showing up that the Cliffordalgebra can be used to describe the internal spaces of fermion and boson fields in an equivalent way, offering correspondingly the explanation for the second quantization postulates e and II A ^ ff. f I Cem II e Cm with respect to !ab. and f. Title Suppressed Due to Excessive Length 195 for fermion and boson fields, is opening the new insight into the quantum field theory, since studies of the interaction of fermion fields with boson fields and of boson fields with boson fields so far looks very promising. The study of properties of the second quantized boson fields, the internal space of which is described by the Clifford even algebra, has just started and needs further consideration. Studying properties of ”basis vectors” in odd dimensional spaces might help to understand anomalies of quantum fields. 11.5 Examples demonstrating properties of Clifford odd and even ”basis vectors” in odd dimensional spaces for d =(1), d =(2 + 1) The spin-charge-family theory, using even dimensional spaces, d =(13 + 1) indeed, offers the explanation for all the assumptions of the standard model, explaining as well the postulates for the second quantization of fermion and boson fields. The internal space of fermions is in this theory described by ”basis vectors” which are superposition of odd products of a’s while the internal space of bosons is described by ”basis vectors” which are superposition of even products of a’s. Subsect. 11.2.2 demonstrates properties of the Cliffordodd and even ”basis vectors” in special cases when d =(1 + 1), d =(3 + 1), and d =(5 + 1). Let us discuss here odd dimensional spaces, which have very different properties: 2 d -1 i. While in even dimensional spaces the Cliffordodd ”basis vectors” have 2 members m in 2 2 d -1 families f, b^m† f , and their Hermitian conjugated partners appear in a separate group of 2 d 2 d 2 -1 families, there are in odd -1 members in 2 d 2 -1 × 2 d 2 dimensional spaces some of the 2 -1 = 2d-2 Cliffordodd ”basis vectors” self adjoint and have correspondingly some of the Hermitian conjugated partners in another group with 2d-2 members. ii. In even dimensional spaces the Clifford even ”basis vectors” i A^m† ;i =(1, 2), f appear in two orthogonal groups, each with 2 d 2 -1 × 2 d 2 -1 members and each with 2 d -1 of them are self the Hermitian conjugated partners within the same group, 2 adjoint. In odd dimensional spaces the Cliffordeven ”basis vectors” appear in two groups, each with 2 d 2 -1 × 2 d 2 -1 = 2d-2 members, which are either self adjoint or have their Hermitian conjugated partners in another group. Not all the members of one group are orthogonal to the members of another group, only the self adjoint ones are orthogonal. bm† iii. While ^have in even dimensional spaces one handedness only (either right f or left, depending on the definition of handedness), in odd dimensional spaces the operatorof handednessisa Cliffordodd object, still commuting with Sab, which is the product of odd number of a’s and correspondingly transforms the Clifford odd ”basis vectors” into Clifford even ”basis vectors” and opposite. Correspondingly are the eigenvectors of handedness the superposition of the Cliffordodd and the Clifford even ”basis vectors”. Correspondingly there are in odd dimensional 196 N. S. Mankoˇc Borˇstnik spaces right handed and left handed eigenvectors of the operator of handedness. Let us illustrate the above mentioned properties of the ”basis vectors” in odd dimensional spaces, starting with the simplest case: d=(1) Thereis one Cliffordodd ”basis vector” ^ b1† = 0 1 and one Clifford even ”basis vectors” 1† ^A i = 1. 1 1† ^A 1† b^ The operator of handedness ..(0+1) = 0 transforms into identity i and 1 1 1† 1† into b^ ^A i . 1 1 The two eigenvectors of the operator of handedness are 11 ( 0 ( 0 - 1) , . + 1) , . 22 with the handedness(+1, -1), that is of right and left handedness. respectively. d=(2+1) There are twice 2d=3-2 = 2 Cliffordodd ”basis vectors”.We chose as the Cartan 01 01 0101 1† 2† 1† 2† ^^^^ subalgebra member S01 of Sab: b=[-i] 2 , b=(+i), b=(-i), b=[+i] 2 , 1 122 with the properties f = 1 ~S01 i = 2 01 1† ^b = [-i] 2 1 01 2† ^b = (+i) 1 f = 2 ~S01 S01 = - i , 2 01 1† ^b= (-i) - i 2 2 03 2† ^i b= [+i] 2 , 2 2 01 01 1† 2† 2† 1† ^ band b^ are self adjoint (up to a sign), b^=(+i) and b^=(-i) are Hermitian 12 12 ^A conjugated to each other. In odd dimensional spaces the ”basis vectors” are not separated from their Hermitian conjugated partners and are correspondingly not well defined. The operator of handedness is (chosen up to a sign to be) ..(2+1) = i 1 2 2 . Therearetwice 2(d=3)-2 = 2 Cliffordeven ”basis vectors”.Wechoose as the Cartan 0101 01 01 1† 1 I subalgebra member S01: = [+i], I ^A 2† 1 = II (-i) 2 , ^A 1† 2 = [-i], II ^A 2† 2 = (+i) 2 , with the properties Title Suppressed Due to Excessive Length 197 S01 S01 01 01 I A^1† II A^1† =[+i] 0 =[-i] 0 12 01 03 I A^2† II A^2† =(-i) 2 -i =(+i) 2 i, 12 0101 01 03 I A^1† =[+i] and II A^1† =[-i] are self adjoint, I A^2† =(-i) 2 and II A^2† =(+i) 2 12 12 are Hermitian conjugated to each other. In odd dimensional spaces the two groups of the Clifford even ”basis vectors” are not orthogonal. Let us find the eigenvectors of the operator of handedness ..(2+1) = i 0 1 2. Since it is the Cliffordodd object its eigenvectors are superposition of Cliffordodd and Cliffordeven ”basis vectors”. It follows 0101 0101 ..(2+1){ [-i] i [-i] 2} = { [-i] i [-i] 2} , 0101 0101 ..(2+1){ (+i) i (+i) 2} = { (+i) i (+i) 2} , 0101 0101 ..(2+1){ [+i] i [+i] 2} = { [+i] i [+i] 2} , 01 0101 01 ..(2+1){ (-i) 2 ± i (-i)} = { (-i) 2 ± i (-i)} , We can conclude that neither Cliffordodd nor Cliffordeven ”basis vectors” have in odd dimensional spaces the properties which they demonstrate in even dimensional spaces. i. In odd dimensional spaces the ”basis vectors” are not separated from their Hermitian conjugated partners and are correspondingly not well defined, that is we can not define creation and annihilation operators as a tensor products of ”basis vectors” and basis in momentum space. 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Mankoˇstnik, ”Spin-charge-family theory is offering next step in understand- c Borˇ ing elementary particles and fields and correspondingly universe”, Proceedings to the Conference on Cosmology, GravitationalWaves and Particles, IARD conferences, Ljubljana, 6-9 June 2016, The 10th Biennial Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields, J. Phys.: Conf. Ser. 845 012017 [arXiv:1409.4981, arXiv:1607.01618v2]. 24. N.S. Mankoˇstnik, ”The attributes of the Spin-Charge-Family theory giving hope c Borˇ that the theory offers the next step beyond the StandardModel”, Proceedings to the 12th Bienal Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields IARD 2020, Prague, 1 - 4 June 2020 by ZOOM. 25. N.S. Mankoˇstnik, ”Matter-antimatter asymmetry in the spin-charge-family theory”, c Borˇ Phys. Rev. D91(2015) 065004 [arXiv:1409.7791]. 26. N. S. Mankoˇstnik, ”How far has so far the Spin-Charge-Family theory succeeded c Borˇ to explain the StandardModel assumptions, the matter-antimatter asymmetry, the appearance of the Dark Matter, the second quantized fermion fields...., making several predictions”, Proceedings to the 23rd Workshop ”What comes beyond the standardmod- els”,4 -c Borˇ 12 of July, 2020 Ed. N.S. Mankoˇstnik, H.B. Nielsen, D. Lukman, DMFA Zaloˇstvo, Ljubljana, December 2020, [arXiv:2012.09640] zniˇ 27. N.S. Mankoˇstnik, D. Lukman, ”Vector and scalar gauge fields with respect to c Borˇ d =(3 + 1) in Kaluza-Klein theories and in the spin-charge-family theory”, Eur. Phys.J.C 77 (2017) 231. 28. N.S. Mankoˇstnik, ”The spin-charge-family theory explains why the scalar Higgs c Borˇ carries the weak charge ± 1 2 and the hyper charge 1 2 ”, Proceedings to the 17th Workshop ”What comes beyond the standardmodels”, Bled, 20-28 of July, 2014, Ed. N.S. Mankocˇ Borˇzniˇ stnik, H.B. Nielsen, D. Lukman, DMFAZalo ˇstvo, Ljubljana December 2014, p.16382[ arXiv:1502.06786v1] [arXiv:1409.4981]. 29. N.S. MankoˇstnikNS, ”The spin-charge-family theoryis explaining the origin c Borˇ of families, of the Higgs and theYukawa couplings”, J. of Modern Phys. 4 (2013) 823 [arXiv:1312.1542]. 30. N.S. Mankoˇstnik, H.B.F. Nielsen, ”The spin-charge-family theory offers under- c Borˇ standing of the triangle anomalies cancellation in the standardmodel”, Fortschritte der Physik, Progress of Physics (2017) 1700046. 31. N.S. Mankoˇstnik, ”The explanation for the origin of the Higgs scalar and for the c Borˇ Yukawa couplings by thespin-charge-family theory”, J.of Mod. Physics 6(2015) 2244-2274, http://dx.org./10.4236/jmp.2015.615230 [arXiv:1409.4981]. 32. N.S. Mankoˇstnik and H.B. Nielsen, ”Why nature made a choice of Cliffordand c Borˇ not Grassmann coordinates”,Proceedingstothe 20th Workshop ”What comes beyond the standardmodels”, Bled, 9-17 of July, 2017, Ed. N.S. Mankoˇstnik, H.B. Nielsen, D. c Borˇ Lukman, DMFAZalo ˇstvo, Ljubljana, December 2017, p. 89-120[arXiv:1802.05554v1v2]. zniˇ 33. N.S. Mankoˇstnik and H.B.F. Nielsen, ”Discrete symmetries in the Kaluza-Klein c Borˇ theories”, JHEP 04:165, 2014 [arXiv:1212.2362]. 34. D. Lukman, N.S. Mankoˇstnik and H.B. Nielsen, ”An effective two dimensionality c Borˇ cases bring a new hope to the Kaluza-Klein-like theories”, New J. Phys. 13:103027, 2011. 35. N.S. Mankoˇstnik, Second quantized ”anticommuting integer spin fields”, sentto c Borˇ arXiv. 200 N. S. Mankoˇc Borˇstnik 36. A. BorˇcBorˇ stnik, N.S. Mankoˇstnik, ”Left and right handedness of fermions and bosons”, J. of Phys. G: Nucl. Part. Phys.24(1998)963-977, hep-th/9707218. 37. A. Borstnik Braˇciˇˇc Borˇ c, N. S. Mankoˇstnik, ”On the origin of families of fermions and their mass matrices”, hep-ph/0512062, Phys Rev. D74073013-28 (2006). 38. T. Kaluza, ”On the unification problem in Physics”,Sitzungsber. d. Berl. Acad. (1918) 204, O. 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Blagojevi´c, Gravitation and gauge symmetries, IoP Publishing, Bristol 2002. 46.M.Breskvar,D.Lukman,N.S.Mankoˇstnik,”OntheOriginof Familiesof Fermions cBorˇ and Their Mass Matrices —Approximate AnalysesofPropertiesof Four FamiliesWithin Approach Unifying Spins and Charges”, Proceedings to the 9thWorkshop ”What Comes Beyond the StandardModels”, Bled, Sept. 16 -26, 2006, Ed. by Norma Mankoˇstnik, c Borˇ Holger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFAZalo ˇstvo, Ljubljana zniˇ December 2006, p.25-50, hep-ph/0612250. 47. G. Bregar, M. Breskvar, D. Lukman, N.S. Mankoˇstnik, ”Families of Quarks and c Borˇ Leptons and Their Mass Matrices”, Proceedings to the 10th international workshop ”What Comes Beyond the StandardModel”,17 -27of July, 2007,Ed. Norma Mankoˇstnik, c Borˇ Holger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFAZalo ˇstvo, Ljubljana zniˇ December 2007, p.53-70, hep-ph/0711.4681. 48. G. Bregar,M. Breskvar,D. Lukman, N.S. Mankoˇstnik, ”Predictions for four families cBorˇ by the Approach unifying spins and charges” New J. of Phys. 10 (2008) 093002, hepph/ 0606159, hep/ph-07082846. 49. G. Bregar, N.S. Mankoˇstnik, ”Does dark matter consist of baryons of new stable c Borˇ family quarks?”, Phys. Rev.D 80, 083534 (2009), 1-16. 50. G. Bregar, N.S. Mankoˇstnik, ”Can we predict the fourth family masses c Borˇfor quarksand leptons?”,Proceedings (arxiv:1403.4441)tothe16thWorkshop ”What comes beyond the standard models”, Bled, 14-21 of July, 2013, Ed. N.S. Mankoˇstnik, c Borˇ H.B. Nielsen, D. Lukman, DMFA Zaloˇstvo, Ljubljana December zniˇ2013, p. 31-51, http://arxiv.org/abs/1212.4055. 51. G. Bregar, N.S. Mankoˇstnik, ”The new experimental data for the quarks mixing c Borˇ matrix arein better agreement with the spin-charge-family theory predictions”, Proceedings to the 17th Workshop ”What comes beyond the standardmodels”, Bled, 20-28 of July, 2014, Ed. N.S. Mankoˇstnik, H.B. Nielsen, D. Lukman, DMFAZalo ˇstvo, Ljubljana c Borˇzniˇ December 2014, p.20-45[arXiv:1502.06786v1] [arxiv:1412.5866]. 52. N.S. Mankoˇstnik, M. Rosina, ”Are superheavy stable quark clusters viable candi- c Borˇ dates for the dark matter?”, International Journalof Modern PhysicsD(IJMPD) 24 (No. 13) (2015) 1545003. Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 201) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 12 AUnified Solution to the Big Problems of the Standard model R. N. Mohapatraa, N. Okadab a Maryland Center for Fundamental Physics and Department of Physics, University of Maryland, College Park, Maryland 20742, USA b Departmentof Physics, Universityof Alabama,Tuscaloosa, Alabama 35487, USA Abstract. We presenta unified model that solves four major problems of the standard model i.e. neutrino masses, originof matter, strongCPproblem and dark matter.We use theAffleck-Dine(AD) mechanismforthis purpose,withthe AD-fieldplayingtheroleof inflaton and where its cosmological evolution leads to the origin of matter. The model relates the neutrino masses to the baryon to photon ratio of the universe. The dark matter in the model is the axion field used to solve the strong CP problem. The model has two testable predictions: (i) a near massless Majorana fermion which contributes to Neff ~ 0:1 in the early universe, which can be tested in the upcoming CMB-S4 experiment, and (ii) the required valueof thereheat temperatureimplies that the lightest neutrino massis so small that it predicts the neutrinoless double beta decay parameter is between2to5 meV. 12.1 Introduction The standard model (SM) despite its experimental successes is an incomplete model. Its major deficiencies are its inability to explain three experimental observa- tions:(i)small neutrino masses;(ii) matter-anti-matter asymmetryintheuniverse; (iii) the dark matterof the universe.Afourth theoreticalproblem with theSMis why strong CP violating parameter . is so small (i.e. . . 10-10). In an attempt to address the first three of these problems, we recently proposed an extension of the SM [1] using the framework of the Affleck-Dine (AD) mechanism [2] for leptogenesis. In this model, a complex scalar field, called AD field here, generates the lepton asymmetry as it evolves from the early stage of the universe. Our model [1] provides an example of how to implement leptogenesis in a minimal modelwith radiative neutrino masses.TheAD fieldalsoplayedtheroleof inflaton whose non-minimal coupling to gravity leads to a viable model of inflation in theearly universe.ThustheADfieldplayedakeyroleinnotonly implementing leptogenesis but also in generating neutrino masses as well as the inflationary expansion.In this paper, we show howa similar buta more economical versionof the model in Ref. [1] can provide an axion solution to the strong CP problem. We work within the invisible axion model framework [6–9], of KSVZ type, where the Peccei-Quinn (PQ) symmetry breaking scale is in the range of 109 - 1012 GeV as required by astrophysical considerations. The PQ symmetry breaking also 202 R. N. Mohapatra, N. Okada provides a lepton number breaking term involving the AD field which is crucial to AD leptogenesis. Our starting point is how to implement leptogenesis in minimal models for small neutrino masses. As is well known, connecting the origin of neutrino masses to the matter-antimatter asymmetry via the mechanism of leptogenesis [10] is an attractive possibility and has been the subject of great deal of activity over the past decades [11, 12]. However, this connection is most compelling only for the caseoftypeIseesaw mechanism[13–17]withtwoorthreeright handed neutrinos. On the other hand, there are other very interesting mechanisms for generating small neutrino masses, such as type II, type III and inverse seesaw as well as loop models (see for some exampleofloop models[18–21] and an exhaustivereviewin Ref. [22]). In the latter class of models, it becomes necessary to add extra particles to implement leptogenesis. These extra particles do not have anything to do with neutrino mass generation but are put in solely to implementleptogenesis. For a discussion of traditional leptogenesis and the need for extra particles, see Ref. [24] for typeII seesaw, Refs. [25,26] for inverse seesaw, and Ref. [23] for loop models. For one class of loop models for neutrino masses, we showed in Ref. [1] that use ofAD mechanismprovidesawayto avoid adding extra particlesto generatethe lepton asymmetry., which in combination with the sphalerons, leads to baryon asymmetry of the universe [27]. (For a recent discussion of AD leptogenesis in the context of minimal type II seesaw models, see Ref. [39].) Our goal in this paper is to provide a new one loop model for neutrino masses, where AD leptogenesis works, without adding extra particles and to show how this model also provides a solution to the strong CP problem. Typically, in the AD mechanism, one relies on the cosmological evolution of a lepton number carrying complex scalar field (called here AD field and denoted here by ), with the Lagrangian of the model explicitly breaking lepton number (L), which plays an essential role in the generation of lepton asymmetry. While the L-breaking term could have any form, we choose it to have a quadratic form in the . field i.e. a 2 term, since with that particular choice, an analytic form for the baryonto entropy ratiocanbe derived.The neutrino massesinthis case arise from the same lepton number breaking 2 term in the Lagrangian. Thus, neutrino masses are a consequence of AD leptogenesis. Of course, neutrinos in this kind of scenario are naturally Majorana type fermions. There are thenrestrictions on the parameters of the model following from phenomenological and cosmological consistency. For example, in the AD leptogenesis models, the L asymmetry created by the AD field typically gets transferred to the SM sector at the inflation reheat temperature TR. So any lepton number washout interactions must decouple at temperature T* with T* TR. Furthermore, one must have TR >Tsph (where Tsph is the sphaleron decoupling temperature) for the lepton asymmetry to be convertedtobaryon asymmetry.Whilethese constraintsputastrongrestrictionon the model parameters, thereis stilla wide rangeofthem where the model works, as we show below. The model in this paperis similar to that of Ref. [1], though somewhat more economical with the neutrino mass arising from a different diagram. As in Ref. [1], we adopt a scheme where the inflaton and the AD fields are one and the same, 12 AUnified Solution to the Big Problems of the Standardmodel 203 unlike many original AD scenarios [2,28–30], thus providing unification of inflation and leptogenesis[31–40].Wefindit convenienttoadoptthe particular scenario proposed in Ref. [36, 37], although we believe it can be extended to other types ofAD modelsas well.We includea complex singlet fieldto implementthePQ solution to the strong CP problem. Adding the axion solution to strong CP in inflation models can lead to complications due to large iso-curvatureperturbations if PQ symmetry is broken during the inflationary period or domain wall problem ifthe scaleis belowthe inflation scale.Weshowhowto avoid theseproblems. Adistinguishing feature of our model is that cosmological consistency requires the existence of a near massless Majorana fermion which contributes Neff ~ 0:1 at the Big Bang Nucleosynthesis (BBN) epoch, This prediction canbe tested in the upcoming CMB-S4 experiment [41].We also show that the constraints on the reheat temperature after inflation predict a rate for the neutrinoless double beta decay, which can provide another test of the model. These two predictions are generic, not dependent on the choice of model parameters and can therefore test the basic framework. This paper is organized as follows: in sec. 2, we present an outline of the model and isolate its symmetries; in sec. 3, we discuss the evolution of the universe in this picture, and discuss leptogenesis and one loop generation of neutrino mass in sec. 4; in sec. 5, we discuss the constraints on the model parameters and provide a benchmark set and in sec. 6, dark matter candidate in the model is discussed; in sec.7,we commenton other possible implicationsofthis model.Sec.8is devoted toa summaryof theresults. 12.2 The model The model is based on the SM gauge group SU(3)c × SU(2)L × U(1)Y. The particle contentis listedinTableI.In additiontotheSM particle content,weintroducethe following new fields i.e. an AD field , whichis anSM singlet scalar and carriesa lepton number -1,a scalar SU(2)L doublet . withhypercharge Y =+1 and lepton number -1, three Majorana fermionic SM singlets i.To them, we add the field complex scalar field , which carries L =-1 and the PQ charge -1 as inTable I. The most general gauge invariant and U(1)PQ × U(1)L invariant Lagrangian of the model (in addition to the straightforwardkinetic terms) is given symbolically by L = Lkin + Linf(, R)- V(, , , H)+ LY. (12.1) Here, LY is thePQ invariantYukawa Lagrangian givenby X 1 LY = YuqHuc Hdc Hec i + + Ydq ~+ Y` ` ~+ YQQQc +(Y)ai ` aiiii + h:c:, 2 i (12.2) and .. 2 V(, , H, )= m jj2 + jj4 + 0()2(2)+ mHy. + h:c. (12.3)  -M2 jj2 + jj4 + mixjj2 jj2 . 204 R. N. Mohapatra, N. Okada Field U(1)PQ SM quantum number L Fermion ` a +1 (1, 2, -1) +1 c e a -1 (1, 1, +2) -1 q +1 (3, 2, +1=3) 0 c u -1 (3 * , 1, -4=3) 0 dc -1 (3 * , 1, +2=3) 0 Q -1 (3, 1, -2=3) +1=2 Qc +2 (1, 3 * , 1, +2=3) +1=2 i 0 (1, 1, 0) 0 Scalars . -1 (1, 2, +1) -1 H 0 (1, 2, +1) 0 . +1 (1, 1, 0) +1 . -1 (1, 1, 0) -1 Table 12.1: Particle content of the model responsible for one loop neutrino mass and dark matter and PQ symmetry. i are new fermionic fields, Q and Qc are new heavy quarksthat help implementing thePQ mechanism. The subscript a goes over lepton flavors and i goes over . flavors with a, i = 1, 2, 3.ThePQchargeofthe different fields are shown in the second column. The SM SU(3)c × SU(2)L × U(1)Y quantum numbers are in the thirdcolumn. Here, . is the field whose imaginary part is the axion field. Linf denotes the non-minimal . coupling to gravity of the form Linf =- 1 (M2 + jj2)R (see, 2P for example, Refs.[42,43])anditplaysacrucialrolein implementing successful inflation, R is the Ricci scalar,and MP = 2:41018 GeVisthereduced Planck mass. Note that the field . has a lepton number (as does )and. being a Majorana fermionhas zero lepton number.Without lossof generality,we can workina basis where the . fields are mass eigenstates We note usingTableIthat the Lagrangian has an exact global symmetry,U(1)PQ as well as a lepton number symmetry U(1)L. The model also has an automatic Z2 symmetry even after U(1)PQ breaking under which the fields , , . are odd and the rest of the fields are even. This Z2 symmetryremains exact and allows for 1 (the lightest among the Z2-odd particles) to be absolutely stable. For subsequent discussion, we assume the following mass hierarchy among the various particles: 11;mH;m` m. . 22;33;m. As we will see below, this allows . to decay only via a three body decay mode that involves the field 1 in the final state i.e. . › ` a + 1 + H. As we show below, this will allow us to relate the reheat temperature TR directly only to the unknown lightest active neutrino mass, which in turn allows us to choose TR appropriately. Once the Field . acquires a vacuum expectation value (vev), it will generate the m2 2 term, with m2 = 0f2 =2. This term breaks lepton number required PQ for neutrino mass generation as well as for AD leptogenesis. The . vev will also give rise to the axion field which prior to the QCD scale will remain as a massless 12 AUnified Solution to the Big Problems of the Standardmodel 205 particle and solve the strong CP problem. Since . field does not have a vev, its imaginary part does not contribute to the axion field. As we showina subsequent section, one loop Majorana masses for all neutrinos are proportional to whereas the baryon to entropy ratio generated by the AD mechanism is inversely proportional to [1,36], therebyrelating the neutrino mass with the lepton asymmetry in a way different from traditional leptogenesis. 12.3 Inflation and evolution of the AD field To discuss inflation in this model, note that there are two scalar singlets. and . unlike the model in Ref. [1] which only had the field . at the epoch of inflation. The field . is the mother-field of the axion and implements the PQ symmetry, as alreadystatedabove.Wecoupleonlyoneofthem non-minimallytogravityi.e. Linf =- 1 (M2 + jj2)R. This kindofa non-minimal gravity coupling emerges 2P naturallywithina supergravity embeddingof the model.Wedo not discuss this here. To discuss the evolution of the two scalars in the early universe, we expand the 11 fields into the radial and polar parts as . = . 'ei. and . = . ei. The . part 22 of the potential in the Einstein frame then looks like: V(', ) VE(', ) '  (12.4) 2 1 + . '2 M2 P with 1 1111 1 2 V(', )= m '2 + '4 - M22 + 4 + 02'2 cos(2. + 2)+ mix2'2 2 4242 4 (12.5) Note the negative sign in front of the . mass, which leadstoPQsymmetrybreaking. During inflation, . ~ MP and asaresult, theeffective potential for . turns out to be 1 10 V() ~ (-M2 )2 + 4 + M2 2 cos(2. + 2). (12.6) . + mixM2 PP 2 42 We see that by setting0 mix and mixM2 >M2 , the mass square of the P . field is now positive.We therefore expect the . field to quickly settle to its minimum at hi = 0 and therefore to play no role in inflation or generating curvature fluctuation. To discuss inflation, we proceed as follows: For . . MP, the potential in the Einstein frame is a constant, which leads to inflation. As the field . rolls down the potential, its value goes down and inflation ends as the slow roll parameters become of order one. After that the effect of the coupling of . to the Ricci scalar becomes unimportant. The angle . can take an arbitrary value when the inflation begins(. = O(1) is naturally assumed), making the real and imaginary parts of 206 R. N. Mohapatra, N. Okada the . fielddifferent.Itisthisdifferencewhichplaysakeyroleinthe development of the baryon asymmetry as the . becomes smaller. After inflation ends, the . field behaves like radiation while the '4 term is dominating the inflaton potential and its value goes down like . ~ a(t)-1, where a(t) is the scale factor. The rest of the story is same as in the paper [36] and concisely explained in Ref. [40]: When the . field gets smaller and reaches its value . 'oscil ~ m=, the '4 term becomes unimportant and the quadratic terms in the Lagrangian dominate . evolution. This leads to a damped harmonic oscillatory behavior of the real and imaginary parts of . with different frequencies due to the presence of the lepton number breaking term m2 2. Using the lepton  asymmetry formula nL ' -Im( ˙ . ), we can then calculate the lepton asymmetry that survives belowthereheat temperature.Thisgivesthe formula discussedinthe next section. The only difference between our case and Ref. [1] is the appearance of the . field as an independent field at this temperature. This is because as . becomes smaller, the mass square of the . field becomes negative and PQ sym- . metry breaks down as . becomes negligible and we get hi = fPQ = M=. The . field thenremains stuck there andeffectively generates the lepton number breaking term m2 2 .  Torealize the scalar field evolution discussed above, the parametersin the scalar potential must be suitably arranged. During inflation, the PQ symmetry is unbroken and hence M2 <mixM2 .Well before the damped harmonic oscillation of P the . field begins, . must be settled down at hi = fPQ to generate the m2 2  2 term. This leads to a condition, M2 = f2 >mix'2 . In . PQ oscil ~ (mix=)m addition, we impose mixf2 m). This 12 AUnified Solution to the Big Problems of the Standardmodel 207 partofthe discussionis similartothatinRef.[1].We choose 1 and H fields to be lighter than the . field. We choose parameters such thatTR = Km. with K<1. This helps to prevent the inverse decay ` + 1 + H › . so that the lepton asymmetry generated by . evolution is transmitted to the SM fields. In the next section, we will see the constraints imposedby thisrequirementon our model. We first note that in such a leptogenesis scenario, the lepton number to entropy ratio is given by [36] T3 nL R ' sin2. ' 10-10 . (12.7) s m2 MP . This formula is valid in our scenario despite the presence of the field . since it gets a vev around 1012 GeV and effectively decouples from the . evolution. An important input into this estimate of nL=s is thereheat temperature TR = Km, which must be less than the AD field mass m, i.e. K<1 as already noted. This implies the following relation between m, and K i.e. m. ' 10-10 MP. (12.8) K3 12.5 Neutrino mass, reheat temperature and washout decoupling In this section, we first look at the one loop neutrino mass generation in our model and then its relation to the reheat temperature and the decoupling temperature T* of the dangerous L-violating washout process that could potentially erase the lepton asymmetry. Our main goal will be to establish that in our model, we can satisfy the essential requirement that Tsph . TR . T. For this purpose, we will assume the following mass hierarchy among the fields, as already stated above, m;22;33 >m. 11. (12.9) We will see later on that the1 mass 11 actually has to be in the eV range or below if it is not to over-close the universe. Neutrino mass The diagram for one loop neutrino massis givenin Fig.1.We then estimate the light neutrino mass as v2 2m2 wk. . X-2 m. = YYT YY T , (12.10) . 4 162m. 2 v m2 wk. where X-2 = 2 , and µ = diag(11;22;33). For the second and third 4 162m. generation neutrinos, this one loop result must give a value of O(10-10) GeV for m. It turns out that for (Y)2a;3a ~ 1, ~ 10-5 , ß ~ 1, m. ~ 106 GeV, and m. = 22 = 33 ~ 106:5 GeV, we get the correct value for the neutrino masses of second and thirdgenerations. The resulting neutrino masses will then fit the oscillation data. The situation for the lightest neutrino mass m1 is however much 208 R. N. Mohapatra, N. Okada Fig. 12.1: Feynman diagram responsible for one loop neutrino mass. Arrows indicate the flow of the lepton number. The lower cross denotes the Majorana mass insertion of ()ii while the upper cross is for the insertion of m2 . smaller as we discuss below. Anyway,the neutrino oscillation fits do not determine the value of m1 . Reheat temperature and m1 Let us now evaluate the reheat temperature in terms of the parameters of the model. For that, we need the decay width of the AD field . whose only decay mode is . › ` a + 1 + H and it is given by 25 X m . ... ' (Y) * (Y)a1. (12.11) a1 4 323 m. a Now using the formula above for neutrino mass, we note that X 2 162m. m1 . X2 m1 (Y) * ;a1Y;a1 ' a1(Y)a1Y * 2 v 2 1 11 wk a † where we have used m. = U * DUwith D. = diag(m1 ;m2 ;m3 ) and MNSMNS the neutrino mixing matrix UMNS. This leads to the important connection between TR and m1 i.e. r 2 1=2 x2 mMP m1 . m1 TR '. XmMP = . , (12.12) 42. 11 vwk 2 11m. m. where x = . Thus as claimed earlier, this TR isrelated to the experimentally un m. determined neutrino observable m1 and can be adjusted to satisfy our constraint Tsph . TR 0 = 15m2=kg. (13.14) M We now wish to crudely estimate the amount of dust that might pile up around a dark matterbubble witha given velocity during the evolutionof the Universe. There are two important effects to be taken into account. First of all the metal densitywashigherinthepastduetothereductioninthe “radius”ofthe Universe by a factor (1 + z)-1 where z is the red shift. Secondly the metallicity was lower in thepastandweusethe linearfitsofDeCiaetal.[22]toitsz dependenceinour estimateof the rateof collectionof metalsby our pearls.We found that the most important time for the rate of collection of metals corresponds to z = 3:3, when the age of the Universe was 1.52 milliardyears. At this time the rate of collecting metals for a given velocity was about 8.4 times bigger than if using the present metallicity and density. So we might crudely estimate the amount of dust being collected by an 8.4 times bigger density of metals than today in the 8.9 times younger Universe, giving effective numbers for the dust settling: “metal density”= 3:71 * 10-31kg=m3 * 8:4 (13.15) eff = 3:1 * 10-30kg=m3 (13.16) = 1:7 * 10-3GeV=c2=m3 . (13.17) For orientation we could first ask how much metal-matter at all could be collected byadustgrain whilealreadyoftheorderof10-7m in size, meaningacross section of 10-14m2 and with a velocity of say 300 km/s = 3 * 105m=s during an effective age of the Universe of 1.52 milliardyears = 4:8 * 1016s.We obtain “available metals” = 3 * 105m=s * 4:8 * 1016 s * 10-14 m 2 * 3:1 * 10-30kg=m3 = 4:4 * 10-22kg (13.18) = 2:4 * 105GeV, (13.19) which is to be compared to what the mass of a (10-7 m)3 large dust particle with say specific weight 1000kg=m3 would be, namely 10-18kg. Sosucha“normal”sizedustgraincouldnotcollectitselfintheaverage conditions in the Universe. However if the grain to be constructed had lower dimension than 3, then the cross section could be larger for the same hoped for volume and thus mass. Decreasing say the thickness in one of the dimensions from the 10-7m to atomic size 10-10m wouldforthesame collectionofmattergivea1000timessmallermass.Thiswould bring such a “normal size” grain close to being just collectable in the average conditions in the Universe. 222 H.B. Nielsen, C. D. Froggatt Our speculated stronger forces than usual due to the big homolumo gap would not help much, because the grain cannot catch the atoms in intergalactic space which it does not come near enough to touch. We shall now estimate the inverse darkness for sucha dust grainof dimension2 or less attached to a dark matter bubble. In this case there is no shadowing of the dust atoms and the parameter “numberthickness” in equation (13.11) becomes unity. Also we estimated that in the main period when the dust attached itself to the dark matter bubbles, we had z = 3:3 and the age of the Universe was 1.52 milliardyears. The density of “metals” at that time was a factor 10-1 times the one today. So the factor 1=600 in equation (13.13) for the “metals” accessible to be caught by the dark matter composite particle becomes Mgrain 1 = 1%=6=10 = . (13.20) M 6000 So taking . jatom = 7 * 105 m 2=kg for the atoms of dust, we obtain our estimate M for the inverse darkness of the dark matter particle composed with a dust grain of dimension2 or less . Mgrain jcomposed = jgrain * (13.21) M MM 2 = 7 * 105 m =kg=6000 (13.22) 2 = 1:2 * 102 m =kg. (13.23) Our expected ratio . jcomposed = 120m2=kg (13.24) M should be compared with the value extracted from the dwarf galaxy data . jCorrea;v!0 = 15m2=kg. (13.25) M 13.5.2 Size of Individual Dark Matter Particles In the approximation of only gravitational interaction of dark matter it is well- known that only the mass density matters, whereas the number density or the mass per particle is not observable. With other than gravitational interactions one could hope that it would be possible to extract from the fits in say our model, what the particle size should be. But the possibility for that in our model is remarkably bad! The Correa measurement yields just the “inverse darkness” ratio . “cross section” = (13.26) M mass Our estimate for the rate of 3.5 keV radiation from dark matter seen by DAMA very crudely -was based on: • The total kinetic energy of the dark matter hitting the Earth per m2 per s (but not on how many particles). 13 Dusty Dark Matter Pearls Developed 223 • The main part of that energy goes into 3.5 keV radiation of electrons. • Estimateofa “suppression” factor for how smalla partof this electron radiation comes from sufficiently long living excitations to survive down to 1400 m into the Earth. None of this depends in our estimate on the size of the dark matter particles (provided it lies inside a very broad range)! If the dark matter particles were so heavy that the number density is so low that the observation over an area of about 1m2 would not get an event through every year, then it would contradict the DAMA data. The rate of dark matter mass hitting a square meter of the Earth is Rate = 300km=s * 0:3GeV=cm3 (13.27) = 3 * 105m=s * 5:34 * 10-22kg=m3 (13.28) = 1:6 * 10-16kg=m2=s (13.29) = 5 * 10-9kg=m2=y (13.30) Taking the DAMA area of observation~ 1m2 we need to get morethan one passage per year and thus M . 5 * 10-9kg (13.31) = 3 * 1018GeV. (13.32) Using the bubble internal mass density as estimated from the 3.5 keV homolumo gap, this upper bound implies that the bubble radius R . 10-7m. Fig. 13.6: Simulated size distribution for dust grains. We can assume a typical grain size (see Figure 13.6) of10-7m, say. Then using the . low velocity limit = 15m2=kg gives M M =(10-7 m)2=(15m2=kg) (13.33) = 7 * 10-16kg. (13.34) 224 H.B. Nielsen, C. D. Froggatt Butif now the dust grainis less than2dimensional, the area fora grain with the 10-7 m same weight asa massive3dimensional one wouldbe more than = 1000 10-10m times bigger, i.e an area bigger than 10-11m2. Correcting for this would give a bigger mass M . 7 * 10-13kg. 13.6 Conclusion We have reviewed and updated our dark matter model in which the dark matter consists of bubbles of a speculated new phase of the vacuum, in which there has collected so much “ordinary” matter that the surface tension of the separation surface between the two types of vacuum can be spanned out. These pearls are here assumedtobe surroundedbya lowerthanthree dimensionalgrainofdust mainly made from atoms of higher atomic weight than hydrogen and helium. We suppose that the Hausdorffdimension of the grain of dust is so low that the interaction between the pearls with their dust corresponds to effectively having no shadowing of the grain atoms by each other (with added up dimensionality less than2).Weusedthegeneral chemical abundancesand estimatedalow velocity inverse darkness of 120m2=kg for our pearls. This is only one order of magnitude larger -thus it essentially agrees with -the value 15m2=kg found by Correa [38]. Thisis summarizedinTable 13.1 as point1. In item2in the table we see that the estimate for the value v0 of the velocity at which the inverse darkness falls significantly down as a function of the velocity is 0:7cm=s if we do not take the hardening of the dust grain seriously, while it is 77km=s if we do take this hardening seriously. From the Correa estimate using the dwarf galaxy data one finds v0 = 220km=s, so only the estimate taking the hardening seriously agrees with experiment. Therestof the itemsinTable 13.1 are other orderof magnitude estimates checking the viabilityof our model. Thus item3estimates the rateof eventsin the DAMALIBRA experiment formulated in terms of the quantity suppression, which denotes the fraction of the excitations made in the dark matter pearls on entering the Earth, which survive down until the pearl reaches the detector. The similar item4 for the XENON1T experiment is now obsolete in as far as the effect found in this experiment was not reproduced after the radon gas was better cleaned away in XENONnT, so it was probably ß decay of 214Pb that was responsible for the previous effect. Item5called “Jeltema” represents the very strange observationof the 3:5keVlinefromtheTycho Brahe supernovaremnant, which shouldnothaveenough dark matter to produce the 3:5keV-line so as to be observed at all. But due to our dark matter particles being excitable by the large amount of cosmic rays in the supernova remnant, we indeed could get agreement with the observed rate of 2:2 * 10-5photons=cm2 coming from the supernova remnant. Asitem6 welistthefitofthe overall factorinthefitby ClineandFreytothe 3:5keV-line sources together with the very energy 3:5keV by one combination fS of our parameters for the model 1=4 . Actually this combination is essentially V the Fermi momentum of the electrons in the highly compressed matter in the 13 Dusty Dark Matter Pearls Developed 225 interior of our our bubbles. This fitting is only sensitive to the density of the matter inside the pearls and does not depend on the size of the pearls at the end. So this successful fit actually originates from earlier articles on our model, when we considered the pearls to be cm-sized and so heavy that an impact inTunguska could have causeda major catastrophe [3]. The last item, item7, justreviews the fact that we found approximately the same 3:5keV at first in three different places. However now after the sad development for our modelin therecent XENONnT experiment [34] onlyin twoplaces, namely inthe satellite etc. observationsof the 3.5 keV X-ray line andin the average energy of the modulating part of the DAMA-LIBRA observed events. InTable13.2wepresent some informationonthe massofthesingledark matter particle mass M (supposedly dominating the mass of the dust grain). Table 13.1: Successes #&exp/th Quantity value related Q. value 1. exp th Dwarf Galaxies inv. darkness = . = jv!0 M 15m2=kg 120m2=kg Mgrain M 2 * 10-5 1:6 * 10-4 2. Dwarf Galaxies exp Velocity par.v0 220km=s 4rdustE 8:1 * 1013kg=s2 th. with hardening 77km=s 4rdustE 1 * 1013kg=s2 th. without hard. 0:7cm=s 4rdustE 400kg=s2 3. DAMA-LIBRA exp 0:041cpd=kg suppression 1:6 * 10-10 th air 0:16cpd=kg 6 * 10-10 th stone 1:6 * 10-5cpd=kg 6 * 10-14 4. Xenon1T exp 2 * 10-4cpd=kg suppression 6 * 10-13 th air 0:16cpd=kg 6 * 10-10 th stone 1:6 * 10-5cpd=kg 6 * 10-14 5. exp th Jeltema &P. counting rate 2:2 * 10-5phs=cm2=s 3 * 10-6phs=s=cm2 . M jTycho . 1%* . * M jnuclear 25:6 * 10-3 cm =kg 28 * 10-4 cm =kg 6. exp th Intensity 3.5 kev N. M2 1023 2cm =kg2 23:6 * 1022 cm =kg2 1=4 fS V 0:6MeV-1 0:5MeV-1 7. Three Energies ast line 3. 5keV DAMA av. en. 3.4 keV Xen. av. en. 3.7 keV 226 H.B. Nielsen, C. D. Froggatt Table 13.2: MassM bounds and estimates Description R R M M Faster than year Corrected . 1:0 * 10-9 m . 3:1 * 10-9 m . 2:1 * 10-15kg . 6:5 * 10-14kg Dust enough . 1:0 * 10-9 m . 2 * 10-15kg Velocity dep. w. E= 4004 . 10-8 m 10-10 m big . 10-13kg . 2 * 10-18kg big DAMA stream . 10-7 m . 5 * 10-9kg Grain size 10-7 m 7 * 10-10 m 7 * 10-16kg Acknowledgements We acknowledge our status as emeriti at respectively Glasgow University and the Niels Bohr Institute, and H. B. N. acknowledges discussions at conferences like of cause the BledWorkshop but also at Corfu. References 1. C. D. Froggatt and H. B. Nielsen, Phys. Rev. Lett. 95 231301 (2005) [arXiv:astroph/ 0508513]. 2. C.D.Froggatt and H.B.Nielsen,Proceedingsof Conference: C05-07-19.3 (Bled 2005); arXiv:astro-ph/0512454. 3. C. D. Froggatt and H. B. Nielsen, Int. J. Mod. Phys. A 30 no.13, 1550066 (2015) [arXiv:1403.7177]. 4. C. D. 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The present work contains a review of some of the work we have done on complex action or non-Hermitian Hamiltonian theory, especiallytheresult that the anti- Hermitian part of the Hamiltonian functions by determining the actual solution to the equations of motion, that should be realized; this means it predicts the initial conditions. It shouldbe stressed thata majorresultof oursis that theeffective equationsof motion will in practice -after long time -be so accurately as if we had indeed a Hermitian Hamiltonian, and so there is at first nothing wrong in assuming a non-Hermitian one. In fact it would practically seem Hermitian anyway.Amajor new pointis that we seekby a bit intuitively arguing to suggest some cosmologically predictions from the mentioned initial conditions predicted:We seek evenby assuming essentially nothing but very generalpropertiesof the non-Hermitian Hamiltonian that we in practice should find a bottom in the (effective Hermitian) Hamiltonianand thattheUniverseat some moment should passthrougha (multiple) saddle point very closely, so that the time spent at it would be very long. Keywords: non-Hermitian Hamiltonian, inflation, weak value PACS: 11.10.Ef, 01.55 +b, 98.80 Qc. 14.1 Introduction It would be very nice to unify our knowledge of the equations of motion, or we could say the time-development, with our knowledge about the initial conditions, orasweshalllookuponithere,which solutionstotheequationsofmotionisbythe initial-condition-physics selected as the one to be realized, the true development. We[9,11–17,20,21,27,28,30–34]andalso Masao Ninomiya[1–8,10]havelong worked on the idea that the action should not be real, but rather complex. It has turned out that such theories of complex action or essentially similarly of non-Hermitian 1 in fact lead to a theory in which • The effect of the non-hermiticity is not seen after appreciable time in the equations of motion, so that effective hermiticity basically came out automatically; and thus this kind of theory is indeed viable! • But the initial conditions is predicted from such theories. 1 The Hamiltonian is not restricted to the class of PT-symmetric non-Hermitian Hamiltonians that were studied in Refs. [22–26]. 14 What givesa “theoryof Initial Conditions”? 229 But it is then of course very important for whether such a hypothesis of of a complex action or equivalently non-Hermitian Hamiltonian can be upheld, whether the action or the Hamiltonian can be arranged in a reasonable way so as to give some initial condition informationsmatching with what we know about the initial conditions having governed the world, the universe. It is the purpose of the present article in addition to reviewing our works on this complex action type of theory to argue even without making any true fitting of the Hamiltonian, except assuming it to have a classical analogue -using in fact a phase space consideration -but rather looking only at an essentially random form of the Hamiltonians, especially the anti-Hermitian part, a not so bad crude picture of the initial condition pops out. In fact what we call this crude success is that the favored or likely initial arrangement becomes that the system -the world -shallpass through and stay very long in saddle points.We namely interpret this prediction as being optimistically the prediction of the world going through an in some sense long stage of the inflation situation. An inflaton field having the value equal to the maximum of the (effective) potential for the inflaton field representsnamelyforeach Fourier componentofthis inflatonfieldasystemsitting at one of its saddle points. So indeed in the phenomenological development of the universe it goes through a state, which is precisely a saddle point, with respect to an infinity of degrees of freedom, namely the various Fourier components. The fact that we predict a very slow going through might be taken as an encouragement by comparison with, thatitisa well-knownproblem, “the slowrollproblem”, that theinflation for phenomenologicalreasons shouldbe kept going longer than expected unless the inflaton effective potential is especially (and somewhat in the models constructed) flat. Flatness should help to make the inflation period longer than it would be “ naturally”. Our long staying prediction might be taken as one of benefits of our model with the complex action seeking to get a long inflation, even witha less flateffective potential. In the following section 14.2 we shall talk about initial conditions and give very crude arguments for a long staying saddle point being favored. In section 14.3 we draw some crude phase space configurations from which we seek to get an idea about which behavior of the development of the mechanical system withthecomplex actionwouldbeto expect.Weendwith favoringthelongstay at the saddle point and going also to a region near a (local) minimum in the effective Hamiltonian, therebyexplaining an effective finding of a bottom of the Hamiltonian. Then, in section 14.4, we allude to our for the belief in our complex action havinga chancetobereallytrue most important derivation: thatyou would not observe any effect (except from the initial conditions) in practice after sufficient time. In section 14.5 we introduce the idea that when one includes the future as one should in our model one may write an expression for the expectation value ofa dynamical variable as theby Aharonov et.al. introduced weak value [18,19]. This is a possible scheme for extracting the to be expected average for experiment. Apriori this kind of weak values are complex, but we have made theorems which prove reality under some assumptions. 230 H.B. Nielsen, K. Nagao 14.2 Hope of Making Theory of Initial Conditions The laws of physics falls basically in the two classes: • The equations of motions including the possible states of the universe and thus the types of particles existing. (it is here we find the StandardModel). • Laws about the initial conditions. Here we may think of the second law of thermodynamics, and perhaps some cosmological laws as the Hubble expansion. Or may be inflation. We have long worked on the hypothesis that the Hamiltonian were not Hermitian. At first one thinks that this would have been seen immediately, but a major result of ours were: For the equations of motion there would be no clean signature of the Hamiltonian not being Hermitian left. The only significant revelation of the imaginary part of the Hamiltonian would be via the initial conditions. This then means that by such a non-Hermitian Hamiltonian theory we potentially has found a theory, that could function as a theory behind the initial state laws, we have at present. 14.3 Intuitive Understanding Let us give the reader an idea about what we have in mind by thinking of a skier with frictionless skies, so that he can onlystop when he has run up the hill and lost the kinetic energy. Then there is some given distribution of the quality of the outlook he can enjoy in different places or of some other sort of attraction which the skier would like to enjoy as long as possible. Itisnotagoodideatojust startata random attractive outlookpointwith splendid outlook, becausetheskierwillmostlikelyfindthatonthehillsideandhewillrush down and thus away from the attractive outlook point quickly. Even arranging to slide firstupwithspeedastojuststopbyloosingthe kinetic energyatan attractive point, might not compete with finding an evena bit less attractive outlook but still very attractive outlook point at the “middle” of a pass, in which one can stand seemingly forever.Just a little accidental slide to one side or the other of course in the pass leads to that he slides down and the attractive outlook place soon gets lost. With quantum mechanics sucha slight leaving the very metastable saddle pointin the pass is unavoidable. So with quantum mechanics the skier has to plan that he cannot be in the saddle point forever, but will slide out some day. Then he has to plan for the next step what is most profitable. Presumably it is best to arrange to find a reasonable attractive outlook place in a little whole in the landscape of only very little less potential energy than the starting situation chosen. Then namely it could be arranged that the skier would only ski little and slowly around when first arriving there. Presumably it would be best to then if possible have arranged togetbackagaintothe first saddlepointinthepassandperhaps cyclicallyrepeat again and againa good trip. 14 What givesa “theoryof Initial Conditions”? 231 Fig. 14.1: This just a skiing terrain, that should really symbolize in our work any stateof the whole Universe.We imaginealittle skier with frictionless skies, which can ski around but his tour is fixed from where and with what velocity he starts. He cannot stop except if he just runs up the hill and runs out of kinetic energy. Fig. 14.2: Now we have put on some red spots which are the regions with the best outlooks or for other reasons the best ones to stay in. Now the skier gets the problem of starting in such a clever way that he manages to stay the longest time in thesethebestregions (markedbyred).Goingjusttoagoodregionat random would probably mean that he very fast would rush out of it and it would only be a short enjoy of the good region. What to do? 14.3.1 Phase space drawings Wenowpresentafewfiguressupposedtobe drawninphase space, ratherthan in a geometrical space with mountains, to illustrate again the considerations which the little skier has to do to get the most glorious outlook for so long as possible. One must think of an integral over time of a quantity measuring the beauty of the outlook now in different “places” in phase space because the outlook beauty degree can of course also depend on the momentum, or say the velocity. In our theory with non-Hermitian or in classical thinking complex Hamiltonian the quantity to be identified with the beauty degree for the skier is of course 232 H.B. Nielsen, K. Nagao the imaginary part ImH of the Hamiltonian. This imaginary part ImH namely enhances the normalization of the wave function describing the the skier or in our model say the universe as it moves along classically in the phase space. Thus the route through phase space which maximizes the time integral over the R imaginary part ImHdt is the one that makes the wave function grow the most. This means that the chance for surviving the tourbythe skier or rather the universe the developmentof whichis describedby the tour has the largest amplitude for R existingatallattheendofthetour,whentheintegral overtime ImHdt is the largest.Itis therefore our theorypredicts that whatreally shall happen most likely istheroutethroughphasespacegivingthis integralthe maximal value.Sowesee that a problem like the one for the little skier is set up. Fig. 14.3: Symbolic Phase Space for Universe, Level curves for ReH Now some are figures formulated in phase space illustrating these consideration: seefigures 14.3,...,14.6.How shouldthe system chooseto move?Tokeepred,or yellow, and avoid turquoise? • It could startintheredto ensurea favorableImHinthe start,butalas,it comes out in the turquoise and spend a lot of time with very unfavorable Im H. • It could choosea not too bad, i.e. e.g yellow, place with high stability so thatit can stay there forever and enjoy at least the yellow! Whatwe wouldliketo learnfrompathfavorableforhighImHintegrated over time? What we want to learn from this consideration: It will usually be favorable withregardtoImHto chooseavery stableplaceto avoidrunningaroundand loosing enormously (the turquoise) with regardto Im H. So this kind of theory predicts: • Preferably Universe should be just around a very stable, locally ground state, it is the vacuum, with the bottom in the Hamiltonian. 14 What givesa “theoryof Initial Conditions”? 233 Fig. 14.4: Symbolic Phase Space for Universe, Level curves for ReH and Color for ImH Imaginary partImHsymbolizedby colors: Red very strongly wished; Yellow also very good, but not the perfect; turquoise strongly to be avoided, bad! Fig. 14.5: Symbolic Phase Space for Universe, Level curves for ReH and Color for ImH Imaginary partImHsymbolizedby colors: Red very strongly wished; Yellow also very good, but not the perfect; turquoise strongly to be avoided, bad! 234 H.B. Nielsen, K. Nagao • Even better might be using a saddle point (there are also more of them and soitmorelikelytobebest)andthen chooseitsothatthereisa stable local ground statenotsofartospend eternity.Sucha saddleisthetipofthe inflaton effective potential. By choosing just that tip in principle it can stand as long as to be disturbed by quantum mechanics. Fig. 14.6: Symbolic Phase Space for Universe, Level curves forReHand Color for ImH Red very stronglywished, the saddle point very good;Yellow also very good, but not the perfect, but on this figure it can circle around more stably in the yellow;turquoisestronglytobe avoided,bad, canbe avoidedby keepingaround the fix points! Results of the intuitive treatment of the non-Hermitian Hamiltonian are as follows: • The world-system should run around so little as possible to avoid the low Im Hplaces (= the unfavorable ones): It should have small entropy (contrary to the intuitive cosmology of Paul Framptons in the other talk). Rather close to stable point (= ground state). So the model predicts there being a bottom in the effective Hamiltonian locally in phase space. • The world-system should stay as long as possible at a saddle almost exactly to keep staying surprisingly long (like a pen standing vertically on its tip in years); but this is in the many degrees of freedom translation a surprisingly long inflation era. That is the problem of the too many e-foldings, which we thus at least claimed to have a feature of the initial condition model helping in the right direction (making inflation longer than the potential that should preferably not be flat indicates). 14 What givesa “theoryof Initial Conditions”? 235 14.4 Main Result Our main result is that you would not discover from equations of motion that the Hamiltonian had an anti-Hermitianpart. The system (the world)would only have significant Hilbert vector components in the states with the very highest imaginary part of the eigenvalues of the non-Hermitian Hamiltonian, since the rest would die out with time. Remember jA(t) > = exp(-iHt)jA(0) >. (14.1) So at least the anti-Hermitian part is near to its (supposed) maximum, and thus at least less significant.We introducea new innerproduct making the Hamiltonian Hnormal, i.e. making the Hermitian and the anti-Hermitian parts commute. So it is unnecessary to assume that the Hamiltonian is Hermitian! It will show up so in practice anyway! 14.5 WeakValue As a result of our thinking of how to interpret the complex action theory we came to the concept already studied by Aharonov et.al. [18, 19], the weak value: Owv(t)= (14.2) or better with time development included: Owv(t)= , (14.3) wherewe have assumed that the states jB(t) > and jA(t) > time-develop according to the following Schr¨odinger equations: d jB(t) > =-iHyjB(t) >, (14.4) dt d jA(t) > =-iHjA(t) >. (14.5) dt Our idea is to use the weak value instead of the usual operator average: Oav(t)= . (14.6) One motivation is that it may give more natural interpretation of functional integrals. Usually the answer to how to use functional integrals is: “You can use it to calculateatime development operator -e.gan S-matrix -andthenusethatto propagate the quantum system in the usual Hilbert space formalism.” The weak value has a more beautiful functional integral expression: R O. * A exp( i S[path])Dpath Bh — Owv(t)= R , (14.7) * A exp(( i S[path])Dpath Bh — 236 H.B. Nielsen, K. Nagao where in the numerator the operator O was inserted at the appropriate time, than the usual operator average. The usual average and the weak value look a priori quite different, but with what we call the maximization principle, that the absolute value of the denominator of Eq.(14.2): | | = | | (14.8) be maximal for fixed normalization of the two states, you may see that (at least for Hermitian Hamiltonian) one gets jB(t) > . jA(t) >. (14.9) In Ref. [13] we have found that one can construct such an inner product jQ that, even if at first the Hamiltonian H is not normal, i.e. if [H, Hy] 6(14.10) = 0, then, with regardto this new inner product, it is [H, HyQ ]= 0. (14.11) The new inner product2 can arrange a normal Hamiltonian. The inner product can be described as composed from the usual one | and a Hermitian operator Q constructed from H. I.e. jQ = jQ means < :::jQ ::. > = < :::jQj::. > . (14.12) One can thus talk abouta Q-Hermitian operator O when it obeys OyQ = O (14.13) where OyQ = Q-1OyQ. (14.14) Remember the point of our new inner product was to make the at first not even normal Hamiltonian at least normal, i.e. the Q-Hermitian and the anti-Q-Hermitian parts commute. It is the idea that the physical observables one should use in a world witha non-Hermitian Hamiltonian are Q-Hermitian. In Ref. [27] we proposed the following theorem “maximization principle in the future-included complex action theory”: Asaprerequisite, assume thata given Hamiltonian H is non-normal but diagonalizable and that the imaginary parts of the eigenvalues of H are bounded from above, and define a modified inner product jQ bymeansofa Hermitian operatorQ arranged so that H becomes 2 Similar innerproducts are also studiedin Refs. [24,25,29]. 14 What givesa “theoryof Initial Conditions”? 237 normal with respect to jQ. Let the two states jA(t) > and jB(t) > time-develop according to the Schr¨odinger equations with H and HyQ respectively: jA(t) > = exp(-iH(t - TA))jA(TA) >, (14.15) jB(t) > = exp(-iHyQ (t - TB))jB(TB) >, (14.16) and be normalized with jQ at the initial time TA and the final time TB respectively: = 1, (14.17) = 1. (14.18) Next determine jA(TA) > and jB(TB) > so as to maximize the absolute value of the transition amplitude | | = | j. Then, provided that an operator O is Q-Hermitian, i.e., Hermitian with respect to the inner product jQ, i.e. OyQ = O, the normalized matrix element of the operator O defined by BA = (14.19) Q becomes real and time-develops under a Q-Hermitian Hamiltonian. We note that this theorem shows that the complex action theory could make predictions about initial conditions. 14.6 Conclusion We have put forwardour works of looking at a complex action or better a non- Hermitian Hamiltonian. Since it would not be easily seen that the Hamiltonian were indeed non-Hermitian after sufficiently long time and only showing itself up as it were the initial conditions that were influenced by the anti-Hermitian part, and even this influence looks promising, we believe that a complex action of non-Hermitian Hamiltonian model like the one described has indeeda good chance to be the truth. Our complex action theory would make predictions about initial conditions.An intuitive useof non-HermitianHsuggested explanation for: Effective bottom in the Hamiltonian; Long Inflation; Low Entropy. One shouldstressthatone should considerita weaker assumptionto assumea non-Hermitian Hamiltonian than a Hermitian one, in as far as it is an assumption that the anti-Hermitian part is zero, while assuming the non-Hermitian Hamiltonian is just allowing the Hamiltonian to be whatever. It is only because we are taught aboutthe Hermitian Hamiltonianfromthe traditionthatwetendto considerita new and strange assumption to take the Hamiltonian to be non-Hermitian. Acknowledgments This work was supportedbyJSPS KAKENHI Grant Number JP21K03381, and accomplishedduringK.N.’s sabbaticalstayin Copenhagen.Hewouldliketothank 238 H.B. Nielsen, K. Nagao the members and visitors of NBI for their kind hospitality and Klara Pavicic for her various kind arrangements and consideration during his visits to Copenhagen. H.B.N. is grateful to NBI for allowing him to work there as emeritus. Furthermore, the authors would like to thank the organizers of Bled workshop 2022 for their kind hospitality. References 1. H. B. Nielsen and M. Ninomiya, Proc. Bled 2006: What Comes Beyond the Standard Models, pp.87-124 (2006) [arXiv:hep-ph/0612250]. 2.H.B. NielsenandM.Ninomiya,Int.J.Mod.Phys.A23,919(2008). 3.H.B. NielsenandM.Ninomiya,Int.J.Mod.Phys.A24,3945(2009). 4. H. B. Nielsen and M. Ninomiya, Prog. Theor. Phys. 116, 851 (2007). 5. H. B. Nielsen and M. Ninomiya, Proc. Bled 2007: What Comes Beyond the Standard Models, pp.144-185 (2007) [arXiv:0711.3080 [hep-ph]]. 6. H. B. Nielsen and M. Ninomiya, arXiv:0910.0359 [hep-ph]. 7. H. B. Nielsen, Found. Phys. 41, 608 (2011) [arXiv:0911.4005[quant-ph]]. 8. H. B. Nielsen and M. Ninomiya, Proc. Bled 2010: What Comes Beyond the Standard Models, pp.138-157 (2010) [arXiv:1008.0464 [physics.gen-ph]]. 9. H. B. Nielsen, arXiv:1006.2455 [physic.gen-ph]. 10. H. B. Nielsen andM. Ninomiya, arXiv:hep-th/0701018. 11. H. B. Nielsen, arXiv:0911.3859 [gr-qc]. 12. H. B. Nielsen, M. S. Mankoc Borstnik, K. Nagao, and G. Moultaka, Proc. Bled 2010: What Comes Beyond the StandardModels, pp.211-216 (2010) [arXiv:1012.0224 [hepph]]. 13. K. Nagao and H. B. Nielsen, Prog. Theor. Phys. 125, 633 (2011). 14. K. Nagao and H. B. Nielsen, Prog. Theor. Phys. 126, 1021 (2011); 127, 1131 (2012) [erratum]. 15. K. Nagao and H. B. Nielsen, Int. J. Mod. Phys. A27, 1250076 (2012); 32, 1792003 (2017)[erratum]. 16. K. Nagao and H. B. Nielsen, Prog. Theor. Exp. Phys. 2013, 073A03 (2013); 2018, 029201 (2018)[erratum]. 17. K. Nagao and H. B. Nielsen, Prog. Theor. Exp. Phys. 2017, 111B01 (2017). 18.Y.Aharonov,D.Z.Albert,andL.Vaidman,Phys.Rev. Lett.60,1351 (1988). 19.Y.Aharonov,S.Popescu,andJ.Tollaksen,Phys.Today63,27 (2010). 20. K. Nagao and H. B. Nielsen, Prog. Theor. Exp. Phys. 2013, 023B04 (2013); 2018, 039201 (2018)[erratum]. 21. K. Nagao and H. B. Nielsen, Proc. Bled 2012: What Comes Beyond the StandardModels, pp.86-93 (2012) [arXiv:1211.7269 [quant-ph]]. 22. C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998). 23.C.M. Bender,S.Boettcher,andP. Meisinger,J.Math.Phys.40,2201(1999). 24. A. Mostafazadeh, J. Math. Phys. 43, 3944 (2002). 25. A. Mostafazadeh, J. Math. Phys. 44, 974 (2003). 26.C.M. BenderandP.D.Mannheim,Phys.Rev.D84, 105038(2011). 27. K. Nagao and H. B. Nielsen, Prog. Theor. Exp. Phys. 2015, 051B01 (2015). 28. K. Nagao and H. B. Nielsen, Prog. Theor. Exp. Phys. 2017, 081B01 (2017). 29.F.G. Scholtz,H.B.Geyer, andF.J.W. Hahne, Ann. Phys. 213,74 (1992). 30. K. Nagao and H. B. Nielsen, Fundamentals of Quantum Complex Action Theory, (Lambert Academic Publishing, Saarbrucken, Germany, 2017). 14 What givesa “theoryof Initial Conditions”? 239 31. K. Nagao and H. B. Nielsen, Proc. Bled 2017: What Comes Beyond the StandardModels, pp.121-132 (2017) [arXiv:1710.02071 [quant-ph]]. 32. K. Nagao and H. B. Nielsen, Prog. Theor. Exp. Phys. 2019, 073B01 (2019). 33. K. Nagao and H. B. Nielsen, Prog. Theor. Exp. Phys. 2022, 091B01 (2022). 34. K. Nagao and H. B. Nielsen, arXiv:2209.11619 [hep-th]. Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 240) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 15 Emergent phenomena in QCD: The holographic perspective GuyF. deT´eramond Laboratorio deF´isicaTe ´ oricayComputacional, Universidadde Costa Rica, 11501 San Jos ´ e, Costa Rica email:gdt@asterix.crnet.cr Abstract. Abasic understandingoftherelevantfeaturesofhadronphysicsfromfirstprinciples QCD has remained elusive and should be understood as emergent phenomena, which depend critically on the number of dimensions of Minkowski spacetime. These properties include the mechanism of color confinement, the origin of the hadron mass scale, chiral symmetry breakingand the pattern of hadronic bound states. Some of these complex issues have been recently addressed in an effective computational framework of hadron structure based ona semiclassical approximation to light-front QCD and its holographic embedding in AdS space. Theframework embodies an underlying superconformal algebraic structure which leads to the introduction of a mass scale within the superconformal group, and determines the effective confinement potential of mesons, baryons and tetraquarks, while keeping the pion massless. This new approach to hadron physics leads to relativistic wave equations similarintheir simplicitytotheSchr¨odinger equationin atomic physics. 15 Emergent phenomena in QCD: The holographic perspective 241 15.1 Introduction The interactions between the fundamental constituents of hadrons, quark and gluons, observed in high energy scattering experiments is described to high precision by Quantum Chromodynamics (QCD), thus establishing QCD as the standard theory of the strong interactions. At large distances, however, the nonperturbative natureof the strong interactions becomes dominant anda basic understandingof the essential featuresofhadron physicsfrom first principlesQCDhasremained an important unsolved problem in the standardmodel of particle physics. Basic hadronic properties are not explicit properties of the QCD Lagrangian but emergent phenomena, among them: The mechanism of color confinement, the origin of the hadron mass scale, the relation between chiral symmetry breaking and confinement, the massless pion vs. the massive proton in the chiral limit, bound statesandthe patternofhadron excitations. Other important aspectsofthestrong interaction, such as the emergence of Regge theory, Pomeron physics and the Veneziano amplitude, were introduced in dual models before the advent of QCD, and should also be considered large distance QCD emergent phenomena. Our present goal is trying to understand how emerging QCD properties would appear in an effective computational framework of hadron structure and its dependence on the dimensionality of physical spacetime. QCD admits an Euclidean lattice formulation [1] which has been established as a rigorous framework to study hadron structureand spectroscopy nonperturbatively. However, dynamical observables in Minkowski spacetime cannot be obtained directly from the Euclidean lattice. Quantum computation of relativistic field theoriesusingthe Hamiltonian formalismin light-front quantization[2]represents a promising venue, but its development is still at the exploratory phase [3]. Other nonperturbative methods based on the Schwinger-Dyson and the Bethe-Salpeter equations, and other approximations and models of the strong interactions are described in Ref. [4]. Recent theoretical developments based on AdS/CFT – the correspondence between classicalgravityina higher-dimensional anti-de Sitter (AdS) space and conformal field theories (CFT) in physical space-time [5], have provided a semi- classical approximation for strongly-coupled quantum field theories, giving new insights into nonperturbative dynamics [6]. This approach provides useful tools for constructing dual gravity models in higher dimensions which incorporates confine- mentand basicQCDpropertiesin physical spacetime.Theresulting gauge/gravity duality is broadly known as the AdS/QCD correspondence, or holographic QCD. Or approach to holographic QCD is based on the holographic embedding of Dirac’s relativistic front form of dynamics [2] into AdS space, thus its name Holographic Light-Front QCD (HLFQCD). This framework leads to relativistic wave equationsin physical space-time, similartotheSchr ¨ odinger or Dirac wave equationsin atomic physics[7–9]. This approach has its originsin theprecise mapping between the hadron form factorsin AdS space [10] andphysical spacetime, which can be carried out for an arbitrary number of quark constituents [11]: It leads to the identification of the invariant transverse impact variable . for the n-parton 242 GuyF.deT´eramond bound state in physical 3+1 spacetime with the holographic variable z, the fifth dimension of AdS. Aremarkable property of HLFQCD is the embodiment of a superconformal algebraic structure which is responsible for the introduction of a mass scale within the algebra.This symmetryalsofixesthe confinement interactionleadingtoamassless pion in the chiral limit (the limit of zero quark masses) and to striking connections betweenthe spectrum of mesons, baryons and tetraquarks [12–17]. Further ex- tensionsof HLFQCDprovide nontrivialrelations between the dynamicsof form factors and quark and gluon distributions [18–20] with pre-QCD nonperturbative approaches such as Regge theory and theVeneziano model. In this introductorypresentationIwillgivean overviewofrelevant aspectsof the semiclassicalapproximationtoQCD quantizedinthelightfront(LF)in1+ 1and3+1spacetime dimensions, followedby the holographic embeddingin AdS5 spaceofthe(3+1) semiclassicalQCD wave equations withan emphasis on the underlying superconformal structure for hadron spectroscopy. Other relevant aspects and applications of the light-front holographic approach have been described in the recent review [21]. 15.2 Critical role of the dimensionality of spacetime and QCD emergent phenomena The number of dimensions of physical spacetime is critical in determining whether hadronicpropertiesare complex emergent phenomena which ariseoutoftheQCD Lagrangian, or can (at least in principle) be computed and expressed in terms of the basic parameters of the QCD Lagrangian [22]. Our starting point is the QCD action in d dimensions with an SU(N) Lagrangian written in terms of the fundamental quark and gluon gauge fields, . and A, Z .. Ga . LS = dd x. — (i Dµ - m) . - 1 Ga , 4 + fabcAb where Dµ = @µ - igTaAa and Ga = @Aa - @Aa Ac , with [Ta;Tb]= . µ ifabc and a, b, c are SU(N) color indices.Asimple dimensional analysis of the QCD action gives [ ] ~ M(d-1)=2 , (15.1) [A] ~ M(d-2)=2 , (15.2) [g] ~ M(4-d)=2 :g (15.3) It follows from g that in 1 + 1dimensions, for example, the QCD coupling g has dimensions of mass, [g] ~ M. In this case, the theory can be solved for any number of constituents and colors using discrete light-cone quantization (DLCQ) methods [23,24]. All physical quantities canbe computedin termsof the basic1+ 1Lagrangian parameters, the coupling and the quark masses, but no emergent phenomena appear. 15 Emergent phenomena in QCD: The holographic perspective 243 In contrast, in 3+1 dimensions the coupling g is dimensionless and, in the limit of massless quarks, the QCD Lagrangian is conformally invariant1. The need for the renormalization of the theory introduces a scale QCD, which breaks the .. 2 conformal invariance and leads to the “running coupling” s = g2()=4. and asymptoticfreedom [25,26] for large valuesof 2. The scale QCD is determined in high energy experiments: Its origin and the emergence of hadron degrees of freedom out of the constituent quark and gluon degrees of freedom of the QCD Lagrangian in the nonperturbative domain remains a deep unsolved problem. 15.3 Semiclassical approximation to light-front QCD + 03 LF quantization uses the null plane x= x+ x= 0 tangent to the light cone as the initial surface,thus withoutreferencetoa specificLorentz frame[2]. Evolution in LF time x+ is given by the Hamiltonian equation . + M2 @P2 LFHEi j i = P-j i;P-j i = j i, (15.4) @x+ P+ for a hadron with 4-momentum P =(P+;P-;P?), P± = P0 ± P3, where the LF Hamiltonian P- isa dynamical generator and P+ and P. are kinematical. Hadron mass spectra can be computed from the LF invariant Hamiltonian P2 = PPµ = P+P- - P2 [9] . P2M2P2j (P)i = M2j (P)i. (15.5) The simple structure of the LF vacuum allows for a quantum-mechanical probabilistic interpretation of hadron states in terms of the eigenfunctions of the LF P Hamiltonian equation P2M2 in a constituent particle basis, j i = njni, writ- n ten in terms of the quark and gluon degrees of freedom in the Fock expansion. In practice, solving the actual eigenvalue problem P2M2 is a formidable computational task fora non-abelian quantum field theory beyond1+1dimensions, and particularly in three and four-dimensional space-time with an unbound particle number with arbitrary momenta and helicities. Consequently, alternative methods and approximations are necessary to tackle the relativistic bound-states in the strong-coupling regime of QCD. 15.3.1 QCD(1+1) The ’t Hooft model [27] in one-space and one-time dimensions, constitutes the first example of a semiclassical Hamiltonian wave equation derived from first principles QCD in light-front quantization [2]. This equation is exact in the large-N limit and leads to the computation of a meson spectrum and light front wave functions in terms of the constituent quark and antiquark, while incorporating chiral symmetry breaking (CSB) and confinement. 2 1 The QED abelian coupling . = e =4. is also dimensionless, but the physical observables in atomic physics can be computed and, in contrast with the proton, depend critically on the constituent masses in the QED Lagrangian. 244 GuyF.deT´eramond InQCD(1+1)gluonsarenot dynamical,therearenogluon self-couplings,and quarks have chirality but no spin. The coupling g has dimension of mass and itisa confining gauge theory for any valueof the coupling.We can express the + 0 QCD LagrangianLin1+1dimensions, withLF coordinates x= x+ x3 and - x= x0 - x3, in the A+ = 0 gauge in terms of the fields ± . R;L and A- . The LF constraint equations imply that there is only one independent degree of freedom, +. The hadron 2-momentum generator P =(P+;P-), P± = P0 ± P3, is then expressed in terms of the field + [24,28,29] with Z  2  mj+a 1j+a + † 2 Pm11P- = dx- + + g, (15.6) i@+ (i@+)2 † for the LF Hamiltonian where j+a = Ta +. From the inverse derivative in the + interactiontermin Pm11(the term withthe coupling) there followsthe potential V Z 2 - V =-g dx-dy-j+a(x -) x - - y j+a(y -). (15.7) The pion mass spectrum can be computed from the LF eigenvalue equation P2M2 for QCD(1+1), namely P+P-j(P+)i = M2 j(P+)i. For the qq —valence state it  leads to [24,29] 22  Z1 mq mq —N dx0 (x)- (x 0) tHE + (x)+ P = M2 (15.8) . (x), x1 - x. (x - x0)2 0 .. 2 the tHooft equation [27] with effective couplingN = gN2 - 1 =2N, where x is the longitudinal momentum fraction of the qq —state. Cancellation of singularities q at x = and x = 1 - for the approximate solution (x) ~ x q (1 - x)ß —tHE leads ..1=2 2 for m=N 1 to q = 3m2 =N and qq r N .. M2 =(mq + m q—)+ O (mq + m q—)2 . (15.9) . 3 p In QCD(1+1) both, the value of the CSB “condensate” h i = f2 N=3 and the  strength of linear confinement depend on the value of the coupling g in the QCD Lagrangian, and are not emerging properties2. 15.3.2 3p1QCD(3+1) In3 +1 dimensions we alsostart withtheQCD LagrangianinLand assume that, to a first semiclassical approximation, gluons with small virtualities are non- dynamical and incorporated in the confinement potential [7]. This approximation entails an important simplification of the full LF Hamiltonian P-, which we express in terms of the dynamical quark field +, ± = . , ± = 0 ± in the A+ = 0 gauge [9] Z 2 2 (ir?)+ m Pm31P- = dx-d2 x. — + + + interactions, (15.10) i@+ 2 The glueball spectrumhas been computedina large-N modelofQCDin1+1dimensions [30]. 15 Emergent phenomena in QCD: The holographic perspective 245 to compute the mass spectrum from the LF eigenvalue Eq. P2M2. For a qq —bound state we factor out the longitudinal X(x) and orbital eiL. de- . iLX(x)()/ pendence from the LF wave function , (x, , )= e2, where 2 = x(1 - x)b2 is the invariant transverse separation between two quarks, with . b?,therelative impact variable, conjugatetotherelative transverse momentum k. with longitudinal momentum fraction x. In the ultra-relativistic zero-quark masslimitthe invariantLF HamiltonianEq.P2M2,with P- givenby Pm31, can be systematically reduced to the wave equation [7]  d2 1 - 4L2 LFWE -- + U() ()= M2(), (15.11) d2 42 where the effective potential U comprises all interactions, including those from higher Fock states. The critical value of the LF orbital angular momentum L = 0 corresponds to the lowest possible stable solution. The LF equation LFWE is relativistic and frame-independent; It has a similar structure to wave equations in AdS provided that one identifies . = z, the holographic variable [7]. 15.4 Higher spin wave equations in AdS The semiclassical LF bound-state wave equation LFWE can be mapped to the equations of motion which describe the propagation of spin-J modes in AdS space [7,8].To examine this equivalence, we start with the AdS action fora tensor- J field J = N1:::NJ in the presence of a dilaton profile '(z) responsible for the confinement dynamics Z . .. '(z) 2 SAdSS = ddxdz ge DMJDMJ - 2 , (15.12) J where g is the determinant of the metric tensor gMN, d is the number of transverse coordinates, and DM is the covariant derivative which includes the affine connection. The variation of the AdS action leads to the wave equation  d-1-2J  '(z)  ze(R)2 AdSWEJ - @z @z + J(z)= M2J(z), (15.13) '(z) d-1-2J 2 ezz after a redefinition of the AdS mass , plus kinematical constraints to eliminate lower spin from the symmetric tensor N1:::NJ [8]. By substituting J(z)= (d-1)=2-J ze-'(z)=2 J(z) in AdSWEJ, we find the semiclassical light-front wave equation LFWE with 11 2J - 3 UvarphiUJ()= '00()+ '0()2 + '0(), (15.14) 24 2. for d = 4 as long as . = z. The precise mapping allows us to write the LF confinement potential U in termsofthe dilatonprofile which modifiestheIRregion of AdS space to incorporate confinement [9], while keeping the theory conformal invariant in the ultraviolet boundary of AdS for z › 0, which corresponds to the 246 GuyF.deT´eramond 4-dimensional physical boundaryof AdS space [21].The separationof kinematic and dynamic components, allows us to determine the mass function in the AdS action in terms of physical kinematic quantities with the AdS mass-radius (R)2 = L2 -(2 - J)2 [7,8]. Asimilar derivation follows from the Rarita-Schwinger action for a spinor field J . N1:::NJ-1=2 in AdS with the result [8]  d2 1 - 4L2 -- + U+() + = M2 +psi1, (15.15) d2 42  d2 1 - 4(L + 1)2 -- + U-() - = M2 -psi2, (15.16) d2 42 RR with . = z, and equal probability d. 2 ()2 d. 2 = -(). The semiclassical LF + wave equations for + and - correspond to LF orbital angular momentum L and L + 1 with 1 + 2L UVU()= V2() ± V 0()+ V(), (15.17) . a J-independent potential, in agreement with the observed degeneracy in the baryon spectrum. 15.5 Superconformal algebraic structure and emergence of a mass scale The precise mapping of the semiclassical light-front Hamiltonian equations to the wave equations in AdS space gives important insights into the nonperturbative structureof bound state equations in QCD for arbitrary spin, but it does not answer the question of how the effective confinement dynamics is actually determined, and how it can be related to the symmetries of QCD itself. An important clue, however,comesfromtherealizationthatthe potential V() inEq.UV playstherole ofthe superpotentialin supersymmetric(SUSY)quantum mechanics(QM)[31].In fact, the idea to apply an effective supersymmetry to hadron physics is certainly not new[32–34],but failedto accountforthe specialroleofthepion.In contrast, as we shall discuss below, in the HLFQCD approach, the zero-energy eigenmode of the superconformal quantum mechanical equations is identified with the pion whichhasno baryonic supersymmetric partner,a pattern whichis observed across the particle families. Supersymmetric QM is based on a graded Lie algebra consisting of two anticom- muting supercharges Q and Q† , fQ, Q} = fQy;Qy} = 0, which commute with the 1 Hamiltonian H = fQ, Qyg, [Q, H]=[Qy;H]= 0. If the state jEi is an eigenstate 2 with energy E, HjEi = EjEi, then,it followsfrom the commutationrelations that the state QyjEi is degenerate with the state jEi for E 60, but for E == 0 we have QyjE = 0i = 0, namely the zero mode has no supersymmetric partner [31]; a key result for deriving the supermultiplet structure and the pattern of the hadron spectrum which is observed across the particle families. 15 Emergent phenomena in QCD: The holographic perspective 247 Following Ref. [13] we consider the scale-deformed supercharge operator R. = 1 Q + S, with K = fS, Sy} the generator of special conformal transformations. 2 † † The generator R. is also nilpotent, fR;R} = fR } = 0, and gives rise to a ;R   † 1 new scale-dependent Hamiltonian G, G = fR;R  g, which also closes under the 2 graded algebra, [R;G] = [R †  ;G] = 0. The new supercharge R. has the matrix representation   00 0r. = R= † , (15.18) RexR. , † . 00 0 r . † f f with r. =-@x + + x.The parameter f is dimensionless and . + x, r = @x + . x x has the dimension of[M2],andthus,amassscaleisintroducedinthe Hamiltonian without leaving the conformal group. The Hamiltonian equation GjEi = EjEi leads to the wave equations  d2 1 - 4(f+)2 2 -- + 2 x + 2. (f-) + = E+, phi1 (15.19) dx2 4x2  d2 1 - 4(f-)2 2 -- + 2 x + 2. (f+) - = E-, phi2 (15.20) dx2 4x2 which have the same structure as the Euler-Lagrange equations obtained from the holographic embedding of the LF Hamiltonian equations, but here, the form of 2 the LF confinement potential, 2x, as well as the constant terms in the potential are completely fixed by the superconformal symmetry [16, 17]. 15.5.1 Light-front mapping and baryons Upon mapping phi1 and phi2 to the semiclassical LF wave equations psi1 and psi2 using the substitutions x 7 › , E 7 › M2;f 7 › L+;+ 7 › - and - 7 › +, we find the result U+ = 22 + 2(L + 1) and U- = 22 + 2L for the confinement potential of baryons [16]. The solution of the LF wave equations for this potential gives the eigenfunctions +() . -() . 1 2 3 2 +L -2=2 eLL (2) (15.21) n +L -2=2LL+1 e (2) (15.22) n with eigenvalues M2 = 4(n + L + 1). The polynomials LL (x) are associated n Laguerre polynomials, where the radial quantum number n counts the number of nodesin the wave function.We comparein Fig. ?? the model predictions with the . measured values for the positive parity nucleons [35] for . = 0:485 GeV. 15.5.2 scMBSuperconformal meson-baryon symmetry Superconformal quantum mechanics also leads to a connection between mesons and baryons [17] underlying the SU(3)C representationproperties, sincea diquark cluster can be in the same color representation as an antiquark, namely 3 . 3 × 3. — 248 GuyF.deT´eramond ..0.485 GeVN(939) N(1720)N(1680) N(2220) N(1440) N(1900) N(1710) n.0n.1n.2n.3012340123456LM2.GeV2. .. ... . .(1232) .(1700).(1620) .(1950).(1920).(1910).(1905) .(2200) .(2420) ..0.498 GeV.(1600) .(1900).(1930) n.0n.10123401234567LM2.GeV2. Fig. 15.1: fig:nucleon-delta Model predictions for the orbital and radial positive-parity nucleons (left) and positive and negativeparity . families (right) compared with the data p. . from Ref. [35]. The values of . are . = 0:485 GeV for nucleons and . = 0:498 GeV for the deltas. The specific connection follows from the substitution x 777 = › , E › M2;. › B M;f 777 › LM-= LB+;+ › M and 2 › B in the superconformal equations phi1andphi2.WefindtheLF meson(M)–baryon(B) bound-state equations  d2 1 - 4L2 M M -- + UM M = M2 M, (15.23) d2 42  d2 1 - 4L2 B B -- + UB B = M2 B, (15.24) d2 42 with the confinement potentials UM = 2 2 + 2M(LM - 1) and UB = 2 2 + MB 2B(LB + 1). The superconformal structure imposes the condition . = M = B and theremark- able relation LM = LB + 1, where LM is the LF angular momentum between the 15 Emergent phenomena in QCD: The holographic perspective 249 quark and antiquark in the meson, and LB between the active quark and spectator cluster in the baryon. Likewise, the equality of the Regge slopes embodies the equivalence of the 3C - 3— C color interaction in the qq —meson with the 3C - 3— C interaction between the quark and diquark cluster in the baryon. The mass spectrum fromMandBis MNspecM2 = 4(n + LM) and M2 = 4(n + LB + 1). (15.25) MB The pion has a special role as the unique state of zero mass and, since LM = 0, it has nota baryon partner. 0246 r,w a2,f2 r3,w3a4,f4024LM = LB + 11-20158872A3M2 (GeV2) D 3–2+ D 1–2- ,D 3–2- D 1–2+ D 11–2+ ,D 3–2+ ,D 5–2+ ,D 7–2+ Fig. 15.2: fig:rho-delta Supersymmetric vector meson and . partners from Ref. [17]. The . experimental values of M2 from Ref. [35] are plotted vs LM = LB + 1 for . ' 0:5 GeV. The . and . mesons have no baryonic partner, sinceit would implya negative valueof LB. 15.5.3 Spin interaction and diquark clusters Embedding the LF equations in AdS space allows us to extend the superconformal Hamiltonian to include the spin-spin interaction, a problem not defined in the chiral limit by standardprocedures. The dilaton profile '(z) in the AdS action SAdS can be determined from the superconformal algebra by integrating Eq. Uvarphi for the effective potential U. One obtains the result '(z)= z2, which is uniquely determined, provided that it depends only on the modification of AdS space. Since the dilaton profile '(z)= z2 is valid for arbitrary J, it leads to the additional term 2. in the LF Hamiltonian for mesons and baryons, which maintains the meson-baryon supersymmetry [36]. The spin = 0, 1, is the total internal spin of the meson, or the spin of the diquark cluster of the baryon partner. The effect of the spin term is an overall shift of the quadratic mass as depicted in 250 GuyF.deT´eramond Fig. ?? for the spectra of the . mesons and . baryons [17]. For the . baryons the 1 total internal spin S is related to the diquark cluster spin by S =+ (-1)L, and 2 therefore, positive and negative . baryons have the same diquark spin, = 1. As a result, all the . baryons lie, for a given n, on the same Regge trajectory, as shown in Fig. ??. 15.5.4 Inclusion of quark masses In the usual formulation of bottom-up holographic models one identifies quark mass and chiral condensates as coefficients of a scalar background field X0(z) in AdSspace[37,38].Aheuristicwaytotakeinto accountthe occurrenceofquark mass terms in the HLFQCD approach is to include the quark mass dependence in the invariant mass which controls the off-shell dependence of the LF wave function[9].This substitutionleads,upon exponentiation,toanatural factorization of the transverse and the longitudinal wave functions, but it is not a unique prescription [21]. This approach has been consistently applied to the radial and orbital excitation spectra of the light , , K, K * and . meson families, as well . as to the N, , , , . * ;. and . * in the baryon sector, giving the value . = 0:523 ± 0:024 GeV [36]. For heavy quarks the mass breaking effects are large. The underlying hadronic supersymmetry, however, is still compatible with the holographic approach and givesremarkable connections across the entire spectrumof light and heavy-light hadrons [39].In particular,the lowest mass meson defining the K, K * ;0, , D, D * ;Ds, B, B * ;Bs and B * families has no baryon partner, conforming to the SUSY mechanism found s for the light hadrons, and depicted in Fig. ??. 15.5.5 Completing the supersymmetric hadron multiplet Fig. 15.3: fig:MBTplet The meson-baryon-tetraquark supersymmetric 4-plet fM;+ ;- ;T } follows from the two step action of the supercharge operator R† : BB  — 3 › 3 × 3 on the pion, followed by 3 › 3 —× 3 —on the negative chirality component of the nucleon. 15 Emergent phenomena in QCD: The holographic perspective 251 Besides the mesons and the baryons, the supersymmetric multiplet . = fM;+ ;- ;T } BB containsa further bosonic partner,a tetraquark, which,as illustratedinFig. ??, fol † lows from the action of the supercharge operator Ron the negative-chirality com . ponentofa baryon [36].Aclear exampleis the SUSY positive parity JP-multiplet + 3 2+ , ;1+ of states f2(1270);(1232);a1(1260) where the a1 is interpreted as a 2 tetraquark. Table 15.1: predPredicted masses for double heavy bosons from Ref. [43]. Exotics which arepredictedtobe stableunderstrong interactionsare markedby (!). quark content JP predicted Mass [MeV] strong decay threshold [MeV] cqcq (!) ccqq0+ 1+ 3660 3870 c. D * D 3270 3880 bqbq (!) bbqq0+ 1+ 10020 10230 b. B * B 9680 10800 (!) bcqq0+ 6810 BD 7150 Unfortunately, it is difficult to disentangle conventional hadronic quark states from exotic ones and, therefore, no clear-cut identification of tetraquarks for light hadrons,orhadronswith hiddencharmorbeauty,hasbeenfound[36,40,41].The situation is, however, more favorable for tetraquarks with open charm and beauty which may be stable under strong interactions and therefore easily identified [42]. InTable ??, the computed masses from Ref. [43] are presented. Our prediction [43] fora doubly charmed stable boson Tcc with a mass of 3870 MeV (second row) has been observed at LHCb a year later at 3875 MeV [44], and it is a member of the + 3 positive parity JP-multiplet 2+ , ;1+ of states c2(3565);cc(3770);Tcc(3875). 2 The possible occurrence of stable doubly beautiful tetraquarks and those with charm and beauty is well founded [42]. 15.6 Summary and outlook Holographic light front QCD is a nonperturbative analytic approach to hadron physics with many applications to spectroscopy and dynamics. In the present overview we have mainly focused on the emerging properties of the holographic QCD approach to describe the hadron spectrum. It originates on a semiclassical approximation to the Hamiltonian equations in light front quantization which leads torelativistic wave equations, similar to the Schr ¨ odinger equation in atomic physics. Remarkably, the LF wave equations can be embedded in AdS space, giving a simple procedure to incorporate arbitrary integer or half-integer spin in the bound state equations. The model embodies an underlying superconformal algebraic structure responsible for the introduction of a mass scale within the superconformal group, and determines the effective confinement potential for mesons, nucleons and tetraquarks. It is an effective supersymmetry, not SUSY 252 GuyF.deT´eramond QCD. Thereisa zero eigenmodeinthe spectrum whichis identified with the pion: It is massless in the chiral limit. There are other aspects and applications of HLFQCD which are not described here but are reviewed in [21]. For example, LF holographic QCD also incorporates important elements for the study of hadron form factors, such as the connection between the twist of the hadron to the fall-offof its current matrix elements for large Q2, and important aspects of vector meson dominance which are relevant at lower energies. It also incorporates features of pre QCD, such asVeneziano model and Regge theory. Further extensions incorporate the exclusive-inclusive connection in QCD and provide nontrivial relations between hadron form factors and quark distributions. Holographic QCD has also been applied successfully to the description of the gravitational form factors, the hadronic matrix elements of the energy momentum tensor, which provide key information on the dynamics of quarks and gluons within hadrons. Holographic QCD also given new insights on the infrared behavior of the strong coupling in holographic QCD, which is described in [45]. 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Deur50Yearsof Quantum Chromodynamics,F.Gross andE. Klempt (editors), Sec. 5.6, Eur. Phys. J. C, to be published. Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 256) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 16 Planetary relationship as the new signature from the dark Universe Zioutas, K.1;Anastassopoulos,V.1;Argiriou, A.1;Cantatore, G.2;Cetin, S.3; Gardikiotis, A.1;4;Karuza, M.5;Kryemadhi, A.6;Maroudas, M.1;4;Mastronikolis, A.7;Ozbozduman, K.8;Semertzidis,Y.K.9;Tsagris, M.1;10;Tsagris, I.1;10 1UniversityofPatras, physics department,PATRAS,Greece, 2University and INFNTrieste, Trieste, Italy,3Istinye University, Istanbul,Turkiye, 4University of Hamburg, Hamburg, Germany, 5University of Rijeka, Rijeka, Croatia, 6Messiah U., Mechanicsburg,PA, USA, 7Department of Physics and Astronomy, University of Manchester, Manchester, UK, 8Bogazici University Physics Department, Istanbul,Turkey, 9IBS/KAIST, Daejeon, Korea, 10Present address: Geneva/Switzerland Abstract. Abstract. Dark Matter (DM) came from unexpected long-range gravitational observations. Even within the solar system, several unexpected phenomena have not conventional explanation. Streaming DM offers a viable common scenario. Gravitational focusing and self-focusing effects, by the Sun or its planets, of DM streams fits as being the underlying process behind otherwise puzzling observations like the 11-year solar cycle, the mysterious heating of the solar corona with its fast temperature inversion, etc. However, unexpected solar activity or the dynamic Earth’s atmosphere and other observations might arise from DM streams. This work is suggestive for an external impact by yet overlooked “streaming invisible matter”, which reconciles investigated mysterious observations. Unexpected planetary relationships exist for the dynamic Sun and Earth’s upper atmosphere; they are considered as multiple signatures for streaming DM. Then, focusing of DM streams could also occur in exoplanetary systems, suggesting for the first-time investigations by searching for the associated stellar activity as a function of the exoplanetary orbital phases. The entire observationally driven reasoning is suggestive for highly cross-disciplinary approaches including also (puzzling) biomedical phenomena like cancer. Favorite candidates from the dark sector are anti-quark nuggets, magnetic monopoles, but also particles like dark photons or the composite pearls. Thus, insisting anomalies /mysteries within the solar system are the as yet unnoticed manifestation of the dark Universe we are living in. Povzetek: Na obstoj temne snovi (DM) so ˇ ze pred skoraj stoletjem opozorila merjenja hitrostikroˇcnem sistemuveˇ zenja zvezd okoli centra galaksije.Vendarje tudiv sonˇc pojavov, ki nimajo razlage in bi jih utegnila povzroˇciti temna snov. Avtor predstavi svojo razlago za nekatere pojave, ki so morda povezane z moˇ cnim strujanjem temne snovi. 16 The Problem of Particle-Antiparticle in Particle Theory 257 16.1 Introduction The discovery of dunkle Materie (DM) by ZWICKY came from unexpected cosmological observations.Todayweknowthatour Universeis dominatedbya mysteriousDM.Its namecomesfromthewidelyused definition,namely:DMdoesnotemitor absorborreflect electromagnetic radiation, making it difficult to detect. Following the observations behind this work, this definition of DM is eventually misleading, because, as we argue in this work, several counter examples might be caused by DM, while, at first sight, contradicting the widely used definitionfor DM. Our working hypotheses are: Planetary (and solar) gravitational effects on non-relativistic “invisible massive particles” are focused on solar and planetary atmospheres; they also might interact “strongly”, while the screening at those placesis negligible comparedtodeep underground locations.With “strongly”is meant that they have large cross section with normal matter and radiation. With time, during planetary alignment with an invisible stream, that cannot be predicted as long as the streams and their velocityremain unknown, activity enhancement should repeat and might be the novel signature for the dark sector. Fortunately for this approach, the gravitational deflection depends on 1/speed2. This favours enormously non-relativistic speeds like the ones widely assumed for the constituents of the dark Universe. This makes any exo-solar planetary systemsofpotentialinterest. Becausetheyalso consistofarelatively large number of orbitinggravitational lenses for DM constituents (whatever they are made of). After all, what counts in gravitational lensing is mainly the velocity of DM partic. In fact, even the Moon can focus DM particles on Earth with velocities up to about 400 km/s covering thusa large fractionofDM phase space [1,2]. The aforementioned planetary gravitational lensing effects within the solar system becomes enormous if DM consists of streams, at least partly. Recent cosmology publications [3] consider fine grained streams of cosmological origin. Thus, to explain unusual or anomalous observations in our vicinity, we early concluded on the existence of streaming DM following the reasoning of this kind of work (see e.g. [4,5]). Interestingly, the suggested streamingDM scenario is supported alsoby cosmological considerations followinga completely differentreasoning[3,4], which was founded on another unbiased approach.Aposteriori we find that both findings based on different input converge towards streaming DM. Figure 1. Schematic view of planetary gravitational focusing of streaming invisible massive particles (IMM) by the Sun. Free fall can be also strong for low-speed particles towardthe Sun [6]. The flux can also be gravitationally modulated by an intervening planet, resulting in a 258 Authors Suppressed Due to Excessive Length specific planetary dependence fora putative signature. The sizeof the planetary orbitsis not to scale. 16.2 Signatures The idea followed in this work goes similarly to the aforementioned reasoningby Zwicky that has led to the discovery of DM on cosmological scales. Namely, the last ~160 years several unexpected energetic observations have been discovered within the solar system defying explanation (see e.g. [5] and references therein). This could be due to the dark Universe, whose manifestation was overlooked forlong time. Drivenby observation,we convergeona classof “invisible” particle candidates from the dark sector, which exclude the parameter phase space of axions and WIMPs following failed direct DM searches since decades. In this work we pinpoint at a simple feature as the common signature from such observations within the solar system. For example, the widely discussed dark sector constituents have a velocity of about 0.001 c (c=velocity of light). As it has been pointed out [4,7,8], streams of “dark” constituents with such velocities canbeefficiently gravitationally focused or deflected withina planetary system like ours, including the Sun and the Moon. The aforementioned energetic observations include the unpredictable flaring Sun, its irradiance and more generally its dynamical behaviour [5] as it is manifested by the widely accepted proxy of the solar radio line (F10.7) at 10.7 cm wavelength. The most energetic planetaryrelationshipis Sun’s slow size variation during onesolarcycle[6]. Because,toliftan1kmthicklayerofthe photosphere(. . 0:1=cm3)by 1km, the required energy of about1030 ergs is enormous. In addition, it is also remarkable the planetary dependence of Sun’s elemental composition, which makes a widely discussed issue more of a riddle within known physics. Similarly, also the planetaryrelationshipof the many elemental magnetic bright points on the solar surface show planetary relationships [5]. In addition to the unexpected solar observables add up a number of nearby terrestrial anomalous phenomena occurring in the atmospherewhile being known since the 1930s. For example, what is beyond ionosphere’s dynamical behaviour showing also planetary relationship [8], i.e., why is there annually about 25% more ionosphere around December than six months apart around June? This anomalyis known since 1937 [9].Two extraordinary facts about the ionosphere are worth mentioning here: A)the ionosphereisthemostouterterrestrialregionthatisdirectlyexposedtoouterspace. Then, any so far “invisible” constituents from the dark Universe may appear up there, providedthey interact“strongly”(=largecross section)withnormal matter.Interestingly, this is possible for DM following recent publications. Then, this requirement has not to be invented for the underlying scenarioof this work (see e.g.,ref. [10]).By contrast, the deep underground direct DM searches address extremely feebly interacting DM constituents due to the screeningof “strongly” interacting dark constituentsby the overhead Earth’s layers. B)We also point out here some cross-disciplinary observationsof high societalrelevance: 1) The not randomly appearing Earthquakes [11]. Probably this happens by some kind of accumulating energy deposition inside the Earth triggering finally an Earthquake. Apparently, it is not necessary for the invisible stream(s) or cluster to provide spatiotemporally the entire energy liberated during an Earthquake. It can be final the external trigger for an Earthquake to occur. Remarkably, during the largest Earthquakes, the ionosphere’s plasma state changes over long distances asit has been observedby the orbiting GPS system that continuously registers the ionospheric plasma for self-calibration purposes. 2) The observed planetary relationships of melanoma appearance [12-14] following the orbital period of planet Mercury. It has also been observed a periodic modulation of the 16 The Problem of Particle-Antiparticle in Particle Theory 259 daily rate of diagnosedmelanoma cases, coinciding with the lunar sidereal periodicity of 27.32 days [14]; this, on its own, points at exo-solar origin, which fits-in the suggested streaming DM scenario. The observations given above have one common feature. Namely, they all show an otherwise unexpected planetaryrelationship. Mostprobably more and moreresults will emerge following this kind of out-of-the-box thinking, and this might allow to corner the microscopic nature of the suspected streams, being not as “invisible” as widely thought to be. Wealsowishtostressherethat followingthereasoning underlyingthiswork,itisinteresting to find out as to whether similar behaviour is encountered in exo-solar planetary systems [15].With nearEarth galactic exo-planetary systemsonemightbeableto establish similar correlations for an exo-planetary system but also a cross-correlation with our solar system. Such observations have the potential to expand our DM horizon within our Galaxy as well as into the dark Universe, establishing the working hypotheses behind such scenarios. 16.3 Summary -conclusion Observationally driven, we conclude in this work that aplanetaryrelationshipisakey signature pointing on its own at exo-solar origin. So far, the only viable explanation we can imagine for a plethora of diverse observations showing planetary dependency, is due to gravitational focusingof streaming “invisible” matter.We tentatively identifyit with constituentsfrom the dark Universe, interacting eventually also with a large cross section with ordinary matter or radiation. At the moment, we only can speculate about the possible particle candidates (see below), which are suggestive for new searches. Implications in ongoing or future DM experiments are obvious. Therefore, we urge all experiments and in particular those searching for direct DM signatures, to perform a statistical re-analysis following the reasoning underlying this work (see ref. [5]]). If a planetary dependency is found also in direct DM searches, this will strengthen the concept of “invisible streams” in our vicinity, which can appear either due to tidal forces in our galaxy or others nearby, or, more probably they can be cosmological in origin [2]. We are aiming to widen the appearance of this type of new signatures being probably still hidden also in other observations. One day we might decipher the properties of the invisible stream(s). Along this line of reasoning emerged the medical observations made with long series data of melanoma diagnoses [12-14]. Surprisingly, the main two planetary signatures appeared so far in medicine are: 1) The 88 days orbital periodicity of Mercury using monthly data from the northern hemisphere (USA) [12], which have been independently confirmed [13]. Inconceivably, the author has overlooked his positive result with most cancer types, and 2) The sidereal geocentric lunar periodicity (=27.32 days) using daily melanoma diagnoses data from the southern hemisphere (Australia) [15]. Interestingly, following the planetary scenario and the possible signatures observed [4,5,8,16] the underlying stream(s) canonlybe exo-solarin origin. Noticeby definition,a sidereal periodicityreferstoa reference frame fixed to remote stars. Of course, a DM stream is of cosmic origin, even ifithappenstobetrappedbythesolarsystemduringits birth.Alsothislast scenariois ofnot minor importancefordirectDM searches,orfor indirect onesin astrophysical/ cosmic observations. In short, a wide diversity of signatures implying planetary relationships may allow to spot the “invisible” components from the dark Universe we are living in. 260 Authors Suppressed Due to Excessive Length Finally, the question arises what canbe the first “invisible candidates” favouredby such investigations. The possible candidates are; a) Anti Quark Nuggets (AQNs) as they have been invented by Ariel ZHITNITSKY (2003) [17-19]. These objects are inspiring many investigations from the origin of the solar corona heating mystery to the direct detection of axions [16]. b) Magnetic monopoles as their interaction with the ubiquitous magnetic fields makes different energy deposition scenarios of potential interest. c) Dark photons,whichcan evenresonantly converttorealphotonsifthelocalplasma density fits-in the rest mass of the hidden photon. Contrary to axions or axion-like particles, the kinetic mixing between real photons with hidden sector photons does not require a magnetic field as catalyst, and this makes them attractive. d) PEARLs [see Holger Nielsen, this conference].We suggest thata quantitative investigation as to whether these composite particles fit-in at least some of the observations made so far, as it has been undertaken already with the AQNs, starting for example with the mysterious solar corona heating and the unpredictable solar Flares, seems as an appropriate first step. e) Some other constituents to be invented yet, remains always an option. Thus, the mostly inspiringparticle constituents fitting-in several observations are Anti- QuarkNuggets, magnetic monopoles and dark photons. Though, more emerging candidates like the pearls (see talk in this conference by Holger Nielsen) are encouraged to investigate whether they fit-in, and, how to identify their possible involvement. Thus, insisting anomalies/mysteries within the solar system are the unnoticed manifestation of the dark Universe we are living in. 16.4 References: [1] Sofue,Y. Gravitational Focusingof Low-Velocity Dark Matter on the Earth’s Surface. Galaxies, 2020, 8, 42; https://doi.org/10.3390/galaxies8020042 . [2] Kryemadhi, A.; Maroudas, M.; Mastronikolis, A.; Zioutas, K. Gravitational focusing effects on streaming dark matter as a new detection concept. Preprint https://doi.org/10.48550/arXiv.2210.07367(2022). [3]Vogelsberger, M.; White, S.D.M. Streams and caustics: The fine-grained structure of . cold dark matter haloes. Mon. Not. R. Astron. Soc. 2011, 413, 1419. https://academic.oup.com/mnras/article/413/2/1419/1070092 . [4] Zioutas, K.;Tsagri, M.; Semertzidis,Y.K.; Papaevangelou,T.; Hoffmann, D.H.H.; Anastassopoulos,V. The11 years solar cycle as the manifestationof the dark Universe. Mod. Phys. Lett.A 2014, 29, 1440008; https://doi.org/10.1142/s0217732314400082. [5] Zioutas,K.; Anastassopoulos,V.;Argiriou,A.; Cantatore, G.; Cetin, S.A.; Gardikiotis, A.; Hoffmann, D.H.H.; Hofmann, S.; Karuza, M.; Kryemadhi, A.; et al. The Dark Universe Is Not Invisible. Phys. Sci. Forum 2021, 2, 10. https://doi.org/10.3390/ECU2021-09313 . 16 The Problem of Particle-Antiparticle in Particle Theory 261 [6] Zioutas, K.; Maroudas, M.; Kosovichev, A. On the origin of the rhythmic Sun’s radius variation. Symmetry 2022, 14, 325; https://doi.org/10.3390/sym14020325 . [7] Hoffmann, D.H.H.; Jacoby, J.; Zioutas, K. Gravitational lensingby the Sunof non-relativistic penetrating particles. Astropart. Phys. 2003, 20, 73. https://doi.org/10.1016/S0927-6505(03)00138-5 . [8] Bertolucci, S.; Zioutas, K.; Hofmann, S.; Maroudas, M. The sun and its planets as detectors for invisible matter. Phys. Dark Univ. 2017, 17, 13; https://doi.org/10.1016/j.dark.2017.06.001 . [9] E.V. Appleton, Regularities and irregularities in the ionosphere, Proc. Roy. Soc. London A162 (1937) 451; http://rspa.royalsocietypublishing.org/content/162/911/451 [10] Emken,T.; Essig,R.; Kouvaris,C.; Sholapurkar,M. Direct Detection of Strongly Interacting SubGeV Dark Matter via Electron Recoils. J. Cosmol. Astropart. Phys. 2019, 9, 70; https://doi.org/10.1088/1475-7516/2019/09/070 . [11] Maroudas, M. PhD thesis, University of Patras 2022. [12] Zioutas,K.;Valachovic,E. Planetary dependenceof melanoma. Biophys. Rev. Lett. 2020, 13, 75; https://doi.org/10.1142/S179304801850008X . [13] Zioutas,K.;Valachovic,E.;Maroudas,M. Responseto Comment on “Planetary Dependence of Melanoma”. Biophys. Rev. Lett. 2019, 14, 11; https://doi.org/10.1142/S1793048019200029 . [14] Zioutas, K.; Maroudas, M.; Hofmann, S.; Kryemadhi, A.; Matteson, E.L. Observation of a 27 Days Periodicity in Melanoma Diagnosis. Biophys. Rev. Lett. 2020, 15, 275; https://doi.org/10.1142/S1793048020500083 . [15] Perryman, M.; Zioutas, K. Gaia, Fundamental Physics, and Dark Matter,https://arxiv.org/abs/2106.15408 .(2021). [16] Zioutas, K.; Argiriou, A.; Fischer, H.; Hofmann, S.; Maroudas, M.; Pappa, A.; Semertzidis,Y.K. Stratospheric temperature anomalies as imprints from the dark universe. Phys. Dark Univ. 2020, 28, 100497; https://doi.org/10.1016/j.dark.2020.100497 . [17] Zhitnitsky, A. “Nonbaryonic” Dark Matter as Baryonic Color Superconductor. JCAP 2003, 310, 10; https://iopscience.iop.org/article/10.1088/1475-7516/2003/10/010 . [18] Zhitnitsky, A. Solar Flares and the Axion Quark Nugget Dark Matter Model. Phys. Dark Univ. 2018, 22, 1; https://doi.org/10.1016/j.dark.2018.08.001 . [19]N. Raza,L. vanWaerbeke,A. Zhitnitsky, Solar Corona HeatingbytheAQNdark matter,Phys.Rev.D, 2018, 98, 103527; arXiv:1805.01897 . Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 262) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 17 Abstractsof talks presentedattheWorkshopand in the Cosmovia forum http://bsm.fmf.uni-lj.si/bled2022bsm/presentations.html https://bit.ly/bled2022bsm Not allthe talks come as articles in this year’s Proceedings, but all the talks can be found on theofficial websiteof theWorkshop and on the Cosmovia forum: https://bit.ly/bled2022bsm. Here are the abstracts of the contributors who did not submit an article. 17.1 T.E. Bikbaev, M.Yu. Khlopov, A.G.Mayorov National research Nuclear University MEPHI, Moscow, and Research Institute of Physics, Southern Federal University, Rostov on Don, Russia Modelling of dark atom interaction with nuclei. Dark atom interaction with nuclei is the crucial long-standing problem of the composite dark matter solution for the puzzles of direct dark matter searches. This solution assumes existenceof stable -2n charged particles boundby Coulomb interaction withn nucleiof primordial helium forming nuclear interacting Bohr-like OHe (n=1) or Thomson-like XHe (n.1) dark atoms.ThepuzzlesofdirectDM searchesarethen explainedbythe annual modulation of low-energy binding of dark atom with nuclei in the DAMA/NaI and DAMA/LIBRA detectors, which cannot be detected in direct WIMP searches for recoil nuclei or electrons from WIMP interaction with the matter in other detectors. The continuous approach to the realistic description dark atom interaction with nuclei by the quantum mechanical accomplishment of the numercial study of classical three body problem both for OHe and XHe is now accompanied by the development of methods to solve the Schroedinger equation for the considered problem. The progress in our studies is reported. Povzetek Avtor predpostavi, da je temna snov iz stabilnih negativno nabitih (-2n) delcev, ki jih pove ˇ ze elektromagnetna sila sila z n jedri ”OHe” ali ”XHe” v atome temne snovi.Stem modelom za temno snovisˇˇ ce pojasnilo, zakaj direktnih meritev temne snovi z detektorjem DAMA/NaI in DAMA/LIBRA niso ponovili drugi detektorji, ki tudi merijo sipanje temne snovi na merilnih aparaturah kot funkcijo gibanja Zemlje okoli Sonca.Avtor poroˇca o napredku pri iskanju stabilnih reˇsitev njihovega modela temne snovi. Title Suppressed Due to Excessive Length 263 This work has been supported by the grant of the Russian Science Foundation No-18-1200213- P https://rscf.ru/project/18-12-00213/ and performed in Southern Federal University. 17.2 A. Chaudhuri1 and J. Das2 1Disciplineof Physics, Indian InstituteofTechnology, Gandhinagar, Gandhinagar, India. 2Department of Physics, University of Delhi, New Delhi, India. Electroweakphase transitionandentropyreleaseinZ2 symmetric extensionofthe Standard Model In this work we consider the simple Z2 symmetric extension to the StandardModel (SM) and proceed to study the nature of electroweak phase transition (EWPT) in the early uni- verse.We show that the natureof the phase transition changesfroma smooth crossoverin the SM to a strong first order with this addition of the real scalar. Furthermore, we show the entropy release in this scenario is higher than that of the SM. This can lead to a strong dilution of frozen out dark matter particles and baryon asymmetry, if something existed before the onset of the phase transition. Povzetek Avtor razˇ siri standarni model tako, da predpostavi simetrijo Z2 ter uporabi ta modelza ˇsibkega faznegaprehodav zgodjem vesolju. Poka ˇ studij elektroˇze, da se narava faznega prehoda razlikuje od elektroˇ sibkega faznega prehoda v standardnem modelu. Sprosti se veˇse gostote nastale temne snovi in do c entropije, kar lahko pripelje do manjˇ zmanjˇcejeta bilaˇzepred faznimprehodom. sane barionske asimetrije, ˇ 17.3 S. R. Chowdhury, M.Yu. Khlopov Research Institute of Physics, Southern Federal University, Rostov on Don, Russia The impact of mass transfer in the formation of compact binary merging. The binary black hole coalescences GW150914 and GW151226 observedby the LIGO started the gravitational wave (GW) astronomy era. It enabled us to investigate gravity in the strong- eldregime.Inordertoresemblethe observations, accurate theoretical modelsarerequired to compare theresults. There are still signi cant uncertainties about the stability of mass transfer and common envelope evolution in formation models involving isolated binary stars. Large binary population simulations have been used to anticipate the sources for GW. Populations can be produced on timescales of days using a binary population synthesis tool that balances physical modelling and simulationspeed.Withthehelpof COSMIC,we simulatethe galactic populationof compact binaries and their GW signals. Based on the metallicity, the nal fate of the population has been estimated. The workof S.R.C was supportedby the Southern Federal University (SFedU) (grant no. P-VnGr/21-05-IF). Theresearchby M.Yu.K. was financially supportedby Southern Federal University, 2020 Project VnGr/2020-03-IF. 17.4 A. Ghoshal Sky Meets Laboratory via RGE: Axions, Peccei-Quinn PhaseTransitions and Gravitational Waves Asa solutiontotheSM hierarchyproblem,wewill discuss model-buildingwith classical scale invariancein 4-dimensionalQFT satisfyingTotal AsymptoticFreedom(TAF):the theory holds up to infifinite energy,where all coupling constants go to zero and is devoid of any Landau poles. Such principlesif beyond thereachof LHC(TeV scale) canbe tested via GravitationalWaves(GW)inLIGO,etc.Asan example,wewill discussaQCDaxioninthe TAF scenario, with strong fifirst order Peccei-Quinn phase transitions and produces GW. Thuswewill concludebypromotingRGEasanovel connectionto complement laboratory searches of BSM with cosmological observables as probes of BSM models. Povzetek Avtor predlaga za reˇ sitev problema hierarhije standardnega modela model z invariantno skalo v ˇzni kvantni teoriji polja s popolno asimptotsko svobodo stiri-razseˇ (TAF), ko so pri neskoncni energiji vse sklopitvene konstante enake niˇ c in ni Landavove singularnosti.Tak ˇ sni privzetki niso merljivi na LHC(z dosegomTeV), sopa opazljivi pri gravitacijskih valovihv experimentih LIGOindrugih.Vtem modelu obravnava av- tor axione v kvantni kromodinamiki, ko pride do moˇ cnih faznih prehodov prvega reda Peccei-Quinnove vrste, ki povzroˇcijo gravitacijske valove. 17.5 M. Ildes IAnalytic Solutions of Scalar Field Cosmology, Mathematical Structures for Early Inflaction and LateTime Accelerated Expansion We study the most general cosmological model with real scalar field which is minimally coupled to gravity. Our calculations are based on Friedmann-Lemaitre-Robertson-Walker (FLRW) background metric. Field equations consist of three differential equations. 17.6 M. Ildes II Analytic Solutionsof Brans-Dicke Cosmology: EarlyIn ationandLateTime Accelerated Expansion Title Suppressed Due to Excessive Length 265 We investigate the most general exact solutions of Brans-Dicke cosmology by choosing the scale factor ”a” as the new independent variable. It is shown that a set of three eld equations can be reduce dto a constraint equation and a rst order linear dierential equation. Comparison of our results with recent observations of type Ia supernovae indicates that eighty-nine percent of present universe may consistof domain walls whilerestis matter. 17.7 S. Kabana Thermal production of Sexaquarks in Heavy Ion Collisions Sexaquarks are a hypothetical low mass, small radius uuddss dibaryon which has been proposedrecently and especially asacandidate for Dark Matter. The low massregion below 2GeV escapes upper limits set from experiments which have searched for the unstable, higher mass H-dibaryon and did not fifind it. Depending on its mass, such state may be absolutely stable or almost stable with decay rate of the order of the lifetime of the Universe therefore making it a possible Dark Matter candidate . Even though not everyone agrees its possible cosmological implications as DM candidate cannot be excluded and it has beenrecently searchedin the BaBar experiment. The assumptionofa light Sexaquark has been shown to be consistent with observations of neutron stars and the Bose Einstein Condensateoflight Sexaquarkshasbeen discussedasa mechanismthatcouldinducequark deconfifinementinthecoreof neutron stars.Sproductioninheavyion collisionsis expected to be much more favorable than in the only experimental search to date, Y › S. › , which is severely suppressed by requiring a low multiplicity exclusive fifinal state. By contrast, parton coalescence and/or thermal production give much larger rates in heavy ion collisions.We usea model which has very successfully described hadron and nuclei production in nucleus-nucleus collisions at the LHC, in order to estimate the thermal production rateof Sexaquarks with characteristics suchas discussedpreviouslyrendering them DM candidates. Weshownewresultsonthe variationoftheSexaquarkproductionrateswithmass,radius and temperature and chemical potentials assumed and their ratio to hadrons and nuclei and discuss the consequences. Povzetek Sexaquarki so hipoteti ˇ cnini dibarioni uuddssz majhno masoin majhnim radijem. Bilinajbi stabilnialiskoraj stabilni,z ˇ zivljenjsko dobo vesolja in zato kandidati za temno snov. Predpostavka o Sexaquarku z majhno maso se je izkazala za skladno z opazovanji nevtronskih zvezd, kjer naj bi Sexaquarki pro ˇ zili razgradnjo kvarkov v jedru nevtronskih zvezd.Verjetnostza nastanek Sexaquarkovpri trkihte ˇcjakot zkih ionov naj bi bila veliko veˇ pri poskusu Y › S. › ,kjersoga iskali doslej.Avtorjipredlaganega poskusa uporabijo za ˇsno opisal nastajanje hadronov in jeder studij poteka poskusa model, ki je zelo uspeˇ v trkih jedro-jedro na pospeˇ sevalniku LHC.Z njim ocenjujejo ali imajo Sexaquarki, ki nastajajo pri toplotni produkciji, znaˇcilnosti, ki jih morajo imeti kandidati za temno snov. Predstavljajo noverezultateo odvisnosti hitrosti nastajanja Sexaquarkovod njihove mase, radija in privzetega kemijskega potenciala v razmerju do hitrosti nastajanja hadronov in jeder. 17.8 A.O.Kirichenko, M.Yu. Khlopov, A.G.Mayorov National research Nuclear University MEPHI, Moscow, and Research Institute of Physics, Southern Federal University, Rostov on Don, Russia Propagation of antinuclei in galactic magnetic field We model the propagation of antihelium particles in the magnetic fields of the Galaxy from a supposed source of antimatter in the Galactic halo in the form of a globular antistellar cluster. The well-known JF12 model (R. Jansson, G. R. Farrar, 2012) with the addition of an irregular component (A. Beck, A. Strong, 2016) was taken as a magnetic field model. The cutoff energy for the penetration of particles into the disk in the total magnetic field of the Galaxy (of the order of 1000 GeV) is estimated. Particles of low energies (less than 100 GeV) are largely suppressed when they try to penetrate the disk region. The observed suppression is similar to the effect of solar modulation, which occurs when cosmic rays penetrate into the heliosphere.Taking into account expected decreasing power law suppression at the high energies in the source convergence of this cut offwith the power law energy dependence favors the energy range which is optimal for search for antihelium component of cosmic rays at the AMS02 experiment. This work has been supported by the grant of the Russian Science Foundation No-18-12 00213-P https://rscf.ru/project/18-12-00213/ and performed in Southern Federal University. 17.9 M. Khlopov National research Nuclear University MEPHI, Moscow, Russia Research Institute of Physics, Southern Federal University, Rostov on Don, Russia Virtual Institute of Astroparticle physics, Paris, France Cosmologicalreflectionof the BSM physics The modern cosmology is based on the BSM physics, involved in the mechanisms of inflation, baryosynthesis and the physical natureof dark matter.To specify the parametersof BSM models methods of multimessenger cosmology are developed with special emphasis on the important role of exotic deviations from the now Standardcosmological paradigm, like macroscopic antimatter in baryon asymmetrical Universe, primordial black holes, structures and inhomegeneities in the dark matter distribution as well asWarmer than Cold dark atom scenario of composite dark matter. Positive evidence for such deviations would strongly restrict possible classes of BSM models and provide determination of BSM parameters with ”astronomical accuracy”. Povzetek Sodobna kozmologija temelji na fiziki, ki presega oba standardna modela. Zahteva razumevanje pojava eksponentnega ˇ sirjenja vesolja (inflacije), bariosinteze in razumevanja, iz ˇcili parametre za za novo teorijo, ki bi presegla cesa je temna snov. Da bi lahko doloˇ oba standardna modela, predlaga avtor modele s posebnim poudarkom na eksotiˇ cnih odstopanjih od standardnega kozmoloˇ skega modela, kot so makroskopska antimaterija Title Suppressed Due to Excessive Length 267 v barionskem asimetriˇcrnih lukenj v zgodnjem vesolju, strukture cnem vesolju, nastanek ˇ in nehomogenostiv porazdelitvi temne snovi, topli temniatomi,kida sestavljajo temno snov. Meritve, ki bi potrdile te modele, bi moˇ cno omejila izbiro predlogov, ki bi pomenili razˇsiritev obeh standardnih modelov. Thisresearch has been supportedby the Ministryof Science and Higher Educationof the Russian Federation under Project ”Fundamental problems of cosmic rays and dark matter”, No. 0723-2020-0040. 17.10 M.Yu. Khlopov1;2, D.Sopin1 1 National research Nuclear University MEPHI, Moscow; 2 Research Institute of Physics, Southern Federal University, Rostov on Don, Russia Primordial asymmetry of new sequential superheavy quarks and leptons New stable family with the Standardmodel electroweak (EW) charges should take part in sphaleron transitions in the early Universe before the phase transition with the EW symmetry breaking. It puts balance between the excess of new quarks and leptons and baryon asymmetry.We consider the asymmetryof superheavy new generation particles (new quarksU,Dand newleptonsE,N) balanced withthe baryon excess.At temperatures above the electroweak phase transition it can be found with the use of system of equations for the chemical potentials and Boltzmann kinetic equation. The work was performed in NRNU MEPHI in the framework of cosmological studies of Prioritet2030 Program 17.11 A. Kleppe SACT, Oslo Mass matrices in a scenario with only one R-handed state According to the Standard Model, before the spontaneous symmetry breaking of the electroweak interactions, the fermions were masslessWeyl particles, such that states with R-handed helicity were completely separated from particles with L-handed helicity. After the symmetry breaking, fermions appear as . = L + R, where the L-handed sector is singled out: only L-handed particles appear in the weak interactions. In our scenario, we take . = L + R very seriously, perceiving . as the sum of two different states L and R,whichremainjustas separateastheywerebeforetheSSB.In addition, the singlet state R is perceived as being the same for all quarks, which means that while the left-handed states take part in charge changing processes, the right-handed states just “stay put”. This assumption has many consequences, and gives rise to mass matrices of a certain, very specific texture. 17.12 A.V. Kravtsova1, M.Yu. Khlopov1;2, A.G. Mayorov1;2 1 National research Nuclear University MEPHI, Moscow 2 Research Institute of Physics, Southern Federal University, Rostov on Don, Russia Interaction of antinuclei with galactic interstellar gas Models of strongly inhomogeneous baryosynthesis in the baryon-asymmetric Universe admitthe existenceof macroscopic domainsof antimatter,which could evolveasa globular cluster of antistars in the halo of our Galaxy. Assuming the symmetry of evolution of the globular cluster of stars and antistars on the basis of symmetry of matter and antimatter properties, such an object could be the source of anthelium nuclei in galactic cosmic rays. This allows us to the prediction of the expected fraction from the luxes of cosmic antinuclei propagation in the magnetic field of the Galaxy, taking into account the inelastic interaction with interstellar matter,in which destructionof antiHe-4 canresultincreationof anti-He3. Assuming that interstellar gas predominantly contains different components of hydrogen we formulate the problem of cosmic ray enrichment by anti-He3, which will be important for interpretation of the coming AMS02 data. The workbyMK andAM has been supportedby the grantof the Russian Science Foundation No-18-12-00213-P https://rscf.ru/project/18-12-00213/ and performed in Southern Federal University. Discussion section The discussion contributions are not arranged alphabetically Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 271) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 18 Discussion of cosmological acceleration and dark energy FelixMLev Artwork Conversion Software Inc. 509 N. Sepulveda Blvd Manhattan Beach CA 90266 USA Email: felixlev314@gmail.com Abstract. The title of this workshop is: ”What comes beyond standardmodels?”. Standard models are based on Poincare invariant quantum theory. However, as shown in the famous Dyson’s paper ”Missed Opportunities”andinmy publications, sucha theoryisa special degenerate case of de Sitter invariant quantum theory.I argue that the phenomenon of cosmological acceleration has a natural explanation as a consequence of quantum de Sitter symmetry in semiclassical approximation. The explanation is based only on universally recognizedresultsof physics and does not involve models and/or assumptions the validity ofwhichhasnotbeen unambiguouslyprovedyet(e.g.,dark energyandquintessence).I also explain that the cosmological constant problem and the problem why the cosmological constant is as is do not arise. Povzetek:Avtor razloˇski pospeˇcjo kvantne de Sitterjeve simetrije zi kozmolˇsek s pomoˇ v polklasiˇzku.Temne energijeindrugih eksotiˇ cnem pribli ˇcnih konceptov njegova razlaga ne vkljuˇcuje. Keywords: quantum de Sitter symmetry; cosmological acceleration; irreducible representations; dark energy 18.1 Introduction The title of this workshop is: ”What comes beyond standardmodels?”. Standardmodels are based on Poincare invariant quantum theory. However, as shown in the famous Dyson’s paper ”MissedOpportunities”andinmypublications,suchatheoryisaspecial degenerate case of de Sitter invariant quantum theory. Theproblemof cosmological accelerationis an example where the approach based onde Sitter symmetry solves the problem proceeding onlyfrom universally recognized results of physics without involving models and/or assumptions the validity of which has not been unambiguously proved yet (e.g., dark energy and quintessence). This problem was 272 FelixMLev considered in my papers published in known journals, and in the book recently published by Springer. My publications are based on large calculations.To understand them, thereaders must be experts not only in quantum theory, but also in the theory of representations of Lie algebras in Hilbert spaces. Therefore, understandingmy results can be a challenge for many physicists. Since the problem of cosmological acceleration is very important and my approach considerably differs from approaches of other authors, in this presentation to the 25thBled workshopIoutlineonlytheideasofmyapproach without calculations. 18.2 History of dark energy This history is well-known. First Einstein introduced the cosmological constant . because he believed that the universe was stationary and his equations can ensure this only if . 6 = 0. But when Friedman found his solutions of equations of General Relativity (GR) with . = 0, and Hubble found that the universe was expanding, Einstein said (according to Gamow’s memories) that introducing . 6 = 0 was the biggest blunder of his life. After that, the statement that . must be zero was advocated even in textbooks. The explanationwasthat, accordingtothe philosophyofGR, mattercreatesa curvatureof space-time, so when matter is absent, there should be no curvature, i.e., space-time should be the flat Minkowski space. That is why when in 1998 it was realized that the data on supernovae could be described only with . 6 = 0, the impression was that it was a shock of something fundamental. However, the term with . in the Einstein equations has been moved from the left hand side to the right hand one, it was declared that in fact . = 0, but the impression that . 6 = 0 was the manifestation of a hypothetical field which, depending on the model, was called dark energy or quintessence. In spite of the fact that, as noted inwide publications(seee.g.,[1]andreferencestherein),their physical natureremainsa mystery, the most publications on the problem of cosmological acceleration involve those concepts. Several authors criticized this approach from the following considerations. GR without the contribution of . has been confirmed with a good accuracy in experiments in the Solar System. If . is as small as it has been observed, then it can have a significant effect only at cosmological distances while for experimentsin the Solar System theroleof sucha small valueis negligible.The authorsof[2] titled”WhyAll ThesePrejudices Againsta Constant?” note that it is not clear why we should think that only a special case . = 0 is allowed. If we accept the theory containing the gravitational constant G, which cannot be calculated and is taken from outside, then why can’t we accept a theory containing two independent constants? Let us note that currently there is no physical theory which works under all conditions. For example, it is not correct to extrapolate nonrelativistic theory to the cases when speeds are comparable to c, and it is not correct to extrapolate classical physics for describing energy levels of the hydrogen atom. GR is a successful classical (i.e., non-quantum) theory for describing macroscopic phenomena where large masses are present, but extrapolation of GR to the case when matter disappears is not physical. One of theprinciples of physics is that a definition of a physical quantity is a description how this quantity should be measured. The concepts of space and its curvature are pure mathematical. Their aim is to describe the motion of real bodies. But the concepts of empty space and its curvature shouldnotbeusedinphysicsbecausenothingcanbe measuredinaspacewhichexistsonly in our imagination. Indeed, in the limit of GR when matter disappears, space remains and hasa curvature (zero curvature when . = 0, positive curvature when >0 and negative 18 Discussion of cosmological acceleration and dark energy 273 curvature when <0)while, since space is only a mathematical concept for describing matter, areasonable approach shouldbe such thatinthis limit space should disappear too. Acommon principleof physicsisthat whena new phenomenonis discovered, physicists should try to first explain it proceeding from the existing science. Only if all such efforts fail, something exotic can be involved. But in the case of cosmological acceleration, an opposite approach was adopted: exotic explanations with dark energy or quintessence were accepted without serious efforts to explain the data in the framework of existing science. 18.3 Elementary particles in relativistic and de Sitter-invariant theories In the problem of cosmological acceleration, only large macroscopic bodies are involved and that is why one might think that for considering thisproblem, there is no need to involve quantum theory. Most works on this problem proceed from GR with additional assumptions the validity of which has not been unambiguously proved yet (see e.g. [1] and references therein). However, ideally, the results for every classical (i.e., non-quantum) problem should be obtainedfrom quantum theoryin semiclassical approximation.We will see that considering the problem of cosmological acceleration from the point of view of quantum theory, sheds a new light on understanding this problem. Standardparticle theory and standardmodels are based on Poincare symmetry where elementary particles are describedby irreducible representations (IRs)of thePoincare group or its Lie algebra. The representation generators of the Poincare algebra commute according to the commutation relations .  [P;P]= 0, [P;M]=-i(P- P), . . .  [M;M]=-i(M+ M- M- M) (18.1) where , . = 0, 1, 2, 3, Pµ are the operators of the four-momentum, M. are the operators -11 -22 -33 of Lorentz angular momenta and . is such that 00 ==== 1 and . = 0 if µ 6 = . Although the Poincare group is the group of motions of Minkowski space, the description in terms of relations (18.1) does not involve Minkowski space at all. It involves only representation operators of the Poincare algebra, and those relations can be treated as a definition of relativistic invariance on quantum level (seethe discussionin[3,3]).In particular, the fact that . formally coincides with the metric tensor in Minkowski space does not imply that this space is involved. In classical field theories, the background space (e.g., Minkowski space) is an auxiliary mathematical conceptfor describingreal fieldsand bodies.In quantum theory,any physical quantity should be described by an operator, but there is no operator corresponding to the coordinate x of the background space. In quantum field theory, Minkowski space is an auxiliary mathematical conceptfor describing interacting fields. Herea local Lagrangian L(x) is used, and x is only an integration parameter. The goal of the theory is to construct the S-matrix in momentum space, and, when this construction has been accomplished, one can forget about space-time background. This is in the spirit of the HeisenbergS-matrix program according to which in quantum theory one can describe only transitions of states from the infinite past when t › -. to the distant future when t › +1. The fact that the S-matrixis the operatorin momentum space does not excludea possibility that, in semiclassical approximation, it is possible to have a space-time description with some accuracy but not with absolute accuracy (see e.g., [3] for a detailed discussion). 274 FelixMLev For example, if p is the momentum operator of a particle then, in the nonrelativistic approximation, the position operator of this particle in momentum representation can be defined as r = i— h@=@p. In this case, r is a physical quantity characterizing a given particle and is different for different particles. Inrelativisticquantummechanics,forconsideringasystemof noninteractingparticles,there isno needto involve Minkowski space.Adescriptionofa single particleis fully defined by its IR by the operators commuting according to Eq. (18.1) while the representation describing several particles is the tensor product of the corresponding single-particle IRs. This implies that the four-momentum and Lorenz angular momenta operators fora system are sumsof the corresponding single-particle operators.In the general case,representations describing systems with interaction are not tensor products of single-particle IRs, but thereis no law that the constructionof suchrepresentations should necessarily involvea background space-time. Inhis famouspaper ”MissedOpportunities”[5]Dyson notesthatde Sitter(dS)andantide Sitter (AdS) theories are more general (fundamental) than Poincare one even from pure mathematical considerations because dS and AdS groups are more symmetric than Poincare one. The transition from the former to the latter is described by a procedure called contraction when a parameter R (see below) goes to infinity. At the same time, since dS andAdSgroupsare semisimple,theyhavea maximum possible symmetryand cannotbe obtained from more symmetric groups by contraction. The paper[5] appearedin1972(i.e.,morethan50 yearsago)and,inviewof Dyson’sresults, a question arises why general theories of elementary particles (QED, electroweak theory and QCD) arestill based on Poincaresymmetry and not dS or AdS ones. Probably,physicists believe that, since the parameter R is much greater than even sizes of stars, dS and AdS symmetries can play an important role only in cosmology and there is no need to use them for describing elementary particles.We believe that this argumentis not consistent because usually more general theories shed a new light on standardconcepts. The discussion in our publications and, in particular, in this paper is a good illustration of this point. Byanalogywithrelativisticquantumtheory,the definitionofquantumdSsymmetryshould not involve dS space. If Mab (a, b = 0, 1, 2, 3, 4, Mab =-Mba)are the operators describing the system under consideration, then, by definition of dS symmetry on quantum level, they should satisfy the commutationrelations of the dS Lie algebra so(1,4), i.e., abcdbd bdMac adbc bcad [M;M]=-i(acM+ - M- M) (18.2) where ab is such that 00 =-11 =-33 = 1 and ab = 0 if a 6 =-22 =-44 = b. The definition of AdS symmetry on quantum level is given by the same equations but 44 = 1. The procedure of contraction from dS and AdS symmetries to Poincare one is defined as M4 follows. If we define the operators Pµ as Pµ = =R where R is a parameter with the dimension length then in the formal limit when R !1, M4µ !. but the quantities Pµ are finite, Eqs. (18.2) become Eqs. (18.1). This procedure is the same for the dS and AdS symmetries and it has nothing to do with the relation between the Minkowski and dS/AdS spaces. In [3,6] it has been proposed the following Definition: Let theoryAcontaina finite nonzero parameter and theoryBbe obtainedfrom theory Ain the formal limit when the parameter goes to zero or infinity. Suppose that, with any desired accuracy,theoryA canreproduceanyresultoftheoryBbychoosingavalueofthe parameter.On the contrary,whenthelimitisalreadytaken,one cannotreturntotheoryA,andtheoryBcannot reproduceallresultsof theoryA. Then theoryAis more general than theoryBand theoryBisa special degenerate caseof theoryA. Asarguedin[3,6],in contrasttoDyson’sapproachbasedonLiegroups,theapproachto symmetry on quantum level should be based on Lie algebras. Then it has been proved 18 Discussion of cosmological acceleration and dark energy 275 that, on quantum level, dS and AdS symmetries are more general (fundamental) than Poincare symmetry, and this fact has nothing to do with the comparison of dS and AdS spaces with Minkowski space. It has been also proved that classical theory is a special degenerate case of quantum one in the formal limit h —› 0, and nonrelativistic theory is a special degenerate case of relativistic one in the formal limit c !1. In the literature the above factsare explainedfrom physical considerationsbut,as shownin[3,6],they canbeproved mathematically by using properties of Lie algebras. Physicists usually understand that physics cannot (and shouldnot) derive that c . 3 · 108m/s and h —. 1:054 · 10-34kgm2/s. At the same time, they usually believe that physics should derive the value of , and that the solution of the dark energy problem depends on this value. However, background space in GR is only a classical concept, while on quantum level symmetryis definedbya Lie algebraof basic operators. The parameters (c, — h, R) are on equal footing because each of them is the parameter of contraction from a more general Lie algebra to a less general one, and therefore those parameters must be finite. In particular, the formal case c = . corresponds to the situation when the Poincare algebra does not exist because it becomes the Galilei algebra, and the formal case R = . corresponds to the situation when the de Sitter algebras do not exist because they become the Poincare algebra. Quantum de Sitter theories do not need the dimensionful parameters (c, — h, R) at all. They arise in less general theories, and the question why they are as are does not arise because the answer is: h —is as is because people want to measure angular momenta in kgm2/s, c is as is because people want to measure velocities in m/s, and R is as is because people want to measure distances in meters. The values of the parameters (c, — h, R) in (kg, m, s) have arisen from people’s macroscopic experience, and there is no guaranty that those values will be the same during the whole history of the universe (see e.g., [3] for a more detailed discussion). The fact that particle theories do not need the quantities (c;h—) is often explained such that the system of units c = h —= 1 is used. However, the concept of systems of units is purely classical and is not needed in quantum theory. It is difficult to imagine standard particle theories without IRs of the Poincare algebra. Therefore, when Poincare symmetry is replaced by a more general dS one, dS particle theories should be based on IRs of the dS algebra. However, as a rule, physicists are not familiarwithsuchIRs.The mathematical literatureonsuchIRsiswidebutthereareonlya few papers where such IRs are described for physicists. For example, an excellent Mensky’s book [7] exists only in Russian. 18.4 Explanation of cosmological acceleration In this section we explain that, as follows from quantum theory, the value of . in classical theory must be non-zero and the question why . is as is does not arise. Considera systemoffree macroscopic bodies, i.e., wedo not consider gravitational, electromagnetic and other interactions between the bodies. Suppose that distances between the bodies are much greater than their sizes. Then the motion of each body as a whole can be formally described in the same way as the motion of an elementary particle with the same mass. In semiclassical approximation, the spin effects can be neglected, and we can consider our system in the framework of dS quantum mechanics of free particles. The explicit expressions for the operators Mab in IRs of the dS Lie algebra have been derived in[8](seealso[3,6,9]).In contrastto standardquantumtheorywherethe mass m of a particle is dimensionful, in dS quantum theory, the mass mdS of a particle is dimensionless. Intheapproximationwhen Poincare symmetry workswithahigh accuracy,these massesin units c = h —= 1 arerelated as mdS = Rm. Also, in dS quantum theory, the Hilbert space of 276 FelixMLev functions in IRs is the space of functions depending not on momenta but on four-velocities v =(v0;v) where v0 =(1 + v 2)1=2. Then in the spinless case, the explicit expressions for the operators Mab are (see e.g., Eq. (3.16) in [3]): . @3 J = l(v);N =-iv0 , E = mdSv0 + iv0(v +) @v @v2 . @3 B = mdSv + i[+ v(v )+ v] (18.3) @v @v2 fM23fM01fM41 where J = ;M31;M12g, N = ;M02;M03g, B = ;M42;M43g, l(v)=-iv × = M40 @=@v and E . The important observation is that, at this stage, we have no coordinates yet. For describing the motion of the particle in terms of coordinates, we must define the position operator. If Poincare symmetry works with a high accuracy, the momentum of the particle can be defined as p = mv and, as noted above, the position operator can be defined as r = i— h@=@p. In semiclassical approximation, we can treat p and r as usual vectors. Then, if E = E=R, P = B=R and the classical nonrelativistic Hamiltonian is defined as H = E - mc 2, it follows from Eq. (18.3) that P2 22 mc r H(P, r)= - (18.4) 2m 2R2 Here the last term is the dS correction to the non-relativistic Hamiltonian. Therepresentation describingafreeN-bodysystemisatensorproductofthe corresponding single-particle IRs. This means that every N-body operator Mab isa sumof the corresponding single-particle operators. Consider a system of two free particles described by the quantities Pj and rj (j = 1, 2). Define standardnonrelativistic variables P = P1 + P2;q =(m2P1 - m1P2)=(m1 + m2) R =(m1r1 + m2r2)=(m1 + m2);r = r1 - r2 (18.5) Here P and R are the momentum and position of the system as a whole, and q and r are therelative momentum andrelative radius-vector,respectively. Then as followsfrom Eqs. (18.4) and (18.5), the internal two-body Hamiltonian is 2 22 qm12cr Hnr(r, q)= - (18.6) 2m12 2R2 where m12 is the reduced two-particle mass. Then, as follows from the Hamilton equations, in semiclassical approximation therelative accelerationis givenby a = rc 2=R2 (18.7) where a and r are therelative acceleration andrelative radius vectorof the bodies,respectively. The fact that the relative acceleration of noninteracting bodies is not zero does not contradict the law of inertia, because this law is valid only in the case of Galilei and Poincare symmetries. At the same time, in the case of dS symmetry,noninteracting bodies necessarily repulse each other. In the formal limit R !1, the acceleration becomes zero as it should be. Equations of relative motion derived from Eq. (18.6) are the same as those derived from GR if . 6 = 0.In particular, theresult (18.7) coincides with thatinGRif the curvatureofdS space equals . = 3=R2, where R is the radius of this space. Therefore the cosmological constant hasa physical meaningonlyon classical level,the parameterof contractionfromdSsymmetryto Poincare one coincides with R and, as noted above,a question why R is as is does not arise. 18 Discussion of cosmological acceleration and dark energy 277 In GR, theresult (18.7) does not depend on how . is interpreted, as the curvature of empty space or as the manifestation of dark energy or quintessence. However, in quantum theory, thereisnofreedomof interpretation.Here R is the parameter of contraction from the dS Lie algebra to the Poincare one, it has nothing to do with dark energy or quintessence and it must be finite because dS symmetry is more general than Poincare one. Every dimensionful parameter cannot have the same numerical values during the whole history of the universe. For example, at early stages of the universe such parameters do not have a physical meaning because semiclassical approximation does not work at those stages. In particular, the terms ”cosmological constant” and ”gravitational constant” can be misleading. General Relativity successfully describes many data in the approximation when . and G are constants but this does not mean that those quantities have the same numerical values during the whole history of the universe. 18.5 Discussion and conclusion In view of the problem of cosmological acceleration, the cosmological constant problem is widely discussed in the literature. This problem arises as follows. One starts from Poincare invariant quantum field theory (QFT) of gravity defined on Minkowski space. This theory contains only one phenomenological parameter — the gravitational constant G, and the cosmological constant . is defined by the vacuum expectation value of the energy-momentum tensor. The theory contains strong divergencies which cannot be eliminated because the theory is not renormalizable. Therefore, the results for divergent integralscanbemade finiteonlywitha choiceofthecutoffparameter.Since G is the only parameterin the theory,areasonable choiceof the cutoffparameterin momentum space is the Planck momentum —hc = 1, G h=lP where lP is the Plank length. In units —= has the dimension 1=length2 and . has the dimension length2. Therefore, the value of . obtained in this approach is of the order of 1=G. However, this value is more than 120 orders of magnitude greater than the experimental one. In view of this situation, the following remarks can be made. As explained in Sec. 18.3, Poincare symmetryisa special degenerate caseofdS symmetryin the formal limit R !1. Here R is a parameter of contraction from dS algebra to Poincare one. This parameter has nothingtodowiththerelationbetween PoincareanddS spaces.Theproblemwhy R is asis doesnot ariseby analogywiththeproblemwhy c and h —are as are. As explained in Sec. 18.4, the cosmological constant . hasa physical meaning onlyin semiclassical approximation and here it equals 3=R2. Therefore the cosmological constant problem and the problem why the cosmological constant is as is do not arise. AsnotedinSec.18.3,thebackground space-timeisonlyamathematicalconceptwhichhasa physical meaning only in classical theory. This concept turned out to be successful in QED. In particular, theresults for the electronand muon magnetic moments agree with experiments with the accuracy of eight decimal digits. However, QED works only in perturbation theory because the fine structure constant is small. There is no law that the ultimate quantum theory will necessarily involve the concept of background space-time. QFTs of gravity (for example, Loop Quantum Gravity) usually assume that in semiclassical approximation, the background space in those theories should become the background space in GR. However, in Sec. 18.4, the result for the cosmological acceleration in semiclassical approximation has been obtained without space-time background and this result is the same as that obtained in GR. Althoughthe physical natureofdark energyremainsamystery,thereexistsawide literature where the authors propose QFT models of dark energy. These models are based on Poincare symmetry with the background Minkowski space. So, the authors do not take into account 278 FelixMLev the fact thatde Sitter symmetryis more general (fundamental)than Poincare symmetry and that the background space is only a classical concept. While in most publications, only proposals about future discovery of dark energy are considered, the authors of [1] argue that dark energy has been already discovered by the XENON1T collaboration. In June 2020, this collaborationreported an excessof electronrecoils: 285 events,53 more than the expected 232 with a statistical significance of 3:5. However,inJuly 2022,a new analysisby the XENONnT collaboration discarded the excess [10]. As shown in Sec. 18.4, the result (18.7) has been derived without using dS space andits geometry (metric and connection).Itis simplya consequenceofdS quantum mechanicsin semiclassical approximation.We believe that thisresultis more important than theresultof GRbecauseany classicalresult shouldbea consequenceof quantum theoryin semiclassical approximation. Therefore, the phenomenon of cosmological acceleration has nothing to do with dark energy or other artificialreasons. This phenomenonispurelya kinematical consequenceofdS quantum mechanics in semiclassical approximation. References 1. S.Vagnozzi,L.Visinelli,P. Brax, A-Ch. Davis andJ. Sakstein: Direct detectionof dark energy: the XENON1T excess and future prospects, Phys. Rev. D104, 063023 (2021). 2. E. Bianchi and C. Rovelli: Why All These Prejudices Against a Constant, arXiv:1002.3966v3 (2010). 3.F.M. Lev:de SitterSymmetry and Quantum Theory, Phys. Rev. D85, 065003 (2012). 4.F.M.Lev: Finite Mathematicsasthe Foundationof Classical MathematicsandQuan- tum Theory.With Application to Gravity and Particle theory. ISBN 978-3-030-61101-9. Springer, Cham, 2020. 5. F. G. Dyson: Missed Opportunities, Bull. Amer. Math. Soc. 78, 635-652 (1972). 6.F.M.Lev: Cosmological Accelerationasa Consequenceof Quantumde Sitter Symmetry, Physics of Particles and Nuclei Letters 17, 126-135 (2020). 7. M.B. Mensky: Method of Induced Representations. Space-time and Concept of Particles. Nauka, Moscow (1976). 8. F.M. Lev: Finiteness of Physics and its Possible Consequences, J. Math. Phys.34,490-527 (1993). 9.F.M. Lev: Could OnlyFermionsBe Elementary?J. Phys. A37, 3287-3304 (2004). 10. E. Aprile,K. Abe,F. Agostini et. al.:Search for New Physics in Electronic Recoil Data from XENONnT, arXiv:2207.11330 (2022). Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 279) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 19 Clifford odd and even objects in even and odd dimensional spaces N. S. Mankoˇc Borˇstnik Department of Physics, University of Ljubljana SI-1000 Ljubljana, Slovenia norma.mankoc@fmf.uni-lj.si Abstract. Ina long seriesof worksIdemonstrated, together with collaborators, that the model named the spin-charge-family theory offers the explanation for all in the standard model assumed properties of the second quantized fermion and boson fields, offering several predictions as well as explanations for several of the observed phenomena. The theory assumesasimple starting actionin even dimensional spaces with d . (13+1) with massless fermions interacting with gravity only. The internal spaces of fermion and boson fields are described by the Cliffordodd and even objects, respectively. This note discusses properties of the internal spaces in odd dimensional spaces, d, d =(2n + 1), which differ essentially from the properties in even dimensional spaces. Povzetek: Vdolgem nizu ˇ clankov sem skupaj s sodelavci pokazala, da ponuja teorija , imenovana spin-charge-family, razlago za vse v standardnem modelu privzete lastnosti fermionskih in bozonskihpolj(vdrugi kvantizaciji),ponujapapolegnapoveditudirazlagoza marsikatero od opaˇskih pojavov.Teorijapredlagapreprosto akcijov sodo-razseˇ zenih kozmoloˇznih prostorih, d . (13 + 1), za brezmasne fermione v interakciji samo z gravitacijskim poljem. Notranje prostorefermionskih in bozonskih polj opiˇ sejo Cliffordovi lihi oziroma sodi objekti. Ta prispevekobravnava lastnosti notranjega prostora fermionov in bozonov v prostorih z lihimi razse ˇ znostimi d, d =(2n + 1). Keywords: Second quantization of fermion and boson fields with Cliffordalgebra; beyond the standard model; Kaluza-Klein-like theories in higher dimensional spaces, Clifford algebra in odd dimensional spaces. 19.1 introduction My working hypothesis is that ”Nature knows all the mathematics”, which we have and possibly will ever invent, and ”she uses it where needed”. Recognizing that there are two kinds of the Clifford algebra objects, a’s and ~a’s [2], each of them of odd and even 280 N. S. Mankoˇc Borˇstnik character,I usethemto describethe internal spacesof fermionand boson fields[5–9]in even dimensional spaces. The Cliffordodd objects, if they are superposition of odd products of a’s, explain in even d = 2n properties of fermion fields. The second kind of the Cliffordodd objects, ~a’s, can be used, after defining their application on the polynomials of a’s (Eq. (7) of my talk in thisProceedings [4]),to equip the irreduciblerepresentationsof odd polynomialsof a’s with the family quantum numbers. The Cliffordeven objects, if they are superposition of even products of a’s, explain in even d properties of boson fields, the gauge fields of the corresponding fermion fields. They do not appear in families [4–9]. In d =(13 + 1) the Cliffordodd objects manifest all the properties of the internal space of fermions — of the observed quarks and leptons and antiquarks and antileptons with their families included — and the Clifford even objects explain the gauge fields of the corresponding fermion fields, as well as the Higgs’ scalars andYukawa couplings. The internal spaceof fermion and boson fields, describedby ”basis vectors” (they are chosen to be eigenvectors of all the members of the Cartan subalgebra members of the Lorentz group in the internal space of fields), demonstrate properties of the postulates of the second quantizationof fermionand boson fields, explaining these postulates[4,6]. Idemonstrate in this note that also in odd dimensional spaces the Cliffordodd and the Cliffordeven objects exist. However, the eigenstates of the operator of handedness are in odd dimensional spaces the superposition of the Cliffordodd and the Cliffordeven objects. This seems to explain the ghost fields appearing in several theories for taking care of the singular contributions in evaluating Feynman graphs. Next section presents the internal spaces, described by the Cliffordodd and the Clifford even ”basis vectors”for fermionand boson fieldsineven dimensional spaces,for d =(1+1) and d =(3 + 1), as well as in odd dimensional spaces, for d =(0 + 1) and d =(2 + 1). This simple casesare chosento easier demonstratethedifferenceinpropertiesin evenandodd dimensional spaces. In Refs. [10–12] from 20years ago the authors discuss the question of q time and d - q dimensions in odd and even dimensional spaces, for any q. Using the requirements that the innerproductof twofermionsis unitary and invariant underLorentz transformations the authors conclude that odd dimensional spaces are not appropriate due to the existence of fermions of both handedness and correspondingly not mass protected. In this note the comparison of properties of fermion and boson fields in odd and in even dimensional spaces are made, using the Cliffordalgebra objects to describe the internal spaces of fermion and boson fields. The recognition of this note might further clarify the ”effective” choice of Nature for one time and three space dimensions. The reader can find more explanation about the properties of internal spaces of fermion and boson fields in even dimensional spaces in my contribution in this proceedings [4]. 19.2 ”Basis vectors” in d = 2n and d = 2n + 1 for n = 0, 1, 2 In Ref. [4–9] the reader can find the definition of the ”basis vectors” as the eigenstates of the Cartan subalgebra of the Lorentz algebra in internal spaces of fermion and boson fields. ”Basis vectors” are written as superposition of the Cliffordodd (for fermions) and the Cliffordeven (for bosons) products of a’s. ”Basis vectors” for fermions have either left or right handedness, . (d) (the handedness is defined in Eq. (19.2), and appear in families Sab i (the family quantum numbers aredeterminedby ~a’s,with ~= 4 { ~a ; ~bg-).The Clifford odd ”basis vectors” have their Hermitian conjugated partners in a separate group. ”Basis 19 Cliffordodd and even objects in even and odd dimensional spaces 281 vectors” for bosons have no families and have their Hermitian conjugated partners within the same group. Properties of the ”basis vectors” in odd dimensional spaces have completely different properties: Only the superposition of the Cliffordodd and the Cliffordeven”basis vectors” have a definite handedness. Correspondingly the eigenvectors of the Cartan subalgebra members have both handedness, ..(2n+1) = 1. 19.2.1 Even dimensional spaces d =(1 + 1), (3 + 1) To simplify the comparison between even and odd dimensional spaces, simple cases for either even or odd dimensional spaces are discussed. The definition of nilpotents and projectors and the relations among them can be found in App. 19.4. d =(1 + 1) There are 4 (2d=2) ”eigenvectors” of the Cartan subalgebra members, Eq. (19.4), S01 and = S01 S01 (Sab i Sab i S01 of the Lorentz algebra Sab and Sab + ~= 4 f a; bg- ~= 4 { ~a ; ~bg-) 01 representing one Cliffordodd ”basis vector” b11† = (+i) (m=1), appearing in one family ^ 01 (f=1) and correspondingly one Hermitian conjugated partner b^1 1 = (-i) 1 and two Clifford 01 01 even ”basis vector” IA11† =[+i] and IIA11† =[-i], each of them is self adjoint. Correspondingly we have two Cliffordodd, Eqs. (19.3, 19.7) 01 01 1† ^b = (+i) , 1 1 ^b 1 = (-i) and two Clifford even 01 01 IA1† = [+i] , 1 IIA1† = [-i] 1 ”basis vectors”. ThetwoCliffordodd”basis vectors”areHermitianconjugatedtoeachother.Imakeachoice that b^11† is the ”basis vector”, the second Cliffordodd object is its Hermitian conjugated partner. Defining the handedness as ..(1+1) = 0 1, Eq. (19.2), it follows, using Eq. (19.5), 1† 1† 1† ^ that ..(1+1) b^ = b^ , which means that bis the right handed ”basis vector”. 11 1 01 ^ We could makea choiceof left handed ”basis vector”if choosingb11† =(-i), but the choice of handedness would remain only one. )† I;IIA1† Each of the two Clifford even ”basis vectors” is self adjoint((I;IIA1 1 † = 1 ). Let us notice, taking into account Eqs. (19.5, 19.9), that 0101 01 11† II fb^1(. (-i)) * A b^ 1 (. (+i))gj oc >= A1 1 † (. [-i])j oc >= j oc >, 01 01 f^1† ^1 b1 (. (+i)) * A b1(. (-i))gj oc >= 0, 01 01 1 It is our choice which one, (+i) or (-i), we chose as the ”basis vector” b^1 1 † and which oneis its Hermitian conjugated partner. The choiceofthe ”basis vector” determines 01 01 the vacuum state j oc >. For b^11† =(+i), the vacuum state is j oc >=[-i] (due to the requirement that b11† j oc > is nonzero), which is the Clifford even object. ^ 282 N. S. Mankoˇc Borˇstnik 0101 01 IA1† ^11 1 (. [+i]) * A b1(. (+i))j oc >= b^1(. (+i))j oc >, 01 01 IA1† 1 1 (. [+i]) b^1(. (-i))j oc >= 0. We find that IA1† IIA1† IIA1† IA1† * A = 0 = * A . 11 11 From the case d =(3 + 1) we can learn a little more: d =(3 + 1) , S12) Thereare 16 (2d=4) ”eigenvectors” of the Cartan subalgebra members(S03;S12)and(S03 of the Lorentz algebras Sab and Sab , Eq. (19.4), in d =(3 + 1). 4 2 4 2 -1,m=(1,2)) members each of the Clifford -1,f=(1,2)) with two(2 Therearetwo families(2 4 4 bm† odd ”basis vectors” ^ f -1 -1 Hermitian conjugated partners b^ m f ina separate ,with2 × 2 2 2 group (not reachable by Sab). 4 4 -1 members of the group of IAm f † -1 There are 2 , which are Hermitian conjugated to × 2 2 2 each other or are self adjoint, all reachable by Sab from any starting ”basis vector IA1 1 † . 4 4 -1 members of IIAm† f , again either Hermitian -1 And there is another group of 2 × 2 2 2 conjugatedtoeachotherorareselfadjoint.Allarereachablefromthe starting vector IIA1 1 † by the application of Sab . Againwecanmakea choiceof eitherrightorleft handedCliffordodd”basis vectors”,but not of both handedness. Making a choice of the right handed ”basis vectors” f = 1f = 2 S03 i S12 S03 S12 1 S03 S12 ~= , ~=- 1 , ~=- i , ~= ;, 22 22 0312 0312 1† 1† i1 ^ b^=(+i)[+] b=[+i](+) 1 222 0312 0312 2† 2† ^^- i - 1 b=[-i](-) b=(-i)[-] , 1 222 we find for the Hermitian conjugated partners of the above ”basis vectors” S03 1 S03 i S~ 03 S~ 12 =- i ;S12 = , = ;S12 =- 1 ;, 22 2 2 0312 0312 1 =(-i)[+] 2 =[+i](-) 22 b^ 1 b^ 1 - i - 1 0312 0312 b^ 2 b^ 2 i1 1 =[-i](+) 2 =(+i)[-] 22 . Let us notice that if we look at the subspace SO(1, 1), with the Cliffordodd ”basis vector” with the Cartan subalgebra member S03 of the space SO(3, 1), and neglect the values of S12 , 03 03 1† 2† we do have b^ 1 =(+i) and b^ 2 =(-i),which have opposite handedness..(1;1) in d =(1 + 1), but they have different ”charges” S12 in d =(3 + 1). In the whole internal space all the Cliffordodd ”basis vectors” have only one handedness. 0312 0312 We further find thatj oc >= . 1([-i][+] + [+i][+]). All the Cliffordodd ”basis vectors” are 2 bm† orthogonal: ^ f * A b^f m 00† = 0. For the Clifford even ”basis vectors” we find two groups of either self adjoint members or with the Hermitian conjugated partners within the same group. The members of one grouparenotreachablebythe applicationof S03 on membersof anothergroup.We have forIAm f † ;m =(1, 2);f =(1, 2) 19 Cliffordodd and even objects in even and odd dimensional spaces 283 S03 S12 S03 S12 0312 03 12 IA1† IA1† 1 =[+i][+] 0 0, 2 =(+i)(+) i1 03 12 0312 IA2† IA2† 1 =(-i)(-) -i -1, 2 =[-i][-] 0 0, and for IIAm f † ;m =(1, 2);f =(1, 2) S03 S12 S03 S12 0312 03 12 IIA1† IIA1† 1 =[+i][-] 0 0, 2 =(+i)(-) i1 03 12 0312 IIA2† IIA2† 1 =(-i)(+) -i 1, 2 =[-i][+] 0 0. The Clifford even ”basis vectors” have no families. IAm† IAm f * Af‘ 0† = 0, forany(m,m’,f,f‘). Even dimensional spaces have the properties of the fermion and boson second quantized fields, as explained in Ref. [4]. 19.2.2 Odd dimensional spaces d =(0 + 1), (2 + 1) In odd dimensional spaces fermions have handedness defined with the odd products of a’s, Eq. (19.2). Correspondingly the operator of handedness transforms the Cliffordodd ”basis vectors” into Clifford even ”basis vectors” and the description of either fermions or bosons with the Clifford even and odd ”basis vectors” have in odd dimensional spaces different meaning than in even dimensional spaces: b^m† f , have 2 2 d -1 i. While in even dimensional spaces the Clifford odd ”basis vectors”, members, m, in 2 2 d -1 families, f, and their Hermitian conjugated partners appear in a d 2 d 2 -1 families, there are in odd dimensional spaces -1 members in 2 separate group of 2 d d -1 -1 = 2d-2 Cliffordodd ”basis vectors” self adjoint and yet they have some of the 2 × 2 2 2 some of the Hermitian conjugated partners in another group with 2d-2 members. A^m† ii. In even dimensional spaces the Clifford even ”basis vectors” i f ;i =(I, II), appear d d -1 -1 members and each with the in two mutually orthogonal groups, each with 2 × 2 2 2 2 d -1 of them are self adjoint. Hermitian conjugated partners within the same group, 2 In odd dimensional spaces the Clifford even ”basis vectors” appear in two groups, each d d -1 -1 = 2d-2 members, which are either self adjoint or have their Hermitian with 2 × 2 2 2 conjugated partners in another group. Not all the members of one group are orthogonal to the members of another group, only the self adjoint ones are. bm† iii. While ^ f have in even dimensional spaces one handedness only (either right or left, depending on the definition of handedness), in odd dimensional spaces the operator of handedness is a Cliffordodd object — the product of an odd number of a’s, Eq. (19.2), (still commuting with Sab)— transforming the Cliffordodd ”basis vectors” into Clifford even ”basis vectors” and opposite. Correspondingly are the eigenvectors of the operator of handedness the superposition of the Cliffordodd and the Clifford even ”basis vectors”, offering in odd dimensional spaces the right and left handed eigenvectors of the operator of handedness. Let us illustrate the above mentioned properties of the ”basis vectors” in odd dimensional spaces, starting with the simplest case: 284 N. S. Mankoˇc Borˇstnik d=(0+1) There is one Cliffordodd ”basis vector”, which is self adjoint 1† 01y† 1 ^ b1 = =(b^ )= b^ and one Cliffordeven ”basis vectors” i A^1† 1 = 1. 1† A1† A1† ^^^ The operator of handedness . (0+1) = 0 transforms binto identity i and i into 1 11 b^11† . The two eigenvectors of the operator of handedness are 1 0 1 0 . ( + 1) , . ( - 1) , 22 with the handedness(+1, -1), thatisof right and left handedness.respectively. d=(2+1) Therearetwice 2d=(3-2) = 2 Cliffordodd ”basis vectors”.Wechoseasthe Cartan subalgebra 01 01 0101 1† 2† 1† 2† ^^^^ member S01 of Sab, Eq (19.4): b=[-i] 2 , b=(+i), b=(-i), b=[+i] 2, with the 1 122 properties f = 1f = 2 S01 i S01 S01 ~= 2 ~=- 2 i , 01 01 1† 1† b^=[-i] 2 b^=(-i)- i 1 22 01 03 2† 2† i ^^ b=(+i) b=[+i] 2 , 1 22 01 01 1† 2† 2† 1† ^ band b^ are self adjoint (up to a sign), b^=(+i) and b^=(-i) are Hermitian conju 12 12 gated to each other. In odd dimensional spaces the Cliffordodd ”basis vectors” describing fermions are not separated from their Hermitian conjugated partners, as it is the case in even dimensional spaces, and do not appear in families. b11† are either self adjoint or have their Hermitian ^ conjugated partners in another family. The operator of handedness is (chosen up to a sign to be) ..(2+1) = i 1 2 2, Eq. (19.2). There are twice 2(d=3)-2 = 2 Clifford even ”basis vectors”. We choose as the Cartan 0101 01 01 II IIII subalgebra member S01:A^1† =[+i], A^2† =(-i) 2 , A^1† =[-i], A^2† =(+i) 2, with the 11 22 properties S01 S01 01 01 I ^A1† = [+i] 1 0 II ^A1† = [-i] 2 0 01 03 I A2† II A2† ^^ 1 =(-i) 2 -i 2 =(+i) 2 i, 0101 01 03 I A1† A1† A2† A2† ^^^^ =[+i] and II =[-i] are self adjoint, I =(-i) 2 and II =(+i) 2 are Hermi 12 12 tian conjugated to each other. 19 Cliffordodd and even objects in even and odd dimensional spaces 285 In odd dimensional spaces the two groups of the Clifford even ”basis vectors” are not orthogonal. Let us find the eigenvectors of the operator of handedness ..(2+1) = i 0 1 2. Since it is the Cliffordodd object its eigenvectors are superposition of Cliffordodd and Clifford even ”basis vectors”. It follows 0101 0101 . (2+1) { [-i] i [-i] 2} = { [-i] i [-i] 2} , 0101 0101 ..(2+1) { (+i) i (+i) 2} = { (+i) i (+i) 2} , 0101 0101 . (2+1) { [+i] i [+i] 2} = { [+i] i [+i] 2} , 01 0101 01 . (2+1) { (-i) 2 ± i (-i)} = { (-i) 2 ± i (-i)} , We can conclude that neither Cliffordodd nor Cliffordeven ”basis vectors” have in odd dimensional spaces the properties which they do demonstrate in even dimensional spaces, thepropertieswhich empowertheCliffordodd ”basis vectors”torepresent fermionsand the Clifford even ”basis vectors” to represent the corresponding gauge fields. i. In odd dimensional spaces the Cliffordodd ”basis vectors” are not separated from their Hermitian conjugated partners, they instead are either self adjoint or have their Hermitian conjugated in another family.We can not define creation and annihilation operators as a tensor products of ”basis vectors” and basis in momentum space so that they would manifest the creation and annihilation operators fulfilling the postulates of the second quantized fermions. In odd dimensional spaces the two groups of the Clifford even ”basis vectors” are not orthogonal, only the selfadjoint ”basis vectors” are orthogonal, the rest of ”basis vectors” have their Hermitian conjugated partners in another group. ii. The Cliffordodd operator of handedness allows left and right handed superposition of Cliffordodd and Clifford even ”basis vectors”. 19.3 Discussion This note discusses the properties of the internal spaces of fermion and boson fields in even and odd dimensional spaces, if the internal spaces are described by the Cliffordodd and even ”basis vectors”, which are the superposition of odd or even products of operators a’s. ”Basis vectors” are arranged into algebraic products of nilpotents and projectors, which are eigenvectors of the Cartan subalgebra of the Lorentz algebra Sab in the internal space of fermions and bosons. The Cliffordodd ”basis vectors”, which are products of an odd number of nilpotents and therestofprojectors,offerin even dimensional spaces the descriptionof the internal space of fermion fields. EachirreduciblerepresentationoftheLorentzalgebrais equippedwiththefamily quantum number determined by the second kind of the Cliffordoperators ~a’s. The Cliffordodd ”basis vectors” anticommute. Their Hermitian conjugated partners appear in a different group. In a tensor product with the basis in ordinary space the ”basis vectors” and their Hermitian conjugated partners form the creation and annihilation operators which fulfil the anticommutationrelations postulated for second quantized fermion fields. In d = 2(2n + 1);n . 7, these creation and annihilation operators, applying on the vacuum state, or on the Hilbert space, offer the description of all the properties of the observed 286 N. S. Mankoˇc Borˇstnik quarks and leptons and antiquarks and antileptons. The massless fermion fields are of one handedness only. The Cliffordeven ”basis vectors”, which are products of an even number of nilpotents and therestofprojectors,offerin even dimensional spaces the descriptionof the internal space of boson fields, the gauge fields of the corresponding fermion fields. The Cliffordeven ”basis vectors” commute. Theydo not appearin families andhave their Hermitian conjugated partners in the same group. In a tensor product with the basis in ordinary space the ”basis vectors” form the creation and annihilation operators which fulfil the commutationrelations postulated for second quantized boson fields fields. In d = 2(2n + 1);n . 7, these creation and annihilation operators offer the description of all the properties of the observed gauge fields as well asof the scalar Higgs’s field, explaining also theYukawa couplings. This way of describing internal space of boson fields with the Cliffordeven ”basis vectors”, although very promising, needs further studies to understand what new it can bring into second quantization of fermion and boson fields. In particular, it must be understood what does it bring if we replace in a simple starting action in d = 2(2n + 1);n . 7 Z A = dxE a d1 ( — p0a )+ h:c. + 2 Z d. ~ dxE ( R + ~R) , . 1 . p0a = fap0. + fp , Efag- , 2E 1 ab1 ~ab ~ p0. = p. - S!ab. - S!ab. , 22 [a b] R = 1 fff(!ab ;ß - !ca. !cb )} + h:c. , 2 ~1!c [a b] R = fff( !~ab ;ß - !~ca. ~b )} + h:c. . (19.1) 2 f [af b] f af b - f bf a . fa !ab. (the gauge fields of Sab)and !~ab. (the gauge fields of S~ ab), manifest in d =(3 + 1) as the known vector gauge fields and the scalar gauge fields taking care of masses of quarks and leptons and antiquarks and antileptons and the weak boson fields 3,if wereplace the covariant derivative p0. Here 2 = , and the two kinds of the spin connection fields, 1 ab1 ~ab ~ p0. = p. - S!ab. - S!ab. 22 in Eq. (19.1) with 2 a aaß f a are inverted vielbeins to e . with the properties e f b = ab;e f a =  , E = det(ea ). Latin indices a, b, ::, m, n, ::, s, t, :. denote a tangent space (a flat index), while Greek indices , , ::, , , ::, , :. denoteanEinstein index(a curved index). Lettersfrom the beginningof boththe alphabets indicatea general index(a, b, c, :. and , , , :. ), from the middle of both the alphabets the observed dimensions 0, 1, 2, 3 (m, n, :. and , , ::),indexesfrom the bottomof the alphabets indicate the compactified dimensions (s, t, :. and , , ::).We assume the signature ab = diagf1, -1, -1, · , -1g. 3 Since the multiplication with either a’s or ~a’s changes the Cliffordodd ”basis vectors” into the Clifford even objects, and even ”basis vectors” commute, the action for fermions can not include an odd numbers of a’s or ~a’s, what the simple starting action of Eq. (19.1) does not. In the starting action a’s and ~a’s appear as 0 a p^a or as SabSab ~ 0 c !abc and as 0 c ~!abc. 19 Cliffordodd and even objects in even and odd dimensional spaces 287 XX m† A ^ I e I Am† I ^ f I Cem Cm f. - = p. - p0. f f. , mf mf m† m† where the relation among I A ^ discussed directly in this article, needs additional study and explanation. While in any even dimensional space the superposition of odd products of a’s, forming the Cliffordodd ”basis vectors”, offer the description of the internal space of fermions with the half integer spins (manifesting in d =(3 + 1) properties of quarks and leptons and antiquarks and antileptons, with the families includedif d =(13 + 1)), the superposition of even products of a’s, forming the Clifford even ”basis vectors”, offer the description of the internal space of boson fields with integer spins, manifesting as gauge fields of the corresponding Cliffordodd ”basis vectors”. The Cliffordodd and even ”basis vectors” exist also in odd dimensional spaces. In this case theirproperties differa lotfrom the ”basis vectors”in even dimensional spaces. The eigenvectors of the operator of handedness arethe superposition of the odd and even ”basis vectors”, offering both handedness, left and right. These basis vectors resembles the ghosts, needed in Feynman diagrams to get read of singularities. This study just starts and needs further comments and understanding. e 19.4 Some useful formulas This appendix contains some equations, needed in this note. More detailed explanations can be found in this proceedings in my talk [4]. The operator of handedness ..d is for fermions determined as follows d Y . (i) 2 , fordeven , aa a . =( ) · d-1 (19.2) a (i) 2 , fordodd , f f. and II A ^ f e I Cem Cm II ef. with respectto !ab. and !~ab , not The Cliffordobjects a’s and~ a’s fulfil the relations bab b f a; g+ = 2= { ~a ; ~g+ , f a ; ~bg+ = 0, (a, b)=(0, 1, 2, 3, 5, · ;d) , † = aa † = aa ( a) a , ( ~a) ~a . (19.3) The choice of the Cartan subalgebra members is made S03 , S12 , S56 · , Sd-1d , · , 031256 d-1d S;S;S, · ;S, 031256 d-1d ~~~~ S;S;S, · ;S, Sab abab . b . = S+ S~ = i (a - ) . (19.4) @b @a Nilpotentsandprojectorsare definedas follows[2,13,14] ab aa ab 1 b1i b (k):= ( a + ) , [k]:= (1 + a ) , (19.5) 2 ik 2k 288 N. S. Mankoˇc Borˇstnik = aabb with k2 . One finds, taking Eq. (19.3) into account and the assumption { ~aB = (-)B i B agj oc >, (19.6) with (-)B =-1, if B is (a function of) an odd products of a’s, otherwise (-)B = 1 [14], j oc > is defined in Eq. (19.8), the eigenvalues of the Cartan subalgebra operators abab abab kk ab ab ~ S(k)= (k) ;S(k)= (k) , 22 abab ab ab kk ab ab S[k]= [k] ;S~ [k]=- [k] . (19.7) 22 The vacuum state for the Cliffordodd ”basis vectors”, j oc >, is defined as d -1 2 2 X b^ m b^m† j oc >= f * A f | 1> . (19.8) f=1 Taking into account Eq. (19.3) it follows ab ab abababab ab ab ab ab (k)= aa [-k]; (k)= -ik [-k], [k]=(-k); [k]= -ikaa (-k) , ab ab abab abab ab ab ~a (k)=-iaa [k]; ~b (k)= -k [k]; ~a [k]= i (k); ~b [k]= -kaa (k) , † ab ab ab abab ab = aa 2 (k) (-k) , ((k))= 0, (k)(-k)= aa [k] , † ab ab ab ab abab [k] =[k] , ([k])2 =[k] , [k][-k]= 0, abab abab ab ab ab ab ab ab (k)[k]= 0, [k](k)=(k) , (k)[-k]=(k) , [k](-k)= 0, † ab ab ab abab ab = aa 2 (k~ ) (-~k) , ((k~ ))= 0, (k~ )(-~k)= aa [k~ ] , † ab ab ab ab abab 2 [k~ ] =[k~ ] , ([k~ ])=[k~ ] , [k~ ][-~k]= 0, abab abab ab ab ab ab ab ab (k~ )[k~ ]= 0, [k~ ](k~ )=(k~ ) , (k~ )[-~k]=(k~ ) , [k~ ](-~k)= 0. (19.9) 19.5 Acknowledgment The author thanks Department of Physics, FMF, University of Ljubljana, Society of Mathematicians, Physicists and Astronomers of Slovenia, for supporting the research on the spin-charge-family theorybyoffering theroom and computer facilities and Matja ˇ z Breskvar of Beyond Semiconductor for donations, in particular for the annual workshops entitled ”What comes beyond the standardmodels”, and N.B. Nielsen, L. Bonora, M. Blgojevic for fruitful discussions which just start and hopefully might continue. References 1. N. Mankoˇstnik, ”Spin connectionasa superpartnerofa vielbein”, Phys. Lett. B292 c Borˇ (1992) 25-29. 19 Cliffordodd and even objects in even and odd dimensional spaces 289 2. N. Mankoˇstnik, ”Spinorand vectorrepresentationsinfour dimensional Grassmann c Borˇ space”, J. of Math. Phys. 34 (1993) 3731-3745. 3. N. Mankoˇstnik, ”Unification of spin and charges in Grassmann space?”, hep-th c Borˇ 9408002, IJS.TP.94/22, Mod. Phys. Lett.A(10)No.7 (1995) 587-595. 4. N. Mankoˇstnik, ”Cliffordodd and even objects, offering description of internal c Borˇ space of fermion and boson fields, respectively, open new insight into next step beyond standardmodel”, contribution in this proceedings . 5. N. S. Mankoˇstnik, H. B. Nielsen, ”How does Cliffordalgebra show the way to the c Borˇ second quantized fermions with unified spins, charges and families, and with vector and scalar gauge fields beyond the standard model”, Progress in Particle and Nuclear Physics, http://doi.org/10.1016.j.ppnp.2021.103890 . 6. N. S. Mankoˇstnik, ”How Cliffordalgebra can help understand second quantization c Borˇ of fermion and boson fields”, [arXiv: 2210.06256. physics.gen-ph]. 7. N. S. Mankoˇstnik, ”Cliffordodd and even objects offer description of internal space c Borˇ of fermions and bosons,respectively, opening new insight intothe second quantization of fields”, The 13th Bienal Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields IARD 2022, Prague, 6 - 9 June [http://arxiv.org/abs/2210.07004]. 8. N.S. Mankoˇstnik, H.B.F. Nielsen, ”Understanding the second quantization of c Borˇ fermions in Cliffordand in Grassmann space”, New way of second quantization of fermions — PartIand PartII, in this proceedings [arXiv:2007.03517, arXiv:2007.03516]. 9. N. S. Mankoˇstnik, ”How do Cliffordalgebras show the way to the second quantized cBorˇ fermions with unified spins, charges and families, and to the corresponding second quantized vector and scalar gauge field ”, Proceedings to the 24rd Workshop ”What comes beyond the standard models”,5 -11 of July, 2021, Ed. N.S. Mankoˇstnik, c Borˇ H.B. Nielsen, D. Lukman, A. Kleppe, DMFA Zaloˇstvo, Ljubljana, December 2021, zniˇ [arXiv:2112.04378] . 10. N.S. Mankoˇstnik, H.B. Nielsen, “Why odd space and odd time dimensions in even c Borˇ dimensional spaces?” Phys. Lett.B486 (2000)314-321. 11. N.S.Mankoˇstnik, H.B.Nielsen, ”Why Nature has madea choiceof one time and c Borˇ three space coordinates?”, [hep-ph/0108269], J. Phys. A:Math. Gen. 35 (2002) 10563-10571. 12. N.S. Mankoˇstnik, H.B. Nielsen, D. Lukman, ”Unitary representations, noncom- c Borˇ pact groups SO(q, d-q) and more than one time”, Proceedings to the 5th International Workshop”WhatComesBeyondthe StandardModel”,13-23ofJuly,2002,VolumeII,Ed. Norma Mankoˇc Borˇstnik, Holger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFA Zaloˇzniˇstvo, Ljubljana December 2002, hep-ph/0301029. 13. N.S. Mankoˇstnik, H.B.F. Nielsen, J. of Math. Phys. 43, 5782 (2002) [arXiv:hep- c Borˇ th/0111257]. 14. N.S. Mankoˇstnik, H.B.F. Nielsen, “Howtogenerate familiesof spinors”, J. of Math. c Borˇ Phys. 44 4817 (2003) [arXiv:hep-th/0303224]. Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 290) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 20 Modules over Clifford algebras as a basis for the theory of second quantization of spinors V.V. Monakhov Ulyanovskaya 1, Saint Petersburg, 198504, Russia Saint PetersburgState University,v.v.monahov@spbu.ru Abstract. We have studied the properties of the fundamental constructions of QFT -algebraic spinors, Clifford vacua generated by primitive idempotents of the Cliffordalgebra of arbitrary even dimension, and the large Cliffordalgebra in the momentum phase space generatedby the creation and annihilation operatorsof spinors. We have proved that a connected Lie group of Lorentz transformations that preserves relations of the CAR algebra of spinor operators of creation and annihilation leads to the appearance of a small Cliffordalgebra. In it, basis Cliffordvectors are gamma operators, whose matrixrepresentation are Dirac gamma matrices, as well as two additional gamma operators corresponding to the internal degrees of freedom of spinors. We have constructed a Lorentz-invariant spinor vacuum operator of the small Clifford algebra from the product of the Clifford vacua operators of the large Cliffordalgebra. Keywords: QFT, RQFT, Cliffordalgebra, CAR algebra, Cliffordmodules, Cliffordvacuum, spinor vacuum, Lie groups, spinors, algebraic spinors, second quantization PACS: 03.70.+k, 03.65.Fd, 11.30.Ly 20.1 Introduction In 1913 Eli Cartan discovered spinors as two-valued irreducible complexrepresentations of simple Lie groups [1]. The importance of spinors in physics was realized after the appearance in 1927 of Pauli’s work on the spin of the electron [2] and in 1928 of the Dirac equation explainingtherelativisticpropertiesofthe electron[3]. BrauerandWeilin1935 laid down an approach to the theory of spinors based on Cliffordalgebras [4]. Pauli in 1940 proved an unambiguous connection between spin and statistics of particles in the presence of Lorentz covariance [5]. He proved that if the vacuum energy is assumed to be zero, then under the requirement that the energy be positive, particles with half- integer spins must satisfy the Fermi-Dirac statistics. And that from the requirement of relativistic causality (commutation of operators of observables at points separated by spacelike intervals) follows the Bose-Einstein statistics for particles with an arbitrary integer spin. Since then, the concepts of “spinor” and “fermion” have been considered identical. Title Suppressed Due to Excessive Length 291 Nevertheless, the term “fermion”is usually used for physical particles witha half-integer spin,and“spinor”for mathematicalobjectsthataretwo-valuedrepresentationsofgroupsof pseudo-orthogonal rotations (we will call them rotations below without indicating pseudo- orthogonality). Since, mathematically, states with half-integerspingreater than1 canbe expressed in terms of the product of an odd number of states with spin1, it suffices to consideronlyspinorswithspin1.Therefore,belowtheword“spinor”will meanaspinor with spin 1. Amathematically rigorous theoryof spinors asrepresentationsofCliffordalgebras was formulatedby Chevalley[6]. Lounestoanda numberofother authors developedthetheory of spinors as elementsof left idealsof Cliffordalgebras([7–10] and so on). Such spinors are called algebraic [8–10]. The theory of algebraic spinors in the modern formulation is the theory of spinor modules. In this article, we will show that the solution of a number of problems, both in the theory of spinors as elements of ideals of Cliffordalgebras, and in the theory of second quantization based on CAR algebras, lies in consideringspinors as elements of a module over Cliffordalgebra. 20.2 Spinor modules Aspinor module (a spinor space) is a module over the Clifford algebra. The theory of such modulesasthemostgeneral mathematicaltheoryofspinorswas developedrelatively recently[11,12]andthereforeisusuallynot familiarto physicistsusingCliffordalgebras. Let M be an Abelian group, K a ring, m, m1;m2 . M, k;k1;k2 . K.Aleft module M over K is an Abelian group with the operation of left multiplication of elements of M byelements of the ring K, satisfying therelations (k1k2)m = k1(k2m), 1m = m, (20.1) k(m1 + m2)= km1 + km2, (k1 + k2)m = k1m + k2m. Fortheright module,therelationsare similar,butthe multiplicationbythe elementsofthe ringis carriedoutontheright.Fora two-sided module, multiplicationbyring elements can be done both on the left and on the right. If the ring K is an algebra, then the module is a module over this algebra. In this case, relations (20.1) define a homomorphism of the algebra K into the module M. An algebra homomorphismisa mapping thatpreserves the basic operations and basicrelationsof the given algebra. In particular, the Cliffordalgebra is a two-sided module over itself. An important consequence of the theory of modules is that there is a one-to-one correspondence (up to isomorphism) between linear representations of any associative algebra and modulesoverthis algebra.This meansthattheresults obtainedfor matrixrepresentationsof algebras are of much greater significance than one might expect – they are applicable to any linearrepresentationsof these algebras.The questionofthe equivalenceor non-equivalence of certain algebraic constructions for the representation of spinors is also removed – they are equivalentif their matrixrepresentations coincide(upto isomorphism). Thus, the matrix algebra generated by the Dirac gamma matrices is equivalent to the corresponding Clifford algebra, and the spinor space in the form of a matrix column is equivalent to the minimal ideal of the Clifford algebra generated using a primitive idempotent. In addition, when trying to create algebraic constructions describing spinors (for example, in [13]), in order to verify the correctness of the algebraic constructions and 292 V.V. Monakhov the physical interpretationof theresults, one should either explicitly check thepresenceof an algebra homomorphism (20.1) or check the corresponding matrix representations [14]. Consider now the application of modules over algebra in physics. Physicists began to actively use the work with modules over algebras after the creation of quantum mechanics. Paul Dirac called the left modules of the algebra of operators of quantum mechanics ket- vectors | >, while the right modules are bra-vectors <. j.Afeature of a one-sided (left or right) module is that the elements of the module can only be added, but not multiplied. Although elements of the left module can be multiplied by elements of the algebra on the left, and elements of the right module can be multiplied on the right. The principle of superposition in quantum mechanics is a manifestation of the fact that state vectors are elements of a module. Working with modules over matrix algebra is used in the theory of Dirac spinors. In matrix representation, the Dirac spinor is a column with four components D = 0 BBBBB@ 1 CCCCCA 123 4 . (20.2) It is a left module over the algebra of 4 × 4 matrices. 20.3 Algebraic spinors An algebraic spinor . in the matrix representation can be given by a 4 × 4 matrix. It has four columns . = 0 BBBBB@ 1 CCCCCA 11 12 13 1 4 21 22 23 2 4 (20.3) . 31 32 33 3 4 41 42 43 4 4 corresponding to the four Dirac spinors 1 = 0 BBBBB@ 1 CCCCCA , 2 = 0 BBBBB@ 1 CCCCCA 0 BBBBB@ 1 CCCCCA , 4 = 0 BBBBB@ 1 CCCCCA . 1 1 1 2 1 3 1 4 2 1 2 2 2 3 2 4 (20.4) 3 = , 3 1 3 2 3 3 3 4 4 1 4 2 4 3 4 4 Aleft ideal of an algebraA isa subalgebra thatis closed under multiplicationby elements of the algebra A. An ideal is called minimal if it does not contain subideals. That is, if it cannot be decomposed into the sum of two or more ideals. The minimal ideal is generated by theproductof all elementsof the algebrabya primitive idempotent.An idempotentis said to be primitive if it cannot be decomposed into two (or more) orthogonal idempotents. In the Cliffordalgebra, the spaces of spinors (minimal left ideals) corresponding to Dirac spinors are generatedby four primitive idempotents Ij, j = 1, 2, 3, 4, having the idempotent property (Ij) 2 = Ij, (20.5) and the orthogonality property IjIk = 0, j 6(20.6) = k. Title Suppressed Due to Excessive Length 293 Inthe matrixrepresentation, matriceswithoneunitelementonthe diagonalcanbe chosen as such idempotents [10]: 1000 0000 0000 0000 0000 0000 0001 Left multiplication of any general matrix (20.3) by idempotents (20.7) leaves nonzero only the columns corresponding to the Dirac spinors (20.4): 1 CCCCA 000 1 CCCCCA 0 BBBB@ 1 CCCCA 0 BBBB@ 1 CCCCA 0000 0000 0010 0 BBBB@ 0 BBBB@ 1 CCCCA 0000 0100 0000 (20.7) I1 ;I2 ;I3 ;I4 = = = = . 0 0 00 0000 0000 0 BBBBB@ 0 BBBBB@ 1 CCCCCA 11 12 0 00 21 22 000 0 00 0 00 I1 , I2 = = , 31 32 0 00 41 42 00 0000 (20.8) 0 BBBBB@ 1 CCCCCA 0 0 000 0 BBBBB@ 1 CCCCCA 10 3 0 1 4 2 4 00 000 23 00 000 I3 = , I4 = . 33 3 4 00 000 43 4 4 00 It is easy to see that further multiplication Ij on the left by an arbitrary number of 4 × 4 matriceskeepsonlycolumn number j nonzero. These columns are left ideals of the algebra, the spinor spaces. In the case of d-dimensional spinors, for even d = 2n, there are 2n columns in a column of 2n components (that is, 2n independent spinors), andthe matrix corresponding to the algebraic spinor has size 2n × 2n. In what follows, we will consider only the case of even d, since in the odd-dimensional case the center of Cliffordalgebra is nontrivial, and the similarity transformation is not an inner automorphism of the algebra. Therefore, in this case,a numberofpropertiesof the spacesof vectors and spinors differfrom those observed physically. For even d,itis possibletopasstothe equivalent matrixrepresentationofthe idempotents of the Cliffordalgebra using the similarity transformation 0 -1 (20.9) I = BIjB , j along with all other elements A of the given matrixrepresentation A0 = BAB-1 , (20.10) for an arbitrary invertible matrix B.Without loss of generality, we can assume that its determinantisequalto1.Moreover,itis obviousthat,asaresultof transformation(20.9), the idempotents retain properties (20.5) and (20.6). It was shownin [10] thatbya similarity transformationof the form (20.10),a setof primitive idempotents of the matrix representation of any complex Cliffordalgebra for even dimensiondcanbereducedtotheform(20.7).Therefore,forspinor modules,itissufficient to consider only idempotents of the form (20.7) and spaces of spinors of the form (20.8). Basis orthonormal vectors ei of the d-dimensional Cliffordalgebra in the matrix representation are usually called d-dimensional Dirac gamma matrices i. The group of Clifford rotationsandreflectionsinthecaseofthe signatureoftheCliffordalgebra(p, q)=(1, d - 1) 294 V.V. Monakhov or (p, q)=(d - 1, 1) is usually called Lorentz d-dimensional group. Here p is the number of basis vectors with a positive signature, and q witha negative one. Due to the fact that rotations groups areisomorphic for(p, q)=(1, d - 1) and (p, q)=(d - 1, 1),andreflections from the full Lorentzgroup are not includedin the transformationsof the Poincar´ e group, we will consider only the signature (p, q)=(1, d - 1). The results obtained can be easily generalized to spaces of even dimension of arbitrary signature. The Cartan subgroup of a connected Lie group is the maximal connected Abelian subgroup of this group. Its Lie algebra is called the Cartan subalgebra of the Lie algebra of the given group. The Cartan subgroup of the Lie algebra of the Lorentz group is the Lie group generated by the maximum possible number of linearly independent commuting elements of the corresponding Cliffordalgebra. In this case, the Cliffordalgebra acts as a universal enveloping algebra for the Lie algebra of the Lorentz group. Each Lie algebra can be uniquely associated with a universal enveloping algebra (up to isomorphism), a Cartan subalgebra can be uniquely associated in the Cliffordalgebra with a subalgebraof the Cliffordalgebra generatedby the maximum possible numberof linearly independent commuting elements of the Cliffordalgebra. It is universal enveloping algebra of the Cartan subalgebra. In the matrix representation, the basis of this algebra can be transformedby a similarity transformationtoa diagonal form.We choosethe generatorsof the Lorentz group 0 3; 1 2 , · ; d-1 d as thebasisof this algebra. They commute, and in the chiral representation of the gamma matrices are diagonal. Using them, we construct 2n idempotents 1 ± 0 3 1 ± 1 2 1 ± d-1 d I03;12;· ;(d-1)d = · . (20.11) 22 2 It is easy to see that idempotents (20.11) will have a form similar to (20.7), but for 2n × 2n matrices, with one unit element on the diagonal and with zero other elements of the matrix. It is also easy to check that if we denote the idempotents (20.11) as Ij, then 2n X 1 = Ij. (20.12) j=1 Idempotents (20.11) were constructed in [16, 17], and decomposition (20.12) for Clifford algebras was obtained in [12]. It was used implicitly in [17] and explicitly by us in [15] to decompose algebraic spinors into spinor modules in RQFT. Note that due to the presence of an imaginary unit in idempotents (20.11), they are admissible only in the complex Cliffordalgebra Cl1;d-1(C) and are inadmissible in the real algebra Cl1;d-1(R). However,the multiplication of elements Cl1;d-1(R) by any of the idempotents (20.11)isa homomorphismof the algebra Cl1;d-1(R). Therefore, the construction of spinor modules over this algebra using themis correct. Similarly,itis notaproblem that Dirac gamma matrices are complex, although the corresponding Cliffordalgebra is real. Corresponding mapping from the Cliffordalgebra to the algebra generated by Dirac gamma matricesisa homomorphism. This matrix algebraisa two-sided module over the Cliffordalgebra. But at the same time, it is impossible to multiply the elements of this matrix algebra by an imaginary unit, since this violates the homomorphism. 20.4 Problems of the theory of algebraic spinors Despite being more general than previous theories of spinors, there are a number of problems in the theory of algebraic spinors. Title Suppressed Due to Excessive Length 295 The first of them is the presence of 2n independent spinors.They have been interpretedina varietyofways,fromcompletelyignoring“extra”spinors[9,18]to consideringthemas independent spinor fields [17] and even interpretingthem as states of fermions and bosons of the StandardModel [13]. In [21] and our work [15], an approach was found to solve this problem, which consists in the fact that these spinors belong to states with different vacua. In [21], the author proposed various expressions for Clifford vacua and indicated the fact that swapping the creation and annihilation operators changes one Clifford vacuum for another.In [15] we madea correct constructionof the Clifford vacua and showedthat they have the properties of spinors. However, as will be shown below, the spinor vacuum has a more complex structure than the Cliffordvacuum. The correct construction of spinor vacua will be given below when considering the properties of the CAR algebra. The second problem in the theory of algebraic spinors is related to conjugate spinors. Each fermion, described by the matrix column , has an antiparticle, which in Dirac’s theory is described by the Dirac conjugate quantity – the matrix row . —=( 0 )+. The matrix column and the matrix row exist in different spaces, and the corresponding states cannot be added. However, algebraic spinors belong to the same Cliffordalgebra, and their matrix representations belong to the same matrix algebra. So the question arises why they can not mix. Moreover, the Dirac conjugationofa matrixof the form (20.3) givesa matrixofa similarform.Thatis,the antispinor stateisa superpositionofspinor states.The solutionof this problem, as we show below, follows from the construction of spinors and spinor vacua in the framework of the CAR algebra. The thirdproblem is related to the impossibility of constructing the spinor vacuum as a scalar.TheCliffordvacuum cannotbeascalar.UnderLorentz transformations,it transforms asaspinor.Duetothepresenceof decomposition (20.12),theunit1inthe one-sidedspinor module is decomposed into 2n spinors and has the property of a spinor. Therefore, it also cannotbea scalar.We will consider thisproblem below. The fourthproblemisrelatedtothe physicsof actually observed fermions. Fermions,asyou know, canbe created and annihilated.To describe theseprocesses, the so-called theoryof second quantization was developed, the mathematical basis of which is the theory of CAR algebras. The study of the CAR algebra together with the Cliffordalgebra corresponding to the Lorentz group will be done next. 20.5 Clifford vacuum In accordance with [19–21], the Clifford vacuum V is built using the creation a + k and annihilation ak operators built from the basic Cliffordvectors, in this case, from the gamma matrices 0 + 3 + 0 - 3 1 - i 2 + - 1 - i 2 a1 = ;a = ;a2 = ;a = ;:::, 2 1 22 2 2 (20.13) - d-1 - d an = ;a = . d-1 - i d + 2 n 2 as a state for which ak V = 0 (20.14) for every k. Note that for d>2, such an operation is possible only in the complex Cliffordalgebra Cl1;d-1(C),orinthe complex module overthereal algebraCl1;d-1(R),orinthereal Clifford algebra Cld=2;d=2(R). It is obvious that V = a1a2 · anA (20.15) 296 V.V. Monakhov where A is an arbitrary nonzero element of the algebra that does not annihilate a1a2 · an. Operators (20.13) satisfy the anticommutation relations fa j ;ak} =  + k j , (20.16) j ;a k } = 0. + + faj;ak} = fa There are various options for specifying the factor A in (20.15) [21]. It was noted in [21] that the action a + j V of the spinor creation operator a + j on the vacuum V for any j should create a state with one spinor. This spinor must belong to the spinor space, that is, the minimal left ideal. Therefore, AV in (20.15) should be represented as a product of some element A1 of the algebra and a primitive idempotent I. That is why V = a1a2 · anA1I. (20.17) Any primitive idempotent can be chosen as I [21]. However, it is natural to require that the action of the vacuum operator on itself leaves the vacuum invariant V V =( V )2 = V . (20.18) Requirement (20.18) means that V must be an idempotent. Let us show the uniqueness of the Clifford vacuum for operators (20.13) under conditions (20.15) and (20.18).We decompose A intoa sumof monomsin termsof the basisof the Cliffordalgebra, for which we use sums and products of operators (20.13). In this case, in the monoms, all operators ak canbe placedto the leftof a + k usingrelations (20.16). Since V V =( V )2 = V = a1a2 · anAa1a2 · anA, (20.19) all monoms, at least one element of which commutes or anticommutes with any of the operators ak, will give a zero contribution on the right side of (20.19). Only + a k does not commute and does not anticommute with ak. Therefore,anonzero contributionto V gives only a + 1 a + 2 · + a n , and we can assume up to a sign that + + + (20.20) V = a1a · ana 1 a2a 2n . From (20.13) it follows that 1 - 0 3 2 1 - i 1 1 - i d-1 d . (20.21) + + + a1a = ;a2a = ;:::;ana = 12 n 2 2 2 Wherein 2 + (aja j ) + = aja j , (20.22) 2 + (a j aj) + = a j aj for each j. Here and below there is no summation over repeated indices. From (20.20), (20.21) and (20.11) we obtain 1 - 0 3 1 - 1 2 1 - d-1 d V = I-03;-12;· ;-(d-1)d = · . (20.23) 22 2 Such a definition is ambiguous due to the arbitrariness in (20.11), (20.13) and (20.23) the signs in front of the gamma matrices. That is, in the choice of which operator to consider as the operator of creation, and which one as the operator of annihilation. In accordance with [15], each of the primitive idempotents (20.11) is a Cliffordvacuum, but in different idempotents the role of some of the creation and annihilation operators has changed. Therefore, in d-dimensional space, where d = 2n, there are 2n independent Clifford vacua. Title Suppressed Due to Excessive Length 297 The idempotents (20.11) are Hermitian. Therefore, when choosing any of them as the Cliffordvacuum, we automatically obtain that all Cliffordvacua operators are Hermitian + (20.24) V = V . Conditions (20.24) and (20.15) imply + = A+ + + + (20.25) A. a · a = a1a2 · n2 a 1 an V That is why + + + (20.26) A = A2a · a n2 a 1 , where A2 is some element of the algebra. From (20.24)-(20.26) we obtain + + + (20.27) V = a1a2 · anA2a · a n2 a 1 , As before, we expandA2 in terms of monoms with operators aj to the left of a + j . The contribution on the right side of (20.27) of all monoms other than 1 with some numerical factor, is equal to zero. Therefore V = a1a2 · + + + ana · a n2 a 1 , (20.28) Taking into account (20.16), we have obtained formula (20.20) up to sign. Based on this, at first glance, conditions (22.11) and (20.24) rather than (22.11) and (20.18) can be used to choose the Clifford vacuum formula. However, when transforming elements of the algebra according to formulas (20.9)-(20.10), conditions (22.11) and (20.18) will be preserved, but condition (20.24) will be violated in the case of non-unitary matrices B. Therefore, to specify the Clifford vacuum operator, one should choose conditions (22.11) and (20.18). Consideraleftideal(thatis,thespinorspace) formedbyleft multiplyingthe elementsof the Cliffordalgebra by the primitive idempotent (20.23). Such a mapping is a homomorphismand definesaCliffordmodule.Init,theunit1oftheCliffordalgebragoesintothe idempotent (20.23), that is, this idempotent is the unit 1m of this module. That is why 1m = V , (20.29) aj1m = 0, 8j. Therefore, we can designate the creation operators as Grassmann variables, and the annihilation operators as derivatives with respect to them j =  , + a j (20.30) . aj = . @j In this case, conditions jj 1m = 6 + 03,andtheHermitianoperators ; j 1m = 0, (20.31) . aj1m = 1m = 0 @j are satisfied. i 0 3 The anti-Hermitian operator S03 = 2 is a d-dimensional boost operator in the plane 2 d S12 = ;:::;Sd-1;d = (20.32) i 1i d-1 22 =  a 298 V.V. Monakhov are d-dimensional spin operators and correspond to rotations in the planes 1; 2, ::, d . These operators have eigenvalues -i=2, -1=2, . . . , -1=2 on the eigenvector (20.15) for any A. That is, Clifford vacuum corresponds to the state with the lowest (lowest sign) half-integer spin. That is, it is a spinor. Obviously, the Clifford vacuum V cannot be a scalar, since the requirement V = 1 is incompatible with conditions (20.13)-(22.11). Therefore, it is not invariant under Lorentz transformations and cannot be considered as a spinor vacuum (or, what is the same one, fermionic vacuum), corresponding to actually observed spinors.We have constructed such a vacuum in the framework of the theory of CAR-algebraic spinors (previously we called them superalgebraic) [22–24]. In this paper, we have studied the properties of spinors based on the theory of CAR algebras. 20.6 Second quantization and CAR algebra The development of the mathematical theory of second quantization followed a parallel branch with the theory of algebraic spinors and practically did not intersect with it. Fermion field quantization based on canonical anticommutation relations (CAR) was introduced by Jordan andWignerin 1928 [25].In modern quantum field theory, the algebraof canonical anticommutationrelations(CAR algebra)is considered as fundamentalin describingthe properties of spinors. Relativistic quantum fieldtheory (RQFT) uses the second quantization method to describe systems with the creation and annihilation of field quanta. Its foundationsfor spinors were formulatedby Schwingerin 1951-1953[26,27]. Mathematical substantiation of the theory of second quantization was developed on the basis of the theoryof CAR algebrasin the worksofG ° arding andWightman [28], Araki andWyss [29], Berezin [30] and so on. RQFT is based on the theory of second quantization and canonical anticommutationrelations, as well as on infinitesimal transformationsof fields and operators. In the modern mathematical interpretation, these are transformations of the Poincar e´ group and the Lie algebra corresponding to it. Anticommutation relations for the fermion creation operator aj(p)+ (with number j and spatial momentum p)and the fermion annihilation operatorak(p 0) (with number k and spatial momentum p 0)acting on the Hilbert space can be written as [31] fak(p)+ ;al(p 0)} = l k (p - p 0), (20.33) 00)+ fak(p);al(p )} = fak(p)+ ;al(p } = 0. (20.34) In fact, the momentum spectrum of fermions in the free state is not continuous, but quasi- continuous discrete, with very small distances between discrete levels. The size L of the flat space (the Universe) is very large, and we can assume that it tends to infinity L !1. In this case, cyclic Born-von Karman boundary conditions can be set, and a discrete spectrum of momenta pi is obtained with an infinitely small momentum step p = 2— h=L › 0. Let us replace equations (20.33)–(20.34) for the continuous spectrum with equations for the discrete quasi-continuous spectrum [22]. In this case, the Dirac delta function in (20.33) must be replaced by its discrete analog, and we obtain discrete relations [22] 1 ki fak(pi)+ ;al(pj)} = l j, (20.35) 3p fak(pi);al(pj)} = fak(pi)+ ;al(pj)+} = 0. (20.36) For operators related to the same value of the spatial momentum pi = pj, they coincide, up to normalization, with conditions (20.16), where the discrete value of the momentum is one of the parts of the particle-type multi-index. Title Suppressed Due to Excessive Length 299 Let us introduce the normalization of the creation and annihilation operators so p Ak(pi)= 3pak(pi), (20.37) that their anticommutation relations fAk(pi)+;Al(pj)} = k l ij, (20.38) fAk(pi);Al(pj)} = fAk(pi)+;Al(pj)+} = 0. (20.39) completely coincide in form with (20.16). Infinite-dimensional algebraof operators (20.37)is called the CAR algebra [28,29]. Let us introduce operators ..+ki = Ak(pi)+ + Al(pi), (20.40) ..-ki = Ak(pi)+ - Al(pi). From (20.40) it follows that (.. )2 = 1, . =+ki, (.. )2 =-1, . =-ki, (20.41) f.. ;.. } = 0, . 6 = . Formulas (20.41) can be generalized as f.. ;.. } = 2 ; ß = diag(+1, -1, +1, -1, . . :). (20.42) Formula (20.42) shows that CAR algebra is an infinite-dimensional countable Clifford algebrawhose operatorsare definedona Hilbertspace.Thisfactiswellknown[32]and is even used as one of the ways to define CAR algebras instead of specifying canonical anticommutationrelations [33].We called this algebra large Cliffordalgebra [24]. Let us find out the transformation laws for operators ak(pi) and ak(pi)+ under Lorentz transformations by analogy with [31], pp.192-193. But, in contrast to [31], we take into account that during boosts, the positive-frequency and negative-frequency components of the spinor must mix. Therefore, relations (20.35)–(20.36) are true only for pi › 0. Let us denote for this case ak(pi)+ = k(pi), . ak(pi)= , @k(pi) p(20.43) k. ..+ki = 3p ((pi)+ ), @k(pi) pk. ..-ki = 3p ((pi)- ). @k(pi) Then . l1 ki { ;(pj)} = l j, @k(pi) 3p (20.44) @. { , = fk(pi);l(pj)} = 0. @k(pi) @l(pj) Relations(20.44)arejust anotherformofrelations(20.42)oftheCARalgebraasaClifford algebra.Werequire that Lorentz boosts transform the creation and annihilation operators for momentum pi into the creation and annihilation operators for another momentum. That 300 V.V. Monakhov is, we require that Lorentz boosts transform the relations (20.44) for momentum pi into relations (20.44) for another momentum. Hence, relations (20.42) of the Cliffordalgebra transform to ;.. } = 2 , are transformed Dirac matrices. 0 0 (20.45) f.. . and .. It follows from the generalized Pauli theorem [34] that matrices .. 0 0 where .. ß 0 . and ... are related by the formula 0 -1 , (20.46) .. = B.. B . where B is some invertible element of the Cliffordalgebra. Consider infinitesimal transformations when B = 1 + dG = e dG , (20.47) where dG is some infinitesimal elementof the large Cliffordalgebra. Denote a commutator of dG with following elements as dG ^=[dG, ], the operator . @k(0) after the boost to the finite momentum pi as bk(pi), and the operator k(0) after the boost as b— k(pi). Then ^G We have obtained infinitesimal transformations of the Lie group. By integrating these transformations,we obtain similar formulasforthe finitevaluesoftherotation angles.In this case, the formulas for the operators after the Lorentz transformation will look like 0 d (20.48) .. . =(1 + dG)..(1 - dG)=(1 + dG^)... ... = e . ^ bk(pi)= e G , @k(0) (20.49) ^G bk(pi)= e andrelations (20.44) will look 1 — bl(pj)} = — k (0). ki l  fbk(pi),  j, 3 (20.50) p — fbk(pi);bl(pj)} = { b— k(pi);bl(pj)} = 0. ^ ^ ^. ^), , . = 0, 1, 2, 3, where gamma operators ^µ are the operator analogs of the corresponding Dirac matrices µ . Therefore, (20.49) can be rewritten as D 0r . 1 G are generated by the operators ^=(^ µ ^. In [23], we proved that the rotations e - 2 . 1 !0r bk(pi)= e 2 , @k(0) (20.51) ^ 0r where !0r are parameters of the boost to the momentum pi. Also, there are two additional gamma operators ^6 and ^7 compared to the Dirac’s theory, which correspond to the internal degrees of freedom of spinors [23, 24, 35–38], and the Cliffordpseudovector ^5, corresponding to the Dirac matrix 5 . The algebra generated by D the gamma operators ^, ^6 and ^7, we called the small Cliffordalgebra [24]. The operator b— k(pi) in (20.51), as it is easy to check, is obtained using the generalized Dirac conjugation of the field operator bk(pi) — bk(pi) = (^ 0bk(pi))+ . (20.52) Thus, we have substantiated the formulas for superalgebraic spinors that we obtained earlier [22–24, 35–38]. As is clear from the above, it is more correct to call them CAR algebraic spinors. 1 — bk(pi)= e k !0r  (0). 2 Title Suppressed Due to Excessive Length 301 20.7 Spinor vacuum Inthetheoryof second quantization,an importantroleisplayedbythespinor (fermionic) vacuum as a state in whichthere are no spinors. In most studies it is assumed that it is unique. However, such an assumption contradicts the theory of CAR algebras. It has been proven that there are an infinite number of physically equivalent vacua, only one of which is the Fock vacuum (in which the particle number operator is meaningful) [28]. However, an explicit algebraic formula for the spinor vacuum has not been obtained in the framework ofthetheoryofCAR algebras.Atthe sametime,inafew attempts[13,17,21], includingour own [15], to explicitly construct an algebraic expression for the spinor vacuum, the authors tried to identify the physical vacuum of spinors with the Cliffordone. This, as shown above, is wrong. In [22], we obtained an explicit expression for the spinor vacuum in terms of the field operators (20.49). Fora state with momentum pi, we introduce the operator 34 Vi =(p)b1(pi) b— 1(pi)b2(pi) b— 2(pi)b3(pi) b— 3(pi)b4(pi) b— 4(pi). (20.53) Since, according to (20.50), ( b— k(pi))2 = 0, then (3pbk(pi) b— k(pi))2 =(3 p)2bk(pi) b— k(pi)( 1 - b— k(pi)bk(pi)) = 3p (20.54) 3 = pbk(pi) b— k(pi). From (20.53) and (20.54) it follows that 2 , that is, Vi is an idempotent. It is easy Vi = Vi to verify that this is the Clifford vacuum (20.23) for the Cliffordalgebra corresponding to a given value of pi. All Vi for different i commute with each other. Therefore, the operator Y V = Vi (20.55) i isan idempotent.Itis invariantwithrespecttoLorentzrotations,sinceClifford vacua Vi simply change the place asa factorin (20.55) under suchrotations. Since bk(pi) b— k(pi) b— k(pi)= 0 11 (20.56) —— bk(pi)bk(pi) b— k(pi)= b— k(pi)( - b— k(pi)bk(pi)) = bk(pi), 3p3p operators bk(pi) play the role of annihilation operators, operators b— k(pi) playthe role of creation operators YY Y bk(pi) V = bk(pi) Vj = Vj bk(pi) Vi Vj = 0, j ji YY Y (20.57) —— — bk(pi) V = bk(pi) Vj = Vj bk(pi) Vi Vj = 60, j ji and N(pi)= 3 pbk(pi)bk(pi) (20.58) — the role of operator of the number of particles with momentum pi N(pi) V = 0 N(pi) b— k(pj) V = 0, i 6 = j, (20.59) — N(pi) b— k(pi) V = bk(pi) V . 302 V.V. Monakhov 20.8 Conclusions We have proved that modules over Cliffordalgebras are a basis for the theory of second quantizationof spinors.We developed the theoryof algebraicspinors as left modules over d-dimensional Cliffordalgebra and showed the presence in it of 2d=2 equivalent Clifford vacua, which differ in the role of 2d=2 creation operators and 2d/2 annihilation operators. We have proved that transformations of the connected Lie group of Lorentz transformations thatpreserverelationsoftheCAR algebraleadtothe appearanceofa smallCliffordalgebra. In it, the basis Cliffordvectors are four gamma operators ^;µ = 0, 1, 2, 3, whose matrix representation areDirac gammamatrices µ ,as well as two additional gamma operators ^6 D and ^7, corresponding to the internal degrees of freedom of the spinors. For CAR algebraic spinors, a spinor (fermionic) vacuum is constructed in explicit form. It is a 4-scalar and is invariant under the Lorentz transformations and gauge transformations corresponding to the internal degrees of freedom of these spinors. References 1. E. Cartan: Lesgroupesprojectifsqui ne laissent invariante aucune multiplicit´ e plane, Bull. Soc. Math. France 41, 53-96 (1913). 2.W. Pauli: ZurQuantenmechanik des magnetischen Elektrons, Zeitschriftf¨ ur Physik 43, 601–632 (1927). 3. P. 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V. Monakhov, A.Kozhedub: Algebra of Superalgebraic Spinors as Algebra of Second Quantizationof Fermions, Geom. Integrability&Quantization 22, 165–187 (2021). 36. V. Monakhov: Spacetime andinner space of spinors in the theory of superalgebraic spinors, J. of Physics: Conf. Series 1557, 12031 (2020). 37. V. Monakhov: Generation of Electroweak Interaction by Analogs of Dirac Gamma Matrices Constructed from Operators of the Creation and Annihilation of Spinors, Bull. of Russian Acad. of Sciences: Physics 84, 1216–1220 (2020). 38.V. Monakhov: The Dirac Sea,TandCSymmetryBreaking, and the SpinorVacuumof the Universe, Universe 7, 124 (2021). Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 304) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 21 Anew view on cosmology, with non-translational invariant Hamiltonian H. B. Nielsen1, M. Ninomiya2 1Bohr Institute, Copenhagen 2Yuji Sugawara Lab., Science and Engineering, Department of Physics Sciences, Ritumeikan university, Japan Abstract. Theideaofthis contributionisto suggestawaytogetridofgravityasadynami- cal space time approximately in cosmology and thus be able to use Hamiltonian formulation ignoring the gravitational degrees of freedom, treating them just as background. Concretely we suggest to use a back groud De Sitter space time and then instead of the usual choice of coordinates leading to a picture in which the Universe Hubble expands, we propose to identify the time translation in the new coordinate system with a Killing form transformation for the De Sitter space time. This then leads to unwanted features like the descripton being formallynot translational invariant,butwehaveinmindjusttogetinasimpleway time translation and its associated Hamiltonian, and shall then in wordgive some ideas of the from this point of view way of looking at the usual cosmology. Keywords: cosmology, Hamiltonian, coordinates PACS: 98.80 Qc, 04.20 -q 21.1 Introduction In quantum mechanics and in analytical mechanics one works with very general mechanical systems using a Hamiltonian formalism, in which the time t is takenasa parameterasa function of which then the state j (t) > is considered. In relativity theory and especially ingeneralrelativitythetime conceptis complicatedbybeingattheendageneral coordinate, which one has to choose, and it cannot be treated correctly unless one includes the gravitational field degrees of freedom. But if we have some ideas developped in analytical mechanics or quantum regi with a simple Hamiltoniannot including gravitationaldegreesoffreedomand wouldlikeata first crude stage to apply it to cosmology, then we would like to be allowed to have at least a crude cosmology,in which the gravitational fieldis considereda static background, so that most importantly an expansion of space can be ignored. Of course one could alternatively introduceasa dynamical variablethesizeofthe Universe, a say,butthatisreally beginning to approximate a dynamical gravity, which it is the purpose of the present idea to avoid. 21 Anew view on cosmology, with non-translational invariant Hamiltonian 305 Let us at least state, that we want in the “central region” in 3-space to have a flat space approximation like one really in a short distance perspective usually work with in the neighborhoodof our MilkyWay. Then other usualrequirements which may notbe so important for making a Hamiltonian description o.k., such as the translational invariance or the associated asssumption, that crudely there is the same density of galaxies etc. all over onaverylargescale,wedonotneed,ifitistroublesometo obtain. Let us take as a first approximation cosmological model the De Sitter space time model. (You may actually choose between taking the cosmological constant either the effective one in the inflation era or the present effective value.) Let us resume and concretize our “model”: • We takeaDe Sitter space time. • We take the time development to be identified with a Killing transformation of the space time approximating the cosmology (i.e. a Killing transformation for the De Sitter space time. • We arrange the MilkyWay to be, where the “new” time translation operation deviates the least from the “usual” FLRW (Friedmann–Lema^itre–Robertson–Walker metric) parametrization “time”. 21.1.1 Why we Like Hamiltonian formalism, butTrouble with Gravity • From quantum mechanics we get (historically ?) accustomed to work with theories describedby a Hamiltonian. • In general relativity the for the Hamiltonian so basic concept, the energy E becomes strongly gauge dependent in the for cosmology interesting situations. • Soitlooksatfirst,thatoneneedsaquantumgravity;butthatisawfull, becausemany colleagues work on that without being even themselves convinced so much. May be string theory is good but not immediately usefull for cosmology? 21.1.2 Our Suggestion: Usea Killing Symmetry for“TimeTranslation”in Approximate Cosmology The main suggestion of the present work/talk is: • Getridofgravitybytakingthe gravitationalfield-the geometry -asonlyabackground field. I.e. do not include gravity in the dynamical degrees of freedom being treated by the Hamiltonian. • But then we need the time translation symmetry to be at least an approximate symmetry of the gravitational degrees of freedom. • So choose an approximately cosmologically correct geometry and identify the “time translation symmetry” with a Killing transformation symmetry of the approximate geometry. 21.2 DeSitter space time To obtain a pictorial image of de Sitter space time we want to present a perspective picture in3dimensionsto illustratethe imbeddingofthe3+1 dimensionalDe Sitterspaceintoa 4+1 dimensional space-time, just for giving the illustration. But to do that we then need to simplyremove2ofthespatial dimentionssoastoreduce4+1to2+1(correpondingtowhat humans can conceive of as perpective drawing): See fig ??. 306 H. B. Nielsen, M. Ninomiya Fig. 21.1: The imbedded 1+1 DeSitter space time made to really represent the physically relevant 4+1 imbedding of the 3+1 dimensional De Sitter space time, whichmaybeacrudelygood cosmology.Herethe coordinates drawnonthefigure are the usual FLRW coordinates,in which the universein the upper(= late time part) is expanding. The lower part will in most sensible models be considered so wrong that we should ignore it. 21.3 Coordinates IntheDe Sittermodelthespacehadata certaintimeintheusualFLRW(fig.??)coodinates a most narrow i.e. least spatial size (radius R)moment of time. This is of course not true if one believes in a genuine Big Bang model, so it is only the time somewhat after that moment of the narrow space that should be taken approximately seriously. Also in the Killing form suggested coordinates as on the figure ?? the region below the narrow neck is of course presumably not to be taken seriosly. Denoting the radius of the universe at the most narrow moment by R we can writein the imbedding coordinates the equation for theDe Sitter spacetime surface as imbeddedin the 4+1 dimensional space time, with the time -like coordinate X0 going upwards on the shown figures. Introducing of course an extra coordinate compared to usual 3+1 space time, say X4 we have the following equation for the imbedded surface to be identified with the universe space time: 12 22 32 42 022 (X)+(X)+(X)+(X)-(X)= R. (21.1) WeputtheMilkyWayatthe maximal valueofX4 for a given value of X0, i.e. indeed, for MilkyWay: p X4 = R2 +(X0)2 for MilkyWay . (21.2) IfwenowwanttokeeptoourwishtoletthetimeattheMilkyWaybethe eigentimethere, then we are forced to both in usual coordinates and in the “new ”ones to have r (dX0)2 -(dX4)2 = 1 along the trackof MilkyWay (21.3) dt2 21 Anew view on cosmology, with non-translational invariant Hamiltonian 307 Fig.21.2:InthisfigureweseebothadeSitterspacetime imbeddingandananti- Desitter space time imbedding. In both figures the time in the front region goes upwards on the figures. But we are in the present article really only interested in the De Sitter space-time in green to the left -and also do not care for the problem of what is wrong with quantum field theories -, and this figure illustrates De Sitter space with 1+1 dimension (instead of 3+1). On the one side of it is drawn a lap with coordinates, illustrating those coordinates we propose here: the coordinate curves going upwardare the time coordinates and as such in the 1+1 dimension eachrepresentapointin space. The more horizontal coordinate lines are “parallel” to the space coordinate and each of the lines represent a moment of time in the “ our ” coordinate system. The region below/earlier than the narrow neck is not to be taken seriouslyin usual cosmology, sinceitwouldbe beforethe smoothed out big bang. 308 H. B. Nielsen, M. Ninomiya This in fact leads to t X4(t)MW = R * cosh (21.4) R t X0(t)MW = R * sinh . (21.5) R where t stands for the time coordinate tu of the usual FLRW coordinates or in the “new” proposal tn. In the “usual” FLRWmodel we keep the equation (21.5) to be valid not only fortheMilkyway,butall over.Usuallyonethen definesthe radiusofthe Universeatthe time tn = t by the equation fora S3-sphere representing the space at that time: 12232422 (X)+(X)+(X)+(X)= a (21.6) so that a = R * cosh tu . (21.7) R In the “new”, heresuggested coordinates, we rather let the “momement of time” cut straight back in “usual” time to the S2-sphere given by “Axis sphere” X0 = X4 = 0 (21.8) 12 22 322 or (X)+(X)+(X)= R. (21.9) That is to say, that X0 (tn) tn = tanh for “new” system. (21.10) X4(tn) R dX0 tn so that for tn fixed = . (21.11) dX4 R Let us also define a distance DisttoMWn from the MilkyWay along the equal time tn in the “new” coordinate system out to a running point counted spacelike by 2 42 02 dDist to MW2 = da+(dX)-(dX)(21.12) n X4 X4 as function of the angle n = arccos = arccos (21.13) X4 MW R * cosh t R n giving Dist to MWn = Rn. (21.14) 401 4 In fact using for fixed tn that (dX)2 -(dX)2 = * (dX)2 (21.15) t cosh2 R 2 42 422tn2 and a +(X)=(XMW )= R(cosh )(21.16) R one gets X4 = cos(n) * R * cosh tn (21.17) R and a = sin(n) * R cosh tn (21.18) R (21.19) 21.3.1 How to consider the Killing transformation in the Imbedding Ifwe considerhowafixedpointinthe“space”inthe‘new’ coordinates moveasfunction of the time tn we can use that it is rotated in the imbedding space time with its indefinite metric -the X0 being a time coordinate -around the three-space given as X4 = X0 = 0. That is to say that the distance to this three-space is constant as long as an event is moved just by progressing the “new” time tt. The rate of running of the local eigentime relative to the coordinate time in our “new” system is thus proportional to the (Lorentz) invariant distance from the three-space to the point. 21 Anew view on cosmology, with non-translational invariant Hamiltonian 309 21.3.2 Developping coordinate transformation Inthe “new” coordinatestheMilkyWay coordinatesinthe imbedding systemaregivenas 40 tn tn (XMW ;XMW )=(R * cosh ;R sinh ) (21.20) TH TH and thus for running point 40tn tn (X;X) = ((cos(n) * R * cosh , cos(n) * R sinh ) (21.21) TH TH Since in the usual FLRW we have X0 = R sinh tu we can put up the equation TH tu tn sinh = cos(n) sinh . (21.22) TH TH (Here we have written TH for Hubble time, a constant parameter of dimension time. The simplest is to take TH = R.) Except at the MilkyWay where tu = tn we have in the whole (half) space tu cos(n) (21.25) and thus u <n, (21.26) wherethedifferencebetweenthetwoanglesthoughgets smallerin absolutevalue(butwe shall see below not relatively) the smaller the . angles, and goes to equality at the Milky Way at thes being zero. Becausethetimedevelopmentinthe“new” schemeisgivenasaKilling transformationthe spatialgeometric structurein this “new” coordinate systemis constant asa functionof the time tn, so that say the radius  2 * R ofthe “half”-spaceremainsofthis valueatall times tn, while the corresponding radiusof the “half”-spacein the “usual” FLRWcoorordinates grows with the time tu as “half”-space radius= . * R * cosh tu u 2TH d“half”-space radiusu 1tu so that the logarithmic derivative = * tanh “half”-space radiusdtu u TH TH tu so TH is Hubble time for tanh . 1. TH 21.3.3 Onlyalapisin both coordinates As one may see from the figure ?? also it is not the whole usual De Sitter space which is described in the “new” coordinates, but rather only a lap, because late in the “new” system one looks the simultaneity surface in the “new” coordinates must still be a space-like 310 H. B. Nielsen, M. Ninomiya surface, and thus seen from the usual system the point motion represented by the tn fixed toa value, mustrun with bigger than light speed. For late tn times however it goes very closetothespeedoflight,except neartheMilkyWay,where tn and tu are approximately equal. But this means that there is no way to get an event so close to (u = 0, tu = 0) that it couldbereachedby a signalfrom (u = 0, tu = 0) represented in the “new” system. This limit means that in order, that an event can be represented in the “new” coordinateswe must have tu . R * ( . - U) approximately for small tu(21.27) 2 tu tu or more exactly: cos(u) * cosh . sinh (21.28) TH TH or 1 . cos u > tanh tu (21.29) TH Dist to MWu tu 1 . cos > tanh . 1for large tu R cosh tu TH TH (21.30) For very late times, i.e. tu !. we have tu tu tanh . 1 - 2 exp(-2 ) (21.31) TH TH ! Dist to MWu Dist to MW2 cos . 1 - 2 * u (21.32) R cosh tu R2 exp(-2 tu ) TH TH (21.33) Inserting these approximations of late time into the inequlity yields Dist to MWu r?(y) see Fig.(1). In order to restore the law of Gravity one adds a ”dark halo” component with: r(y) V2(y)- r?(y) V? 2 (y) rh(y)= (22.4) V2(y)- V? 2 (y) and: V2(R)= V? 2 (R)+ Vh 2 (R) (22.5) in Eq (5), for simplicity, we have neglected the small contribution of the HI gaseous disk R 4h(R)R2dR and we have: Vh 2 (R)= G R where h(R) is the DM halo density. From the above equations the DM halo density reads:   -12MD2 1 + 2 rh(y) h(y)= GV(y)- v ?(y) (22.6) RD 4R2 y2 D and can be determined, once we measure RD and V(R) and we estimate MD sufficiently well. Fig. 22.1: M33: the profile of the stellar disk contribution to the circular velocity does not coincide with that of the latter(r > r?). [3]. 22 Anew Paradigm for the Dark Matter Phenomenon 317 It is well known that the dark matter reveals itself also in the other types of galaxies (see, e.g.[4])andthatthepresenceofthis non-interacting massive particleis necessarytoexplain a number of cosmological observations such as the rate of the expansion of the universe, the anisotropies in the Cosmic Background Radiation, the evolution of the large scale structures and the existence itself of galaxies (e.g. see [6]). The starting point to account for all this has been to postulate the ubiquitous presence in the Universe of massive particles that emit radiation at a level totally negligible with respect to that emitted by the Standard Model (SM) particles. Then, this particle, necessarily outside the SM, unlike its particles, is hiddentous also whenit aggregatesin vast amounts.We take this dark particle optionasa foundation of Physics and Cosmology. However, it is important to stress that this does not automatically determine the mass or the nature of such a particle. Furthermore, the present statusof ”darkness” means that the particle hasa very small, but not necessarily zero, self- interactions or interactions with the SM particles and this can have various cosmological, physical and astrophysical consequences. 22.2 The Standard Paradigm The next step has beentoprovide the particle witha theoretical scenario. Let us introduce the concept of the Paradigm for the Dark Matter Phenomenon. Here, for Paradigm we intendasetofpropertiesthattheactualDM scenariomustpossessandthat,inturn,reveals the natureof the particle. After the first ”detections”ofDMin the Universe,a Paradigm has, indeed, emerged lasting until today. According to this, the scenario behind the DM Phenomenon must have the following properties: 1)it connects the (new) Dark Matter physics with the (known) physics of the Early Universe; it introduces in a natural way the required massive dark particle and relates it with the value of the cosmological mass density of the expanding Universe. 2)itis mathematically describedbya very small numberof parameters andby a very well known and specific initial conditions, while having, at the same time, a strong predictive power on the evolution of the structures of the Universe. Furthermore, these latter can be thoroughly followedbyproper numerical simulations. 3) its (unique) dark particle can be detected by experiments and observations with present technology. 4) it sheds light on issues of the StandardModel particle physics. 5) it provides us with hints for solving long standing big issues of Physics. In other words, the ruling paradigm heads us towards scenarios for the dark matter phenomenon that are very beautiful, and hopefully towards the most beautiful one, where beauty is in the sense ofsimplicity, naturalness, usefulness, achieving expectations and harmonically extending our knowledge. For definiteness and clarity of the discussion, we name it as: ”The Apollonian paradigm”. Let us point out that here we just name concepts emerged and solidified in the mid 80’ and that since then have served as lighthouses in the investigation of the DM mystery. This Paradigm has straightforwardly led the Cosmologists to one particular scenario: the well known CDM scenario (e.g. [6]).Not only the Apollonian paradigm has identified the possible scenario for the dark particle, but it is directly responsible for large part of its claimed successes, so that, to adopt the above scenario or to adhere to the originating paradigm is the same thing. Finally, the CDM scenario is rather unique: in the past 30 years no other scenario has emerged with such complete Apollonian status. . stays for the Dark Energy having 70% of the total energy of the Universe and CDM for Cold Dark Matter. Cold refers to the fact that the dark matter particles move very slow compared to the speed of light. Dark means that these particles, in normal circumstances, 318 P. Salucci do not interact with the ordinary matter via electromagnetic force but very feebly with a cross section of the order of 3 × 10-26 cm 2 characteristicof theWeak Force. This specific value of the cross section inserted in the Physics of the early Universe, makes the predicted WIMP (Weak Interacting Massive Particles) relic densitycompatible with the observed value of about 3 × 10-30g=cm3 (e.g. [6]). It is well known that in this scenario the density perturbations evolve through a series of halos mergings from the smallest to the biggest in mass and the final stateisa matrioskaof halos with smaller halos inside bigger ones.Very distinctively, these dark halos show an universal spherical spatial density [7]: s NFW (r)= , (22.7) (r=rs)(1 + r=rs)2 where rs isa characteristic inner radius, and s the related density. Notably, this scenario confirms its beauty resulting extremely falsifiable since in all the Universe and throughout its history, therelated dark component creates structures with the same one configuration. The well known evidence is that no such dark particle has been detected in the past 30 years. This has occurred in experiments at underground laboratories, searching for the soft scatter of these particles with particular nucleus, in particle collisions at LHC collider with a general search for Super-Symmetric partners or more exotic invisible particles to be seen as missing momentum of unbalanced events; in measurements at space observatories as gamma rays coming from dense regions of the Universe where the dark particle annihilates with its antiparticle (see e.g.[?,8]). Furthermore, the current upper limits for the energy scaleofSuSy,as indicatedbyLHC experiments,rulesoutthe NeutralinoastheDM particle. It is, however, important to notice that, in the attempts made so far, only WIMP particles havebeenthoroughly searched.The searchfor particlesrelatedtootherDM scenarioshas been very limited and almost no blind searches have been performed. Thus, the lack of detection of the dark particle so far, in no way indicates that this does not exist, but only the failure of certain detection strategies related to particular scenarios. In recent years, at different cosmological scales, observational evidence in strong tension with the above scenario has emerged (e.g. [5]). Here, we focus on the distribution of dark matter in galaxies, a topic for which the failure of the CDM scenario is the most eventful ([9]). Dark Matter is, in fact, located mostly in galaxies that come with very large ranges of total masses, luminosities, sizes, dynamical state and morphologies. This diversity of the properties of their luminous components is an asset for the investigation of their dark components. 22 Anew Paradigm for the Dark Matter Phenomenon 319 Fig. 22.2: Stacking of 1000 individual RCs in 11 luminosity bins. The coadded curves V(R=Ropt)=V(Ropt) (points with errorbars) are fitted with the URC model (solid line):a cored DM halo(dashed line) +a Freeman Disk (dotted line) ([13, 16]) 22.3 The cored DM halos The rotation curves of disk systems are well measured from the Doppler measurements of theH. and the 21cm galaxy emission lines. They extend in many cases well beyond the stellardiskedgeandin some caseoutto20%ofthehalosize.By investigating several thousands RCs covering: a) all the morphologies of the disk systems: normal spirals, dwarf Irregulars and low surface brightness galaxies and b) all the magnitudes from the faintest to the most luminous objects, one finds that the RCs, from the center of the galaxies out to the edge of the dark matter halo, combine in an Universal Rotation Curve (see [4]). That is, in order of retrieving the galaxy dark and luminous mass distributions from their circular velocity V(R), the latter canberepresentedby an unique function VURC(R=RD, Mag, c, T ), where RD is the disk length scale of Eq. (2), Mag is the magnitude, c indicates how compact isof the distributionof light andTthe galaxy morphology [10,11,13–15]). Vcoadd(R=RD;Pi) the coadded velocitydata(andtherelatedr.m.s.)(seeFig.2)are obtained by stacking with a proper procedure a large number of individual RCs in bins of the 320 P. Salucci observed quantity(ies) Pi, (e.g. Mag and T). VURC(R=RD;Pi)(The ensemble of solid lines in Fig. 2) is an analytical function found to fit the above Vcoadd data (see [13]). Vcoadd isa crucial kinematical quantity, any values and their r.m.s. would take, moreover, since the latter are found very small, they are good templates of the large majority of individual RCs. On the other hand, the function VURC allows one to interpret the Vcoadd data in terms of a universal mass model. Remarkably, all the RCs identifier quantities belong to the stellar component of the galaxies despite that the dark component dominates the mass distribution. This is a first indication ofa direct coupling between the dark and luminous components. The proposed mass model features the following two components: the above stellar disk of mass MD asa free parameter anda dark halo with the Burkert density distribution [16]: B(r)= 02(22.8) (1 + r=r0)(1 +(r=r0)) The latter has2free parameters: the central density 0 and the core radius r0 that marks the edgeoftheregioninwhichtheDMdensityisroughly constant.Thismodelwellreproduces the coadded RCs [13–16], so as the individual RCs of disk galaxies (see also [4]) and it is dubbed as theURC model.Notably, its success faces the failureof the NFW halo+stellar disk mass model in the coadded RCs [17]. Fig. 22.3: The halo density from Eq. (6) blue as function radius R for galaxies with different values of log Vopt. The disagreement with the predicted NFW one red is evident Importantly, the same outcome occurs also for the control sample of high quality individual RCs(e.g.[19–22,26]).Thisdisagreementis serious,model independentand emergesdirectly 22 Anew Paradigm for the Dark Matter Phenomenon 321 from Eq(6) in combination with: MD ' (0:72-0:95r)G-1V2 Ropt [18] with Ropt . 3:2 RD opt (see Fig. 5). Furthermore, in the framework of CDM cuspy halos model, we also find implausible best-fitting values for the masses of the stellar disk and dark halo and for the two structural parameters of the NFW halo (e.g. [17]). This raises strong doubts about the collisionless status of the DM particles in galaxies, a fundamental aspect of the CDM scenario. Moreover, at radii r >> r0, the density profile of the dark matter in disk galaxies returns to be that of the collisionless particles [11](see Fig. 4). This fits well with the above observational scenario: in the external regions of halos, the luminous and dark matter are so rarefied that, in the past 10 Gyrs, had no time to interact among themselves, also if this was physically allowed. Thus, on the scale of the halo’s virial radius, the standardphysicsof galaxy formationis notin tension withthe observed distributionofdark matter.Differently,onthescaleofthe distributionofthe luminous com- ponent,the observationsimplythattheDMhalo densityhaveundergonetoa significant and not yet well understood evolution over the Hubble time (see also [27]) Fig.22.4:TherelationshipamongtheDMandLMstructural parameters 0;r0;MD (see [11]). Log-units: M , kpc, g=cm3 . Therefore,the mass distributionofadiskgalaxyis described,in principle,byone parameter belonging to the luminous world and two to the dark world which represent structural quantities (not defined in the standard CDM scenario). In disk galaxies and extraordinary observational evidence adds up: the three parameters r0, 0 and MD result well correlated among themselves(seeFig.5,[11]andFig.11in[4]).This, obviously, cannot occurinthe 322 P. Salucci standard CDM scenario; then, we focus on this evidence ad we directly investigate the structural physical properties of disk galaxies. Fig. 22.5: The density of the DM halos today (blue) and the primordial one (red) as function of radius and mass. The agreement of the two profiles at outer radii revealsa time evolutionof theDMin the centralregionsof the halos. Log-units: kpc g=cm3 1011M 22.4 Unexpected relationships We remark that the properties of the internal structure of the disk galaxies, at the basis of this work, have been previously discovered and independently confirmed (references in this work and thereview [4]). Here we use them as motivation forproposinga paradigm shift in the way we investigate the dark matter mystery. 22.4.1 Central halo surface density The quantity 0 . 0r0, i.e. the central surface density of the DM halo, is found constant in objects of any magnitude and disk morphology (see Fig. 6) [16, 28–30] (see also [4]): log 0 = 2:2 ± 0:25 (22.9) M pc-2 this means that 0, the value of the DM halo density at the center of galaxy, is inversely proportional to the size r0 of theregionin which the densityis about constant. This seems to imply that the dark particle possesses some form of self-interaction of unspecified nature. 22 Anew Paradigm for the Dark Matter Phenomenon 323 1.41.61.82.02.22.4-0.50.00.51.0LogVopt(km/s) LogRd(1.41.61.82.02.22.40.00.51.01.52.02.5LogVopt(LogRc(Figure12.LSBsrelationshipsbetweenthestellardiscscalelength,theDMcoreradiusandthecentralDMcoredensityversustheopticalvelocitywiththeirbestfitfunctionsinthefirst,secondandthirdpanelrespectively. Figure13.Relationshipbetweenthestellardiscscalelengthandthestellardiscmasswiththebestfitfunction. 0.50.00.51.01.5LogMd(Msun) LogRd(kpc) LSBs72 individual LSBsLSBs fit910111213-LogRc(kpc)0.6-0.4-0.20.00.2-0.4-0.20.00.20.4LogCDMLogC* 72 individual LSBs36 individual dwarf discsLSBs fitdwarf discs fit1.41.61.82.02.22.42.61.01.52.02.53.0LogVopt(Fig. 22.6: The dark halo central surface density 0 as a function of the reference velocity Vopt in disk systems and in the giant elliptical M87 22.4.2 DM core radii vs. disk length scales Amazingly, r0 tightly correlates with the stellar disc scale length RD [13–15,24,25]: log r0 =(1:38 ± 0:15) log RD + 0:47 ± 0:03 (22.10) see Fig. (7). This relationship, found for the first time in a large sample of Spirals by [13], is present alsoinLSBsand DwarfIrregularsandinthegiant ellipticalM87(seeFig.7). Overall,it extends in objects whose luminosities span over five orders of magnitudes. Then, the size of the region in which the DM density does not (much) change with radius results related with the size of the stellar disk RD. It is very difficult to understand such tight correlation between very different quantities without postulating that dark and luminous matter are able to interact more directly than via the gravitational force. Fig. 22.7: DM halo core radius rc (= r0 for Burkert profile) vs. the stellar disk length scale RD (from Eq. (2)) in galaxies of different morphology . 22.4.3 Stellar disks vs. DM halos compactness Similar mysterious entanglement emerges also from the evidence that, in galaxies with the same stellar disk mass, the more compact is the stellar disk, i.e. the larger is the value of MD=R2 , the more compact is the 2-D projected DM core region, i.e. the larger is the D value of Mh=r2 0 (seeFig.8)([14],Fig.(15)in[15],.Moreoverthe stellarandtheDM surface brightness, once averagedinside r0, are found to be proportional [23]). Dark and luminous world seem to have communicated in an unknown language. 22.4.4 Total vs. baryonic radial accelerations Also without assuming a-priori thepresenceofa darkhaloin galaxies, this emerges and results mysteriously entangled with the baryonic component. V2(y)=y . g is the radial accelerationofa point massinrotational equilibriumata distance y from the center of a disk galaxy and Vb 2 (y)=y . gb is its baryonic (stellar) component. In spiral galaxies we find: g(y) >gb(y), that calls fora dark component, but also: g = g(gb): the two accelerations are relatedbya tightrelationship [31]. 325 Fig. 22.8: Compactness of the stellar disk vs that of the cored DM region Including in the game also dwarf Irregulars and Low Surface Brightness galaxies, the above relation gains an other parameter, the radius y . R=RD and form a surface log g = g(log gb;y) around which, withina very smallr.m.s. distanceof 0.04 dex, the accelerations at all radii and in all galaxies lay [32] (see Fig. (9)). The origin of this surface of hybrid dark-luminous quantities is difficult even to frame in a pure collisionless scenario. 22.4.5 The crucial role of r0 The relationships above indicate the quantity r0 as the radius of the region inside which the DM–LM interaction has taken place so far. Here, we show a direct support for such identification. In the case of self-annihilating DM the number of interactions per unit of time has a dependence on the DM halo density given by: KSA(R)= 2 DM(R), here we take KC(R) . DM(R)?(R) asthe analogue quantityinascenariowith DM-baryons interactions. KC has no physical meaningina collisionlessDM particle scenario.From the above URC mass model we get: -47:50:3 2 -6 KC(r0) ' const = 10gcm. (22.11) 326 P. Fig. 22.9: The relationship in dwarf and LSB galaxies among the total and the baryonic accelerations at y and y. We see in Fig.(10) that the kernelKC(R), ata same physical radius R, varies largely among galaxies of different mass, and, in each galaxy, varies largely at different radii. But, at R ' r0 and only there, this quantity takes the same value in allgalaxies. In the scenario of interacting dark matter, this clearly suggests the radius r0 as the edgeof theregion inside which interactions between dark matter particles anda StandardModel particles havetaken place so far, flattening the original halo cusp. 22.5 Anew Paradigm Dark Matter particles have been thought, as their main characteristic, to interact with the rest ofthe Universe essentially only by Gravity. However, in such a framework, the properties of the mass distribution in galaxies do not make much sense. Observations, therefore,appeartostronglycallforanew interaction, negligibleontimescalesoftheorder of the galaxy free fall time, like the WIMP one, but, unlike the latter, able to modify the dark halo density distribution withina timescaleaslongastheageofthe Universe. 22 Anew Paradigm for the Dark Matter Phenomenon 327 Fig. 22.10: DM(r)LM(r) as function of radius and halo mass ( yellow). DM(r0)LM(r0) resides, for all the objects, inside the two red planes. Shown (blue)also the analogue of the the dark particle annihilation,DM(r)2. Log Units: 2 -6 M , kpc, gcm. It is certain that the impact of these observational evidences goes beyond the falsification of the CDM WIMP scenario and proceeds till to rule out the entire Apollonian paradigm from which such scenario has emerged. The defining criteria of the paradigm, in fact, appear unaccountable by the above evidences. Thus, the spectacular DM–LM entanglement found in galaxies, allied with the fact that the WIMPparticle has escaped detection, becomea strong motivation for a change of the Paradigm with which to approach the dark matter Phenomenon and determine the nature of the dark particle and its Cosmological History. Reflectinguponthe failureofthecurrent paradigmwerealizethatit stemsfromtheadopted correlation between truth and beauty in solving the DM mystery. Instead, the observational properties of the dark and luminous matter in galaxies tell another story that bends towards thetriumphof ugliness. Observationalrelationshipsandgalaxyproperties seemto indicate scenarios with a large number of free unexplained parameters, with no much predictive power,no obvious connection with known Physics, let alone with the theoretically expected new physics and no help in resolving well known big problems of Physics, but actually an addition of new ones. Then, we need a new Paradigm that opens the road to ”ugliness” and prefers scenarios with properties unacceptable by the Apollonian Paradigm. Many philosophers have expressed their opinion for this situation, but it is fair to acknowledge thatF. Nietzschehasbeen obsessedbythe conceptsofbeautyand uglinessinrelationto those of truth and falsity, so we name after him the proposed new Paradigm, that allows 328 P. Salucci the building of scenarios that seriously follow the observational evidences how ugly the former and the latter can appear. Let us to remind that to work in the CDM scenario yields to clear advantages : -The underlying Physics is solid, undisputed and rather simple but also capable to lead one to new results in the fields of Cosmology and Physics of the Elementary particles. -In this scenario the initial conditions and the general knowledge at the basis of any new investigation is well-known and generally agreed upon. -The scenario has inbuilt a clear agenda for the investigation of dark matter mystery which is already in use in the scientific community and that fostersa global spiritofresearch.-The scenario hasa fundamental and straightforwardconnection with ”state of the art” computer simulations, observations and experiments andrequires always better performances. Therefore, to abandon such scenario has important consequences in the investigation of the DM phenomenon. In the complexity that the new scenario may have. In understanding and in generally agreeing on its basic physics, in the contribution that computer simulations, observations and experiments may provide in the investigation of the DM Mystery. Given this, it is simply not possible to sneak away from the CDM to some other scenario without performing a deep reflection on what we are leaving, why and what we are looking for. Then, in order to value and protectbybiasesthe seemingly exotic, mysteriouslyentangledandadhoc scenariosthatthe observations seem to indicate, we claim that an explicit switch to the Nietzschean paradigm is necessary. Within this new Paradigm the exploration of DM mystery proceeds according to the following loop: available observations suggest usa scenariowhich, once verifiedby other purposely planned observations, is thought to provide us with the nature of the dark particle and the theoretical background of the DM Phenomenon that, once we arrive at this point, certainly will appear very beautiful. 22.6 Conclusions Here, we have motivated and proposed that, in the investigation of the complex and entangled world of the phenomenon of the Dark matter in galaxies we take a new and tailored approach. In detail, while abandoningthe failing CDM scenario, we must be poised to search for scenarios without requiring that: a) they naturally come from (known) ”first principles”b)theyobeytotheOccam razorideac)theyhavethebonustoleadus towards the solution of presently open issues of the SM of the Elementary particles. On the otherside, such search must:i) follow the observations and the experiments wherever they may lead ii) consider the possibility that the Physics behind the Dark Matter phenomenon be disconnected from the Physics we know and iii) does not comply with the usual canons ofbeauty.Finally,forthegoalofthisworkwithrespecttothe scientific community,itis irrelevant whether such a search is undertaken after an individual convincement or to followa generally agreed new paradigm. References 1. Rubin,V.C.,FordJr,W.K.and Thonnard,N., Rotationalpropertiesof21SC galaxies ApJ ,1980, 238, 471–487 2. Freeman, K.C, . On the Disks of Spiral and S0 Galaxies ApJ, 1970, 160,811 3. Corbelli,E. and Salucci,P. The extendedrotation curve and the dark matter haloof M33 MNRAS, 2000, 311, 441-447 22 Anew Paradigm for the Dark Matter Phenomenon 329 4. Salucci,P. The distributionof dark matterin galaxies AARv, 2019, 27,2 5. Abdalla,E. etal Cosmology intertwined:Areviewof the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies JHEAp ,2022, 34, 49A2022/06 6. Kolb,E. andTurner,M. The Early Universe, AddisonWesley, 1990 7. NavarroJ.F.Frenk,C.S.and White,S.D.M.,TheStructureofColdDark Matter Halos ApJ, 1996, 462, 563 8.Arcadi,G.etalThewaningoftheWIMP?Areviewofmodels,searches,and constraints European Phys.J.C, 2018, 78, 203 9. Salucci,P.,Turini,N.,Di PaoloC. Paradigms and scenarios for the dark matter phenomenon Universe, 2020, 6, 118 10. Persic, M, Salucci,P. The universal galaxy rotation curve ApJ, 1991, 368, 60-65 11. Salucci,P. etal The universal rotation curveof spiral galaxies–II. The dark matter distribution out to the virial radius MNRAS, 2007, 378, 41–47 12.YegorovaIA, SalucciP(2007)The radialTully–Fisherrelationfor spiral galaxies— MNRAS ,2007, 377, 507 13. Persic,M., Salucci,P.andStel,F.,The universalrotation curveofspiral galaxies—I. The dark matter connection MNRAS, 1966, 281, 27–47 14.DiPaolo,C,Salucci,PandErkurt,A.,The universalrotation curveoflow surface brightness galaxies -IV. The interrelation between dark and luminous matter MNRAS, 2019, 490, 5451-5477 15. Karukes,E.V., SalucciP. The universalrotation curveof dwarf disc galaxies MNRAS ,2017, 465, 4703–4722 16. Salucci,P. and Burkert,A., Dark matter scalingrelations ApJ ,2000, 537,9 17. Dehghani,R., Salucci,P., Ghaffarnejad,H., Navarro-Frenk-White dark matterprofile and the dark halos around disk systems AA ,2020, 643, 161 18. Persic,M. Salucci,P. Mass decompositionof spiral galaxiesfrom disc kinematics MN ,1990, 245, 577 19. Gentile, G. et al The cored distribution of dark matter in spiral galaxies MNRAS ,2004, 351, 903-922 20. Oh, S. etal Dark and Luminous Matter in THINGS Dwarf Galaxies AJ ,2011, 193, 45 21. Karukes,E.V., Salucci,P., GentileP. The dark matter distributionin thespiral NGC 3198 out to 0.22 Rvir AA, 2015, 578, 13 22. LelliF,McGaughSS, SchombertJMSPARC: mass modelsfor175disk galaxieswith Spitzer photometry and accurate rotation curves. AJ ,2016, 152, 157 23. Gentile G. et al Universality of galactic surface densities within one dark halo scale- length Nature ,2009, 461, 627 24. De Laurentis, MF, Salucci.P The accurate mass distribution of M87, the Giant Galaxy with imaged shadow of its supermassive black hole, as a portal to new Physics ApJ ,2022, l929, 17 25. Donato,F, Gentile, G., SalucciPCores of dark matter haloes correlate with stellar scalelengths MNRAS ,2004, 353, 17-22 26. SalucciPThe constant-densityregionofthe dark halosof spiral galaxies. MNRAS, 2001, 320,1 27. Sharma,GSalucci,P. vandeVenG. Observationalevidenceof evolving dark matter profiles at z=1 AA ,2022, 659, 40 28. Burkert, A. The Structureand Dark Halo CoreProperties of Dwarf Spheroidal Galaxies ApJ ,2015, 808, 158 29. Kormendy J, Freeman KC Scaling laws for dark matter halos in late-type and dwarf spheroidal galaxies.. IAU 220S, 2004, 377 330 P. Salucci 30. Donato,FetalAconstant dark matter halo surface densityin galaxies MNRAS ,2009, 397, 1169-1176 31. McGaugh S., LelliF. and Schombert J. Radial Acceleration Relation in Rotationally Supported Galaxies PhRvL, 2016, 117 ,201101 32. Di Paolo, C. , Salucci,P. , Fontaine JP The radial acceleration relation (RAR): crucial cases of dwarf disks and low-surface-brightness galaxies ApJ, 2019, 873, 106 Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 331) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 23 On the construction of artificial empty space Elia Dmitrieff elia@quantumgravityresearch.org Abstract. We consider principles of four-dimensional tessellation model of physical vac- uum,and suggesta conceptof experimentalhardware equipment intendedtoreproduce some of its properties. Povzetek Avtor uporabi princip ˇznega teselacijskega modela fiziˇ stiri-razseˇcnega vakuuma, da predlaga model osnovih fermionov in ustreznih umeritvenih polj standarnega modela. • The basic structure for our modelis the4D space-fillingby 26-cell ‘satori’ polytopes. It is theVoronoi diagramof tesseractgrid Z4 having all its nodes shifted in four orthogonal directions on length of 2 1 along the crystallographic axes. Each node is considered havinga parity,either even or odd,according to product of its coordinates in the original tesseract lattice. • Wepostulate the changeable and discrete electricalcharge of node, and its initial equality to the parity bit. Doing so, we can consider the grid asa kindof memory, storing the data locally in nodes, one bit per node. • While the parity of nodes is fixed by their position in the lattice, their charge can be exchanged between immediate neighbors. This exchange produces a pair of anti- structural defects. Being separated, these opposite-charged defects, having also oppositeparity, arerecognizedasa particleand anti-particle.Itreflectstheknown natural charge-parity (CP) symmetry.Tobe exact, this symmetry needs alsoaTranslation operation, since opposite defects exist in different (even or odd) sub-lattices. • The electrical charge associated with node is ± 1 6 e. Consequential inverses of several node’s charge changes the total charge with steps of 3 1 e, which is in accordance with known particle charges. The single defect carries ± 1 3 e, corresponding to down quarks/anti-quarks. The double defect is an up quark with ± 2 / 3e, and the triple defect is a charged lepton with e. The estimated physical scale (cell radius), based on Higgs expectation value, is about 10-21 m. • The data in grid is stored both in nodes and edges. The edges are more conservative than nodesin sense that modifications(rewritings)of edges are associated with more energetic processes, like Universe formation, Big Bang, inflation, bariogenesis, vacuum phase transitions. In the modern Universe (maybe excepting black holes) the geometrical structureofedgesis fixed.It determines the emergent space dimension count, its 332 Elia Dmitrieff symmetry, and therulesof node data exchanges. However, the orientationof edgeis not so conservative because it is determined by the data in nodes thaw it connects • The node data is represented by charge bits. It is more volatile than edges. It describes individual particlesand definestheir behavior.Thenodedatais assumedtobe1bit per node. Or, one may say that the node is one bit and the whole tessellation is a form of a binary memory. • The compactification of grid together with charge-parity concept can explain why the observable space is isotropic while grids, that are supposed to be the background structure of space, are not. For 4D space curled with the minimal radius into 5D cylinder, oneof4dimensionseffectively vanishes and the space looks likeitis flat3D. However, the space is doubled in the following sense: every 3D point corresponds to two points on different surfaces of the cylinder. They are seen from different sides, so the parity difference is compensated, and opposite charges cancel each other. As the result,the electricalcharge(in absenceof defects)isexactly0everywhere,sonodesare not observable, and the projected space appears isotropic and empty. • However, all nodes still exist in the 4D space and they can participate in time clock movement and cellular-automaton-like evolution. Defects cause the de-compensations of charges. They are observable as particles (charged or not charged) on the ’empty’ background. • The 4D tesseract grid translation unit, as well as the unitof ’satori’ structure, has the equal lengths of its main diagonal and of the edge. Also, the both ends of the main diagonal have the same parity. So, this grid can be compactified in two ways using the same radius.Itis possible that thereare other waysof compactification. • The cellular automaton’s cell may be equipped with a simple hardware circuit to apply the evaluatingrule locally. Runningthe evaluation asynchronouslyis supposedtohave some competitiveeffects that cannotbe achieved when using dedicated CPU thatruns simpleprogramruleforallthecellsin turn.Thesimplecircuitapproach allowstoget ridof computationalresources limitationsandof lowering performanceonbig arrays. • In the 26-cell ’satori’ filling, there are no straight (geodesic) paths, but there are forks with equal angles. They need a choice to be made. In backwarddirection, they are junctions.So, eachpathisa combinationof junctions and forks with some rate, that is always non-negative. The proper time for the propagating defect may be effectively defined on the basis of this forks count. Zero proper time, corresponding to the light- speed movement, is presumably caused by no forks on 5D helical paths having the maximal possiblepitch.Withthis definitionofthepropertime,there cannotbeany tachyons. • There is another geometrical time that is the compactified dimension. It is of very short lengththatisone translationunit.TheTsymmetryis connectedtoreflectioninthis direction. not with theproper time (thatisa positive count). The movement alongitin both directionsis asfree asin other three dimensions. But sinceitis compactified,both results have just minor differences that are observable as rare cases of CP symmetry violations. • The compactificationofthe4thdimensiondownto3Dmakesthemodelingmucheasier. It allows using existing 3D hardware technology. Exploiting the edge conservatism, a fixed 3D circuit of PLA chips can be built. This hardware model would not be able to reproduce gravity, black holes and other special cases without additional tricks, but it might be useful in demonstration of propagation, scattering and decay of macroscopic artificial particles. Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 334) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 24 Virtual Institute of Astroparticle physics asthe online platform for studiesof BSM physics and cosmology MaximYu. Khlopov1;2;3 1 Centre for Cosmoparticle Physics ”Cosmion” National Research Nuclear University MEPHI”, 115409 Moscow, Russia 2 Virtual Institute of Astroparticle physics, 75018, Paris, France 3 Institute of Physics, Southern Federal University Stachki 194, Rostov on Don 344090, Russia Abstract. The relaxation of pandemia conditions is not complete and the meetings in per- sonaretobe still accompaniedby online sessions, leadingto theirhybrid forms.The unique multi-functional complexofVirtual Instituteof Astroparticle Physics (VIA) operating on website http://viavca.in2p3.fr/site.html, provides the platform for online virtual meetings. Wereview VIA experienceinpresentation online for the most interesting theoretical and experimental results, participation online in conferences and meetings, various forms of collaborative scientific work as well as programs of education at distance, combining online videoconferenceswith extensive libraryofrecordsofprevious meetingsand Discussionson Forum. Since 2014 VIA online lectures combined with individual work on Forum acquired the form of Open Online Courses. Aimed to individual work with students the Course is not Massive, but the account for the number of visits to VIA site converts VIA in a specific tool for MOOC activity. VIA sessions, beingatraditional partof BledWorkshops’program, have convertedat XXV BledWorkshop ”What comes beyond the Standardmodels?” into the hybrid format, combining streaming of the presentations in the Plemelj House (Bled, Slovenia) with distant talks, preserving the traditional creative nonformal atmosphere of BledWorkshop meetings.We openly discuss the stateof artof VIA platform. Keywords: astroparticle physics,physics beyond the Standardmodel, e-learning, e-science, MOOC 24.1 Introduction Studies in astroparticle physics link astrophysics, cosmology, particle and nuclear physics and involve hundredsof scientificgroups linkedbyregional networks (like ASPERA/ApPEC [1,2]) and national centers. The excitingprogressin these studies will have impact on the knowledge on the structure of microworld and Universe in their fundamental relationship andonthe basic, still unknown, physical lawsof Nature(see e.g.[3,4]forreview).The Title Suppressed Due to Excessive Length 335 progress of precision cosmology and experimental probes of the new physics at the LHC and in nonaccelerator experiments, as well as the extension of various indirect studies of physics beyondthe Standardmodel involve with necessity their nontrivial links.Virtual InstituteofAstroparticle Physics(VIA)[5] wasorganizedwiththeaimtoplaytheroleof an unifying and coordinating platform for such studies. Starting from the January of 2008 the activity of the Institute took place on its website [6] in a form of regular weekly videoconferences with VIA lectures, covering all the theoretical and experimental activitiesin astroparticle physicsandrelated topics.The libraryofrecords of these lectures, talks and their presentations was accomplished by multi-lingual Forum. Since 2008 there were 220 VIAonline lectures, VIA has supported distant presentations of 192 speakers at 32 Conferences and provided transmission of talks at 78 APC Colloquiums. In 2008 VIA complex was effectively used for the firsttime for participation at distance inXIBledWorkshop[7].SincethenVIA videoconferencesbecamea naturalpartofBled Workshops’programs, openingthe virtualroomof discussionstothe world-wide audience. Its progress was presented in [8–20]. Here the current state-of-art of VIA complex, integrated in 2009 -2022 in the structure ofAPC Laboratory,ispresentedinorderto clarifythewayin which discussionof open questions beyond the standardmodels of both particle physics and cosmology were sup- portedbythe platformofVIA facilityatthehybrid MemorialXXVBledWorkshop.The relaxation of the conditions of pandemia, making possible offline meetings, is still not complete, preventing many participants to attend these meetings. In this situation VIA videoconferencing became the only possibility to continue in 2022 traditions of open discussions at Bled meetings combining streams of the offline presentations and support of distant talks and involving distant participants in these discussions. 24.2 VIA structure and activity 24.2.1 The problem of VIA site The structure of the VIA site was based on Flash and is virtually ruined now in the lack of Flash support. This original structure is illustrated by the Fig. 24.1. The home page, presentedonthisfigure, containedthe informationonthecomingandrecordsofthe latest VIA events. The upper line of menu included links to directories (from left to right): with general informationonVIA(AboutVIA); entrancetoVIAvirtualrooms(Rooms);thelibrary ofrecords andpresentations(Previous), which containedrecordsof VIA Lectures(Previous › Lectures),recordsof online transmissionsof Conferences(Previous › Conferences), APC Colloquiums (Previous › APC Colloquiums), APC Seminars (Previous › APC Seminars) and Events (Previous › Events); Calendarofthepastand futureVIA events(All events) and VIA Forum (Forum). In the upper right angle there were links to Google search engine (Search in site) and to contact information (Contacts). The announcement of the next VIA lecture and VIA onlinetransmission of APC Colloquium occupied the main part of the homepage with the recordof the most recent VIA events below. In the announced time of the event (VIA lecture or transmitted APC Colloquium) it was sufficient to click on ”to participate” on the announcement andto Enter as Guest (printing your name)in the correspondingVirtualroom. The Calendar showed theprogramof future VIA lectures and events. The right column on the VIA homepage listed the announcements of the regularly up-dated hot news of Astroparticle physics and related areas. In the lackof Flash support this systemof linksisruined, but fortunately,they continue to operate separately and it makes possible to use VIA Forum, by direct link to it, as well asdirectinksto virtualroomof adobeConnectusedforregularLaboratory meetingsand 336 MaximYu. Khlopov Fig. 24.1: The original home page of VIA site Title Suppressed Due to Excessive Length 337 SeminarandtoZoom(seeFig24.2).The necessitytorestoreallthelinks withinVIAcomplex is a very important task to revive the full scale of VIA activity. Another problem is the necessityto convert .flv filesofrecordsinmp4 format. 24.2.2 VIA activity In 2010 special COSMOVIA tours were undertaken in Switzerland (Geneva), Belgium (Brussels,Liege)andItaly(Turin,Pisa,Bari,Lecce)inordertotest stabilityofVIA online transmissions from different parts of Europe. Positive results of these tests have proved the stabilityofVIAsystemand stimulatedthis practiceatXIIIBledWorkshop.Therecordsof the videoconferences at the XIII BledWorkshop were put on VIA site [21]. Since 2011 VIA facility was used for the tasks of the Paris Center of Cosmological Physics (PCCP), chaired by G. Smoot, for the public program ”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 meetingin sucha virtualway. The simplicityofVIA facilityforordinary userswas demonstratedatXIVBledWorkshopin 2011.Videoconferences at thisWorkshop had no special technical support except forWiFi Internet connection and ordinary laptops with their internal webcams and microphones. This test has proved the ability to use VIA facility at any place with at least decent Internet connection. Of course the quality of records is not as good in this case as with the use of special equipment, but still it is sufficient to support fruitful scientific discussion as can be illustrated by the recordof VIA presentation ”New physics and its experimental probes” givenbyJohn EllisfromhisofficeinCERN(seetherecordsin[22]). 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-Oxfordnetwork meeting as well as to provide online transmissions from the lecturesat Science Festival2012in University Paris7.VIA communicationhaseffectively resolvedtheproblemofreferee’s attendanceatthe defenceofPhD thesisby MarianaVargas in APC. The referees made their reports and participated in discussion in the regime of VIA videoconference. In 2012 VIA facility was first used for online transmissions from the Science Festival in the University Paris 7. This tradition was continued in 2013, when the transmissionsof meetingsat Journ ´eveloppement Logiciel (JDEV2013) ees nationalesduD´ at Ecole Politechnique (Paris) were organized [24]. 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 [23]. In 2014 the 100th anniversaryof oneof the foundatorsof Cosmoparticle physics,Ya.B. Zeldovich, was celebrated.Withthe useofVIA M.Khlopov could contributetheprogrammeof the ”Subatomic particles, Nucleons, Atoms, Universe: Processes and Structure International conferencein honorofYa.B. Zeldovich 100th Anniversary” (Minsk, Belarus)by his talk ”Cosmoparticle physics: the Universe as a laboratory of elementary particles” [25] and the programme of ”ConferenceYaB-100, dedicated to 100 Anniversary ofYakov Borisovich Zeldovich” (Moscow, Russia) by his talk ”Cosmology and particle physics”. In 2015 VIA facility supported the talk at distance at All Moscow Astrophysical seminar ”Cosmoparticle physics of dark matter and structures in the Universe” by Maxim Yu. Khlopov and the work of the Section ”Dark matter” of the International Conference on Particle Physics and Astrophysics (Moscow, 5-10 October 2015). Though the conference room was situated in Milan Hotel in Moscow all the presentations at this Section were givenat distance(byRita BernabeifromRome,Italy;byJuanJose Gomez-Cadenas, Paterna, 338 MaximYu. Khlopov Fig. 24.2: The current home page of VIA site Title Suppressed Due to Excessive Length 339 UniversityofValencia, Spain andby Dmitri Semikoz, Martin Bucher and Maxim Khlopov from Paris) and itsproceeding was chairedby M.Khlopovfrom Paris.In the endof 2015M. Khlopov gave his distant talk ”Dark atoms of dark matter” at the Conference ”Progress of Russian Astronomy in 2015”, held in SternbergAstronomical Institute of Moscow State University. In 2016 distant online talks at St. PetersburgWorkshop ”Dark Ages and White Nights (Spectroscopy of the CMB)” by Khatri Rishi (TIFR, India) ”The information hidden in the CMB spectral distortions in Planck data and beyond”, E. Kholupenko (Ioffe Institute, Russia) ”Onrecombination dynamicsof hydrogen and helium”, Jens Chluba (Jodrell Bank Centre for Astrophysics,UK) ”Primordialrecombination linesofhydrogen and helium”,M. Yu. Khlopov (APC and MEPHI, France and Russia)”Nonstandardcosmological scenarios” andP.de Bernardis(La Sapiensa University,Italy) ”Balloon techniquesforCMBspectrum research” were given with the useof VIA system.At the defenseof PhD thesisbyF.Gregis VIA facility made possible for his referee in California not only to attend at distance at the presentation of the thesis but also to take part in its successive jury evaluation. Since 2018 VIA facility is used for collaborative work on studies of various forms of dark matter in the framework of the project of Russian Science Foundation based on Southern Federal University (Rostov on Don). In September 2018 VIA supported online transmission of 17 presentations at the Commemoration day for Patrick Fleury, held in APC. The discussion of questions that wereput forwardin the interactive VIA events is continued and extended on VIA Forum. 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. Discussions in English on Forum are arranged along the following directions: beyond the standardmodel, astroparticle physics, cosmology, gravitational wave experiments, astrophysics, neutrinos. After each VIA lecture its pdf presentation togetherwithlinktoitsrecordandinformationonthe discussionduringitare putinthe correspondingpost,whichoffersa platformto continue discussioninrepliesto this post. 24.2.3 VIA e-learning, OOC and MOOC One of the interesting forms of VIA activity is the educational work at distance. For the last eleven years M.Khlopov’s course ”Introduction to cosmoparticle physics” is given in the formofVIA videoconferencesandtherecordsofthese lecturesand theirpptpresentations are put in the corresponding directory of the Forum [26]. Having attended the VIA course of lectures in order to be admitted to exam students should put on Forum a post with their small thesis. In this thesis students are proposed to chose some BSM model and to study the cosmological scenario based on this chosen model. The list of possible topics for such thesis is proposed to students, but they are also invited to chose themselves any topic of their own on possible links between cosmology and particle physics. Professor’s comments and proposed correctionsareputinaPostreplysothat students should continuouslypresent on Forum improved versions of work until it is accepted as admission for student to pass exam.Therecordof videoconferencewiththeoral examisalsoputinthe corresponding directory of Forum. Such procedure provides completely transparent way of evaluation of students’ knowledge at distance. In2018thetesthas startedfor possible applicationofVIA facilitytoremote supervisionof student’s scientific practice. The formulation of task and discussion of progress on work are recorded and put in the corresponding directory on Forum together with the versions of student’s report on the work progress. 340 MaximYu. Khlopov Since 2014 the second semester of the course on Cosmoparticle physics is given in English and convertedinanOpen Online Course.It was aimedto developVIA systemasa possible accomplishment for Massive Online Open Courses (MOOC) activity [27]. In 2016 not only students from Moscow, but also from France and Sri Lanka attended this course. In 2017 students from Moscow were accompanied by participants from France, Italy, Sri Lanka and India [28]. The students pretending to evaluation of their knowledge must write their small thesis, present it and, being admitted to exam, pass it in English. The restricted number of online connections to videoconferences with VIA lectures is compensated by the wide-world access to their records on VIA Forum and in the context of MOOC VIA Forum and videoconferencing system can be used for individual online work with advanced participants. Indeed Google Analytics shows that since 2008 VIA site was visited by more than 250 thousand visitors from 155 countries, covering all the continents by its geography (Fig.24.3).Accordingtothis statisticsmorethanhalfofthese visitors continuedto enterVIA site after the first visit. Still the form of individual educational work makes VIA facility most Fig.24.3: GeographyofVIAsite visits accordingtoGoogleAnalytics appropriate for PhD courses and it could be involved in the International PhD program on Fundamental Physics, which was plannedtobe startedonthe basisofRussian-French collaborative agreement. In 2017 the test for the ability of VIA to support fully distant education and evaluation of students (as well as for work on PhD thesis and its distant defense) was undertaken. Steve Branchu from France, who attended the Open Online Course and presented on Forum his small thesis has passed exam at distance. The whole procedure, starting from a stochastic choice of number of examination ticket, answers to ticket questions, discussion by professors in the absence of studentand announcement of resultof examtohimwasrecordedandputonVIAForum[29]. In2019in additionto individual supervisoryworkwith studentstheregular scientificand creative VIA seminar is in operation aimed to discuss the progress and strategy of students scientific work in the field of cosmoparticle physics. In 2020 the regular course now for M2 students continued, but the problems of adobe Connect, related with the lack of its support for Flash in 2021 made necessary to use the platform of Zoom, This platform is rather easy to use and provides records, as well as whiteboardtools for discussions online canbe solvedby accomplishmentsof laptopsby Title Suppressed Due to Excessive Length 341 graphic tabloids. In 2022 the Open Online Course for M2 students was accompanied by special course ”Cosmoparticle physics”, given in English for English speaking M1 students. 24.2.4 Organisation of VIA events and meetings First tests of VIA system, described in [5,7–9], involved various systems of videoconferenc- ing.Theyincludedskype,VRVS,EVO,WEBEX, marratechandadobe Connect.Intheresult of these tests the adobe Connect system was chosen and properly acquired. Its advantages were: relatively easy use for participants, a possibility to make presentation in a video contact betweenpresenterand audience,a possibilitytomakehighqualityrecords,tousea whiteboardtools for discussions, the option to open desktop and to work online with texts in any format. The lack of support for Flash, on which VIA site was originally based, made necessary to use Zoom, which shares all the above mentioned advantages. Regular activityof VIA asa partof APCincluded online transmissionsof all the APC Colloquiums and of some topical APC Seminars, which may be of interest for a wide audience. Online transmissions were 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, sittingin the conferenceroom, changed them followingpresenter, directing simultaneously webcam on the presenter and the audience. If the advanced uploading was not possible, VIA streaming was used -external webcam and microphone are directed to presenter and screen and support online streaming. This experience has found proper place in the current weakeningof the pandemic conditions andregular meetingsinreal canbe streamed. Moreover, such streaming can be made without involvement of VIA operator, by direction of webcam towards the conference screen and speaker. 24.2.5 VIA activity in the conditions of pandemia The lack of usual offline connections and meetings in the conditions of pandemia made the use of VIA facility especially timely and important. This facility supports regular weekly meetings of the Laboratory of cosmoparticle studies of the structure and dynamics of Galaxy in Institute of Physics of Southern Federal University (Rostov on Don, Russia) and M.Khlopov’s scientific -creative seminar and their announcements occupied their permanent positiononVIA homepage(Fig.24.2),whiletheirrecordswereputinrespective place of VIA forum, like [31] for Laboratory meetings. TheplatformofVIA facilitywasusedforregular Khlopov’s course”IntroductiontoCosmoparticle physics” for M2 students of MEPHI (in Russian) and in 2020 supported regular seminars of Theory group of APC. The programme of VIA lectures continued to present hot news of astroparticle physics and cosmology, like talk by Zhen Cao from China on the progress of LHAASO experi- mentor lecturebySunnyVagnozzifromUKontheproblemof consistencyofdifferent measurements of the Hubble constant. Theresultsofthis activityinspiredthe decisiontoholdin2020XXIIIBledWorkshop online on the platform of VIA [19]. The conditions of pandemia continued in 2021 and VIA facility was successfully used to provide the platform for various online meetings. 2021 was announcedbyUNESCO as A.D.Sakharov year in the occasion of his 100th anniversary VIA offered its platform for various events commemorating A.D.Sakharov’s legacy in cosmoparticle physics. In the frameworkof1 Electronic Conference on Universe ECU2021), organizedby the MDPI journal ”Universe” VIAprovided the platform for online satelliteWorkshop ”Developing A.D.Sakharov legacy in cosmoparticle physics” [32]. 342 MaximYu. Khlopov Fig. 24.4: M.Khlopov’s talk ”Multimessenger probes for new physics in the light of A.D.Sakharovlegacyin cosmoparticlephysics”atthe satelliteWorkshop ”Developing A.D.Sakharov legacy in cosmoparticle physics” of ECU2021. 24.3 VIA platform at Hybrid Memorial XXV BledWorkshop VIA sessions at BledWorkshops continued the tradition coming back to the first experience at XI BledWorkshop [7] and developed at XII, XIII, XIV, XV, XVI, XVII, XVIII, XIX, XX, XXI and XXII BledWorkshops[8–18]. They becamearegular but supplementary partof theBledWorkshop’sprogram.Inthe conditionsof pandemiait becametheonlyformof Workshop activity in 2020 [19] and in 2021 [20]. During the XXV Bled Workshop the announcement of VIA sessions was put on VIA home page, giving an open access to the videoconferences at theWorkshop sessions. The preliminary program as well as the corrected program for each day were continuously put on Forum with the slides andrecordsof all the talks and discussions [33]. VIA facility tried to preserve the creative atmosphere ofBled discussions. The program of XXV BledWorkshop combined talkspresentedin Plemelj Housein Bled, which were streamed by VIA facility, as the talk ”Understanding nature with the spin-charge-family theory, New way of second quantization of fermions and bosons” by Norma Mankoc- Borstnik (Fig. 24.5) with talks given in the format videoconferences ”Recent results and empowered perspectivesof DAMA/LIBRA-phase2”by R.Bernabei, (Fig. 24.6),from Rome University, Italy(seerecordsin [33]). DuringtheWorkshoptheVIA virtualroom was open, inviting distant participantstojoin the discussion and extending the creative atmosphere of these discussions to the worldwide audience. The participants joined these discussions from different parts of world. The talk ”A Nietzschean paradigm for the dark matter phenomenon” was given by Paolo Salucci from Italy (Fig. 24.7). R.Mohapatra and S. Brodsky gave their talks from US and E.KiritsisfromfromParis. The online talks were combined withpresentationsin Bled such as ”Dusty dark matter pearls developed” by H.B. Nielsen (Fig. 24.8) or ”Cosmological reflection of the BSM physics” by M.Y. Khlopov (Fig. 24.9). Title Suppressed Due to Excessive Length 343 Fig. 24.5: VIA stream of the talk ”Understanding nature with the spin-charge- family theory, New way of second quantization of fermions and bosons” by Norma Mankoc-Borstnik at XXV BledWorkshop Fig.24.6:VIAtalk ”Recentresultsand empowered perspectivesof DAMA/LIBRAphase2” by R.BernabeifromRomeatXXVBledWorkshop The distant VIA talks highly enriched the Workshop program and streaming of talks from Bled involved distant participants in fruitful discussions. The use of VIA facility has providedremotepresentationof students’ scientific debutsin BSM physics and cosmology. The records of all the talks and discussions can be found on VIA Forum [33]. VIA facility has managed to join scientists from Mexico, USA, France, Italy,Russia, Slovenia, India and many other countries in discussion of open problems of physics and cosmology beyond the Standardmodels. 24.4 Conclusions The Scientific-Educational complexofVirtual Instituteof Astroparticle physicsprovides regular communication between different groups and scientists, working in different sci 344 MaximYu. Khlopov Fig. 24.7: VIA talk ”A Nietzschean paradigm for the dark matter phenomenon” by Paolo Salucci at XXV BledWorkshop Fig.24.8:VIAstreamoftalk”Dustydark matterpearls developed”byHolgerBech Nielsen at XXV BledWorkshop entific 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 cosmology, particle physics, astroparticle physics and related topics. It can help in the elaboration of strategy of experimental particle, nuclear, astrophysical 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 aspectsof VIA activity can evolveina specific tool for International PhD program for Fundamental physics. Involvement of young scientists in creative discussions wasan important aspectofVIA activityatXXVBledWorkshop.VIA applications cangofar Title Suppressed Due to Excessive Length 345 Fig. 24.9: VIA stream of talk ”Cosmological reflection of the BSM physics” by MaximYu. Khlopov at XXV BledWorkshop beyond the particular tasks of astroparticle physics and give rise to an interactive system of mass media communications. VIA sessions, which becamea natural partofaprogramof BledWorkshops, maintainedin 2022 the platform for online discussions of physics beyond the StandardModel involving distant participants from all the world in the fruitful atmosphere of Bled offline meeting. This discussion can continue in posts and post replies on VIA Forum. The experience of VIA applicationsatBledWorkshopsplays importantroleinthe developmentofVIA facility as an effective tool of e-science and e-learning. One can summarize the advantages and flaws of online format of Bled Workshop. It makes possible to involve in the discussions scientists from all the world (young scientists, especially) free ofthe expenses related with meetings in real (voyage, accommodation, ...), but loses the advantage of nonformal discussions at walks along the beautiful surrounding of the Bled lake and other places of interest. The improvement of VIA technical support by involvement of Zoom provided better platform for nonformal online discussions, but in no case canbe the substitute foroffline Bled meetings and its creative atmosphereinreal, whichhasrevivedattheofflineXXVBledWorkshop.Onecan summarizethatVIA facility provides important toolof theoffline BledWorkshop, involving world-wide participantsin its creative and open discussions. Acknowledgements The initialstepofcreationofVIA was supportedby ASPERA.I expressmy tributeto memoryofP.Binetruyand S.Katsanevasandexpressmy gratitudeto J.Ellisfor permanent stimulating support, to J.C. Hamilton for early support in VIA integration in the structure of APC laboratory, to K.Belotsky, A.Kirillov, M.Laletin and K.Shibaev for assistance in educational VIA program, to A.Mayorov, A.Romaniouk and E.Soldatov for fruitful collaboration, to K.Ganga, J.Errard, A.Kouchner and D.Semikoz for collaboration in development of VIA activity in APC, to M.Pohl, C. Kouvaris, J.-R.Cudell, C. Giunti, G. Cella, G. Fogli andF. DePaolis for cooperation in the tests of VIA online transmissionsin Switzerland, Belgium and Italy and to D.Rouable for help in technical realization and support of VIA 346 MaximYu. Khlopov complex. The work was financially supported by Southern Federal University, 2020 Project VnGr/2020-03-IF.Iexpressmy gratitudetotheOrganizersofBledWorkshopN.S.Manko cˇ Borˇ stnik, A.Kleppe and H.Nielsen or cooperation in the organization of VIA online Sessions at XXV BledWorkshop.I am grateful toT.E.Bikbaev for technical assistance and help.I am grateful to Sandi Ogrizek for creation of compact links to VIA Forum. References 1. http://www.aspera-eu.org/ 2. http://www.appec.org/ 3. M.Yu. Khlopov: Cosmoparticle physics,World Scientific, NewYork -London-Hong Kong -Singapore, 1999. 4. M.Yu. Khlopov: Fundamentals of Cosmic Particle Physics, CISP-Springer, Cambridge, 2012. 5. M.Y. Khlopov, Project ofVirtual Institute of Astroparticle Physics, arXiv:0801.0376 [astro-ph]. 6. http://viavca.in2p3.fr/site.html 7. M.Y. Khlopov, Scientific-educational complex -virtual instituteof astroparticle physics, 981–862008. 8. M.Y. Khlopov,Virtual Institute of Astroparticle Physics at BledWorkshop, 10177– 1812009. 9. M.Y. Khlopov, VIAPresentation, 11225–2322010. 10. M.Y. Khlopov, VIA Discussions at XIV BledWorkshop, 12233–2392011. 11. M.Y..Khlopov,Virtual Instituteof astroparticle physics:Science and education online, 13183–1892012. 12. M.Y. .Khlopov,Virtual Instituteof Astroparticle physicsin online discussionof physics beyond the Standardmodel, 14223–2312013. 13. M.Y. .Khlopov,Virtual Instituteof Astroparticle physics and ”What comes beyond the Standardmodel?” in Bled, 15285-2932014. 14.M.Y. .Khlopov,Virtual InstituteofAstroparticlephysicsand discussionsatXVIIIBled Workshop, 16177-1882015. 15. M.Y. .Khlopov,Virtual Institute ofAstroparticle Physics — Scientific-Educational Platform for Physics Beyond the StandardModel, 17221-2312016. 16. M.Y. .Khlopov: Scientific-Educational Platform ofVirtual Institute of Astroparticle Physics and Studies of Physics Beyond the StandardModel, 18273-2832017. 17. M.Y. .Khlopov: The platformofVirtual Instituteof Astroparticle physicsin studiesof physics beyond the Standardmodel, 19383-3942018. 18. M.Y. .Khlopov: The PlatformofVirtual Instituteof Astroparticle Physics for Studiesof BSM Physics and Cosmology, Journal20249-2612019. 19. M.Y. .Khlopov:Virtual Institute of Astroparticle Physics as the Online Platform for Studies of BSM Physics and Cosmology, Journal21249-2632020. 20. M.Y. .Khlopov: Challenging BSM physics and cosmology on the online platform of Virtual Institute of Astroparticle physics, Journal22160-1752021. 21. http : ==viavca:in2p3:fr=whatcomesbeyondthestandardmodelsxiii:html 22. http : ==viavca:in2p3:fr=whatcomesbeyondthestandardmodelsxiv:html 23. http : ==viavca:in2p3:fr=pohlmartin:html 24. In http://viavca.in2p3.fr/ Previous -Events -JDEV 2013 25. http : ==viavca:in2p3:fr=zeldovich100meeting:html 26. In https://bit.ly/bled2022bsm Forum-Discussion in Russian -Courses on Cosmoparticle physics Title Suppressed Due to Excessive Length 347 27. In https://bit.ly/bled2022bsm Forum -Education -From VIA to MOOC 28. In https://bit.ly/bled2022bsm Forum -Education -Lectures of Open Online VIA Course 2017 29. In https://bit.ly/bled2022bsm Forum -Education -Small thesis and exam of Steve Branchu 30. http : ==viavca:in2p3:fr=johnellis:html 31. In https://bit.ly/bled2022bsm Forum -LABORATORYOF COSMOPARTICLE STUDIES OF STRUCTURE AND EVOLUTION OF GALAXY 32. In https://bit.ly/bled2022bsm Forum -CONFERENCES -CONFERENCES ASTROPARTICLE PHYSICS -The Universe of A.D. Sakharov at ECU2021 33. In https://bit.ly/bled2022bsm Forum -CONFERENCES BEYOND THE STANDARD MODEL -XXV BledWorkshop ”What comes beyond the Standardmodel?” Proceedings to the 25th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ::. (p. 348) VOL. 23, NO.1 Bled, Slovenia, July 4–10, 2022 25 Apoem WhenImet Elia Dmitrieffin Bled,he told me thathe hasa datchaby the Lake Baikal. “Really?”Isaid, “o,Iwould love to visit you and see the Baikal!” He smiledandnodded,andIthoughtoftheTrans-Siberianrailway,andmadeavagueplan to one day visit him. Some years later, when the warin Ukraine started,Iwasin total shock, and soon thereafter, Iwent intoa depression.Ithoughtof the Ukrainians, andIthoughtof Elia, andIthought ofallmy Russians friends.AndthenIwrotethispoem. Voyage to Irkutsk IthoughtIone day would pay you a visit, take the train many days through the large waste land woods We would go to your datcha chop some wood, make a fire, drink our tea and discuss the meaning of charge! By the hearth you would tell me your binary codes of the innermost parts, of your Glasperlenspiel, while over the house the moon would sail, and mirror its light in the Lake Baikal, And you’d say: If you pray, pray for healing and reason, and if you believe, believe in the good! 25 a poem 349 Iwill,Iwould say, butIwant you to tell me a story where all goes well! Iwant it to be in a grandiose land where oceans are pure, since pollution is banned. Andthere mustbea housewithagardenandtrees with apples and plums, and flowers and bees And children shall play in the neighbouring wood and nobody barks, and people are good! Do you hear,Iwill come and visit you soon! Then we’ll talk about binary codes. When the devils are gone and the light is restored, and the roads are repaired and the flowers all bloom, Iwill visit you soon, very soon! Astri Kleppe BLEJSKE DELAVNICE IZFIZIKE, LETNIK 23, ˇISSN 1580-4992 ST.1, BLED WORKSHOPSIN PHYSICS, VOL.23,NO.1,ISSN1580-4992 Zbornik 25. delavnice ‘What Comes Beyond the StandardModels’, Bled, 4. -10 julij 2022 [Virtualna delavnica 11.-12. julij] Proceedings to the 25th workshop ’What Comes Beyond the StandardModels’, Bled, July 4.–10., 2022[VirtualWorkshop, July 11-12 2022] Uredili Norma Susana Mankoˇc Borˇstnik, Holger Bech Nielsen in Astri Kleppe Izid publikacije je finanˇ cno podprla Javna agencija za raziskovalno dejavnost RS iz sredstev drˇcih znanstvenih zavnega proraˇcuna iz naslova razpisa za sofinanciranje domaˇ periodiˇcnih publikacij Brezplaˇzence cni izvod za udeleˇ Tehniˇcni urednik Matjaˇsnik z Zaverˇ Zaloˇzilo: DMFA – zaloˇstvo, Jadranska 19, 1000 Ljubljana, Slovenija zniˇ Natisnila tiskarna Itagraf v nakladi 130 izvodov Publikacija DMFA ˇ stevilka 2160