BLEJSKEDELAVNICEIZ FIZIKE LETNIK 22, . ST.1 BLED WORKSHOPSIN PHYSICS VOL.22,NO.1 ISSN1580-4992 Proceedings to the 24thWorkshop What Comes Beyond the Standard Models Bled, July 5–11, 2021 [VirtualWorkshop] [July 5.–11. 2021] Edited by Norma Susana Manko.c Bor.stnik Holger Bech Nielsen Dragan Lukman Astri Kleppe DMFA– ZNI. ZALO.STVO LJUBLJANA, DECEMBER 2021 The 24thWorkshop What Comes Beyond the Standard Models, 5.– 11. July 2021, Bled [VirtualWorkshop, 5.–11. July 2021] 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 (Matja.z 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 Manko.c Bor.stnik 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 24thWorkshop “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-Body Problems in Hadronic and Atomic Physics (July 7–14, 2001), Vol.2(2001) No.1 . Quarks and Hadrons (July 6–13, 2002),Vol. 3(2002) No.3 . Effective Quark-Quark Interaction (July 7–14, 2003),Vol. 4(2003) No.1 . Quark Dynamics (July 12–19, 2004),Vol. 5(2004) No.1 . Exciting Hadrons (July 11–18, 2005),Vol. 6(2005) No.1 . Progress in Quark Models (July 10–17, 2006),Vol. 7(2006) No.1 . Hadron Structure and Lattice QCD (July 9–16, 2007),Vol. 8(2007) No.1 . Few-Quark States and the Continuum (September 15–22, 2008), Vol.9(2008) No.1 . Problems in Multi-Quark States (June 29–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 1 Virtual talks A. Addazi, L. Bonora, S. Kabana, E. Kiritsis, R. Mohapatra, Q. Shafi . . . . . . . . 1 2 Type IIB moduli stabilisation, inflation and waterfall fields I. Antoniadis, O. Lacombe, G. K. Leontaris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4 The multicomponent dark matter structure and its possible observed manifestations V. Beylin, V. Kuksa, M. Bezuglov, D. Sopin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5 Numerical simulation of Bohr-like and Thomson-like dark atoms with nuclei T.E. Bikbaev, M.Yu. Khlopov, A.G. Mayorov. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 6 Supersymmetric and Other Novel Features of Hadron Physics from Light-Front Holography S. J. Brodsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7 Charge asymmetry of new stable quarks in baryon asymmetrical Uni­verse A. Chaudhuri, M. Yu. Khlopov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 8 Entopy release in Electroweak Phase Transition in 2HDM A. Chaudhuri,M.Yu. Khlopov,S. Porey ................................ 104 9 Gravitational waves in the modified gravity S. Roy Chowdhury, M.Yu. Khlopov .................................... 114 10 Representing rational numbers and divergent geometric series by binary graphs E. Dmitrieff ......................................................... 123 11 Neutrino masses within a SU(3) family symmetry and a 3+5 scenario Albino Hernandez-Galeana ........................................... 135 12 BSM Cosmology from BSM Physics M.Yu. Khlopov...................................................... 152 13 Statistical analyses of antimatter domains, created by nonhomoge­neous baryosynthesis in a baryon asymmetrical Universe M.Yu. Khlopov,O.M. Lecian ......................................... 160 14 Researchingof magnetic cutofffor local sourcesof charged particles in the halo of the Galaxy A.O. Kirichenko, A.V. Kravtsova, M.Yu. Khlopov, A.G. Mayorov ........... 170 15 Mass asa dynamical quantity M. Land ............................................................ 177 16 New way of second quantization of fermions and bosons N.S. Manko.c Bor.stnik ................................................ 190 17 The achievements of the spin-charge-family theory so far N.S. Manko.stnik c Bor................................................. 226 18 Novel String Field Theory and Bound State, Projective Line, and sharply 3-transitive group H. B. Nielsen, M. Ninomiya ........................................... 257 19 Atomic Size Dark Matter Pearls, Electron Signal H.B. Nielsen, C.D.Froggatt ............................................ 278 20 Galactic model with a phase transition from dark matter to dark energy I. Nikitin ........................................................... 300 21 Ultraviolet divergences in supersymmetric theories regularized by higher derivatives K.V. Stepanyantz .................................................... 332 Virtual Institute of Astroparticle Physics (CosmoVia)348 22 Challenging BSM physics and cosmology on the online platform of Virtual Institute of Astroparticle physics M.Yu. Khlopov...................................................... 349 Postscriptum .................................................. 365 1 Preface in English and Slovenian Language The series of annual workshops on ”What Comes Beyond the StandardModels?” startedin1998withtheideaofNormaandHolgerfororganizingareal workshop, in which participants would spend most of the time in discussions, confronting different approaches and ideas. All this time we have been looking for answers to the questionofwhatthelawsof natureare.Andwe learnedalot.ThisyearinJuly the 24th workshop took place. Workshops have always taken placein the picturesque townof Bledby the lake of the same name, surrounded by beautiful mountains and offering pleasant walks and mountaineering. Except for the last two years, 2020 and 2021, when workshop has again taken place in July,but without personal conversations all day and late at night, even between veryrelaxing walks and mountaineering due to COVID-19 pandemic.Wehave, however,averylong traditionof videoconferences (cosmovia), enabling discussions and explanations with laboratories all over the world. This enabled usto have these two yearsa virtual workshop,resembling Bled workshops as much as possible. In our very open minded, friendly, cooperative, long, tough and demanding discussions several physicists and even some mathematicians have contributed. Most of topics that have been presented and discussed in our Bled workshops deal withthe proposals for explaining physics beyond the so far accepted and experimentally confirmed both standardmodels — in physics of fermion and boson fields and cosmology — in order to understand the origin of assumptions of both standardmodels and be consequently able to propose new theories, models and to make predictions for future experiments. Although in all these years most of participants were theoretical physicists, many of them with their own suggestions how to make the next step beyond the accepted models and theories, experts from experimental laboratories were and are very appreciated, helping a lot to understand what do measurements really tell and which kind of predictions can best be tested. Also in the last two years we tried to keep our habit of (long) presentations (with breaks and continuations over several days), followed by very detailed discussions. The authors of the articles worked hardand with enthusiasm already before the presentations and as well when preparing the articles for this Proceedings in such a short time. However, as lectures and especially discussions over the Internet are more ex­hausting than live, many issues remain open, unresolved, also undefined and undiscussed. And we did not succeed to continue the discussions over the Internet after the workshop, even though we tried, because of several reasons, one of them was that the computer of one of the organizers broke down. Herearesome questions that we have notreally discussed yet but have just started discussing: How efficient are models offering a small next step beyond both standardmodels, suggesting experiments which could test such a model, to be able to explain all the observations so far, or at least many of them? There are several contributions with such proposals presented in this Proceedings, most of them trying to explain what does the dark matter consist of. Would the confrontation of these models with string theories, for example, or with the spin-charge-family theory, which offers the explanation for all the assumptions of both standardmodels, offering as well the explanation for several phenomena observed so far, with the dark matter and the matter/antimatter asymmetry included, the theory is presented in this Proceedings, help to understand our universe better and also to easier propose relevant experiments having correspondingly more chance to be the right next step beyond both standardmodels? Combining knowledge, ideas and hardwork could increase our opportunities to recognize the real next step beyond the two standardmodels and to suggest trustable experiments. In particular if experimentalists would be involved in discussions. Experiments are expensive. Although the black holes are experimentally well confirmed objects, the quantum mechanics of black holes is notreally known. This knowledgeis needed forheavy black holes as well as if we accept the possibility that the space-time is larger than (3 + 1), asitisin string theories, in Kaluza-Klein theories and in the spin-charge-family theory (with fermions interacting with gravity only) with space-time (13 + 1)or larger (appearing in two contributions in this paper), or might be even infinite (since zero and infinite are easy to be accepted, all other possibilities need the explanation). Do we understand what in this context the primordial black holes, discussed in this and last year Proceedings, mean and do they appear before or after the electroweak phase transition? Is in the time of the formation of the primordial black holes space-time already (3 + 1)?What happens inside the primordial black holes and what happens within the very massive experimentally confirmed black holes?Can string theories within M-theory help to understand the quantum gravity even in the context that the internal space of fermions and bosons describe the Cliffordalgebra objects? If Naturedoes use the Cliffordalgebra objects to describe internal space of fermions and bosons, what explains the second quantizationpostulates for fermions and bosons, as explainedin one contribution in this proceedings, can the quantum mechanics of black holes be easier understood? Would thenovel string theory, discussedinthisproceedings, where noninteracting objects representing strings, are themselves bound states ofstrings and might explain bound states of objects with the high symmetry, be helpful as well for describing the heavy black hole objects? Even under the assumption that the internal space of fermion and gravitational fields are described with the Cliffordalgebra objects? It does happen (after having a vision and after a hard work) that a new way of treating quantum mechanics of bound systems, this time the superconformal quantum mechanics and light-front holography used in hadron physics, presented in this proceedings, opens a new understanding of dynamics and symmetries of bound states. To understand the start and the starting expansion of our universe the knowledge of quantum gravity and the knowledge of the internal space of fermions and bosons is needed. Some periods and some phenomena in the expansion of our universe can be explained in the context of string theories, as it is the period of the inflation described with one article in this proceedings. When has the inflation taken place and how is it connected with the today non observed extra dimensions? Do we have besides the ordinary matter also domains of antimatter in our uni­verse? What are properties of the antimatter? Do domains of antimatter contain mostly the dark matter?What is the interaction of matter with antimatter on the bor­derofboth domains?In the spin-charge-family theory the laws and the interactions are the same — for fermions, antifermions and dark matter. In several talks these problems were discussed, some of them presented only as a talk on the website on http://bsm.fmf.uni-lj.si/bled2 and on Forum of Cosmovia as https://bit.ly/bled2021bsm . Symmetries play the essential role on all levels of physics, on the level of elemen­tary fermion and boson fields, of cosmology and also of matter of all kinds. In the theories assuming more than (3 + 1) dimensions with fermions which interact with gravity only, like there are the Kaluza-Klein theories, the spin-charge-family theoryis alsoof this kind,as wellas string theories,the symmetryoriginsin the Lorentz invariance of spacetime, manifesting also in the internal space of boson and fermion fields. At observable (that is low) energies the Lorentz symmetry of higher dimensions manifests in (3 + 1)spacetime (after breaking the starting symmetry of spacetime) the symmetry of the internal space of fermion and boson fields only, which usually is described by the group theoretical methods. The symmetries of elementary fermion and boson fields are discussed in several contributions talks of Bled 2021 workshop, manifesting that all these different understanding of symmetries have a strong overlap. Some talks about symmetries appear in these proceedings, the others can only be found on the follow-up pageof theofficial websiteof theWorkshop: http://bsm.fmf.uni-lj.si/bled2021bsm/presentations.html, and on the Cosmovia Forum https://bit.ly/bled2021bsm . There are several other topics, discussed in this proceedings, like i. What is indeed the origin of masses of fermions and gauge fields? ii. What modes of gravitational waves can be observed? iii. How far can we interpret experiments correctly if we accept the standard model only? iv. The DAMA/LIBRA experiment, measuring the collisions of the dark matter particles with the ordinary matter, reports on the newest results of the annual modulation of data together with the measurements, collected in more than 10 years. v. Choosing the action for the assumed laws of Nature to predict experiments one must be able to calculate properties of systems accurately enough. One must be ableto evaluatetherenormalizability,the anomalies,foranyproposed theory.The reader can find some answers to these questions in this proceedings. This year neither the cosmological nor the particle physics experiments offered much new, as also has not happened in the last three years, which would offer new insight into the elementary fermion and boson fields and also into cosmological events, althoughalotofworkandefforthavebeenputin. However, there are more and more cosmological evidences, whichrequire the new step beyond both standardmodels, the one of the elementary fermion and boson fieldsandof cosmology.The understandingthe universethroughthe cosmological theories and theories of the elementary fermion and boson fields have, namely, so far never been so dependent on common knowledge and experiments in both fields. Although cosmovia served the discussions all the time (and we are very glad that we did have in spite of pandemic also the 24th workshop), it was not like previous workshops. Discussions were fiery and sharp, at least during some talks. But this was not our Bled workshop. Effective discussions require the personal presence of the debaters, as well as of the rest of participants, which interrupt the presentations with questions all the time. And let us add also this year that due to the on line presentations we have students participants, who otherwise would not be able to attend the Bled conference, the travel expenses are too high for them. The organizers hope that the virus will be defeated at least up to next year,although the data are not supporting our hope. Let our hope be valid for all over the world, especially for theyoung generation, as well as for the BledWorkshop 2022, so that we will in July next year meet at Bled. Since, as every year, also this year there has been not enough time to mature the discussions into the written contributions, only two months, authors can notreally polish their contributions. Organizers hope that this is well compensated with fresh contents. Thereadercanfindallthetalksand soonalsothewholeProceedingsontheofficial websiteof theWorkshop: http://bsm.fmf.uni-lj.si/bled2021bsm/presentations.html, and on the Cosmovia Forum https://bit.ly/bled2021bsm . The organizers are thanking Dragan Lukman for his excellent technical support to more then twenty years of Bled workshops, entitled ”What comes beyond the standardmodels”, in particular for his excellent work done on proceedings. In July of this year we learned how small the step is between to be and not to be. The memberof editors Dragan Lukman, our friend and the man whorecognized clearly the essential problems of our planet, is not among us any longer. He left us after this year workshop due to the hart attack.We are missing him very much, also during the preparation of the proceedings, although we copied his way of preparing the proceedings, using his styles. In memory of Dragan, we added a short summaryof his work and Astri’s song, which saysa lot about Dragan. The organizing committee thanks Astri Kleppe, who offered to take over Dragan’s work on the proceedings when our hardship was greatest. The organizing committee thanks also Ana Braci..stnik Bra.c c and Anamarija Bor.ci.who have done the translations of English abstracts to the Slovenian language. Letus concludethisprefacewithaheartfeltandwarmthanktoalltheparticipants, present via videoconference, for their lectures and especially for the very prolific discussions and, nevertheless, an excellent atmosphere.We are very sorry that some of participants could not prepare their talks as contributions to the proceed­ings. Norma Manko.c Bor.stnik, Holger Bech Nielsen, MaximY. Khlopov, (the Organizing comittee) Norma Manko.c Bor.stnik, Holger Bech Nielsen, Dragan Lukman, Astri Kleppe (the Editors) Ana Bra.ci.c, Anamarija Bra.ci.c Bor.stnik (the translators into Slovenian) Ljubljana, December 2021 2 Predgovor (Preface in Slovenian Language) Vsakoletne delavnice z naslovom ,,Kako prese. ci oba standardna modela, koz­molo.sibkega” (”What Comes Beyond the StandardModels?”) sta skega in elektro.postavila leta 1998 Norma in Holger z namenom, da bi udele .crpnih zenci v iz.diskusijah kriti .cali razli.casu smo iskali cno soo.cne ideje in teorije.V vsem tem .odgovore na vprasanje kak.sni so zakoni narave. In se veliko nau.cili. To leto je stekla 24. delavnica. Delavnice domujejo v Plemljevi hi . si na Bledu ob slikovitem jezeru, kjer prijetni sprehodiin pohodi na .znosti cudovite gore, ki kipijo nad mestom, ponujajo prilo .in vzpodbudo za diskusije.Takoje bilo vsedo zadnjih dveh let. Tudi zadnji dve leti, v letu 2020 in 2021, sta bili delavnici v juliju, vendar nam je tokrat covid-19 onemogo.canje v Plemljevi hi. cil sre.si.Tudi diskutirali nismo med hojo okoli jezera ali med hribolazenjem.Vendar namje dolgoletna isku. snjas “cosmovio” — videopovezavami z laboratoriji po svetu — omogo. cila, da je tudi letos stekla Blejska delavnica, tokrat prek interneta. K na. sim zelo odprtim, prijateljskim, dolgim in zahtevnim diskusijam, polnim iskrivega sodelovanja,je prispevalo veliko fizikovin celo nekaj matematikov.V ve.zenci poskusili razumeti in pojasniti pred­ cini predavanj in razprav so udelele .postavke obeh standadnih modelov, elektro. sibkega in barvnega v fiziki osnovnih delcev in polj ter kozmolo. skega, predpostavke in napovedi obeh modelov pa vskla­diti z meritvami in opazovanji, da bi poiskali model, ki prese . ze oba standardna modela, kar bi omogo.cilo zanesljivej.se napovedi za nove poskuse. .cina udele .cnih fizikov, mnogi z lastnimi idejami kako Cepravje ve.zencev teoreti .narediti naslednji korak onkrajsprejetih modelovin teorij,so .sli se posebej dobrodo.predstavniki eksperimentalnih laboratorijev,ki nam pomagajovodprtih diskusijah razjasniti resni .cilo meritev in nam pomagajo razumeti kak. cno sporo.sne napovedi so potrebne,dajih lahkos poskusi dovolj zanesljivopreverijo. Tudiv zadnjih dveh letih smo posku. sali ohraniti navado,da sobilepredstavitve dolge, ker so jih udele .sanji, da bi bili privzetki in pred­ zenci prekinjali z vpra.postavke jasni. Predavanja so se zato po dveh urah prekinila in se nadaljevala naslednje dni. Avtorji prispevkovsotrdoinz navdu. senjem delali, da so pripravili predavanja, in dasovtako kratkem .clankezata zbornik. casu pripravili .Ker pa so predavanja preko interneta bolj naporna kot v predavanja v . zivo, so mnoga vpra.sanja ostala odprta, nerazjasnena, tudi nedefinirana in nere.sena. Ni nam uspelo nadaljevati pogovorov preko interneta po kon.cetudi cani delavnici, .smo posku.cni, med njimi sesutje ra. sali. Razlogi so bili razli.cunalnika ene(ga) od organizatorjev. Med vpra. sanji,kismojihodprli,paonjihnismouspelizares razpravljati,so: Kakou. cinkoviti so lahko modeli, ki ponudijo majhen naslednji korak glede na oba standardna modela, da bi nato predlagali izvedbo poskusov, ki naj povedo ali so taki modeli v skladu z naravo, pri iskanju odgovorov na vsa odprta vpra. sanja, ali vsaj na del odprtih vpra.sajo sanj? Kar nekaj prispevkovv tem zborniku,ki posku.pojasniti,iz.cesa utegne biti temna snov,jete vrste. Ali bi bilo smiselno in bi zmogli primerjati te predloge, denimo, s teorijami strun ali s teorijo spinov, nabojev, dru.zin,ki .sanja obeh stan­ ze odgovori na odprta vpra.dardnih modelov in ponudi tudi napovedi, ki jih je potrebno preveriti, za temno snov in tudi za druga kozmolo.ska opa. zenja. Zdru.zenega dela bi lahko pove.znosti, da pre­ zevanje znanja, idej in vlo .calo mo .poznamo,kajjepravinaslednjikorak,kiprina .sanja sa odgovore na odprta vpra.v fiziki osnovnih fermionskih in bozonskih polj in kozmologiji ter bi pomagalo predlagati zaupanjavredne poskuse,kibodo domnevepotrdili, posebej, . ce bi pri diskusijah tvorno sodelovali tudi experimentalci. Experimenti so dragi. .crne luknje eksperimentalno potrjeni objekti, kvantna mehanika . Ceprav so .crnih lukenj vresnicini znana.Vendarjeto znanjepotrebno, .znost,da ce sprejmemo mo.je prostor-.c kot (3 + 1)-razse. cas ve.zen, kotto domnevajo teorije strun in Kaluza-Kleinove teorije, da je njegova razse.c, kot domneva znost morda celo (13 + 1)ali ve.teorija spin-charge-family (s fermioni, ki interagirajo samo z gravitacijsko silo, z dvema prispevkoma v tem zborniku), ali kot domneva tudi teorija strun, ali pa je lahko neskon.c in neskon.znosti cen, saj je ni.cno enostavno sprejeti, vse druge mo .potrebujejo pojasnila. Kako v tem kontekstu razumeti primordialne . crne luknje? Ali se pojavijo po elektro.ze prej? Ali sibkem faznem prehodu? Ali nastanejo .tedajprostor-.zeu.zen?Inkajse dogaja znotrajteh cas .cinkuje kot (3 + 1)-razse.primordialnih .crnih lukenj?Ali crnih lukenj?Kaj pa se dogaja znotraj zelo masivnih.lahko teorije strunv kontekstu M-teorije pomagajo razumeti kvantno gravitacijo tudi,.ca Cliffordova algebra? ce notranji prostor fermionov in bozonov dolo.Ali bi teorija, imenovana nova teorja strun, poro. cilo je najti v tem zborniku, s strunami iz inertnih objektov, ki so dejansko strune vezane v struno z veliko stopnjo simetrije, bila sprejemljiva tudi, . cebinotranje stopnje fermionov in bo­zonov dolo . cali Cliffordovi objekti? Bi bile take strune koristne tudi za razumevanje kvantne mehanike zelo masivnih . crnih lukenj? Zgodi se, da nov na.sen cin obravnavanja kvantne mehanike vezanih sistemov,kakr.jeuporaba superkonformne kvantne mehanikein holografije ”light-front”vhadron-ski fiziki, omogo. ci nov pogled in novo razumevanje dinamike in simetrij vezanih stanj. Poro.cilo o tem prina.sa zbornik. Za razumevanje nastanka in za.siritve na. cetne.sega vesolja sta kvantna gravitacija in poznavanje notranjega prostora fermionov in bozonov potrebno orodje. Vsaj nekatera obdobja .ce razlo. siritve, obdobje inflacije denimo,je mogo.zitiv kontekstu teorije strun,o.cemer pori.ca en prispevek. Ali imamo v na.cajne snovi tudi domene antisnovi? Je anti- sem vesolju poleg obi.snov prete .sne so interakcije snovi s temno snovjo?Otem zno iz temne snovi?Kak.diskutirajo avtorji nekaterih prispevkovv tem zborniku.Teorija spina, nabojev in dru.zin gradi na predpostavki, da so zakoni gibanja enotni — za snov, za antisnov in za temno snov. Simetrije igrajo bistveno vlogo na vseh ravneh fizike: v kozmologiji, v fiziki os­novnih fermionskih in bozonskih polj, tudi v fiziki vseh vrst snovi. V teorijah, ki predpostavijo da ima prostor ve.razse. c kot (3 + 1) znost, in da interagirajo fermioni samo z gravitacijskimi bozoni — take so Kaluza-Kleinove teorije, tudi teorija spina-nabojev-dru.zin,patudi teorije strun —je izvor simetrijev Lorentzovi invariantnosti prostor-.cuje tudi notranji prostor fermionov casa, ki vklju.in bozonov. Pri opa.cetne simetrije) dolo . zenih (nizkih) energijah (po zlomitvi za.ca lastnosti prostora z razse . znostimi d> (3 + 1)notranji prostor fermionskih in bozonskih polj, kar opazimo v d =(3 + 1)-razse.casu kot simetrije, znem prostor-.ki jih lahko opi.semo tudi z metodami teorije grup. Simetrije osnovnih fermionskihin bozonskihpoljso obravnavanevnekaj prispevkih. Koliko skupnega imajo razli.cni pristopi pa bi bilo potrebno in koristno raziskati. Naj omenimo.se nekateredruge teme,kjih prispevkiv zborniku obravnavajo: i. Kaj je pravi vzrok, da imajo fermioni in nekatera bozonska polja maso? ii. Kako dolgo . se lahko pravilno interpretiramo rezultate poskusov z uporabo samo standardnega modela? iii. Experiment DAMA/LIBRA prin.sa poro.ciloo zadnjihrezultatih meritev letne modulacije trkov delcev temne snoviz obi . cajno snovjo v njihovih merilnih aparat­urah, povzema pa tudi vse dolgoletne meritve. iv.Ko izberemo model, moramovmodelu znati primerjatirezultate meritev dovolj natan. cno. Moramo vedeti alije teorijarenormalizabilna, ali ima anomalije,in kako se ra.sanja posku. cunov lotiti.Tudi na taka vpra .sa odgovoriti edenod prispevkov. Takokotvpreteklihtrehletihtuditoletoniso eksperimentiv kozmologijiin fiziki osnovih fermionskih in bozonskih polj ponudili rezultatov, ki bi omogo . cili nov vpogledv fiziko osnovnih delcevinpolj, .zenegaveliko ceprav je bilo vanje vlo .truda. Vse ve.skih meritev, za katere se zdi, da jih standardni model cje tudi kozmolo.os­novnih fermionskih in bozonskih polj ne more pojasniti in vse bolj kozmolo. ske meritve in opa . zanja ter experimentalne meritvev fiziki osnovnih fermionskihin bozonskih polj dolo . cajo iskanje teorije, ki lahko pojasni vse predpostavke stan­dardnega modela, pa tudi vsa nova kozmolo.zanja in vse nove meritve ter ska opa.predlaga prave experimente. .cas,takokotjebilonavseh Cetudije cosmovia poskrbela,daso diskusije tekle ves .delavnicah doslej, blejskih diskusijv . zivo diskusije po internetu niso mogle nado­mestiti. Diskusije so bile ognjevite in ostre, vsaj pri nekaterih predavanjih, vendar potrebujejo u.salcev, ki cinkovite diskusije osebno prisotnost diskutantov in poslu .z vpra.studentom internet ne sanji poskrbijo,daje debata razumljiva vsem.Tudi .more nadomestiti dobrega u.citelja. Organizatorji upamo,dabovsajdo naslednjega letaviruspremagan, . cetudi ta trenutek na.cnimi podatki. Naj na. se upanje ni podprto s statisti.se upanje velja za ves svet,za mlado generacijopa . se posebej, pa tudi za Blejsko delavnico 2022, da bo steklav . zivo na Bledu. Ker je vsako leto le malo casa.od delavnice do zaklju. cka redakcije, manj kot dva meseca, avtorji ne morejo dovolj skrbno pripravti svojih prispevkov, vendar upamo, da to nadomesti sve.zina prispevkov. Bralec najde zapise vseh predavanj in kmalu tudi leto. snji zbornik na uradnem naslovu Delavnice na medmre . zju: http://bsm.fmf.uni-lj.si/bled2021bsm/presentations.html, in na Cosmovia Forum https://bit.ly/bled2021bsm . Zahvaljujemo se Draganu Lukmanu za odli .cno podporo ve. cno tehni.c kot dva­jsetletnim blejskim delavnicamznaslovom ”Kakoprese . ci oba standardna modela”, ter za tehni. cno pripravo zbornikov. Letos smo izvedeli, kako majhen je korak med biti in ne biti. .skega odbora Dragan Lukman, na.clovek, Clan uredni.s prijateljin .ki je jasno prepoznaval probleme na.zbe, ni ve. se dru.c med nami. Zapustil nas je kmalu za tem, ko se je kon.snja Blejska delavnica. Imel je sr. cala leto.cni napad. Pogre.se posebej zdaj medpripravo zbornika, . samo ga, .ceprav namje zapustil tehni. cno znanje priprave zbornika.. Draganu v spomin smo dodali kratek povzetek njegovega dela ter Astrino pesem, ki veliko pove o Draganu. Organizacijski odbor se zahvaljuje Astri Kleppe, ki se je ponudila, da prevzame Draganovo delo na zborniku,koje bila na.aa stiska najve.cja. Zahvaljujemo se tudi Ani Braci..stnik Bra.c za prevode an- c in Anamariji Bor.ci.gle.skega tekstav sloven.s.cino. Naj zaklju.cno in toplo zahvalo vsem zencem, cimo ta predgovor s prisr.udele.prisotnim preko videokonference, za njihova predavanja in . se posebno za zelo plodne diskusije in kljub vsemu odli.cno vzdu.sje. Zelo namje .enci niso utegnili pripraviti polegpredavamj zal, da nekateri udele.tudi zapis teh predavanj v obliki prispevkov. Norma Manko.c Bor.stnik, Holger Bech Nielsen, MaximY. Khlopov, (Organizacijski odbor) Norma Manko.c Bor.stnik, Holger Bech Nielsen, Dragan Lukman, Astri Kleppe (uredniki) Ana Bra.ci.c, Anamarija Bra.ci.c Bor.stnik (prevodi v sloven.s.cino) Ljubljana, grudna (decembra) 2021 Proceedings to the 24th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ... (p.1) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 1 Virtual talks A. Addazi, L. Bonora, S. Kabana, E. Kiritsis, R. Mohapatra, Q. Shafi http://bsm.fmf.uni-lj.si/bled2021bsm/presentations.html https://bit.ly/bled2021bsm 1.1 Virtual talks Becauseof the pandemic, the BledWorkshop has now been virtual for the two last years, 2020 and 2021. Not all the 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: http://bsm.fmf.uni-lj.si/bled2021bsm/presentations.html https://bit.ly/bled2021bsm. Some of the talks are only available online, namely: A. Addazi:The multicomponent dark matter structure and its possible observed manifestations. L. Bonora:HSYang-Mills-like models: I review the attempt to construct massless gauge field theories in Minkowski spacetime that go under the name of HS-YM.Ipresent their actions and their symmetries.Imotivate their gravitational interpretation.In particularIshow how to recover the local Lorentz invariance, which is absent in the original formulation of the theories. ThenIproposea perturbative quantizationin the so-calledfrozen mo[1]mentum frame.Idiscuss physical and unphysical modes and show how to deal with them. FinallyI uncover the gauge symmetry hidden under such unphysical modes. This requires a nonlocal reformulation of the theory, which is, however, characterized by an augmented degree of symmetry. Povzetek:Avtor pregledno poro. ca o teorijah z imenom HS-YM, ki v prostoru Minkovskega obravnavajo brezmasna umeritvena polja. Predstavi privzete akcije in simetrije ter njihovo gravitacijsko interpretacijo. Poka. ze, kako obnoviti lokalno Lorentzovo invariantnost, ki je v prvotni formulaciji teorij ni. Predlaga kvanti­zacijo v teoriji motenj v tako imenovanem okviru zamrznjene gibalne koli. cine. Predstavi fizikalne (opazljive)in nefizikalne dele poljinpredlaga, kakoz njimi ravnati. Razkrije umeritveno simetrijo, skritov nefizikalnih delih polj,kipajoje A. Addazi, L. Bonora, S. Kabana, E. Kiritsis, R. Mohapatra, Q. Shafi opaziti v okviru nelokalne reformulacije teorij z ve.cjo stopnjo simetrije. S. Kabana:Sexaquarks, the unexpected Dark Matter candidate: Sexaquarks are a hypothetical low mass, small radius uuddss dibaryon which has been proposed recently and especially asa candidate for Dark Matter. The low massregion below 2GeV escapes upper limits set from experiments which have searched for the uuddss dibaryon and did not find 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 andit has beenrecently searchedin the BaBar experiment.We use a model which has very successfully described hadron and nuclei production in nucleus-nucleus collisions at the LHC in order to estimate the thermal production rate of Sexaquarks with characteristics such as discussed previously rendering themDM candidates.Weshowresultsonastudyofthe variationoftheSexaquark production rates with mass, radius and temperature and chemical potentials as­sumed and their ratio to hadrons and nuclei and discuss the interdependences and their consequences. These estimates are important for future experimental searches and enrich theoretical estimates in the multiquark sector. Povzetek:.cni dibarioni uuddss z majhno maso, velikost­ Sest-kvarki so hipoteti .nega reda 2GeV in majhnim polmerom. Zato jih je te. zko izmeriti. Ker je njihov razpadni.senkotje starostna. cas zelo dolg, vsaj tolik.sega vesolja, se zde primerni kandidati za temno snov (DM). .se niso na. Cetudijihdoslej .sli, iskalisojihtudiv eksperimentu BaBar, ostajajo kandidati za temno snov, vsj izklju.ce. citi jih ni mogo.Avtorji prispevka uporabijo model, ki uspe.se nastanek hadronov in jeder sno opi.pri trkih dveh jederv velikem hadronskem trkalniku (LHC). . Studirajo verjetnost za tvorbo . sest-kvarkov v odvisnosti od mase, radija, temperature nastanka in izbirekemi. cnega potencialainjo primerjajozverjetnostjoza nastanekhadronovin jeder. Rezultati ne le bogatijo teoreti.znih vezanih stanjih ve. cno vedenje o mo .cjega .znosti za nove experimente. stevila kvarkov, ampak ponudijo mo . E. Kiritsis:Coleman de Luccia transitions, and their implications for Quantum Field Theories in De Sitter space. R. Mohapatra:The Next Symmetry of Nature: B-L as a gauge generator of electroweak interactions were proposed forty years ago. The discovery of neutrino mass has made pursuing its phenomenological and experimental implications more interesting.In this talkIfocusona minimal model of gauged B-L symmetry and show how the Higgs field that breaks this symmetry can provide a candidate for dark matter which very weakly coupled to matter.Ialsopresent how sucha versionof B-L canbe testedin ongoing LHC experiments.Ithen discuss possible grand unificationof the weakly coupled B-L Title Suppressed Due to Excessive Length in an SO(10) in five dimensions and its test in the proton decay experiment. Reference: R. N. Mohapatra and N. Okada, Phys.Rev. D10111, 115022 (2020) Povzetek: Simetrija B-L in ustrezno umeritveno polje sta bila predlagana kot pojasnilo nastanka elektro.zepred . sibkega polja .stiridesetimi leti.Po odkritju,da imajo tudi nevtrini maso, se je zanimanje za to polje in za morebitne napovedi v zveziz njim znova obudilo.Avtorpredstavi minimalni model simetrije B-Lin poka.si to simetrijo, zagotovi kandidata za ze, kako lahko Higgsovo polje, ki poru.temno snov,kije zelo .cajno snovjo.Predstavi, kakoje mogo. sibko povezanaz obi.ce polje B-L opaziti pri poskusih na LHC. Predlaga tudi veliko poenotenje polja B-L, kije . sibko povezanozSO(10)vpetihdimenzijahin prispevek poenotenjav poskusu, ki meri razpad protona. Q. Shafi:Quest for Unification: Grand Unified Theories (GUTs) provide a compelling framework for unifying the strong, weak and electromagnetic interactions.Iwillreview gauge andYukawa unification, dark matter, proton decay, topological defects and inflation in super-symmetric and non-supersymmetric GUTs. Someimplicationsof merging grand unification and the weak gravity conjecture are briefly discussed. Povzetek:Velike poenotene teorije (GUT) ponujajo okvir za poenotenje mo. cnih, . sibkihin elektromagnetnih interakcij.Avtorpreglednopredstavi poenotenje umer­itvenih polj inYukawinih sklopitev, temno snov, protonski razpad, topolo. ske okvarein inflacijov supersimetri.cnih teorijah poenotenja. cnih in nesupersimetri.Na kratko obravnava tudi nekatere posledice, ko velikemu poenotenju pridruzi.tudi.sibko gravitacijsko polje. Proceedings to the 24th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ... (p. 4) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 2 Type IIB moduli stabilisation, inflation and waterfall fields I. Antoniadis1.. email:antoniad@lpthe.jussieu.fr 2 O. Lacombe1 , email:osmin@lpthe.jussieu.fr G. K. Leontaris3 email: leonta@uoi.gr ´place Jussieu, 75005 Paris, France 2 Center for Gravitational Physics,Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan 3 Physics Department, University of Ioannina, Ioannina 45110, Greece 1Laboratoirede PhysiqueTh´eoriqueet Hautes Energies, Sorbonne Universit´e, CNRS,4 Abstract. We present a string realisation of the hybrid inflationary scenario within type IIB effective string theory constructions and a geometric configuration of intersecting D7 branes.Ametastablede Sitter minimumis ensuredby perturbative logarithmic corrections and D-term contributions from abelian factors associated with the D7 branes. The inflaton is identified with the internal volume modulus whereas possible waterfall fields correspond to excitations of open strings attached to the magnetised D7 branes. Incorporating contri­butions of these fields in the scalar potential, inflation stops and the metastable vacuum settlestoa minimumwiththe observed tuneablevalueofthe cosmological constant. Povzetek:Avtorji predstsavijo hibridni model inflacije vesolja. Uporabijo efektivno teorijo strun vrste IIBz branami D7. Pois..cne popravke,ki hkratis prispevki . cejo logaritmi.clenaD za “brane” D7 zagotovijo metastabilni de Sitterjev minimum. Notranji volumski modul dolo.cenaz vzbujenimi stanji odprtih strun, ca napihovanje vesolja, ”polja slapov” pa so dolo.ki so pritrjena na magnetizirane brane D7. Inflacijo vesolja ustavijo, ko vgradijo ta polja v skalarno polje, ki minimum pove .sko konstanto. ze s kozmolo. 2.1 Introduction At present, String Theory formulated in ten or eleven dimensions appears to be the only promising candidate for a consistent quantum theory of the four known fundamental forces and their interactions. Compactification of the higher dimensional theoryto four spacetime dimensions entails an immense numberof string vacua dubbed as the string landscape. Numerous Effective Quantum Field Theories, on the other hand, have been built to describe the low energy physics and make cosmological predictions. Amongst the most important features such a .. Presenter 2 Type IIB moduli stabilisation, inflation and waterfall fields theory should possess,isa positivetiny cosmological constant . ˜ 10-120 M4 Planck in order to account for the dark energy suggestedby cosmological observations. The simplestwaytorealisethe dark energy scenarioisto introducea scalar field . with a potential V(.), which displays a minimum value equal to the cosmological constant Vmin(.0)= ., at some suitable point .0. There is a significant ongoing debate, however, on whether the string landscape contains any de Sitter vacua which comply with the prediction of positive .. Recent Swampland conjectures [1], in particular, suggest that the first and second derivatives of V(.)must satisfy '' the inequalities |.V|/V = c or min(.i.jV)= -c (in Planck units) where c, c are positive constantsoforder one.If theseinequalitiesaretrue, some apparently consistent (anomaly free) theories in four dimensions do not have an ultra-violet completion and cannotbe derivedfrom string theory.In other words, they belong tothe Swampland 1. Puttingitdifferently, startingfroma successfulEffective Field Theory weakly coupled to gravity which describes adequately the known physics phenomena, we cannot always embed it in the string theory landscape. The above considerations have far reaching consequences both in cosmology and particlephysics[5].Here,we mentionafew implicationson otherwisevery success­ful cosmological scenarios. For example, it is rather obvious that the Swampland criteria summarised in the aforementioned inequalities contradict the assumption that the cosmological constant can account for the dark energy of the universe. Furthermore, slowroll inflationisinconsistent with these criteria. Instead, there are suggestions [5] that quintessence models where the cosmological constant varies over time satisfy current observational constraints. If this scenario prevails, the present acceleration phase eventually will terminate whereas the expansion of the universe will come to an end in the distant future. The ensuing years since their formulation, Swampland conjectures have faced increased scrutiny. Most of the criticism focused on the assumed heuristic argu­ments, and the neglected role of string quantum corrections. Indeed, the latter are anticipated to be essential for the final form of the effective scalar potential in the resulting field theory model after compactification. This presentation will focus on investigations of de Sitter vacua and the realisation of inflation in type IIB superstring theory. These investigations will take place assuming a geometric configuration of intersecting D-brane stacks with magnetic fluxes [6]. At the same time, we will consider the effects of a new four-dimensional Einstein-Hilbert term (localised in the internal space) which is generated from higher derivative terms inthe ten-dimensionalstringeffective action[7,8].Thissetup induceslogarith-mic corrections to the scalar potential via loop effects [9]. Minimisation of the whole scalar potentialof the theory fixes the internal volumeK¨ahler modulus, V, whereas the ratios of the worldvolumes along the three D7-brane stacks are fixed by virtue of D-term contributions and their parameters depending on the quan­tised magnetic fluxes. In addition, slow-rollinflation can be realised considering the (canonically normalised) inflaton field to be proportional to the logarithm of the internal volume V. Furthermore, the open string spectrum associated with the D7 brane stacks playsa significantrole. One can fix magnetic fluxes and brane sep­arations so that charged open string states have positive squared-masses, except 1 Forreviews and furtherreferences see [2–4] I. Antoniadis, O. Lacombe, G. K. Leontaris for oneof them which becomestachyonic when V becomes less than some critical value. It turns out that this state can be identified with a waterfall field which can beusedtostopthe inflationaryphaseanddeepenthe vacuum.Ageneralisationof this scenario with several waterfall fields shows that the model can accommodate the present dark energy. 2.2 Type IIB moduli stabilisation We briefly introduce the basic geometric setup and the moduli field content.We considera six-dimensional compactification ona Calabi-Yau (CY) threefold within a type IIB framework in the presence of quantised 3-form fluxes. Deformations of the compactification correspond to massless scalars which do not acquire tree-level potential and do not affect the four-dimensional action. Such scalars are the dilaton field F, theK¨ ahler moduli Ti, the complex structure (CS) ones za, moduli correspondingto brane deformationsandsoon.We furtherintroducea two index antisymmetric tensor denoted with Bµ. (the Kalb-Ramond field) and the p-form potentials Cp,p = 0, 2, 4. The C0 potential and the dilaton field, define i the usual axion-dilaton combination S = C0 + ie-F . C0 + where gs is the gs string coupling. At the effective theory level, there are two basic ingredients: the superpotentialof the moduli fields and theK¨ahler potential. To construct the superpotential one introducesp-form field strengths Fp = dCp-1, H3 := dB2 and defines G3 := F3 - SH3. In terms of these, the fluxed induced superpotential W0 is given, at the classical level, by the well-known formula [10]: Z W0 = G3 ..(za), (2.1) where .(za)isa holomorphic 3-form.It turns out that the perturbative superpo­tential W0 isa holomorphic function which depends on the axion-dilaton modulus S, and the CS moduli za. Imposing the supersymmetric conditions, the moduli za,S canbe stabilised.On the contrary, theK ¨ ahler moduli, do not participate in the perturbative superpotential and thus remain completely undetermined at this stage. The second ingredientis theK¨ ahler potential which depends logarithmically on the various moduli fields through the expression: Z K0 =-2 ln(V)-ln(-i. . .¯), (2.2) where V is the volume of the 6d internal CY manifold X6, in string units. The effective potential is computed from (2.2) using the standardsupergravity formula X K Veff = e DIW0KIJ¯ DJ¯ W0 -3|W0|2 , (2.3) I,J where DI = .I + KI ahler covariant derivative. At the classical level is theK¨this potential vanishes identically due to its no-scale structure, and appropriate 2 Type IIB moduli stabilisation, inflation and waterfall fields supersymmetric (flatness) conditions for the dilaton and the CS moduli. It is thus impossible to stabilise theK ¨ ahler moduli at this level. These moduli can be stabilised when quantum correctionsbreaking the no-scale structureof theK ¨ ahler potential are included. Severalwaystofixthisproblemhave appeared overthelasttwo decades.Afirst approach[11,12]wasbasedonthe inclusionof non-perturbative superpotential P terms of the form Wnp ~ Aie-aiTi . The coefficients Ai may depend on the i complex structure moduli, and the exponential factors on theK ¨ ahler ones Ti. The parameters ai may arise form gaugino condensation on D-brane stacks and for 2p the SU(N)case, they are of the form ). The above ingredients can stabilise the N K¨ ahler fields, however the potential acquires an anti-de Sitter (AdS) vacuum [11]. Apossible solution to this problem [12] is to uplift the vacuum by taking into account contributions from D3 branes. Therearetwo issuesregarding this solution. Firstly, in order to obtain an AdS minimum the coefficients W0,Ai and ai require unnatural fine-tuning. Secondly, these contributions rely on non-perturbative effects which cannot be controlled at the full string level. Some improvements of the original models, however, have appeared using nilpotent chiral multiplets [13], whichleadtoanewmechanismfor upliftingthe vacuainthestring landscape[14]. Adifferentwayto stabilisethe moduliis basedonLargeVolume Scenario(LVS) ' [15]. This proposal takes advantage of the leading a corrections to theK ¨ ahler potential (together with the non-perturbative contributions) which ensure an AdS solutionintheLargeVolume Limitbut avoid tuning W0 in (2.1) at extremely small values. Uplifttoade Sitter(dS) vacuum canberealisedthrough D-terms. Perturbative moduli-dependent corrections in weakly coupled string theory, on the other hand, are fully controllable and therefore more reliable. However, not all types of corrections are suitable for moduli stabilisation. Ordinary perturbative ' expansions, either in a or in powers of the weak string coupling gs, fail to generating a (meta)stable dS minimum in a controllable way. This is the well-known Dine-Seibergproblem which we now describe in brief. When perturbative moduli-dependent quantum corrections are includedin theK ¨ ahler potential they induce contributions to the scalar potential, V(ti)where ti are the imaginary partsof theK ¨ ahler moduli Ti and are associated with the internal volume. The validity of perturbation theory implies that such corrections should vanish for ti .8 implying also the vanishing of the scalar potential V(ti)t.8 . 0. If the zero at infinity is reached from negative values, then, for non-contrived scalar potentials V(ti), this implies an AdS minimum which is not acceptable. Thus, the vanishing of the potential atinfinity should be approached from positive values. Again, for reasonable V(ti), this implies that there shouldbe somewherea maximumbeforeadS minimumis formed.Thesethreeshapesareplottedinfigure 2.1. The potential on the right-hand side exhibits local minimum and maximum and its shape suggests that there should be two competing terms of different functional dependence on t. While previously considered perturbative corrections do not share this property at large volumes, a possible exception known from field theory are logarithmic corrections similar to those in the Coleman-Weinberg mechanism [16]. I. Antoniadis, O. Lacombe, G. K. Leontaris Fig.2.1: Leftfigure:Vanishingof V(t)from 0- happens for potentials with an AdS minimum. Middle: Large t behaviour of V(t)with power law correction ~ 1 . tn The potential on the right-hand side exhibits local minimum and maximum. The above observation shows the way to overcome the difficulties in superstring constructions.Werecall that string theory hasareach structure including non-perturbative objects such as D-branes which open up possibilities to construct realistic cosmological models. Another ingredient, of particular interest in the present study, comes from high order curvature terms in the ten dimensional effective action. These elements are sufficient to generate loop corrections which induce new contributions to theK ¨ ahler potential K, break its no scale invariance and stabilise the moduli.We will describein short how perturbative logarithmic corrections are generated with the above constituents. The low-energy expansion of the type IIB superstring action contains fourth order terms in the Riemann curvature, R4, which do not receive any perturbative correc-tionsbeyondoneloop[7,17,18].Upon compactificationtoourfour dimensional spacetime M4, these one-loop corrections induce a novel Einstein-Hilbert (EH) term R(4). Its coefficient is proportional to the Euler characteristic ., defined on X6 by Z 3 . = R .R .R · 4p3 X6 Observing that . contains three powers of R, we deduce that the effective EH term R(4) (originating from R4)is only possible in four dimensions. Furthermore, such an EH term can be viewed as a vertex localised at certain points in the six-dimensional bulk where . acquires non-zero values, emitting closed strings (gravitons).We thus study the case of three-graviton scattering involving two massless gravitons and a Kaluza-Klein (KK) excitation propagating towards a D7­brane stack. The sum over the KK modes corresponds to a propagation that takes place in a two-dimensional bulk space transverse to the D7 stack, see Figure 1.2. Consequently, this process yields logarithmic contributions breaking the no-scale invarianceof theK ¨ ahler potential[6,9].Taking these logarithmic contributions into account the final effective action (obtained in the T6/ZN orbifold limit) contains [9] Z 1 4.(2). X 2FR. k S . e -2FR(10)+ 1 - eTk log R(4). (2.4) (2p)3 (2p)3 w k=1,2,3 M4×X6 M4 Here, Tk is the brane tension of the k-th stack, Rk the size of the two-dimensional . space transverse to the D7-stack and w an ‘effective’ localisation width of the 2 Type IIB moduli stabilisation, inflation and waterfall fields vv ' graviton vertex, given by w = ls/N with ls = a the fundamental string length [8]. From the correction terms (2.4)in the4dreduced action we canreadilyextract the corresponding induced termsin theK ¨ ahler potential. For simplicity we assume the same tension for all three brane stacks, so that Tk = T = e-FT0, and for each Kahler modulus ¨Tk we denote tk = ImTk. For D7-brane stacks with orthogonal co­ v volumes, the internal volume is simply V = ahler potential t1t2t3, and the theK¨takes the form v K =-2 ln ( t1t2t3 + . + . ln (t1t2t3))= -2 ln(V + . + . ln V). (2.5) Computations for the orbifold and smooth CY cases show that the parameters . and . are givenby[8,9]  2 1. p2 gs for orbifolds 3 . = - gsT0., with . =- × , (2.6) 2 4.(3) for smooth CY In (2.6) tree-level contributions for the orbifold case have not been included, since the .(3). correctiontotheEH term vanishes[7,8].The identity .(2)= p2 has also 8 been used in the orbifold action (2.4). Fig.2.2: Non-zero contributionfrom 1-loop; 3-graviton scattering amplitudeof2massless gravitonsand1KKmode correspondingtoaclosedstringpropagationin 2-dimensions towardsa D7 brane. 2.3 Inflationary phase From (2.3) we can readily compute the F-part of the scalar potential VF.To this end, we assume that all complex structure moduli are stabilised and the fluxed induced superpotential W0 can be taken as a constant, while for convenience we . introduce the new parameter µ = e 2. . The exact expression for VF can thus be written as 3.W2 2(. + 2V)+(4. -V)ln(µV) 0 VF = , (2.7) .42 (V + 2. ln(µV))(6.2 + V2 + 8.V + .(4. -V)ln(µV)) I. Antoniadis, O. Lacombe, G. K. Leontaris v where . = 8pGN is the reduced Planck length. In the large volume limit, VF takes the simplified form 3W2 0 VF =(. + 2.(lnV -4))+··· (2.8) 2.4V3 By virtue of the logarithmic term the potential (2.8) acquires a global minimum, althoughthisisan anti-de Sitter vacuum.Yet,aD-part contributiontothe scalar potential comes from the existence of universal U(1)factors associated with the three D7-brane stacks. In the large world-volume limit this contribution takes the form d1 d2 d3 VD =+++ ··· (2.9) .4t3 .4t3 .4t3 123 where the di for i = 1, 2, 3 are model-dependent constants related to U(1)Fayet-Iliopoulos (FI) terms. Forthe subsequent discussionitis usefultoreplacethe dependenceofthe potential onK¨ ahler moduli with the canonically normalised fields.We identify them witha logarithmic function of the volume and two perpendicular directions defined in terms of ti ratios.We alsorecall that we considera simple setup with “orthogonal” v D7-brane stacks, such that V = t1t2t3. The new basis then reads: 2 . = ln(V), (2.10) 3 1 t1 u = log , (2.11) 2t2 v 3 t1 t2 v = log . (2.12) 6t3 t3 In terms of these, the total scalar potential Veff = VF +VD in the large volume limit is v v . .( 6. -4)+. 3W2 Veff ˜ 0 e -3 3 2 2.4 v - 6. v vv e + d1e - 3v-3u + d2e - 3v+3u + d3e 2 3v. (2.13) .4 In the inflationary scenario that we will discuss shortly, the field . defined in 2.10 will play the role of the inflaton. In order to examine its evolution during the v inflation era, we need first to stabilise the three moduli u, v, V = e 2 3 . and derive the constraintsinorderto ensureadS vacuum.We first minimise Veff withrespect to the two transverse fields u, v, and find their values at the minimum: 1 d1 1 d1d2 u0 = ln ,v0 = v ln . (2.14) d2 6d2 63 3 Substituting back into (2.13) we obtain the simple expression v 3 v C 3 3 3 -3 . . , (2.15) V(.). - . -4 + q + se 22 e .4 2 2 . 2d C = -3W0 2 . > 0, d = 3(d1d2d3) 1 3 (2.16) ,q = ,s = . 2 2.9W0 . Afew comments are in order. First, in order to ensure a dS vacuum, the parameter . must be negative, hence the coefficient C is positive. Moreover, the parameter d, related to the D-term part of the potential, is always positive. Furthermore, increasing of the value of the parameter q shifts the local extrema towards larger volumes. Finally, s is the only free parameter of the model. It acquires negative values, hence the total coefficient of the last term is positive and is expected to uplift the minimum of the potential to positive values. To study inflation and compute the slow-roll parameters we need to determine the extrema of the potential with respect to the inflaton field . [19]. Thus we take the firstand second derivativesofthe potentialwithrespectto . and obtain v 3 v 3 C 13 3 3 ' (.) -3 . . , (2.17) . + q - V 3 + se 22 = e 2.4 2 3 v 3 v 27 C 14 2 3 3 '' (.) -3 . . . (2.18) - . + q - V + se 22 = e 2.4 2 33 Requiring the vanishing of the first derivative, V ' (.)= 0, we obtain two solutions which are expressed in terms of the two branches W0 and W-1 of the LambertW function (product logarithm): .- = 2 - 3 q - 13 3 -x-1+ W0 -e , (2.19) .+ = 2 - 3 q - 13 3 -x-1+ W-1 -e . (2.20) The new parameter x introduced in the above solutions is defined by 16 3q-16 -x . -e -log(-s) (2.21) x = q - s = . 3 while .- is the local minimum and .+ the local maximum. Large volumes can be achieved at weakcoupling for q<0, implyinga negative Euler number .<0, see 2.6 and 2.16. Notably, most of the important quantities are expressed through simple analytical forms in terms of x. For example, the slow-roll parameter . depends only on x through the LambertWfunction: '' (.-/+) -x-1) V1 + W0/-1(-e .(.-/+)= =-9. (2.22) 2 V(.-/+)+ W0/-1(-e-x-1) 3 Similarly, the distance between the two extrema is .+ -.- = 2 W0 -e -x-1-W-1 -e -x-1>0. (2.23) 3 I. Antoniadis, O. Lacombe, G. K. Leontaris The parameter x thus clearly plays a significant role. For the critical value xc . 0.072132 the potential at the minimum vanishes, V(.-)= 0, which corresponds to a Minkowski minimum. Below this critical value, in the region 0xc it displays an AdS minimum. For x<0 the two branches of the Lambert function join and the potential loses its local extrema. The potential for the three regimes described above is depicted in 2.3. Having determined the region of the parameter x which is consistent with dS minima, we are now ready to study cosmological implications and in particular inflationary observables.Wefirst find that some well-known inflationary scenarios such as slow-roll inflation hilltop, cannotberealisedin ourrestricted model.We can easily adjust the value of the slow-roll parameter . (which depends only on x) by varying x . (0, xc), so that inflation starts near the maximum, and the modes exit horizon with the required value of the spectral index. It is found, however, that the slow-roll parameters ., . remain much less that unity all the way down the slope, hence inflation does not stop, and as a result an unacceptably large number of e-folds is generated. As we describe below, in order to study more general inflationary scenarios, we will scan the x parameter space. For each value of x, we can solve the evolution equation for the Hubble parameter and derive therelevant parameters to study the eventual inflationary stage. Before entering the details of such a procedure, we thus recall a few basic equations regarding the evolution of the expansion of the Universe and the inflationary epoch assuming a single scalar field . in the standardFriedmann-Lemaˆitre-Robertson-Walker (FLRW) background. The Friedmann equations for an expanding Universe are 1 3H2 = ..2 + .2V(.), (2.24) 2 2H .=-..2 , (2.25) a . where, as usual H(t)= , represents the Hubble parameter. The equation of a motion for the scalar fieldreads . ¨ + 3H. .+ .2V ' (.)= 0. (2.26) 2 Type IIB moduli stabilisation, inflation and waterfall fields .dH . Changing variable H = ., equation (2.25) yields d. dH = H = 1. . (2.27) ' (.)- . d. 2 Using (2.24) and expressing . .as a function of H and V, we obtain the Hubble parameter evolution equation: 1 H ' (.)= ±v 3H2(.)-.2V(.). (2.28) 2 The exact forms of the slow-roll parameters ., . are [20] '' (.) HH .H ' (.) 2 .(.)= 2 ,.(.)=- = 2, (2.29) H(.) H2 H(.) '' V (.) while in the slow-roll limit they acquire the usual forms .(.) ˜ V(.) , and V (.) 2 .(.)˜ 1 ' . From the first expression of . in (2.29), we obtain 2V(.) a ¨ = 1 -., (2.30) aH2 so that .<1 is the natural criterium characterising inflation, a phase with ¨ a>0. Finally, the number of e-folds N is given by tend 1 . d. N = Hdt = vv . (2.31) . t 2 .end As mentioned above, one can investigate inflationary possibilities through a scan of the x parameter in the following way. The value of x determines the shape of the inflaton scalar potential V(.), which enters the evolution equation (2.28) for the Hubble parameter. For a given value of x, solving this equation thus allows to compute the slow-roll parameters and number of e-folds, through 2.29 and 2.31, and study the inflationary phase. Theabovescangaverisetoanovel scenariowheremostofthee-foldsare obtained near the minimum. In this scenario, the inflaton starts rolling down from a point close to the maximum towards the minimum of its potential with zero initial speed. If .(.+) < -0.02, because at the inflection point the second derivative V '' (.)changes sign, the inflaton will pass through the point where .(.*)=-0.02 beforeit crosses the inflection point.We can then choose the parameter x so that 60 e-folds are obtainedfrom this point to minimum. Thus, in order to reproduce the observational data, the initial position of the inflaton has to be higher than the inflection point, where . is negative, so that . =-0.02 is taken at the horizon exit. Inordertorealisethis scenario,wehave solved numericallythe evolution equation (2.28) for various values of x, starting near the maximum with vanishing initial speed for the inflaton. The required number of e-folds, N* . 60 are achieved for x . 3.3 10-4 while the two extrema of the potential are found at .- = 4.334 and .+ = 4.376. The e-folds are computed from the horizon exit .* . 4.354 I. Antoniadis, O. Lacombe, G. K. Leontaris at which .(.*) =-0.02, down to the minimum .-. Is it worth observing that the corresponding inflaton field displacement .. . 0.02, is much less than one in Planck units. Hence it corresponds to small field inflation, and as such is compatible with the validity of the effective field theory. Finally,this model predicts an inflation scale H* . 5 × 1012 GeV and a ratio of tensor to scalar perturbations r . 4 × 10-4 . 2.4 Waterfall fields and hybrid inflation Up to this point, we have explained how in the simple geometric set up of three D7-brane stacks we can ensureK ¨ ahler moduli stabilisationinadS vacuum and investigatedthe conditionstorealise inflation.We foundthat logarithmic radiative corrections and brane magnetisations generate a scalar potential with a very shallow dS minimum, which can realise inflation with the required 60 e-folds collected near the minimum (as opposed -for example-to the case of hilltop scenario). However, the tight constraints imposed by the various requirements entail a metastable minimum with a cosmological constant much larger than the one observed today.Adetailed consideration shows that this false vacuum of the so-obtained scalar potential is suggestive for a solution through hybrid inflation [21] where a waterfall field ends the inflation phase and settles to a lower (true) vacuum with the anticipated value of the cosmological constant. Such a waterfall field is realised by a scalar field with effective mass depending on the valueofthe inflaton.If this field becomes tachyonic undera certain critical value for the inflaton, it generates the waterfall direction of the scalar potential. Within the present geometric configuration, potential waterfall field candidates are the various states associated with the excitations of open strings with endpoints attached to D7 brane stacks. The scalar components of these states may receive supersymmetric positive square masses from brane separation orWilson lines, and non-supersymmetric contributions due to the presence of the worldvolume magnetic fields generating the D-termsrequired for moduli stabilisation. In the following, we briefly describe how these fields contribute to the materialisa­tion of this scenario in the context of a Z2 × Z2 orbifold.We assumea factorised 6-torus into three 2-tori T 6 = T2 × T 2 × T 2 spanning the internal dimensions (45), (67) and (89) respectively. The model under consideration consists of three D7 brane stacks, which we denote with D71,D72 and D73. Each of them spans four internal dimensions and is localised in the remaining two. This setup can be considered as dual to the configuration of the D9 and D5 branes as in the toroidal orbifold model describedin the literature [22,23]. Thisis shown schematicallyin the following table where we imposeT-duality along (45) dimensions. (45) (67) (89) D71 · ×× D91 -. D72 ×× · D52 D73 ×·× D53 (45) (67) (89) × × × · × · · · × 2 Type IIB moduli stabilisation, inflation and waterfall fields We usea cross× to represent the D7 world-volume spanning the corresponding torus, and a dot · to indicate the transverse directions where the D7 brane is localised. As motivated above, we can introduce magnetic fields H(i) , on the a-th stack a D7a R and in the i-th torus Ti2 . They are subject to the Dirac quantization condition (i) H(i) k(i) m= 2pn(i) , leading to the magnetic field quantisation 2pH(i) Ai = , aaa aa (i)(i) where 4p2Ai is the T2 area. Here m, nare the winding numbers and the flux i aa quanta and we defined the ratio k(i) = n(i) /m(i) . Q. The magnetic fields modify a aa the world-sheetactionbyintroducingboundaryterms[24,25]andshiftthemodes of the charged oscillators by .(i) 1 ' H(i) = Arctan(2pa ). (2.32) a qaa p where qa = ±1, 0 are the U(1)charges of the open string endpoints. The mass spectrum can be extracted, either from the field theory mass formula or from vacuum amplitudes, and one sees that when magnetic fields are intro­duced intothe D7-brane configuration, tachyonic states may appear in the spec­trum [25,26].In general, one can eliminate themby introducing appropriate brane separations orWilson lines. To be concrete, we consider magnetic fields on eachD7 stack, denoted by a circled cross . as the following table. (45) (67) (89) D71 · .× D72 ×·. D73 .× · Three different kinds of states appear. The first two describe strings with both endpoints on the “same” stack D7i -D7i which are either neutral (attached to the same brane, hence with opposite endpoints charges) or doubly charged (stretching between the brane and its orientifold image). The last ones are mixed states D7i ­D7j, with i .j. Dueto thepresenceof magnetic fields, the massless statesof the = original orbifold model are modified. The masses of the D7i -D7i doubly charged ' 2 states read am=-2|.(j) |2 whereas those of the D7i -D7j states are of the form i (3)(2)(2)(1)(1)(3) (|.|-|.|), (|.|-|.|)and (|.|-|.|). 2113 32 Observing the above mass formulae, it can be deduced that tachyonic states indeed appear in the spectrum [25, 26]. The only way to eliminate all three potential tachyons along the D7-brane intersections(D7i –D7j mixed states) is to choose |.(2) |.(3) |.(1) | = | = |. On the other hand, in order to uplift the tachyons on the 123 D7i –D7i sectors, we can introduce distance separations between branes and their images in the direction orthogonal to their worldvolume, orWilson lines i.e. constant background gauge fields on unmagnetised worldvolume tori. In the Table belowwepresenta configuration keepingonly one potential tachyonic state that can play the role of the waterfall field:2 2 The following definitions are introduced: the discreteWilson linesin the dual lattice are expressed as Ak = akxR*kx + akyRk *y , with akx,aky . Q. The D7k brane position xk I. Antoniadis, O. Lacombe, G. K. Leontaris (45) (67) (89) D71 · .× D71 -. D72 ×·. D72 D73 .× · D73 (45) (67) (89) ·. ×A1 ×· ±x2 . .×A3 · We introduce discreteWilson lines along the thirdtorusT2 for the D71 stack and 3 along the second torus T2 for the D73 stack, while we separate the D72 stack from 2 its orientifold image in its transverse directions. Next, we denote the Ai tori areas ' (i = 1, 2, 3)as power fractions of the total volume Ai = ariV1/3, with r1r2r3 = 1 and Ui the corresponding complex structure moduli. Then, the masses for the doubly charged states in the three brane stacks are found to be [19] 2|k(2) 2 | a ' 2 11 am ˜ -+ , (2.33) 11 pr2V1/3 r3V1/3 2|k(3) | ' 22 + y2r2V1/3 am 22 ˜ - , (2.34) pr3V1/3 2|k(1) 2 | a ' 2 33 am ˜ -+ , (2.35) 33 pr1V1/3 r2V1/3 where a1, a3 and y2 are functions of the complex structure moduli Ui defined in2.By choosing appropriately a1, a3 with respect to the values of the magnetic fluxes |k(2) | and |k(1) |, one can eliminate the D71 -D71 and D73 -D73 tachyons. 13 For ai = 1/2, typical for Z2 orbifolds, this requires flux numbers smaller than wrapping numbers. On the other hand, the D72 -D72 state becomes tachyonic at andbelowa criticalvalueofthevolumethatcanbechosentobeinthevicinityof the minimum of the potential, as required for the waterfall field, denoted by f- in the following. We turn now to the scalar potential. The magnetic fields contribute through a D-term of the form X g2 X |fn |2 2 U(1)a n VD =+ q a+ ··· .aa 2 an 22 U(1)a U(1)2 = X g.2 + g.2 + 2|f+|2 -2|f-|2 + ··· 2 + ··· , (2.36) a 22 a=1,3 where in the second line contributions only from the tachyonic field and its charge conjugate are taken into account. Wehave also explained that the tachyonic scalar,coming from strings stretching be­tween the D72 brane stack and its image,mayreceivea positive mass contribution duetothebrane position.Intheeffectivefieldtheory,this contributionis described by a trilinear superpotential obtained by an appropriate N = 1 truncation of an N = 4 supersymmetric theory. The physical mass for the canonically normalised fields can be computed from the physicalYukawa couplings, derived from the xy xy as xk = x k Rkx + xk Rky with x k,xk . Q, while Rik · R*il = dlk. For later use, we also xy 4|x -iUx|2 |aky+iUakx|2 define yk(U)= kkand ak(U)= . Re(U) Re(U) 2 Type IIB moduli stabilisation, inflation and waterfall fields supergravity action, and can be expressed as [27] Wtach = Yijk fifjfk, where Yijk areYukawa coefficients expressedin termsof theK¨ ahler metrics of related matter fields. Their volume dependence can be worked out and the final form of the coupling is 1/2 .3 A2 Wtach = gs f2f+f-, (2.37) a 'V 2 which induces a scalar potential F-part of the form VF . m|f+|2 + |f-|2 x2 22 with m= y2(g/.2V)A2/a '. In addition to this mass-squared terms, the F-term x2 s scalar potential also contains quartic terms. They can be worked out and the leading term in the scalar potential for the tachyonic scalar is found to be of the 2 form VF . .2m|f-|4 . x2 The effective scalar potential includes the D-term and F-term contributions and its final form is achieved after the minimisation procedure whose details can be found in [19] . Neglecting, in particular, the massive f+ field, the scalar potential receives the simplified form C ln V -4 + q 3s 1.(V) 2 V(V,f-)=- -+ m (V)|f-|2 + |f-|4 , (2.38) .4 V3 2V2 2 Y4 2 where the explicit forms of the volume dependent mass m and quartic coupling Y . are given in terms of integers representing magnetic fluxes [19] and other string parameters. The final dependence of V(V,f-)on the two fields has been written in the form of the hybrid scenario [21] scalar potential. In this form it is even clearerthattheroleofthe waterfall fieldis playedbythe scalar field f- associated with the state stretching between the D72 brane and its orientifold image. Its mass 2 squared mdepends on the internal volume V, directly related to the inflaton, Y and turns negative when the internal volume acquiresa critical value.Awaterfall direction is thus generated, as in the hybrid scenario. This mechanism leads to a new lower minimum. It has been found [19] that when only a single tachyon is involved, the amount of reduction falls short to explain the observed value of dark energy of our Universe. This situation can be remedied within our model by introducing more tachyons, coming from the two other D7-brane stacks and from a fourth magnetised stack, parallel to one of the initial stacks. These additional tachyons contribute negatively to the scalar potential and are sufficient to achieve the present value of the cosmological constant. Apart from (or instead of) these contributions, one should of course expect new physics at low energies, leading to other phase transitions that affect the scalar potential. Hence, the precise tuning of the vacuum energy within our high energy model shouldberegarded asaproof of principle. 2.5 Conclusions In this presentation we have discussed aspects of perturbative corrections in the weak string couplingregime and large volume compactifications within the frameworkoftypeIIBstringtheory.Wehave consideredageometric configuration of intersecting D7-brane stacksand investigatedtheroleof logarithmic corrections I. Antoniadis, O. Lacombe, G. K. Leontaris which are present by virtue of local tadpoles induced by localised gravity kinetic terms. Such termsare generatedfromthe dimensionalreductionofthe R4 terms in the effective ten dimensional action and arise only in four spacetime dimensions. We have shown that in this string theory context, metastable de Sitter vacua can be ensured together withK¨ahler moduli stabilisation. Subsequently, we have examined the possibility of realising the mechanism of cos­mological inflation.We haveshown that the inflationary scenario canbe naturally implemented when the internal volume modulus is considered to be the inflaton field. The effective scalar potential contains only a single free parameter, whose value is fixed in order to meet the inflationary conditions and in particular the requirement of 60 e-folds which, in our construction, are collected near the mini­mum, while the horizon exit occurs near the infection point. These requirements, however, lead toa very shallow potential with its minimum much larger than the known value of the cosmological constant. Toresolve this discrepancy,we have suggested thata string versionof the hybrid inflationary scenario could be realised where possible waterfall fields could be identified with some of the charged string states stretching between the branes and their orientifold images. In the effective theory, the (volumed dependent) masses squaredof such excitations consistofpositive contributionsfrom brane separations and possible negative ones when worldvolume magnetic fields areturned on.With suitable conditions on various quantities such as magnetic fluxes and geometric characteristics, tachyonic states may appear. For illustrative purposes, we have presented a simple scenario where a tachyonic field arises, with its mass squared turning negative as soon as the internal volume acquiresa critical value. Thisis exactlywhatisrequiredfora waterfall field.More specifically,intheeffectivefield theory, states of the kind described above induce specific contributions to the F-and D-terms of the effective potential. When these contributions are included in the total scalar potential [19], the tachyonic field can indeed play the role of the waterfall field, providing in this way an explicit string realisation of the hybrid inflationary scenario. 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Proceedings to the 24th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ... (p. 20) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 3 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.Ye†5 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 Laboratory of Particle Astrophysics IHEP, Chinese Academy of Sciences, Beijing, PR China Abstract. Heretheresults obtainedbyanalysingothertwoannualcyclesofDAMA/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 s 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 s;the modulation amplitude of thesingle-hit scintillation events is: (0.01014±0.00074)cpd/kg/keV,the measured phase is (142.4±4.2)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 . 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 razli. cne konfig­uracije in vsebnosti merilcev ter poskrbeli za njihovo .cinkovitost.Vposkusu cistost in u . ... F. Montecchia alsoDip.diIng. Civilee Informatica, Universit‘adi RomaTorVergata, Rome, Italy † Z.P.Ye also Universityof Jinggangshan, Jiangxi, China DAMA/LIBRA–phase2(. 250kg visoko radijsko . cistega NaI(Tl)) uporabljajo drugo gen-eracijovfotopomno .cinkovitostjoin najsodobnej . zevalkz visoko kvantnou.so elektroniko, karjimje omogo.zali energijski prag,do kateregaso meritve . cilo, da so zni .se 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 s C.L.(stopnja zanesljivosti)v energijskem obmo .cju med cju(1–6) KeV.Venergijskem obmo.(2 -ze s poskusoma DAMA/NaI in DAMA/LIBRA–phase1, 6) KeV, kjer so podatki zbrani .paje C.L. (stopnja zanesljivosti) za polno izpostavljenost (2,86 ton × leto) enaka 13,7 s. 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 povzro. cili delci temne snovi. Noben drug dogodek, v zadnjih desetletjih so jih predlali kar nekaj, ni v skladu z izmerjenimi rezultati. 3.1 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 following requirements: 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 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 featuressuchas energyresolutionand acceptanceefficiency 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 annualcycles. 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 s = 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 (mostlyoutside 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 sealingsystem 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 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], allowusto controlandto maintaintherun­ning conditions stableata level better than1% alsoin DAMA/LIBRA–phase2(see e.g. Ref. [21, 23]). Duringphase2thelightresponseofthe detectorstypically 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. 3.2 Eight DAMA/LIBRA–phase2 annual cycles Table 3.1 summarizes the details of the DAMA/LIBRA–phase2 annual cycles including the last two released 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) (a -ß2)= 0.355 well different from 0.5 (i.e. the detectors were not being operational evenly throughout the year), cannotbe usedfor the annual modulationstudies;however,ithasbeenusedforother purposes[6,13].Thus(see Table 3.1) the considered annual cycles of DAMA/LIBRA–phase2are eight for an exposure of 1.53 ton×yr. The cumulative exposure, when considering also the former DAMA/NaI and DAMA/LIBRA–phase1, is 2.86 ton×yr. 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 efficiencyfornoiserejection 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. 3.2.1 The annual modulation of the residual rate In Fig. 3.1 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 Table 3.1: Details about the annual cycles of DAMA/LIBRA–phase2. The mean value of the squared cosine is a = .cos2.(t - t0). and the mean value of the cosine is ß = .cos.(t -t0). (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, (a -ß2), is . 0.5 fora detector being operational evenly throughout the year. DAMA/LIBRA–phase2 Period Mass Exposure (a -ß2) annual cycle (kg) (kg×day) 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 kg×day . 1.53 ton×yr 0.501 DAMA/NaI+DAMA/LIBRA–phase1+DAMA/LIBRA–phase2: 2.86 ton×yr 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–phase2residual rates of the single-hit scintillation events are reported in Fig. 3.2. 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. 3.1) have been fitted 2p 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 3.2.InTable 3.2 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 com­Fig. 3.1: 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 2p 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. 2p 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. 3.1. pleteness,werecall thata slight energy dependenceofthephase couldbe expected (see e.g. Refs. [38,58,59,61–63]),providing intriguing information on the natureof Dark Matter candidate and related aspects. Table 3.2: Modulation amplitude,A, obtained by fitting the single-hit residual rate of DAMA/LIBRA–phase2,asreportedinFig.3.1,andalso includingtheresidual ratesof the former DAMA/NaI and DAMA/LIBRA–phase1.It was obtainedby 2p 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. 2p A (cpd/kg/keV) T = . (yr) t0 (days) C.L. DAMA/LIBRA–phase2: 1-3 keV (0.0191±0.0020) 1.0 152.5 9.7 s 1-6 keV (0.01048±0.00090) 1.0 152.5 11.6 s 2-6 keV (0.00933±0.00094) 1.0 152.5 9.9 s 1-3 keV (0.0191±0.0020) (0.99952±0.00080) 149.6±5.9 9.6 s 1-6 keV (0.01058±0.00090) (0.99882±0.00065) 144.5±5.1 11.8 s 2-6 keV (0.00954±0.00076) (0.99836±0.00075) 141.1±5.9 12.6 s DAMA/LIBRA–phase1+phase2: 2-6 keV (0.00941±0.00076) 1.0 152.5 12.4 s 2-6 keV (0.00959±0.00076) (0.99835±0.00069) 142.0±4.5 12.6 s DAMA/NaI+DAMA/LIBRA–phase1+phase2: 2-6 keV (0.00996±0.00074) 1.0 152.5 13.4 s 2-6 keV (0.01014±0.00074) (0.99834±0.00067) 142.4±4.2 13.7 s 3.2.2 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. 3.3 shows the distribution of the percentage variations of R90 withrespecttothe mean valuesforallthe detectorsin DAMA/LIBRA–phase2. It shows a cumulative gaussian behaviour with s . 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 annual cycles (seeTable 3.3). This also excludes thepresenceof any background 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 s far away from the measured values. Fig.3.3: 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 3.3: 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 s)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 (0.12±0.14) (0.0032±0.0017) 3 -(0.08±0.14) (0.0016±0.0017) 4 (0.07±0.15) (0.0024±0.0015) 5 -(0.05±0.14) -(0.0004±0.0015) 6 (0.03±0.13) (0.0001±0.0015) 7 -(0.09±0.14) (0.0015±0.0014) 8 -(0.18±0.13) -(0.0005±0.0013) 9 (0.08±0.14) -(0.0003±0.0014) Similar results are obtained when comparing the single-hit residuals in the (1–6) keV with those in other energy intervals; for example Fig. 3.4 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). Fig. 3.4: 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 3.3 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. 3.5 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 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 wellcompatible 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. Fig. 3.5: 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 each class of events the data as collected in a single annual cycle and by using in both cases the same identical hardware and the same identical software procedures. The initialtimeofthefigureis takenonAugust7th. 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,nobackgroundprocessableto mimictheDM 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]). 3.3 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 Fig. 3.6: 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 the peaks are expected to fall for the (2–6) keV energy interval. In the frequency range far from the signal for the (2–6) keV energy region and for the whole (6–14) keVspectrum,theupperlimitofthe shadedregion(90%C.L.)canbe calculatedto be 10.8 (continuous lines, green on-line). eachtimebin,aproceduredetailedinRef.[64]hasbeen applied.Fig.3.6showsthe whole power spectra up to the Nyquist frequency and the zoomed ones: a clear peak correspondingtoa periodof1yearis evident for the 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]. Fig. 3.7: 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. 3.6 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 3.6.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 reported in Fig 3.6. 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. Their power spectrum in the frequency 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. 3.7. 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 correspondingtoa1year period,no otherpeak is statistically significant either in the low and high energy regions. 3.4 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 -µijk 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 µijk =[bjk + Si(Ek)]Mj.ti.E.jk. 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 energy bin, .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 ton×yr) are plotted in Fig.3.8;thedata below2keVreferonlytothe DAMA/LIBRA–phase2 exposure (1.53 ton×yr). 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 Fig. 3.8: Modulation amplitudes, Sm, for the whole data sets: DAMA/NaI, DAMA/LIBRA–phase1 and DAMA/LIBRA–phase2 (total exposure 2.86 ton×yr) 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 forthe 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 donefortheother 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-.Sm. effects. Therefore, the variable x = has been considered to verify that the s Sm are statistically well distributedin the16 energy bins(.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, s are the errors associated to Sm and .Sm. are the mean values of the Sm averaged over the detectors and the annual cycles for each considered energy bin. Defining .2 = Sx2, 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 ascribedto statistical fluctuations,letus ascribeittoa 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 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. 3.5 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) (3.1) = 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 cumulatively the data of DAMA/NaI, DAMA/LIBRA–phase1 and DAMA/LIBRA–phase2 the obtained 2s contours in the plane (Sm,Zm)for the (2–6) keV and (6–14) keV energy intervals are shown in Fig. 3.9–left while the obtained 2s contours in the plane (Ym,t *)are depicted in Fig. 3.9–right. Moreover, Fig. 3.9 also shows only for DAMA/LIBRA–phase2 the 2s contours in the (1–6) keV energy interval. The best fit valuesin the considered cases(1s errors) forSm versusZm and Ym versus t * arereportedinTable 3.4. Fig. 3.9: 2s 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 3.4: Best fit values(1s 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. 3.9. 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. (3.1) 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 energy intervals and the phase agrees with that expected for DM induced signals. No modulationispresent above6keV and the phaseis undetermined. 3.6 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.3.10.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. Fig. 3.10:As Fig. 3.8; 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 dedicated hardware upgrade of DAMA/LIBRA–phase2 is 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. 3.7 Conclusions DAMA/LIBRA–phase2 confirms a peculiar annual modulation of the single-hit scintillation events in the (1–6) keV energy region satisfying all the many require­ments of the DM annual modulation signature; the cumulative exposure by the former DAMA/NaI, DAMA/LIBRA–phase1 and DAMA/LIBRA–phase2 is 2.86 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 the DM signal; 5) the modulation is present only in the single-hit events, while it is absent in the multiple-hit ones as expected for the DM signal; 6) the measured modulation amplitude in NaI(Tl) target of the single-hit scintillation events in the (2–6) keV energy interval, for which data are also availableby DAMA/NaI and DAMA/LIBRA–phase1, is: (0.01014 ± 0.00074)cpd/kg/keV (13.7 s C.L.). No systematicorsideprocessesableto mimicthe signature,i.e.abletosimulta­neously 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 13.7s 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 increas­ingprecisionandwillallowustodisentanglewithlargerC.L.amongdifferentDM candidates, DM models and astrophysical, nuclear and particle physics scenarios. 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Sopin1 email: sopindo@mail.ru 1Southern Federal University, Russia, Institute of Physics, Russia 2 Laboratory of Theoretical Physics, JINR, Dubna, Russia Abstract. In the framework of hypercolor extension of the StandardModel having vector-like hyperquarks and two stable dark matter candidates originated from different hyper-currents, we consider some effects which result from reactions with participation of the dark matter components. Namely, there are decays of charged hyperpions into leptons and neutral component, annihilation andtransitions of heavy dark matter candidates into the light ones. In the last case, low energy photon radiation from intermediate charged states is possible. This type of the dark matter luminescence is analyzed in more detail. Povzetek:Avtorji uporabijo model, ki kvarkom in leptonom standardnega modela doda nove vektorske ”hiper kvarke”in dva kandidata za temno snov. . Studirajo razpade nabitih hiperpionov na nabite leptone in nevtrine, anihilacijo in prehode masivnih kandidatov temne snovi v manj masivne. Pri tem se lahko pojavi fotonsko sevanje nizkih energij, ki ga podrobneje analizirajo PACS: 12.60-i, 96.50.S-,95.35.+d. 4.1 Introduction Experimental evidence of symmetry breaking mechanism in the StandardModel (SM) and the existence of Higgs bosons with the properties predicted have even worsened, in some sense, the situation in high-energy physics. From this time the SM has acquired the features of a complete and closed theory, without being such. This obviously means that the SM is only a kind of limit of a more general theory that should give solutions of open problems of the SM. Unfortunately, the inspiring idea of supersymmetry (SuSy) does not manifest itself in experiments at a scale of ~ 1 TeV, whichreduces the potentialof the theory, althoughit does not completely close it. Then, the search for ideas to solve the SM problems leads to analysisof the extension options for the SM.Alotof variantsof the generalization Title Suppressed Due to Excessive Length and extension of the SM have been discussed in literature, in particular, those that offer different ways of explaining the structure and properties of dark matter (DM). At the moment, the DM existence is a firmly established fact, which is confirmed by many astrophysical observations. About a quarter of the universe mass is the DM and it plays a crucial role in the evolution of galaxies. Despite the fact that the existenceof this substance has been known for an almosta century, and that it literally surrounds us, we still do not have the faintest idea what it is. All that we have is a set of hypothetical essences for its explanation, they are neutralinos from SuSy, axions, sterile neutrinos, inert Higgses, primordial black holes, manifestations of modified dynamics and last but not list WIMPs. The most popular option is considered to be the WIMPs and they will be discussed in this work. It should be added that all efforts to catch the DM particles directly do not successful up to now, and indirect methods to see some and measure any signals of processes with the DM participation become much more important [1–7]. Multicomponent models of the DM have become the object of attention and study mainlyinrecent decade[8,9] because,onthe one hand, various variantsofthe SM extension were proposed suggesting some possible types of the DM carriers. On the other hand, there are unexplained astrophysical phenomena that can be better interpreted and explained within the frameworkofa multicomponentDM scenario. These phenomena are, in particular, monochromatic photon signals of unknownoriginfromGalaxy centerwith energiesuptotensofGeV[10,11]and some featuresof spectraof cosmic leptons (positron excess, for instance) [13,31]. In accordance with these two aspects, it is possible to divide the proposed DM variants into some two classes. In the first one, a multicomponent DM is resulted froma specific symmetry, which extendsandgeneralizes theSMgroupof symme­try and consequentlyintroduces some additional degreesoffreedomproviding stability for the part of them. What manifestations and specific effects can be in-ducedbythese new particles -it dependsonthepropertiesofthe model symmetry and interactions. But the second type of scenarios introduces new Dark Matter candidates aiming the explanation of the observed physical phenomena, for exam­ple, the positron fraction excess in cosmic-ray spectra (leptophilic models [14, 15]) ortointerpretphotonsignals(gamma emissionfrom Galactic centerregion)asthe result of annihilation of DM particles. Certainly, to stabilize new objects the initial SM symmetry should be also modified, for instance, using discrete symmetries. Namely, imposingaZN symmetry(it canresultfrom spontaneouslybrokenU(1)) provides simultaneously an existence of several stable scalar fields as the DM candidates [16]. As another example, the renormalizable extension of the SM by a scalar, pseu­doscalar and a singlet fermion fields is considered where the DM has a fermion and a scalar components [17, 18]. It allows to explain photon signal with energy in a keV region by the light scalar decay, and 130-GeV photons emerging, for example, as a consequence of heavy fermions annihilation. Another approach to introduce and use a two-component Dark Matter is to add a neutral Majorana fermion and a neutral scalar singlet interacting with the SM fields through the Higgs portal. Fermion, however, interacts at the tree level asYukawa particle. V. Beylin,V. Kuksa,M. Bezuglov,D. Sopin And again, in various regions for the mass of the scalar, the photon signal can be interpreted as result of the scalars annihilation [19]. As the most obvious case, some co-existence of axions with neutralino or wino [20, 21] allows to keep the SuSy scale near 1 TeV(however, it is difficult to provide the necessary value of the DM relic density). There also suggested also an interesting way to use Exceptional Supersymmetric StandardModel (E6SSM) [22], where two DM components arise from the set of Higgs superfields due to discrete symmetries again. The DM candidates can be built from any suitable ”matter” -additional scalars, fermions, even strongly interacted objects, Higgses portal, dark atoms, the DM can be presented by elementary particles or some compound states. In any case, these DM candidates should be neutral and stable, and channels and steps of their production in direct experiments at colliders together with their indirect manifestations in astrophysics phenomena or observed by space telescopes and at ground observatories (IceCube, LHAASO etc.) are analyzed carefully in a lot of papers[23–27,29–31]andalsoin[32,33,35–44]. Certainly,all possible scenarios consider two aspects of the DM physics: theoretical validity and self-consistency of the model, and (qualitative and quantitative) description of the observed specific effects. An existence of several DM components substantially increases the number of reactions with the DM participation and allows to predict some interesting chan­nels of its manifestations. The most important for such predictions is the structure of the DM sector in the model and features of the dark matter interactions with the SM particles and between the DM components.To clarify this possibility, we consider hypercolor model with additional heavy fermions (hyperquarks) in confinement, which can produce a set of composite states, hyperhadrons, in the framework of s-model at some high scale [45–48] in an analogy with low-energy quark-meson theory. So, in the model a number of pseudo-Nambu-Goldstone (pNG) particles emerges, they acquire masses after the chiral symmetry breaking. There arise fifteen pNG states (and their chiral partners) which are connected with corresponding H-quark currents. The model includes almost standardHiggs boson which is (slightly) mixed with scalar s-meson. Specifically, the model contains several neutral stable particles they can be inter­preted as the DM candidates. In more detail, these states and their main charac­teristics will be presented in Section 2. Section3will be devoted to analysis of some new effect induced by the complex structure of the DM sector -radiation of photons in transitions of one of the DM component into another one. In Conclusion we discuss some possible application of this effect. 4.2 Hyperquark scenario: features of the dark matter sector TheSM content and possibilities canbe extendedby introducinga new fermion sector in confinement using, for example, an extra gauge symmetry SU(2)tc. Be­sides, an additional SU(2)w symmetry should be to ensure electroweak interaction of new fermions (H-quarks) with the SM fields. Then, the hypercolor model in its minimal form contains only one doublet (with zero hypercharge) of heavy Title Suppressed Due to Excessive Length Dirac H-quarks doublet and keeps the standardHiggs boson. It, however, will mix with scalar s˜-meson generated by extra singlet scalar field, which is necessary for a spontaneous symmetry breaking. As a result, the new fields can acquire masses. Note, the mixing between scalars should be small to ensure the stability of precisely measured SM parameters (in other words, oblique corrections of Peskin Tackeuchi should be sufficiently small). In an analogy with the low-energy hadron QCD-based theory, H-quarks should form H-hadrons, which can be described in the H-s- model with an effective Lagrangian. So, there arises (due to global SO(4) symmetry breking) a set of pseudo-Nambu-Goldstone (pNG) states: a triplet of pseudoscalar H-pions and one neutral H-baryon along with its antiparticle. More exactly, H-baryon is a diquark state having an additive conserved quantum number, H-pions possess a multiplicative conserving quantum number [48,49]. Importantly, neutral states, p˜0 and B0 ,B0,arestable. Consequently,they canbe interpretedastheDM candidates ¯with equal masses at the tree level. Because we are mostly interested to analyze the stable candidates properties, we do not consider heavier unstable H-hadrons and H-mesons (see, however, study of their mass spectrumat lattice[50,51]inthe same gauge Sp(4)theory). Then, we need to know tree-level masses of lowest states (H-pions and H-baryon), mass of H-sigma and its v.e.v.; all of them are supposed to be O(TeV). The angle of mixing, ., between H-sigma and the Higgs boson, asit dictatesby Peskin -Tackeuchi parameters for the model, should be such that sin . . 0.1. There were calculated both the mass splitting between neutral and charged states in H-pion triplet (induced by electroweak loops only) and between the lowest states of different origin (H-pions and H-baryons). In the last case, the mass splitting depends on some renormalization scale because of different H-quark currents generating these H-states, so, the mass splitting can be as much as tens of GeV. Electroweak mass splitting in H-pions triplet is well known and it is ˜ 160 GeV. So, in this minimal scenario there are three stable particles possibly constituting dark matter: neutral stable H-baryon along with its antiparticle (we will consider them as the one component) and the lightest neutral H-pion. To estimate their masses, there was used known way of analysis of the dark matter density evolution to its modern value. Namely, there were written down five equations for each DM component taking into account charged H-pions that are decayed eventually into neutral one, i.e. so-called, co-annihilation processes were also considered for H-baryons and H-pions. Numerical solution of the system of equations demonstrates that the correct value of the DM abundant corresponds to some areas in the parameter plane-of H-pions and H-sigma mases. Despite of the DM candidates masses estimation (approximately, they are in the region 0.8 -1.2 TeV), it was found that in all permitted areas of parametersB0-baryons dominate in the DM density [52]. The reason of this asymmetry for the DM components contributions into the total density follows from asymmetry of their interactions with the SM matter: H-pions have EW channels, but H-baryons do not participate in tree level EW interactions, they use (pseudo)scalar exchanges through Higgs boson and/or s˜-meson instead. It is an important consequence of different origin for these DM components providing slower burnout of B0 component in comparison with H-pion component. Remind, both DM components were considered initially as having equal masses, mass splitting in the H-pion triplet is defined only EW loops and it is small. However, one-loop mass splitting between p˜0 and B0 can be as high as 10 -15 GeV depending on s˜- meson mass and value of renormalization parameter, µ. Again, it is due to the different structures of H-quark currents with which these components are associated. Corresponding mass splittings are demonstrated in Fig.1a,b. Note, mass of s˜-mesonis near the value whichis dictatedby relation 22 m˜ 3 · mresulted from zero H - s ˜mixing. s ˜p ˜ a) b) Fig.4.1:Mass splitting betweenDM componentsin dependenceonrenormaliza­tion scale: a) B0 is heavier;b)˜p0 is heavier. For nonzero mass splitting between the components, there occur an interesting process of the heavier DM component transformations into the lighter one. It can result to some effects, which are specific for suitable scenarios of multicomponent DM. Here, we will consider the case when B0 is heavier than p˜0, it is that the scenario when the tree level process of annihilation of heavy B0B0 pair into H-pions can be accompanied with some final state radiation (FSR). 4.3 Transitions between DM components and an effect of luminescence To discuss possible.-radiation in the transitions between dark matter compo­nents, we firstly analyzed the ratio of cross section of B0B0 pair annihilation into H-pions (see diagrams in Fig.2a) to the total cross section of B0B0 pair annihilation into all possible SM final states. Denoting this ratio as a, we consider its values in the p ˜- s ˜plane for various sets of model parameters: scale of renormalization (it also determines the mass splitting between B0 and p ˜states), mixing angle, and the vacuum shift fora heavy scalar field. Some regions of a values are shown in Fig.3; in all cases it is possible to find an areas where a parameter is sufficiently large, a = 10. Fortunately, in these regions H-pions and B0-H-baryons masses are ~ 1 TeV, as it also follows from kineticsoftheDMburnout.Thus,itis possibletofixasuitable intervalofthe DM components and the sigma meson masses, at which the BB-pair transition into charged unstable H-pions dominates. Title Suppressed Due to Excessive Length a) b) Effect of FCR occurs just in the reaction B0B0 . p˜+ p˜- + . (see diagrams in Fig. 2b) with subsequent decays of charged H-pions, p˜+ . p˜0 + l.l. These charged states decay through strong and EW channels producing neutral stable p˜0 and pair of lepton plus (anti)neutrino; corresponding widths [48] are: G(p˜± . p˜0p±)= 6 · 10-17 GeV,tp = 1.1 · 10-8 s, ctp ˜ 330 cm; G(p˜± . p˜0l±.l)= 3 · 10-15 GeV,tl = 2.2 · 10-10 s, ctl ˜ 6.6 cm. (4.1) Now, for the differential cross section we get the following expression: dsv(B0B0 . p˜+ p˜-.) 4aesv(B0B0 . p˜+ p˜-) = · (4.2) dE. M22 pMBE.-m Bp ˜ -2MB(MB -E.)MB(MB -E.)-m2 + p˜ 2MB(MB -E.) 2 (2MB(E. -MB)+m )log[-1]. p˜MB(MB -E.)+MB(MB -E.)-m2 p ˜ So, possibility of radiation from (unstable) charged components (of H-pion triplet) is a specifics of the SM extensions with a complex structure resulting to the V. Beylin,V. Kuksa,M. Bezuglov,D. Sopin multi-component DM. If there are suitable channels of interaction, heavier DM component can transform into the light one, but for the FSR (or virtual internal bremsstrahlung) an occurrence of this transition should have an intermediate stage with some charged states. Here is exactly the same case. Cross sections for different values of the DM component masses, mixing angle, mass of s˜-meson and scale of H-symmetry breaking are presented in Fig.5 and 6. Here are shown also total cross sections demonstrating an obvious s-channel resonance near M s ˜in Fig. 7. . With an integration of the differential cross section from energies~ (0.1 -0.2)GeV up to 2 · .m, we have found the total cross section, its values are shown in Fig.5 in dependence on M s ˜for various mass splittings and masses of B0. Obviously, there are some features of the effect considered. First, the resonance structure is manifested at M s ˜due to s-channel contributions, and the total cross section is prac­tically independent on s˜-meson mass when its value is = (2.0 -2.5)TeV.We also estimate total flux of photons using stot values between 10-28 and 10-26 cm3/s and the Navarro-Frank-White profile for the dark matter density, .NFW(l, ., .). We used also known astrophysical J-factor forthe Galaxy center, namely,we take the angularresolution ~ 1. and the value of J ˜ 10-21 . Title Suppressed Due to Excessive Length Fig.4.5:Totalcross sectionfortheprocessif transition betweentheDM components with FSR, values of parameters are depicted in figures. Then, the values of total gamma-flux of low-energy photons produced by transi­tions between the dark matter components near Galaxy center are the following: -2 -1 F(E.)˜ (0.9 -1.5)· (10-14 -10-12)cm s . (4.3) However, J-factor for the Galaxy center can increase up to an order or two if the parameter . inNFWprofile changesfrom1to1.4to simulatetheDMspikeinthe DM distribution near the GC. Then, the flux also increases up to two orders. In this minimal scenario the mass splitting is much lower than the DM masses, so, there arises diffuse photons with energies in a narrow limited area. Certainly, these photonsareonlyan admixturefor(monochromatic) radiationfromtheDM annihilation into photon pairs. This luminescence of the DM is, however, too small to explain the whole excess of GeV photons from GC. Note also that scenarios witha complexDM sector structure shouldbe analyzed carefully in the case of .m ~ MDM: the DM candidates can be freezed out at different temperatures, so, they can be produced at different stages and contribute separately to features of evolution processes. Of course,thepossibleeffectofsmallphotonicfluxfromregionswiththeincreased DM density is specific because it does not lead to the resorption of dense DM clumps.Total mass and the particle number density does not change practically V. Beylin,V. Kuksa,M. Bezuglov,D. Sopin in this process. Indeed, there takes places also an ”ordinary” annihilation of DM components into two photons orinto pairsofSM particles with subsequent photon emission from final or intermediate bosons, leptons and quarks. However, monochromatic photons with energies of the order of the DM masses areseparated by an energy gap in the full spectrum of emitted photons. Unfortunately,a large backgroundisproducedby diffuse FSRfrom theSM parti­cles; the total gamma flux can be noticeably larger than the indicated effect. So, analysis of the photons spectrum at GeV energies is a difficult task. Indeed, detection and selection of a (nearly constant) photonic component with energies of the order of (1 -10)GeV can indicate the presence of some structure in theDM mass spectrum, or the possibilityoftransitions between exited levelsin the spectrum of states, as it can occur in the hadronic DM scenario [53]. This specific radiation also should be collimated with a some (small) angular aperture if it comes from some “point sources” – GC, dwarf galaxies, subhaloes or other types of DM regions with high density. If the DM clump not very far (~ 0.1 pc)from our space telescopes, the low-energy limited flux of photons can be seen andrecognized. Note, an inverse case when the neutral H-pions are heavier also should be con­sidered, however, annihilation into the (lighter) B0-components with radiation of photons is difficult to ensure in this case -diffuse photons production takes place mostly due to VIB from H-pions and/or H-quarks loops and corresponding cross section should be smaller. 4.4 Conclusion As some additional considerations, it should be noted that stable DM candidates can be produced from H-quark-gluon plasma at early stages at large temperatures. Besides, due to high scale of H-vacuum condensates, H-hadronization should occur before the QCD hadronization, so the photons from transitions between various H-states can contribute significantly to total density of radiation. This process can maintain the plasma temperature as a kind of delay mechanism that prevents cooling during expansion, in accordance with the Le Chatelier principle. This type of annihilation induced by transition between the DM components, is interesting also from the point of view of the DM accumulation inside massive objects – red giants, white dwarfs and the possible dark stars at early stage. In this case, photons, leptons, and neutrinos generated during the transition between components will heat up the interior of the gravitationally coupled system more slowly than the annihilationofDM intoSM particles woulddo (thisreaction,of course, also takes place, but witha noticeably smaller cross section for some model parameter values). In the case, the dark star life time in the relatively “cold” state should icrease. Moreover, if such reactions dominate, the gravitating mass of the object also will changes slowly. Energies of the photons emitted from such objects will be distributed over significantly different regions separated by a gap of the order of the DM mass. Such an analysis would be reasonable for (early) dark stars with long lifetimes. Their thermonuclear heating is actually replaced by an energy Title Suppressed Due to Excessive Length release during the annihilationofDM particles. The discussedeffect showsthat thepresenceofa specific complex structureofDM states canbe important for the dark stars study. Namely, the luminosity of dark stars can be provided also by low-energy component whichis inducedby transitions between theDM objects within the stars. Thus, it was considered the scenarioin which one DM component can be ef­fectively transform into another through intermediate stage of charged H-pions production and decay. Certainly, it is possible an annihilation of B0B0 pair into standardquarks and gauge bosons (via scalar exchanges or loops), but it turned outthat thereisaregionof parameters where the cross section for annihilation into hyperpions dominates. Note, the lifetime of charged H-pions is larger than the lifetime ofgauge bosons. So, the emission of photons from intermediate charged states, in principle, could be observed. The (small) flux of photons is proportional to the squared DM density, so the most interesting should be to study intensity of such radiation from the GC (from where an increased flux of low-energy photons is observed) or from probable DM clumps. Note also, the DM number densityin these processes does not change, intermediate charged H-pions produce neutral stable H-pions together with the low-energy secondaries such as leptons and neutrinos. Apart from the low-energy characteristic radiation with small flux, the effect obviously leads to the burn out of heavier DM component transforming it to another one. This process, however, is very slow, since the DM concentration is low. BothDM components are practicallyin equilibrium, so thata small changein their concentrations is hardly noticeable. 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(p. 53) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 5 Numerical simulation of Bohr-like and Thomson-like dark atoms with nuclei T.E. Bikbaev1 M.Yu. Khlopov1,2,3 e-mail: khlopov@apc.univ-paris.fr A.G. Mayorov 1 National Research Nuclear University MEPhI, 115409 Moscow, Russia 2 Institute of Physics, Southern Federal University, Russia, Stachki 194 Rostov on Don 344090, Russia 3 Universit´ede Paris, CNRS, Astroparticuleet Cosmologie, France; F-75013 Paris, France Abstract. The puzzles of direct dark matter searches can be solved in the scenario of dark atoms, which bind hypothetical, stable, lepton-like particles with charge -2n, where n is any natural number, with n nuclei of primordial helium.Avoid experimental discovery because they form with primary helium neutral atom-like states OHe (X – helium), called ”dark” atoms. The proposed solution to this problem involves rigorous proof of the ex­istence of a low-energy bound state in the dark atom interaction with nuclei. It implies self-consistent account for nuclear attraction and Coulomb repulsion in such an interaction. Weapproachthe solutionofthisproblemby numerical modelingtorevealthe essenceof theprocessesof dark atom interaction with nuclei.We start with the classical three-body problem, to which the effects of quantum physics are added. The numerical model of the dark atom interaction was developed for O-- having a charge of -2, bound with He in Bohr-like OHe dark atom and for -2n charged X bound with na-particle nucleus in the Thomson-like atom XHe. The development of our approach should lead to the solution of the puzzles of direct dark matter searches in the framework of dark atom hypothesis. Povzetek:Rezulate neposrednih meritev temne snovi je mogo.ziti s temnimi atomi, ce razlo.ki pove.cne, stabilne, leptonu podobne delce z nabojem -2n, zejo hipoteti.n je poljubno nar­avno. stevilo, z n jedri helija v nevtralne atome, ki jih avtorji imenujejo ”temni atomi” OHe. Znumeri.stevanjem kvantnih popravkov cnim modeliranjem skupka treh teles in z upo.is..ca in so stabilna. Za. cejo dokaz, da so vezana stanja takih ” temnih atomov” mogo.celi so s studijem.” temnega atoma” O-- , z nabojem -2, ki se pove. zes helijevim jedromv OHe, zdaj.studirajo tudi kompleksnej.se ”temne atome”. PACS: 02.60.-x; 02.70.-c; 12.60.-i; 36.10.-k; 98.80.-k 5.1 ”Dark” atoms XHe If dark matter consists of particles, then they are predicted beyond the Standard Model. In particular, it is assumed that stable, electrically charged particles can exist [1–3]. Stable negatively charged particles can only have a charge of -2 – we T.E. Bikbaev, M.Yu. Khlopov, A.G. Mayorov will denote themby O-- (in the general case -2n, where n is any natural number, we will denote them by X)[4]. In this paper, we investigate a composite dark matter scenario [5–7]. Hypothetical stable O--(X)particles avoid experimental discovery because they form neutral atom-like states OHe (X –helium) with primordial helium called ”dark” atoms [8]. Since all these models also predict the corresponding +2n charged antiparticles, the cosmological scenario should provide a mechanism for their suppression, which, naturally, can take place in the charge-asymmetric case, corresponding to an excess of -2n charged particles [1]. The electric charge of the excess of these particles is compensated by the corresponding excess of positively charged baryons. So the electroneutrality of the Universe is preserved. Hence, positively charged antiparticles can effectively annihilate in the early universe. There are various models predicting such stable -2n charged particles [9–11]. A”dark” atomisa system consistingof-2n charged particles (in the case n = 1, this is O--), bound by the Coulomb force with n 4He nuclei. The structure of bound state depends on the value of a ˜ ZaZoaAmpRnHe parameter, where a is fine structure constant, Zo and Za – are the charge numbers of particle X and n nuclei of He, respectively, mp – is the proton mass, A is the mass number of n –nucleus He, and RnHe is the radius of the corresponding nucleus. For 0RHe, R3 . FXHe(RXHe) XHe = 4e2n2 (5.2) - .RXHe for RXHe 14 GeV2 for the |[ud]u. twist-3 Fock state with orbital angular momentum L = 0 and Q2 > 16 GeV2 for the later onset of CT for its L = 1 twist-4 component. See fig. 6.11 Note that LF holography predicts equal quark probability for the L = 0 and L = 1 Fock states. Color transparency is thus predicted to occur at a significantly higher Q2 for baryons (Q2 > 14 GeV2), than for mesons Title Suppressed Due to Excessive Length (Q2 > 4 GeV2). This is consistent with a recent test of color transparency at JLab which has confirmed colortransparency for the the p and . [74]; however, the measurements in this experiment are limited to values below the range of Q2 where proton color transparency is predicted to occur. Fig. 6.12:Two-stagecolor transparency and transmissionprobabilityof theproton ina nuclear mediumfromLF Holography. Remarkably,color transparency for the production of an intact deuteron nucleus in eA . d + X(A-2) quasi-exclusivereactions shouldbe observedat Q2 > 50 GeV2 . This can be tested in ed . ed elastic scattering in a nuclear background. It has been speculated [75] that the “Feynman mechanism”, where the behavior of the struck quark at x ~ 1 in the proton LFWF plays a key role for hardexclu­sive hadronic processes does not predict color transparency. However, LF wave 22 P .k+m .ii functions are functions of the invariant mass so that their behavior at ixi large k. and large x are correlated. Thus color transparency occurs for scattering amplitudes involving both the large transverse momentum and large x domains. The three-dimensional LF spatial symmetry of LFWFs also leads to the exclusive-inclusive connection,relatingthe countingrulesforthe behaviorofform factorsat large Q2 and structure functions at xbj . 1. S. J. Brodsky 6.9 Removing Renormalization Scale Ambiguities 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) [76], which generalizes the conventional Gell-Mann-Low method 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 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. The key problem in making precise perturbative QCD predictions is the uncer­tainty in determining the renormalization scale µ of the running coupling as(µ2). The purpose of the running coupling in any gauge theory is to sum all terms involving the ß function; in fact, when the renormalization scale is set properly, all non-conformal ß .0 terms in a perturbative expansion arising from renor- = malization are summed into the running coupling. The remaining terms in the perturbative seriesarethen identicaltothatofa conformal theory;i.e.,the corre­sponding theory with ß = 0. 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 µ0. The PMC thus sums all of the non-conformal terms associated with the QCD ß function, thus providing a rigorous method for eliminating renormal­ization scale ambiguities in quantum field theory. Other important properties of thePMCarethattheresulting seriesarefreeofrenormalonresummationproblems, 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 method used in lattice gauge theory. The number of active flavors nf in the QCD ß function is also correctly determined.Wehavealso showedthatasingleglobalPMCscale,validatleading order, can be derived from basic properties of the perturbative QCD cross section. Wehavegivenadetailed comparisonofthesePMCapproachesbycomparingtheir predictions for three important quantities Re+e, Rt and GH.b ¯ up to four-loop b pQCD corrections [76]. 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 Title Suppressed Due to Excessive Length pQCD predictions with less residual scale dependence. The PMC thus greatly improves the reliability and precision of QCD predictions at the LHC and other colliders [76]. 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. + in ee- annihilation, using the PMC to set the pQCD renormalization scale vs. conventional methods. An essential property of renormalizable SU(N)]/U(1) gauge theories, is “Intrinsic Conformality,” [77]. 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 general properties of the PMC.We have also introduced a new method [77] to identify the conformal and ß terms which can be applied either to numerical or to theoretical calculations and in some cases allows infi­niteresummationof the pQCD series,The implementationof the PMC8 can significantly improve the precision of pQCD predictions; its implementation in multi-loop analysis also simplifies the calculation of higher orders corrections in a S. J. Brodsky 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 s to approximately1 s. The PMC has also been used to precisely determine the QCD running coupling constant as(Q2)over a wide range of Q2 v from event shapes for electron-positron annihilation measured at a single energy s [78]. The PMC method has also been applied to a spectrum of LHC processes including Higgsproduction,jetshapevariables,and final states containingahigh pT photon plus heavy quark jets, all of which, sharpen the precision of the StandardModel predictions. 6.10 Is the Momentum Sum RuleValid for Nuclear Structure Functions? Sum rules for deep inelastic lepton-hadron scattering processes are analyzed using the operator product expansion of the forwardvirtual Compton amplitude, assuming it depends in the limit Q2 .8 on matrix elements of local operators such as the energy-momentum tensor. The moments of the structure function and other distributions can then be evaluated as overlaps of the target hadron’s light-front wave function, as in the Drell-Yan-West formulae for hadronic form factors[17,79–81].TherealphaseoftheresultingDIS amplitudeanditsOPE matrix elements reflects the real phase of the stable target hadron’s wave function. The “handbag” approximation to deeply virtual Compton scattering also defines the “static” contribution [82,83] to the measured parton distribution functions (PDF), transverse momentum distributions, etc. Theresulting momentum, spin and other sumrulesreflectthepropertiesofthehadron’s light-front wavefunction. However, final-state interactions which occur after the lepton scatters on the quark, can give non-trivial contributions to deep inelastic scattering processes at leading twist andthus survive at high Q2 and high W2 =(q + p)2 . For example, the pseudo-T­odd Sivers effect [84] is directly sensitive to the rescattering of the struck quark. Similarly, diffractive deep inelastic scattering involves the exchange of a gluon after the quark has been struck by the lepton [85]. In each case the corresponding DVCS amplitude is not given by the handbag diagram since interactions between the two currents are essential. These “lensing” corrections survive when both W2 and Q2 are large since the vector gluon couplings grow with energy. Part of the phasecanbe associatedwithaWilsonlineasan augmentedLFWF[86]whichdo not affect the moments. ' The cross section for deep inelastic lepton-proton scattering lp . lp ' X includes a diffractive deep inelastic (DDIS) contribution in which the proton remains intact witha large longitudinal momentum fraction xF > 0.9 greater than 0.9 and small transverse momentum. The DDIS events, which can be identified with Pomeron exchange in the t-channel, account for approximately 10% of all of the DIS events. Diffractive DIS contributes at leading-twist (Bjorken scaling) and is the essential Title Suppressed Due to Excessive Length component of the two-step amplitude which causes shadowing and antishadow­ing of the nuclear PDF [87–90]. It is important to analyze whether the momentum and other sum rules derived from the OPE expansion in terms of local operators remain valid when these dynamicalrescattering correctionstothenuclearPDFare included. The OPE is derived assuming that the LF time separation between the virtual photons in the forwardvirtual Compton amplitude . * A . . * A scales as 1/Q2. However, the propagation of the vector system V produced by the diffrac­tive DIS interaction on thefront face andits inelastic interaction with the nucleons in the nuclear interior V + Nb . X are characterized by a longer LF time which scales as 1/W2. Thus the leading-twist multi-nucleon processes that produce shad­owing and antishadowingina nucleus are evidently notpresentin the Q2 .8 OPE analysis. Thus, when one measures DIS, one automatically includes the leading-twist Bjorken-scaling DDIS events as a contribution to the DIS cross section, whether or not the final-stateproton is explicitly detected. In such events, the missing momentum fraction in the DDIS events could be misidentified with the light-front momentum fraction carried by sea quarks or gluons in the proton’s Fock structure. The underlying QCD Pomeron-exchange amplitude which produces the DDIS events thus does not obey the operator product expansion nor satisfy momentum sumrules–thequarkandgluon distributions measuredinDIS experimentswillbe misidentified, unless the measurements explicitly exclude the DDIS events [88,91] The Glauber propagation of the vector system V produced by the diffractive DIS interaction on the nuclear front face and its subsequent inelastic interaction with the nucleons in the nuclear interior V + Nb . X occurs after the lepton interacts with the struck quark. Because of the rescattering dynamics, the DDIS amplitude acquires a complex phase from Pomeron and Regge exchange; thus final-state rescattering corrections lead to nontrivial “dynamical” contributions to the measured PDFs; i.e., they involve the physics aspects of the scattering process itself [92]. The I = 1 Reggeon contribution to diffractive DIS on the front-face nucleon leads to flavor-dependent antishadowing [89, 93]. This could explain why the NuTeV charged current measurement µA . .X scattering does not appear to show antishadowing in contrast to deepinelastic electron nucleus scattering [90]. Again, the corresponding DVCS amplitude is not given by the handbag diagram since interactions between the two currents are essential to explain the physical phenomena. It should be emphasized that shadowing in deep inelastic lepton scattering on a nucleus involves nucleons at or near the front surface; i.e, the nucleons facing the incoming lepton beam. This geometrical orientation is not built into the frame-independent nuclear LFWFs used to evaluate the matrix elements of local currents. Thus the dynamical phenomena of leading-twist shadowing and antishadowing appearto invalidatethesumrulesfor nuclear PDFs.The samecomplications occur in the leading-twist analysis of deeply virtual Compton scattering . * A . . * A on a nuclear target. S. J. Brodsky 6.11 Summary Light-Front Hamiltonian theory provides a causal, frame-independent, ghost-free nonperturbative formalism for analyzing gauge theories such as QCD. Remarkably, 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 e2 , 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 as(Q2)in the nonperturbative domain. When one introduces super-conformal algebra, theresult is “Holographic LF QCD” which not only predicts a 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 - 3C LF bound-state equation. Thus tetraquarks are compact ¯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-exclusivereactions traversinga nuclear medium and asymmetric intrinsic heavy-quark distributions Q(x).Q(x), appearing at high = ¯ x in the non-valence higher Fock states of hadrons. 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Schmidt andM.D. Sievert, “Single-Spin Asymmetries in Semi-inclusive Deep Inelastic Scattering and Drell-Yan Processes,” Phys. Rev.D 88, no.1, 014032 (2013) doi:10.1103/PhysRevD.88.014032 [arXiv:1304.5237 [hep-ph]]. 93. S. J. Brodsky and H. J. Lu, “Shadowing and Antishadowing of Nuclear Structure Functions,” Phys. Rev. Lett. 64, 1342 (1990) doi:10.1103/PhysRevLett.64.1342 Proceedings to the 24th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ... (p. 94) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 7 Charge asymmetry of new stable quarks in baryon asymmetrical Universe A. Chaudhuri1 email: arnabchaudhuri.7@gmail.com M.Yu. Khlopov2 email: khlopov@apc.in2p3.f 1 Department of Physics and Astronomy, Novosibirsk State University, Russia 2 Instituteof Physics, Southern Federal University, Russia; Universit ´ e de Paris, CNRS, Astroparticule et Cosmologie, France; and National Research Nuclear University ”MEPHI”, Russia Abstract. Effects of electroweak phase transition (EWPT) in balance between baryon excess and the excessofstable quarksof new generationis studied.With the conservationof SU(2) symmetry and other quantum numbers, it makes possible sphaleron transitions between baryons, leptons and newof leptons and quarks.Adefiniterelationship between the excess relative to baryon asymmetry is established. In passingby we also show the small, yet negligible dilution in the pre-existing dark matter density due the sphaleron transition. Povzetek:Avtorji privzamejo model, ki dopus.. ca poleg doslej izmerjenih kvarkov in lep­tonov tudi novo dru .studirajo v elektro . zino kvarkov in leptonov.Vtem modelu .sibkem prehodu ravnovesje med prese.zno prve dru . zkom barionov prete .zine kvarkov in leptonov in med barioni privzetedru.cju,ko se .stevila zine kvarkov.Vobmo.se ohranjajo vsa kvantna .sistema,.cinkeprehoda sfalerona med kvarkiin leptoni standardnega modela studirajou .in kvarki in leptoni privzete dru. zine ter vpliv prehoda na gostoto temne snovi. Opazijo majno, skoraj zanemarljivo, razred. citev njene gostote. 7.1 Introduction The matter-antimatter asymmetry, otherwise known as the baryon asymmetry of the universe (BAU) has been the focus of physicists for many a decade [1–3]. Various models have been developed to answer the question, ranging from the grand unified theory (GUT) to electroweak phase transition (EWPT). Irrespective of the mechanisms, the preexisting asymmetry is diluted by the baryon number violating mechanisms in the electroweak theory. This is due to the violation of the baryon and lepton number and the non-trivial topological structure of the Yang–Mills theory. The possibility to convert baryons into anti-leptons and the reverse exists in electroweak theory. The difference between the baryon and lepton numbers (B-L) is conserved, even though individually, the quantum numbers are violated. Hence, it is important to know about the transition rates of such processes. Sphalerons are generally associated with saddle points [4], and is interpreted as the peak energy configuration, thus the transitions between vacua are associated witha violationof Baryon(and lepton) number. EWPT canbeof firstorder, second orderora smoothcrossover.Withinthe frameworkoftheSM,itisa smooth crossover. However, BSM physics can lead to any of the three. The order of the phase transition can affect the outcome of the process. Entropy production and, in return, the dilution of preexisting frozen out species and baryon asymmetry can be some of them. Although baryon excess can be created at the time of electroweak symmetry breaking, it is preserved during the first order phase transition. In the second order, sphalerons can wipe out the total asymmetry created, but in first order, only the asymmetry created in the unbroken phase is wiped out. Arecent overview of physics beyond the standardmodel and its cosmological signatures can be found in [20], whereit was shown that from the lack of supersym-metric particles at the LHC and from the positive results of the directly searched Weakly Interacting Massive Particles (WIMPs), the list of dark matter candidates can be strongly extended. The model of dark atoms of dark matter deserves special attention in this list, in light of its possibility to propose a nontrivial solution for the puzzles of direct dark matter searches, explaining the positive results of the DAMA/LIBRA experiment by the annual modulation of the low energy binding of dark atoms with sodium nuclei, which can be elusive in other experiments for direct WIMP searches. In this approach, dark atoms represent a specific form of asymmetric, strongly interacting dark matter, being an atom-like state of stable -2 (or -2n)charged particlesofanew origin boundedbya Coulomb interaction with (correspondingly, n)nuclei of primordial helium (see [17] for recent review and references). This explanation implies the developmentofa correct quantum mechanical description of the dark atom interaction with nuclei, which is now under way [19]. Even though there are several models predicting ±2 (±2n)-charged stable species, [8–11,11–15,15–18],inthisworkwerestrictourselftothe4th generationfamily as an extension to the standardmodel (SM) and proceed to study the electroweak phase transition (EWPT). The simplest charge-neutral model is considered here; also, we consider that EWPT is of the second order. In passing by, we show the dilution of pre-existing frozen out dark matter density in the presence of the 4th generation. The paper is organized as follows: In the next section, we talk about the 4th generation family, defining a definite relationship between the value and sign of the 4th generation family excess, relative to the baryon asymmetry, which is due to the electroweak phase transition and possible sphaleron production being established. Thisis followedbythe calculationofthedilution factorofpre-existing dark matter density, followedby a general conclusion. 7.2 Abrief review of4th generation The fourth generation is of theoretical interest in the context of sphaleron transition, electroweak symmetrybreakingandlargeCP violatingprocessesinthe 4 ×4 CKM matrix, whichmayplayacrucialrolein understandingthebaryon asymmetryin A. Chaudhuri,M.Yu. Khlopov the universe. Thus, there are significant ongoing efforts to search for the fourth generation. In this work, we consider the stable 4th generation, which is basically constrainedby contributionsofvirtual 4th generation particlesin the Higgs boson decay rates, in the precision tests of standardmodel parameters, as well as by the LHC searches for R-hadrons, which mimic stable 4th generation stable hadrons. These constraints can still leave some room for the existence of such a family and an explanationof the puzzlesof direct dark matter searchesby dark atoms formed ¯ with primordial helium by (UU ¯ U¯ )antiquark clusters. Due to the excess of U¯,only-2 charge or neutral 4th generation species arepresent in the universe. Indeed, stable antiquarks can form a (U ¯U ¯U¯)cluster and a small ¯ U ¯¯ fraction of neutral Uu with ordinary u-quark. In principle, (Uu¯)baryon should alsobe stable,butinabaryon asymmetrical universe,its interactionwithordinary ¯ baryons leads to its destruction in two Uu mesons. 4He, formed during the Big Bang nucleosynthesis, completely screens Q-- charged hadrons in composite [4HeQ--]“atoms”. If this 4th family follows from string phenomenology, we have new charge(F)associated with 4th family fermions. Principally,F should be the only conserved quantity but to keep matters simple, an analogy with WTC model ' is made and we assume two numbers: FB (for 4th quark) and L (for 4th neutrino). Detailed calculationsofWTCweremadein[8,14]andmostofthe terminology were kept the same as the above mentioned papers. As the universe expanded and the temperature decreased and the quantum num­ber violating processes ceased to exist, the relation among the particles emerging fromtheprocess(SM+4th generation) followedthe followingexpression: ' 3(µuL + µdL )+µ + µUL + µDL + µL = 0. (7.1) ' here, µ is the chemical potential of all the SM particles, µL is the chemical potential of the new species leptons and µUL and µDL are that of the 4th generation quarks; see [14]. The number densities follow, respectively, for bosons and fermions: m n = g*T3 µ f( ), T T (7.2) and m n = g*T3 µ g( ), T T (7.3) where f and g arehyperbolic mathematical functions and g* is the effective degrees offreedom, which are givenby the following: 8 1 1 2z2 + 2 f(z)= x cosh-2 xdx, (7.4) 4p20 2 and 1 1 8 22 + 2 g(z)= x sinh-2 zxdx. (7.5) 4p20 2 The number density of baryons follows the following expression: nB -n B ¯ B = . (7.6) gT 2/6 As the main point of interest is the ratio of baryon excess to the excess of the stable 4th generation, the normalization cancels out, without loss of generality. Let us defineaparameter s, which,respectively,for fermions and bosons aregiven Tc by the following: m s = 6f , Tc (7.7) and m s = 6g . (7.8) Tc is the transition temperature and is given by the following: 2MW(Tc) . Tc = B(). (7.9) aW ln(Mpl/Tc) aW In the above equation, MW is the massofW-boson, Mpl is the Planck mass and . is the self-coupling. The function B is derived from experiment and takes the value from 1.5 to 2.7. The new generation charge is calculated to be the following: FB = 2 (sUL µUL + sUL µDL + sDL µDL ), (7.10) 3 where FB corresponds to the anti-U( U¯)excess. For detailed calculations, please see [14]. The SM baryonic and leptonic quantum numbers are expressed as the following: B =[(2 + st)(µuL + µuR)+3(µdL + µdR)] (7.11) and L = 4µ + 6µW (7.12) where in Equation (7.11), the factor 3 of down-type quarks is the number of families. For the 4th generation lepton family, the quantum number is given by the following: ' '' L = 2(s. + sUL )µ. ' L + 2sUL µW +(s. -sUL )µ0 (7.13) ' where . isthe newfamilyof neutrinos originatingfromthe extensionofSM,and µ0 is the chemical potential from the Higgs sector in SM. DuetothepresenceofasingleHiggs particle,thephase transitionisofthe second order. The ratio of the number densities of the 4th generation to the baryons is determinedby the following: .FB 3 FB mFB = . (7.14) .B 2B mp The electrical neutrality and negligibly small chemical potential of the Higgs sector is the result of the second order phase transition. The ratio of the number density of the 4th generation to the baryon number density can be expressed as a function A. Chaudhuri,M.Yu. Khlopov of the ratio of the original and new quantum numbers. In the limiting case of the second order EWPT, we obtain the following: ' FB sUL (21 + s. ) L 29 + 5s. L ' '' - =(17 + s. )+' + ' . (7.15) B3(18 + s. ) 3 B3s. B In the following Figure 7.1, the predicted relationship between the frozen out ¯ excess of U-antiquarks and baryon asymmetry is shown as a function of U-quark mass m. The minimal mass m can be determined from the R-hadrons searchat the LHC as 1 TeV.Thepredicted contributionofdarkatoms,inwhich(U ¯U ¯U¯)arebound with primordial helium nuclei, can explain the observed dark matter density at m ~ 3.5 TeV, which is compatible with the above mentioned experimental lower limit. Fig.7.1:Theratioofdark matterandbaryon densitiesasa functionoftheU-quark mass(m). This ratio is frozen out at the critical temperature of the assumed second order EWPT T = Tc = 179 GeV. At the U-quark mass m ˜ 3.5 TeV, the predicted densityof dark atoms, formedby( U ¯U ¯U¯)bound with primordial helium nuclei, can explain the observed dark matter density. ¯for the second order EWPT. Hence, we establish a relationship between the baryon excess and the excess of U 7.3 Dilution of Pre-Existing Dark Matter Density The thermodynamic quantity, entropy density, is a conserved quantity in the initial stage of universe expansion, especially when the primeval plasma is in thermal equilibrium with a negligible chemical potential. As soon as the universe enters into the state of thermal non-equilibrium, i.e., when G>H, where G is the reaction rate and H is the Hubble parameter, the conservation law breaks down, and entropy starts pouring into the plasma; this can dilute the pre-existing baryon asymmetry and dark matter density. There are many instances of entropy production, such as primordial black hole evaporation [21], electroweak phase transition within the standardmodel and thetwoHiggs doubletmodel[3,6,11].Apartfromthese,thefreezeoutofdark matter density might lead to entropy production, which in turn, can dilute the pre-existing dark matter density. The Lagrangian theory consists of the Langrangian of the standardmodel (SM) and the interaction terms of the 4th generation fermionic family. It is given by the following: L = LSM + L4th , (7.16) where LSM is givenby the following: X 1 µ.µ.† LSM = g.µ.... -U.(.)+ ig.µ.j ...j -Uj(.j). (7.17) 2 j The CP violating potential of the theory is as follows: .T2.2 X mj(T) U.(.)= .2 -.22 + hj . (7.18) 4 2T j here, . = 0.13 is the quartic coupling constant and . is the vacuum expectation values, which is ~246 GeV in the SM. T is the plasma temperature and mj(T )is the mass of the .j-particle at temperature T;see [25]. To calculate the dilution factor, it is necessary to compute the energy and the pressure density of the plasma, using the energy–momentum tensor. Following the detailed calculation in [6] and assuming that the universe was flat in the early epoch with the metric gµ. = (+, -, -, -), we have the following: 4p2g* . + P = ..2 + T4 . (7.19) 3 30 In order to proceed with the calculation of the dilution factor, the EWPT transition temperature needs to be calculated first. The transition temperature is derived using the following expression: V(. = 0, T = Tc)= V(. = ., T = Tc). (7.20) here, . is the vacuum expectation value and Tc is the transition temperature. In Equation (7.20), substituting the values of the standardmodel particles and the minimal allowed masses of the 4th generation particle, which can be estimated from the R-hadrons search at the LHC as1TeV, Tc is found to be ~179 GeV.With a rangeof allowed values, one can obtaina rangeof Tcs and study the nature of the EWPT and other related properties, but that is beyond the scope of the current paper.Withproper data and tools, this analysis willcertainlybe madein the near future. The last term in Equation (7.19) arises from the Yukawa interaction between fermions. The Higgs field starts to oscillate around the minimum, which appears during the phase transition. Particle production from this oscillatingfield causes damping. The characteristic time of decay is equal to the decay width ofthe Higgs bosons. If it is large in comparison to the expansion, and thus the universe cooling rate, then we may assume that the Higgs bosons essentially live in the minimum of the potential. This was clearly discussed in [11]. To calculate the entropy production, it is necessary to solve the evolution equation for energy density conservation as follows: . .=-3H(. + P). (7.21) In Figure 8.1, both the dilution of the pre-existing dark matter (blue line) and the entropy production in the presence of 4th generation lepton family (black line) are shown. It is clear that since the sphaleron transition is of the second order, the net dilution and entropy production (~18%)are somewhat low compared to the scenarios of the first order. Again, the presence of a single Higgs field makes the phase transition of the second order. 7.4 Conclusions In the present paper, we have deduced a definite relationship between the value and sign of the 4th generation family excess and baryon asymmetry due to the sphaleron effects frozen out at the electroweak phase transition, as is clear from Equation (7.15) and Figure 7.1. At the transition temperature, Tc = 179 GeV and the mass of the stable U-quark of the new family m ˜ 3.5 TeV, the predicted density of the dark atoms can explain the observed cosmological dark matter density. This value and experimental constraints on the contribution of new electroweakly interacting fermions appeal to the involvement of additional Higgs bosons, whose existence can influence the value of Tc and, correspondingly, the determination of the mass of the U-quark, for which dark atoms explain the dark matter density. Being beyond the scope of the present work, such a self-consistent analysis of the models with new stable quarks, accompanied by an extended Higgs sector, can open up a new specific direction of studies of BSM physics. The search for new physics and dark matter has been an ongoing areaofresearch for decades. Even though there are many multi-Higgs models, there are few multi-charged models present. The theory of the 4th generation can serve to leap toward new physicsin theframeworkof heterotic string phenomenology.As seenfrom the work, just like the standardmodel, we can link the stable quarks of the new generationparticleswiththebaryon asymmetrytheoretically.Theexistenceofnew stable quarks with the SM electroweak charges can follow from other unifying schemes(in the approach [11,15]in particular); the important conclusionof our work is that balancing baryon asymmetry with sphaleron transitions can provide an excess of U ¯ antiquarks, forming a (U ¯ U ¯ U¯ )‘core’ of dark atoms in which it is bound by a Coulomb force with primordial helium. The possibility of dark atoms extends the listof possible dark matter candidates,predictedin such models.To make new quarks with electroweak charges compatible with the data on the Higgs boson decay rates, their coupling to the SM Higgs boson should be suppressed, and they should acquire their mass from coupling to other Higgs bosons [26]. It would imply the accomplishment of models with new stable generations with SM electroweak chargesby multi-Higgs models, openingupaprobeof studying the Higgs and electroweak symmetry breaking sectors in a rigorous manner. Dark matter candidates in the form of bounded dark atom can emerge from this ¯ model, due to the excess of U withinthe primordialHe nuclei.We have considered only the lightest and most stable particles and also took into account only the second order phase transition. The dilution of pre-existing dark matter density was calculated; in the present scenarios, the dark matter density was reduced by ~18%. Acknowledgements The work of A.C. is funded by RSF Grant 19-42-02004. The research by M.K. was supported by the Ministry of Science and Higher Education of the Russian Federation under Project ”Fundamental problems of cosmic rays and dark matter”, No. 0723-2020-0040. References 1. Dolgov, A.D.; Zeldovich,Y.B. Cosmology and elementary particles. Rev. Mod. 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Rev.D 2008, 77, 065002, doi:10.1103/PhysRevD.77.065002. 9. Sannino,F.;Tuominen,K. Orientifold theory dynamics and symmetrybreaking. Phys. Rev.D 2005, 71, 051901, doi:10.1103/PhysRevD.71.051901. 10. Hong,D.K.;Hsu, S.D.H.; Sannino,F. CompositeHiggsfromhigherrepresentations. Phys. Lett.B 2004, 597, 89, doi:10.1016/j.physletb.2004.07.007. 11. Dietrich, D.D.; Sannino,F.;Tuominen, K. Light composite Higgs boson from higher representations versus electroweak precision measurements: predictions for CERN LHC. Phys. Rev.D 2005, 72, 055001, doi:10.1103/PhysRevD.72.055001. 12. Dietrich, D.D.; Sannino,F.;Tuominen, K. Light composite Higgs and precision elec­troweak measurements on theZ resonance:An update. Phys. Rev.D 2006, 73, 037701, doi:10.1103/PhysRevD.73.037701. 13. Gudnason, S.B.; Kouvaris,C.; SanninoF.Towards working technicolor:Effective theo­ries and dark matter. Phys. Rev.D 2006, 73, 115003, doi:10.1103/PhysRevD.73.115003. 14. Gudnason, S.B.; Kouvaris,C.; Sannino,F. Dark matterfrom new technicolor theories. Phys. Rev.D 2006, 74, 095008, doi:10.1103/PhysRevD.74.095008. 15. Belotsky, K.M.; Khlopov, M.Y.; Shibaev, K.I. Stable quarks of the 4th family? In Physics of Quarks: New Research; HorizonsinWorld Physics 16. Khlopov, M.Y. Composite dark matter from the fourth generation. JETP Lett. 2006, 83, 1, doi:10.1134/S0021364006010012 . 17. Belotsky, K.M.; Khlopov, M.Y.; Shibaev, K.I. Stable matter of 4th generation: Hidden in the Universe and close to detection? arXiv 2006, arXiv:astro-ph/0602261. 18. Belotsky, K.M.; Khlopov, M.Y.; Shibaev, K.I. Composite Dark Matter and its Charged Constituents. Gravit. Cosmol. 2006, 12, 1–7, doi:10.1134/S0202289312020028. 19. Mankoc-Borstnik, N. Unification of spins and charges in Grassmann space? Mod.Phys. Lett. 1995, 10, 587–596. 20. Mankoc Borstnik, N.S. The spin-charge-family theory is explaining the origin of families, of the Higgs and theYukawa couplings. J. Mod. Phys. 2013, 4, 823. 21. Chaudhuri, A.; Dolgov, A. PBH evaporation, baryon asymmetry and dark matter. arXiv 2020, arXiv:2001.11219. 22. Chaudhuri, A.; Dolgov, A. Electroweak phase transition and entropy release in the early universe. JCAP 2018, 2018, 032, doi:10.1088/1475-7516/2018/01/032. 23. Chaudhuri, A.; Khlopov, M.Y. Entropy production due to electroweak phase tran­sition in the framework of two Higgs doublet model. Physics 2021, 3, 275–289, doi:10.3390/physics3020020. 24. Chaudhuri, A.; Khlopov, M.Y.; Porey, S. Effects of 2HDM in electroweak phase transition. Galaxies 2021, 9, 45, doi:10.3390/galaxies9020045. 25. Melo, I. Higgs potential and fundamental physics. Eur. J. Phys. 2017, 38, 065404, doi:10.1088/1361-6404/aa8c3d. 26. Khlopov, M.Y.; Shibaev, R.M. Probes for 4th generation constituents of dark atoms in Higgs boson studies at the LHC. Adv. High Energy Phys. 2014, 2014, 406458. Proceedings to the 24th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ... (p. 104) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 Entopy releasein Electroweak PhaseTransitionin 2HDM A. Chaudhuri1 email: arnabchaudhuri.7@gmail.com M.Yu. Khlopov2 email: khlopov@apc.in2p3.fr S. Porey1 email: shiladityamailbox@gmail.com 1Department of Physics and Astronomy, Novosibirsk State University, Russia 2 Instituteof Physics, Southern Federal University, Russia; Universit´ede Paris, CNRS, Astroparticule et Cosmologie, France; and National Research Nuclear University ”MEPHI”, Russia Abstract. Electroweak phase transition in the simplest extension of the standardmodel namely two Higgs doublet model and entropy production within this framework is studied. We have considered severalbenchmark points which were called using BSMPT,a C++ pack­age, within the limit of vev/TC > 0.2 are studied, and corresponding entropy productions are shown in this paper. Povzetek: Povzetek:Avtorji obravnavajo pove.sibkem faznem canje entropijepri elektro.prehoduv modelu,ki raz. siri standardni modelz dvemaHiggsovima skalarjema. Prispevek prinasarezultate teh ra.cunov za ve ..cnih to. c referen.ck, ki so jih poiskali z uporabo BSMPT, paketa C++, znotraj meje vev/TC > 0.2. 8.1 Introduction For a successful explanation about the origin of excess baryons over antibaryons in the universe through electroweak baryogenesis (EWBG), a strong first-order electroweak phase transition (EWPT) in the early universe is necessary. Cosmic EWPT happened when the hot universe cooled down enough in the primeval time so that the potential of the Higgs field got and settled at a non-zero minimum and in consequence, the symmetry of the theory SU(2)L × U(1)Y broke to U(1)em. At the time of first-order EWPT, bubbles of the broken phase originate and baryon-antibaryon asymmetry generates outside the wall of the bubbles of the broken phase. However, after the discovery of the standardmodel (SM) Higgs boson, itiswidely knownthatEWPTinSMwithasingleHiggs fieldisjusta smooth cross-over. Therefore, for a successful EWBG, a theory of EWPT in beyond SM (BSM) is needed [1]. On the other side, ~ 26.5%of the total energy density of the universe is contributed by the dark matter (DM) whose mysterious nature has not been unveiled till now. Although, primordial black holes and MACHOs which are considered as one of the viable baryonic DM candidates, it is now clear that they are unable to contribute completely to the DM energy density of the universe. There are theories about multicharged extension of the standardmodel like dark atoms which can be viable dark matter candidates, [2]. But there are no experimental evidences as of now. Not only about the baryogenesis, but thereis also no irrefutable theory in SM about nonbaryonic DM particle which can successfully explain all the observations. Similar to the above mentioned facts, there are many limitations of the SM. Thus scientists are desperately searching for experimental evidence of BSM. For them the recent result from Fermilab about gµ - 2 for muon may be a ray of hope. gµ is the gyromagnetic ratio of muon which is defined as the ratio of magnetic moment to the angular moment of muon and whose value is 2 from tree-level calculation. If we define aµ =(gµ - 2)/2, then higher order loop corrections from SM gives aµ = 116, 591, 810(43) × 10-11 wherethe value measuredfrom Fermilab is 16, 592, 061(41) × 10-11 which differs from SM at 3.3s level [3]. This contradictionis actually buttressed thepreviously claimedresultfrom the E821 experiment at Brookhaven National Lab (BNL). There are numerous explanation for this anomalous result including the existence of BSM. Among all the BSM theories, the two Higgs doublet model (2HDM) is one of the most popular theories which not only exhibits strong first-order EWPT for the proper choice of parameter space but also provides the minimal phenomeno­logical description of some effects of the supersymmetric model predicting two Higgs boson doublet. In addition to that, this model can produce dark matter particles [4] and gives a satisfactory explanation for the gµ anomaly of muon [5] if the parameter space is properly chosen. At or around the epoch of EWPT the energy density of the universe was dom­inated by relativistic species with negligible chemical potential. In addition to that, the universe was almost always in thermal equilibrium except some special epochs. Thus entropy density per comoving volume of the relativistic plasma was conserved. However, EWPT is a strongly thermally non-equilibrium process and thus there is a possibility that entropy might have been generated during this cosmic process. In this work, we explored the increase in entropy during the epoch of EWPT in the real type-I 2HDM framework.We have shown that entropy density per comoving volume increases if EWPT happens as a first-order phase transition in 2HDM model. The article is arranged as follows: In the next section the Lagriangian of the model alongwiththeresultsaregiven.Ageneric conclusion followsandinthe appendix the detailed potential is mentioned. 8.2 Lagrangian density of the model The Lagrangian densityof EWPT theoryinreal type-I 2HDMis givenby L = Lgauge,kin + Lf + LYuk + LHiggs -V(F1,F2,T) (8.1) where Lgauge,kin, Lf and LYuk are the kinetic energy term of gauge bosons(Wa and Ba with a = 0, 1, 2, 3), kinetic energyof fermions andYukawa interaction termof fermionswithHiggs bosons.These termsare definedin[6,7]andalso discussed in Appendix A. Throughout this article, all the Greek indices used in super or sub-scriptrun from 0 to 3 and Latin indices from 1 to 3 if not mentioned otherwise. LHiggs incorporates the kinetic term of the Higgs field and their interaction with the gauge bosons. Thus LHiggs = {(.µ + iWµ)Fa } † {(.µ + iWµ)Fa} (8.2) where a = 1, 2 for two Higgs field, i = g and g ' (-1), iWµ = +igTkWkµ + ig ' YBµ, and are coupling constants, Ti is the generator of SU(2)L (left-Chiral), which is alsoa formof Pauli matrices, and Y is the hyper-charge generator of the U(1). The total CP-conserving potential for our 2HDM model considered is V(F1,F2,T)= Vtree(F1,F2)+VCW(F1,F2)+VT (T)+Vdaisy(T) (8.3) The tree-level potential can be written as 1 + F .1 2 † † † † † * 2 2 2 Vtree(F1,F2)F2 -FF1 + FFF2 + m 12FF1 F1 =m m m 11221212121 2 F 2 F + .3 F+ .4 FF 1 † † † † † .2 F2 F1 F2 F2 F1 + 21212 2 (8.4) 1 2F 2 1 + . * F 5 † † .5 F2 F1 + . 12 2 2 22 2 mm , and m can be estimated from the following formula 12, 1122 2 m = 1002 GeV2 (8.5) 12 1 2 3 2222 m = -2.1v 1 + 4m2 -2.4v1v -(8.6) 11 12v2 -2.3v1v 22 -.5v1v 2 v1v 2.5 4v1 1 2 2 2232 m = 4m21v|2 -2.4v 1v2 -2.2v -v (8.7) 22 12v1 -2.3v 1v2 -.5v 21v2.5 4v2 2 The value of mcan alter for different parameter space. These formulas are valid 12 since .5 is real and .6 = .7 = 0. The .1-5 can be calculated from the parameter space as 2 22 mcos a2 + msin a2 -mtan ß H h12 l1 = , (8.8) v2 cos ß2 222 msin a2 + mcos a2 -mtan ß-1 Hh 12 l2 = , (8.9) v2 sin ß2 22 2 (m-m)sin a cos a + 2m2 m H± cos ß sin ß - Hh12 l3 = , (8.10) v2 sin ß cos ß 22 (m-2m2 m H± )sin ß cos ß + A 12 l4 = , (8.11) v2 sin ß cos ß 22 m12 -mA sin ß cos ß l5 = . (8.12) v2 sin ß cos ß 2 22 where v is the standardmodel expectation value, v= v+ v2, tan ß = v2/v1 and 1 cos(ß -a). 0 leadstoSMresult.The details aboutthe parameter spaceof mH, mh, mH± can be found in the recent works [8–10]. The Coleman–Weinberg correction to the potential ­ 2 X nj (-1)2sj 4 mj (v1,v2) VCW (v1 + v2)= m j (v1,v2) log -cj (8.13) 2 64p2 µ j 2 The values of nj, sj, cj and different mass-values mj (v1,v2)are mention in Ap­pendixBand µ = 246 GeV. Temperature correction of potential and its series expansion in Landau gauge are T4 .X m j2 (v1,v2) X m j2 (v1,v2) .VT = njJB + njJF .(8.14) 2p2 .T2 T2 j=bosons j=fermions 22 p4T4 p2 T4JB =- + T2 m 2 - T(mm 4 ln + ··· , m p 2)3/2 - 1m (8.15) T 4512 6 32abT2 2 p22 m7p4T4 1m T4JF =- T 2 m 2 - m 4 ln + ··· , (8.16) T 36024 32 afT2 where ab = 16af = 16p2 exp(3/2 - 2.E)with .E being the Euler-Mascheroni constant. The daisy term is defined as P - T M2 (v1,v2,T)3/2 2 (v1,v2)3/2 Vdaisy(T)= 12p i -mi (8.17) i=1 Details about the M2 (v1,v2,T)termcanbe foundin[12,13].Actually, wewillsee i later that all these terms will be taken care of the software package we have used for this work. At sufficiently high temperature, the total potential of eq.(8.3) has only one mini­mum at .F1. = .F2. = 0 and there is no symmetry breaking. The critical tempera-ture(Tc)is defined as the temperature at which if the temperature drops down, the total potential gets a second minimum at (Fa,min)= {.F1. = v1, .F2. = v2}. For simplicity, we are assumingin this workthat bothof the Higgs field F1 and F2 get the second minimumat the same temperature Tc at the same time. Thus V (F1 = 0, F2 = 0, Tc)= V (F1 = v1,F2 = v2,Tc). (8.18) As soon as the Higgs potential gets a non-zero minimum, the other relativistic particles starts to gain mass and becomes non-relativistic. Thereaction rate among them and also with photon becomes comparable with the Hubble parameter and thus decouples from relativistic plasma. The mass of the particle and coupling constant determine the decoupling temperature. For instance, top quark decouples earlier than electron or other quarks. Now, at the time of EWPT the universe can be assumed as perfectly homogeneous and isotropic and thus we can neglect the spatial partial derivatives of the Higgs fields. Therefore, when the Higgs fields start to oscillate around their minima (Fa,min)then energy density . and pressure P are g*p2 . = F.2 T4 . (8.19) a,min + Vtot(F1,F2,T)+ 30 1g*p2 P = F.a,2 min -Vtot(F1,F2,T)+ T4 (8.20) 3 30 Thelasttermsineq.(8.19)andeq.(8.20)arisefromtheYukawa interactionbetween fermions and Higgs bosons and from the energy density of the fermions, the gauge bosons, and the interaction between the Higgs and gauge bosons. g* depends on the effective number of particles present in the relativistic soup at or near the EWPT. It’s value in our model is greater than the value in SM. Since the oscillation of Fa around Fa,min is small compared to Hubble expansion, we can neglect the time derivative of F.a,min [11] for simplicity in this work. Again, entropy density per comoving volume is defined as . + P 3 s = a (8.21) T which is conserved for relativistic species with negligible chemical potential. From eq.(8.19) and eq.(8.20) we get 4g*p2 . + P = 2F.2 T4 (8.22) a,min + 3 30 As discussed earlier, g* will change with the decoupling process and thus s for relativistic plasma will increase for our considered scenario. Then the increase in entropy can be calculated using conservation of energy momentum tensor . .=-3H (. + P) (8.23) To solve eq.(8.23), we have used BSMPT [13, 14], a C++ package to calculate the vacuum expectation value (VEV) of the total potential, value of the total potential at VEV for different temperatures including Tc.We have chosen the parameter in such a way so that VEV/Tc > 0.02.We have considered five sets of benchmark values and the corresponding figures are shown in Fig. 8.1 Table 8.1: 2HDM Benchmark points for entropy production BM 1 BM 2 BM 3 BM 4 BM 5 mh [GeV] 125 ” ” ” ” mH [GeV] 500 ” ” 485 90 mH± [GeV] 500 485 485 485 200 mA [GeV] 500 500 485 485 300 tan ß 2 2 2 10 10 cos (ß - a) 0 0.00 0.07 0.1 0 2 m 12 GeV2 105 105 105 23,289.6 801.98 .1 0.258 0.258 1.28 3.9 0.258 .2 0.258 .258 0.002 0.22 0.258 .3 0.258 -0.23 0.21 3.9 1.31 .4 0 0.49 0.244 0 0.3 .5 0 0 0.244 0 -1.35 Tc 161.36 153.27 168.61 230.18 135.38 vev/Tc 1.4 1.25 1.7 1.86 1.06 ds/s[%] 57 53 59 70 37 8.3 Conclusion As seen from Fig.8.1, the entropy productions for some benchmark points are shown here.Aproper difference canbe noticedfrom the standardmodel scenario. As seenin [11], the entropyreleasedis around 13%and in the present scenario, we see that the production is considerably higher. This is because first-order phase transitionasseenin2HDMcanreleasemoreentropycomparedtosmooth crossover or second-order in the case of the standardmodel. The massive scalar particles in 2HDM contribute considerably to this production as well. Acknowledgements The work of S.P. and A.C. is funded by RSF Grant 19-42-02004. The research by M.K. was supported by the Ministry of Science and Higher Education of the Russian Federation under Project ”Fundamental problems of cosmic rays and dark matter”, No. 0723-2020-0040. References 1. D. E. Morrissey and M. J. Ramsey-Musolf, New J. Phys. 14 (2012), 125003 doi:10.1088/1367-2630/14/12/125003 [arXiv:1206.2942 [hep-ph]]. 2. A. Chaudhuri and M. Yu. Khlopov, Universe 7, (2021) 8, 275 doi:10.3390/universe7080275 [arXiv: 2106.11646 [hep-ph]]. 3. B. Abi et al. [Muon g-2], Phys. Rev. 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Dolgov, JCAP 01 (2018), 032 doi:10.1088/1475-7516/2018/01/032 [arXiv:1711.01801 [hep-ph]]. 12. J. Bernon, L. Bian and Y. Jiang, JHEP 05 (2018), 151 doi:10.1007/JHEP05(2018)151 [arXiv:1712.08430 [hep-ph]]. 13. P. Basler and M. Muhlleitner, Comput. Phys. Commun. 237 (2019), 62-85 ¨doi:10.1016/j.cpc.2018.11.006 14.P. Basler,M.M¨uller, Comput. Phys. Commun. 269 (2021), 108124 uhlleitner andJ.M ¨doi:10.1016/j.cpc.2021.108124 [arXiv:2007.01725 [hep-ph]]. 15.P. Basler,M. Krause,M. Muhlleitner,J.Wittbrodt andA. Wlotzka, JHEP02 (2017), 121 doi:10.1007/JHEP02(2017)121 [arXiv:1612.04086 [hep-ph]]. AppendixA The kinetic energy termof gauge bosons, kinetic energyof fermions andYukawa interaction term of fermions with Higgs bosons are X ¯//Lf = i .L D.L + .¯R D.R(8.24) .=Q,L,u,d,l * LYuk =-yee¯ RFa† LL + y eL¯ LFa† eR + ··· (8.25) 1Gjµ. 1FBµ. Lgauge,kin =- Gj - FB (8.26) µ.µ. 44 where . is the fermionic field, subscript L (R)is for the left (right) chiral field. The sum in eq.(8.25) is also over quarks. ye is the complex constant and -g.jkl Gj = .µWj -..Wj Wk Wl (8.27) µ..µ µ. FB = (8.28) µ. .µB. -..Bµ (j) ' (j) /= .µ(.µ D.+ igWµ + ig YL,RBµ).(8.29) L,R L,R 8.4 Appendix B: Masses of new Scalars  5 , (i = W±, Z, .) 6 ci = (8.30) 3 , otherwise 2 Bosons ni si m(v)2 h 1 1 eigenvalues of 8.42 Higgs H 1 1 eigenvalues of 8.42 Higgs A 1 1 eigenvalues of 8.42 Higgs G0 1 1 eigenvalues of 8.42 Goldstone H± 2 1 Eq.8.34 Charged Higgs G± 2 1 Eq.8.35 Charged Goldstone ZL 1 1 Eq.8.32 Higgs ZT 2 2 Eq.8.32 Higgs WL 2 1 Eq.8.31 Higgs WT 4 2 Eq.8.31 Higgs .L 1 2 Eq.8.33 .T 2 2 Eq.8.33 2 m = W 2 g2 v . 4 (8.31) 2 m = Z 2 '2 g + g 2 v . 4 (8.32) 2 m = 0. . (8.33) 2 1 MC 1 MC 2 + MC 2+ MC -MC 2 m ¯H± = 11 + MC + 4 121311 . (8.34) 2222 22 11 2 m ¯= MC 22+- 4 MC 2 + MC 2+ MC 2 . G± 11 + MC -MC (8.35) 12131122 22 where 1 c1 = 12.1 + 8.3 + 4.4 + 3 3g2 + g '2(8.36) 48 1 24 2 c2 = 12.2 + 8.3 + 4.4 + 3 3g2 + g '2+ m (T = 0) t 48 v22 2 + 1m (T = 0) (8.37) b 2v22 where mt(T = 0)= 172.5Gev and mb(T = 0)= 4.92GeV. For our case (v3 = 0), 22 vv 2 12 MC 11 = m 11 + .1 + .3 (8.38) 22 22 vv 2 21 MC = m + .3 (8.39) 22 22 + .2 22 v1v2 2 MC =(.4 + .5)-m (8.40) 12 12 2 MC = 0 (8.41) 13 Masses of h, H and A are the eigen values of the matrix M¯N = MN(8.42) For our case (v3 = 0), 3.1 .3 + .4 1 22 22 MN = mv v.5v (8.43) 11 11 + 1 + 2 + 2 2 22 .1 .3 + .4 1 22 22 MN 22 = m 11 + v 1 + v 2 - .5v 2 (8.44) 2 22 3.2 1 22 2 MN = mv 2 +(.3 + .4 + .5)v (8.45) 33 22 + 1 22 .2 1 22 2 MN = mv 2 +(.3 + .4 -.5)v (8.46) 44 22 + 1 22 MN = 0 (8.47) 12 2 MN =-m +(.3 + .4 + .5)v1v2 (8.48) 13 12 MN = 0 (8.49) 14 MN = 0 (8.50) 23 2 MN =-m (8.51) 24 12 + .5v1v2 MN = 0 (8.52) 34 Fermions ni si mf(T = 0) e µ t 4 4 4 1 2 1 2 1 2 yev vk 2 yµv vk 2 vyt vk 2 lepton lepton lepton u c t d s b 12 12 12 12 12 12 1 2 1 2 1 2 1 2 1 2 1 2 yuv vk 2 ycv vk 2 vyt vk 2 ydv vk 2 vys vk 2 vyb vk 2 quark quark quark quark quark quark Proceedings to the 24th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ... (p. 114) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 9 Gravitational waves in the modified gravity S. Roy Chowdhury1 email:roic@sfedu.ru M. Khlopov1,2 email:khlopov@apc.in2p3.fr 1Research Institute of Physics, Southern Federal University, Russia 2Universit´ e de Paris, CNRS, Astroparticule et Cosmologie, France; F-75013 Paris, France, and Center for Cosmoaprticle Physics Cosmion, National ResearchNuclear University “MEPHI”, 31 Kashirskoe Chaussee, 115409 Moscow, Russia Abstract. We have takena modified versionof the Einstein Hilbert action,f(R, T.)gravity under consideration, where T. is the energy-momentum tensor trace for the scalar field under consideration. The structural behaviour of the scalar field considered varies with the form of the potential. The number of polarization modes of gravitational waves in modified theories has been studied extensively for the corresponding fields. There are two additional scalar modes, in addition to the usual two transverse-traceless tensor modes foundin generalrelativity:a massive longitudinal mode anda massless transverse mode (the breathing mode). Povzetek:Vspremenjeni razli’v cici Einstein-Hilbertove akcije,f(R, T.), gravitacije, kjerje T. sled tenzorja energije in gibalne koli.cine za obravnavano skalarno polje,.studirajo pojav gravitacijskih valov. . Stevilo polarizacij gravitacijskih valovv razli.cicah teorije za izbrana skalarna polja je bilo doslej dobro raziskano. Poleg obi. cajnih dveh transverzalnih tenzorskih polarizacij gravitacijskih valov, kijihpredvidisplo.sna teorijarelativnosti, obstajata.sedva dodatna skalarna na.cina,kiju avtorja opazita,kospreminjata obliko skalarnega potenciala: masivni vzdolzni na..cin in brezmasni pre.cni (dihalni) na.cin. 9.1 Introduction The FLRW metric is an exact solution to Einstein’s equations, achieved under the implication of space homogeneity and isotropy. It has been well recognized for satisfactorily explaining several other observational evidence about our Uni­verse, including the distribution of large-scale galaxies and the near-uniformity of the CMB temperature [1]. The FLRWmetric [2] underpins the existing accepted cosmological model, which is quite good at likely fitting continued application data sets and trying to explain measured cosmic acceleration. The fact that the cosmological space-time metric differs from the FLRWmetric would have massive consequences for inflation theory as well as fundamental physics. Alternative explanations of gravity have long been considered to prevent a few ofthe contradictionsin conventional cosmology[3,4].Apotential substituteis the f(R, T)gravity,createdrecentlyby Harkoetal.[5].The latest identification of gravitational waves (GWs) by the Advanced LIGO group has opened up a massive door to analyse the Universe. [6–8]. Apart from directly detecting GWs with LIGO/VIRGO interferometers, one could use the informal identificationof GWs by assessing the substantial reduction of the orbital period of stellar binary configuration. Detecting nano-Hertz GWs with a pulsar timing array includes timing various millisecond pulsars, which seem to be extremely stable celestial clocks, accordingtoJenet[9].This connectioniseffectedbytheangular distance (.)between both the two pulsars, as well as the polarization of GW and graviton mass, according to C(.)[10]. The range of the GW, including its polarization modes, is based on the theories. In the radiative domain, the polarization and dispersion of GWs in vacuum are two critical features of GWs that distinguish between the authenticityof gravity theories.GWs can also have up to six conceivable polarization states in substitute metric theories, four more than GR permits. Hou et al. [11] carried out a detailed analysis of the polarization mode for the Horndeski theory. Using GWs polarization, Alves et al. [12] investigatedthe f(R) framework. In f(R)gravity metric methodology, the model, including other f(R) theoretical models, confirms the effectiveness of scalar degrees of freedom. There isa scalar modeof polarizationof GWs existsin theory. This polarization mode appears in two different states: a massive longitudinal mode and a transverse massless breathing mode with non-vanishing trace [13]. Capozziello and Laurentis [14] find the palatini formalism, conformal transformations and find the new polarization states for gravitational radiation for the higher order of extended gravity (f(R)= R +aR2)Later on, Alves et al. [15] studied for f(R, T)and f(R, T.) theoretical models, . In this article, we studied the polarization modes based on the potential, which is a function of the scalar field under the framework of modified gravity f(R, T.)for the vacuum system. In Sec. 9.2 we developed the basic formalism of the modified gravity. The scalar field structure and equation of motion is developed in Sec. 9.3. Polarization modes using Newman-Penrose (NP) formalism is analyzed Sec. 9.4. Andin Sec. 9.5 we conclude theresults. 9.2 Basic formalism of the modified gravity In the context of modified gravity [5], for the vacuumed system, the total action including the scalar field can be introduced in the following manner, Z v S = d4 x -gf(R, T.)+L(., .µ.), (9.1) where R stands for the Ricci scalar, while T. is the trace of the scalar field’s energy-momentum tensor. The field’s action,withgasthe metric’s determinantand signature(-,+,+,+).We use geometric units with the formula G = c = 1. Following that, we considered L(., .µ.)= L.. Here L. is the standard La­grangian density forareal scalar field(.), as follow [16], L. = 1 .a..a. -V(.). (9.2) 2 Aself-interacting potential is represented byV(.). In this theory, matter fields have a relatively limited coupling to gravity and no coupling to the scalar field. The stress-energy tensor can define as v 2d(-gL) T . µ. =- v . (9.3) -g dgµ. We assumed that the Lagrangian densityL is free of its derivatives and is only µ.conditional on the metric tensor modules g. Therefore, the energy-momentum tensor of the scalar field is T. µ. = 1 gµ..a..a. -gµ.V(.)-.µ...., 2 (9.4) and the corresponding trace is given by T. = .a..a. -4V(.). (9.5) The generalized formof the Einstein fieldequationin vacuumin the involvement of scalar fieldis obtainedby varying the gravitational field’s actionSconcerning the metric tensor components, gµ., and then on integration as follow, f1 fRRµ. - gµ. = T. -fT gµ.L. (9.6) µ. + fT T. µ. 22 Here, fR = fR(R, T.)and fT = fT (R, T.)denotes .f(R, T .)/.R and .f(R, T .)/.T . , respectively. We assume that the modified gravity function f(R, T .)is given by f(R, T.)= R + ßT. , ß is an arbitrary constant. The field equation immediately takes the following form, Gµ. = 1 [T. µ. + gµ.ßT. -2ß.µ....]. 2 (9.7) 9.3 Scalar Field On contraction and simplification, the Eq. (9.6) the Ricci scalar of can be obtained as follows, 1 R =- [4ßT . + T. -2ß.µ..µ.] (9.8) 2 The equation of motion for the scalar field can be found from the covariant diver­gence of the field Eq. (9.7) as follows, .V (1 + 2ß).. +(1 + 4ß)= 0. (9.9) .. Since we are considering the vacuum system, we consider the potential in the following form, 2 V(.)= 1µ .2 + 1..4 , (9.10) 24 where, µ and . are real constants. We limited ourselves to first-order termsin.. The thirdterm of Eq. (9.9) disappear asaresultof this estimation. V is being expanded around the non-null minimum value V0. . = .-.0 can be used to expand the field. Following identical approach as before, we encounterit with such assumptions and first-orderrestrictions, 1 + 4ß .V .. += 0. (9.11) 1 + 2ß .. The field equations in the linear region has been investigated and leads to the solution in the following form, ' .(x)= . + .1 exp (iq.x .), (9.12) Solution of the scalar field corresponds to the above equation can be written as in Eq. (9.12) with, 2 µ+ ..2 ' 0 . = .0 - .0, (9.13) µ2 + 3..2 0 and 1 + 4ß µ2 qµq=(µ + 3..2 ) . (9.14) 0 1 + 2ß The variation of the effective mass(m.)with the coupling constant is shown in the Fig. 20.2. The restricted range is for -0.50 = ß = -0.25, from Eq. (9.14). 1/2 1 + 4ß 2 E = ± q 2 +(µ + 3..2 ) (9.15) 0 1 + 2ß The first-order minimally coupled scalar field exposes an effective cosmological constant, as follows: V0 . = 4ß + 1. (9.16) 2 With. being a positive constant, the potential in Eq. (9.10) could be categorized 2 into two situations: (i) µ2 >0, and (ii) µ<0. This is what the universe needs to be stable. While the minimum scalar field for µ2 <0 is non-zero, the effective cosmological constant is non-zero. The cosmological constant that is effective is 44 1 µµ . =- ß + . (9.17) 2. 4. 2 The steady minimum of the scalar field is zero for µ>0, which causes the effective cosmological constant(.)to be zero. 9.4 Polarization modes of the modified gravity Newman-Penrose formalism The Newman-Penrose(NP)[17,18] methodisusedtofind additional polarization modes; further informationis availableinthereferences[19,20].Tetradsarea com­bination of standardized linearlyindependent vectors (et,ex,ey,ez)that could be used to describe the NP quantities that correlate to all of the six polarization modes of GWs at any spatial position. The NP tetrads k, l, m, m¯. can be used to recognize these vectors. The actual null vectors are as follows: 11 (9.18) k = v (et + ez),l = v (et -ez), 22 And the other two complex null vector are, 11 m = v (ex + iey), m ¯= v (ex -iey). (9.19) 2 2 -k.l = m.m ¯= 1, Ea =(k, l, m, m¯). While all other dot product vanishes. In the NP notation, the indefinable components of the Riemann tensor R.µ.. are definedby ten componentsof theWely tensor(.’s), nine components of the traceless Ricci tensor(F’s), anda curvature scalar(.). They are reduced to six by some symmetrical and differential properties: .2,.3,.4 and F22 arereal and .3 and .4 are complex. These NP variables are associated with the following components of the Riemann tensor in the null tetrad basis: .2 =- 1Rlklk ~ longitudinal scalar mode, 6 .3 =- 1Rlkl ¯~ vector-x&vector-y modes, m 2 .4 =-Rl ¯m ~ +, × tensorial mode, ml ¯ F22 =-Rlmlm ¯~ breathing scalar mode. (9.20) The additional nonzero NP variables are F11 = 3.2/2, F12 = F21 = .3 and . = .2/2, respectively. All of them can be defined base on the variables in Eq. (9.20). The group E(2), the group of the Lorentz group for masslessparticles, can be used to classify these four NP variables .2,.3,.4, and F22 based on their trans­formation properties. Only .2 is invariant, and the amplitudes of the four NP variables are not observer-independent, according to these transformations. The absence (zero amplitude) of some of the four NP variables, on the other hand, is not dependent on the observer. The following relations for the Ricci tensor and the Ricci scalar hold: Rlklk = Rlk, Rlklm = Rlm, Rlklm ¯= Rlm¯, 1 Rl ¯m Rll, ml ¯= 2 R =-2Rlklk = 2Rlk. (9.21) Following Eq. (9.6), the Ricci tensor can be written as, 1 Rµ. = µ. ..µ.] (9.22) [aRgµ. + gµ.f(T.)+T. -2fT .µ 2a Using Eq. (9.20) and Eq. (9.21), one finds the following Ricci tensors: = m == m 0. Rlklk .0, Rlml ¯.0, Rlklm Rlkl ¯= From the above relation and Eq. (9.20), one finds the following NP quantities: .2 .0;.3 = 0;.4 = .0 = .0, and F22 = Thus we get four polarization modes for the GW:+,× tensorial mode, breathing scalar mode and longitudinal scalar mode. 9.5 Conclusion The theoretical foundations of modified gravity, a new approach intended to address and find solutions to the shortcomings and discrepancies of GR, are outlinedinthisreport.These issues primarily manifest themselvesat infraredand ultraviolet ranges, i.e., cosmological and astrophysical scales on the one hand and quantum scales on the other. The stability analysis of the scalar field varies depending on the circumstances of potential, and we have taken into account the spontaneous symmetry breaking analogous potential for our structure. The scalar field’s behaviour varies identi­fication and characterization of the critical parameter (µ2). The stable minimum value of the scalar field for µ2 >0 is zero,resultingina zeroeffective cosmological constant(.). For µ 2 <0,the minimum scalar field would be non-zero, and the effective cosmological constant is non-zero as well. The variation of potential is shown in Fig. 20.3. The µ2 >0 variationis showninred, whereasin blue coloured, the interpretation of µ2 <0 is shown. The scalar field Lagrangian is taken in conjunction may emerge a new set of Friedmann equations. Due to a mathematical constraint, the effective mass has a finite discontinuity. It is found for the range -0.50 = ß = -0.25 effective mass is discontinuous. The variation is shown in Fig. 20.2. The post-Minkowskian constraint of modified gravity,the problem of gravitational radiation, also deserves careful consideration. When the gravitational action is just not Hilbert–Einstein, new polarizations emerge: in general, massive, massless, and ghost modes must be considered, whereas, in GR, only massless modes and two polarizations are present. This result necessitates a rethinking of GW physics. If GWs have nontensorial polarization modes, as mentioned, an analyzed signal, such as a stochastic cosmological background of GWs, would be an integration of each of these modes. In Einstein’s General Relativity, the plus and cross modes of polarization are quite common. The plus mode is depicted by P+ = Rtxtx + Rtyty, the cross mode by P× = Rtxty, the vector-x mode by Pxz = Rtxtz, the vector-y mode by Pyz = Rtytz, and the longitudinal mode by Pl = Rtztz, and the transverse breathing mode by 11 Pb = Rtxtx +Rtyty. For the form of potential V(.)= µ2.2 + 4 ..4, in the frame 2 of modified gravity f(R, T.)= R + ßT., we obtain four polarization modes of GWs exists:+,× tensorial mode, breathing scalar mode and longitudinal scalar mode, respectively. Acknowledgements The southern federal university supported the work of SRC (SFedU) (grant no. P-VnGr/21-05-IF). SRC is also thankful to Ranjini Mondol of IISc, Bangalore, for the fruitful discussion to improve the manuscript. References 1. Planck CollaborationandP.A.R.Adeetal.,A&A 594, A13 (2016). 2. S. Cao et al., Scie. Reports 94, 11608 (2019). 3.T. Clifton,P.G.Ferreira,A. Padilla, andC. Skordis, Phys. Reports 513,1(2012). 4.T. Padmanabhan, Phys.Reports 380, 235 (2003). 5.T. Harko,F.S.N. Lobo,S. Nojiri, andS.D. Odintsov, Phys. Rev.D 84, 024020 (2011). 6. B.P. Abbott et al. (LIGO Scientific Collaboration andVirgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016). 7. B.P. Abbott et al. (LIGO Scientific Collaboration andVirgo Collaboration), Phys. Rev. Lett. 119, 161101 (2017). 8.B.P. Abbottetal.(LIGO Scientific CollaborationandVirgo Collaboration),Astrophys.J. L848, L12 (2017). 9.F.A.Jenet,G.B. Hobbs,K.J. Lee, andR.N. Manchester, Astrophys.J.625, L123 (2005). 10. K.J. Lee,F.A. Jenet, andR.H. Price, Astrophys.J. 685, 1304 (2008). 11.Y. Hou,S. Gong andY. Liu, Eur. Phys.J.C 78, 378 (2018). 12. M. Alves,O. Miranda, andJ.de Araujo, Phys. Lett.B 679, 401 (2009). 13. D.J. Gogoi andU.DevGoswami, Eur. Phys.J.C 80, 1101 (2020). 14. S. Capozziello,M.De Laurentis, Physics Reports 509, 167 (2011). 15. M.E.S. Alves,P.H.R.S. Moraes,J.C.N.de Araujo, andM. Malheiro, Phys. Rev.D 94, 024032 (2016). 16.P.H.R.S. Moraes andJ.R.L. Santos, Eur. Phys.J.C 76, 60 (2016). 17. E. Newman and R. Penrose, J. Math. Phys. 3, 566 (1962). 18. E. Newman and R. Penrose, J. Math. Phys. 4, 998 (1963). 19.D.M.Eardley,D.L.Lee,andA.P. Lightman,Phys.Rev.D 8, 3308 (1973). 20. D. M. Eardley et al., Phys. Rev. Lett. 30, 884 (1973). Proceedings to the 24th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ... (p. 123) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 10 Representing rational numbers and divergent geometric seriesby binary graphs E. Dmitrieff email: elia@quantumgravityresearch.org, elia@linden.institute Irkutsk State University Abstract. We consider digital (mostly binary) representations of rational numbers as ex­plicit finite graphs, meaning digits as nodes, and expressing their positioning with directed edges. Following the symmetries of graphs, we found out that some divergent geometric series also can be represented this way as finite graphs. They manifest arithmetical proper­ties that allow us to map them onto the set of negative fractions with odd denominators. Their values appear the same withresultsof sum formula S = 1-1b continued to the range of common rates b exceeding1.We suppose that these seriesin fact converge when they are expressed declarative as graphs, while the divergence is connected to the character of the sum operator, that can be in some cases avoided. Povzetek: Avtorpredstavi racionalna (ve.stevila kot kon . cinoma binarna) racionalna .cne grafe, kjer .cajo vozleinrobovi usmerjenost. Ugotovi,da lahko nekatere diver- sevilke dolo .gentne vrste predstavi s kon.cajo preslikavo na mno. cnimi grafi s simetrijo, ki omogo.zico negativnih ulomkovz lihimi imenovalci. Njihove vsote so izrazljivez ena .1-1b ,ko b cbo S = prese. zevrednost1.Avtor domneva,date vrste,predstavljenezgrafi, dejansko konvergirajo, divergentnostpa pove .cinom iskanja vsote vrste, . zezna.cemursejev nekaterih primerih mogo.ce izogniti. 10.1 Overview of numeric representations and graphs Representation of integer and rational numbers by sequences of digits is the common way to visualize, express and manipulate them. Real numbers in practical applications are usually approximated by rationals, that also assumes using their digital representations. The way the numbers being written is following some rules. They establish rela­tionships between digits that canbe,in principle, expressed explicitly.Wereview heretherulesof writing numbers focusingonexpressingtheminthe declarative style, as graphs. 10.1.1 Digital representation of a number The number a is mapped to the ordered sequence of digits ai, usually written asa string from left to right. Digits are elements of a finite set isomorphic to Zb, where b is called the base of the representation. By default, the base b = 10 and the digit set is {0, 1, ..9}. It is assumed that the represented number can be calculated as a sum of base’s powers with digits’ numeric values as factors: 8i=-8 Tomark the place of digita0 that corresponds to the 0th power of the base, the point X i (10.1) b··· == a...aaaa...aaaaa.-----. 1231012innnnorcommasymbolisused.Thepointcanbeomitted.Inthiscase,itisimplicitly 10.1.2Calculabilityofrepresentations Notethatthesumwithinfinitestartingindexcan becalculateddirectlysince not thestartingdigitintheposition isunreachable. 8 -However,inmostpracticalcases,theleading,ortrailingdigits,orbothareall assumed after the last written digit, so the number expressed this way is an integer. zeroes, so they are omitted. The remaining finite number of ’significant’ digits are m aibi can be calculated following (10.1), producing n usually written. This sum therepresented rational number. The infinite sum with the most significant digit th but without the less significant one also can be calculated iterating from the mpower down: X X X X m8n aibi = a-ib-i = lim a-ib-i = a. (10.2) n.8 i=-8 i=-mi=-m X It converges to the represented number, that is in this case real. The calculation canbe stopped when the desiredprecisionisreached. On another hand, the sum having no most significant digit 8m aibi = lim aibi = 8 (10.3) m.8 i=ni=n is not limited from above and diverges. In most cases it is treated as having no meaning, but in some applications it is possible to re-normalize some infinities, obtaining finite results by using special tricks. 10.1.3 Splitting into integer and fractional parts There is other way to make the sum (10.1) calculable, that has the advantage of symmetry. Namely, the sequenceof digitsis splittedin two partsby the point. The left part, starting with 0th power, is integer, and the right part, starting with -1th power, is fractional: X X X 888 a = aibi = aibi + a-ib-i . (10.4) i=-8 i=0i=1 Both parts are sequences of digits starting at the point. The calculations can be performed recurrently with the same initial power b0 that is 1, and with the same algorithm. It includes finding current power of b, multiplying it to the current digit’s value and accumulating the sum. The difference is just in the b power modification on each iteration, that is the post-multiplication by b for integer, and the pre-division by b for fraction part. Note that there is exactly one bit of information that is required to determine whether the part is integer or fractional. Also note that the integer parts sequences should have leading zeros on the left to avoid divergence, while the fractional part may end with either trailing zeroes or nines (for binary numbers, ones)since both converge. 10.1.4 Non-equivalence of numbers and their representations The tradition of representing numbers in the form of digital symbol chains has become so well established, that it is sometimes perceived as the only conceivable or natural way of expressing a particular number. In practice, in the mind digital strings often actuallyreplace original numbers that theyrepresent. However, there are some differences between them: • The map of numbers to digital sequences is not bijective. Leading zeroes for integers, and also trailing zeroes for finite rationals, can be freely added or eliminated without affecting the represented value. Being omitted in writing, they appear back when they are needed to perform a digit-wise calculations. • Itis known thatin decimalrepresentations(b = 10)the least significant digit, that resides just before the trailing zeroes, can be decreased by one together withreplacingof all these zeroes with nines, for instance: 5.3840000 . . . = 5.3839999 . . . (10.5) It can be proved by taking the limit of the sum in (10.1). So, the integers, and also fractional rational numbers with denominators of 10n,n . Ncanbe decimal-representedintwodifferent ways.Zeroes(thatare omitted) are preferred to nines for convenience reasons. Representations with other bases also have this effect. In case of base=2 trailing zeros and ones are the only possible casesofrepeating digits. • The negative numbers have norepresentationsin the sense noted above.Itis obvious because all the factors are non-negative. Instead, conventional repre­sentationsof positive numbersare used,precededbythe extra(non-digital) minus sign, for instance ”-0.155”. This sign has a meaning of subtraction operation 0 - a, that shouldbe applied to this positive number to make it negative. Thisproblem becomes seriousin digital computers, that have native digits0 and1butdonothaveany native minussign. Luckilythisproblemhasbeen gracefully solvedbyimplementingthe so-called two’s complement encoding [1], that involves leading ones that is the binary equivalent of leading nines. In our opinion, this solution is not just a useful trick but it sheds some light on the mathematical nature of numbers. 10.1.5 Loops in rational numbers representations Unlike the irrationals, rational numbers with denominators other than 10n are represented by infinite sequences of digits (that are finite-periodic). These se­quences are conventionally written using some extra non-digital symbols, for instance 2 1 = 2.1428571428 . . . = 2.142857 =.2.(142857). In vague writing, and 7 also oftenin calculatorsand computerstheyare implicitlyroundedortruncated without any notice to the nearest finite decimal number, that sometimes may cause computational errors. These additional parenthesis or lines applied to help keeping the representations finite and writable.Without them the loops and infinite chains would not be writable on paper or in computer memory. So these representations are either non-writable infinite sequences, or they are not pure sequences of digits if they are decorated by extra non-digital symbols to make them finite. 10.1.6 Graphs Graphs are mathematical objects that correspond to the common-sense concept of entities interconnected to each otherby somerelations. Thegraphisasetofnodes,andthenodescanbe explicitlyrelatedtoeach others by edges. Both nodes and edges can be weighted, i.e. carry some additional information. Generally, graph may consist of one or more connectivity components, nodes of which are connected toeach otherthrough somepathofedges(maybe,throughin­termediate nodes), but nodes from different components have no such connection. Weconsider herethe special, limited kind of graphs that have all the edges directed (weighted with one bit of information about direction), and all the nodes also weighted with one binary digit. Such a graph we call a binary graph [3]. In order to represent numbers, we restrict the freedom of graphs a little more, by requirement that each node has one and only one outgoing edge, meaning that when standing on the particular node, the next node is always known. The edge must end on some node. When it ends on the same node, it is the shortest loop: (10.6) With this restriction, the graph has well-known structure [2]. Each connectivity component has the only loop with trees on its nodes. The loop can be of one node and the tree can have a form of the linear sequence. 10.2 Converting numbers into graphs We see that number representations in form of strings of digits have the structure similarto linear graphs.The digits correspondingto nodesarefollowedby(related to) other digits: Here we convert numbers of several kinds into graphs. Inthis paper, we consider only binary graphs as representations of numbers, so the rules for the conversion are the following: • the base b = 2, so the digits are bits with values in the set Z2 = {0, 1}, • all graph nodes are weighted, and the weight is always one explicit bit -either 0or 1. • the implicit connections between digits in the sequence areexpressed explicitly by directed edges, • edges follow the symmetrical calculation scheme of (10.4), starting from the point towards the positions of higher powers by magnitude. (10.9) As noted before, each node is equipped with an outgoing edge. Thisrequirementfixesthepossiblestructureofgraph connectivity componentthat nowcan containaforestofoneormoredirectedtreesor sequences, terminatedby one loop. To satisfy the requirement, the last node of the graph (10.9) should also be ter­minated by a loop that represent trailing zeroes. The loop can be the minimal, containing one zero node. 10.2.1 Positive integers To build the binary-graph representation of a positive integer number, we sim­ply treat the bits of the binary representation as nodes, and explicitly apply the edges between them, in the order of the bits in the string. The leading zeros are represented by one additional self-cycled zero-bit: We can see that any linear binary graph with the zero terminating loop of one node correspondstothe binaryrepresentationofan integer number. It is not strong bijection since one could ’unwind’ one or more zeroes from the loop into the linear part without affecting the numeric value. 10.2.2 Positive binary fractions There is the mirror symmetry between finite integers and fractional parts of the representation. Since that they are linear, the difference is only in the the Next operation. Accordingto thissymmetry,the graphs correspondingto integers also can be mapped to positive binary fractions, 0 . a<1, with the denominator 2n . We re-write the graph (10.10) from left to right, as fractions are usually written, to show this correspondence: 7 = .0111(0)2 . (10.12) 16 10 10.2.3 Negative integers According to the two’scomplement encoding consideration, negative integers can be represented by performing the subtraction of their magnitude from zero (ignoring the carry bit). The leading zeroes turns into the leading ones. The same result is achievedby inversionof all the bits and incrementingby one. This representations is focused on the arithmetic: the addition and subtraction are performed the same way as with positive numbers, so they can be mixed in expressionsand computations.For instance, adding1to-1gives0: -1 + 1 =(1). +(0)1. = . . . 1111. + . . . 0001. =[carrybit]. . . 0000. = 0 (10.13) The carry bit gets ”lostin the infinity”.Inthe graph form thereis no infinity and after two bit additions the calculation ends and the carry bit is discarded. Subtracting positive numbers from the binary representations of 0 or -1, we always get representations with leading ones, or graphs with the looped one. Thus, negative integers can be mapped to linear binary graphs terminated with the loop of node 1. 10.2.4 Reflected negative integers are again the positive binary fractions Making the same reflection as with positive integers (that is, in fact, just rewriting from left to right) we do not get representatios of negative fractions. Instead, we get another representations of positive binary fractions, this time with trailing 1s. To get the same value, two bits are inverted: the bitin the loop, zero to one, and the last bit before loop -one to zero. 7 = .0110(1)2 . (10.15) 16 10 10.2.5 Non-binary fractions In case the loop consist of two or more nodes, the corresponding binary fraction is periodic, so it maps to geometric series that converges to some fraction with the denominator that is not the power of 2: 8 X 31 15 == . .011(01)2 + 4n 88 12 i=1 (10.16) Each loopofNnodes gives fractions with the denominator 2N -1. Probably there always exists suchNthat 2N -1 has an arbitrary divisor d, to ensure that each rational fraction 0 . a<1 can be represented by the finite binary graph. If the fraction has even denominator, the representation must be shifted right according with the power of two in the denominator. 10.2.6 Irrational numbers Irrational numbersin their digitalrepresentations have infinite tailof non-repeating digits on the right side. Corresponding binary graphs would also be infinite, hav­ingnoloops.Wedidnotfocusonthemsinceweinterestednamelyin closed forms. Tobe practicallyusable,theymayberoundedupintofinitegraphs(representing rational fractions)by switching the endof some node’s outgoing edge, insteadof the next node, to some previous one or to self, forming a loop. So the rounding has here the literal meaning of closure the sequence into the circle. The length of this loop determines denominator of the rational number, as noted above (10.2.5).Taking in account the local quasi-periodicity of digits, one can choose optimal place and loop length for rounding, getting rational approxima­tions with the size and precision balanced. For instance, the binary representation of p ˜ 11.00100 10000 11111 10110 10101 00010 00100 00101 10100... (10.18) can be cut in any place and looped back. Doing so after the 3rd position and looping back to the 1st one, we exploit therepeating sequence (001). Here 1 . 1 7 is the valuerepresentedby 001 and 7 = 23 -1 follows from the length of the loop. 1 The corresponding rational value is 3 += 3.(142857). 7 Looping to self in the 7th position we get the finite binary fraction with trailing 9 201 zeros: 11.001001(0)= 3 += = 3.140625. 64 64 Therepeating1s startingfrom the 11th position allow to get 144 1 11.0010010000(1)= 3 ++ , or (10.19) 1024 1024 11.0010010001(0)= 3 + 145 = 3.1416015625. (10.20) 1024 By cutting the tail in the 20th position, where the pattern 01 repeats three times, 74235 . 1 androunding2nodes back, we get the loop (01)and theresultis 3 ++ 3 54288 11 ˜ 3.1415923. 220 3 10.2.7 Reflections of non-binary fractions We consider here the reflections of graphs (10.17) corresponding to the fractions. Rewritten right to left, theymap to infinite geometric series that now have the 1 common ratio b = 2 instead of . These series are divergent, but representing 2 binary graphs are still finite. They terminate with the loop that contains different bits (Thus, the minimal length of the loop is2because loops 11 and 00 can be replacedby1and0,respectively). Consider the following examples: 8 X . (10)110.2 = 6 + 82 · 4n = 8 n=1 (10.21) Here 6 = 0 · 20 + 1 · 21 + 1 · 22 comes from the linear part of the graph, 8 = 23 arises becausetheloopbeginsinthe3rdplace(shiftbyone positiontotheleft is multiplication by 2), 2 is the value of bits 1 and 0 in the loop and the 4 in the powerbaseisused becausetheloopsizeis2andeach iterationis shiftedleftfor two positions, so that is the multiplication by 4. 8 X = 5 · 8n = 8 (10.22) . (101)10.2 2 + 4 n=1 Again, 2 is linear part, the loop is shifted left twice, so the factor before sum is 22 = 4, the loop "101" is 5 and the loop length of three digits shifts this 5 three times left on each iteration. We can see that despiteof the divergenceofthe series, that comesfrom the loop, and the formal infinite sum as the result, the graphs of such a form keep their individuality. The set of such graphs is as rich as set of rationals in the segment (0;1]. 10.2.8 Arithmetic of divergent series in the graph form Consider two ”divergent” graphs of the kind 10.2.7: 8 X . (10). = . . . 10101010. =(2 · 4n)= 8 (10.23) n=0 and 8 X . (01). = . . . 01010101. = 4n = 8 (10.24) n=0 They are the same (consisting of the loop of1 and0 only), excepting that the starting point points to different nodes. BinaryrepresentationofAcanbeproducedfromBby shifting left once, so A = 2B. (10.25) They canbe added withthe usualway. Since thereisno carry,theresultis obvious: A + B (11). = . . . 1111. . -1. . (10). +(01). =(1). = (10.26) In the binary representation it is an infinite sequence of leading ones, that is mapped to -1 (10.2.3). Combining with (10.25) we get A ,B . -1 . -2 33 The same result we get when applying the formula of convergent geometric series sum, S = 1/(1 -q): 8 X B = 4n = 1 =- 1. (10.27) 1 -43 n=0 Addition of graphs representing negative numbers can be more complicated in case of different loop lengths. It corresponds to the addition of fractions with different denominators. To perform this addition, one must first turn the denominators into the form 2N - 1, that would be its multiple. Then both operands are denormalized by explicitrepetitionof their loops, making them equal-sized. Maximal lengthofloop is the product of lengths of the operands’ loops: 23 63 -- =-- . (0110). +(011). 57 157 =(011001100110). +(011011011011). = (10.28) 3393 29 =(110101000001). . - =- . 4095 35 In this example 4-loop and 3-loop,repeated3and4times,become 12-loops, that corresponds to denominator 212 -1 = 4095. Adding 12-loops node-by-node with carrying, we get the result in the form of 12-loop. 35 (10.29) In the case of carry in the loop, that occurs to the pointed node, it must be avoided by unwinding turns of the chain: 46 126 -- =-- . (1100). +(110). 57 157 =(110011001100). +(110110110110). = (10.30) =(101010000011)101010000010. = 23 =(110101000001)0. . -1. 35 The carry bit that goes from the ”most significant” bit of the loops sum, is added to the ”least significant” bit of it, again. This process is not repeated since there are zeroes inside, stopping the carry. But the carry bit must not be added to the first instance of the loop, that follows the point.To avoidit, one turnis unwinded,andinitthe zerointhe first position is kept. We can see that there is no need to unwind the whole turn, so eleven nodes can be winded back to the loop. This avoiding of addition of carry bit is in fact the same operation as subtraction of one, anditisreflectedin theresult. 10.3 Discussion From examples above we can see that negative rational numbers, which can be represented by the single-component graphs having non-trivial loop, are not limitedby range -1.1 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 (M 21 + M 22 )Y + 1 Y - 1 + M - 2 Y + 3 Y - 3 2122 Y + 2 Y + M M+ 4M23 g 22 21 1 2 1 1 2 (M 21 ) v 2Z1 Z2 (11.7) 3 2121 Z 22 + M Zg 2H + + .2 21 .2 22 2H 21 22 (11.8) MM = = , Z1 Z2 21 M M Z1 21 v 3 22 21 M+4M Table 11.2:Z1 -Z2 mixing mass matrix Diagonalization of the Z1 -Z2 squared mass matrix yield the eigenvalues 2 M1 v Z2 3 3 2 2 2 2 2 12 12 12 12 22 22 22 22 12 12 - 2 + 2 M= M+ M-(M-M)2 + MM3 2 M= M+ M+ (M-M)2 + MM3 (11.9) (11.10) and finally 2 + 2 - 2 - - 3+ 32 2- 2+ 22 1- 1+ 12 22 1ZZ(M+ M)YY+ MYY+ MYY+ M+ M2 2 2 + , (11.11) where Z1 cos . sin . Z- = Z2 -sin . cos . Z+ (11.12) v 2 1 3 Mcos . sin . = (11.13) 4 M 41 + M 22 (M 22 -M 21 ) 11.4 Electroweak symmetry breaking The ”Electroweak Symmetry Breaking” (EWSB) is achieved by the Higgs fields F ui and F di , which transform simultaneously as triplets under SU(3)and as Higgs doublets with hypercharges -1 and +1 under the SM,respectively, explicitly: ..o u ...+ d ..- .o .1 ..1 . ...... ...o u . .+ d . Fu = . . ,Fd = .. .. ..- ...o . .2 ..2 .. . . u . ....o ...+ d . ..- ....o 3 3 with the VEV’s vu1 0 .... v 1122 0 vd1 .... .... .... v vu2 0 . .. .Fd. = . ... . 2 0 .. .. 1v v 12 vd2 .. .. .Fu. = , v . vu3 . . 0 . .1..1. v 2 2 0 vd3 The contributions from .Fu. and .Fd. generate the W and Zo SM gauge boson masses 22 '2 g(g+ g ) 22 22 (v + v )W+W- +(v + v )Z2 (11.14) 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 u 1u 2u 3u , d 1d2d3d2 22 v = vu + vd ˜ 246 GeV. 11.5 Fermion masses 11.5.1 Dirac see-saw mechanisms 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 and the corresponding vector-like fermionsU,D,EandN: The scalars and fermion content allow the gauge invariantYukawa couplings for quarks and charged leptons Hu .o Fu Uo + hu .o .1 Uo + hu .o .2 Uo + MU Uo Uo + h.c q R.1 uRL .2 uRL LR Hd .o Fd Do + hd .o .1 Do + hd .o .2 Do + MD Do Do + h.c q R.1 dRL .2 dRL LR He .o Fd Eo + he .o .1 Eo + he .o .2 Eo + ME Eo Eo + h.c l R.1 eRL .2 eRL LR MU ,MD ,ME are free mass parameters and Hu ,Hd He ,hf hf , f = u, d, e are .1 .2 coupling constants. When the involved scalar fields acquire VEV’s, we get for To charged leptons in the gauge basis .o =(e,µo,to,Eo)L,R, the mass terms L,R .¯ o Mo.o + h.c, where LR ...... 0 00He vd1 0 00He vd2 Mo = (11.15) . 0 00He vd3 he .1 he 12.2 0ME . It is worth to notice that completed analogous tree level mass matrices are obtained for u anddquarks Mo is diagonalizedby applyinga biunitary transformation .o = Vo L,R L,R .L,R. T Vo Mo Vo = Diag(0, 0, -.3,.4) (11.16) LR TT Vo MoMoT Vo = Vo MoT Mo Vo = Diag(0, 0, .23,.2 ), (11.17) LLRR 4 where .3 and .4 are the nonzero eigenvalues, .4 being the fourth heavy fermion mass, and .3 oftheorderofthetop, bottomandtau massforu,dande fermions, respectively.We see from Eqs.(11.16,11.17) that from tree level there exist two massless eigenvalues associated to the light fermions: 11.6 Neutrino masses Now we describe the procedure to generate neutrino masses 11.6.1 Tree level Dirac neutrino masses With the fieldsof particles introducedin the model, we may write the Diractype gauge invariantYukawa couplings .¯ o No .¯ o .¯ o .¯ o hDl FuR + h1. .1 NoL + h2. .2 NoL + h3. .3 No L + MD N¯o No + h.c (11.18) LR hD, h1, h2 and h3 areYukawa couplings, and MD a Dirac type, invariant neutrino mass for the sterile neutrinos No . After electroweak symmetry breaking, we L,R T obtain in the interaction basis .o =(.o ,.o ,.to ,No)L,R, the mass terms .L,R eµ hD v1 .¯eLo + v2 .¯µLo + v3 .¯tLo NRo No No + h1 .1 .¯o .o .o L + MD ¯NRo + h.c. (11.19) eR + h2 .2 ¯µR + h3 .3 ¯tRL 11.6.2 Tree level Majorana masses: Since No ,Table 1, are sterile neutrinos, we may also write left and right handed L,R Majorana type couplings hL .¯o Fu(No )c + mL N¯o (No )c + h.c (11.20) lLLL and .¯o )c .¯o )c .¯o )c h1R . .1 (NoR+ h2R . .2 (NoR+ h3R . .3 (NoR + mR N¯ o (No )c + h.c , (11.21) RR respectively. After spontaneous symmetry breaking, we also get the left handed and right handed Majorana mass terms hL v1 .¯eLo + v2 .¯µLo + v3 .¯tLo (NLo )c + mL N¯Lo (NLo )c + h.c. , (11.22) + h1R .1 .¯oeR + h2R .2 .¯oµR + h3R .3 .¯otR(NoR)c + mR N¯ o (No )c + h.c. (11.23) RR T Thus, in the basis .o = .o ,.o ,.o ,NLo , (.o )c , (.o )c , (.o )c , (No )c , the . eLµLtLeRµRtRR Generic 8 × 8 tree level Majorana mass matrix for neutrinos Mo ,fromTable 11.3, ..¯o Mo (.o )c , read . .. .000a1 0 00a1 .000a2 0 00a2 .. .000a3 0 00a3 . .. .a1 a2 a3 mL b1 b2 0mD . Mo = (11.24) . b1 ß1 . 000 000 . .. .000b2 0 00ß2 . .. .. 000 0 0000 .. a1 a2 a3 mD ß1 ß2 0mR Diagonalization of M. (o) , Eq.(11.24), yields four zero eigenvalues: T ooo Uo Mo Uo = Diagonal(0, 0, 0, 0, m5o ,m 6,m 7 ,m ) (11.25) ... 8 .o eL .o µL .o tL No L (.o )c eR (.o )c µR (.o )c tR (No )c R(.o (.o (.o (NoL)c .o .o .o No eL)c µL)c tL)c eR µRtR R 0 0 0 hLv1 0 0 0 hDv1 0 0 0 hLv2 0 0 0 hDv2 0 0 0 hLv3 0 0 0 hDv3 hLv1 hLv2 hLv3 mL h1.1 h2.2 0 MD 0 0 0 h1.1 0 0 0 h1R.1 0 0 0 h2.2 0 0 0 h2R.2 0 0 0 0 0 0 0 0 hDv1 hDv2 hDv3 MD h1R.1 h2R.2 0 mR Table 11.3:Tree Level Majorana masses 11.7 One loop neutrino masses: 11.7.1 One loop Dirac Neutrino masses After the breakdown of the electroweak symmetry, neutrinos may get tiny Dirac mass terms from the generic one loop diagram in Fig. 1, The internal fermion line in this diagram represent the tree level see-saw mechanisms, Eqs.(11.18-11.23). The vertices read from the SU(3)family symmetry interaction Lagrangian gH gH iLint = 2 (.¯o .µ.o -.¯to .µ.ot )Z1µ + v ( .¯o .µ.o -2.¯o .µ.o + .¯ot .µ.to )Z2µ ee eeµµ 23 gH+++ + v .¯o .µ.o Y+ .¯o .µ.o + .¯o .µ.o + h.c. (11.26) eµ1 et Y2 µt Y3 2 The contribution from these diagrams may be written as 2 aHH cY m.(MY)ij ,aH = g, (11.27) p 4p X oo m Uo Uo f(MY,m ), (11.28) m.(MY)ij = k .ik .jk kk=5,6,7,8 M2 M2 M2 o YY Y f(MY,mk)= M2 o2 ln o2 ˜ ln o2 ,M2Y >> mo 2 k valid for neutrinos. -mmm Ykk k 11.7.2 One loop L-handed and R-handed Majorana masses Neutrinos also obtain one loop corrections to L-handed and R-handed Majorana massesfrom the diagramsof Fig.2and Fig.3,respectively.Asimilarprocedure as for Dirac Neutrino masses, leads to the one loop Majorana mass terms iLjR. M= MD,mL,mR .¯o eL .¯o µL .¯o tL N¯ o L .o .o .o No eR µR tRR D. 15 D. 16 00 D. 25 D. 26 00 D. 35 D. 36 D. 37 0 0 0 00 aH Table 11.4: One loop Dirac mass terms .o .o p D. ij ¯iL jR iLjL Thus, in the .o basis, the one loop contribution for neutrinosread . .o eL .o µL .o tL No L .o .o .o No eL µL tL L L. 11 L. 12 L. 13 0 L. 12 L. 22 L. 23 0 L. 13 L. 23 L. 33 0 0 000 Table 11.5: One loop L-handed Majorana mass terms a H p L. ij .¯iLo (.jLo )T .o eR R. 55 R. 56 0 0 .o µR R. 56 R. 66 0 0 .o tR 0 0 0 0 No R 0 0 0 0 .o .o .o eR µRtR NRo Table 11.6: One loop R-handed Majorana mass terms H ap R. ij .¯iRo (.jRo )T iRjR . L. 11 L. 12 L. 13 0D. 15 D. 16 00 . . . .L. 12 L. 22 L. 23 0D. 25 D. 26 00 . .. .. .. L. 13 L. 23 L. 33 0D. 35 D. 36 D. 37 0 . . .. .. .0 0 000000 . .. Mo = aH , (11.29) 1. . . p .D. 15 D. 25 D. 35 0R. 55 R. 56 00 . .. .. . . .D. 16 D. 26 D. 36 0R. 56 R. 66 00 . .. .. .. 0 0D. 37 00 0 00 . . .. .. 0 0 000000 where,afterusingtherelationshipscomingfromthezeroentriesof Mo .,eq.(11.24); ooo T Mo = Uo Diagonal(0, 0, 0, 0, m5o ,m 6,m 7 ,m )Uo , (11.30) .. 8. and in the limit M2 >> mo2 , we may write: Yk 1 L. ij = Fij , i, j = 1, 2, 3 3 -1 D. 15 = 1 F15 + 1 F26 , D. 16 = F16 , 32 6 -1 D. 25 = F25 , D. 26 = 1 F26 + 1 F15 , 6 32 1 -1 -1 D. 35 = 6 F35 , D. 36 = F36 , D. 37 = 2 (F15 + F26) 6 111 R. 55 = F55 ,R. 56 = F56 ,R. 66 = F66 333 where o2 o2 o2 mmm Uo Uo8 Uo8 Uo8 Fij = ln + Uo ln + Uo ln (11.31) .i5 .j5 o2 .i6 .j6 o2 .i7 .j7 o2 mmm 567 11.7.3 Neutrino mass matrix up to one loop Finally, we obtain the Majorana mass matrix for neutrinos up to one loop T ooo = Uo Mo U.o + Diag(0, 0, 0, 0, m5o ,m 6,m 7 ,m ), (11.32) M..1. 8 11.7.4 (VCKM)4×4 and (VPMNS)4×8 mixing matrices Within this scenario, the transformation from massless to physical mass fermion eigenfields for quarks and charged leptons is (1)(1) .o = Vo Vand .o = Vo V.R , L LL .L RRR and for neutrinos .o = Uo U1 ..; . .. T U1 M. U1 = Diagonal(.1,.2,.3,.4,.5,.6,.7,.8) (11.33) .. Recall now that vector like fermions,Table1,are SU(2)L weak singlets, and hence, they do not couple to W boson in the interaction basis. So, the coupling of L- TooT handed up and down quarks; fo =(u ,c ,to)L and fo =(do ,s o,bo)L, to uL dL the W charged gauge boson is W+µ v gf¯o fo uL.µdL 2 (1)(1) = v g.¯ uL [(Vo )3×4]T (Vo )3×4 .µ, (11.34) uL VuL dL VdL .dL W+µ 2 with g the SU(2)L gauge coupling. Hence, the non-unitary VCKM of dimension 4 × 4 is identified as (1)(1) (VCKM)4×4 = uLuL dL )3×4 [(Vo V)3×4]T (Vo V(11.35) dL (1)(1)(1)(1) [Vo (Vo (Vo uL VuL ]3×4 = uL)3×4 (VuL )4×4 , [VdLo VdL ]3×4 = dL)3×4 (VdL )4×4 Similar analysis of the coupling between active L-handed neutrinos and L-handed charged leptons to W boson, leads to the lepton mixing matrix (1) )3×4]T (Uo U1 (UPMNS)4×8 = eL V.(11.36) [(Vo )3×8 eL . (1)(1) [Vo =(Vo )3×4 (V)4×4 , (Uo U1 =(Uo )3×8 (U1 eL VeL ]3×4 eLeL ..)3×8. .)8×8 11.8 Numerical results for Neutrino masses and mixing in a 3+5 scenario Wereportherenumericalresultsforlepton massesandmixing,attheMZ scale [15] The input values for the horizontal boson masses, Eq.(8), and the coupling constant of the SU(3)family symmetry are: M1 = 5.3 × 103 TeV ,M2 = 3.3 × 105 TeV ,aH = 0.05 , (11.37) p .1 = 3352.7 TeV , .2 = 103 .1 ,gH = 2.23561 Horizontal gauge bosons from the SU(3)family symmetry introduce flavor chang­ing couplings, and in particular mediate .F = 2 processes attree level. The above high scales and heavy boson masses provide the proper suppression of Ko - K¯o and Do - D¯ o meson mixing from the tree level exchange diagrams mediated by the SU(3)horizontal gauge bosons. 11.8.1 Charged leptons: Tree level: Mo = e .. . 0 0 0 2670.25 0 0 0 11902.6 0 0 0 16264.7 1.21882 × 1010 -2.32202 × 109 0 6.07835 × 1010 .. . MeV , 0 -19.9797 -83.226 -16.9884 0.6408 71.9782 293.027 59.814 MeV -0.8544 168.853 -1712.54 480.432 -2.74 × 10-7 0.000054 0.000755 6.20 × 1010 Me = the charged lepton masses (me ,mµ ,mt ,ME )=(0.486031 , 102.717 , 1746.17 , 6.20 × 1010 )MeV Mixing matrix: (1) Vo VeL = eL VeL : 0.986458 0.0744614 -0.146138 4.30921 × 10-8 0.00276675 -0.898433 -0.439101 1.93334 × 10-7 -0.163991 0.43275 -0.886473 2.62497 × 10-7 0 5.68933 × 10-8 3.23887 × 10-7 1 .... .... ....... . 148 Albino Hernandez-Galeana 11.8.2 Neutrino masses and Lepton (UPMNS)4×8 mixing: Tree levelMo ., eq.(11.24):in eV 0 0 0 975.261 0 0 0 13.2472 0 0 0 4601.39 0 0 0 62.502 0 0 0 5663.49 0 0 0 76.9286 975.261 4601.39 5663.49 800. 1404. 2188.33 0 22500. 0 0 0 1404. 0 0 02.73 × 107 0 0 0 2188.33 0 0 0 1.24382 × 107 0000 0 000 13.2472 62.502 76.9286 22500. 2.73 × 107 1.24382 × 107 0 1.81238 × 109 Mo , eq.(11.29):in eV 1. -0.57491 -2.71249 -3.33859 0 -0.755918 0.148816 0 0 -2.71249 -12.7979 -15.7519 0 -3.18585 0.62145 0 0 -3.33859 -15.7519 -19.3877 0 -3.9212 0.864196 -0.0806809 0 00 0000 00 -0.755918 -3.18585 -3.9212 0 -284546. -129640. 0 0 0.148816 0.62145 0.864196 0 -129640. -59065.7 0 0 00 -0.08068090 0 0 0 0 00 0000 00 M., eq.(11.32):in eV 00 0 -0.0470918 0.00670865 0.00603167 0.0648786 -0.00107364 0 0 0.0515474 0.0490276 -0.00671814 -0.00655172 0.0368801 -0.00061029 0 0.0515474 0 -0.0116693 -0.0442476 -0.0419316 3.5622 × 10-3.4263 × 10 -6 -9 -0.0470918 0.0490276 -0.0116693 -3.45879 -6.3112 -5.99272 2.35948 -0.0390377 0.00670865 -0.00671814 -0.0442476 -6.3112 -7121.58 -12.9471 -873.148 14.4492 0.00603167 -0.00655172 -0.0419316 -5.99272 -12.9471 7892.83 897.758 -14.8564 0.0648786 0.0368801 3.5622 × 10-6 2.35948 -873.148 897.758 -839967. 5684.58 -0.00107364 -0.00061029 -3.4263 × 10-0.0390377 14.4492 -14.8564 5684.58 1.8128 × 10 -99 .... .... ..... ....... ..... .... ...... ..... .... .... .... ..... . . . .......................... .......................... Neutrino masses: (m1 = 0.000645302 , m2 = 0.0510146 , m3 = 0.0517498 , m4 = 3.45909 , m5 = 7120.68 , m6 = 7893.8 , m7 = 839969 , m8 = 1.81288 × 109)eV Squared neutrino mass differences: 22 m-m= 2.602 × 10-3 eV2 21 22 m-m= 7.555 × 10-5 eV2 32 22 m4 -m1 = 11.965 eV2 Neutrino mixing: U. = Uo U1 .. . ........................... 0.97728 0.123928 -0.106511 -0.035932 -0.211124 0.529448 -0.536153 -0.103884 -0.000103928 -0.448468 0.452078 -0.1536 -0.0000170002 9.508 × 10-6 -0.0000155169 -0.00116041 -0.00557906 0.00505439 -0.00311974 -0.407139 0.0122451 -0.0110938 0.00684744 0.893615 0.0129939 -0.709263 -0.704815 -0.00358262 1.45741 × 10-9 6.46089 × 10-10 -1.15813 × 10-10 9.37014 × 10-9 0.0945439 -0.0893912 3.50955 × 10-6 7.30727 × 10-9 0.446074 -0.421754 0.000016186 3.44794 × 10-8 0.549035 -0.519105 0.000019873 4.24383 × 10-8 -0.687182 -0.726481 -0.0021559 0.0000124295 -0.0576333 0.0524664 0.909874 0.015054 0.122879 -0.11889 0.41455 0.00685878 6.22084 × 10-6 5.30567 × 10-6 00 0.0000333177 0.0000346837 -0.0165428 0.999863 . ......................... (UPMNS)4×8 lepton mixing matrix: . .......... 0.963479 0.19726 -0.180688 -0.0105438 0.262405 -0.660521 0.669404 0.0241869 -0.0500208 0.146962 -0.149764 0.187029 -9 -8 -8 -8 1.26831 × 10-1.00207 × 101.04226 × 10-6.19523 × 10 0.00446092 -0.00421887 2.47814 × 10-7 3.442 × 10-10 -0.156133 0.147619 -5.68064 × 10-6 -1.20681 × 10-8 -0.696392 0.658429 -0.000025237 -5.38282 × 10-8 2.34435 × 10-7 -2.21655 × 10-7 00 . ......... 11.9 Conclusions Wehavereportedanupdated numericalanalysisforneutrino massesandmixing in a 3+5 scenario, within a local SU(3)Family symmetry model, which combines tree level ”Dirac see-saw” mechanisms and radiative corrections to implement a successful hierarchical spectrum, for charged fermion masses and mixing. The massof the activeSM neutrinos anda sterile neutrino with massofa few eV’s come out from the application of the see-saw approximation. Weupdate numericalresults for neutrinos andreport the non-unitary(UPMNS)4×8 lepton mixing matrix. 11.10 Acknowledgements It is my pleasure to thank the organizers N.S. Mankoc-Borstnik, H.B. Nielsen, M. Y. Khlopov, and all participants for this year stimulatingVirtual BledWorkshop 2021. This work was partially supportedby the ”Instituto Polit ´ ecnico Nacional”, Grant from COFAA. References 1. A. Hernandez-Galeana, Rev. Mex. Fis. Vol. 50(5), (2004) 522. hep-ph/0406315. 2. A. 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Zhi-zhongXing,HeZhangandShunZhou,Phys.Rev.D 86, 013013 (2012). Proceedings to the 24th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ... (p. 152) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 12 BSM Cosmology from BSM Physics M.Yu. Khlopov e-mail: khlopov@apc.univ-paris7.fr Virtual Instituteof Astroparticle physics, Universit´ e de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France; Center for Cosmoparticle physics “Cosmion”, National Research Nuclear University MEPhI, 115409 Moscow, Russia, and Research Institute of Physics, Sousthern Federal University, Stachki 194, Rostov on Don 344090, Russia Abstract. Now Standard . CDM cosmology is based on physics Beyond the Standard Model (BSM), which in turn needs cosmological probes for its study. This vicious circle of problems canberesolvedby methodsof cosmoparticle physics,in whichcosmological mes­sengers of new physics provide sensitive model dependent probes for BSM physics. Such messengers, which are inevitably present in any BSM basis for now Standardcosmology, leadto deviationsfromthe Standardcosmologicalparadigm.Wegive briefreviewof some possible cosmological features and messengers of BSM physics, which include balancing of baryon asymmetry and dark matter by sphaleron transitions, hadronic dark matter and exotic cosmic ray components,a solution for puzzlesof direct dark matter searchesin dark atom model, antimatter in baryon asymmetrical Universe as sensitive probe for models of inflation and baryosynthesis and its possible probe in AMS02 experiment, PBH and GW messengers of BSM models and phase transitions in early Universe. These aspects are discussed in the general framework of methods of cosmoparticle physics. Povzetek: Kozmologija CDM temelji na fiziki, ki standardni model raz. siri (BSM), ven­dar so zato potrebne metode, primerne za opazovanje novih fenomenov, ki jih nove teorije ponujajo.Avtor ponudi kratekpregled nekaterih kozmolo .sirijo stan­ skih modelov, ki raz.dardni model in napovedujejo uravnote. zenje barionske asimetrijein temne snovispre hodi sfalerona, hadronsko temno snovin eksoti.cnih. cne komponente kozmi.zarkov. Ponuja tudi razlago za dosedanje neujemanje rezultatov razli. cnih poskusov, ki merijo sipanje delcev temne snovi na merilnih aparaturah, razlago za morebitni obstoj antisnoviv barionskem asimetri.cutljiva sonda za preizku. cnem vesolju, ki utegne biti ob.sanje modelov za inflacijo vesolja in bariosinteze ter mo . zna sonda v eksperimentu AMS02, PBH in GW ter za fazne prehodev zgodnjem vesolju. 12.1 Introduction The now Standard model of elementary particles appeals to its extension for recovery of its internal problems and/or embedding in the framework of unified description of the fundamental natural forces (see [23] for recent review). Such extensions are unavoidable in the fundamental physical basis for now Standard cosmological scenario, involving inflation, baryosynthesis and dark matter/energy [2,3,3–5,8,9,20].ProbesfortheBSM physics, underlying now standardcosmology, inevitably imply methods of cosmoparticle physics of cross disciplinary study of physical, astrophysicaland cosmological signaturesof new physics[3,8].Here we discuss some development of these methods presented at the XXIV Bled Workshop ”What comes beyond the Standardmodels?” with special emphasis on the cosmological messengers of new physics, which can find positive evidence in the experimental data and thus acquire the meaning of signatures for the corresponding BSM models, specifying their classes and ranges of parameters. If confirmed, such cosmological signatures should find explanation together with the basic elements of the modern cosmology. Therefore, the approach, which pretends on the unified descriptionof Nature [11,15] should not onlyreproduce the Standardmodel of elementary particles and propose BSM features, which provide realistic description of inflation, baryosynthesis and dark matter, but should be in possession to confront possible signatures of new physics, which can go beyond the standardcosmological paradigm. Cosmological messengersof new physics canhelptoremove conspiracyofBSM physics, related with absence of its experimental evidence at the LHC, as well as conspiracy of BSM cosmology, reflected in concordance of the data of precision cosmology with now standard .CDM cosmological scenario [12]. Multimessenger cosmological probes can provide effective tool to study new physics at very high energy scale [13, 14]. Signatures for new physics play especially important role inthese studies.TheycanstronglyreducethepossibleclassofBSMmodelsand provide determination of their parameters with high precision. We consider such possible signatures in the direct searches of dark matter (Section 12.2.1), in gravitational wave signals from coalescence of massive black holes and searches for antinuclear component of cosmic rays (Section 12.3.We spec­ify open questions in their confrontation with the corresponding messengers of BSM physics.We discussis the conclusive Section 12.4 there signatures and their significance in the context of cosmoparticle physics of BSM physics and cosmology. 12.2 Signatures of dark matter physics 12.2.1 Dark atom signature in direct dark matter searches The highly significant positive result of underground direct dark matter search in DAMA/NaI and DAMA/LIBRA experiments [15] can hardly be explained in the frameworkof the Standardcosmological paradigmofWeakly Interacting Massive Particles (WIMP), taking into account negative results of direct WIMP searches by other groups (see [23] for review and references). Though these apparently contradicting results may be somehow explained by difference of experimental strategy and still admit WIMP interpretation, their non-WIMP interpretation seems much more probable, making the positive results of DAMA group the signature for dark atom natureof cosmological dark matter [14,16,17,23]. The idea that dark matter can be formed by stable particles with negative even charge -2n bound in dark atoms with n primordial helium nuclei can qualitatively explain negative results of direct WIMP searches based on the search for nuclear recoil from WIMP interaction [14, 16, 23]. Dark atom interaction with matter is determined by its nuclear interacting helium shell, so that cosmic dark atoms slow downin terrestrial matter and cannot cause significantrecoilin underground experiments. However,in the matter of underground detector dark atoms can form low energy(fewkeV)bound stateswith nucleiof detector.The energyreleasein such binding possess annual modulationdueto adjustmentoflocal concentration of dark atoms to the incoming cosmic flux and can lead to the signal, detected in DAMA experiment. Dark atoms represent strongly interacting asymmetric dark matter, since the corresponding models assume excess of -2n charged particles over their +2n antiparticles. Such excess can naturally be related with baryon asymmetry, if mul­tiple charged particles possess electroweak charges and participate in electroweak sphaleron transitions. It is shown in [18] that the excess of -2n charged particles in modelofWalkingTechnicolor (WTC) and U ¯antiquarks (with the charge -2/3 of new stable generation can be balanced with baryon asymmetry and explain the observed dark matter density by dark atoms. The open question is whether such balance, which shouldalso take place in the case stable 5th generation in the approach [11], can lead to the sufficient excess of u¯5 antiquarks to implement the idea of dark atoms in this case. Pending on the value of -2n charge, multiple charged constituents of dark atoms form either Bohr-like OHe atoms, binding -2 charged particles with primordial helium nucleus, or Thomson-like XHe atoms for n>1. In the first case, double charged particles may be either composite, being formed by chromo-Coulomb binding in cluster U ¯U ¯U ¯of stable antiquarks U ¯with charge -2/3, or -2 stable technileptons or technibaryons. Heavy quark clusters have strongly suppressed interaction with nucleons, while techniparticles behave as leptons. It leads to rather peculiar properties of dark atom -they have a heavy lepton or lepton-like core and nuclear interacting helium shell, which determines their interaction with baryonic matter. Though interactionof dark atoms withnucleiare determinedby their helium shell and thus don’t involve parameters of new physics, the problem needs develop­ment of special methods for its solution. The approach of [19], assumed continuous extension of a classical three body problem to realistic quantum-mechanical de­scription, taking into account finite size of interacting nuclei and helium shell, in order to reach self-consistent account for Coulomb repulsion and nuclear at­traction, which can lead to creation of a shallow potential well with low energy bound state in dark atom -nucleus interaction. The developmentof this approach is presented in [20] for both Bohr-like and Thomson-like atoms. However, it be­comes clear that probably the correct quantum-mechanical description should start from very beginning from quantum-mechanical nature of dark atom and numerical solutions for Schrodinger equation for dark-atom -nucleus quantum system. Development of self-consistent quantum-mechanical model of dark-atom interaction with nuclei and will makepossible interpretationof theresults [15]in terms of signature of dark atoms. 12.2.2 Multimessenger probes for decaying dark matter Development of large scale experimental facilities like IceCube, HAWC, AUGER and LHAASO provides multimessenger astronomical probes for cosmological messengers of superhigh energy physics [21]. The complex of LHAASO can pro­vide unique measurement of ultra high energy photons, being in some cases most sensitive probe for existence of messengers of new physics at ultra-high energy scales. Superheavy decaying dark matter may be one of such messengers. Its decay products may contain ultrahigh energy neutrinos, photons, charged leptons and quarks. Sensitivity of LHAASO for the measurement of dark matter decay time forDMdecayingtoquarksis demonstratedonFig.22.9,takenfrom[22].Yellow band shows the range of decay times for which DM decays give sizable contribu­tion to the IceCube neutrino signal [23]. Blue and gray shaded regions show the existing bounds imposedby HAWC [24] and ultra-high-energy cosmic ray experi­ments [25]. and dashed curves are from the HAWC search of the DM decay signal in the Fermi Bubble regions [26]. It makes possible to confront multimessenger cosmological probes with the data of multimessenger astronomy. 12.3 Signatures for strong primordial inhomogeneities 12.3.1 Massive PBHs Strong inhomogeneity of early Universe can lead to formation of primordial black holes (PBH). Such inhomogeneity may result from BSM physics at superhigh energy scales and thus even absence of positive evidence of PBH existence can provide important tool to probe allowed parameters of new physics at these scales [21,27]. Formed within the cosmological horizon, which was smallin the early Universe, it can seem that PBHs should have mass much smaller, than Solar mass M.. However, the mechanisms of PBH formation can provide formation of PBHs with stellar mass, and even larger than stellar up to the seeds for Active Galactic nuclei (AGN) [28–30]. LIGO/VIRGO detected gravitational wave signal from coalescence of black holes with masses exceedingthe limitofpair instability(50M.). Therefore black holes of such mass cannot be formed in the evolution of first stars. It has put forward thequestionontheirprimordialorigin[31,32]andmaybe consideredassignature for BSM physics, underlying formation of such massive PBHs. In the approach [28] massive PBHs are formed in the collapse of closed walls originated from succession of phase transitions of breaking of U(1) symmetry and their mass is determined by the scale f of spontaneous symmetry breaking at the inflationary stage and scale . of successive explicit symmetry breaking. Therefore confirmation of primordial origin of massive PBHs would strongly narrow the choice of models of very early Universe and its underlying physics. 12.3.2 Cosmic antinuclei as probe for matter origin Baryon asymmetryof the Universereflects absenceof macroscopic antimatterin the amount comparable with baryonic matter within the observed Universe. Its originisrelated with the mechanismof baryosynthesis,in which baryon excess is created in very early Universe. However, inhomogeneous baryosynthesis can lead not only to change of the value of baryon excess in different regions of space, but in the extreme case can change sign of this excess, giving rise to antimatter, producedinthe sameprocess,in whichthe baryonic matter wascreated[3,9,11, 13,17,33]. Antimatter domains shouldbesufficientlylargeto survivein matter surrounding and it implies also effect of inflation in addition to nonhomogeneous baryosynthesis. It means that the prediction of macroscopic antimatter, surviving to the present time, involves rather specific combination of necessary conditions and correspondingly specific choice of BSM model parameters. The choice of BSM model parameters determines the forms of macroscopic an­timatter in our Galaxy. Antimatter domain can evolve in the way, similar to the baryonic matter and form antimatter globular cluster in our Galaxy [3,39]. The an-tibaryon density may be much higher, than the baryonic density and then specific ultra-dense antibaryon starscanbe formed[12],Inanycase,thepredicted fraction of antihelium nuclei in cosmic rays from astrophysical sources is far below the sensitivity of AMS02 experiment, making positive results of cosmic antihelium signature of macroscopic antihelium in our Galaxy. The possibility of confirmation of first indications to the antihelium events in AMS02 makes necessary to study in more details evolution of antibaryon domains in baryon asymmetrical universe [42,43]in the contextof modelsof inhomoge­neous baryosynthesis. It makes necessary to study expected composition and spectrum of cosmic antinuclei from antimatter globular cluster [44], as well as to consider more general question on propagation in galactic magnetic fields of antinuclei from local source in galactic halo [45] 12.4 Conclusions There are some hints to new phenomena in the observational data [46–48]. The deviationsfromthe standardcosmologicalmodelmayberelatedwiththe modified gravity [49, 50], leading beyond the Standardmodel of all the four fundamental interactions. Then one can expect additional types of polarization of gravitational waves [51]. Such hints are not at such high significance level as the results of DAMA experiments [15], but they can strongly extend the list of multimessenger probes of BSM physics. Constraints on such exotic phenomena, as PBHs or antimatter in baryon asym­metrical Universe exclude rather narrow ranges of BSM model parameters. Signa­tures for such phenomena make these ranges preferential, strongly reducing the class of BSM models and fixing their parameters with high precision. 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Atom. Nucl. 63, 233 (2000). 39. M.Yu. Khlopov:An antimatter globular clusterin our Galaxy -aprobe for the originof the matter, Gravitation and Cosmology, 4, 69-72 (1998). 40. S.I. Blinnikov, A.D. Dolgov, K.A. Postnov: Antimatter and antistars in the universe and in the Galaxy, Phys. Rev.D 92, 023516 (2015). 41.V. Poulin,P. Salati,I. Cholis,M. Kamionkowski,J. Silk: Wheredo the AMS-02 anti-helium events comefrom? Phys. Rev.D 99, 023016 (2019). 42. M.Yu.Khlopov, O.M.Lecian: Analyses of Specific Aspects of the Evolution of An­timatter Glubular Clusters Domains. Astronomy Reports, 65, 967–972 (2021) DOI: 10.1134/S106377292110019X 43. M.Khlopov, O.M.Lecian: Statistical analyses of antimatter domains, created by non-homogeneous baryosynthesisina baryon asymmetricalUniverse. BledWorkshopsin Physics, 22 (2021), arXiv:2111.14114 This issue 44. M.Yu. Khlopov, A.O. Kirichenko, A.G. Mayorov: Anihelium flux from antimatter glob­ular cluster, BledWorkshops in Physics, 21, 118-127 (2020), arXiv:2011.06973 [astro-ph.HE]. 45. A.O.Kirichenko, A.V.Kravtsova, M.Yu.Khlopov and A.G. Mayorov Researching of magnetic cutofffor local sources of charged particles in the halo of the Galaxy. Bled Workshops in Physics,22 (2021), arXiv:2112.00361. This issue 46. Z. Arzoumanian et al [NANOGrav Collaboration]: The NANOGrav 12.5-year Data Set: Search For An Isotropic Stochastic Gravitational-Wave Background,Astrophys. J. Lett. 905, L34 (2020). 47. S.Vagnozzi: New physicsin lightof the H0 tension:An alternative view, Phys. Rev.D 102, 023518 (2020). 48. S.A. Levshakov, M.G. Kozlov, I.I. Agafonova: Constraints on the electron-to-proton mass ratio variation at the epoch of reionization, Mon.Not.Roy.Astr. Soc. 498,3624–3632 (2020). 49. P. Kroupa, M. Haslbauer, I. Banik, S.T. Nagesh, J. Pflamm-Altenburg: Constraints on the star formation historiesof galaxiesin the Local CosmologicalVolume, Mon. Not. Roy. Astr. Soc. 497, 37 (2020) 50. K.-H. Chae,F. Lelli, H. Desmond, S.S. McGaugh,P. Li, J.M. Schombert:Testing the Strong Equivalence Principle: Detection of the External Field Effect in Rotationally Supported Galaxies, Astrophys. J. 904 (2020) arXiv:2009.11525. 51. S. Roy Chowdhury, M.Yu. Khlopov: Gravitational waves in the modified gravity. Bled Workshops in Physics,22, 127-135 (2021), arXiv:2111.07704. This issue Proceedings to the 24th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ... (p. 160) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 13 Statistical analyses of antimatter domains, created by nonhomogeneous baryosynthesis in a baryon asymmetrical Universe M.Yu. Khlopov1 email: khlopov@apc.in2p3.fr O.M. Lecian2 email: orchideamaria.lecian@uniroma1.it 1Institute of Physics, Southern Federal University, Russia; National Research Nuclear University ”MEPHI”, 115409 Moscow, Russia, and Universit ´ e de Paris, CNRS, Astroparticule et Cosmologie, France; 2 Sapienza University of Rome, Faculty of Medicine and Pharmacy, V.le Regina Elena, 324-00185 Rome, Italy Abstract. Within the framework of scenarios of nonhomogeneous baryosynthesis, the formation of macroscopic antimatter domains is predicted in a matter-antimatter asymmet­rical Universe. Thepropertiesof antimatter within the domains are outlined;the matter-antimatter boundary interactions are studied. The correlation functions for two astrophysi­cal objects arecalculated. The theoretical expression in the limiting process of the two-points correlation function of an astrophysical object and an antibaryon is derived. Povzetek: Avtorja uporabita model nehomogene bariosinteze za nesimetri.cno vesolje. . Studirata nastanek domen antisnovi, lastnosti antisnovi v domenah ter interakcijo med snovjo in antisnovjo na mejah domen. Izra. cunala sta korelacijske funkcije dveh astrofizikalnih objektovvnehomogenemvesolju. Prika .cnerelacije za limitneprocese dvoto. zeta teoreti.ckovnih korelacijskih funkcij astrofizikalnega objekta in antibariona. Keywords: General Relativity, Dark matter. 13.1 Introduction The origin of the baryon asymmetry of the Universe is explained in the now Stan-dardcosmology by the mechanism of baryosynthesis. If baryon excess generation is nonhomogeneous, the appearance of domains with antibaryon excess can be predicted in baryon asymmetrical Universe. In such non-trivial baryosynthesis frameworks, we study evolution of antimatter domains according to their dependence on the size and antimatter densities within them. The boundary conditions for antimatter domains are determined through the interaction with the surrounding baryonic medium. Within the analysis, new classifications for antibaryon domains, which can evolve in antimatter globular clusters, are in order. Differences mustbe discussed within therelativistic framework chosen, the nu-cleosynthesis processes, the description of the surrounding matter medium, the confrontation with the experimental data within the observational framework. The space-time-evolution of antimatter domains and the correlation functions are described within the nucleon-antinucleon boundary interactions. The manuscript is organized as follows. We consider formation of antibaryon domains in the spontaneous CP-symmetry-breaking scenario (Section 13.2) and in the model of spontaneous baryosynthesis. Evolutionofsuch domainsisdeterminedby nucleon-antinucleon interactionat the boundaries of antimatter domains. We deduce correlation functions for the celestial objects, predicted in these scenar­ios. Them manuscript is organized as follows. In Section 13.2, the symmetry-breaking scenario have been recalled. In Section 13.3, the cosmological implications have been studied. In Section 13.4, the spontaneous baryosynthesis process has been analyzed. ISection 13.5, different antimatter spacetime distributions have been presented. In Section 13.6, antimatter interactions have been studied. Is Section 13.7, nucleon-antinucleon interactions have been codified. In Section 13.8, correlation function for celestial objects have been analytically calculated. Is Section 13.9, brief outlook and perspectives have been outlined. Concluding remarks end the paper. 13.2 Symmetry-breaking scenario The symmetry-breaking scenarios have been stusied in [1]-[5]. The spontaneous CP violation is described [1] after the Lagrangean potential density 2 V(.1,.2,.)=-µ (.+ .1 + .+ .2)+.1[(.+ .1)2 +(.+ .2)2]+2.3(.+ 11212 1 .1)(.+ .2)(.+ .2)+2.4(.+ .2)(.+ .1)+.5[(.+ .2)2 + h.c.]+ 21 121 2 .6(.+ .1 + .+ .2)(.+ .2 + .+ .1)-µ.+. + d(.+ 1212 2 .)2 + 2a(.+.)(.+ .1 + .+ .2)+2ß[(.+ .)(.+.1)+(.+ .)(.+.2)]. 121 2 (13.1) Asaresult,aneffective low-energy electroweak SU(2). U(1)theory is achieved, allowing foraGUT spontaneous CP violation. The formation of vacuum structures separatedfromtherestofthe matter universeby domain walls follows.The sizeof the domainsis calculatedtogrow withthe evolutionoftheUniverse.The behavior is calculated not to affect the evolution of the Universe if the volume energy .(˜V) density of the walls for .(˜V)~ s2 T 4/h˜, . with h ˜value of the scalar coupling constant. In Eq. (13.1), for the three effective scalar fields, the CP violation is achieved with complex vev’s and vacuum domain structures appear with opposite CP viola­tion sign: walls are predicted to be massive, and the size of walls is predicted to grow [1]. ACP-invariant Lagrangian density can be assumed of the form [2] L =(..)2 -.2(.2 -.2)2 + .¯ (i. -m -ig.5.). (13.2) in which the vacuum is characterized by the values <.>= s., with s = ±1. Therotation . . eia.5 ., with tg2a =-gs./m induces the appearance in the Lagrangian density of two terms with opposite CP symmetry. The sign of the phase depends on s, with . = 1, for which 1 = .. For the Lagrangian density [6] 1 gmg2s. 2 L =(..)2 - m ..2 -4s.2..3 -.2.4 + .¯ (i. ^-M -i.5. - .).. (13.3) 2 MM a CP violation can be achieved after the substitution . = . + s.. For the Lagrangian potential 2 V(.)=-m .. * . + ..(. * .)2 + V0, (13.4) f f with . = v e ia ,a U(1)symmetry breaking is achieved, with . = a/f. 2 The domain wall problem can be solved after the Kuzmin-Shaposhnikov-Tkachev mechanism. 13.3 Implications in cosmology Several phenomena can be looked for following the described mechanisms. The research for antinuclei in cosmic rays is analyzed as a possible outcome of the model. Theresearch for annihilationproducts constitutesa further verificationprocedure for the theoretical framework. In particular, annihilation at rest on Relativistic background is to be studied. The annihilation of small-scale domains is a further investigation theme. It can be achieved within the thin-boundary approximation. Moreover, at different times, the diffusion of the baryon charge is determined after different processes. 13.4 Spontaneous baryosynthesis Aspontaneous baryosynthesis allowing for the possibility of sufficiently large domains through proper combination of effects of inflation and baryosynthesis is described after the choice of fields f . = v e ., for which the variancereads 2 H3t < d. >= . (13.5) 4p2f2 as in [7], [8], [9], [10], [11]. This way,the probability for the existence of antimatter particles is set. The number of objects N˜(t)- N˜(t0)is calculated as Z t. N˜(t)- N˜(t0)= P(.)ln .d.(t), (13.6) t0 with P(.)including variance. The evaluation of the number of antibaryons is performed after the use of the quantity H˜, i.e. the Hubble-radius function, and after the definition of effective quantities .feff, i.e. the effective (time-dependent) phase function, and g..MPl feff = f1 +(Nc -N), (13.7) 12p. i.e. the effective phase, with N the e-foldings at inflation. The following consequences are extracted. If the density is so low that nucleosynthesis is not possible, low density antimatter domains contain only antiprotons (and positrons). High density antimatter domains contain antiprotons and antihelium. Heavy elements can appear in stellar nucleosynthesis, or in the high-density antimatter domains. Strong non-homogeneity in antibaryons might imply (probably as a necessary condition) strong non-homogeneity for baryons, and produce some exotic results in nucleosynthesis. 13.5 Spacetime antimatter statistical distributions It is possible to specify the standarddeviation for the field . for different spacetime antimatter statistical distributions. 13.5.1 Binomial spacetime antimatter distribution In the hypothesis antibaryons are described as followinga binomial [12] statistical spacetime distribution, the number of antibaryons N ˜contained in an antimatter domainreads X 1.ta 2 (-2) LueHc(tc-t0) -eH0t0 k N˜(k)- N˜0(k).· ln (t)k-3 (k!)(1 -k)! .t0 .feff(t;ta,ti,t0) 4p2 l k (13.8) which is described after the effective quantities feff and those defined after the Hubble function H. 13.5.2 Poisson space-time antimatter statistical distribution In the hypothesis antibaryons are described as following a Poisson statistical space-time distribution, the number of antibaryons N ˜contained in an antimatter domainreads X k Hc(tc-t0) -H0t0 k kne.ta 2 (-2) Lue ˜e ˜ N˜(k)- N˜0(k)(.t).· ln (t)k-3 , n! .t0 .feff(t;ta,ti,t0) 4p2 l nk (13.9) which is described after the effective quantities feff and those defined after the Hubble function H. 13.5.3 Bernoulli spacetime antimatter distribution In the hypothesis antibaryons are described as following a Bernoulli statistical spacetime distribution, the number of antibaryons N ˜contained in an antimatter domainreads 1.ta 2 (-2) LueHc(tc-t0) -eH0t0 k N˜(k)- N˜0(k).· ln (t)k-3 (k!)(1 -k)! .t0 .feff(t;ta,ti,t0) 4p2 l (13.10) which is described after the effective quantities feff and those defined after the Hubble function H. 13.6 Antimatter domains and antibaryons interactions At the radiation-dominated era within the cosmological evolution, the dominant contribution to the total energy is due to photons. In the case of low density antimatter domains, the contribution of the density of antibaryons .is smaller than the contribution dueto the radiation .. even at the B matter-dominated stage. In a FRWUniverse, within its thermal history, for T < 100keV, only photons as a dominant components are considered. At the matter-dominated era and following within a non-homogeneous scenario, .DM >.B, with . = .(x). Thecreationofhigh density antibaryon domains canbe accompaniedby similar increase in baryon density in the surrounding medium. Therefore outside high density antimatter domain baryonic density may be also higher than DM density .B(x)>.DM(x). In the case of low density antimatter domains: the total density is such that .+ .., and .<.., with .dm >.B BB 13.7 Nucleon-antinucleon interaction studies Within the framework of the studies of nucleons-antinucleons interaction, several schematizations are possible. In the case of proton-antiproton annihilation probability, the limiting process and the theoretical formulation can be studied. Let P(p¯)be the probability of existence of one antiproton of mass mp, with mp being the proton mass, in the spherical shell of section rI, of (antimatter)-density .I, delimiting the antimatter domain, in which the interaction takes place P(p¯)= 3Nmp/(rI.I). This way, the interaction probability reads P˜i = P ˜p¯.(d.c.i) i.e. it constitutes the probability of antiproton p ¯interaction with a proton p in a chosen i annihilation channel a.c., possibly also depending on the chemical potential. Let .t be the time interval considered, under the most general hypotheses (most stringent constraint), .t ± dt, .t . tU . 4 · 1017s, with tU age of the universe, dt to be set according to the particular phenomena considered. This way ¯ p,i(t, .t). P ¯ i.e. the probability of antiproton interaction, i.e. antiproton-proton annihilation (density), reads 1 P ¯¯P ¯P˜i (13.11) p,i . p .t As a second study, the nucleon-antinucleon interaction (annihilation) probabilities are evaluated after the antinucleus M interaction probability P M,j(t, .t)through the annihilation chan- ¯¯ ¯ nel(s) k as 1 ¯ M,k(t, .t). P ˜(13.12) P ¯ P ¯¯ A A,k. .t In these examples, all the probabilities are normalized as [t-1]. The studies of nucleons-antinucleons interactions are to be further specified for non-trivial Relativistic scenarios such as perturbed FRWwith the thermal history of the Universe, i.e., also, according to the Standard Cosmological Principle. The non-trivial Relativistic scenarios are schematized as at large scales asymptoti­cally isotropic and homogeneous. Further specifications can be in order in the case of non-trivial nucleosynthesis, possibilities of surrounding media, antibaryon-baryon annihilation. In the latter ¯ case, the most stringent constraint follows after P evaluated for present times in the description of reducing density in the limiting process of a low-density antimatter domain. 13.7.1 An example In the example of low-density antimatter domains, non-interacting antiprotons are described, boundary interactions are taken onto account, and interaction with surrounding medium can be considered. In particular, low-density antimatter domains canbe surroundedby low-density matter regions. 13.8 Correlation functions Two-point correlation functions C˜2 for two antimatter domains a1 and a2 of size > 103M. each can be within the present framework analytically calculated. More in detail, on (homogeneous, isotropic) Minkowski-flat background, and under the hypothesis antimatter densities . = N/V following a Poisson space­ ˜time statistical distribution. The two-point correlation function C˜2 is defined as dC˜2(a1,a2)= .2(1 + .(|.ra1a2 |))dV1dV2 (13.13) where .(|.ra1a2 |)=|.ra1a2 | (13.14) defines the estimator, and .ra1a2 the distance of the two antimatter domains. Given two antimatter domains a1 and a2 of volume Val = 4 pr3 , separated of a 3l distance |.ra1a2 |the correlation function is analytically integrated as ˜˜11 ˜2k 4k-4 C2(a1,a2)= 2pn˜(n, k;.feff,H;.t)|.ra1a2 | + Hc t(13.15) r2 r1 evaluated at the present time t, with H˜c the effective Hubble-radius function. 13.8.1 An example: the two-point correlation functions for an antimatter domain and another object Itispossibleto considerthe limiting example of the correlation function between an antimatter domain a1 and an antibaryon a3. The Davis-Peebles estimator for the macroscopic objects described in terms of density distribution and temperature distribution reads ˜ Nbin Dl(|.r |) .l,l ' = -1, (13.16) ' N ˜Dl (|.r |) with N ˜number of antibaryons in a low-density antimatter domain, where the antimatter is assumed to be distributed according to a Poisson space-time statisti­cal distribution, and N˜bin the number of antibaryons in a low-density antimatter domain where the antimatter is distributed according to a binomial space-time statistical distribution, the quantity Dl(|.r |)indicates the number of pairs of low-density appropriate-mass antimatter domains within the geodesics (coordinate) interval distance dr dr ' r - ,r + , the quantity D (| .r |)indicates the number of pairs of objects 22 l between an antimatter domain and an(Poisson-distributed)antibaryon on the coordinate geodesics. For the Davis-Peebles estimator n˜bin(n, k;.feff,H˜;.t) Dl(|.r |) ' .l,l = -1 (13.17) ˜' n˜(n, k;.feff,H;.t) D (|.r |) l ˜2k within the use of statistical estimators, the time dependence Hc t4k-4 is sup-nbin(n,k;.feff,H;.t) ˜ pressed, and the time dependence is expressed after the ratio ˜, n˜(n,k;.feff,H˜;.t) i.e. on the different statistical antimatter space-time distributions and on their dependence on the H ˜Hubble-radius function, and on the .feff effective (time-dependent) phase function. 13.8.2 Hamilton estimator The Hamilton estimator .˜l,l takes into account the difference in distances among the Binomial distribution ' and the Poisson distribution. 13.9 Outlook and perspectives In the case antimatter domains are described to be separated in a small angular distance, the Rubin-Limber correlation functions [13], [14] for small angles can be used. An analysisof the metricrequiringa time evaluation after the timeof the surface of last scattering can be analyzed also for different metrics, as in [15] 13.10 Concluding remarks Prediction of macroscopic antimatter in baryon asymmetrical Universe is based on rather specific choiceof parametersof baryosynthesis.To make antimatter do­mains sufficiently large to survive in the baryon matter surrounding a nontrivial combination of baryosynthesis and inflation are needed. It may look like we study a highly improbable and very exotic case. However, on the other hand, positive evidence for existence of macroscopic antimatter in our Galaxy, which may appear in the searches of cosmic antinuclei in AMS02 [22] would strongly favor models, predicting antimatter domains in baryon asymmetric Universe, and would make possible to select the narrow classes of models of inflation and baryosynthesis, as well as to specify their parameters with high precision [20]. In view of this possibility we started to develop in the present work statistical analysis of possible space distribution of antimatter domains with the account for their evolution. Confrontation of the predicted distribution of antimatter domains withthe ob­servational data would be important for multimessenger test of the models of nonhomogeneous baryosynthesis. The observable signatures of this distribution is the important direction of our future studies. In particular, the most probable forms of the evolved antimatter in our Galaxy should be clarified in this analysis. It should be noted that the mechanisms of generation of antibaryon excess in baryon asymmetrical Universe may be accompanied by formation of domain walls at the border of antimatter domains. If these closed walls start to dominate, before they enter the horizon, the corresponding domains, surrounded by walls would become closed worlds, separating from our Universe. This open question is another challenge for our future analysis Acknowledgements The work by MK has been supported by the grant of the Russian Science Founda­tion (Project No-18-12-00213-P). References 1.V.A. Kuzmin, M.E. Shaposhnikov, I.I. Tkachev, Matter-antimatterdomainsin the Uni-verse:a solutionof the vacuum wallsproblem, Phys. Lett. B105,1(1981). 2. Ya.B. Zeldovich, L.B. Okun, L.Yu. Kobzarev, Cosmological Consequences of a Sponta­neousBreakdownofa Discrete Symmetry, Zh. Eksp.Teor. Fiz. 40,1(1974). 3.V.M. Chechetkin, M.G. Sapozhnikov, M.Yu. Khlopov,Ya.B. Zeldovich, Astrophysical aspects of antiproton interaction with 4He(antimatterin the universe) Phys. Lett. B 118, 329 (1982). 4. V.M. Chechetkin, M.G. Sapozhnikov, M.Yu. Khlopov,Antiproton interactions with light elementsasatestofGUT cosmology,Riv.N.Cim. 5,1(1982). 5. V.A. Kuzmin, I.I. Tkachev, M.E. Shaposhnikov, Are There Domains of Antimatter in the Universe?, Zh. Eksp.Teor. Fiz. Lett. 33, 557 (1981). 6. M.Yu. Khlopov, S.G. Rubin, A.S. Sakharov, Possible origin of antimatter regions in the baryon dominated universe, Phys. Rev. D62, 083505 (2000)­ 7. M.Yu. Khlopov, S.G. Rubin, A.S. Sakharov, Possible origin of antimatter regions in the baryon dominated universe, Phys. Rev. D62, 083505 (2000). 8. M.Yu. Khlopov, S.G. Rubin, A.S. Sakharov, Antimatter regions in the baryon domi­nated universe, 14th Rencontres de Blois on Matter-Anti-matter Asymmetry [hep­ph/0210012]. 9. A.D. Dolgov et al., Baryogenesis During Reheating in Natural Inflation and Comments on Spontaneous Baryogenesis, Phys. Rev. D56, 6155 (1997). 10. A.G. Cohen and D.B. Kaplan, Thermodynamic generationof thebaryon asymmetry, Phys. Lett. B199(1987) 251. 11. A.G. Cohen and D.B. Kaplan, Spontaneous baryogenesis, Nucl. Phys. B308(1988) 913. 12.J. Bernoulli,Ars Conjectandi,Opus Posthumum. AcceditTractatusde Seriebus infinitis, et Epistola Gallice scripta de ludo Pilae recticularis; Impensis Thurnisiorum, Fratrum, Basel (1713). 13. V.C. Rubin, Fluctuations in the space distribution of galaxies, Proc. Natl. Acad. Sci. USA 40, 541 (1954). 14. D.N. Limber, The Analysisof Countsof the Extragalactic NebulaeinTermsofa Fluctu­ating Density Field. II., Astrophys. J. 119, 655 (1954). 15. A. Pontzen, A. Challinor, Linearization of homogeneous, nearly-isotropic cosmological models, Class. Quant. Grav. 28, 185007 (2011), Eq. (52). 16. A.D. Dolgov: Matter and antimatter in the universe, Nucl. Phys. Proc. Suppl. 113, 40 (2002). 17. A. Dolgov, J. Silk: Baryon isocurvature fluctuations at smallscales and baryonic dark matter, Phys. Rev.D 47, 4244 (1993). 18. A.D. Dolgov, M. Kawasaki, N. Kevlishvili: Inhomogeneous baryogenesis, cosmic anti­matter, and dark matter, Nucl. Phys.B 807, 229 (2009). 19.V. Poulin,P. Salati,I. Cholis,M. Kamionkowski,J. Silk: Wheredo the AMS-02 anti-helium events comefrom? Phys. Rev.D 99, 023016 (2019). 20. M.Y. Khlopov. What comes after the Standard Model? Prog. Part. Nucl. Phys. 116, 103824, (2021). Proceedings to the 24th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ... (p. 170) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 14 Researching of magnetic cutofffor local sources of charged particles in the halo of the Galaxy A.O. Kirichenko2 e-mail: aokirichenko@yandex.ru A.V. Kravtsova2 M.Yu. Khlopov1,2,3 A.G. Mayorov2 1 Institute of Physics, Southern Federal University, Russia 2 National Research Nuclear University MEPHI, 115409 Moscow, Russia 3 Universit´ede Paris, CNRS, Astroparticuleet Cosmologie, F-75013 Paris, France Abstract. Models of highly inhomogeneous baryosynthesis of the baryonic asymmetric Universe allow for the existence of macroscopic domains of antimatter, which could evolve ina globular clusterof antimatter starsin our Galaxy.We assume the symmetryof the evolutionofaglobular clusterof starsand antistars basedonthe symmetryoftheproperties of matter and antimatter. Such object canbea sourceofa fractionof antihelium nucleiin galactic cosmic rays. It makes possible to predict the expected fluxes of cosmic antinuclei with useof knownpropertiesof matter star globular clustersWe have estimated the lower cutoffenergy for the penetration of antinuclei from the antimatter globular cluster, situated in halo, into the galactic disk based on the simulation of particle motion in the large-scale structureof magnetic fieldsintheGalaxy.Wehave estimatedthe magnitudeofthe magnetic cutofffor the globular cluster M4. Povzetek:Vmodelih, ki predpostavijo zelo nehomogeno bariosintezo asimetri. cnega vesolja, se lahko pojavijo makroskopske domene antisnovi.V na. si galaksiji bi domena antisnovi lahko nastala v kroglasti kopici zvezd iz antisnovi in bi bila izvor antihelijevih jeder v galakti.cnih.cekroglaste cnih kozmi.zarkih.Avtorji prilagodijo spremembo simetrije rasto .kopice zvezd in antizvezd tako, da se lastnosti ujemajo s poznanimi lastnostmi snovi. Od tod napovedo tok kozmi.si cnihantijeder.Ssimulacijo gibanja delcevvmagnetnih poljihvna.galaksiji ocenijo mejno energijo,pri kateri . se lahko antijedra iz kroglaste kopice antisnovi, kise nahajav haloju galaksije,prodrejov galakti. cnini disk.Tirezultati so namenjeni iskanju kozmi.cnega antihelija v poskusu AMS02. 14.1 Introduction Today, our knowledge of the chemical composition of galactic cosmic rays is being enhancedbyprecision experimentssuchasPAMELA[1],BESS[2]and AMS-02[3] in near-Earth orbit. Along with common components such as protons or helium nuclei, there is no doubt about the presence of antiprotons in the cosmic ray flux, and a search for heavier anti-nuclei is also underway. For the first time experimentally antinuclei were discovered at accelerators, which contributed to the development of theoretical models suggesting the existence of antimatter in the Universe and, in particular, in our Galaxy [4]. According to them, antimatter is usually classified into three groups: • Relic or primary. • Secondary. • From exotic sources. There is a classical mechanism of particle production, and antiparticles, as we understand it, are born as a secondary component(for example, positrons, an­tiprotons, antideuterons or antihelium [5]). Butnevertheless, modern models of cosmic ray generation suggest their formation and acceleration after supernova explosions in termination shocks [6] and when propagating in the interstellar medium, the fraction of various components in cosmic rays changes as a result of nuclearreactions with interstellar gas [5]. In addition, secondary particles or antiparticles are born that are initially absent from the sources, for example, positrons, antiprotons, antideuterons or antihelium. Also, the creation of galactic antiprotons as a result of annihilation or decay of massive hypothetical dark matter particles or during the evaporation of primordial black holes is not excluded [7]. In this case, the calculated fluxes can exceed the flux of the secondary component. Antideuteronscanbe formedinthe same mechanisms,butwithalesserprobability, they have not yet been detected in cosmic rays [5]. Antihelium was not found either: in the case of its secondary origin, the calculated ratio of the fluxes of antihelium and helium nuclei is small and does not exceed ~ 1012-1014 [5]. Detection of antinuclei above this value would indicate the existence of primordial antimatter,preservedfromthe momentoftheBigBang[8,9].The creationof4antinucleonsat once witharelatively smallrelative momentumin processes in exotic sources is unlikely. Today, primary antimatter could exist in the form of antimatter domains, which are not excluded in models of inhomogeneous baryosynthesis, taking the form of clusters of anti-stars or antigalaxies [10]. 14.2 Globular cluster of antistars Based on the symmetry of matter and antimatter [4], it is possible to indicate the expected parameters of a globular cluster of antistars. That is, a globular cluster will have the same set of properties as an ordinary cluster of ordinary stars. This approach assumes similar initial conditions and similar evolution of antimatter and matter. One should note that the mechanisms of nonhomogeneous baryosynthesis may lead to difference in the conditions within antimatter domain and in ordinary baryonic matter. In particular, the approach [11, 12] predicts much higher antibaryon density in antimatter domains, than in surrounding baryonic matter, what leads to prediction of ultra dense antibaryonic objects in the Galaxy and their specific effects [12]. Here we follow the approach of [13], which assumes similar conditions in antimatter domains as in the surrounding matter, and elaborate the prediction of antihelium flux from antistars accessible to the AMS02 experiment [3], which follows from this similarity. Aglobular clusterof starsisagroupof stars that gatherin the shapeofa sphere and orbit around the core of the Galaxy. The stars turn out to be gravitationally bound, which, in fact, is the reason for such a shape of these clusters. Globular clusters are localized far from most other objects in the Galaxy -in the halo. They are much denser than open clusters, and they are also older and contain more stars.In the MilkyWay, the numberof globular clusters about 150. The birth regions of such clusters are the dense interstellar medium. However, no star formation is currently observed in globular clusters. All dust and gas have long been ”blown out” from the clusters. This confirms the opinion that globular clusters are the oldest objectsin theGalaxy [14]. Orbiting the outskirtsof the galaxy, globular clusters take severalhundred million years to complete one orbit. At the center of the cluster, the highest density is achieved -on the order of 100-1000 stars/pc3. For comparison, the density of stars near the Sun is 0.14 stars/pc3. Globular clusters have a low metallicity due to the fact that they are composed of first generation stars. Which once again confirms the opinion that globular clusters are old clusters [15] 14.3 Cosmic antihelium propagation in interstellar medium After generation and acceleration in the source, cosmic ray particles enter the interstellar medium, where they change their original trajectory, ”entangled” in the magnetic fields of the Galaxy and can leave it. The propagation of cosmic rays in the modern view is of a diffusion nature. The GCR confinement time before leaving the Galaxy is inversely proportional to the diffusion coefficient, i.e. decreases with increasing energy. For particles with energies of 1–2 GeV, it is ~ 4 · 107 years. During this time, they manage to fill the halo of the Galaxy and, although the substance in the Galaxy is generally very rarefied, they also pass through a thickness of matter of about 10 g/cm2. For high-energy particles, the distance traveled sharply decreases and, for example, at an energyof10TeVis 0.1-0.4 g/cm2, and the lifetime is ~ 106 years [5,16]. At present, attempts are being made to calculate the fluxes of galactic CRs. Solving this problem requires knowledge of the structure and size of the Galaxy, the location and power of the sources, the location of the Solar System, and the properties of the interstellar medium. CR propagation in the Galaxy is seriously determined by the structure of magnetic fields. The regular field lines lie in the galactic plane and approximatelyrun along the spiral arms. The average amplitude of the field strength is (2-3)·10-6 G. The magnetic field also exists in the halo, but its structureis not exactly known.It shouldbe noted that currently thereisa numerical implementationoftheleakyboxmodelintheformofasetof GALPROP programs. GALPROP is a numerical solver of the diffusion equation taking into account a detailed description of the distributions of the interstellar gas and the galactic magnetic field [5]. The approach used in this work differs from the work of the GALPROP software package, and instead of solving the transport equation, individual particles are traced in interstellar space and we also take into account the parameters of the interstellar medium and the structure of magnetic field. 14.4 Boris -Bunemann tracing method Now there are various software packages that perform tracing of particles in electromagneticfields[17].In1970Boris[18]proposeda convenientwaytosolve the equations of motion of particles in electromagnetic fields, this method is now widely known as the Boris method. De facto, it is the standard for modeling particle motion in plasma. The method solves the classical equation of motion in an electromagnetic field specifiedbyvectors .E and .B (vectors of electric and magnetic fields, respectively). Further,the electric fieldiseliminatedbyredefiningthe variablesandthe equation is reduced to describing only the rotational motion in the magnetic field. Then Bunemann introduces some additions to Boris’s algorithm that increase the ac-curacyof the method.Today, thereis an implementationof the method with the inclusionofrelativistic corrections [19]. 14.4.1 Using of method To makethe method convenientto use,a softwarepackage wascreated that allows you to transfer all the necessary parameters to the function for calculating the trajectory of the environment. Below is a list of them. • Particle initial coordinates and directional distribution. • Particle type and energy. • Configuration of magnetic and electric fields. • Characteristics of temporal and spatial steps for numerical solution. • Conditions for saving trajectories, interrupting the tracing algorithm. It is also important to note that for the development of the software package, it is possible to determine the interstellar medium for calculating the absorption of cosmic rays, and a program for calculating such interactions. 14.4.2 Helium antinucleus tracing The following initial conditions were chosen for tracing helium antinuclei: • The initial coordinates correspond to the globular cluster M4 with the position (-5.9, -0.3, 0.6) kpc in the galactic coordinate system [17]. • The angular distribution of particles at the point of birth is isotropic. • The energyof particles variedfrom10GeVto10TeV. • The structure of regular component of magnetic field of the Galaxy is taken from publication [18]. 14.5 Results Figure1shows examplesof trajectoriesof helium antinuclei launched towards the plane of the galactic disk with different energies. Particles with an energy of 100 GeV did not penetrate into the plane of the disk, deflected and flew away into intergalactic space. Particles of higher energy penetrated into the galactic disk and at an energy close to1TeV they had the opportunity to leave it. InFig.2,theline connectingthepointswitherrorsshowsthe fractionof antihelium that fell into the galactic disk 300 pc thick, depending on the particle energy. The obtained dependence was smoothed taking into account the error and the energies were determined at which the smoothed curves intersect the 0.25 level, i.e. width at half-height of the graph (with increasing energy, the graph tends to a value of~ 0.5, which corresponds to the geometric factor of the disk plane from the point with the coordinates of the M4 cluster). The obtained energy – magnetic cutoff energy is 100 ±10 GeV. This means that the flux of helium antinuclei from the hypothetically globular cluster of antistars M4 will be largely suppressed at energies less than ~ 100 GeV, but will not be completely suppressed. 14.6 Conclusion Helium antinuclei were traced with [24] from the M4 cluster, hypothetically con­sisting of antistars, to the plane of the galactic disk. The characteristic energy of magnetic cutoffis determined, below which it is difficult for particles to penetrate into the disk. Predictions of antihelium flux would strongly depend on the interfer­ence of the initial spectrum, which is expected to be falling down with energy and magnetic cutoff, which redices the lower energy part of the spectrum in galactic disk. The result will be used to interpret the experimental data on antinuclear fluxes obtainedby thePAMELA and AMS-02in near-earth orbit. Thepreliminary indicationsto possible detectionof antihelium eventsin AMS02 experiment, which cannot be explained as secondaries from astrophysical sources [22], if confirmed, would become serious evidence for existence of forms of pri­mordial antimatter in our Galaxy. It will favor Beyond the StandardModel (BSM) physics, which can support creation and survival of antimatter domains in baryon asymmetrical Universe, and provide a sensitive probe for parameters of the cor­responding models [23]. Whatever is the actual form of antimatter objects in our Galaxy, propagation of antinuclei from these sources would inevitably involve their diffusion in galactic magnetic fields, studied in the present paper. Acknowledgements The workbyAK andAM has been supportedby the grantof the Russian Science Foundation (Project No-18-12-00213-P). References 1. O. Adriani,G. Barbarino:Ten yearsofPAMELAin space, Rivista del Nuovo Cimento 10, 473-522 (2017). 2. K. Abe et al.: Search for Antihelium with the BESS-Polar Spectrometer, Phys. Rev. Lett. 108, 131301 (2012). 3. K.M. Belotsky,Y.A. Golubkov, M.Y.Khlopov, R.V. Konoplich, A.S. Sakharov: Anti-helium flux as a signature for antimatter globular cluster in our Galaxy, Phys. Atom. Nucl. 63, 233 (2000). 4. L. Boyle, K. Finn and N.Turok: CPT-Symmetric Universe, Phys. Rev. Lett. 121(25), 251301 (2018). 5. A.W. Strong, I.V. Moskalenko: Propagation of cosmic-ray nucleons in the Galaxy, The Astrophysical Journal 509 , 212-228 (1998 6. N.Tomassetti,A. Oliva: Secondary antinucleifrom supernova remnants and back­ground for dark matter searches, 35th International Cosmic Ray Conference -ICRC2017 301, 271, 2017. ). 7.F.W. Stecker, A.J.Tylka: The cosmic-ray antiproton spectrumfrom dark matter anni­hilation and its astrophysical implications: a new look, Astrophysical Journal 336 L51 (1989), doi: 10.1086/185359 . 8. M.Y. Khlopov: Fundamentals of Cosmoparticle Physics CISP-Springer, Cambridge, UK, (2012). 9. A.D. Dolgov: Matter and antimatter in the universe, Nucl. Phys. Proc. Suppl. 113, 40 (2002). 10. M.Yu. Khlopov:An antimatter globular clusterin our Galaxy -aprobe for the originof the matter, Gravitation and Cosmology, 4, 69-72 (1998). 11. A.D. Dolgov, M. Kawasaki, N. Kevlishvili: Inhomogeneous baryogenesis, cosmic anti­matter, and dark matter, Nucl. Phys.B 807, 229 (2009). 12. S.I. Blinnikov, A.D. Dolgov, K.A. Postnov: Antimatter and antistars in the universe and in the Galaxy, Phys. Rev.D 92, 023516 (2015). doi:10.1103/PhysRevD.92.023516. 13. M.Y. Khlopov, S.G. Rubin, A.S. Sakharov: Possible origin of antimatter regions in the baryon dominated Universe, Phys. Rev.D 62, 083505 (2000). Place,Year. 14. M. Paul: Star Clusters. Encyclopedia of Astronomy and Astrophysics,CRC Press, Boca Raton, 2014. 15. J.S. Kalirai, H.B. Richer: Star clusters as laboratories for stellar and dynamical evolution, Royal society publishing 368, 755-782 (2010). 16. A.W. Strong, I.V. Moskalenko: Secondary antiprotons and propagation of cosmic rays in the galaxy and heliosphere, The Astrophysical Journal 564, 280-296 (2001). 17. Yuting Ng, et. al.: Introduction to motion of charged parti­cles in Earth’s magnetosphere, (2013) URL: https://www.s.u­tokyo.ac.jp/en/utrip/archive/2013/pdf/06NgYuting.pdf 18. J.P. Boris: The acceleration calculation from a scalar potential, Plasma Physics Laboratory, Princeton University, MATT-152, (1970) URL: https://www.osti.gov/biblio/4168374 19. S. Zenitani,T. Umeda: On the Boris solver in particle-in-cell simulation, Physics of Plasmas 25, 112110 (2018). 20. http://gclusters.altervista.org 21. C.J. Nixon,T.O. Hands: The originof the structureof large–scale magnetic fieldsin disc galaxies, Noticesof the Royal Astronomical Society3 477, 3539–3551 (2018). 22.V. Poulin,P. Salati,I. Cholis,M. Kamionkowski,J. Silk: Wheredo the AMS-02 anti-helium events comefrom?, Phys. Rev.D 99, 023016 (2019). 23. M. Khlopov: What comes after the Standardmodel? Progress in Particle and Nuclear Physics 116, 103824 (2021) 24.V. Golubkov,A. Mayorov: Software for Numerical Calculationsof ParticleTrajectories in the Earth’s Magnetosphere and Its UseinProcessingPAMELA Experimental Data, Bull.Russ.Acad.Sci.Phys. 85, 383-385 (2021). Proceedings to the 24th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ... (p. 177) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 15 Mass asa dynamical quantity M. Land email: martin@hac.ac.il Hadassah College Jerusalem Abstract. The StandardModel (SM) ascribes the observed mass of elementary particles to an effective interaction between basis states defined without mass terms and a scalar potential associated with the Higgs boson. In the relativistic field theory that underlies the SM, mass itself, understood as the Lorentz-invariant squared 4-momentumofa particle or field,is fixed a priori, imposing a constraint on possible momentum states. Stueckelbergintroduced an alternative approach, positing antiparticles as particles evolving backwardin time, thusrelaxing the mass shell constraint for individual particles. FurtherworkbyPironand Horwitz establisheda covariant Hamiltonian mechanicsonan unconstrained 8D phase space, leading to a gauge field theory that mediates the exchange of mass between particles, while the total mass of particles and fields remains conserved. In a recently developed extension of general relativity, consistent with this approach, the spacetime metric evolves in a manner that permits the exchange of mass across spacetime through the gravitational field. Mechanisms thatrestrict mass exchange between particles have also been identified. Nevertheless, mass exchange remains possible under certain circumstances and may have phenomenological implications in particle physics and cosmology. Povzetek Standardni model pripi. se maso osnovnih delcev interakciji delcevs skalarnim poljem — Higgsovim bozonom.Vkvantni teoriji polja, na kateri standardni model gradi, je masa dolo.cine delca ali polja, kar cena s kvadratom Lorentzove invariantne gibalne koli.omeji njuno gibalno koli.cino. Stueckelbergov privzetek,da so antidelci delci,ki potujejov . casu nazaj, omejitev masne lupine za posamezne delce odstrani. Delo Pirona in Horwitza uveljavi kovariantno Hamiltonovo mehaniko na neomejenem 8D faznem prostoru, kar vodi do teorije umeritvenega polja, ki posreduje izmenjavo mase med delci in polji, medtem ko se skupna masa delcev in polj ohrani. Vnedavno razviti raz.sne teorijerelativnosti,kijeskladnazomenjenim pristopom, siritvi splo.se metrika prostora-.cin, ki dovoljuje izmenjavo mase preko gravitacijskega casa razvija na na .polja pod dolo.studira fenomenolo . cenimi pogoji.Avtor .ke posledice te omejitvev fiziki delcev in kozmologiji. 15.1 Introduction The Standard Model (SM) of elementary particles is a locally gauge invariant relativistic quantum field theory with particular choices for the basis states and the gauge group. While efforts to move beyond the SM usually begin by gener­alizing the algebraic structure of the gauge fields, other work has focused on the underlying framework of relativistic dynamics. In this paper, we present such an approach pioneered by Stueckelbergin 1941 in his work on antiparticles. In this approach, particle mass is treated as a dynamical quantity, leading to gauge theories in which fields and particles may exchange mass, just as they exchange energy-momentum.Wereview theresulting classical and quantum electrodynam­ics, indicating mechanisms that maintain particle masses at their familiar on-shell values. Recent work on general relativity and gravitation is also outlined. The notion of an elementary particle characterized by a fixed mass can be traced back to the 1897 discovery by Thomson [1] that cathode rays are composed of discretebodieswithafixedcharge-to-massratio,andthe1909 Millikan-Fletcher[2] oil-drop experimentindicatinga minimum electron charge.Today, the measured mass uncertainty of an electron is on the order of .m . 10-8 [3] and so it is conventional to write single-electron equations of the type duµ m = eFµ.u. i./-eA/-m. = 0 (15.1) dt for fixed m and metric signature .µ. =(-, +, +, +) . (15.2) The fixed mass shell pµpµ =-m 2 c 2 is expressed in scattering by writing µ2 d3p d4pdppµ + mc 2= (15.3) 22 2p2 + mc for the momentum space measure. This picture is, of course, complicated by the SM, which defines a Lagrangian containing no mass terms for elementary states, finding effective mass terms through interaction with the symmetry-broken Higgs boson. Intriguingly, the effective masses of the composite nucleons p and n are sharper(.m ~ 10-10)than the masses of their constituentu and d quarks (.m ~ 25%) [3]. The assumption of fixed masses is associated with a number of issues in physics, including flavor oscillations, the problem of time, and missing mass/energy in cosmology [4]. 15.2 The Stueckelberg-Horwitz-Piron (SHP) framework AdifferentapproachwasproposedbyStueckelberg[5]inhisworkon antiparticles. Pair creation and annihilation are described in QED by the Feynman diagram in Figure 1a, showing an intermediate electron state propagating backwardin time with E<0, but observed as a positron with E>0 propagating forwardin time. In the quantum picture, the electron jumps from forwardtimelike propagation to backwardtimelike propagation, remaining on its mass shell throughout. But Stueckelbergproposed this model of pair processes in classical electrodynamics, for a continuously evolving spacetime trajectory xµ(t)as shown in Figure 1b. In such a curve, x.0(t)= dx0/dt must vanish for some t, and so x.µ(t)must cross the spacelike region twice. Figure 1a Figure 1b Stueckelbergobserved that x.2(t)= x.µ x.µ isadynamical quantity for this trajectory, and so all eight components of xµ(t)and x.µ(t)must be independent. Moreover, since x.2 changes sign, the evolution cannot be parameterized by the proper time v of the motion ds =-x.2dt. The evolution parameter t must be external to the spacetime manifold, much as Newtonian time t is external to space. HorwitzandPiron[6]wereledtoa similar modelwhen constructinga covariant canonical mechanics with non-trivial interactions.Writinga classical Lagrangian system on 8D unconstrained phase space 1 d.L .L µ L = Mx.µ x.µ + ex.Aµ (x)-V -= 0 (15.4) 2 dt .x.µ .xµ one obtains the generalized Lorentz force µ + Gµ. ..eFµ. . M x¨x .x = x. -.µV (15.5) .. with field strength and conjugate momentum .L Fµ. = .µA. -..Aµ pµ == Mx.µ + eAµ (x) . (15.6) .x.µ Transforming to the manifestly covariant Hamiltonian mechanics µ K = x.pµ -L = 1 (pµ -eAµ)(pµ -eAµ)+V (15.7) 2M leads to the classical and quantum equations of motion .K .K µµ x.= p.=- i.t.(x, t)= K.(x, t). (15.8) .pµ .xµ Generalizing the classical central force problem as V(x1,x2)= V(.) . =(x1 -x2)2 -(t1 -t2)2 (15.9) Horwitzand Arshansky[7] obtained solutionsforrelativistic scatteringand bound states, Horwitz and Land studied radiative transitions, selectionrules, perturba­tion theory, Zeeman and Stark effects, and bound state decay [8], and Horwitz demonstrated entanglement and interference in time [9]. The physical picture [10] that emerges from Stueckelberg’s unconstrained mechan­ics can be summarized as an upgrade of nonrelativistic classical and quantum mechanicsin which Galilean symmetryisreplaced with Poincar´e symmetry: .. Newtonian time t External time t + . . + .. dxj dx. µ Unconstrained x i , -. Unconstrained x, dt dt ++ . . .. Scalar Hamiltonian H Scalar Hamiltonian K This covariant canonical mechanics inherits many methods and insights of Newto­nian mechanics, so for example, from the Poisson bracket .F.G .G.F dF .F {F, G}=- = {F, K}+ (15.10) .xµ .pµ .xµ .pµ dt .t it follows that .H .K = 0 . conserved energy -. = 0 . conserved mass .t .t which for a free particle can be seen from 1pµ µµµ 2 K = ppµ -. x.= ,p.= 0 -. x .= constant . (15.11) 2M M As discussed by Horwitz, Arshansky, and Elitzur [11], this framework formalizes a distinction between two aspects of time: the time t is one of four spacetime coordinates xµ characterizing the location of a single event, while the time t represents the chronological order of multiple events. The physical spacetime event xµ(t)is understood as an irreversible occurrence at time t so that for t2 >t1,event xµ(t2)occurs after xµ(t1)and cannot change it. This changes the significance of a closed timelike curve, resolving the grandfather paradox. The proverbial traveler revisiting at t3 >t2 the spacetime locations xµ(t)of his grandfather’s trajectory as it evolved from t1 to t2 >t1, may add events xµ(t3)but cannot alter that life trajectory as it has irreversibly occurred. More generally, the 4D block universe M(t)occurs at t, evolving to the infinitesimally close the 4D block universe M(t +dt)under motion generatedby the Hamiltonian K. Because K isa Lorentz scalar and t is external, we expect no conflict with general diffeomorphism invariance. 15.3 Classical Off-Shell Electrodynamics The origin of the scalar potential V in (15.4) can be understood by requiring invariance under t-dependent gauge transformations [12], leading to a theory with five gauge fields, Aµ(x). aµ(x, t)and a5(x, t). The maximal gauge freedom of the classical action [13] Z Z d µ dt L -. dt L + . (x, t)= dt [L + x..µ. + .t.] (15.12) dt suggests a coupling with a pure gauge field with components .µ. and .t.. Introducing the notation µ, . = 0, 1, 2, 3 and a, ß = 0, 1, 2, 3, 5 (15.13) 50 and writing x= c5t in analogyto x= ct, werewrite the classical interaction as e eee µµ5a x.Aµ (x)-V(x)-. x.aµ (x, t)+ x .a5 (x, t)= x .aa (x, t) (15.14) c ccc now invariant under 5D gauge transformations aa . aa + .a. (x, t). The La­grangian a L = 1Mx.µ x.µ + ex .aa (x, t) (15.15) 2c admits the conserved 5-current a ja (x, t)= cx .d4 (x -X (t)) .aja = .µjµ + .5j5 = 0 (15.16) which can be related to the Maxwell current by observing that under appropriate boundary conditions Z Jµ(x)= dtjµ (x, t)-. .µJµ = 0. (15.17) This integral is called concatenation, understood as the sum of contributions g(x, t)to G(x)along the worldline, under g(x, ±8)= 0. The interaction (15.14) appears to be 5D symmetric, but this symmetry is broken to vector and scalar 5 representations of O(3,1), because x.= c5 « c isa constant and nota dynamical quantity. The Lorentz force [14] found from the Euler-Lagrange equations are ee ß. Mx¨µ = x .fµß =(x .fµ. -c5f5µ) (15.18) cc d 1 µ .e µ - Mx.xµ = c5 x.f5µ (15.19) dt2 c with 5D field strength faß = .aaß -.ßaa a, ß = 0,1,2,3,5 . (15.20) From (15.19) we see that eµ (x, t)= f5µ (x, t)= .5aµ - .µa5 induces mass ex­change. The field fµ. (x, t)becomes the Maxwell field Fµ. (x)under concatenation, decoupling from f5µ. Expanding the interaction term Z X.a aa -. d4 xX.a(t)d4 x -X(t)aa(x, t) (15.21) we define the sharp current, a delta function on 4D spacetime, . ja(x, t)= cXa(t)d4 x -X(t)(15.22) whichby (15.17)recovers the standardMaxwell current Z Jµ(x)= c dtX.µ(t)d4 x -X(t). (15.23) To complete the electromagnetic action, we introduce a kinetic term for the electro­magnetic field Z e ds1 Sem d4xdt 10 2 ja(x, t)aa(x, t)- faß(x, t)F(t -s)faß (x, s) c.4c (15.24) where the interaction kernel d '' (t) 1 c5 2 F(t)= d (t)-(..)2. = 1 + (15.25) 2c smooths the sharp current (15.22). The constant . has dimensions of time and serves as a correlation length along the worldline. The scalars jaaa and faßfaß suggest a 5D symmetry containing O(3,1), either O(4,1) or O(3,2). Although any 5 higher symmetryisbrokenby x.= constant and by the d '' (t)term in F(t), it is convenientto introducea formal5D metric gaß -. .aß = diag(-1, 1, 1, 1, s) .55 = s = ±1 (15.26) flat for raising the 5-index in faß. Since faßfaß = fµ.fµ. +2.55f µ f5µ we mayregard 5 s = .55 as simply the relative sign of the vector-vector kinetic term, with no geometric significance. The interaction kernel is invertible as Z -i.t d.e1 -|t|/.. f(t)= .F-1(t)= . = e (15.27) 2 2p 1 +(...)2. Z ds dt f (t -s)F (s)= d(t) f (t)= 1 (15.28) .. so that variation of the electromagnetic action with respect to aa(x, t)provides the field equations Z ee .ßfaß (x, t)= ds f (t -s)ja (x, s)= ja (x, t) (15.29) f cc .afß. + ..faß + .ßf.a = 0 (identically) (15.30) with source current ja (x, t)smoothed along the worldline by convolution of f ja (x, t)with the inverse interaction kernel f. These are known as pre-Maxwell equations, and when writtenin 4+1 (spacetime+ t)components 1.e e ..fµ. - f5µ jµ f5µ j5 = f .µ= c5 .tc c f (15.31) 1. .µf.. + ..f.µ + ..fµ. = 0..f5µ -.µf5. + fµ. = 0 c5 .t are comparableto Maxwell’s equationsin3+1(space+time) componentswhere f5µ plays the role of the electric field sourced by j5 in Gauss’s law, and fµ. is a f magnetic field induced by “curl” and t dependence of f5µ.Writing Z aß f(x, t)= dsF(t -s)faß (x, s) (15.32) F . translation invariance of the action leads to the Noether symmetry aß efßaaß 1 a. 1 aßd. =- T= ffß - gf(15.33) .aTF c2 jaF c F. 4 F fd. where T µ. is the energy-momentum tensor, and the terms T5a represent mass FF density in the field and the flow of mass into spacetime. Inserting the current (15.22) into (15.33) and using the second Lorentz force equation (15.19), we find d Z d4d Z d41 xT5µ + Mx.µ= 0 xT55 -s Mx .2= 0 (15.34) dt dt2 demonstrating that the total energy-momentum and mass of the particle and field are conserved [14]. As was seen for the current, concatenation of the pre-Maxwell equations leads to the Maxwell equations .ßfaß (x, t) = e ja f (x, t) . .. . .. ..Fµ. (x) = e Jµ (x) cc .. Z ---------. dt .. (15.35) .[afß.] = 0.[µF..] = 0 . representing a sum of microscopic contributions at each t to the Maxwell fields at a given spacetime point. The pre-Maxwell equations lead to the 5D wave equation .µ .µ s.2 eja aa a .ß.ß a =(.µ+ .t.t)a =(.µ+)a =- (x, t) (15.36) tf 2 cc 5 with Green’s function [15] 1c5 .1 GP(x, t)=- d(x 2)d(t)- .(-sxa xa) v = (15.37) 2p 2p2 .x2 -sxaxa = GMaxwell + GCorrelation (15.38) where Gcorrelation is smaller than GMaxwell by c5/c and drops offfaster with distance. Notice that GCorrelation has spacelike support for s =-1 and timelike support for s =+1. Under concatenation GMaxwell(x, t)goes over to the standard Maxwell Green’s function and GCorrelation vanishes. A “static” source event evolving along the x0-axis in its rest frame as X (t)= (ct, 0, 0, 0) induces for an observer on the parallel trajectory x(t) =(ct, x) a Yukawa-type potential [16] -|x|/..c a 0(x, t)= e 1e (15.39) 4p|x| 2. with photon mass spectrum m.c 2 ~ ./... Using the accepted experimental error for photon mass .m. ~ 10-18 eV leads to . > 104 seconds. The constant . can be seen as a correlation time along the worldline, the width of the ensemble of events contributing to the pre-Maxwell current and potential [17]. In the limit . . 0 the kinetic term in the action (15.3) reduces to faßfaß, the photon mass 0 spectrum goes to infinity, and a becomes a delta function. In the limit . . 8 the photon mass spectrum vanishes and the pre-Maxwell system reduces to Maxwell electrodynamics. TheLi ´ enard-Wiechert potential for an arbitrary source event Xµ (t)at an observation point xµ similarly leads to the Maxwell formula multiplied by the factor f (t -tR)where tR is the retarded time found from 2 [x -X(tR)]= 0. Comparing the Lorentz forces for e-/e+ and e-/e- scattering leads to an experimental bound on c5 « c [18]. 15.4 Mass interactions and mass stability Asimple model of mass variation is a uniformly moving particle undergoing a stochastic perturbation x = ut . ut + X(t)when entering a dense distribution of charged particles [19]. If the typical short distance between charge centers is d then the particle will encounter charges periodically, with a short characteristic period d/ |u|, leading to a high characteristic frequency .0 = 2p |u|/d. Expanding the perturbation in a Fourier series X in.0tµ0 t X (t)= Re an ea= ad s, sn= ad cs , sn(15.40) nnnn µ with normalized coefficients s~ 1 and some macroscopic factor a . 1.We obtain 0 a small perturbation of position |Xµ (t)|~ad, but a velocity perturbation X µ iein.0t x.µ (t)= u+ a |u| Re 2pn sµ (15.41) n n of macroscopic scale a |u|.Writing the particle mass as m(t)=-Mx.2/c2 leads to a macroscopic mass perturbation X m -. m 1 + .m .m = 4pa |u|Re ns t iein.0t (15.42) n mm n which could persist when the particle leaves the charge distribution. One possible reason that we do not see such mass perturbations more frequently isa self-interactionthat tendstorestore masstoits familiar on-shellvalue.We 0 consider a particle with arbitrary x.0(t)in its rest frame, where x¨.0 entails = 2 mass variation through -Mx.= Mc2 t.2(t). Along the worldline, the particle may interact at time t * >t with the field it produces at t, but of course GMaxwell, the leading term in the Green’s function, vanishes on .X(t * ,t)= X(t *)-X(t)= c(t(t *)-t(t), 0), the timelike separation. Nevertheless, from (15.37) we see that GCorrelation has support for -sxaxa >0, which is this case is the condition 2 22 5 -.X2 + c (t * -t)2= c t (t * )-t (t)2 - c (t * -t)2>0 (15.43) 52 c when s =+1. Expanding t(t)inaTaylor series, one finds that condition (15.43)is satisfied if and only if t ¨ .= 0, which in the rest frame implies mass variation. Now suppose that a particle evolves uniformly as t = t until t = 0 when it makes a sudden jump to t =(1 + ß)t. The field strength acting on the particle at t * >0 contains only the component 2 f50 ˜ e1 c5 Q ß, (15.44) 32 4p2 c2 (ßt*)c 5 2 c 5 where Q ß, is a positive function that vanishes for ß = 0 or c5 = 0. The 2 c Lorentz force is then 0 Mx ¨=-c5ef50 = . .. .. 0 ,t * <0 2 c 5 .e2 1 (15.45) - Q ß, ,t * >0 3 4p2 c5 (ßt*) 2 c i Mx ¨=-c5ef5i x.i = 0 (15.46) and 2 d 1 .e2 c c - Mx .2= ef5µ x.µ =-ecf50 t .=- Q ß, 5 t .(15.47) dt2 4p2 c2 (ßt*)3 c2 5 which acts as a restoring force, damping the mass towardits on-shell value and vanishing on shell. Another approach [20] describes the particle as a statistical ensemble with both an equilibrium energy and an equilibrium mass, controlledby temperatures and chemical potentials, assuring asymptotic states with the correct mass. The ther­modynamic properties are found from the microcanonical ensemble, where both energy and mass are parameters of the distribution.Acritical point in the free energy emerges from equilibrium requirements of the canonical ensemble (where total system massis variable),and equilibriumrequirementsofthegrand canonical ensemble (where a chemical potential arises for the particle number). Because par­ticle mass is controlled by a chemical potential, asymptotic variations in the mass arerestoredtoagiven valuebyrelaxation, satisfyingthe equilibrium conditions. 15.5 Off-Shell Quantum Electrodynamics Transforming the classical Lagrangian (15.15) to Hamiltonian form, we are led to the Stueckelberg-Schrodinger equation and global gauge invariance providing the conserved current .aja = 0 with 4-vector part c5 i..t + e a5 . (x, t)= c 1 e e µpµ - apµ - aµ . (x, t) 2M c c (15.48) with local 5D gauge invariance ie.(x,t)/.c .(x, t). e .(x, t) aa(x, t). aa(x, t)+.a.(x, t) (15.49) jµ i. . * ie µ.µ ie µ. * =- .µ - a. -. + a(15.50) 2Mc c and event density j5 = c5 |.(x, t)|2 representing the probability of finding an event at a spacetime point x at time t. The quantum Lagrangian is 1. faß L = . * (i.t + ea5). - . * (-i.µ -eaµ)(-i.µ -eaµ). - fF (15.51) aß 2M 4 where fF (x, t)is defined in (15.32), which admits Jackiw first order constrained aß quantization [21] by introducing .µ = f5µ. Because a.5 does not appear in the Lagrangian, path integration over a5 inserts the Gauss law constraint d(.µ.F - µ e. * .)which may be solved to eliminate longitudinal modes. Feynman rules may be read from the unconstrained Lagrangian 11 µ Fµ L = i. * ..- . * (-i.µ -ea. µ)(-i.µ -ea).+ a. µ . + s.2 a (15.52) .t. 2M 2 as matter and photon propagators 1 -i µ. - kµk. -i1 g(15.53) 1 (2p)5 p2 -P -i. k2 k2 + .2 -i. 1 + .2.2 2M along with three and four particle vertex factors e i(p + p ' ). (2p)5d4(p -p ' -k)d(P -P ' -.)12 2M (15.54) -ie2 ' (2p)5gµ.d4(k -k ' -p + p)d(-. + . ' + P ' -P) M which conserve total energy-momentum and mass. The matter propagator Z i(k·x-.t) d4kd. e d4k 2 i(k·x- 1 k+i.) 2M G(x, t)= = i.(t) e (15.55) 1 (2p)5 2M k2 -. -i. (2p)4 enforcesretarded causalityin t, so that there are no matter loops, just as there are no grandfather paradoxes.ThisexpressionwaspreviouslyfoundbyFeynman[22] for the Klein-Gordon equation, leading to the Feynman propagator by extracting a stationary eigenstate of the mass operator -i..t as 8 Z d4k ik·x 2/2M)te dte-i(m G(x, t)= = 2M .F(x). (15.56) 1 -8 (2p)4 (k2 + m2)-i. 2M We see that the interaction kernel inherited from the classical electromagnetic term provides the natural mass cut-off 1 + .2K2-1 which renders the theory super-renormalizable. The cross-section for elastic scattering is nearly identical to the Klein-Gordon case, but the pole is slightly shifted away from 0o for non-zero mass exchange between the outgoing particles (expressed as an undetermined hyperangle, much as the scattering angle is undetermined in on-shell QED) [23]. 15.6 General relativity with t-evolution As describedinSection15.2,theSHP frameworkposesablock universe M(t)that evolves to a block universe M(t + dt)under a Hamiltonian K.We thus expect that the spacetime metric gµ.(x, t) should similarly evolve to gµ.(x, t + dt). To find field equations that prescribe this evolution, welook for hints from the development of the pseudo-5D off-shell electromagnetic field equations, which 5 differ from Maxwell equations written in five dimensions because excluding xfrom the dynamical degrees of freedom breaks any 5D symmetry [13, 24]. Just as there is no Lorentz force for x ¨5, there must be no geodesic equation for x ¨5 in curved spacetime. In standard 4D general relativity (GR), the invariance of the squared interval dx2 = .µ.dxµdx. =(x2 -x1)2 between neighboring events (an instantaneous displacement) is a geometrical statement, characterizing the manifold M. For events X1 =(x1,c5t1)and X2 =(x2,c5 (t1 + dt)) belonging to M(t)and M(t + dt)the squared interval dx(t) 2 dXadXa = dx + dt + sc2 dt2 = gaß (x, t)dxadxß (15.57) 5 dt suggests a pseudo-5D metric gaß (x, t), analogous to the pseudo-5D electromag­netic field faß (x, t). The evolution of gaß (x, t)differs from a standardmetric defined in 5D, because it combines 4D geometrical symmetries of M(t)with the scalar dynamical symmetry of Hamiltonian K.To preserve the constraint x5 = c5t we expand the classical Lagrangian as 11 1 a .ßµ ..µ L = Mgaß(x, t)x .x = Mgµ. x.x + Mc5 gµ5 x.+ Mc52 g55 (15.58) 22 2 to obtain four geodesic equations and an identity . µs s2 G µ . x¨+ Gµ x.. x.+ 2c5Gµ x.+ c= 0 .s 5s 555 a 0 = x ¨+ G a x .ß x .. -. (15.59) ß. . 5 x¨= c.5 = 0 and the Hamiltonian 11 K = pµ x.µ -L = Mgµ. x.µ x .. - Mc25 g55 = L -Mc25 g55 (15.60) 22 from which we find dK1 .gµ. 1 .g55 . =- Mx.µ x .- Mc2 (15.61) 5 dt2 .t2 .t showing that particle mass is not generally conserved along geodesics. We define the event density in spacetime n(x, t) and mass density .(x, t)= Mn(x, t), leading to the event current ja (x, t)= x.a(t).(x, t) and continuity equation .aja = 0 with covariant derivative .µ defined in the standard manner and .5 = .5. Current conservation leads to conservation of the analogously defined mass-energy-momentum tensor Taß = .x.a x.ß. If we write the standard Einstein field equations in 5D and study the linearized form for weak gravitation gaß ˜ .aß +haß,we obtaina wave equation that canbe solved using theGreen’s function (15.37). However, the metric perturbation found from a “static” source in its rest frame includes h00 = 2hij = h55, which deviates from the expected structure, h00 = hij » h55.To determine the correct modification of the field equations we choose a form that preserves the 5D symmetries of the Ricci tensor Raß, but breaks the apparent 5D symmetry in the relationship between Raß and Taß. In trace-reversed form, we write [13] 8pG 1 ¯ Raß = Taß - g¯aß Tg¯µ. = gµ. g¯5a = 0 (15.62) 4 c2 where T ¯ = g¯µ.Tµ., which for the “static” source in the weak field approximation leads to 2Gm 2Gm -1 c2 5 gµ. =- 1 - 2,1 - dij g55 = s + o 2 (15.63) 2 crcr c consistent witha spherically symmetric Schwarzschild metric.A sourcein its rest frame with mass varying arbitrarily as x.0(t)= c[1 + a (t)/2]leads to a t­dependent perturbation. The geodesic equations for a test particle in this space undergoes a nonlinear x0 acceleration, and satisfiesa radial equation d1 1L2 GM 1 GM d R.2 +- 1 + a (t) =- a (t) (15.64) dt2 2M2R2 R 2 2Rdt with conserved angular momentum L = MR2 ... The term in brackets on the LHS is the Hamiltonian in these coordinates, indicating that the mass of the test particle isnot conservedwhenthemassofthesourcevaries.Thissimpleexamplesuggests that a source particle of varying mass can transfer mass across spacetime to a test mass moving geodesically under the influence of the metric field induced by the source [24]. Decomposing these field equation (15.62) into 4+1 form [13], analogous to the 3+1 decomposition used in the ADM formalism [25], we find that the 20 spacetime components Rµ. are unconstrained second order evolution equations, while the five components Ra5 are constraints that propagate at first order in .t. Moreover, from T ¯ = T - g55T55, the mass density T55 sourced by g55 and not necessarily constant, is seen to play the role of a small cosmological term .. 15.7 References References 1. ThomsonJJ1897 Phil. Mag. 44 293–396 2. MillikanRA1910 Science 32 436–448 3. ZylaPAandetal (ParticleDataGroup)2020 Progress of Theoretical and Experimental Physics 2020 4. ShiY2021 Force, metric, or mass: Disambiguating causesof uniform gravity(Preprint 1908.02159) 5. StueckelbergE1941 Helv. Phys. Acta 14 321–322 (InFrench); StueckelbergE1941 Helv. Phys. Acta 14 588–594 (In French) 6. HorwitzLand PironC1973 Helv. Phys. Acta 48 316–326 7. HorwitzLand LavieY1982 Phys. Rev.D 26 819–838;ArshanskyRand HorwitzL 1989 J. Math. Phys. 30 213;ArshanskyRand HorwitzL1988 Phys. Lett.A 131 222–226; ArshanskyRand HorwitzL1989 J. Math. Phys. 30 66;ArshanskyRand HorwitzL1989 J. Math. Phys. 30 380;ArshanskyR1986 The classicalrelativistic two-bodyproblem and symptotic mass conservation.TelAviv UniversitypreprintTAUP 1479-86 8. LandM, ArshanskyR and HorwitzL 1994 Found. Phys. 24 563–578; LandM and HorwitzL1995 J. Phys.AMath. Gen. 28 3289–3304;LandMand HorwitzL2001 Found. Phys. 31 967–991 9. Horwitz L and Arshansky R I 2018 Physics Letters A 382 1701–1708 (Preprint 1707.03294) 10. LandMand HorwitzLP2020 Relativistic classical mechanics and electrodynamics (Morgan and Claypool Publishers) 11. HorwitzL, ArshanskyRand ElitzurA1988 Found. Phys. 18 1159 12. SaadD, HorwitzLand ArshanskyR1989 Found. Phys. 19 1125–1149 13. LandM2020 Symmetry 12 ;LandM2021Journal of Physics: Conference Series 1956 012010 14. LandMand HorwitzL1991 Found. Phys. Lett. 461 15. LandMand HorwitzL1991 Found. Phys. 21 299–310 16. LandM1996 Found. of Phys. 27 19 17. LandM2017 Entropy 19 234 ISSN 1099-4300 18. LandM2017 Journal of Physics: Conference Series 845 012024 19. LandM2017 Journal of Physics: Conference Series 845 012025 20. HorwitzLP2017 Journal of Physics: Conference Series 845 012026 21. JackiwRURL https://arxiv.org/pdf/hep-th/9306075.pdf 22. FeynmanR1950 Phys. Rev. 80 440–457 23. LandMand HorwitzL2013 J. Phys. Conf. Ser. 437 012011 24. LandM2019 Astronomische Nachrichten 340 983–988 25. ArnowittRL, DeserSand MisnerCW2004 General Relativity and Gravitation 40 1997– 2027 Proceedings to the 24th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ... (p. 190) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 16 New way of second quantization of fermions and bosons N.S. Manko.c Bor.stnik email: norma.mankoc@fmf.uni-lj.si Department of Physics, University of Ljubljana SI-1000 Ljubljana, Slovenia Abstract. This contribution presents properties of the second quantized not only fermion fields but also boson fields, if the second quantization of both kinds of fields origins in the description of the internal space of fields with the ”basis vectors” which are the superposi­tion of odd (when describing fermions) or even (when describing bosons) products of the Cliffordalgebra operators .a’s. The tensor products of the ”basis vectors” with the basis in ordinary space forming the creation operators manifest the anticommutativty (of fermions) or commutativity (of bosons) of the ”basis vectors”, explainingthe second quantization postulates of both kinds of fields. Creation operators of boson fields have all the properties of the gauge fields of the corresponding fermion fields, offering a new understanding of the fermion and boson fields. Povzetek: Prispevek razlo. zi drugo kvantizacijo ne le fermionskih ampak tudi bozon­skih polj. Notranji prostor fermionov opi. sejo ”osnovnimi vektorji”, ki so superpozicija produktov lihega . stevila Cliffordovh operatorjev .a, bozonski”osnovnimi vektorji” pa so superpozicijaproduktov sodega . stevila Cliffordovih operatorjev .a. Kreacijski in ani­hilacijski operatorji, ki so tenzorski produkt kon .stevila ”osnovnih vektorjev” in cnega .zvezno neskon .stevila komutirajo .cajnem prostoru, ”podedujejo” cnega.cih vektorjevv obi.antikomutativnost ali komutativnost od ”osnovnih vektorjev”. Posledi . cno fermionska stanja antikomutirajo in bozonska komutirajo, kar razlo . zi postulate druge kvantizacije za fermionska in bozonska polja in ponudi nov pogled za lastnosti obeh vrst polj. 16.1 Introduction Inalong seriesofworks[19,20,23,25,26,28,29,31]Ihavefound, togetherwiththe collaborators([1,26,32,34,35,37] and thereferences therein), with H.B. Nielsen and in long discussions with participants during the annual workshops ”What comes beyond the standard models”, the phenomenological success with the model named the spin-charge-family theory: The internal space of fermions are in this model described with the Cliffordalgebra objects of all linear superposition of odd products of .a’s in d =(13 + 1). Fermions interact with only gravity — with the vielbeins and the two kinds of the spin connection fields (the gauge i Sab 1 fields of Sab =(.a.b -.b.a)and ˜=(.˜a .˜b - .˜b .˜a)1). Spins from higher 44 1 If there are no fermion present the two kinds of the spin connection fields are uniquely described by the vielbeins [36]. dimensions, d> (3 + 1), described by .a’s, manifest in d =(3 + 1)as charges of the standard model quarks and leptons and antiquarks and antileptons, appearing in (two times four) families, the quantum numbers of which are determined by the second kindofthe Cliffordalgebra object S˜ab’s. Gravity in higher dimensions manifests as the standard model vector gauge fields as well as the scalar Higgs and Yukawa couplings[1,5,23,25,26,26–29,31,32,34,35,37],predicting new scalar fields, whichoffer the explanation besides for higgs scalar andYukawa couplings also for the asymmetry between matter and antimatter in our universe and for the dark matter (represented by the stable of the upper group of four families), predicting a new family — the fourth family to the observed three. In this contributionIshortlyrepeat the descriptionof the internal spaceof the second quantized fermion fields with the odd products of the Cliffordoperators .a’s, what leads to the creation operators for fermions without postulating the sec­ond quantization requirements of Dirac [7–9]. The creation operators for fermions, which are superposition of tensor products of the ordinary basis and the ”basis vectors” describing the internal space of fermions, anticommute, explaining corre­spondingly the postulates of Dirac, offering also a new understanding of fermion fields([1]andreferences therein).Thecreation operatorsof fermions appear in families, carrying either left or right handedness, their Hermitian conjugated partners belong to another set of Cliffordodd ”basis vectors” carrying the opposite handedness. The main part of this contribution discusses properties of the second quantized boson fields, which are the gauge fields of the corresponding second quantized fermion fields. The internal space of bosons is described by the superposition of even products of .a’s. The boson fields correspondingly commute. The cor­responding creation operators and their Hermitian conjugated partners belong to the same set of ”basis vectors”, carrying all the quantum numbers in adjoint representations. They interact among themselves and with the corresponding fermion fields. In Sect. 16.2 the anticommuting Grassmann and Cliffordalgebras are presenting and the relations among them discussed. The ”basis vectors” are defined as the egenvectors of the Cartan subalgebra of the Lorentz algebra for the Grassmann and the two Cliffordalgebras for odd and for even products of algebras members, and their anticommutation or commutation relations presented, Subsect. 16.2.1. The reductionofthetwokindsoftheCliffordalgebrastoonlyonemakestheClifford odd ”basis vectors” anticommuting, giving to different irreduciblerepresentations of the Lorentz algebra the family quantum numbers, Subsect. 16.2.2.To make understanding of the properties of the Clifford odd and Clifford even ”basis vectors” easier in Subsect. 16.2.3 the case of d =(5 + 1)-dimensional space is chosen and the ”basis vectors” of odd, 16.2.3, and even, 16.2.3, Cliffordcharacter are presented in details and then generalized to any even d, Subsect. 16.2.4. In Sect. 16.3 the creation operators of the second quantized fermion and boson fields are discussed, as well as their Hermitian conjugated partners. In Sub-sect. 16.3.1 the simple action for fermion interacting with bosons and for cor­responding bosons, as assumed in the spin-charge-family theory is presented. Sect. 17.5 reviews shortly what one can learn in this contribution. Both algebras, Grassmann and Clifford, offer ”basis vectors” for the description of the internal space of fermions [1, 19, 20] and the corresponding bosons with which fermions interact. The oddness or evenness of ”basis vectors”, transfered to the creation operators, which are tensor products of the finite number of ”basis vectors” and the (continuously) infinite number of momentum (or coordinate) basis, and to their Hermitian conjugated partners annihilation operators, offers the second quantization of fermions and bosons without postulating the second quantized conditions [7–9] for either the half integer spin fermions or integer spin bosons, enabling the explanation of the Dirac’s postulates. Further investigations are needed in both case, for the boson case in particular, although promising, the time for this study was too short. 16.2 Grassmann and Clifford algebras To describe the internal space of fermions and bosons one can use either the Grassmann or the Cliffordalgebras. In Grassmann d-dimensional space there are d anticommuting operators .a , {.a,.b}+ = 0,a =(0, 1, 2, 3, 5, .., d),andd anticommuting derivatives withrespect .. to .a ,, { . , }+ = 0, offering together 2 · 2d operators, the half of which ..a ..a ..b are superposition of products of .a and another half corresponding superposition . of ..a . {.a,.b}+ = 0, { ., . }+ = 0, ..a ..b {.a,. }+ = dab , (a, b)=(0, 1, 2, 3, 5, ··· ,d). (16.1) ..b Defining [32] .. .aa .aa (.a)† = , leads to ()† = .a . (16.2) ..a ..a .a and . are, up to the sign, Hermitian conjugated to each other. The identity ..a is the self adjoint member of the algebra. We make a choice for the complex . properties of .a, and correspondingly of , as follows ..a {.a} * =(.0,.1 , -.2,.3 , -.5,.6 , ..., -.d-1,.d), . ...... .. {} * =( ,, - ,, - , , ..., - , ). (16.3) ..a ..0 ..1 ..2 ..3 ..5 ..6 ..d-1 ..d In d-dimensional space of anticommuting Grassmann coordinates and of their Hermitian conjugated partners derivatives, Eqs. (17.3, 16.2), there exist two kinds of the Cliffordcoordinates (operators) — .a and .˜a — both expressible in terms .a . of .a and their conjugate momenta p = i [20]. ..a .a =(.a + . ),.˜a = i (.a - . ), ..a ..a 1 .1 .a =(.a -i.˜a), =(.a + i.˜a), 2 ..a 2 (16.4) offering together 2 · 2d operators: 2d of those which are products of .a and 2d of those which are products of .˜a.Taking into account Eqs. (16.2, 16.4) it is easy to prove that they form two independent anticommuting Cliffordalgebras, Refs.([1] and references therein) {.a,.b}+ = 2.ab = {.˜a ,.˜b}+ , {.a ,.˜b}+ = 0, (a, b)=(0, 1, 2, 3, 5, ··· ,d), .aa .a .a)† .aa .˜a (.a)† = , (˜= , (16.5) with .ab = diag{1, -1, -1, ··· , -1}. While the Grassmann algebra can be used to describe the ”anticommuting integer spin second quantized fields” and ”commuting integer spin second quantized fields” [1,25], the Cliffordalgebras describe the second quantized fermion fields, if the superposition of odd products of .a’s or .˜a’s are used. The superposition of even products of either .a’s or .˜a’s describe the commuting second quantized boson fields. The reduction, Eq. (17.9) of Subsect. (16.2.2), of the two Cliffordalgebras — .a’s and .˜a’s — to only one is needed — .a’s are chosen — for the correct description of the internal space of fermions. After the decision that only .a’s are used to describe the internal space of fermions, the remaining ones, .˜a’s, are used to equip the irreducible representations of the Lorentz group (with the infinitesimal i generators Sab = {.a,.b}-)with the family quantum numbers in the case that 4 the odd Clifford algebra describes the internal space of the second quantized fermions. It then follows that the even Cliffordalgebra objects, the superposition of the even products of .a’s, offer the description of the second quantized boson fields, which are the gauge fields of the second quantized fermion fields, the internal space of whichare describedbytheoddCliffordalgebraobjects.Thiswillbe demonstrated in this contribution. 16.2.1 ”Basis vectors” determined by superposition of odd and even products of Clifford objects. d There are members of the Cartan subalgebra of the Lorentz algebra in even 2 dimensional spaces. One can choose S03 , S12 , S56 · , Sd-1d , ·· , S03 ,S12,S56 , ··· ,Sd-1d , S˜03 S˜12 S˜56 S˜d-1d ,,, ··· ,, Sab Sab Sab =+ ˜. (16.6) Let us look for the ”eigenstates” of each of the Cartan subalgebra members, Eq. (16.6), for each of the two kinds of the Cliffordalgebras separately, .aa .aa 1 k1 1i k1i Sab Sab (.a + .b)=(.a + .b), (1 + .a.b)=(1 + .a.b), 2 ik 22 ik 2k 22k .aa .aa 1 k1 1i k1i S˜ab .a .b).a .b),S˜ab .a .˜b).a .˜b), (˜+ ˜=(˜+ ˜(1 + ˜=(1 + ˜(16.7) 2 ik 22 ik 2k 22k k2 = .aa.bb. The proof of Eq. (17.7) is presented in Ref. [1], App. (I). .aa .aa Let us introduce for nilpotents 1 (.a+.b), (1 (.a+.b))2 = 0 and projectors 2ik 2ik 1i i1i (1 + .a.b), (1 (1 + .a.b))2 =(1 + .a.b)of both algebras the notation 2k2k 2k † ab .aa ab abab abab ab 1 (.a .b),.aa (-k)= .aa (k):=+ (k)= (-k), ((k))2 = 0, (k)[k] 2 ik † ab abab ab ab abab [k]: = 1 (1 + i.b), [k] =[k], ([k])2 =[k], [k][-k]= 0, .a2k abab abab ab ab ab ab ab ab (k)[k]= 0, [k](k)=(k), (k)[-k]=(k), [k](-k)= 0, ab .aa ab † ab ab 1 .aa (k˜): =(.˜a + .˜b), (k˜)= (-˜k), ((k˜))2 = 0, 2 ik † ab abab ab ab 1i [k˜]: =(1 + .˜a .˜b), [k˜] =[k˜], ([k˜])2 =[k˜], 2k abab abab ab ab ab ab ab ab (k˜)[k˜]= 0, [k˜](k˜)=(k˜), (k˜)[-˜k]=(k˜), [k˜](-˜k)= 0, (16.8) Statement 1. One can define ”basis vectors” to be eigenvectors of all the members of the Cartan subalgebras as even or odd products of nilpotents and projectors in any even dimensional space. Due to the anticommuting properties of the Clifford algebra objects there are anticommuting and commuting ”basis vectors”. The anticommuting ”basis vec­tors” contain an odd products of nilpotents, at least one nilpotent, the rest are then projectors. Let us denote the Cliffordodd ”basis vectors” of the Clifford .a kind as b^m† , where m and f determine the mth member of the fth irreducible f representation.We shall denoteby b^m =(b^m† )† the Hermitian conjugated part- ff bm† nerof the ”basis vector” ^. The ”basis vectors” of the Clifford .˜a kind would f b^m† correspondinglybe denotedby ˜and b^˜m . ff Itisnotdifficulttoprovethe anticommutationrelationsoftheCliffordodd ”basis vectors” and their Hermitian conjugated partners for both algebras([1,19] and references therein). Let us here present only the one of the Cliffordalgebras — .a’s. b^m f * A |.oc > = 0 |.oc >, b^m†* A |.oc > = |.mf >, f {b^m ,b^m '' }* A+|.oc > = 0 |.oc >, ff {b^m† ,b^m† }* A+|.oc > = |.oc >, (16.9) ff bm† bm ' where * A represents the algebraic multiplication of ^and ^among them­ ' ff selves and with the vacuum state |.oc > of Eq.(17.10), which takes into account Eq. (17.5), d -1 2 2 X b^m b^m† |.oc >= f * A |1 >, (16.10) f f=1 for one of the members m, anyone, of the odd irreducible representation f, with |1>, which is the vacuum without any structure — the identity. It follows that b^m ||.oc >= 0. The relations are valid for bothkinds of the odd Cliffordalgebras, f weonly havetoreplace b^m† by b^˜m† and equivalently for the Hermitian conjugated ff partners. The Cliffordodd ”basis vectors” almost fulfil the second quantization postulates for fermions. There is, namely, the property, which the second quantized fermions must fulfil in addition to the relations of Eq. (16.9). If the anticommutation rela­tions of ”basis vectors” and their Hermitian conjugated partners would fulfil the relation: ' † ' {b^mf ,b^mf ' }* A+|.oc > = dmm dff ' |.oc >, (16.11) for either .a or .˜a, then the corresponding creation and annihilation operators would fulfil the anticommutation relations for the second quantized fermions, explaining the postulates of Dirac for the second quantized fermion fields. For d-1d any b^m and any b^m ' † this is not the case. It turns out that besides b^m=1 = (-) ' ff f=1 5612 03 d-1d 561203 ··· (-)(-)(-i), for example, also b^m ' = (-) ··· (-)[+][+i]and several others ' f bm=1† give, when applied on ^ f=1 , nonzero contributions. There are namely 2 2d -1 -1 too many annihilation operators for each creation operator which give, applied on the creation operator, nonzero contribution. The problem is solvable by the reduction of the two Clifford odd algebras to only one[1,5,36,37]asitispresentedin subsection 16.2.2:If .a’s are chosen to determine internal space of fermions, the remaining ones, .˜a’s, determine then quantum numbers of each family (describedby the eigenvalues of S˜ab of the Cartan subalgebra members). Correspondingly the creation and annihilation operators, expressible as tensor products, * T , of the ”basis vectors” and the basis in ordinary (momentum or coordinate) space, fulfil the anticommutation relation for the second quantized fermions. Let me point out that the Hermitian conjugated partners of the ”basis vectors” belongtodifferentirreduciblerepresentationsofthe correspondingLorentzgroup than the ”basis vectors”. This can be understood, since the Cliffordodd ”basis ab vectors” have always odd numbers of nilpotents, so that an odd number of (k)’s ab transforms under Hermitian conjugation into (-k)’s, which can not be the member of the ”basis vectors”, since the even generators of the Lorentz transformations transform always even number of nilpotents, keeping the number of nilpotents always odd. It is different in the case of the Cliffordeven ”basis vectors”, since an ab even number of (k)’s, transformed with the Hermitian conjugation into en even ab number of (-k)’s belongs to the same group of the ”basis vectors”. Statement 2. The Clifford odd 2 d 2 -1 members of each of the 2 d 2 -1 irreducible representa­ tions of ”basis vectors” have their Hermitian conjugated partners in another set of 2 d -1 ·2 d -1 ”basis vectors”. Each of the two sets of the 2 d 2 -1 × 2 d 2 -1 Clifford even ”basis vectors” has their Hermitian conjugated partners within the same set. The Clifford even ”basis vectors” commute. Let us denote the Clifford even ”basis vectors”, described by .a’s,by .Thereisnoneedto denote their Hermitian A^m† f conjugated partners by A^m , since in the even Cliffordsector the ”basis vectors” f and their Hermitian conjugated partners appear within the samegroup.We shall manifest this in the toy model of d =(5 + 1). In the Clifford even sector m and f are just two indexes: f denotes the subgroups within which the ”basis vectors” do not have the Hermitian conjugated partners (Subsect. 16.2.3, Eq. (16.21)). We shall need also the equivalent ”basis vectors” in the Cliffordeven part of the A^˜m† kind ˜.a’s. Let these ”basis vectors” be denoted by . f These commuting even Cliffordalgebra objects have interestingproperties.Ishall discuss the properties of even and odd ”basis vectors” in Sects. 16.2.3, 16.2.4, first in d =(5 + 1)-dimensional space, then in the general case. 16.2.2 Reduction of the Clifford space The creation and annihilation operators of an odd Cliffordalgebra of both kinds, of either .a’s or .˜a’s, turn out to obey the anticommutation relations for the second quantized fermions, postulated by Dirac [1], provided that each of the irreducible representations of the corresponding Lorentz group, describing the internal space of fermions, would carry a different quantum number. But we know that a particular member m has for all the irreducible representa­tions the same quantum numbers, that is the same ”eigenvalues” of the Cartan subalgebra (for the vector space of either .a’s or˜.a’s), Eq. (17.8). Statement 3. The only possibility to ”dress” each irreducible representation of one kind of the two independent vector spaces with a new, let us say ”family” quantum number, is that we ”sacrifice” one of the two vector spaces. Let us ”sacrifice” .˜a’s, using .˜a’s to define the ”family” quantum numbers for each irreducible representation of the vector space of ”basis vectors of an odd prod­ucts of .a’s, while keeping the relations of Eq. (17.5) unchanged: {.a,.b}+ = .aa .a .aa 2.ab = {.˜a ,.˜b}+, {.a ,.˜b}+ = 0, (.a)† = , (.˜a)† = .˜a , (a, b)= (0, 1, 2, 3, 5, ··· ,d). We thereforepostulate: Let ˜.a’s operate on .a’s as follows [20,29,31,32,35] {.˜aB = (-)B i B.a}|.oc >, (16.12) with (-)B =-1, if B is (a function of) an odd products of .a’s, otherwise (-)B = 1 [35], |.oc > is defined in Eq. (17.10). Statement 4. After the postulate of Eq. (17.9) ”basis vectors” which are superposition of an odd products of .a’s obey all the fermions second quantized postulates of Dirac, presented in Eqs. (16.11, 16.9). We shall see in Sect. 16.2.3 that the Cliffordeven ”basis vectors” obey the bosons second quantized postulates. After this postulate the vector space of .˜a’s is ”frozen out”. No vector space of .˜a’s needs to be taken into account any longer for the description of the internal space of fermions or bosons, in agreement with the observed properties of fermions. .˜a’s obtain the role of operators determining properties of fermion and boson ”basis vectors”. Let me add that we shall still use S˜ab for the description of the internal space of fermion and boson fields, Subsects. 16.2.3, 16.2.3, 16.2.3. S˜ab’s remain as operators. One finds, using Eq. (17.9), ab ab abab ab ab ab abab ab (k˜)(k)= 0, (-˜k)(k)=-i.aa [k], (k˜)[k]= i (k), (k˜)[-k]= 0, abab ab ab ab ab ab ab abab [k˜](k)=(k), [-˜k](k)= 0, [k˜][k]= 0, [-˜k][k]=[k] . (16.13) Taking into account anticommuting properties of both Cliffordalgebras,.a’s and .˜a’s, it is not difficult to prove the relations in Eq. (16.13). 16.2.3 Properties of Clifford odd and even ”basis vectors” in d =(5 + 1) To make discussions easier let us first look for the properties of ”basis vectors” in d =(5 + 1)-dimensional space. Let us look at: i. internal space of fermions as the superposition of odd products of the Cliffordobjects .a’s, ii. internal space of the corresponding gauge fields as the superposition of even products of the Clifford objects .a’s. Choosing the ”basis vectors” to be eigenvectors of all the members of the Cartan subalgebra of the Lorentz algebra and correspondingly the products of nilpotents andprojectors (Statement1.) one findsthe ”basis vectors”presentedinTable 16.1. The table presents the eigenvalues of the ”basis vectors” for each member of the Cartan subalgebra for the group SO(5, 1). The oddI group (is chosen to) present the ”basis vectors” describing the internal space of fermions. Their Hermitian conjugated partners arethen the ”basis vectors” presented in the group odd II. The evenI and even II represent commuting Cliffordeven ”basis vectors”, repre­senting bosons, the gauge fields of fermions. We shall analyse both kinds of ”basis vectors” through the subgroups of the SO(5, 1)group. The choices of SU(2)× SU(2)× U(1)and SU(3)× U(1)subgroups of the SO(5, 1)group will also be discussed just to see the differences in properties from the properties of the SO(5, 1)group. InTable 16.1 thepropertiesof ”basis vectors” arepresented asproductsof nilpo­ 22 ab ab abab ab tents (+i)( (+i)= 0)and projectors [+]( [+] = [+]). ”Basis vectors” for fermions contain an odd number of nilpotents, ”basis vectors” for bosons contain an even ab ab number of nilpotents. In both cases nilpotents (+i)and projectors [+] are chosen to be the ”eigenvectors” of the Cartan subalgebra. Eq. (16.6), of the Lorentz algebra. The ”basis vectors”, determining the creation operators for fermions and their Hermitian conjugated partners, b^m† and b^m , respectively, as we shall see in Sub- ff sect. 16.2.3 they are superposition of odd products of .a, algebraically anticom-mute, due to the properties of the Cliffordalgebra {.a,.b}+ = 2.ab = {.˜a ,.˜b}+ , {.a ,.˜b}+ = 0, (a, b)=(0, 1, 2, 3, 5, ··· ,d), (.a)† .aa .a .a)† .aa .˜a = , (˜= , .aa .aa .a.a = ,.a(.a)† = I, .˜a .˜a = ,.˜a(.˜a)† = I, (16.14) where I represents the unit operator. ”Basis vectors” of odd products of .a’s in d =(5 + 1). Let us see in more details properties of the Cliffordodd ”basis vectors”, analysing them also with respect to two kinds of the subgroups SO(3, 1)× U(1)and SU(3)× U(1)of thegroup SO(5, 1), with the same number of Cartan subalgebra members in all three cases, d = 3.We use the expressions for the commuting operators for the subgroup 2 SO(3, 1)× U(1) 11 N3 (S12 ± iS03),N˜3 N3 S12 ± iS˜03), ±(= N3 ):= ±(= ˜):= (˜(16.15) (L,R)(L,R) 22 and for the commuting generators for the subgroup SU(3)and U(1) t3 1 (-S12 -iS03),t8 1 (-iS03 + S12 -2S13 14), := = v 2 23 1 (-iS03 + S12 + S56) t4 := - . (16.16) 3 The corresponding relations for t˜3 ,t˜8 and t˜4 can be read from Eq. (16.16), if re-Sab placing Sab by S˜ab. Recognizing that Sab =+ S˜ab one reproduces all the relations for the corresponding .t and N ±3 . The rest of generators of both kinds of subgroups of the group SO(5, 1)can be found in Eqs. (17.26, 17.28) of App. 16.7. bm† InTable 16.2 the properties of the odd ”basis vectors” ^are presented with f respect to the generators of the group i. SO(5, 1)(with 15 generators, 3 of them forming the corresponding Commuting among subalgebra), ii. SO(4)× U(1)(with 7 generators and 3 of the corresponding Cartan subalgebra members) and iii. SU(3)× U(1)(with 9 generators and 3 of the corresponding Cartan subalgebra members), together with the eigenvalues of the commuting generators. These ”ba­sis vectors” are alreadypresented asa partofTable 16.1.They fulfil together with their Hermitian conjugated partners the anticommutation relations of Eqs. (16.9, 16.11). The right handed, G(5+1) = 1, fourthpletof the fourth familyofTable 16.2 canbe foundin the first four linesofTable 16.5if only the d =(5 + 1)part is taken into account. The left handed fourthpletof the fourth familyofTable 16.4 canbe found in four lines from line 33 to line 36, again if only the d =(5 + 1)part is taken into 16 New way of second quantization of fermions and bosons 199 Table 16.1: 2d = 64 ”eigenvectors” of the Cartan subalgebra of the Cliffordodd and even algebrasin d =(5 + 1)-dimensional space are presented, divided into four groups. The first group, odd I, is chosen to represent ”basis vectors”, named dd b^m† -1 -1 2 ,appearing in2 2 = 4 ”families”(f = 1, 2, 3, 4),each ”family” with2 = 4 f ”family” members(m = 1, 2, 3, 4). The second group, odd II, contains Hermitian conjugated partners of the first group for each family separately, b^m =(b^m† )†. The ff b^m† S03 S˜12 S˜56), ”family” quantum numbers of , that is the eigenvalues of (˜,, are f written above each ”family”. The properties of anticommuting ”basis vectors” are discussed in Subsects. 16.2.3, 16.2.4. The two groups with the even number of .a’s, even Iand even II, have their Hermitian conjugated partners within their own group each. The two groups which are products of even number of nilpotents and evenoroddnumberofprojectorsrepresentthe ”basis vectors”forthe correspond­ing boson gauge fields. Their properties are discussed in Subsecs. 16.2.3 and 16.2.4. G(5+1) and G(3+1) represent handedness in d =(3 + 1)and d =(5 + 1)space calculated as products of .a’s, App. 16.9. '' basis vectors '' m f = 1 ( i , - 1 , - 1 ) 2 2 2 03 12 56 f = 2 (- i , - 1 , 1 ) 2 2 2 03 12 56 f = 3 (- i , 1 , - 1 ) 2 2 2 03 12 stackrel56 f = 4 ( i , 1 , 1 ) 2 2 2 03 12 56 S03 S12 S56 G(5+1) G(3+1) m† odd I ^bf 1 03 12 56 (+i) [+] [+] 03 12 56 [+i] [+] (+) 03 12 56 [+i] (+) [+] 03 12 56 (+i) (+) (+) i 2 1 2 1 2 1 1 2 [-i](-)[+] (-i)(-)(+) (-i)[-][+] [-i][-](+) - i 2 - 1 2 1 2 1 1 3 [-i][+](-) (-i)[+][-] (-i)(+)(-) [-i](+)[-] - i 2 1 2 - 1 2 1 -1 4 (+i)(-)(-) [+i](-)[-] [+i][-](-) (+i)[-][-] i 2 - 1 2 - 1 2 1 -1 03 12 56 03 12 56 03 12 56 03 12 56 G(5+1) odd II ^bm f 1 (-i)[+][+] [+i][+](-) [+i](-)[+] (-i)(-)(-) -1 2 [-i](+)[+] (+i)(+)(-) (+i)[-][+] [-i][-](-) -1 3 [-i][+](+) (+i)[+][-] (+i)(-)(+) [-i](-)[-] -1 4 (-i)(+)(+) [+i](+)[-] [+i][-](+) (-i)[-][-] -1 even I m (- 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 G(5+1) G(3+1) 1 [+i](+)(+) (+i)[+](+) [+i][+][+] (+i)(+)[+] i 2 1 2 1 2 1 1 2 (-i)[-](+) [-i](-)(+) (-i)(-)[+] [-i][-][+] - i 2 - 1 2 1 2 1 1 3 (-i)(+)[-] [-i][+][-] (-i)[+](-) [-i](+)(-) - i 2 1 2 - 1 2 1 -1 4 [+i][-][-] (+i)(-)[-] [+i](-)(-) (+i)[-](-) i 2 - 1 2 - 1 2 1 -1 even II m ( 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 G(5+1) G(3+1) 1 [-i](+)(+) (-i)[+](+) [-i][+][+] (-i)(+)[+] - i 2 1 2 1 2 -1 -1 2 (+i)[-](+) [+i](-)(+) (+i)(-)[+] [+i][-][+] i 2 - 1 2 1 2 -1 -1 3 (+i)(+)[-] [+i][+][-] (+i)[+](-) [+i](+)(-) i 2 1 2 - 1 2 -1 1 4 [-i][-][-] (-i)(-)[-] [-i](-)(-) (-i)[-](-) - i 2 - 1 2 - 1 2 -1 1 account. Statement 5. In a chosen d=dimensional space there is the choice that the ”basis vectors” are right handed. Their Hermitian conjugated partners are correspondingly left handed. One could make the opposite choice, likeinTable 16.4. Then the ”basis vectors”ofTable 16.2 wouldbe the Hermitian conjugated part­ners to the left handed ”basis vectors” ofTable 16.4. For the left handed ”basis vectors” the vacuum state |.oc >, Eq. (17.10), chosen as the P b^m * A b^m† , has ff f bm to be changed, since the vacuum state must have the property that ^f |.oc >= 0 200 N.S. Manko.c Bor.stnik m=(ch,s)† ^ Table 16.2: The basic creation operators, ”basis vectors”— b(each is a f product of projectors and an odd product of nilpotents, and is the ”eigenvector” of , S12 S03 S12 all the Cartan subalgebra members, S03 , S56 and˜,˜, S˜56, Eq. (16.6)(ch (charge), the eigenvalue of S56,ands (spin), the eigenvalues of S03 and S12,explain S03 S12 S56) index m, f determines family quantum numbers, the eigenvaluesof( ˜,˜,˜ — are presented for d =(5 + 1)-dimensional case. This table represents also the eigenvalues of the three commuting operators N3 and S56 of the subgroups L,R SU(2)× SU(2)× U(1)and the eigenvalues of the three commuting operators t3,t8 and t4 of the subgroups SU(3)× U(1), in these two last cases index m represents the eigenvalues of the corresponding commuting generators. G(5+1) = m=(ch,s)† m=(ch,s) -.0.1.2.3.5.6 , G(3+1) = i.0.1.2.3. Operators b^and b^fulfil ff the anticommutationrelationsof Eqs. (16.9, 16.11). f = (ch, s) m ^m=(ch,s)† fb 03S 12S 56S 3+1G 3 LN 3 RN 3t 8t 4t S˜03 S˜12 S˜56 I () (, -) (-) (-, -) 1 21 21 21 2, , 1 21 21 21 21234 03 12 56 (+i) [+] | [+]03 12 56 [-i] (-) | [+] 03 12 56 [-i] [+] | (-) 03 12 56 (+i) (-) | (-) --i 2 i 2i 2 i 2 --1 2 1 2 1 21 2 -1 21 21 21 2- 11-1-1 001 21 2- 1 21 2001 2 - 001 2 1 2 - 03 3 3 11 1 2 2 -v v v 1 2 1 6 1 6 1 6 - i 2i 2i 2i 2 1 21 21 21 2---- 1 2 1 2 1 2 1 2 ---- II () 1 2, 1 21 03 12 56 [+i] (+) | [+] i 2 1 2 1 2 1 0 1 2 0 0 1 2 - i 2 - 1 2 1 2 - 03 12 56 (, -) 1 21 22 (-i) [-] | [+] -i 2 1 2 - 1 2 1 0 1 2- 0 3 1-v 1 6 i 2 - 1 2 1 2 - 03 12 56 (-) 1 2, 1 23 (-i) (+) | (-) i 2 - 1 2 1 2- -1 1 2 0 1 2 - 3 1 2 v 1 6 i 2 - 1 2 1 2 - 03 12 56 (-, -) 1 21 24 [+i] [-] | (-) i 2 1 2- 1 2- -1 1 2- 0 1 2 3 1 2 v 1 6 i 2 - 1 2 1 2 - 03 12 56 III () 1 2, 1 21 [+i] [+] | (+) i 2 1 2 1 2 1 0 1 2 0 0 1 2- i 2- 1 2 - 1 2 (, -) 1 21 22 03 12 56 (-i) (-) | (+) i 2- 1 2 - 1 2 1 0 1 2- 0 3 1-v 1 6 i 2- 1 2 - 1 2 (-) 1 2, 1 23 03 12 56 (-i) [+] | [-] i 2 - 1 2 1 2- -1 1 2 0 1 2 - 3 1 2 v 1 6 i 2- 1 2 - 1 2 03 12 56 1 2(-- , ) 1 24 [+i] (-) | [-] i 2 1 2- 1 2- -1 1 2- 0 1 2 3 1 2 v 1 6 i 2- 1 2 - 1 2 IV 1 2, 1 2() 1 03 12 56 (+i) (+) | (+) i 2 1 2 1 2 1 0 1 2 0 0 1 2 - i 2 1 2 1 2 03 12 56 1 21 22(, -) [-i] [-] | (+) i 2- 1 2 - 1 2 1 0 1 2- 0 3 1-v 1 6 i 2 1 2 1 2 1 2, 1 23(-) 03 12 56 [-i] (+) | [-] i 2 - 1 2 1 2- -1 1 2 0 1 2 - 3 1 2 v 1 6 i 2 1 2 1 2 1 21 24(-, -) 03 12 56 (+i) [-] | [-] i 2 1 2- 1 2- -1 1 2- 0 1 2 3 1 2 v 1 6 i 2 1 2 1 2 bm† b^m† and ^>= . f |.oc f One can notice that: i. The family members of ”basis vectors” have the same properties in all the families, independently whether one observes the group SO(d -1, 1)(SO(5, 1)in the case of d =(5 + 1))or of the subgroups with the same number of commuting operators(SU(2)×SU(2)×U(1)or ×SU(3)×U(1)in d =(5+1)case). The families carrydifferentfamily quantum numbers.Thisistrueforright,(G(5+1) = 1), and for left(G (5+1) =-1),representations. ii. The sumof all the eigenvaluesof allthe commuting operators over the 2 2d -1 family members is equal to zero for each of 2 2d -1 families, separately for left and separately for right handed representations, independently whether the group SO(d -1, 1)(SO(5, 1))or the subgroups(SU(2)× SU(2)× U(1)or ×SU(3)× U(1)) are considered. iii. The sum of the family quantum numbers over the four families is zero as well. iv. The properties of the left handed family members differ strongly from the right handed ones.Itis easytorecognize thisin our d =(5 + 1)case when looking at SU(3)× U(1)quantum numbers since the right handed realization manifests the ”colour”propertiesof ”quarks” and ”leptons” and the left handed the ”colour” properties of ”antiquarks” and ”antileptons”. v. For a chosen even d there is a choice for either right or left handed family members. The choice of the handedness of the family members determine also the vacuum state for the chosen ”basis vectors”. Let me add that the ”basis vectors” and their Hermitian conjugated partners ful­fil the anticommutation relations postulated by Dirac for the second quantized fermion fields. When forming tensor products, * T , of these ”basis vectors” and the basis of ordinary, momentum or coordinate, space the single fermion creation and annihilation operators fulfil all the requirements of the Dirac’s second quantized fermion fields, explaining therefore the postulates of Dirac, Sect. 16.3. ”Basis vectors” of even products of .a’s in d =(5 + 1) The Clifford even ”basis ab vectors”, they are products of an even number of nilpotents, (k), and the rest ab up to d of projectors, [k], commute since even products of (anticommuting) .a’s 2 commute. Let us see in more details several properties of the Clifford even ”basis vectors”: A. The properties of the algebraic, * A, application of the Clifford even ”basis vectors” on the Cliffordodd ”basis vectors” b^m† ,presentedinTable 16.2, teaches f us that the Clifford even ”basis vectors” describe the internal space of the gauge fields of f . b^m† A.i. bm‘† Let ^represents the m‘th Clifford odd I ”basis vector”(thepartofthecreation f‘ operators which determines the internal part of the fermion state) of the f‘th family and let f denotes the mth Clifford even II ”basis vector”of the fth irreducible A^m† Sab Sab representation with respect to Sab — but not with respects to Sab =+ ˜, which includes all 2 A^m† bm‘† on ^ ff‘ d 2 -1 × 2 d 2 -1 members. Let us evaluate the algebraic products for any (m, m ' )and (f, f ' ). Taking into account Eq. (17.8) andTables (16.1,16.2) one can easily evaluate the 1† Am† ' † algebraic products ^on b^m for any (m, m ' )and (f, f ' ). Starting with b^one ' ff 1 1† finds the non zero contributions only if applying A^m† , m =(1, 2, 3, 4)on b^ 31 03 1256 A^m† ^1† 3 * A b1 (= (+i)[+][+]) : 03 1256 03 1256 1† 1† A^1† (= [+i][+][+]) * A b^(= (+i)[+][+]) . b^, 31 1 03 1256 03 1256 A^2† 1† 2† ^ (= (-i)(-)[+]) * A b. b^(= [-i](-)[+]) , 3 11 03 1256 03 1256 A3† 1† 3† ^(= (-i)[+](-)) * A b^. b^(= [-i][+](-)) , 3 11 03 1256 03 1256 A4† 1† 4† ^(= [+i](-)(-)) * A b^. b^(= (+i)(-)(-)) . (16.17) 3 11 The products of an even number of nilpotents and even or an odd number of projectors,representedby evenproductsof .a’s, applyingon family membersofa particular family, obviously transform family members, representing fermions of one particular family, into the same or another family member of the same family. A^m ^1† All the rest of ,f .= 3, applying on b, give zero for any family f. f1 Let us comment the above events, concerning only the internal space of fermions and, obviously, bosons: If the fermion, the internal space of which is described 1† by Cliffordodd ”basis vector” b^, absorbs the boson A^1 (with S03 = 0, S12 = 13 1† 0, S56 = 0), its ”basis vector” b^remains unchanged. 1 1† The fermion with the ”basis vector” b^, if absorbing the boson with A^2 (with 13 S03 1† =-i, S12 =-1, S56 = 0), changes its internal ”basis vector” b^into the 1 ^2† -i -11 ”basis vector” b(which carries now S03 = ,S12 = , and the same S56 = 1 222 1† as before). The fermion with ”basis vector” b^absorbing the boson with the ”basis 1 A^3 ^3† vector” changes its ”basis vector” to b, while the fermion with the ”basis 31 1† vector” b^absorbing the boson with the ”basis vector” A^4 changes its ”basis 13 4† ^Let us see how do the rest of A^m , m =(1, 2, 3, 4), f =(1, 2, 3, 4) change the vector” to b1 . f bn† properties of ^, n = 2, 3, 4. 1 Itis easyto evaluateif taking into accountEq . (17.8) that 03 1256 Am† 2† ^^ 4 * A b1 (= [-i](-)[+]) : 03 1256 03 1256 A1† 2† 1† ^^ (= (+i)(+)[+]) * A b(= [-i](-)[+]) . b^, 4 11 03 1256 03 12 56 A2† 2† 2† ^(= [-i][-][+]) * A b^. b^(= [-i](-)[+]) , 4 11 03 12 56 03 1256 A3† 2† 3† ^^. ^ (= [-i](+)(-)) * A bb(= [-i][+](-)) , 4 11 03 1256 03 12 56 2† 4† A^4† (= (+i)[-](-)) * A b^. b^(= (+i)(-)(-)) , 4 11 03 1256 A^m† ^3† 2 * A b1 (= [-i][+](-)) : 03 1256 03 12 56 3† 1† A^1† (= (+i)[+](+)) * A b^(= [-i][+](-)]) . b^, 211 03 12 56 03 1256 3† 2† A^2† (= [-i](-)(+)) * A b^. b^(= [-i](-)[+]) , 2 11 03 1256 03 12 56 3† 3† A^3† (= [-i][+][-]) * A b^. b^(= [-i][+](-)) , 2 11 03 1256 03 12 56 1† 4†' A^4† (= (+i)(-)[-]) * A b^. b^(= (+i)(-)(-)) , 2 11 03 1256 A^m† ^4† b(= (+i)(-)(-)) : 1 * A1 03 1256 03 1256 4† 1† A^1† (= [+i](+)(+)) * A b^(= (+i)(-)(-)) . b^, 11 1 03 1256 03 1256 4† 2† A^2† (= (-i)[-](+)) * A b^. b^(= [-i](-)[+]) , 1 11 03 1256 03 1256 4† 3† A^3† (= (-i)(+)[-]) * A b^. b^(= [-i][+](-)) , 1 11 03 1256 03 12 56 4† 4† A^4† (= [+i][-][-]) * A b^. b^(= (+i)(-)(-)) . (16.18) 1 11 All the rest of A^m , applying on b^n† , give zero for any other f except the one f1 presented in Eqs. (16.17, 16.18). We can repeat this calculation for all four family membersb^m‘† of any of families f‘ f‘. concluding A^m† * A b^1† . b^m† , 3 ff Am† 2† bm† ^^. ^ * A b, 4 ff 3† A^m† * A b^. b^m† , 2 ff Am† 4† bm† ^* A b^. ^. (16.19) 1 ff The recognition of this subsection concerns so far only internal space of fermions, not yet its dynamics in ordinary space. Let us interpret what is noticed: Statement 6. Afermion with the ”basis vector”b^m‘† , ”absorbing” one of the commuting f‘ A^m† Clifford even objects, , transforms into another family member of the same family, to f b^m† , changing correspondingly the family member quantum numbers and keeping the f same family quantum number orremains unchanged. The applicationof the Cliffordeven ”basis vector” A^m† on the Cliffordodd ”basis f vector” does not cause the changeof the familyof the Cliffordodd ”basis vector”. A.ii. Weneed to know the quantum numbers of the Cliffordeven ”basis vectors”, which obviously manifest properties of the boson fields since they bring to the Clifford odd ”basis vectors” — representing the internal space of fermions — the quantum numbers which cause transformation into another fermion withadifferent Clifford odd ”basis vectors” of the same family f. The Clifford even ”basis vectors” do not cause the change of the family of fermions. Let us point out that the Clifford odd ”basis vectors” appear in 2 2d -1 families with 2 2d -1 family members in each family, four members in four families in the d =(5 +1)case, while the Hermitian conjugated partners belong to another group of 2 d 2 -1 × 2 d 2 -1 Cliffordodd ”basis vectors”, (to oddII inTable 16.1), while the Clifford even ”basis vectors” have their Hermitian conjugated partners within the same group of 2 d 2 -1 × 2 d 2 -1 members (appearing in our treating case in evenII inTable 16.1). Since we foundin Eqs. (16.19, 16.17, 16.17) that the Clifford even ”basis vector” transforms the Cliffordodd ”basis vector” into another memberof the same family, changing the family members quantum numbers for an integer, they must carry the integer quantum numbers. One can seeinTable 16.1 that the membersof thegroup evenII, for example, are Hermitian conjugated to one another in pairs and four of them are self adjoint. Correspondingly † has no special meaning, it is only the decision that all the Am† Cliffordeven ”basis vector” are equipped with †: ^. f A^m† Let us thereforecalculate the quantum numbers of ,wherem and f distinguish f among different Cliffordeven ”basis vectors” (with f which does notreally denote Sab the family, since Sab =+ S˜ab defines the whole irreducible representation of 2 d 2 -1 × 2 d 2 -1 ”basis vectors”) with the Cartan subalgebra operators Sab = Sab + S˜ab, presented in Eqs. (16.6). InTable 16.3 the eigenvaluesof the Cartan subalgebra membersof Sab are pre­sented, as well as the eigenvalues of the commuting operators of subgroups SU(2)×SU(2)×U(1),that is the eigenvalues of(N L3 , N R3 , S03),and ofSU(3)×U(1), that is the eigenvalues of(t3,t8,t4), expressions for which can be found in Sab Eqs. (16.15, 16.16) if one takes into account that Sab =+ S˜ab. The alge­braic application of any member of a group f on the self adjoint operator (denoted inTable 16.3by .)of this groupf, gives the same member back. The vacuum state of the Clifford even ”basis vectors” is correspondingly the normalized sum of all the self adjoint operators of these Clifford even group evenII. Each of^Am† when applying on sucha vacuum state gives the same A^m† . ff 03 1256 03 1256 03 1256 03 1256 >= ([+i] [-] [-] + [-i] [+] [-] + [+i] [+] [+] + [-i] [-] [+]) . 1 (16.20) |.oceven 2 Thepairsof ”basis vectors” A^m† , which are Hermitian conjugated to each other, f areinTable 16.3 pointed outby the same symbols. Thispropertyis independent of the group or subgroups which we choose to observe properties of the ”basis Am† vectors”. If treating the subgroup SU(3)× U(1)one finds the 8 members of^, f which belong to the group SU(3)forming octet which has t4 = 0, six of them ap­pear in three pairs Hermitian conjugated to eachother, two of them are self adjoint members of the octet, with eigenvalues of all the Cartan subalgebra members equal to zero. There are also two singlets with eigenvalues of all the Cartan subalgebra membersequaltozero.Andthereisthe sextet,withthreepairswhicharemutually Hermitian conjugated. One can notice that the sum of all the eigenvalues of all the Table 16.3: The ”basis vectors” A^m† , each is the product of projectors and an f even number of nilpotents, and is the ”eigenvector” of all the Cartan subalgebra , S12 members, S03 , S56, Eq. (16.6), are presented for d =(5 + 1)-dimensional dd case. Indexes m and f determine 2 2 -1 × 2 2 -1 different members A^m† . In the f thirdcolumn the ”basis vectors” Am† which are Hermitian conjugated partners ^ f 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 with m = 4 and f = 3. The sign . denotes the Am† Am† ”basis vectors” which are self adjoint (^)† =^. This table represents also ff the 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 t3,t8 and t4 of the subgroups SU(3)× U(1). f m * ^m† Af S03 S12 S56 N3 L N3 R t3 t8 t4 03 12 56 I 1 .. [+i] (+) (+) 03 12 56 0 1 1 1 2 1 2 - 1 2 - 1 v 2 3 - 2 3 2 . (-i) [-] (+) 03 12 56 -i 0 1 1 2 - 1 2 - 1 2 - 3 v 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 v 2 3 - 2 3 2 . [-i] (-) (+) 03 12 56 0 -1 1 - 1 2 - 1 2 1 2 - 3 v 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 - 1v 3 2 3 3 • 03 12 56 (-i) [+] (-) -i 0 -1 1 2 - 1 2 - 1 2 1 v 2 3 2 3 03 12 56 4 .. [+i] (-) (-) 0 -1 -1 - 1 2 - 1 2 1 2 1 v 2 3 2 3 IV 1 .. 03 12 56 (+i) (+) [+] i 1 0 0 1 0 1v 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 v 2 3 0 03 12 56 4 . (+i) [-] (-) i 0 -1 - 1 2 1 2 1 2 3 v 2 3 0 Cartan subalgebra members over the 16 members A^m† is equal to zero, indepen- f dent of whether we treat the group SO(5, 1),SU(2)×SU(2)×U(1),orSU(3)×U(1). A.iii. In A.i. we saw that the application of A^m† on the fermion ”basis vectors” b^m† ff transforms the particular member bm† to one of the members of the same family f, ^ f changing eigenvaluesof the Cartan subalgebra members for an integer.We found 2 -1× 2 -1 in A.ii the eigenvalues of the Cartan subalgebra members for each of 2 dd (equal to 16 in d =(5 + 1)) Am† , recognizing that they do have properties of the ^ f boson fields. Itremainstolookforthe behaviouroftheseClifford even”basisvector”when they apply on each other. Let us denote the self adjoint member in each group of ”basis vectors” of particular f as Am0† f ^ .We easily see that ' † ^ ' ) ' Am† , Am ff ^ ^ if (m, m ., . m, . f. = , . f, { }- = 0, m0 or m = m0 = m Am0† Am† ff ^ Am† f ^ * A (16.21) = ' † Am† f ^ Am f ^ ' ) Two ”basis vectors”andof the same f and of (m, m m0 are orthog- = ., the algebraicproduct, * A, of which gives onal. ' † Am† f ^ Am ^f ^ The two ”basis vectors”and ' A4† A1† 22 ^ A1† 1 ^ nonzero contribution, like ^ , ”scatter” into the thirdone, or anni- * A = ^ A3† A2† 44 2To generate A2† 2 ^ hilate into vacuum |.oceven >, Eq. (16.20), like * A = . creation and annihilation operators the tensor products, * T , of the ”basis vectors” or coordinate, space is needed. ' ^ Am† f ^ , as well as of the ”basis vectors”, with the basis in ordinary, momentum bm† f † ^A m f Am† f ^ ' ) Statement 7. Two ”basis vectors”andof the same f and of (m, m 0 = .are orthogonal. The two ”basis vectors” with nonzero algebraic product, * A,”scatter” into the third one, or annihilate into vacuum. B. Let us point out that the choice of the Cliffordodd ”basis vectors”, odd I, describing the internal space of fermions, and consequently the choice of the Cliffordeven ”basis vectors”, even II, describing the internal space of their gauge fields, is ours. If we choose inTable 16.1 odd II to represent the ”basis vectors” describing the internal space of fermions, then the corresponding ”basis vectors” representing the internal space of bosonic partners are those of even I. For a different choice of handedness of the Cliffordodd ”basis vectors” for de­scribing fermions — makinga choiceof the left handedness insteadof the right handedeness —Table 16.2 shouldbereplacedbyTable 16.4 and correspondingly also A.i., A.ii., A.iii. should be rewritten. For an even d there is a choice for either right or left handed family members. The choice of the handedness of the family members determine also the vacuum state for the chosen ”basis vectors” for either — Cliffordodd ”basis vectors” of fermions or for the corresponding Clifford even ”basis vectors” of the corresponding gauge boson fields. m0 C. The Clifford even ”basis vectors” ^ Am† f , representing the boson gauge fields ^ bm† of each family within the family f ^ to the corresponding Cliffordodd ”basis vectors” bm† , have the properties that f they transform Cliffordodd ”basis vectors” members. There are the additional Clifford even ”basis vectors” ^A ˜m† f which trans­ form each family member of particular family into the same family member of someof therest families. ^ 2Iuse ”scatter” in quotation marks since the ”basis vectors” ^ Am† f determine only the internal spaceof bosons,asalsothe ”basis vectors” bm† f determine only the internal space of fermions. ^˜m† These Clifford even ”basis vectors” A are products of an even number of f nilpotents and of projectors, which are eigenvectors of the Cartan subalgebra S03 S˜12 S˜56 ˜ operators ˜, , ,..., Sd-1d. The table likeTable 16.3 should be prepared and their properties described as in the case of A.i., A.ii., A.iii..Ashort illustration ^˜m† is to help understanding the role of these Clifford even ”basis vectors” A . f m† ^ Let us use for the Clifford even ”basis vectors” A ˜the same arrangement with f products of nilpotents and projectors as the one, chosen for the Cliffordeven ”basis vectors” A^m† in the case of d =(5 + 1)inTable 16.3, except that now nilpotents f S03 S˜12 S˜56 and projectors are eigenvectors of the Cartan subalgebra operators ˜,, , ab ab and are correspondingly written in terms of nilpotents (k˜) and projectors [k˜]. The application of these nilpotents and projectors on nilpotents and projectors 03 12 1† b^m† ^˜˜ appearingin arepresentedinEq. (16.13). Makinga choiceof A (= [+˜i] (+) f1 56 03 1256 4† ˜0, S12 1, S56 (+)), with quantum numbers (S03 === 1), on b^(= (+i) (-) (-)) 1 i with the family members quantum numbers (S03 = ,S12 =-1 ,S56 =-1 )and 222 S03 i S˜12 -1 S˜56 -1 the family quantum numbers (˜= , = , =)it follows 222 03 1256 03 12 56 03 1256 ^˜1† ˜˜˜^4† 4† A (= [+i] (+) (+)) * A b(= (+i) (-) (-)) . b^(= (+i) [-] [-]) . (16.22) 1 14 03 1256 4† 4† (S03 ^ b(= (+i) [-] [-]) carry the same family members quantum numbers as b^= 41 i ,S12 =-1 ,S56 =-1 ))but belongs to the different family with the family 222 S03 i S12 1 S56 1 quantum numbers (˜= , ˜= , ˜=). 222 The detailed analyse of these last two cases B. and C. will be studied after this Bled proceedings. A^m† We can conclude that the Cliffordeven ”basis vectors” f : a. Have the quantum numbers determined by the Cartan subalgebra members of Sab + ˜A^m† the Lorentz group of Sab = Sab. Applying algebraically, * A, on the f Cliffordodd ”basis vectors” b^m† , A^m† transform these ”basis vectors” to another f3 ones with the same family quantum numbers, b^m‘† . f b. In any irreducibly representation of Sab A^m† appear in pairs, which are Hermi- f tian conjugated to each other or they are self adjoint. A^m† c. The self adjoint members define the vacuum state of the second quantized f boson fields. d. Applying A^m† algebraically to each other these commuting Cliffordeven ”basis f vector” forming another Cliffordeven ”basis vector” or annihilate into the vacuum. e. The choice of the left or the right handedness of the ”basis vectors” of an odd Cliffordcharacter, describing the internal space of fermions, is ours. The left and the right handed ”basis vectors” of an odd Cliffordcharacter arenamely Hermitian conjugatedtoeachother.Withthe choiceofthe handednessofthe fermion ”basis vectors” also the choice of boson Cliffordeven ”basis vectors” — which are their corresponding gauge fields — are chosen. 03 1256 m† 1† ^˜^˜˜˜ f. Thereexist the Cliffordeven ”basis vectors” A (like A (= [+i] (+)(1˜)))which f1 b^m† transform the Cliffordodd ”basis vectors” , representing the internal space of f fermions, into the Cliffordodd ”basis vectors” b^m† with the same family member ' f m belonging to another family f‘. 16.2.4 ”Basis vectors” describing internal space of fermions and bosons in any even dimensional space In Subsect. 16.2.3 the properties of the ”basis vectors”, describing internal space of fermionsand bosonsinatoy modelwith d =(5 + 1)are presented in order to simplify (to make moreillustrative) the discussions on the properties of the Clifford odd ”basis vectors” describing the internal space of fermions and the Cliffordeven ”basis vectors” describing the internal space of corresponding bosons, the gauge fields of fermions. The generalization to any even d is straightforward. For the description of the internal spaceof fermionsIfollow here Ref. [1]. a. The ”basis vectors” offering the description of the internal space of fermions, ab b^m†' , must contain an odd product of nilpotents (k), 2n + 1, in d = 2(2n + 1), f ab ' (d '' '' n =(0, 1, 2, . . . , 1 -1), and the rest is the product of n projectors [k], n = 22 d ' -(2n + 1). Nilpotents and projectors are chosen to be ”eigenvectors” of the d 22 members of the Cartan subalgebra. After the reduction of the two kinds of the Cliffordalgebras to only one, .a’s, the generators Sab of the Lorentz transformations in the internal space of fermions d 2 -1 family members for each of 2 d 2 -1 families, describedby .a’s, determine the 2 while S˜ab’s determine the d 2 numbers (the eigenvalues of the Cartan subalgebra members of the 2 2d -1 families). bm† The Clifordodd ”basis vectors” ^obey the postulates of Dirac for the second f quantized fermion fields ' ' {b^m b^m † dmm f , ' }* A+ |.oc > = dff ' |.oc >, f {b^mf ,bmf '' }* A+ |.oc > = 0 · |.oc >, ^ {b^m† ,b^m '' † }* A+ |.oc > = 0 · |.oc >, ff b^m† > = |.m >, * A |.oc f f ^ bmf * A|.oc > = 0 · |.oc >, (16.23) with(m, m ')denoting the ”family” members and(f, f ')denoting ”families”,* A represents the algebraic multiplication of b^m† with their Hermitian conjugated f bm† bm‘ objects b^m , with the vacuum state |.oc >, Eq. (17.10), and ^or ^among f ff‘ themselves. It is not difficult to prove the above relations if taking into account Eq. (17.5). bm† The Cliffordodd ”basis vectors” ^’s and their Hermitian conjugated partners f b^m f ’s appear in two independent groups, each with 2 d 2 -1× 2 d 2 -1 members, Her­ mitian conjugated to each other. d d -1 -1 members to take It is our choice which one of these two groups with 2 × 2 22 †m as ”basis vectors” b^’s. Making the opposite choice the ”basis vectors” change f handedness. A^m† b. The ”basis vectors” for bosons, ,must contain an even number of nilpotents f ab ' '' (k), 2n '. In d = 2(2n + 1), n =(0, 1, 2, . . . , 1 (d -1)), the rest, n , are projectors 22 ab '' (d [k], n = -(2n ' )). 2 The ”basis vectors” are either self adjoint or have the Hermitian conjugated part­ ners within the same group of 2 d 2 -1× 2 d 2 -1 members. They do not form families, m and f only notea particular ”basis vector”. Oneof the members of particular f is selfadjoint and participates to the vacuum state which has 2 2d -1 summands, Eq. (16.20). The Clifford even ”basis vectors” A^m† commute, { A^m† , A^m ' † }- = 0, if both have f ff the same index f and none of them or both of them are self adjoint operators. ' { A^m† , A^m ' † 0, if (m, m ' ) .m0 or m = m0 = m, . f, ff }- == A^m† * A A^m0† = A^m† , . m, . f. (16.24) ff f The two ”basis vectors”, A^m† and A^m ' † , the algebraic product, * A, of which gives ' ff nonzero contribution, ”scatter” into the thirdone, or annihilate into the vacuum |.oceven >. Quantum numbers of A^m† are determined by the Cartan subalgebra members of f Sab Sab the Lorentz group Sab =+ ˜. bm† Ifa fermion with the ”basis vector” ^”absorbs” one of the commuting Clifford f even objects, A^m‘† ,it transforms into another family member of the same family,to ' f b^m ' † , changing correspondingly the family member quantum numbers, keeping f the family quantum number the same, orremains unchanged. The remaining group of 2 d 2 -1 × 2 d 2 -1 Clifford even ”basis vectors”, presented in Table 16.1 do not influence the chosen Cliffordodd ”basic vectors”, but rather their Hermitian conjugated partners b^m . f ^˜m† There are the even ”basis vectors” A , the nilpotents and projectors of which f ab ab m† are (k˜), [k˜], respectively. These ”basis vectors” A ^˜, if applying on the Clifford f bm† b^m† odd ”basis vectors” ^, transform these ”basis vectors” into ”basis vectors” ' ff belonging to different family f, while the family member quantum number m remains unchanged. Exchanging theroleof the Cliffordodd ”basis vector” and their Hermitian b^m† f conjugated partners b^m (what means in the case of d =(5 + 1)the exchange of f odd I, which is right handed, with odd II, whichis lefthanded,inTable 16.1), not bn† only causes the change of the handedness of the new ^, but also the change of f the role of the Clifford even ”basis vectors” (what means in the case of d =(5 + 1) the exchange of even II with even I). 210 N.S. Manko.c Bor.stnik 16.3 Second quantized fermion and boson fields with internal space described by Clifford algebra After the reduction of the Cliffordspace to only the part determined by .a’s, the ”basis vectors”, which are superposition of odd products of .a’s, determine the internal space of fermions. The ”basis vectors” are orthogonal and appear in even dimensional spaces in 2 d 2 -1 families, each with 2 d 2 -1 family members. Quantum numbersof family membersare determinedby Sab, quantum numbers of families are determinedby .˜a’s, or better by S˜ab’s. .˜a’s anticommute among themselves and with .a’s,astheydid beforethereductionoftheCliffordspace,Eq. (17.11). ”Basis vectors” b^m† , determining internal space of fermions, are in even dimen- f sional spaces products of an odd number of nilpotents and an even number of projectors, chosen to be eigenvectors of the d Cartan subalgebra members of the 2 Lorentz algebra Sab,Table 16.2. There are 2 2d -1 2 × 2 d -1 Hermitian conjugated bm† partners of ”basis vectors”, denoted by b^m (=(^)†. It is our choice which one of ff these two groups of 2 d 2 -1 × 2 d 2 -1 members are ”basis vectors” and which one are their Hermitian conjugated partners. These two groups differ in handedness as canbe seeninTable 16.1,if observing odd I and odd II, as well as if we compare Table 16.2 andTable 16.4. The Clifford odd anticommuting ”basis vectors”, describing the internal space of fermions, obey together with their Hermitian conjugated partners the postulates of Dirac for the second quantized fermion fields, Eq. (17.11). The Clifford even products of .a’s (with the even number of nilpotents) form twice d -1 × 2 d 2 -1 ”basis vectors”, A^m† f , describing properties of bosons, Table 16.3.Each 2 2 of the two groups arecommuting objects due to the fact that even number of .a’scommute. Also the Clifford even ”basis vectors” are chosen to be the eigenvectors of the Sab Sab Cartan subalgebraoftheLorentzgroup, this time determinedby Sab =+ ˜, Eqs. (16.19, 16.21). While the Cliffordodd ”basis vectors” and their Hermitian con­jugated partners form two independent groups, the Clifford even ”basis vectors” have their Hermitian conjugated partners within each of the two groups. The choice of the ”basis vectors” among the two groups of the Clifford odd products of nilpotents and projectors for the description of the internal space of fermions, distinguishing also in handedness and other properties (Table 16.2 and Table 16.4) made as well the choice foe the Clifford even ”basis vectors” describing the corresponding boson fields.We noticeinTable 16.1 that the choice of odd I for the description of the internal space of fermions makes even II to be the corresponding boson field. The remaining group of the 2 d 2 -1 × 2 d 2 -1 Clifford even ”basis vectors”, presented inTable 16.1 as even I are not the boson partners to the chosen Cliffordodd odd I ”basic vectors”,but ratherto their Hermitian conjugatedpartners b^m , presented as f odd II in the sameTable 16.1. The creation operators, either for creating fermions or for creating bosons, must have besides the ”basis vectors” defining the internal space of fermions and bosons also the basis in ordinary space in momentum or coordinate representation.Ifollow here shortly Ref. [1]. Letus brieflypresenttherelations concerningthe momentumor coordinatepartof the single particle states. The longer version is presented in Ref.([1] in Subsect. 3.3 and in App. J) † b^ ^ |.p> = |0p >, < .p |=<0p | † .p b.p , .p ' ' )=<0p |b^.b^ p < .p |.p> = d(.p -.p |0p >, ' leading to † d(p . ' -.p), (16.25) ^^ b .b ' p .p = where the normalization <0p |0p >= 1 to identity is assumed. While the quan- ^ik tized operators .p and .x ^commute {p^,p^j}- = 0 and {x^,x^l}- = 0, this is not the i case for {p^,x^j}- = i.ij. It therefore follows † >)† † ^^. b x ^^ < .p |.x> = |0.x >= (<0.x | <0. p ||0. p b.x b. p b. p † † † {b^ b^ p, ' p, }- = 0, {b^.b^.p }- = 0, {b^.b^ }- = 0, , ' ' . p .p . p † † † {b^ b^ ' }- = 0, {b^.x,b^.x }- = 0, {b^.x,b^ }- = 0, , ' ' .x .x .x while † .x 1 1 † i.p·.x -i.p·.x {^p,b^ b., , {b^.x,b^ , (16.26) }- = }- = e e . p (2p)d-1 Statement 8. While the internal spaceof either fermionsor bosons has the finite degrees (2p)d-1 of freedom — 2 d 2 -1 × 2 d 2 -1 — the momentum basis has obviously continuously infinite degrees of freedom. m bm† A^m† Let us use the common symbol a^for both ”basis vectors” ^and . And f ff let be taken into account that either fermion or boson second quantized states are 0 solving equationsof motion, whichrelate p0 and .p:p= |.p|. Then the solution of the equations of motion can be written as the superposition of the tensor products, * T , of a finite number of ”basis vectors” describing the internal space of a second m quantized single particle state, a^, and the continuously infinite momentum basis f X {aˆ s† f sm (.p)= c f (.p) b^ † . p * T a ^ m† f }|vacc > * T |0. p >, (16.27) m where .p determines the momentum in ordinary space and s determines all the rest of quantum numbers. The state written here as |vaco > * T |0.> is considered p as the vacuum for a starting single particle state from which one obtains the other single particle states by the operators, like b^ † . p , which pushes the momentum by an amount .p and the vacuum for either fermions |.oc >, Eq. (17.10), or bosons >, Eq. (16.20). |.oceven The creation operators for fermions can be therefore written as X {bˆs† f † * T b^m† f }|.oc > * T |0. p >, (16.28) b^ sm (.p)= c f (.p) .p m X { A^s† f Csm (.p)= f (.p) b^ † * T A^m† f }|.oceven > * T |0. p > . (16.29) p m b^m† Since the ”basis vectors” , describing the internal space of fermion, and f their Hermitian conjugated partners do fulfil the anticommuting properties of bs† bs† Eq. (17.11), then also ˆ(.p)and (ˆ(.p))†, Eq. (17.12), fulfil the anticommutation ff . † . p and anticommutativity of ”basis vectors”. The ”basis vectors” for fermions bring to the second quantized fermions, that is to the creation and correspondingly to the annihilation operators operating on the † b^ (b^)† = b^-. relations of Eq. (17.11) due the commutativity of operators = -. pp vacuum state, the anticomutativity and 2 d 2 -1 × 2 d 2 -1 quantum numbers of family members and of families for each of continuously 8 many .p. The fermion single particle states therefore already anticommute. The 2 d 2 -1 × 2 d 2 -1 Clifford even ”basis vectors” A^m† f , appearing in pairs which are Hermitian conjugatedto each other, fulfilthe commutingpropertiesofEq. (16.21), transfering these commuting properties also to 2 d 2 -1 × 2 d 2 -1 members of A^s† f (.p), As† Eq. (17.12), for any of continuously 8 .p, so that ^(.p)fulfil the commutation f . Am† relations of Eq. (16.21) according to commutativity properties of operators ^ p Statement 9. The odd products of the Clifford objects .a’s offer the ”basis vectors” to describe the internal space of the second quantized fermion fields. The even products of the Clifford objects .a’s offer the ”basis vectors” to describe the internal space of the second quantized boson fields. They are the gauge fieldsof the fermion fields describedby the odd Clifford objects. Statement 9.a The description of the internal space of fermions with the odd Clifford algebra explains the second quantization postulates of Dirac. The quantized single fermion states anticommute. A^s† The (.p)”basis vectors” bring to the second quantized bosons, that is to the f creation operators and annihilation operators, appearing in pairs or as self ad­joint operators, operating on the vacuum state, the commutativity properties and . d -1 × 2 d 2 -1 quantum numbers, explaining properties of boson particles. The or­ 2 2 † dinary basis, b^, brings to the creation operators the continuously infinite degrees . p of freedom. Statement 9.b The description of the internal space of bosons with the even Clifford algebra explains the second quantization postulates for gauge fields. The quantized single boson states commute. Let us represent here the anticommutation relations for the creation and annihila­ bs† bs tion operators of the second quantized fermion fields ^(.p)and ^(.p)by taking ff into account Eq. (17.11) {^' ^' ^^^^ bb b bb b s† dss (.p)}+ |.oc > |0.p > = dff ' d(.p ' -.p)|.oc (p . ' ), s > |0.p >, f‘ f ' {^{^' † bb (p . ' ), s s f(.p)}+ |.oc > |0.p > = 0 |.oc > |0p .>, f‘ s† f (p . s ' ), (.> |0.> = > |0.>, p)}+ |.oc p 0 |.oc p ' f s† f (.p)|.oc > |0.p > = |.fs (.p)> s f(.p)|.oc > |0.p > = 0 |.oc > |0p .> 0 |p |= |.p|. (16.30) s† f (.p, p0)) and their Hermitian conjugated partners anni- The creation operators ^^ hilation operators bb 0)), creating and annihilating the single fermion state, s (.p, p f respectively, fulfil when applying on the vacuum state, |.oc > |0.>, the anti- p commutation relations for the second quantized fermions, postulated by Dirac (Ref. [1], Subsect. 3.3.1, Sect. 5). The anticommutation relations of Eq. (17.14) are valid also if we replace the vac­uum state, |.oc > |0.p >, by the Hilbert space of Cliffordfermions generated by the tensor product multiplication, * TH , of any number of the Cliffordodd fermion statesof all possible internal quantum numbers and all possible momenta(thatis s † of any number of ^ b A^s† The commutation relations among boson creation operators (.p)can be written f as ' '' '' '' † A^s† A^s ' † fsss ff‘f A^s ' ) { (.p), (.p ' )}- = d(.p -.p. (16.31) '' ff f Let us present an example with .p =(0, 0, p3, 0, 0)and the choice A^3† (.p) and 1 A^2† (.p ' ), takenfromTable 16.3, one finds 2 { A^3† (.p), A^1† (.p ' )}- =-d(.p -.p ' ) A^2† (.p). (16.32) 12 1 , S12 , S56 One can notice that the sums over eachof the quantum numbers(S03 , N L3 , N 3 ,t3,t8 , R of the left hand side are equal tothe corresponding quantum numbers on the right hand side. The study of properties of the second quantized bosons with the internal space of which is described by the Clifford even algebra has just started and needs further consideration. Let us point out that when breaking symmetries, like in the case of d =(5 + 1) into SU(2)× SU(2)× U(1), one easily sees that the same, either the right or the leftrepresentationsappear withinthesame,onlytheright,Table16.2,oronlythe left,Table 16.4,representation, manifesting the right (left)hand fermions and the left (right) handed antifermions [24]. The same observation demonstrates also Table16.5, in whichin each octet ofu-quarks and d-quarks of any colour and in the octet of colourless leptons the left and the right members of fermions and antifermions appear. (.p)of any (s, f, .p)), Ref.([1], Sect. 5.). f 16.3.1 Simple action for fermion and boson fields Let the space be d = 2(2n +1)-dimensional. The spin-charge-family theory proposes d =(13+1)-dimensional space, or larger,so that the ”basis vectors”. describing the internal space of fermions and bosons, offers the properties of the observed quarks and leptons and their antiquarks and antileptons, as well as the corresponding boson fields, as we learn in thic contribution. The action for the second quantized massless fermion and antifermion fields, and the corresponding massless boson fields in d = 2(2n + 1)-dimensional space is therefore Z dd1 ¯ A = xE (..a p0a.)+h.c. + 2 Z ddxE (aR + a ˜R˜), p0a = faap0a + 1 {pa, Efaa}- , 2E 1Sab1S˜ab ˜ p0a = pa - .aba - .aba , 22 R = 1 {fa[afßb] (.aba,ß -.caa .cbß)}+ h.c. , 2 1fßb] (˜ R ˜= {fa[a.aba,ß - .˜caa .˜cbß)}+ h.c. . (16.33) 2 faafßb -fabfßa Here 3 fa[afßb] = . It is proven in Refs. [26, 38] that the spin connection gauge fields manifest in d =(3 + 1)as the ordinary gravity, the known vector gauge fields and the scalar gauge fields, offering the (simple) explanation for the origin of higgs assumed by the standard model, explaining as well theYukawa couplings. 16.4 Conclusions In the spin-charge-family theory the Cliffordalgebra is used to describe the internal space of fermion fields, what brings new insights, new recognitions about proper-tiesof fermion and boson fields([1] andreferences therein): The use of the odd Cliffordalgebra elements .a’s to describe the internal space of fermionsoffers not only the explanationfor all the assumptionsof the standard model, with the appearance of the families of quarks and leptons and antiquarks and antileptons included, but also for the appearance of the dark matter in the universe, for the explanation of the second quantized postulates for fermions of 3 a aaß faa are inverted vielbeins to e a with the properties e afab = dab,e afßa = da, E = det(e aa). Latin indices a, b, .., m, n, .., s, t, .. denotea tangent space(a flat index), while Greek indices a, ß, .., µ, ., ..s, t, .. denoteanEinstein index(a curved index). Lettersfrom the beginningof boththe alphabets indicatea general index(a, b, c, .. and a, ß, ., .. ), from the middle of both the alphabets the observed dimensions 0, 1, 2, 3 (m, n, .. and µ, ., ..),indexesfrom the bottomof the alphabets indicate the compactified dimensions (s, t, .. and s, t, ..).We assume the signature .ab = diag{1, -1, -1, ··· , -1}. Dirac, for the matter/antimatter asymmetry in the universe, and for several other observed phenomena, making several predictions. This article is the first trial to describe the internal space of bosons while using the even products of Cliffordalgebra objects .a’s. Although this study of the internal space of boson fields with the even Clifford algebra objects needs further considerations, yet the properties demonstrated in this paper are at least very promising. Let merepeat briefly whatIhope that we have learned. i. There are two kinds of the anticommuting algebras, the Grassmann algebra, offering in d-dimensional space 2 · 2d operators, and the two Cliffordalgebras, each with 2d operators. The Grassmann algebra operators are expressible with the operators of the two Cliffordalgebras and opposite, Eq. (16.4), and opposite. ThetwoCliffordalgebrasare independentofeach other,Eq. (16.5), formingtwo independent spaces. ii. Either the Grassmann algebra or the two Cliffordalgebras can be used to de­scribe the internal space of anticommuting objects, if the odd products of operators are used to describe the internal space of these objects, and of commuting objects, iftheevenproductsof operatorsareusedto describethe internalspaceofthese objects. iii. The ”basis vectors” can be found in each of these algebras, which are eigenvec­tors of the Cartan subalgebras, Eq. (16.6), of the corresponding Lorentz algebras Sab , Sab and S˜ab, Eq. (17.7). iv. After the reduction of the two Cliffordalgebras to only one — .ab’s — as­suming how does .˜a apply on .a: {.˜aB = (-)B i B.a}|.oc >, with (-)B =-1, if B is (a function of) an odd products of .a’s, otherwise (-)B = 1, there remain twice 2 d 2 -1 iredduciblerepresentations of Sab, each with the 2 d 2 -1 members. .˜a’s operate on superposition of products of .a’s. v. The ”basis vectors”, which are superposition of odd products of .a’s, can be arranged to fulfil the anticommutationrelations, postulatedby Dirac, explaining correspondingly the anticommutation postulates of Dirac, Eqs. (16.9, 16.11). v.a. The Cliffordodd 2 d 2 -1 members of each of the 2 d 2 -1 irreducible representa­ tions of ”basis vectors” have their Hermitian conjugated partners in another set d d -1 -1 ”basis vectors”,Tables (16.1, 16.2). The two setsof ”basis vectors” of 2 ·2 22 differin handedness,Tables (16.2, 16.4). v.b. It is our choice which set we use to describe the creation operators and which one to describe the annihilation operators. Correspondingly we have either left or right handed creation operators. v.c. The family members of ”basis vectors” have the same properties in all the families. The sum of all the eigenvalues of allthe commuting operators over the d -1 family members is equal to zero for each of 2 d 2 -1 families, separately for left 2 and separately for right handedrepresentations. The sumof the family quantum numbers over the four families is zero. vi. The Cliffordeven ”basis vectors”, which are superposition of even products of .a’s, commute. vi.a. The Clifford even ”basis vectors” have their Hermitian conjugated partners d d -1 -1 members,Table 16.3, or are self adjoint. within the same group of 2 ×2 22d d -1 -1 ”basis vectors” vi.b. Each of the two groups of the Clifford even 2 × 2 22 applies algebraicallyononlyoneofthetwoCliffordodd ”basis vectors”,(inTa­ble 16.1 Clifford even II ”basis vectors” apply on Clifford odd I ”basis vectors”), conserving the quantum numbers of the internal space. vi.c. The Cliffordeven ”basis vectors”, applying algebraically on the Cliffordodd ”basis vectors”, transform the Cliffordodd ”basisvector” into another memberof the same family, Eqs. (16.17, 16.18, 16.19). vi.d. The Cliffordeven ”basis vectors” have obviously the quantum numbers of theadjointrepresentationswithrespecttothe fundamentalrepresentationofthe Cliffordodd partnersof the Clifford even ”basis vectors”,Table 16.3. vi.e. The sum of all the eigenvalues of all the Cartan subalgebra members over the members of Clifford even ”basis vectors” is equal to zero, independent of the choice of the subgroups (with the same number of the Cartan subalgeba), Table 16.3. Am† Am ' † ^^ vi.f. Two Clifford even ”basis vectors”( and )of the same f and of ff (m, m ' ).= m0 are orthogonal. The two ”basis vectors” with non zero algebraic product, * A, ”scatter” into the thirdone, or annihilate into the vacuum,. vi.g. The superposition of products of even number of .˜a’s transform the member oftheCliffordodd ”basis vector”of particularfamilyintothe samefamily member of another family. vii. The creation and annihilation operators for either the Clifford odd or the Clifford even fields, contain besides the corresponding ”basis vectors” also the basis in ordinary, coordinate or momentum, space, Eqs. (17.12, 16.28, 16.29). vii.a. The tensor products, * T , of the ”basis vectors” describing the internal space of fermions or bosons and the basis in ordinary space have the properties of cre­ation and annihilation operators for either fermion or boson fields, defining the states when applying on the corresponding vacuum states, Eqs. (17.10, 16.20). vii.b. While the internal space of either fermions or bosons has the finite degrees of freedom — 2 d 2 -1 × 2 d 2 -1 — the momentum basis has obviously continuously infinite degrees of freedom. Correspondingly the single particle states have contin­uously infinite degrees of freedom. vii.c. There are the ”basis vectors” describing the internal spaces of either fermions or bosons, which bring commutativity or anticommutativity to creation and anni­hilation operators. vii.d. The single particle states described by applying the Cliffordodd creation operators on the vacuum state, anticommute, while the single particle states de­scribed by applying the Clifford even creation operators on the vacuum state commute. The same rules are valid also when applying creation operators on the corresponding Hilbert spaces, Ref. (nh2021RPPNP), Sect. 5. vii.e. Fermion fields describedbyusing the Cliffordodd creation operators interact with exchange of the corresponding boson fields described by the Clifford even creation operators, Eq. (16.19). Bosons fields interacts on both ways, with boson fields (if the corresponding two ”basis vectors” have non zero algebraic product, * A), as well as with fermions. vii.f. The application of the creation operators with the Clifford even ”basis vec­tors”, in which all the .a’s arereplacedby .˜a’s, on the fermion creation operators, transform the fermion creation operator to another one, belonging to different family with the unchanged family members of the ”basis vectors”, Subsect. (16.2.4, part b.). Let me conclude this contribution by saying that so far the description of the internal space of the second quantized fermions with the Cliffordodd ”basis vec­tors” offers a new insight into the Hilbert space of the second quantized fermions (although there are still open questions waiting to be discussed, like it is the ap­pearance of the Dirac sea in the usual approaches), the equivalent description of the internal space of the second quantized boson fields with the Clifford even ”basis vectors” needs, although to my opinion very promising, a lot of further study. 16.5 Eigenstates of Cartan subalgebra of Lorentz algebra The eigenvectors of Sab and S˜ab in the space determinedby .a’s is as follows .aa .aa 1 k1 Sab (.a + .b)=(.a + .b), 2 ik 22 ik 1i k1i Sab (1 + .a.b)=(1 + .a.b), 2k 22k .aa .aa 1 k1 Sab ˜(.˜a + .˜b)=(.˜a + .˜b), 2 ik 22 ik 1i k1i S˜ab (1 + .˜a .˜b)=-(1 + .˜a .˜b). (16.34) 2k 22k .aa.bb with k2 = . The proof of the first two equations of Eq.(16.34) goes as follows, a .b is assumed: = .aa .aa.bb i .a.b1 i1 (-.aa.b k1 (.a -.aa i (.a + .b)= + .a)= .b). 22ik22 ik 22 k i ii1 .aa.bb)k1 i .a.b1 (1 + .a.b)=(.a.b - i =(1 + .a.b). 22k 22k 22k Forprovingthesecondtwo equationsitmustberecognizedthatafterthereduction of the Cliffordspace to only the part spent by .a’s, that is after requiring {.˜aB = (-)B i B.a}|.oc >, with (-)B =-1, if B is (a function of) an odd product of .a’s, otherwise (-)B = 1 [35], the relations of Eq. (16.5) remain unchanged. One can see this as follows (I follow here Ref. [1], Statement 3a. of App.I) {.˜a ,.˜b}+ = 2.ab = .˜a .˜b + .˜b .˜a = .˜ai.b + .˜bi.a = i.b(-i).a + i.a(-i).b = 2.ab . {.˜a,.b}+ = 0 = .˜a.b + .b .˜a = .b(-i).a + .bi.a = 0. Taking this into account it follows .aa .aa .aa Sab1i i1 i1 ˜(.a+.b)= .˜a .˜b1 (.a+.b)=(.a+.b).b.a = (-.aa.b+ 2ik 22ik 22ik 22 .aabb .aa .k1 .a)=(.a + .b), ik 22ik Sab1i i1i i1 i .aa.bb)1 ˜(1 + .a.b)=(1 + .a.b).b.a = (-.a.b + =-k (1 + 2k22k 22 k 22 i .a.b), k .aa.bb where it is taken into account that k2 = . 218 N.S. Manko.c Bor.stnik 16.6 Clifford odd and even ”basis vectors” continue InTable 16.2 the Cliffordodd ”basis vectors”of the right handedness were chosen for the description of the internal space of fermions in d =(5 + 1)-dimensional space, notedinTable 16.1 as odd I. If we make a choice of odd II for the Cliffordodd ”basis vectors” inTable 16.1, and take the odd I as their Hermitian conjugated partners, then these ”basis vectors” are left (not right) handed and havepropertiespresentedinTable 16.4. We can compare their properties by the properties of the right handed ”basis vectors” appearinginTable 16.2. The twogroups odd I and odd II are Hermitian conjugatedtoeachother.Weclearlyseeif comparingbothtables,Table16.2and m=(ch,s)† ^ Table 16.4: The ”basis vectors”, this time left handed — b(each is a f product of projectors and an odd number of nilpotents, and is the ”eigenstate” of , S12 S03 S12 all the Cartan subalgebra members, S03 , S56 and˜,˜, S˜56, Eq. (16.6)(ch (charge), the eigenvalue of S56,ands (spin), the eigenvalues of S03 and S12,explain S03 S12 S56) index m, f determines family quantum numbers, the eigenvaluesof( ˜,˜,˜ — are presented for d =(5 + 1)-dimensional case. Their Hermitian conjugated m=(ch,s) ^ partners — bf — canbe foundinTable 16.2 as ”basis vectors”. This table represents also the eigenvalues of the three commuting operators N3 and S56 of L,R the subgroups SU(2)× SU(2)× U(1)and the eigenvalues of the three commuting operators t3,t8 and t4 of the subgroups SU(3)× U(1), in these two last cases index m represents the eigenvalues of the corresponding commuting generators. G(5+1) -1, G(3+1) m=(ch,s)† =-.0.1.2.3.5.6 == i.0.1.2.3. Operators b^and f m=(ch,s) b^f fulfil the anticommutationrelationsof Eqs. (16.9, 16.11). f = (ch, s) m f b ^m=(ch,s)† S03 S12 S56 G3+1 3 LN 3 RN 3t 8t 4t S˜03 S˜12 S˜56 I 1 2, 1 21() 03 12 56 (-i) (+) | (+) i 2 - 1 2 1 2 -1 1 2 0 1 2- 3 1 2 -v 1 6- i 2 - 1 2 1 2 I 1 21 22(, -) 03 12 56 [+i] [-] | (+) i 2 1 2 - 1 2 -1 1 2- 0 1 2 3 1 2 -v 1 6- i 2 - 1 2 1 2 I 1 2, 1 23(-) 03 12 56 [+i] (+) | [-] i 2 1 2 1 2- 1 0 1 2 0 3 1v 1 6- i 2 - 1 2 1 2 03 12 56 I 1 21 21(-, -) (-i) [-] | [-] i 2- 1 2- 1 2- 1 0 1 2- 0 0 1 2 i 2 - 1 2 1 2 03 12 56 II 1 2, 1 21() [-i] [+] | (+) i 2 - 1 2 1 2 -1 1 2 0 1 2- 3 1 2 -v 1 6 - i 2 1 2 - 1 2 03 12 56 II 1 21 22(, -) (+i) (-) | (+) i 2 1 2 - 1 2 -1 1 2- 0 1 2 3 1 2 -v 1 6 - i 2 1 2 - 1 2 II 1 2, 1 23(-) 03 12 56 (+i) [+] | [-] i 2 1 2 1 2- 1 0 1 2 0 3 1v 1 6 - i 2 1 2 - 1 2 II 1 21 24(-, -) 03 12 56 [-i] (-) | [-] i 2- 1 2- 1 2- 1 0 1 2- 0 0 1 2 i 2 1 2 - 1 2 III 1 2, 1 21() 03 12 56 [-i] (+) | [+] i 2 - 1 2 1 2 -1 1 2 0 1 2- 3 1 2 -v 1 6 - i 2 1 2 1 2 - 03 12 56 III 1 21 22(, -) (+i) [-] | [+] i 2 1 2 - 1 2 -1 1 2- 0 1 2 3 1 2 -v 1 6 - i 2 1 2 1 2 - 03 12 56 III 1 2, 1 23(-) (+i) (+) | (-) i 2 - 1 2 1 2- 1 0 1 2 0 3 1v 1 6 - i 2 1 2 1 2 - 03 12 56 III 1 21 24(-, -) [-i] [-] | (-) i 2- 1 2- 1 2- 1 0 1 2- 0 0 1 2 i 2 1 2 1 2 - IV 1 2, 1 21() 03 12 56 (-i) [+] | [+] i 2 - 1 2 1 2 -1 1 2 0 1 2- 3 1 2 -v 1 6- i 2- 1 2- 1 2 - IV 1 21 22(, -) 03 12 56 [+i] (-) | [+] i 2 1 2 - 1 2 -1 1 2- 0 1 2 3 1 2 -v 1 6- i 2- 1 2- 1 2 - IV 1 2, 1 23(-) 03 12 56 [+i] [+] | (-) i 2 1 2 1 2- 1 0 1 2 0 3 1v 1 6- i 2- 1 2- 1 2 - IV 1 21 24(-, -) 03 12 56 (-i) (-) | (-) i 2- 1 2- 1 2- 1 0 1 2- 0 0 1 2 i 2- 1 2- 1 2 - Table 16.4, that they do differ in properties. In particular the difference among these two kinds of ”basis vectors” is easily seen in the SU(3)× U(1)subgroup, that is in (t3,t8,t4)values. InTable 16.5 one finds the left and the right handed contentof oneof the families, the fourth ones,presentedinRef.[1],Table5,if d =(5 +1)is taken as the subspace of the space d =(13 + 1). 16.7 Some useful relations in Grassmann and Clifford space, needed also in App. 16.8 The generator of the Lorentz transformation in Grassmann space is defined as follows [20] Sab .b -.b .a) =(.a pp {Sab Scd}- = = Sab + S˜ab ,, ˜0, (16.35) where Sab and S˜ab are the corresponding two generators of the Lorentz transfor­mations in the Cliffordspace, forming orthogonal representations with respect to each other. We makea choiceof the Cartan subalgebraof the Lorentz algebra as follows S03 , S12 , S56 · , Sd-1d , ·· , S03· ,Sd-1d ,S12,S56 , ·· , S12 S˜03 , ˜,S˜56 , ··· ,S˜d-1d , if d = 2n . (16.36) We find the infinitesimal generators of the Lorentz transformations in Clifford space Sab iSab† .aa.bbSab =(.a.b -.b.a), = , 4 S˜ab iS˜ab† .aa.bb S˜ab =(.˜a .˜b - .˜b .˜a), = , (16.37) 4 where .a and .˜a are defined in Eqs. (16.4, 16.5). The commutation relations for Sab , Sab Sab Sab either Sab or Sab or ˜=+ ˜, are {Sab S˜cd}- = , 0, {Sabi(.adSbc + .bcSad -.acSbd -.bdSac), ,Scd}- = Sab S˜cd}- = i(.ad S˜bc + .bc S˜ad -.ac S˜bd -.bd S˜ac) {˜,. (16.38) The infinitesimal generators of the two invariant subgroups of the group SO(3, 1) can be expressed as follows .1 (S23 ± iS01 N±(= .:= ,S31 ± iS02,S12 ± iS03). (16.39) N(L,R)) 2 The infinitesimal generators of the two invariant subgroups of the group SO(4) are expressible with Sab , (a, b)=(5, 6, 7, 8)as follows t1 1 (S58 -S67+ S68 .:= ,S57 ,S56 -S78), 21 (S58 + S67+ S78), .t2 := ,S57 -S68,S56 (16.40) 2 while the generators of the SU(3)and U(1)subgroups of the group SO(6)can be expressedby Sab , (a, b)=(9, 10, 11, 12, 13, 14) t3 1 {S9 12 -S10 11 ,S9 11 + S10 12,S9 10 -S11 12 .:= , 2 S9 14 -S10 13,S9 13 + S10 14 ,S11 14 -S12 13 , S11 13 + S12 14 1 (S9 10 + S11 12 -2S13 14)}, , v 3 (S9 10 + S11 12 + S13 14) t4 := - 1. (16.41) 3 d The group SO(6)has d(d-1) = 15 generators and = 3 commuting operators. 22 The subgroups SU(3)×U(1)have the same number of commuting operators, ex­ pressed with t33 , t38 and t4, and 9 generators, 8 of SU(3)and one of U(1). The 1 {S9 12 rest of 6 generators, not included in SU(3)×U(1), can be expressed as + 2 S10 11,S9 11 -S10 12 , S9 14 + S10 13,S9 13 -S10 14,S11 14 +S12 13,S11 13 -S12 14 . t23 The hyper charge Y can be defined as Y =+ t4 . Sab The equivalent expressions for the ”family” charges, expressedby ˜, follow if Sab in Eqs. (17.26 -17.28) Sab arereplacedby ˜. Let us present some useful relations from Ref. [23]. abab abab ab abab ab ab ab (k)(k)= 0, (k)(-k)= .aa [k], (-k)(k)= .aa [-k], (-k)(-k)= 0, abab ab abab abab ab ab ab [k][k]=[k], [k][-k]= 0, [-k][k]= 0, [-k][-k]=[-k], abab abab ab ab ab ab ab ab (k)[k]= 0, [k](k)=(k), (-k)[k]=(-k), (-k)[-k]= 0, abab ab abab abab ab ab ab (k)[-k]=(k), [k](-k)= 0, [-k](k)= 0, [-k](-k)=(-k) . (16.42) 16.8 One family representation in d =(13 + 1)-dimensional space with 2 d -1 members representing quarks and leptons 2 and antiquarks and antileptons in the spin-charge-family theory InTableTable so13+1. the ”basis vectors”of one irreduciblerepresentation, one family, of the Cliffordodd basis vectors of left handedness, G(13+1), is presented, including all the quarks and the leptons and the antiquarks and the antileptons of 16 New way of second quantization of fermions and bosons 221 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 c1 R u c1 R u c1 R dc1 R d c1 d dc1 c1 L u c1 L u c2 R u c2 R u dc2 R c2 R dc2 L dc2 L dc2 L u c2 L u c3 R u c3 R u c3 R dc3 R dc3 L d dc3 L c3 L u c3 L u 03 12 56 78 910 1112 1314 (+i) [+] | [+] (+) || [-] (+) [-] 03 12 56 78 910 1112 1314 [-i] (-) | [+] (+) || [-] (+) [-] 03 12 56 78 910 1112 1314 (+i) [+] | (-) [-] || [-] (+) [-] 03 12 56 78 910 1112 1314 [-i] (-) | (-) [-] || [-] (+) [-] 03 12 56 78 910 1112 1314 [-i] [+] | (-) (+) || [-] (+) [-] 03 12 56 78 910 1112 1314 - (+i) (-) | (-) (+) || [-] (+) [-] 03 12 56 78 910 1112 1314 - [-i] [+] | [+] [-] || [-] (+) [-] 03 12 56 78 910 1112 1314 (+i) (-) | [+] [-] || [-] (+) [-] 03 12 56 78 910 1112 1314 (+i) [+] | [+] (+) || [-] [-] (+) 03 12 56 78 910 1112 1314 [-i] (-) | [+] (+) || [-] [-] (+) 03 12 56 78 910 1112 1314 (+i) [+] | (-) [-] || [-] [-] (+) 03 12 56 78 910 1112 1314 [-i] (-) | (-) [-] || [-] [-] (+) 03 12 56 78 910 1112 1314 [-i] [+] | (-) (+) || [-] [-] (+) 03 12 56 78 910 1112 1314 - (+i) (-) | (-) (+) || [-] [-] (+) 03 12 56 78 910 1112 1314 - [-i] [+] | [+] [-] || [-] [-] (+) 03 12 56 78 910 1112 1314 (+i) (-) | [+] [-] || [-] [-] (+) 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 2 - 1 2 1 2 - 1 2 1 2 - 1 2 1 2 - 1 2 1 2 - 1 2 1 2 - 1 2 1 2 - 1 2 1 2 - 1 2 0 0 0 0 - 1 2 - 1 2 1 2 1 2 0 0 0 0 - 1 2 - 1 2 1 2 1 2 1 2 1 2 - 1 2 - 1 2 0 0 0 0 1 2 1 2 - 1 2 - 1 2 0 0 0 0 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 0 0 0 0 0 0 0 0 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 2 3 2 3 - 1 3 - 1 3 1 6 1 6 1 6 1 6 2 3 2 3 - 1 3 - 1 3 1 6 1 6 1 6 1 6 2 3 2 3 - 1 3 - 1 3 - 1 3 - 1 3 2 3 2 3 2 3 2 3 - 1 3 - 1 3 - 1 3 - 1 3 2 3 2 3 .R .R eR eR eL eL .L .L 03 12 56 78 910 1112 1314 (+i) [+] | [+] (+) || (+) (+) (+) 03 12 56 78 910 1112 1314 [-i] (-) | [+] (+) || (+) (+) (+) 03 12 56 78 910 1112 1314 (+i) [+] | (-) [-] || (+) (+) (+) 03 12 56 78 910 1112 1314 [-i] (-) | (-) [-] || (+) (+) (+) 03 12 56 78 910 1112 1314 [-i] [+] | (-) (+) || (+) (+) (+) 03 12 56 78 9 10 11 12 13 14 - (+i) (-) | (-) (+) || (+) (+) (+) -1 - 1 2 - 1 2 0 0 0 - 1 2 - 1 2 -1 03 12 56 78 9 10 11 12 13 14 - [-i] [+] | [+] [-] || (+) (+) (+) -1 1 2 1 2 0 0 0 - 1 2 - 1 2 0 03 12 56 78 910 1112 1314 (+i) (-) | [+] [-] || (+) (+) (+) 1 1 1 1 -1 -1 1 2 - 1 2 1 2 - 1 2 1 2 - 1 2 0 0 0 0 - 1 2 1 2 1 2 1 2 - 1 2 - 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 0 0 -1 -1 - 1 2 - 1 2 -1 -1 -1 ¯ c1 ¯ d ¯ c1 ¯ d ¯ c1 u ¯ ¯ c1 u ¯ ¯ c1 R ¯ d ¯ c1 R ¯ d ¯ c1 R u ¯ ¯ c1 R u ¯ |a .i > (7,1)(6) (Anti)octet,G= (-1) 1, G=(1)- 1 of (anti)quarks and (anti)leptons 03 12 56 78 910 1112 1314 (+i) [+] | [+] (+) || (+) [-] [-] 03 12 56 78 910 1112 1314 [-i] (-) | [+] (+) || (+) [-] [-] 03 12 56 78 910 1112 1314 (+i) [+] | (-) [-] || (+) [-] [-] 03 12 56 78 910 1112 1314 [-i] (-) | (-) [-] || (+) [-] [-] 03 12 56 78 910 1112 1314 [-i] [+] | (-) (+) || (+) [-] [-] 03 12 56 78 910 1112 1314 - (+i) (-) | (-) (+) || (+) [-] [-] 03 12 56 78 910 1112 1314 - [-i] [+] | [+] [-] || (+) [-] [-] 03 12 56 78 910 1112 1314 (+i) (-) | [+] [-] || (+) [-] [-] 03 12 56 78 910 1112 1314 [-i] [+] | [+] (+) || [-] (+) (+) 03 12 56 78 910 1112 1314 (+i) (-) | [+] (+) || [-] (+) (+) 03 12 56 78 910 1112 1314 - [-i] [+] | (-) [-] || [-] (+) (+) 03 12 56 78 910 1112 1314 - (+i) (-) | (-) [-] || [-] (+) (+) 03 12 56 78 910 1112 1314 (+i) [+] | [+] [-] || [-] (+) (+) 03 12 56 78 910 1112 1314 - [-i] (-) | [+] [-] || [-] (+) (+) 03 12 56 78 910 1112 1314 (+i) [+] | (-) (+) || [-] (+) (+) 03 12 56 78 910 1112 1314 [-i] (-) | (-) (+) || [-] (+) (+) (3,1) G 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 12 S 1 2 - 1 2 1 2 - 1 2 1 2 - 1 2 1 2 - 1 2 1 2 - 1 2 1 2 - 1 2 1 2 - 1 2 1 2 - 1 2 13 t 0 0 0 0 - 1 2 - 1 2 1 2 1 2 0 0 0 0 1 2 1 2 - 1 2 - 1 2 23 t 1 2 1 2 - 1 2 - 1 2 0 0 0 0 1 2 1 2 - 1 2 - 1 2 0 0 0 0 33 t 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 - 1 2 38 t 1 v 23 1 v 23 1 v 23 1 v 23 1 v 23 1 v 23 1 v 23 1 v 23 4 t 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 Y 2 3 2 3 - 1 3 - 1 3 1 6 1 6 1 6 1 6 Q 2 3 2 3 - 1 - 1 - 1 - 1 2 3 2 3 2 2 2 2 2 2 2 2 - - - - - - - - 1 v 3 1 v 3 1 v 3 1 v 3 1 v 3 1 v 3 1 v 3 1 v 3 1 v 3 1 v 3 1 v 3 1 v 3 1 v 3 1 v 3 1 v 3 1 v 3 1 23 - v 1 23 - v 1 23 - v 1 23 - v 1 23 - v 1 23 - v 1 23 - v 1 23 - v - 1 6 - 1 6 - 1 6 - 1 6 - 1 6 - 1 6 - 1 6 - 1 6 1 3 1 3 - 2 3 - 2 3 - 1 6 - 1 6 - 1 6 - 1 6 1 3 1 3 - 2 - 2 1 3 1 3 - 2 - 2 222 N.S. Manko.c Bor.stnik i |a.i > G(3,1) S12 t13 t23 t33 t38 t4 Y Q (Anti)octet, G(7,1) = (-1) 1 , G(6) = (1) - 1 of (anti)quarks and (anti)leptons 42 ¯ ¯ d c2 L 03 12 56 78 9 10 11 12 13 14 (+i) (-) | [+] (+) || (+) [-] (+) -1 - 1 2 0 1 2 1 2 - 1 v 2 3 - 1 6 1 3 1 3 43 ¯ ¯u c2 L 03 12 56 78 9 10 11 12 13 14 - [-i] [+] | (-) [-] || (+) [-] (+) -1 1 2 0 - 1 2 1 2 - 1 v 2 3 - 1 6 - 2 3 - 2 3 44 ¯ ¯u c2 L 03 12 56 78 9 10 11 12 13 14 - (+i) (-) | (-) [-] || (+) [-] (+) -1 - 1 2 0 - 1 2 1 2 - 1 v 2 3 - 1 6 - 2 3 - 2 3 45 ¯ ¯ d c2 R 03 12 56 78 9 10 11 12 13 14 (+i) [+] | [+] [-] || (+) [-] (+) 1 1 2 1 2 0 1 2 - 1 v 2 3 - 1 6 - 1 6 1 3 46 ¯ ¯ d c2 R 03 12 56 78 9 10 11 12 13 14 - [-i] (-) | [+] [-] || (+) [-] (+) 1 - 1 2 1 2 0 1 2 - 1 v 2 3 - 1 6 - 1 6 1 3 47 ¯ ¯u c2 R 03 12 56 78 9 10 11 12 13 14 (+i) [+] | (-) (+) || (+) [-] (+) 1 1 2 - 1 2 0 1 2 - 1 v 2 3 - 1 6 - 1 6 - 2 3 48 ¯ ¯u c2 R 03 12 56 78 9 10 11 12 13 14 [-i] (-) | (-) (+) || (+) [-] (+) 1 - 1 2 - 1 2 0 1 2 - 1 v 2 3 - 1 6 - 1 6 - 2 3 49 ¯ ¯ d c3 L 03 12 56 78 9 10 11 12 13 14 [-i] [+] | [+] (+) || (+) (+) [-] -1 1 2 0 1 2 0 1v 3 - 1 6 1 3 1 3 50 ¯ ¯ d c3 L 03 12 56 78 9 10 11 12 13 14 (+i) (-) | [+] (+) || (+) (+) [-] -1 - 1 2 0 1 2 0 1v 3 - 1 6 1 3 1 3 51 ¯ ¯u c3 L 03 12 56 78 9 10 11 12 13 14 - [-i] [+] | (-) [-] || (+) (+) [-] -1 1 2 0 - 1 2 0 1v 3 - 1 6 - 2 3 - 2 3 52 ¯ ¯u c3 L 03 12 56 78 9 10 11 12 13 14 - (+i) (-) | (-) [-] || (+) (+) [-] -1 - 1 2 0 - 1 2 0 1v 3 - 1 6 - 2 3 - 2 3 53 ¯ ¯ d c3 R 03 12 56 78 9 10 11 12 13 14 (+i) [+] | [+] [-] || (+) (+) [-] 1 1 2 1 2 0 0 1v 3 - 1 6 - 1 6 1 3 54 ¯ ¯ d c3 R 03 12 56 78 9 10 11 12 13 14 - [-i] (-) | [+] [-] || (+) (+) [-] 1 - 1 2 1 2 0 0 1v 3 - 1 6 - 1 6 1 3 55 ¯ ¯u c3 R 03 12 56 78 9 10 11 12 13 14 (+i) [+] | (-) (+) || (+) (+) [-] 1 1 2 - 1 2 0 0 1v 3 - 1 6 - 1 6 - 2 3 56 ¯ ¯u c3 R 03 12 56 78 9 10 11 12 13 14 [-i] (-) | (-) (+) || (+) (+) [-] 1 - 1 2 - 1 2 0 0 1v 3 - 1 6 - 1 6 - 2 3 57 ¯eL 03 12 56 78 9 10 11 12 13 14 [-i] [+] | [+] (+) || [-] [-] [-] -1 1 2 0 1 2 0 0 1 2 1 1 58 ¯eL 03 12 56 78 9 10 11 12 13 14 (+i) (-) | [+] (+) || [-] [-] [-] -1 - 1 2 0 1 2 0 0 1 2 1 1 59 ¯.L 03 12 56 78 9 10 11 12 13 14 - [-i] [+] | (-) [-] || [-] [-] [-] -1 1 2 0 - 1 2 0 0 1 2 0 0 60 ¯.L 03 12 56 78 9 10 11 12 13 14 - (+i) (-) | (-) [-] || [-] [-] [-] -1 - 1 2 0 - 1 2 0 0 1 2 0 0 61 ¯.R 03 12 56 78 9 10 11 12 13 14 (+i) [+] | (-) (+) || [-] [-] [-] 1 1 2 - 1 2 0 0 0 1 2 1 2 0 62 ¯.R 03 12 56 78 9 10 11 12 13 14 - [-i] (-) | (-) (+) || [-] [-] [-] 1 - 1 2 - 1 2 0 0 0 1 2 1 2 0 63 ¯eR 03 12 56 78 9 10 11 12 13 14 (+i) [+] | [+] [-] || [-] [-] [-] 1 1 2 1 2 0 0 0 1 2 1 2 1 64 ¯eR 03 12 56 78 9 10 11 12 13 14 [-i] (-) | [+] [-] || [-] [-] [-] 1 - 1 2 1 2 0 0 0 1 2 1 2 1 Table 16.5: The left handed(G(13,1)=-1 [23]) multiplet of spinors — the members of the fundamental representation of the SO(13, 1) group, manifesting the subgroup SO(7, 1) of the colour charged quarks and antiquarks and the colourless leptons and antileptons — is presented in the (3,1) 13 ± 1 massless basis using the techniquepresentedin Refs. [23,31,34,35].It contains the left handed(G=-1)weak(SU(2)I)charged(t= , 2 23 (3,1) Eq. (17.27)), and SU(2)II chargeless(t= 0, Eq. (17.27)) quarks and leptons and the right handed(G= 1)weak(SU(2)I)chargeless and SU(2)II charged(t= ± 1 )quarks and leptons, both with the spinS12 up and down(± 1 , respectively). Quarks distinguish from leptons only 23 22 i 3338) 1 1122 in the SU(3) × U(1) part: Quarks are tripletsof three colours(c =(t,t= [( 1, v ), (- 1, v ), (0, - v )], Eq. (17.28)) 2323 3 41 4 carrying the ”fermion charge”(t= , Eq. (17.28)). The colourless leptons carry the ”fermion charge”(t=- 1 ). The same multiplet contains also the 62 left handed weak(SU(2)I)chargeless andSU(2)II charged antiquarks and antileptons and the right handed weak(SU(2)I)charged andSU(2)II chargeless antiquarks and antileptons. Antiquarks distinguish from antileptons again only in the SU(3) × U(1) part: Antiquarks are antitriplets, carrying 4 41 23 62 electromagnetic charge is Q =(t13 + Y). The vacuum state, on which the nilpotents andprojectors operate, is presented in Eq. (17.10). The reader can find thisWeylrepresentation alsoin Refs. [25], [26], [31] and thereferences therein. the ”fermion charge”(t=- 1 ). The anticolourless antileptons carry the ”fermion charge”(t= ). Y =(t+ t4) is the hyper charge, the the standard model. The needed definitions of the quantum numbers are presented in App. 16.7. InTables16.1,16.2,16.3a simpletoy modelfor d =(5 + 1)-dimensional space is discussed, and the properties of fermions (appearing in families) and their gauge boson fields (the vielbeins and the two kinds of the spin connection fields) analysed. The manifold (5 +1)was suggested to break either into SU(2)× SU(2)× U(1)or to SU(3)×U(1)to study properties of the fermion and boson second quantized fields, with second quantization origining in the anticommutativity or commutativity of ”basis vectors”. Here only one family of ”basis vectors” is presented to see that while the starting ”basis vectors” can be either left or right handed, the subgroups,of the starting group contain left and right handed members, as it is SU(2)× SU(2)× U(1)of SO(5 + 1)in the toy model. The breaks of the symmetries, manifesting in Eqs. (17.26, 17.27, 17.28), are in the spin-charge-family theory caused by the condensate and the nonzero vacuum expectation values (constant values) of the scalar fields carrying the space index (7, 8)(Refs. [23, 31] and the references therein), all originating in the vielbeins and the two kinds of the spin connection fields. The space breaks first to SO(7, 1) ×SU(3)× U(1)II and then further to SO(3, 1)× SU(2)I × U(1)I ×SU(3)× U(1)II, what explains the connections between the weak and the hyper charges and the handedness of spinors. 16.9 Handedness in Grassmann and Clifford space The handedness G(d) is one of the invariants of the group SO(d), with the infinites­imal generators of the Lorentz group Sab, defined as G(d) Sa1a2 · Sa3a4 Sad-1ad = aea1a2...ad-1ad ··· , (16.43) with a, which is chosen so that G (d) = ±1. In the Grassmann case Sab is defined in Eq. (16.6), while in the Clifford case i Sab|a Eq. (16.43) simplifies, if we take into account that Sab|a.=b = .a.b and˜.= =b 2 i ˜.a ˜.b , as follows 2 G(d) : = (i)d/2 Y v .aa.a), ( if d = 2n . a (16.44) Acknowledgment The author N.S.M.B. thanks Department of Physics, FMF, University of Ljubljana, Society of Mathematicians, Physicists and Astronomers of Slovenia, for supporting theresearch onthe spin-charge-family theorybyoffering theroom and computer facilities and Matja . zBreskvar of Beyond Semiconductor for donations, in particular for the annual workshops entitled ”What comes beyond the standardmodels”. References 1. N. Manko.stnik, ”Spin connectionasa superpartnerofa vielbein”, Phys. Lett. B292 c Bor.(1992) 25-29. 2. N. Manko.stnik, ”Spinorand vectorrepresentationsinfour dimensional Grassmann c Bor.space”, J. of Math. Phys. 34 (1993) 3731-3745. 3. N. Manko.stnik, ”Unification of spin and charges in Grassmann space?”, hep-th c Bor.9408002, IJS.TP.94/22, Mod. Phys. Lett.A(10)No.7 (1995) 587-595. 4. N. S. Manko.stnik, H. B. Nielsen, ”How does Cliffordalgebra show the way to the c Bor.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 . 5. N.S. Manko.stnik, H.B.F. Nielsen, ”Understanding the second quantization of c Bor.fermions in Cliffordand in Grassmann space”, New way of second quantization of fermions — PartIand PartII, in this proceedings [arXiv:2007.03517, arXiv:2007.03516]. 6. N.S. Manko.stnik, H.B.F. Nielsen, ”Understanding the second quantization of c Bor.fermions in Cliffordand in Grassmann space” New way of second quantization of fermions — PartIand PartII, Proceedings to the 22nd Workshop ”What comes beyond the standard models”,6 -14of July, 2019, Ed. N.S. Manko.stnik, H.B. Nielsen,D. Lukman, DMFA c Bor.Zalo.stvo, Ljubljana, December 2019, [arXiv:1802.05554v4, arXiv:1902.10628]. zni. 7. P.A.M. Dirac Proc. Roy. Soc. (London), A117(1928) 610. 8. H.A. Bethe, R.W. Jackiw,”Intermediate quantum mechanics”, NewYork:W.A. Benjamin, 1968. 9. S.Weinberg, ”The quantum theoryof fields”, Cambridge, Cambridge UniversityPress, 2015. 10. N.S. Manko.stnik, H.B.F. Nielsen, ”New way of second quantized theory of c Bor.fermions with either Clifford or Grassmann coordinates and spin-charge-family theory ” [arXiv:1802.05554v4,arXiv:1902.10628]. 11. D. Lukman, N. S. Manko.stnik, ”Properties of fermions with integer spin de- c Bor.scribed with Grassmann algebra”, Proceedings to the 21st Workshop ”What comes be­yond the standardmodels”,23of June -1of July, 2018, Ed. N.S. Manko.stnik, H.B. c Bor.Nielsen, D. Lukman, DMFAZalo .stvo, Ljubljana, December 2018 [arxiv:1805.06318, zni.arXiv:1902.10628]. 12. N.S. Manko.stnik, H.B.F. Nielsen, J. of Math. Phys. 43, 5782 (2002) [arXiv:hep- c Bor.th/0111257]. 13. N.S. Manko.stnik, H.B.F. Nielsen, “How to generate families of spinors”, J. of Math. c Bor.Phys. 44 4817 (2003) [arXiv:hep-th/0303224]. 14. N.S. Manko.stnik, ”Spin-charge-family theory is offering next step in understand- c Bor.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]. 15. N.S. Manko.stnik, ”The attributes of the Spin-Charge-Family theory giving hope c Bor.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. 16. N.S. Manko.stnik, ”Matter-antimatter asymmetry in the spin-charge-family theory”, c Bor.Phys. Rev. D91(2015) 065004 [arXiv:1409.7791]. 17. N. S. Manko.stnik, ”How far has so far the Spin-Charge-Family theory succeeded c Bor.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 Bor. 12 of July, 2020 Ed. N.S. Manko.stnik, H.B. Nielsen, D. Lukman, DMFA Zalo.stvo, Ljubljana, December 2020, [arXiv:2012.09640] zni. 18. N.S. Manko.stnik, D. Lukman, ”Vector and scalar gauge fields with respect to c Bor.d =(3 +1)in Kaluza-Klein theories and in the spin-charge-family theory”, Eur. Phys.J.C 77 (2017) 231. 19. N.S. Manko.stnik, ”The spin-charge-family theory explains why the scalar Higgs c Bor.carries the weak charge ± 12 and the hyper charge ± 12 ”, Proceedings to the 17th Workshop ”What comes beyond the standardmodels”, Bled, 20-28 of July, 2014, Ed. N.S. Mankoc.Bor.zni. stnik, H.B. Nielsen, D. Lukman, DMFAZalo .stvo, Ljubljana December 2014, p.163­82[arXiv:1502.06786v1] [arXiv:1409.4981]. 20. N.S. Manko.stnikNS, ”The spin-charge-family theoryis explaining the origin c Bor.of families, of the Higgs and theYukawa couplings”, J. of Modern Phys. 4 (2013) 823 [arXiv:1312.1542]. 21. N.S. Manko.stnik, H.B.F. Nielsen, ”The spin-charge-family theory offers under- c Bor.standing of the triangle anomalies cancellation in the standardmodel”, Fortschritte der Physik, Progress of Physics (2017) 1700046. 22. N.S. Manko.stnik, ”The explanation for the origin of the Higgs scalar and for the c Bor.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]. 23. N.S. Manko.stnik and H.B. Nielsen, ”Why nature made a choice of Cliffordand c Bor.not Grassmann coordinates”,Proceedingstothe 20th Workshop ”What comes beyond the standardmodels”, Bled, 9-17 of July, 2017, Ed. N.S. Manko.stnik, H.B. Nielsen, D. c Bor.Lukman, DMFAZalo .stvo, Ljubljana, December 2017, p. 89-120[arXiv:1802.05554v1v2]. zni. 24. N.S. Manko.stnik and H.B.F. Nielsen, ”Discrete symmetries in the Kaluza-Klein c Bor.theories”, JHEP 04:165, 2014 [arXiv:1212.2362]. 25. N. S. Manko.stnik, Second quantized ”anticommuting integer spin fields”, sent c Bor.to Manuscript ID: symmetry-1300624 the thirdtime 30.09. 2021, getting ManuscriptID: symmetry-1423760 as the answer, waiting for theresponse. 26. A. Bor.cBor. stnik, N.S. Manko.stnik, ”Left and right handedness of fermions and bosons”, J. of Phys. G: Nucl. Part. Phys.24(1998)963-977, hep-th/9707218. Proceedings to the 24th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ... (p. 226) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 17 The achievements of the spin-charge-family theory so far N.S. Manko.c Bor.stnik Department of Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia Abstract. Fifty years ago, the standard model offered an elegant new step towards under­standing elementary fermion and boson fields, making several assumptions, suggested by experiments. The assumptions are still waiting for an explanation. There are many proposals in the literature for the next step. The spin-charge-family theory, proposing a simple starting action in d = (13 + 1)-dimensional space with fermions interacting with the gravity only (the vielbeins and the two kinds of the spin connection fields), is offering the explanation for not only all by the standard model assumed properties of quarks and leptons and antiquarks and antileptons, with the families included, of the vector gauge fields, of the Higgs’s scalar andYukawa couplings, of the appearance of the dark matter, of the matter-antimatter asymmetry, making several predictions, but explains as well the second quantization postulates for fermions and bosons by using the oddand the even Clifford algebra ”basis vectors” to describe the internal space of fermions and bosons, respectively. Consequently the single fermion and single boson states alreadyanticommute and commute,respectively.Ipresentin this talka very short overviewof the achievement of the spin-charge-family theory so far, concluding with presenting not yet solved problems, for which the collaborators are very welcome. Povzetek:Pred petdesetimi leti jestandardni model, zgrajen na predpostavkah, porojenih iz rezultatov poskusov,ponudil eleganten nov korakkrazumevanju osnovnih fermionskihin bozonskih polj.Vliteraturije velikopredlogov,ki pojasnjujejopredpostavkein ponujajo nov korak.Teorija spin-charge-family, ki predlaga preprosto za. cetno akcijo v d = (13 + 1)­razse.znje in znemprostoru,v katerisi fermioni izmenjujejo samo gravitone (vektorske sve .dve vrsti spinskih povezav), ponuja razlago ne le za vse predpostavke standardnega modela — za vse lastnosti kvarkov in leptonov ter antikvarkov in antileptonov, ki se pojavljajo v dru. zinah,za umeritvenavektorskapolja,za Higgsove skalarjeinYukawesklopitve—am­pak tudi za pojave v vesolju kot so temna snov, nesimetrija med snovjo in antisnovjo, ponudi vrsto napovedi, ponudi pa tudi pojasnilo za postulate za drugo kvantizacijo za fermione in bozone. Opis notranjega prostora fermionov in bozonov z liho in sodo Cliffordovo alge-bro poskrbi,da fermionska stanja antikomutirajo, bozonskapa komutirajo.Vpredavanju ponudim kratek pregled dosedanjih dose .cku pa zkov spin-charge-family teorije, v zaklju.predstavim odprta vpra.sanja. Pri iskanju odgovorov nanje vabimksodelovanju. 17.1 Introduction Thereview article[1]presentsa short overviewof mostofthe achievementsofthe spin-charge-family theory so far.Ishall make useof this article whenpresentingmy talk. Fifty years ago the standard model offered an elegant new step towards understand­ing elementary fermion and boson fields by postulating: a. The existence of massless fermion family members with the spins and charges in the fundamentalrepresentationof thegroups, a.i. the quarks as colour triplets and colouress leptons, a.ii the left handed members as the weak doublets, the right handed weak chargeless members, a.iii. the left handed quarks differing from the left handed leptons in the hyper charge, a.iv. all the right handed mem­bers differing among themselves in hyper charges, a.v. antifermions carrying the corresponding anticharges of fermions and opposite handedness, a.vi. the fami­lies of massless fermions, suggested by experiments and required by the gauge invariance of the boson fields (there is no right handed neutrino postulated, since it would carry none of the so far observed charges, and correspondingly there is also no left handed antineutrino allowed in the standard model). b. The existence of massless vector gauge fields to the observed charges of quarks and leptons, carrying charges in the adjoint representations of the corresponding charged groups and manifesting the gauge invariance. c. The existence of the massive weak doublet scalar higgs, c.i. carrying the weak charge ± 1 2 and the hyper charge ± 1 2 (as it would be in the fundamental represen­ tation of the two groups), c.ii. gaining at some step of the expanding universe the nonzero vacuum expectation value, c.iii. breaking the weak and the hyper charge and correspondingly breaking the mass protection, c.iv. taking care of the properties of fermions and of the weak bosons masses, c.v. as well as the existence of theYukawa couplings. d. The presentation of fermions and bosons as second quantized fields. e. The gravitational field in d =(3 + 1)as independent gauge field. (The standard model is defined without gravity in order that it be renomalizable, but yet the standard model particles are ”allowed” to couple to gravity in the ”minimal” way.) The standard model assumptions have been experimentally confirmed without raising any severe doubts so far, except for some few and possiblystatistically caused anomalies 1, but also by offering no explanations for the assumptions. The last among the fields postulated by the standard model, the scalar higgs, was detected in June 2012, the gravitational waves were detected in February 2016. 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 [2,4–18, 59]. At least SO(10)-unifying theories offer the explanation for the postulates from a.i. to a.iv, partly to b. by assuming that to all the ”fermion” charges there exist the corresponding vector gauge fields — but does not explain the assumptions a.v. up to a.vi., c. and d., and does not connect gravity with gauge vector and scalar fields. Inalong seriesof works with collaborators([19–23,25,26,28–32,38]andthe references therein), we have found the phenomenological success with the model named the spin-charge-family theory, with fermions, the internal space of which is described with the Cliffordalgebra of all linear superposition of odd products of 1Ithink here on the improvedstandard model, in which neutrinos have non-zero masses, and the model has no ambition to explain severe cosmological problems. .a’s in d =(13 + 1), interacting with only gravity([38] andreferences therein). The spins of fermions from higher dimensions, d> (3 + 1), manifest in d = (3 + 1)as charges of the standard model, gravity in higher dimensions manifest as the standard model gauge vector fields as well as the Higgs’s scalar andYukawa couplings [26,31]. Let be added that one irreducible representation of SO(13, 1)contains, if looked from the point of view of d =(3+1),all the quarks and leptons and antiquarks and antileptons and just with the properties, required by the standard model, including the relation between quarks and leptons and handedness and antiquarks and antileptons of the opposite handedness, as can be reed in Table 5 of App. D, appearing in the contribution of the same author in this Proceedings [33]. All that in the standard model had to be assumed (extremely effective ”read” from experiments and also from the theoretical investigations) in the spin-charge-family theory appear as a possibility from the starting simple action, Eq. (17.15), and from the assumption that the internal space of fermions are describedby the odd Cliffordalgebra objects. One can reed in my second contribution to this Proceedings [33] that the descrip­ tion of the internal space of fermions with the odd Cliffordalgebra operators .a’s offers the explanation for the observed quantum numbers of quarks and leptons and antiquarks and antileptons while unifying spin, handedness, charges and families. The ”basis vectors” which are superposition of odd products of operators .a’s, appear in irreducible representations which differ in the quantum numbers determinedby ˜.a’s. The simple starting action of the spin-charge-family theory offers the explanation for not only the properties of quarks and leptons and antiquarks and antileptons, but also for the vector gauge fields, scalar gauge fields, whichrepresent higgs and explain theYukawa couplings, and for the scalars, which cause matter/antimatter asymmetry, the proton decay, while the appearance of the dark matter is explained by the appearance of two groups of the decoupled families. It appears, as it is explained in my second contribution to this Proceedings [33], that the description of the internal space of bosons fields (the gauge fields of the fermion fields described by the Cliffordodd ”basis vectors”) with the Clifford even ”basis vectors” explains the commutativity and the properties of the second quantized boson fields, as the description of the internal space of fermion fields with the Clifford odd ”basis vectors” explains the anticommutativity and the properties of the second quantized fermion fields. The description of fermions and bosons with the Cliffordodd and Clifford even ”basis vectors”, respectively, makes fermions appearing in families, while bosons do not. Both kinds of ”basis vectors” contribute finite number, 2 d 2 -1 × 2 d -1 2 , degrees of freedom to the corresponding creation operators, while the basis of ordinary space contribute continuously infinite degrees of freedom. Is the way proposed by the spin-charge-family theory the right way to the next step beyond the standard model?The theory certainly offers a different view of the properties of fermion and boson fields and a different view of the second quantization of both fields than that offered by group theory and the second quantization by postulates. It has happened so many times in the history of science that the simpler model has shownup asa more ”powerful” one. My working hypotheses is that the laws of nature are simple and correspondingly elegant and that the many body systems around the phase transitions look to us complicated at least from the point of view of the elementary constituents of fermion and boson fields. To this working hypotheses belong also the description of the internal space of fermions and bosons with the Cliffordalgebras and the simple starting action for the(second quantized) massless fermions interactingwiththe(second quantized) 2 massless bosons, representing gravity only — the vielbeins and the two kinds of the spin connection fields, the gauge fields of the two kinds of the generators of the Lorentz transformations Sab(= i (.a.b -.b.a)) and S˜ab(= i (.˜a .˜b - .˜b .˜a)). 22 In Sect. 17.2 I shall very shortly overview the Clifford algebra description of the internal space of fermions, following Ref. [1], and bosons (explained in my additional contribution to this Proceedings [33]), after the reduction of the two independent groups of Cliffordalgebras to only one. In Sect. 17.3 the definition of the creation and annihilation operators as tensor productsof the ”basis vectors” definedby the Cliffordalgebra objects and basisin ordinary space is presented. In Sect. 17.4 the simple starting action of the spin-charge-family theory is presented and the achievements of the theory so far discussed. In Sect. 17.5 the open problems of the spin-charge-family theory are presented, and the invitation to the reader to participate. 17.2 Clifford algebra and internal space of fermions and bosons Ifollow here Ref. [1], Sect.3and also my second contribution to this Proceed­ings [33], Sect. 2. Single fermion states are functions of external coordinates and of internal space of fermions. If Mab denote infinitesimal generators of the Lorentz algebra in both Lab abaa. spaces, Mab =+ Sab, with Lab = xpb - xp, p= i , determining .xa operators in ordinary space, while Sab are equivalent operators in internal space of fermions, it follows {Mabi{Mad.bc + Mbc.ad -Mac.bd -Mbd.ac}, ,Mcd}- = {Mab ba ,p c}- =-i.ac p + i.cb p, {Mabi{Sad.bc + Sad -Sac.bd -Sbd.ac}, ,Scd}- = (17.1) while the Cartan subalgebra operators of the Lorentz algebra are chosen as M03 ,M12,M56,...,Md-1d , (17.2) 2 Since the single fermion states, described by the Cliffordodd ”basis vectors”, anticom-mute due to the anticommuting properties of the Cliffordodd ”basis vectors” and the single boson states, described by the Clifford even ”basis vectors”, correspondingly commute there are only the second quantized fermion and boson fields. and willbe usedto define the basisin both spaces as eigenvectorsof the Cartan subalgebra members. The metric tensor .ab = diag(1, -1, -1, . . . , -1, -1)for a =(0, 1, 2, 3, 5, . . . , d)is used. There are two kinds of anticommuting algebras, the Grassmann algebra .a’s and . p.a’s(= ’s), in d-dimensional space with d anticommuting operators .a’s and ..a . with d anticommuting derivatives ’s, ..a {.a,.b}+ = 0, { ., . }+ = 0, ..a ..b . {.a, }+ = dab , (a, b)=(0, 1, 2, 3, 5, ··· ,d), ..b (.a)† .aa )† .aa.a = ., ( . = , (17.3) ..a ..a where the last line was our choice [32], and the two anticommuting kinds of the Clifford algebras .a’s and .˜a’s 3 are expressible with the Grassmann algebra operators and opposite .a =(.a + . ),.˜a = i (.a - . ), ..a ..a 1 .1 .a =(.a -i.˜a), =(.a + i.˜a), (17.4) 2 ..a 2 offeringtogether 2 · 2d operators: 2d of those which are products of .a and 2d of those which areproducts of .˜a,the same number of operators as of the Grassmann algebra operators. The two kinds of the Cliffordalgebras anticommute, fulfilling the anticommutationrelations {.a,.b}+ = 2.ab = {.˜a ,.˜b}+ , {.a ,.˜b}+ = 0, (a, b)=(0, 1, 2, 3, 5, ··· ,d), (.a)† .aa .a .a)† .aa .˜a = , (˜= , .aa .aa .a.a = ,.a(.a)† = I, .˜a .˜a = ,.˜a(.˜a)† = I, (17.5) where I represents the unit operator. The two kinds of the Cliffordalgebra objects are obviously independent. 3 The existenceof the two kindsof the Cliffordalgebrasis discussedin [19,20,22,34,35]. The corresponding infinitesimal Lorentz generators are then Sab for the Grass-mann algebra, and Sab and S˜ab for the two kinds of the Cliffordalgebras. i (.a.b -.b Sab = .a), 4 S˜ab i =(.˜a .˜b - .˜b .˜a), 4 Sab = i (.a . -.b . ), ..b ..a {Sab S˜ab}- Sab Sab + S˜ab , = 0, = , {Sab-i (.ae .b -.be ,.e}- = .a), {Sab .e}- =-i (.ae .b -.be .a), ,p pp {Sabi(.bc.a -.ac ,.c}- = .b), Sab i(.bc .˜a -.ac ˜ {˜,.˜c}- = .b), {Sab Sab ,.˜c}- = 0, {˜,.c}- = 0. (17.6) Thereader can finda more detailed informationin Ref.[1]in Sect.3. It is useful to choose the ”basis vectors” in each of the two spaces to be products of eigenstates of the Cartan subalgebra members, Eq. (17.2), of the Lorentz algebras, (Sab i (.a.b - .b.a),S˜ab i .a .˜b - .˜b ˜ = =(˜.a)). The ”eigenstates” of each of 44 the Cartan subalgebra members, Eqs. (17.4, 17.5), for each of the two kinds of the Cliffordalgebras separately can be found as follows, .aa .aa 1 k1 1i k1i Sab Sab (.a + .b)=(.a + .b), (1 + .a.b)=(1 + .a.b), 2 ik 22 ik 2k 22k .aa .aa 1 k1 1i k1i S˜ab .a .b).a .b),S˜ab .a .˜b).a .˜b), (˜+ ˜=(˜+ ˜(1 + ˜=(1 + ˜(17.7) 2 ik 22 ik 2k 22k k2 = .aa.bb. The proof of Eq. (17.7) is presented in App. (I) of Ref. [1], Statement .aa .aa 2a. The Clifford”basis vectors” — nilpotents 1 (.a +.b), (1 (.a +.b))2 = 0 2 ik2 ik i i1i and projectors 1 (1+ .˜a .˜b), (1 (1+ .˜a .˜b))2 =(1+ .˜a .˜b)— of both algebras 2k2k 2k are normalized, up to a phase, as described in the contribution of the same outhor in this Proceedings [33]. Both, nilpotents and projectors, have half integer spins. It is useful to introduce the notation for the ”eigenvectors” of the two Cartan subalgebras as follows, Ref. [34, 35], † ab .aa ab abab abab ab .aa (k): = 1 (.a + .b), (k)= (-k), ((k))2 = 0, (k)(-k)= .aa [k] 2 ik † ab abab ab ab abab [k]: = 1 (1 + i.b), [k] =[k], ([k])2 =[k], [k][-k]= 0, .a2k abab abab ab ab ab ab ab ab (k)[k]= 0, [k](k)=(k), (k)[-k]=(k), [k](-k)= 0. (17.8) ab ab The corresponding expressions for nilpotents (k˜)and projectors [k˜]follows if we replace in Eq. (17.8) .a’s by .˜a’s, the same relation k2 = .aa.bb is valid for both algebras. Let us notice that the ”eigenvectors” of the Cartan subalgebras are equivalent and the eigenvalues are the same in both algebras: Both algebras have projectors and ab abab ab abab nilpotents:(([k])2 =[k], ((k))2 = 0),(([k˜])2 =[k˜], ((k˜))2 = 0). 2d -1 members In eachof the twoindependent algebras we have twogroupsof 2 which are eigenvectors of all the Cartan subalgebra members, Eq. (17.2), appearing in 2 2d -1 irreducible representations which have an odd Cliffordcharacter — they are products of an odd number of .a’s(.˜a’s). These two groups are Hermitian conjugatedtoeachother.Wemakea choiceofoneofthetwogroupsoftheClif­ bm† fordodd ”basis vectors” and name these ”basis vectors” ^ f , m describing 2 2d -1 membersof one irreduciblerepresentation, f describing one of 2 2d -1 irreducible representations. The 2 d 2 -1× 2 d 2 -1 members of the second group, Hermitian conju­ bm† bm† gated to ^, are named as b^m =(^)† . f ff There are besides two Cliffordodd groups in each of the two algebras .a’s and .˜a’s, also two Clifford even groups. They are superposition of an even number d d -1 Clifford even ”basis vectors” A^m† f -1 of .a’s(.˜a’s).Inamed these two 2 × 2 22 B^m† Am† b^m† and ,respectively. ^represent gauge vectors of , on which they operate. ff f B^m† operate on b^m .Idiscuss their properties in my second contribution of this ff Proceedings [33]. bm† The ”basis vectors” of an odd Cliffordcharacter, ^, and their Hermitian conju- f gated partners, b^m , fulfil the postulates for second quantized fermions of Dirac, f ifwereduce both Cliffordalgebrastoonly one[?,37, 38], while keeping all the relations, presented in Eq. (17.5), valid. Let us make a choice of .a’s and postulate the application of .˜a’s on B which is a superposition of any products of .a’s as follows {.˜aB = (-)B i B.a}|.oc >, (17.9) with (-)B =-1, if B is(a functionof) an oddproductsof .a’s, otherwise (-)B = 1 [35], |.oc > is defined in Eq. (17.10). (Sects. (2.1, 2.2 in [33]) and Sects. (3.2.2, 3.2.3 in [1])). The vacuum state |.oc > is defined as follows d -1 2 X 2 bm† >= b^m ^|1 >, (17.10) |.oc f * A f f=1 for one of the members m, anyone, of the odd irreducible representation f, with |1>, which is the vacuum without any structure, the identity, * A means the bm† algebraic product. It follows that b^m * A |.oc >= 0 and ^* A |.oc >= |.m >. f ff bm† After the postulate of Eq. (17.9) ”basis vectors” ^which are superposition of an f odd products of .a’s (represented by an odd number of nilpotents, the rest are projectors) obey all the fermions second quantization postulates of Dirac. Thereare S˜ab whichdresstheirreduciblerepresentations withthe family quantum numbers S03 S˜12 S˜56 ˜ of the Cartan subalgebra members(˜, , ,..., Sd-1d), Eq. (17.2). ' † ' bm b^m dmm {^f , f ' }* A+ |.oc > = dff ' |.oc >, {b^mf ,b^mf '' }* A+ |.oc > = 0 · |.oc >, bm† b^m {^f , f ' † }* A+ |.oc > = 0 · |.oc >, ' bm† ^|.oc > = |.m >, f * A f b^mf * A|.oc > = 0 · |.oc >, (17.11) with(m, m ')denoting the ”family” members and(f, f ')denoting ”families”,* A †m ^ represents the algebraic multiplication of b^and bm with the vacuum state ff |.oc > of Eq. (17.10) and among themselves, taking into account Eq. (17.5). Ref.([33], Sects. 2.4 and3)presents the starting studyofpropertiesof the second quantized boson fields, the internal space of which is represented by the ”basis vectors” A^m† which appear as the gauge fields of the second quantized fermion f fields the internal space of which is described by the ”basis vectors” b^m† . f We pay attention on even dimensional spaces,d = 2(2n + 1)or d = 4n, n = 0, only. . 17.3 Creation and annihilation operators Here Sect. 3.3 of Ref. [1] is roughly followed. bs† bs Describing fermion fields as the creation ˆ(.p)and annihilation ˆ(.p)operators ff2 -1× 2 2 -1 operating on the vacuum state we make tensor products, * T , of 2 dd bm† Cliffordodd ”basis vectors” ^and of continuously infinite basis in ordinary f † space determinedby b^ p X bs† {ˆ(.p)= f b^ † b^m† f (17.12) ms f(.p) * T |0.p }|.oc > >, c * T .. p m 0 where .p determines the momentum in ordinary space with p= |.p|and s deter­mines all the rest of quantum numbers. The state |.oc > * T |0.> is considered as p the vacuum for a starting single particle state from which one obtains the other single particle state by the operators, b., which pushes the momentum by an ^ p amount .p,ina tensorproduct with b^m† .We have f † b^^ p |.p> = |0p >, < .p |=<0p |b.p , . '† b^ p ' )=<0p |b^. < .p |.p> = d(.p -.p ' |0p >, p leading to † d(p . . b^b^ p ' -.p), (17.13) = ' . p since we normalize <0p |0p >= 1 to identity. bm† The ”basis vectors” ^which are products of an odd number of nilpotent, the f rest to d are then projectors, anticommute, transferring the anticommutativity 2 bs† to the creation operators ˆ(.p) and correspondingly also to their Hermitian f conjugated partners annihilation operators bˆs (.p), Eq. (17.12). The creation and f annihilation operators then fulfil the anticommutation relations of the second quantized fermions explaining the postulates of Dirac {b^s ' (.' ), b^s† dss '' ' -. p (.p)}+ |.oc > |0.p > = dff d(.pp)|.oc > |0.p >, f‘ f {b^s ' (p .' ), b^s (.> |0.> = > |0.>, f‘ fp)}+ |.oc p 0 |.oc p ' † bs† {b^s ' (p .' ), ^(.> |0.> = > |0.>, ff p)}+ |.oc p 0 |.oc p b^s† (.p)|.oc > |0.p > = |.fs (.p)> f b^s f(.p)|.oc > |0.p > = 0 |.oc > |0p .> 0 |p |= |.p|. (17.14) Statement The description of the internal space of fermions with the superposition of odd products of .a’s, that is with the clifford odd ”basis vectors”, not only explains the Dirac’s postulates of the second quantized fermions but also explains the appearance of families of fermions. Ref. [33] is offering the explanation for the second quantized commuting boson fields (described by the ”basis vectors” of an even number of nilpotents, the rest are projectors), they are the gauge fields of the anticommuting fermion fields (described by the ”basis vectors” of an odd number of nilpotents). 17.4 Achievements so far of spin-charge-family theory Here Sects. (6, 7.2.2 and 7.3.1) of Ref. [1], which review shortly the achievements so far of the spin-charge-family theory, are followed. The main new achievement of this theory in the last few years is the recognition thatthe descriptionoftheinternal spaceof fermion fields withthe Cliffordalgebra objects in d> (3 + 1)not only offers the explanation for all the assumptions of the standard model for fermion and boson fields, with the appearance of families for fermion fields and the properties of the corresponding vector and scalar gauge fields included, but also get to know, that the anticommuting property of the inter­nal space of fermions takes care of the second quantization properties of fermions, so that the second quantized postulates are not needed. The second quantized properties of fermions origin in their internal space and are transferred to creation and annihilation operators. This year contribution to Proceedings Ref. [33] offers the recognition that also commuting properties of the second quantized boson fields origin in the internal space of bosons. Describing the internal space of bosons by the Clifford even ”basis vectors”, written in terms of the Clifford even number of .a’s, these Clifford even ”basis vectors”, A^m† , applying on fermion states transform the ”basis vectors” b^m† either ff into another ”basis vectors” b^m ' † with the same family quantum number f, or if f m† ^bm ' † bm† written in terms of the Clifford even number of .˜a’s, A ˜, transform ^to ^, f ff‘ keeping the family member quantum number m unchanged and changing the family quantum number to f‘.4 This topic, started in Ref. [33], needs further study. 4 The first operation happensif the internal spaceof bosonsis describedby ”basis vectors” 03 1256 d-1d A^m† which are even products of nilpotents of the kind f =(-i)(-)[+] ··· [+]), in this The spin-charge-family theory proposes a simple action for interacting second quantized massless fermions and the corresponding gauge fields in d =(13 + 1)­dimensional space as Z dd1 ¯ A = xE (..a p0a.)+h.c. + 2 Z ddxE (aR + a ˜R˜), p0a = faap0a + 1 {pa, Efaa}- , 2E 11 SabS˜ab ˜ p0a = pa - .aba - .aba , 22 R = 1 {fa[afßb] (.aba,ß -.caa .cbß)}+ h.c. , 2 1fßb] (˜ R ˜= {fa[a.aba,ß - .˜caa .˜cbß)}+ h.c. . (17.15) 2 faafßb -fabfßa Here 5 fa[afßb] = . This simple action in d =(13 + 1)-dimensional space, i. in which massless fermions interact with the massless gravitation fields only (with the vielbeins and the two kinds of the spin connection fields, the gauge fields Sab of Sab and ˜, respectively), ii. together with the assumption that the internal space of the second quantized fermions are described by the Cliffordodd ”basis vectors” (what explains after the break of symmetries at low energies the appearance of quarks and leptons and antiquarks and antileptons of the standard model and the existence of families, predicting the number of families [46]), iii.and the internal space of the second quantized boson fields are described by the Clifford even ”basis vectors”, offers the explanations for iv. not only all the assumptions of the standard model — for properties of quarks and leptons and antiquarks and antileptons (explaining the relations among spins, handedness and charges of fermions and antifermions [23, 44]) and for the appear-anceof familiesof quarksand leptons[34,35,42], v. for the second quantized postulatesof Dirac [36,37], vi. for the appearance of the vector gauge fields to the corresponding fermion fields [26], particular case two nilpotents form ”basis vectors”, the second operation happens if ab ab ab cd all the nilpotents (k)and projectors [k]are replaced by the corresponding (k˜)and [k˜], respectively. 5 a aaß faa are inverted vielbeins to e a with the properties e afab = dab,e afßa = da, E = det(e aa). Latinindices a, b, .., m, n, .., s, t, .. denotea tangent space(a flat index), while Greek indices a, ß, .., µ, ., ..s, t, .. denoteanEinstein index(a curved index). Lettersfrom the beginningof boththe alphabets indicatea general index(a, b, c, .. and a, ß, ., .. ), from the middle of both the alphabets the observed dimensions 0, 1, 2, 3 (m, n, .. and µ, ., ..), indexes from the bottom of the alphabets indicate the compactified dimensions (s, t, .. and s, t, ..).We assume the signature .ab = diag{1, -1, -1, ··· , -1}. vii. for the appearance of gauge scalars explaining the interactions among fermions belongingtodifferent families[26,28,29,31,39–41,46],and correspondinglyofthe appearanceof the higgs scalar andYukawa couplings, viii. predicting the number of families — the fourth one to the observed three [46], ix. predicting the second group of four families the stable of which explains the appearance of the dark matter [23, 45], x. predicting additional gauge fields, xi. predicting additional scalar fields, which explain the existence of matter-antimatter asymmetry [25], and several others. The manifold M(13+1) breaks at high scale . 1016 GeV or higher first to M(7+1)× M(6) due to the appearance of the scalar condensate (so far just assumed, not yet proven that it appears spontaneously) 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 bringing masses (of the scale of break . 1016 GeV or higher) to all the gauge fields, which interact with the condensate [25]. Since the left handed spinors — fermions — couple differently (with respect to M(7+1))to scalar fields than the right handed ones, the break can leave massless and mass protected 2((7+1)/2-1)(= 8)families [49]. The rest of families get heavy masses 6. The manifold M(7+1)× SU(3)× U(1)breaks furtherbythe scalarfields,presented in Sect. 17.4.2, to M(3+1)× SU(3)× U(1)at the electroweak break. This happens since the scalar fields with the space index (7, 8), Subsubsect. 17.4.2, they area part ofasimple starting action Eq.(17.15), gain the constant values (the nonzerovacuum expectation values independent of the coordinates in d =(3 + 1)). These scalar fields carry with respect to the space index the weak charge ±1 and the hyper 2 charge ±1 [23,25], Sect. 17.4.2, just asrequiredby the standard model, manifesting 2 withrespect to S˜ab and Sab additional quantum numbers. Let us point out that all the fermion fields (with the families of fermions and the neutrinos forming the condensate included), the vector and the scalar gauge fields, offering explanation forby the standard model postulated ones, origin in the simple starting action. The starting action, Eq. (17.15), has only a few parameters. It is assumed that the .ab coupling of fermions to .abc’s can differ from the coupling of fermions to ˜c’s, The reduction of the Clifford space, Eq. 17.9, causes this difference. The additional breaks of symmetries influence the coupling constants in addition. Thebreaksof symmetriesis under consideration for quitea long time and has not yet been finished. 6Atoy model [49,52,53] was studied ind =(5 + 1)-dimensional space with the action presented in Eq. (17.15), The break from d =(5 + 1)to d =(3 + 1)× an almost S2 was studiedfora particular choiceof vielbeinsandfora classofspin connection fields. While the manifold M(5+1) breaks into M(3+1) times an almost S2 the 2((3+1)/2-1) families remain massless and mass protected. Equivalent assumption, although not yet proved how doesitreally work,is made also for the d =(13 + 1)case. This studyisinprogress quite some time. All the observed properties of fermions, of vector gauge fields and scalar gauge fields follow from the simple starting action, while the breaks of symmetries influence the properties of fermion and boson fields as well. 17.4.1 Properties of interacting massless fermions as manifesting in d =(3 + 1) before electroweak break One irreducible representation of SO(13, 1)includes all the left handed and right handedquarksandleptonsand antiquarksand antileptonsasonecanseeinTable5 of Ref. [33] in this Proceedings or inTable7of Ref. [1]. In both tables fermion ”basis vectors”arerepresentedbyodd numbersof nilpotentsand theirproperties analysed from the point of view of the standard model subgroups SO(3, 1)×SU(2)× SU(2)× SU(3)× U(1) of the group SO(13, 1). Quarks and leptons as well as antiquarks and antileptons appear with handedness as required by the standard model. One easily notices that quarks and leptons have the same content of the subgroup SO(7, 1), distinguishing only in SU(3)× U(1)content of SO(6):all the quarks, left and right handed, have the ”fermion” t4 equal to 1 and appear in three colours, 6 all the leptons, left and right handed, have t4 equal to -1 and are colourless. 2 Also antiquarks and antileptons have the same content of the subgroup SO(7, 1) (which is different from the one of quarks and leptons), and differ in SU(3)× U(1) content of SO(6), all the antiquarks, left and right handed, have t4 equal to -1 6 1 and appear in three anticolours, all the antileptons have t4 equal to and are 2 anticolourless. Let us notice also that since there are two SU(2)weak charges the right handed neutrinos and the left handed antineutrinos have non zero the second SU(2)II weak charge and interact with the SU(2)II weak field. Both have the standard model hyper charge Y = t4 + t23 equal to zero. Let me point out that this particular propertyareoffered alsobythe SO(10)unifying model [59], but with the manifold M(3 + 1)decoupled from charges. (Comments can be found in Ref. [1], Sect. 7). The expressions for the generators of the Lorentz transformations of subgroups SO(3, 1)× SU(2)× SU(2)× SU(3)× U(1)of the group SO(13, 1)can be found in App. 17.6 (also in Eqs. (39-41) of Ref. [33] or in Eqs. (85-89) of Ref. [1]). The condensate,presentedinTable 17.2(Table6of Ref. [1]), makes oneof the two weak SU(2)fields massive and causes the break of symmetries from M(13+1) to M(7+1) × SU(3)× U(1)[49, 52, 53], leaving only two decoupled groups of four 7+1 -1 families massless, 2 2 = 8. The reader can find these two groups of families in Table 17.1(fromTable5of Ref.[1]). bm† Table 17.1 presents ”basis vectors”(^, Eq. (17.11)) for eight families of the f 1 right handed u-quark of the colour (1 , v )and the right handed colourless .­ 2 23 lepton. The SO(7,1) content of the SO(13, 1)group are in both cases identical, they distinguish only in the SU(3)and U(1)subgroups of SO(6). All the members of any of these eight families ofTable 17.1 follows from either the u-quark or the .-lepton by the application of Sab. Each family carries the family quantum numbers, determined by the Cartan subalgebra of S˜ab in Eq. (17.2) and presented inTable 17.1. The two groups of families are after the break of symmetries decoupled since Ni ˜j t1i t2j}- Ni t1,2 j}- {˜L,NR}- = 0, .(i, j), {˜, ˜= 0, .(i, j), {˜L,R, ˜= 0, .(i, j), while {Sab Scd }- , ˜= 0, since {.a ,.˜a}- = 0, Eq. (17.5). c1† Table 17.1: Eight families of the ”basis vectors”b^m† , of u ^— the right handed fR v 1 u-quark with spin and the colour charge (t33 = 1/2, t38 = 1/(23)), appearing 2 inthefirstlineofTable7inRef.[1],orTable5inRef.[33]—andofthe colourless † right handed neutrino .^of spin 1 , appearing in the 25th lineofTable7in Ref.[1], R2 orTable5 in Ref. [33] — are presented in the left and in the right part of this table,respectively.Tableistakenfrom[31]. Familiesbelongtotwogroupsoffour families, one(I)isa doublet withrespectto(N.˜L and .t˜1)and a singlet with respect to(N.˜R and .t˜2), App. 17.6 (Eqs. (85-88)of Ref. [1]), the othergroup(II)is a singlet withrespect to( .˜˜N˜R and .t˜2). All the NL and .t1)anda doublet withrespectto(.families followfromthe starting onebythe applicationofthe operators( N˜± , R,L † t˜(2,1)±). The generators(N± , t(2,1)±)transformu ^to all the members of one R,L1R family of the same colour charge. The same generators transform equivalently the † right handed neutrino .^to all the colourless members of the same family. 1R ˜t13 ˜t23 ˜N3 ˜N3 ˜t4 L R I I I I c1† ^u R 1 c1† ^u R 2 c1† ^u R 3 c1† ^u R 4 03 12 56 78 9 10 11 12 13 14 (+i) [+] | [+] (+) || (+) [-] [-] 03 12 56 78 9 10 11 12 13 14 [+i] (+) | [+] (+) || (+) [-] [-] 03 12 56 78 9 10 11 12 13 14 (+i) [+] | (+) [+] || (+) [-] [-] 03 12 56 78 9 10 11 12 13 14 [+i] (+) | (+) [+] || (+) [-] [-] † ^.R 1 † ^.R 2 † ^.R 3 † ^.R 4 03 12 56 78 9 10 11 12 13 14 (+i) [+] | [+] (+) || (+) (+) (+) 03 12 56 78 9 10 11 12 13 14 [+i] (+) | [+] (+) || (+) (+) (+) 03 12 56 78 9 10 11 12 13 14 (+i) [+] | (+) [+] || (+) (+) (+) 03 12 56 78 9 10 11 12 13 14 [+i] (+) | (+) [+] || (+) (+) (+) - 1 0 - 1 0 - 1 2 2 2 - 1 0 1 0 - 1 2 2 2 1 0 - 1 0 - 1 2 2 2 1 0 1 0 - 1 2 2 2 II II II II c1† ^u R 5 c1† ^u R 6 c1† ^u R 7 c1† ^u R 8 03 12 56 78 9 10 11 12 13 14 [+i] [+] | [+] [+] || (+) [-] [-] 03 12 56 78 9 10 11 12 13 14 (+i) (+) | [+] [+] || (+) [-] [-] 03 12 56 78 9 10 11 12 13 14 [+i] [+] | (+) (+) || (+) [-] [-] 03 12 56 78 9 10 11 12 13 14 (+i) (+) | (+) (+) || (+) [-] [-] † ^.R 5 † ^.R 6 † ^.R 7 † ^. R 8 03 12 56 78 9 10 11 12 13 14 [+i] [+] | [+] [+] || (+) (+) (+) 03 12 56 78 9 10 11 12 13 14 (+i) (+) | [+] [+] || (+) (+) (+) 03 12 56 78 9 10 11 12 13 14 [+i] [+] | (+) (+) || (+) (+) (+) 03 12 56 78 9 10 11 12 13 14 (+i) (+) | (+) (+) || (+) (+) (+) 0 - 1 0 - 1 - 1 2 2 2 0 - 1 0 1 - 1 2 2 2 0 1 0 - 1 - 1 2 2 2 0 1 0 1 - 1 2 2 3 Itis the assumption that the eight familiesfromTable 17.1remain massless after the break of symmetry from SO(13, 1)to SO(7, 1)× SO(6), made after we proved forthetoymodel[49,52]thatthebreakofsymmetrycanleavesome familiesof fermions massless, while the rest become massive. But we have not yet proven the masslessness of the 2 7+1 -1 families after the break from SO(13, 1)to SO(7, 1)× 2 SO(6). The break from the starting symmetry SO(13, 1)to SO(7, 1)× SU(3)× U(1)is supposed to be caused by the appearance of the condensate of two right handed neutrinos with the family quantum numbers of the upper four families, that is of thefour families,whichdonot containthethreesofar observed families,atthe energy of = 1016 GeV. This condensateispresentedinTable 17.2. To see how do gravitational fields— vielbeins and the two spin connection fields, Sab the gauge fields of Sab and ˜, respectively — contribute to dynamics of fermion fields and after the electroweak break also to the masses of twice four families and the vector gauge field let us rewrite the fermion part of the action, Eq. (17.15), in the way that the fermion action manifests in d =(3 + 1), that is in the low energy regime before the electroweak break, by the standard model postulated Table 17.2: The condensate of the two right handed neutrinos.R,with the quantum numbers of the VIIIth family,Table 17.1, coupled to spin zero and belonging to a triplet with respect to the generators t2i, is presented, together with its two 0, t23 partners. The condensate carries .t1 == 1, t4 =-1 and Q = 0 = Y. t4 t23 N3 N˜3 ˜˜ The triplet carries ˜=-1,˜= 1 and ˜= 1, = 0, Y = 0, Q = 0. The RL family quantum numbersof quarks and leptons arepresentedinTable 17.1. The definition of the operators .t1 , .t˜1 , .t2 , .t˜2,t4 ,t˜4,N3 N˜3 ,N3 N˜3 , Q, Y, ˜Y can be R, L, Q, ˜ RL found in App. 17.6 (and in Ref. [1], Eqs. (85-88) or in Eqs. (39-41) of Ref. [33]). state S03 S12 t13 t23 t4 Y Q t13 ˜ t23 ˜ ˜t4 ˜Y ˜Q ˜N3 L ˜N3 R (|.VIII >1 |.VIII 1R 2R >2) 0 0 0 1 -1 0 0 0 1 -1 0 0 0 1 (|.VIII VIII 1R >1 |e 2R >2) VIII VIII (|e 1R >1 |e 2R >2) 0 0 0 0 -1 -1 -1 0 0 0 -1 -1 -2 -2 0 0 1 1 -1 0 0 0 -1 0 0 0 1 1 properties, while manifesting the properties which make the spin-charge-family theorya candidatetogobeyondthe standard model: i. The spins, handedness, charges and family quantum numbers of fermions are determined by the Cartan subalgebra of Sab and S˜ab, and the internal space of fermionsis describedby the Clifford”basis vectors” b^m† . f ii. Couplings of fermions to the vector gauge fields, which are the superposi­ tion of gauge fields .stm, Sect. 17.4.2, with the space index m =(0, 1, 2, 3)and Sab (Sab.cd with charges determined by the Cartan subalgebra of Sab and ˜e = i(.ad .bc -.bd ee.ac)and equivalently for the othertwo indexesof.cde gauge fields, manifesting the symmetry of space (d - 4)), and couplings of fermions to the scalar gauge fields [19,20,23,29,31,38,41,42,45,46] with the space index s = 5 and the charges determined by the Cartan subalgebra of Sab and S˜ab (as explained in the case of the vector gauge fields), and which are superposition of .abt either .sts or ˜s, Sect. 17.4.2 X AitAiAAi ¯ Lf = ..m(pm - g ). + m A,i X ¯ { ..s p0s .}+ s=7,8 X { ..t p0t .}, ¯ (17.16) t=5,6,9,...,14 '' 1 Sss" .s ' 1 S˜ab ˜1 Stt" .t1 S˜ab ˜ where p0s = ps - s"s - .abs, p0t = pt - ' t"t - .abt, 22 22 ' S˜ab) with m . (0, 1, 2, 3), s . (7, 8), (s ,s") . (5, 6, 7, 8), (a, b) (appearing in ' 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 mem­bers of all the 2 7+1 -1 = 8 families. 2 The first line of Eq. (17.16) determines in d =(3+1)the kinematics and dynamics of fermion fields, coupled to the vector gauge fields [23,26,31]. 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, Sect. 17.4.2. P Ai The operators tAi (tAi = c ab Sab , Sab are the generators of the Lorentz a,b transformations in the Cliffordspace of .a’s) are presented in Eqs. (17.27, 17.28) of App. 17.6. Theyrepresent the colour charge, .t3, the weak charge,.t1,and the hyper t4 +t23 charge, Y = , t4 is the ”fermion” charge, originating in SO(6). SO(13, 1), t23 belongs together with .t1 of SU(2)weak to SO(4)(. SO(13 + 1)). One fermion irreducible representation of the Lorentz group contains, as seeninTable7 of Ref.[1]orinTable5of Ref.[33], quarks and leptons and antiquarks and antileptons, belongingtothefirstfamilyinTable 17.1. Let usrepeat again that the SO(7, 1)subgroup content of the SO(13, 1)group is the same for the quarks and leptons and the same for the antiquarks and antileptons. Quarks distinguish from leptons, and antiquarks from antileptons, only in the SO(6). SO(13, 1)part, that is in the colour (t33,t38)part and in the ”fermion” quantum number t4. The quarks distinguish from antiquarks, and leptons from antileptons, in the handedness, in the SU(2)I (weak), SU(2)II, in the colour part and inthe t4 part, explaining the relation between handedness and charges of fermions and antifermions, postulated in the standard model 7. All the vector gauge fields, which interact with the condensate, presented in Table 17.2, become massive, Sect. 17.4.2. Thevector gauge fields not interacting with the condensate — the weak, colour, hyper charge and electromagnetic vector gauge fields — remain massless,in agreement withby the standard model assumed gauge fields before the electroweak break 8. Afterthe electroweakbreak, causedbythe scalar fields,theonly conservedcharges t13 t4 are the colour andthe electromagnetic charge Q =+ Y (Y =+ t23). All the rest interact with the scalar fields of the constant value. The second line of Eq. (17.16) is the mass term, responsible in d =(3 + 1)for the masses of fermions and of the weak gauge field (originating in spin connection fields .stm).The interaction of fermions with the scalar fields with the space index s =(7, 8)(to these scalar fields particular superposition of the spin connection .ab fields .abs and all the superposition of ˜s with the space index s =(7, 8) and (a, b)=(0, 1, 2, 3)or (a, b)=(5, 6, 7, 8)contribute), which gain the constant values in d =(3 + 1), makes fermions and antifermions massive. The scalar fields, presented in the second line of Eq. (17.16), are in the standard model interpreted as the higgs and the Yukawa couplings, Sect. 17.4.2, predicting in the spin-charge-family theory that there must exist several scalar fields 9. These scalar gauge fields split into two groups of scalar fields. One group of two triplets and three singlets manifests the symmetry (SO(3,1),L)SU(2)SO(4),L) SU(2)× ( 7 Ref. [30] points out that the connection between handedness and charges for fermions and antifermions, both appearingin the same irreduciblerepresentation, explains the triangle anomalies in the standard model with no need to connect ”by hand” the handedness and charges of fermions and antifermions. .st .st 8 The superposition of the scalar gauge fields ˜7 and ˜8, which at the electroweak break gain constant values in d =(3 + 1), bring masses to all the vector gauge fields, which couple to these scalar fields. 9 The requirement of the standard model that there exist theYukawa couplings, speaksby itself that there must exist several scalar fields explaining theYukawa couplings. ×U(1). The other group of another two triplets and the same three singlets mani­fests the symmetry SU(2)SO(3,1),R) SU(2)(×U(1). (× SO(4),R) ' The three U(1)singlet scalar gauge fields are superposition of .s ' ts, s =(7, 8), ' t (s ,t ' )=(5, 6, ··· , 14), with the sums of Ss '' arranged into superposition of t13 , t23 and t4. The three triplets interact withbothgroupsof quarks and leptons and antiquarks and antileptons [39–41, 45–48]. Families of fermions fromTable 17.1, interacting with these scalar fields, split as well into two groups of four families, each of these two groups are coupled to one of the two groups of scalar triplets while all eight families couple to the same three singlets. The scalar gauge fields, manifesting SU(2)L,R, are SU(2)L,R × the superposition of the gauge fields .˜abs, s =(7, 8), (a, b)= either (0, 1, 2, 3)or (5, 6, 7, 8), manifesting as twice two triplets. 17.4.2 Vector and scalar gauge fields before electroweak break The second line of Eq. (17.15) represents the action for the gauge fields Agf Z Agf = ddxE (aR + a ˜R˜), 1 {fa[a R = fßb] (.aba,ß -.caa .cbß)}+ h.c. , 2 ˜{fa[afßb] (˜.c R = 1.aba,ß - .˜caa ˜bß)}+ h.c. . (17.17) 2 It is proven in Ref. [26] that the vector and the scalar gauge fields manifest in d =(3 + 1), after the break of the starting symmetry, as the superposition of spin connection fields, when the space (d -4)manifest the assumed symmetry. fßa and ß a aa afßafa da ea are vielbeins and inverted vielbeinsrespectively, ea = da, eb = b, a E = det(ea). Varying the action of Eq. (17.17) with respect to the spin connection fields the expression for the spin connection fields .abe follows 1 ee .ab = {e a .ß(Efa [afßb])-eaa .ß(Efa [bfße])-eba.ß(Efa[efßa])} 2E 1 ¯ + {.(.e Sab -.[aSb] e).} 4 11 d {de ¯ -[ e a.ß(Efa [dfßb])+ ..dSdb .] a d -2E 1 d ¯ - de [ e a.ß(Efa [dfßa])+ ..dSd .]}. (17.18) ba E Replacing Sab in Eq. (17.18) with S˜ab, the expression for the spin connection fields .˜abe follows. ee If there are no spinors (fermions) present, . = 0, then either .ab or .˜ab are uniquely expressed with the vielbeins. Spin connection fields .abe represent vector gauge fields to the corresponding fermion fields if index e is m =(0, 1, 2, 3). If e = 5 the spin connection fields manifest in d =(3 + 1)as scalar gauge fields. It is proven in Ref. [26] 10 that in spaces with the desired symmetry the vielbein can be expressed with the gauge fields, X fs tAs A.A m = ., m A X tAisAi t = c st (est fst -ett fss)x, st X AAi Ai = c st .stm , m st X tAi Ai = c st Sst , st {tAifAijktAk ,tBj}- = idAB. (17.19) The vector gauge fields AAi of tAi represent in the spin-charge-family theory all the m observed gauge fields, as well as the additional non observed vector gauge fields, which interacting with the condensate gain heavy masses. The scalar (gauge) fields, carrying the space index s =(5, 6, . . . , d), offer in the spin-charge-family for s =(7, 8)the explanation for the origin of the Higgs’s scalar andtheYukawa couplingsofthe standard model, while scalars with the space index s =(9, 10, . . . , 14)offer the explanation for the proton decay, as well as for the matter/antimatter asymmetry in the universe. .ab ' In the scalar gauge fields besides .sts also ˜s contribute. Ai The explicit expressions for cab, and correspondingly for tAi, and AAi , are a written in Sects. 4.2.1. and 4.2.2 of Ref. [1]. 2.aVector gauge fields. All the vector gauge fields are in the spin-charge-family theory expressible with the spin connection fields .stm as X AAi Ai = c st .st (17.20) mm , s,t * tAi AAi Sab.ab * with = m, means that summation runs over (a, b) A,i m a,b respecting the symmetry SO(7, 1)× SU(3)× U(1), with SO(7, 1)breaking further to SO(3, 1)× SU(2)I × SU(2)II. The vectorgauge fieldsarenamely analysedfromthepointofviewofthe possibly observed fields in d =(3 + 1)space: besides gravity, the colour SU(3), the weak SU(2)I,the secondSU(2)II and the U(1)t4 -the vector gauge field of the ”fermion” quantum number t4 . 10Wepresentedin Ref.[26]theproof, thatthe vielbeinsfsm (Einstein index s = 5, m = 0, 1, 2, 3)lead ind =(3 + 1)to the vector gauge fields, which are the superposition of tAs Ai the spin connection fields .stm: fsm = P A.A .t x t, with AAi = P c st .stm, m A m s,t when the metric in (d -4), gst, is invariant under the coordinate transformations x s ' = P s eAi mAist EssttAist Esst x +(x )c (x )and c = tAis, while tAis solves the A,i,s,t s,t +DttAi 0 (Ds tAi .s tAi tstAi Killing equation: Ds tAi = =-Gt '' ). And similarly also for ts ttt the scalar gauge fields. Due to the interaction with the condensate the second SU(2)II (one superposition of the thirdcomponent of SU(2)II and of the U(1)t4 vector gauge fields and the rest two components of the SU(2)II vector gauge field) become massive, while the colour SU(3), the weak SU(2)I, the second superposition of the thirdcomponent of SU(2)II and the U(1)t4 , forming the hyper charge vector gauge field, remain massless. That is: All the vector gauge fields, as well as the scalar gauge fields of Sab and of S˜ab, which interact with the condensate, become massive. The effective action for all the massless vector gauge fields, the gauge fields which do not interact with the condensate and remain therefore massless, before the R FAi electroweak break, equal to d4x {-1 mn FAimn }, with the structure constants 4 fAijk concerning the colour SU(3), weak SU(2) and hyper charge U(1)groups [26]. All these relations are valid as long as spinors and vector gauge fields are weak fields in comparison with the fields which force (d -4)space to be (almost) curled, Ref. [50]. When all these fields, with the scalar gauge fields included, start to be comparable with the fields (spinors or scalars), which determine the symmetry of (d -4)space, the symmetry of the whole space changes. The electroweak break, caused by the constant (non zero vacuum expectation) values of the scalar gauge fields, carrying the space index s =(7, 8), makes the weak and the hyper charge gauge fields massive. The onlyvector gauge fields which remain massless are, besides the gravity, the electromagnetic and the colour vector gauge fields — the observed three massless gauge fields. 2.b. Scalar gauge fields in d =(3 + 1). The starting action of the spin-charge-family theory offers scalar fields of two kinds: a. Scalarfields,takingcareofthe massesofquarksandleptonshavethespaceindex s =(7, 8)and carry with respect to this space index the weak charge t13 = ±1 2 and the hyper charge Y = ±1 ,Table17.3,Eq. (17.23).Withrespecttotheindex Ai, 2 P Ai tAi Ai ˜ determined by the relation tAi = cabSab and ˜= cab Sab, that ab ab is with respect to Sab and S˜ab, they carry charges and family charges in adjoint representations. b. There are in the starting action of the spin-charge-family theory, Eq. (17.15), scalar fields, which transform antileptons and antiquarks into quarks and leptons and back. They carry space index s =(9, 10, . . . , 14), They are withrespect to the space index colour triplets and antitriplets, while they carry charges tAi and t˜Ai in adjoint representations. FollowingRefs.[1,31,38]Ishallreviewbothkindsofscalar fields. 11 2.b.i Scalar gauge fields determining scalar higgs andYukawa couplings Making a choice of the scalar index equal to s =(7, 8)(the choice of (s = 5, 6) would also work) and allowing all superposition of ˜˜, while with respect to . a ˜bs 11 Let us demonstrate how do the infinitesimal generators Sab apply on the spin connections fields .bde (= fae .bda)and ˜b ˜(= fa ˜b ˜), on either the space index . ˜de e . ˜dae or any d)Sab Ad...e...g i (.ae Ad...b...g -.be of the indices (b, d, b, ˜˜= Ad...a...g)(Section IV. and AppendixBin Ref. [31]). Q .abs only the superposition representing the scalar gauge fields As , AY and A4 , ss s =(7, 8)(or any three superposition of these three scalar fields) may contribute. Let us use the common notation AAi for all the scalar gauge fields with s =(7, 8), s independently of whether they originate in .abs — in this case Ai =(Q,Y, t4)— or in . ˜˜˜. All these gauge fields contribute to the masses of quarks and leptons abs and antiquarks and antileptons after gaining constant values (nonzero vacuum expectation values). ˜˜ AAi .˜1 ˜.˜N L ˜.˜2 ˜.˜N R ˜ represents (AQ ,AY ,A4 ,A ,A s ,A ,A s ), s ssss s .t1 .˜.t2 .˜ tAi represents (Q,Y, t4 , ˜,NL, ˜,NR). (17.21) Here tAi represent all the operators which apply on fermions. These scalars with the space index s =(7, 8), they are scalar gauge fields ofthe generators tAi and t˜Ai, are expressible in terms of the spin connection fields, App. 17.6 (Ref. [31], Eqs. (10, 22, A8, A9)). All the scalar fields with the space index (7, 8)carry with respect to this space index the weak and the hyper charge(±1 , ±1 ),respectively, all having therefore 22 properties as required for the higgs in the standard model. 1 To make the scalar fields the eigenstates oft13 =(S56 -S78)and to check their 2 +t23 (1 (S56+S78)-1 (S9 10+S11 12+S13 14)) properties withrespectto Y (= t4= 23 and Q (= t13 +Y)we need to apply the operators t13 , Y and Q on the scalar fields with the space index s =(7, 8), taking into account the relation Sab Ad...e...g = i (.ae Ad...b...g -.be Ad...a...g). Letusrewritethe secondlineofEq. (17.16),payingno attentiontothe momentum ps ,s . (5, . . . , 8), when having in mind the lowest energy solutions manifesting at low energies. X AAi ¯ ..s (-tAi ). = s s=(7,8),A,i X 78 78 - . { (+) tAi (AAi -iAAi )+ (-)(tAi (AAi + iAAi )}., ¯ 78 78 A,i 78 AAi (±)= 1 (.7 ± i.8 ), :=(AAi ± iAAi ), (17.22) 78 78 2 (±) with the summation over A and i performed, with AAi representing the scalar s Q4 ˜.˜1 ˜.˜2 ˜.˜N˜R '' fields(As , AsY , As4 )determined by.s ' ,s ,s , as wellas(A˜s, As, As, A s and .˜ A ˜Ns L ), determined by ˜.a,b,s ,s =(7, 8). The application of the operators t13 , Y and Q on the scalar fields(AAi ± iAAi ) 78 withrespect to the space index s =(7, 8), gives t13 (AAi 1 (AAi ± iAAi )= ±± iAAi ), 78 78 2 Y (AAi 1 (AAi ± iAAi )= ±± iAAi ), 78 78 2 Q (AAi ± iAAi )= 0. (17.23) 78 Q Since t4 , Y, t13 and t1+,t1- give zeroif applied on(As , AY and A4 )(withrespect ssto the quantum numbers(Q, Y, t4)), and since Y, Q, t4 and t13 commute with Q the family quantum numbers, one sees that the scalar fields AAi (=(As , AY , ss ' ˜ ˜4 ˜˜Q .˜1 ˜.˜2 ˜.˜N˜R .˜N˜L (AAi AY , A, A s , A, A, A , A )), s =(7, 8), rewritten as AAi = ± iAAi ), ssssss 78 78 (±) are eigenstates of t13 and Y, having the quantum numbers of the standard model Higgs’s scalar. These superposition of AAi are presented inTable 17.3 as two doublets with 78 (±) respect to the weak charge t13, with the eigenvalue of t23 (the second SU(2)II 1 charge) equal to either -1 or +,respectively. 22 -1 Table 17.3: The two scalar weak doublets, one witht23 = and the other with 2 t23 1 =+, both with the ”fermion” quantum number t4 = 0, are presented. In this 2 table all the scalar fields carry besides the quantum numbers determinedby the space index also the quantum numbers A and i from Eq. (17.21). The table is taken from Ref. [31]. name superposition t13 t23 spin t4 Q AAi 78 (-) AAi 56 (-) AAi + iAAi 7 8 AAi + iAAi 5 6 1 +2 -1 2 -1 0 0 0 2 -1 0 0 -1 2 AAi 78 (+) AAi 56 (+) AAi -iAAi 7 8 AAi -iAAi 5 6 -1 2 1 +2 1 +0 0 0 2 1 +0 0 +1 2 It is not difficult to show that the scalar fields AAi are triplets as the gauge fields 78 (±) . of the family quantum numbers(.˜N˜L, .t2 , .t˜1 or singlets as the gauge fields of NR, ˜t13 '' + t23 Q =+ Y, Q =-tan2 .1Y +t13 and Y =-tan2 .2t4 . Table 17.1 represents two groups of four families. It is not difficult to see that N˜± L and t˜1± transform the first four families among themselves, leaving the second group of four families untouched, while N˜± and t˜2± do not influence the first R four families and transform the second four families among themselves. All the scalar fields with s =(7, 8)”dress” the right handed quarks and leptons with the hyper charge and the weak charge so that they manifest charges of the left handed partners. The mass matrices 4 × 4, representing the application of the scalar gauge fields on fermions of each of the two groups, have the symmetry SU(2)× SU(2)× U(1)of the form as presented in Eq. (17.24) 12. The influence of scalar fields on the masses of quarks and leptons depends on the coupling constants and the masses of the 12 The symmetry SU(2)× SU(2)× U(1)of the mass matrices, Eq. (17.24), is expected to remain in all loop corrections [47]. -a1 -ae db e* -a2 -ab d a = . ... .. Ma . d* b* a2 -ae b* d* e* a1 -a . , (17.24) with a representing family members — quarks and leptons [39–41, 46, 48]. In Subsect. 17.4.3 the predictions of the spin-charge-family theory following from the symmetry of mass matrices of Eq. (17.24) are discussed. The spin-charge-family theory treats quarks and leptons in equivalent way. The Q differences among family members occur due to the scalar fields(Q · A78 ,Y · (±) Q · A4 A 78 ,t4 78 )[46, 48]. (±)(±) Twice four families ofTable 17.1, with the two groups of two triplets applying each on one of the two groups of four families and one group of three singlets applying on all eight families, i. offer the explanation for the appearance of the Higgs’s scalar andYukawa couplingsof the observed three families,predicting the fourth family to the observed three families and several scalar fields, ii. predict that the stable of the additional four families with much higher masses that the lower four families contributes to the dark matter. 2.b.ii Scalar gauge fields causing transitions from antileptons and antiquarks into quarks and leptons [25] Besides the scalar fields with the space index s =(7, 8)which manifest in d = (3 + 1)as scalar gauge fields with the weak and hyper charge ±1 and ±1 , re­ 22 spectively, and which gaining at low energies constant values cause masses of families of quarks and leptons and of the weak gauge field, there are in the start­ing action, Eqs. (17.15, 17.16), additional scalar gauge fields with the space index t =(9, 10, 11, 12, 13, 14).Theyarewithrespecttothespace index t either triplets or antitriplets causing transitions from antileptons into quarks and from antiquarks intoquarksandback.These scalar fieldsareinEq.(17.16)presentedinthethird line. These scalar fields are offering the explanation for the matter/antimater asymme­tryin the universe, and mightberesponsible forproton decay and lepton number nonconservation.Thereaderiskindlyasktoreadthe article[25],forashortreview one can see the Refs. [1,23]. 17.4.3 Predictions of spin-charge-family theory Let me say that the fact that the simple starting action, Eq. (17.15) — in which fermions interact with gravity only (the vielbeins and the two kinds of the spin connection fields), while the internal spaces of fermions and bosons are describ­able by the ”basis vectors” which are superposition of odd or even products of Cliffordalgebra operators .a’s, respectively — offers the explanation for all the assumptions of the standard model and for the second quantized postulates for fermions and bosons, while unifying all the so far known forces, with gravity included, predicting new vector gauge fields, new scalar gauge fields and new families of fermions, gives a hope that the spin-charge-family theory is offering the right next step beyond the standard model. i. The existence of the lower group of four families predicts the fourth family to the observed three, which should be seen in next experiments. The masses of quarks of these four families are determined by several scalar fields, all with the properties of the scalar higgs, some of them of which might also be observed. The symmetry [46, 47], Eq. (17.24), and the values of mass matrices of the lower ˜ ˜N ˜ four families are determined with two triplet scalar fields, A.˜1 and A .˜L , and 78 78 (±)(±) Q three singlet scalar fields, A, AY , A4 , Eq. (17.21), explaining the Higgs’s 78 78 78 (±)(±)(±) scalar andYukawa couplings of the standard model, Refs. [23, 27, 31, 46, 48] and references therein. Any accurate 3 × 3 submatrix of the 4 × 4 unitary matrix determines the 4 × 4 matrix uniquely. Since neither the quark and (in particular) nor the lepton 3 × 3 mixing matrix are measured accurately enough to be able to determine three complex phases of the 4 × 4 mixing matrix, we assume (what also simplifies the numerical procedure) [39–41, 45, 46] that the mass matrices are symmetric and real and correspondingly the mixing matrices are orthogonal.We fitted the 6 free parameters of each family member mass matrix, Eq. (17.24), to twice three measured masses(6)of each pair of either quarks or leptons and to the6 (from the experimental data extracted) parameters of the corresponding 4 × 4 mixing matrix. Ipresent here the old results for quarks only, taken from Refs. [46]. The accuracy of the experimental data for leptons are not yet large enough that would allow any meaningful prediction 13. It turns out that the experimental [54] inaccuracies are for the mixing matrices too large to tell trustworthy mass intervals for the quarks masses of the fourth family members 14. Taking into account the calculations of Ref. [54] fitting the experimental data (and the meson decays evaluations in the literature as well as our own evaluations) the authors of the paper [46] very roughly estimate that the fourth family quarks masses might be pretty above 1 TeV. Since the matrix elements of the 3 × 3 submatrix of the 4 × 4 mixing matrix de­pend weakly on the fourth family masses, the calculated mixing matrix offers the prediction to what values will more accurate measurements move the present ex­ 13 The numerical procedure, explained in the paper [46], to fit free parameters of the mass matrices to the experimental data within the experimental inaccuracy of the mixing matrix elements of the so far observed quarks (the inaccuracy of masses do not influence theresults very much)is tough. 14We have notyet succeededtorepeatthe calculationspresentedin Refs.[46] withthe newest data from Ref. [55]. Let us say that the accuracy of the mixing matrix even for quarksremainsinRef.[55]farfromneededtopredictthe massesofthefourthtwoquarks. For the chosen masses of the four family quarks the mixing matrix elements are expected to slightly change in the direction proposed by Eq. (17.25). perimental data and also the fourth family mixing matrix elements in dependence of the fourth family masses, Eq. (17.25): Vud will stay the same or will very slightly decrease; Vub and Vcs, will still lower; Vtd will lower, and Vtb will lower; Vus will slightly increase; Vcd will (after decreasing) slightly rise; Vcb will still increase and Vts will (after decreasing) increase. In Eq. (17.25) the matrix elements of the 4 × 4 mixing matrix for quarks are pre­sented, obtained when the 4 × 4 mass matricesrespectthe symmetryofEq. (17.24) while the parameters of the mass matrices are fitted to the(exp)experimental data [54], Ref. [46]. The two choices of the fourth family quark masses are used in the calculations: mu4 = md4 = 700 GeV(scf1)andmu4 = md4 = 1 200 GeV (scf2). In parentheses, ()and [], the changes of the matrix elements are presented, which are due to the changes of the top mass within the experimental inaccuracies: with the mt =(172 + 3 × 0.76)GeV and mt =(172 -3 × 0.76), respectively (if there are one, two or more numbers in parentheses the last one or more numbers aredifferent,ifthereisnoparenthesesno numbersaredifferent) [arxiv:1412.5866]. scf1 0.22534(3) 0.97335 0.04245(6) 0.00349(60) scf2 0.22531[5] 0.97336[5] 0.04248 0.00002[216] exp 0.0084 ± 0.0006 0.0400 ± 0.0027 1.021 ± 0.032 scf1 0.00667(6) 0.04203(4) 0.99909 0.00038 scf2 0.00667 0.04206[5] 0.99909 0.00024[21] scf1 0.00677(60) 0.00517(26) 0.00020 0.99996 scf2 0.00773 0.00178 0.00022 0.99997[9] |V(ud)| = . (17.25) Let me conclude that according to Ref. [46] the masses of the fourth family lie much above the known three. The larger are masses of the fourth family the larger are Vu1d4 in comparison with Vu1d3 and the more is valid that Vu2d4 < Vu1d4 , Vu3d4 Vu1d3 , Vu2d4 = coef. * (A, C;B, D) (18.26) where (A, C;B, D)= “anharmonic raatio” (18.27) AB * CD AB AD = = : . CB * AD CB CD (18.28) µ. µ. or more precisely: = The form given by the anharmonic ratio is specified by the Møbius invariance un­derreal numbers, becauseonlythe anharmonic ratiosare invariantforfourpoints. Here the pairs of letters like AB meansthedifferenceofthe fieldFcoordinatefor B minus that for A. However, the coefficient coef. is ana priori unspecified constant. Whatthe value shallbeisonlyspecifiedbythe initialand final state information onthebound stateswehadtoimpose.Inthepictureoftherebeingashort distance or high energy interaction underneath the coef. would inherit from such a short distance interaction. In the string theory with the open strings identified with our ' bound states this coefficient wouldbe givenby the Regge-slope a , coef. . 1. (18.30) ' a It should be in mind that the scale symmetry is only broken by these initial and final state informations. ' So the a energy scale in theVeneziano model in our picture comes in via the initial and final state information, only. 18.10 The Cyclic ordering partly violates the full Møbius symmetry We have to mention what formally looks like a little problem: In our novel string field theory we had to impose the condition (18.20), which is not invariant under shift of orientation along the cyclically ordered chain. Formally such a condition would break the symmetry under half of the Møbius transformations.To have this condition consistent with the symmetry we should only keep those Møbius transfromations, which leave the orientation along the chain orin the wordsof thepresent article along theprojectiveline intact. This means that formally the group does not act 3-transitively, but only 3-transitively modulo the cyclic orientation. But from what we could call an estetic point of view this only orientation keeping subgroupisquiteniceinasfarasit indeedlies insidethefull Møbiusgroupasa topologically seprate part, a component. It might be needed in our building of the bound states from a high energy theory to let this one have sufficient breaking of its symmetries so as to deliver such bound states that the orientation gets fixed. 18.11 Conclusion Conclusion We have proposed an approximation applicable hopefully to some boundstates: that they have so many constituents with so equally divided momenta -or better Bjorken x’s ˜ 0 [2] -that we can ignore the scattering of the constituents, when the bound states scatter. (This means the constituents are in the approximation free, and thus the bound state not truly bound) Conclusion Details • RequiringHigh Symmetryin formof 3-transitive symmetry operationwe ex­pected -like Zassenhaus -the constituents to form a structure like a projective line F . {8}for a field F. The srting is the case F = Ri.e. the field is the real number field.(Topologically the projective line is a circle.) • We suggest that such string theory might be used when the approximation of many constituents with little momentum each becomes good. (of course string theory historically started as attempt to describe hadron physics [19]) • The p-adic theory[8]ofVeneziano modelis suggestively incorporated. Acknowledgement One of us (H.B.N.) acknowledges the Niels Bohr Institute for allowing him to work as emeritus and for partial economic support. Also thanks food etc. support from the Corfu conference and to Norma Mankoc Borstnik for asking for a way to get meaningful quantum field theoriesin higher than4dimensions. The thinking on hadronic like bound states could namelybe looked upon as an attempt to find such a scheme using bound states as the theory behind the particles for which to make the convergent theory. M. Ninomiya acknowledgesYukawa Instituteof Theoretical Physics, Kyoto Uni­versity, and also the Niels Bohr Institute and Niels Bohr International Academy for giving him very good hospitality during his stay. M.N. also acknowledges at Yuji Sugawara Lab. Science and Engeneering, Department of physics sciences Rit­sumeikan University, Kusatsu Campus for allowing him as a visiting Researcher. References 1. Yichiiro Nambu for earlir work Prog.Theoretical Phys. 5[4] 614 (1950) July-August. H. Bethe and E. Salpeter, “A relativistic Equation for bound state problem”, Phys. Rev. 84, 1232 (1951) 2. J. Bjorken “Inelastic Electron-Proton and .-Proton Scattering and the Structure of Nucleon, Phys.Rev.185 (1969) 3. H.B. Nielsen andM. Ninomiya,ANewTypeof String Field Theory,inProceedings of the 10thTohwa International Symposiumin String Theory, July 3-7, 20011 Fukuoka Japan AIP conference Proc vol. 607 p. 185-201; arXiv: hep-th 0111240.v1, Nov.2001 H. B. Nielsen and M. Ninomiya, An idea of New String Field Theory -Liberating Right and Left movers,inProceedingsofthe 14thWorkshop, “What Comes Beyond the StandardModels”Bled July 11-21, 2011, eds. N. M. Borstnik, H. B. Nielsen and D. Luckman arXiv 1112. 542 [hep-th] H. B. Nielsen and M. Ninomiya, “A Novel String Field Theory Solving String Theory by Liberating Left and Right Movers’, JHEP 202P 05131.v2 (2013) ’ 4. As for bosonc string field theory in the light-cone gauge: M. Kaku and K. Kikkawa, Phys. Rev.D10(1974)110; M. Kaku and K. Kikkawa, Phys. Rev. D10(1974)1110; M. Kaku and K. Kikkawa, Phys-Rev. D10(1974) 1823: S. Mandelstam, Nucl. Phys. B64 (1973)205; E. Cremmer and Gervais, Nucl. Phys., B90 (1975) 410 Witten type mid-point interaction of covariant string field theory for open string: E.Witten, Nucl. Phys. B268 (1986) 253. 5.R.GilesandC.B.Thorn,AlatticeAproachtoStringTheory,Phys.Rev.D16(1977)366 C.B. Thorn, On the Derivation of Dual Model from Field theory, Phys. Lett. 70B (1977) 85 C. B. Thorn, On the Derivation of Dual Models from Field Theory 2, Phys. Rev. D17 (1978) 1073 C. B. Thorn, Reformulating string theory with 1/N expansion, In Moscow 1991. Pro­ceedings, Sakharov Memorial Lecturein Physics,Vol1* 447 hep-th/9405069 O. Bergman, C. B. Thorn, String Bit Model for Superstring, Phys. Rev. D52 (1995) 5980 C. B. Thorn, Space from String Bits, JHEP11 (2014) 110 6. H. B. Nielsen (Bohr Inst.) and M. Ninomiya (Ritsumeikan U.) (2019) Contribution paper to 22ndWorkshop on What comes Beyond the StandardModelsp. 232-236 H. B. Nielsen(Bohr Inst.) and M. Ninomiya (Ocami, Osaka City U.), “Novel String Field theory with also Negative Energy Constituents/Objects givesVeneziano Amplitude”, JHEP 02(2018) 097, e-print: 1705.01739[hep-th] H.B. Nielsen andM.Ninomiya, “An Object Modelof String Field Theory and Deriva­tion of Veneziano Amplitude, published in Proc. of Corfu 2016 (2017) 134, arXiv 1705.01739. H. B. Nielsen and M. Ninomiya, “Instructive Review of Novel SFT with non-interacting constituents objects and the generalization to p-adic theory”, Corfu Summer Institue 2019 “School andWorkshops on Elementary Particle Physics and Gravity” (Corf 2019) 31. August -25. September Corfu Greece, arXiv 2006.09546. 7. E.g.: J.L. Gervais “Operator Expression for the Koba-Nielsen Multi-VenezianoFormula and Gauge Identities” Nuclear Physics B21 (1970) 192-204. Masatsugu Minami “Modular Group and Non-cyclic Symmetry of the Veneziano Formula”,Progressof Theoretical Physics,Vol.53,No.1,January 1975 8. PeterFreund andMOlson, “Non-archimedian strings”, Physics LettersB1992 (1987) I.Volovich, “p-adic space-time and string theory”, Math. Phys. 71 574 -576 (1987) P.Freund andE.Witten, “Adelic string Amplitudes”, Phys. LetterB199191 (1987) 9. “p-Adic, AdelicandZeta Strings” Branko Dragovich, Instituteof Physics,P.O.Box57, 11001 Belgrade, SERBIA 10. Holger B. Nielsen, Masao Ninomiya, “Dirac Sea for Bosons. I: -Formulation of Negative Energy Sea for Bosons”Progressof Theoretical Physics,Volume 113, Issue3, March 2005, Pages 603-624, https://doi.org/10.1143/PTP.113.603 Holger B. Nielsen, Masao Ninomiya, “Dirac Sea for Bosons. II: -Study of the Naive Vacuum Theory for theToy ModelWorld Prior to Filling the Negative Energy Sea” —Progressof Theoretical Physics,Volume 113, Issue3, March 2005, Pages 625-643, https://doi.org/10.1143/PTP.113.625 11. C. Jordan, Recherches sur les substitutions, J. Math. Pures Appl. (2) 17 (1872), 351-363 12. Marshall Hall “Ona theoremof Jordan”, https://projecteuclid.org download pdf1 euclid.pjm 1103044881 13. C. Jordan, Journal de Math´ees (1872),Volume: 17, page ematiques Pures et Appliqu ´351-367, ISSN: 0021-7874 14. S.Weinberg, Physica A96, 327 (1979), StevenWeinberg, PoS(Proceedingsof Science) (CD09)001 “Effective Field Theory, Past and Future”. 15. Norma Mankoc Borstnik, See e.g. (other) contribution(s) in the present Proceedings to the 24th workshop on “What comes beyaond the StandardModels” in Bled. 16. Streater,R.F.;Wightman,A.S. (1964). “PCT, Spin and Statistics, and All That.” New York:W.A. Benjamin. OCLC 930068; Bogoliubov,N.; Logunov,A.;Todorov,I. (1975). “Introductionto Axiomatic Quantum Field Theory.” Reading, Massachusetts:W.A. Benjamin. OCLC 1527225. Araki, H. (1999). “Mathematical Theory of Quantum Fields”. OxfordUniversity Press. ISBN 0-19-851773-4. 17. CambridgeTogetsin Mathematics Book24 (English Edition);they are the five sporadic simple groups called M11, M12, M23, and M24, which multiple transitive groups on 11, 12, 22, 23 arzt objects 18. Holger B. Nielsen, Masao Ninomiya,arXiv:1211.1454v3 [hep-th] 28 May 2013 OIQP-12­10 “A Novel String Field Theory Solving String Theory by Liberating Left and Right Movers” 19. Nambu,Y. (1970). ”Quark model and the factorizationof theVeneziano amplitude.” In R. Chand (ed.), Symmetries and Quark Models: Proceedings of the International Conference held atWayne State University, Detroit, Michigan,June 18–20, 1969 (pp. 269–277). Singapore:World Scientific. Nielsen,H.B.”An almostphysicalinterpretationofthedualNpoint function.”Nordita preprint (1969); unpublished. Mentioned in: XVth INTERNATIONAL CONFERENCE ON HIGH ENERGY PHYSICS, KIEV, 1970. Susskind, L (1969). ”Harmonic oscillator analogy for the Veneziano ampli­tude”. Physical Review Letters. 23 (10): 545–547. Bibcode:1969PhRvL..23..545S. doi:10.1103/physrevlett.23.545. Susskind,L(1970). ”Structureof hadrons impliedby duality”. Physical ReviewD.1(4): 1182–1186. Bibcode:1970PhRvD...1.1182S. doi:10.1103/physrevd.1.1182. Proceedings to the 24th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ... (p. 278) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 19 Atomic Size Dark Matter Pearls, Electron Signal H.B. Nielsen1.. email: hbech@nbi.dk C.D.Froggatt2 email: Colin.froggatt@glasgow.ac.uk 1 Niels Bohr Institute, Copenhagen, Denmark 2 Glasgow University, Glasgow, UK Abstract. We seek to explain both the seeming observation of dark matter by the seasonal variation of the DAMA-LIBRA data and the observation of “electron recoil” events at Xenon1T in which the liquid-Xe-scintillator was excited by electrons -in excess to the expected background -bythe same dark matter model. In our model the dark matterconsists ofbubblesofanewtypeof vacuumcontainingordinaryatomicmatter,saydiamond,under highpressure ensuredby the surface tensionof the separation surface (domain wall). This atomic matteris surroundedbya cloudof electrons extending almost outto atomic size.We also seek to explain the self interactions of dark matter suggested by astronomical studies of dwarf galaxies and the central structure of galaxy clusters. At the same time we consider the interaction with matter in the shielding responsible for slowing the dark matter down to a low terminal velocity, so that collisions with nuclei in the underground detectors have insufficient energy to be detected. Further we explain the “mysterious” X-ray line of 3.5 keV from our dark matter particles colliding with each other so thatthe surfaces/skins unite.Eventhe3.5keVX-ray radiationfromtheTycho supernovaremnantis explainedas our pearls hitting cosmic rays in the remnant. Whatthe DAMA-LIBRAand Xenon1T experimentsseeis supposedtobeourdark matter pearls excitedduringtheir stoppinginthe shieldingortheair.Themostremarkable support for our type of model is that both these underground experiments see events with about 3.5 keV energy, just the energy of the X-ray line. Wegetagood numerical understandingofthefittedcross section over massratioofself interacting dark matter observed in the study of dwarf galaxies. Also the total energy of the dark matter pearls stopped in the shield is reasonably matching order of magnitudewise with the absolute observation rates of DAMA-LIBRA and Xenon1T, although the proposed explanation of their ratio requires further development. It shouldbestressedthat acceptingthatthedifferentphasesofthe vacuumcouldberealized inside the StandardModel, our whole scheme could be realized inside the StandardModel. So then no new physics is needed for dark matter! Povzetek:Avtorja predlagata novo vrsto vakuuma, ki bi lahko pojasnila neujemanje med meritvama poiskusov DAMA-LIBRAin Xenon1T.Vnjunem modelu imajo lastnosti temne snovi mehur.cajno snov, recimo diamant,podvisokim tlakom, cki v vakuumu, ki vsebuje obi.kije posledica povr.cajno snovjoin oblakom elektronov, velikosti sinske napetosti med obi.atoma.S tem modelom razlo.zita, zakaj je interakcija med konstituenti temne snovi tako .. SpeakerattheWorkShop“What comesbeyondthe StandardModels”inBled. majhna, kako se temna snov upo.cajno snovjov .citu galaksije. casni pri interakciji z obi.s.Posledi. cno ima temna snov majhno hitrost pri trkih z jedri v podzemskih detektorjih v soglasjuzrezultati obehposkusov. Podporo svojemupredlogu vidita avtorja tudiv tem, da pojasni hkrati rentgensko . crtoz energijo3,5keVv sevanju supernoveTychoinda experimenta DAMA/LIBRA in Xenon1T izmeritatrke temne snovi prav pri energiji 3,5 KeV. Sipalna amplituda in gostota njune temne snovi se ujemata z izmerjenima, pri tem pa njun model ostaja znotraj standardnega modela. 19.1 Introduction Fora long time we have worked ona dark matter model[1–8],in which the dark matter consisted of cm-size pearls which were in fact bubbles of a new vacuum type surroundedbya skin causedby the surface tensionof this new vacuum. This skin kept a piece of usual atomic matter highly compressed inside the bubble. In fitting data with this model the most and almost only successful fit consisted in that we fitted, withacommon parameter,both the overall rate and the very 3.5 keV energyof theXray line originally observedin several galaxy clusters, Andromeda and the MilkyWay Center[9–14] and supposedly comingfrom dark matter. But nowit turned out that this successful fittingrelation between the 3.5 keV energy and the overall rate of the X-ray radiation only depends on the density of the pearls or equivalently the fermi momentum or energy of the electrons kept inside the pearls, but not on the absolute size of the pearls. Thus we could change the model to make the pearl sizes much smaller, as we shall do in this article, so that they are e.g. now rather of atomic size. Really we shall let the pearls be of radius rcloud 3.3MeV = 5 * 10-12m. But even such small pearls get stopped to some extent by the shielding into which they must penetrate to reach the underground experiments like the DAMA-LIBRA and Xenon experiments looking for dark matter. Using an astronomical observation based model by Correa [15] especially, we shall construct a rather definite picture of our pearls from which we estimate that the pearls hitting the earth actually get stopped presumably in the atmosphere, butifnottherethenatleastintheearth shielding.Thepearlstherebylosesomuch speed that it becomes quite understandable that the Xenon-experiments, looking for nuclei being hit by them and causing scintillation in fluid xenon, will not see any such events. However the DAMA-LIBRA experiment [16,17] would not distinguishifitisa nucleus thatis hit or some energyisreleased which causes the scintillator to luminesce. So only the DAMA-LIBRA experiment would be able to get a signal if the dark matter, e.g. our pearls, could be somehow excited and emit their excitation energy when they pass through the detector. In our model we shall indeed suggest that the pearls get excited and emit their energy by electron emission. That would not be easy to distinguish for DAMA-LIBRA but would still of course come with seasonal variation1 so that it would be observed as dark matterby DAMA-LIBRA. Whetherthe emissionisvia electronsor nuclei would not matter. But for the xenon-experiments such electron emission was effectively 1We note however that the ANAIS experiment has failed to see an annual modulation with NAI(Tl) scintillators and their results [18] are incompatible with the DAMA-LIBRA results at 3.3s. not counted fora long time, but now ratherrecently the Xenon1T experiment has actually observed an excess of “electron recoil events”. So they have now in fact seen an electron emission somehow. Weshallseeinsection19.7thatboththe excessofelectronrecoileventsinXenon1T [19]andthe events seenby DAMA-LIBRA[16,17]havethe energyofeach event remarkably enough centering aboutthe energy value 3.5 keV of the mysterious X-ray line found astronomically! This coincidence of course strongly suggests that these events from DAMA-LIBRA and Xenon1T are related to dark matter particles that can be excited precisely by this energy 3.5 keV. In our earlierpapers[5–7] we have already connectedthe excitabilityof our pearls by just this energy 3.5 keV and especially the emission of photons (or here in the presentworkalso electrons)withjustthis energywithagapinthesingle particle electron spectrum of the pearls caused by what we call the homolumo-gap effect. Avery serious warning, which needs an explanation in order to rescue our model, is delivered by the fact that if as we now suggest the Xenon1T electron recoil event excess is coming from just the same decay of dark matter excitations as the DAMA-Libra observation, then these two experiments ought a priori to see equally many events, say per kg. However, DAMA-LIBRA sees 250 times as many events as Xenon1T sees excess events. We shall postpone this question to a later article in detail, but the hope for now is that the Xenon1T experiment has the observed decaying pearls falling through a fluid, namely the fluid xenon, while the scintillatorin DAMA-LIBRAisa solid made from NaI(Tl). The pearls are likely to form a little Xe-fluid bubble around them and flow or fall through the xenon-fluid, while they will much more easily get caught so as to almost sit still or only move much slower through the NaI scintillator. If so the pearls with their supposed excitations would spend much more time in the DAMA-LIBRA NaI than in a corresponding volume of xenon-liquid. In the following section 19.2 we describe how the particles making up the dark matter in our model are imagined to be bubbles of the size R = rcloud 3.3MeV = 5 * 10-12m withheavy atomic matter inside, whichis surroundedouttoa radius rcloud 3.5keV = 5 * 10-11m by electrons. Here the quantities 3.3 MeV and 3.5 keV in the subscripts are the numerical electric potentials felt by an electron at the distances mentioned.Aspecialpointtonoteinthis sectionalreadypresentinthe earlier articles about the big pearls is the homolumo-gap effect, causing a band or gap in the energy levels without any single particle electron eigenstates. The width of this gap is fitted to the 3.5 keV line in the observed X-ray spectrum from galaxy clusters, the MilkyWay Center etc. [9–11]. Next in section 19.3 we briefly review astronomical observations and modelling of the dark matter, which suggests the idea that dark matter interacts with itself (strongly interacting dark matter SIDM). It is only when the corresponding cross section s is dividedby the particle mass M do we havea combination that has any s chance of being observed by its effects on the atomic matter. In fact this ratio M matches well with the atomic physics structure of our pearls including the cloud of electrons outside the bubble itself. In section 19.4 we list a series of numerical successes of our model for the dark matter, hopefullymakingthereaderseethatthereisreally somereasonforitbeing at least in some respects correct. In section 19.5 we restress that our dark matter pearls get stopped and at the same time excited, mainly to emit quanta of energy 3.5 keV, in the air and/or in the shielding above the experiments. According to our best estimates they get stopped already about 53 km up in the air. It is the braking energy from this slowing down that is supposed to feed the excitations. Aspecial estimation, based on energy considerations, of whether the number of events seenby DAMA-LIBRA andbythe Xenon1T electronrecoil excess areofa reasonable order of magnitude is put forwardin section 19.6. The success of such an estimation has to be rather limited in as far as the rates of the two observations -that should have been the same if we do not include the possibility of faster or slower motion through the detectors -deviateby a factorof 250. In section 19.7 we call attention to the perhaps most remarkable fact supporting a major aspect of our model: That the energy per event for both DAMA-LIBRA and the Xenon1T-electronrecoil excess centers around 3.5 keV, just the energyof the photonsin the mysterious X-rayline seenin galactic clusters mentioned above!So all three effects should correspond to the emission of an electron or photon due to the same energy transition inside dark matter. Finally in section 19.8 we conclude and provide a short outlook. 19.2 Pearl Dark Matter Atomic Size Pearls, Electronic 3.5 keV Signal We sketch the structure of our small dark matter pearls in Figure 1. • In the middle is a spherical bubble of radius R ˜ rcloud 3.3MeV ˜ 5 * 10-12 m. (19.1) Here rcloud 3.3MeV denotes the radius where the electron potential is 3.3 MeV, which is identified with the Fermi energy Ef of the electrons in the bulk of the pearl -i.e. inside the radius R.We estimated the value Ef = 3.3 MeV in previous papers[6–8]by fitting the overall rateof the intensityof the 3.5 keV line emittedbygalactic clustersandtheveryfrequency3.5keVofthe radiation in our model. • The outer radius rcloud 3.5keV ˜ 5 * 10-11 m (19.2) is where the electron potentialis 3.5 keV.By our storyof the “homolumo gap”: the electron density crudely goes to zero at this radius. (It gradually falls in the range between rcloud 3.3MeV and rcloud 3.5keV ). The electron density and potential in the pearls • Due to an effect, we call the homolumo-gap effect [5,20], the nuclei in the bubble region and the electrons themselves become arranged in such a way as to prevent there from being any levels in an interval of width 3.5 keV. So, as illustrated in Figure 19.2, outside the distance r3.5keV = rcloud 3.5keV from the center of the pearl at which the Coulomb potential is ~ 3.5 keV deep there are essentially(~ in the Thomas-Fermi approximation) no more electrons in the pearl-object. • The radius r3.3MeV = rcloud 3.3MeV at which the potential felt by an electron is 3.3 MeV deep, is supposed to be just the radius to which the many nuclei inside the pearl (whichreplace the single nucleusin ordinary atoms)reach out. So inside the bubble the potential is much more flat. • The energy difference between the zero energy line and the effective Fermi surface, abovewhichtherearenomoreelectrons,isoforder3.5keV,the energy so crucial in our work. • Since in the Thomas-Fermi approximation there are no electrons outside roughly the radius r3.5keV = rcloud 3.5keV , this radius will give the maxi­mal cross section, even for very low velocity sv.0. The homolumo gap effect. Letus considerthespectrumofenergylevelsfortheelectronsinapieceofmaterial, e.g. one of our pearls, and at first assume that the positions or distributions of the charged particles in the material are fixed. Then the ground state is just a state built e.g. as a Slater determinant for the electrons being in the lowest single electron states, so many as are needed to have the right number of electrons. But now,if the charged particles can be moved due to their interactions, the ground state energy could be lowered by moving them so that the filled electron state levels get lowered. So we expect introducing sucha “backreaction” will lower the filled states. When the filled levels get moved downwards, then the homo = “highest occupied molecular orbit”levelwillbeloweredandits distancetothenextlevel,thelumo (= lowest unoccupied molecular orbit), will appear extended on the energy axis. We believe that we can estimate the homolumo-gapEH. Using the Thomas-Fermi approximation -or crudely just some dimensional ar­gument where the fine structure constant has the dimension of velocity -we calculated the homolumo gap in highly compressed ordinary matter for relativis­tic electrons: EH a v )3/2 ~ ( 2pf (19.3) c where pf = Fermi momentum (19.4) a c = 1 137.03... (19.5) v (the 2 comes from our Thomas-Fermi calculation). It is by requiring this homolumo-gap to be the 3.5 keV energy of the X-ray line mysteriously observed by satellites from clusters of galaxies, Andromeda and the MilkyWay Centerthatwe estimatethe Fermi-energytobe Ef ˜ pf = 3.3 MeV in the interior bulk of the pearl. Brief summary of theoretical ideas underlying our dark matter pearls • Principle Nothing but StandardModel! (Seriously it would mean not in a BSM-workshop.) • New Assumption Several PhasesofVacuum with Same Energy Density; this is the so-called Multiple Point Principle [5,6,21–26]. • CentralPart BubbleofNewPhaseofVacuumwithe.g. carbonunderveryhigh pressure, surrounded by a surface with tension S (= domain wall) providing the pressure. • Outer part Cloud of Electrons much like an ordinary atom having a nucleus withachargeofordertentoa hundred thousand(Z ˜ 5 * 104 effectively). 19.3 Non-gravitational Interactions The collisionless cold dark matter model provides a good description of the large scale structure of the Universe. However there are various problems at small scales [27,28]for the hypothesis that dark matter only has gravitational interactions. Originally Spergel et al [29] suggested that the lack of a peak or cusp in the center of galaxy clusters, as expected for cold dark matter with purely gravitational interactions, required self interacting dark matter with a relatively large cross s section. Therelevantparameterisin fact the cross section per mass and for the M s cores in galaxy clusters, where the collision velocity is v ~ 1000 km/s, a value ~ M 0.1 cm 2/g is needed. The self interaction can of course be velocity dependent and s the cores in spiral galaxies where v ~ 100 km/s require ~ 1cm2/g. In dwarf M galaxies around our MilkyWay, where darkmatter moves more slowly v ~ 30 s km/s, larger cross section to mass ratios ~ 50 cm2/g are needed. M Recently Correa [15] made a study of the velocity dependence of self interacting dark matter. In particular she analysed the MilkyWay dwarf galaxies and her results are displayed in Figure 19.5. The extrapolation of Correa’s fit to the data s towards zero velocity points to the ratio . 150 cm2/g. This ratio can be taken M as an experimental estimate of the impact area over the mass as seen for very soft collisions. In our model the cross section in this low velocity limit is given by the extent out to which the electrons surrounding our pearls reach. This range of extension of electrons is supposed to be given by the requirement that the electron binding energy is of the order of the homolumo gap value 3.5 keV. So we denote this radius by rcloud 3.5keV . Similarly the radius of the bubble containing the nucleons inside our dark matter pearl corresponds toa radius rcloud 3.3MeV atwhichthe potentialforthe electronis-3.3MeV(=Fermi energyofthe electrons). Thehigh velocityhardcollisionsof our pearls, supposedtoresultinthe unification of two pearls into a single pearl, correspond to interactions between the bubble skins with a cross section of order pr2 cloud 3.3MeV . We will now consider the electric potential for our pearl using the Thomas-Fermi approximation for a heavy atom [30–32]. In this approximation the Coulomb potentialof the “nuclear” chargeZis multipliedby the Thomas-Fermi screening function .(r/b)where b = 0.88 a0 (19.6) Z1/3 and a0 is the Bohr radius. The skin of the bubble or “nucleus” of the pearl mainly acts on the nucleons or rather nuclei. So the electrons spread out and an appreciable part, say half of them, are outside the central part of the pearl inside the skin. Therefore the effective charge Z of the central part of the pearl or bubble of the new phase is e.g. one half of the number of protons inside the skin. Assuming also s M function of the collision velocity v in dwarf galaxies from reference [15]. that there are about equally many neutrons and protons inside the central part, the mass of the pearl is then given order of magnitudewise by M = 4mN * Z, where mN is the nucleon mass. In the Thomas-Fermi approach we are then led to the following equations for rcloud 3.5keV and rcloud 3.3MeV : a * Z * .(rcloud 3.5keV /b)= 3.5 keV (19.7) rcloud 3.5keV a * Z * .(rcloud 3.3MeV /b)= 3.3 MeV (19.8) rcloud 3.3MeV b = 0.88 * a0 (19.9) Z1/3 (19.10) We identifyrcloud 3.5keV with the radius of the electron cloud and rcloud 3.3MeV with the skin radius R of the pearl. It is going to be an important success of our model that we get a similar value for R ˜ rcloud 3.3MeV using another method to calculate it.We shall use s |v.0 = 150 cm2/g M (19.11) and 2 s = p * r cloud 3.5keV (19.12) to determine the mass M. Then using the formula for the mass of a pearl in terms of the radius R and the Fermi momentum [7,8] M = 8 * (R * pf)3 , (19.13) mN 9p we can calculate another value for R. In our updated contribution to the Bled Proceedings from last year [8] we estimated a pearl mass of M ~ 105 GeV. So we take here Z = 5.3 * 104 asa typical chargein the central part of the pearl, for which then b = 1.24 * 10-12m.Using numerical values for the Thomas-Fermi screening function in the paper [33], we obtain from (19.7) the radius of the electron cloud to be rcloud 3.5keV = 4.96 * 10-11 m (19.14) s Then assuming the low velocity ratio = 150 cm2/g we obtain M p * (4.96 * 10-11m)2 M = (19.15) 150 cm2/g = 5.2 * 10-19 g (19.16) = 3.1 * 105 mN (19.17) Asa sideremark notice that, using ourproposedruleof taking Z to be a quarter of the number M/mN, we would get Z = 8 * 104 to be compared with our input here 5.3 * 104, which is very well consistent within a factor 2. -1 Next using (19.13) with pf = 3.3 MeV = 1.6 * 1013 m (R * pf)3 = 3.1 * 105 * 9p (19.18) 8 = 10.9 * 105 (19.19) v -1 R * 1.6 * 1013 m = 3 10.9 * 105 = 102 (19.20) giving R = 102 -1 1.6 * 1013m (19.21) = 6.4 * 10-12 m. (19.22) This is to be compared with the Thomas-Fermi value obtained from (19.8) using the numerical values for .(r/b)in [33] R = rcloud 3.3MeV = 3.66 * 10-12 m. (19.23) These two different estimates of the radius rcloud 3.3MeV at which the potential is 3.3MeV essentially coincidetothe accuracyofour calculation;they deviatebya factor of order unity 6.4/3.7 =1.7. So we could claim that formally our model is able s to predict the low velocity limit |v.0 in agreement with the value 150 cm2/g M estimatedfromthestudyof dwarf galaxiesaroundtheMilkyWay. We shall take the average of the two values (19.22) and (19.23) as our best estimate of the bubble skin radius: rcloud 3.3MeV = 5.0 * 10-12 m (19.24) and from (19.14) we have the radius of the electron cloud rcloud 3.5keV = 5.0 * 10-11 m. (19.25) We note that these two radii differ byan order of magnitude, which means that s the quantity for our pearls should differ by two orders of magnitude between M low velocities and high velocities, as astronomical observations indicate is the case for self interacting dark matter [15]. 19.4 Achievements s • Low velocity |v.0 cross section to mass ratio. The a priori story, that dark M matter has only gravitational interactions seems not to work perfectly: Espe­cially in dwarf galaxies (around our MilkyWay) where dark matter moves relatively slowly an appreciable self interaction cross section to mass ratio s M is needed. According to the fits in [15] this ratio has the low velocity limit s |v.0 = 150 cm2/g.We may say our pearl-model “predicts” this ratio in M order of magnitude. • Can make the Dark Matter Underground Searches get Electron Recoil Events Most underground experiments are designed to look for dark matter particles hitting the nuclei in the experimental apparatus, which is then scintillating so that such hits presumed to be on nuclei can be seen. But our pearls are excitedin sucha way that they send out energetic electrons (rather than nuclei) and this does not match with what is looked for, except in the DAMA-LIBRA experiment. In this experiment the only signal for events coming from dark matter is a seasonal variation due to the Earth running towards or away from the dark matter flow. • The Intensity of 3.5 keV X-rays from Clusters etc. We fit the very photon-energy 3.5 keV and the overall intensity from a series of clusters, a galaxy, and fS the MilkyWay Center[8] with one parameter .1/4 = 0.6 MeV-1 . .V • 3.5 keV Radiation from theTycho Supernova Remnant. Jeltema and Profumo [34] discoveredthe3.5KeVX-ray radiationcomingfromtheremnantofTycho Brahe’s supernova, which was unexpected for sucha small source.We have a scenario givingthe correct order of magnitude for the observed intensity in our pearl model: supposedly our pearls are getting excited by the high intensity of cosmic rays in the supernova remnant [8]. fS 2 Even though we can use only the one parameter .1/4 = , it is nice to know .V pf the notation: .V = “ difference in potential for a nucleon between the inside and the outside of the central part of the pearl” ˜ 2.5 MeV (19.26) .fS = R Rcrit estimated to be ˜ 5 (19.27) where R = “actual radius of the new vacuum part” ˜ rcloud 3.3MeV (19.28) and Rcrit = “ Radius when pressure is so high that nucleons are just about being spit out” (19.29) The subscript fS on the parameter .fS indicates that the surface tension S is fixed independent of the radius R. • DAMA-rate Estimating observation rate of DAMA-LIBRA from kinetic energy of the incoming dark matter as known from astronomy. • Xenon1T Electron recoil rate Same for the electronrecoil excess observed by the Xenon1T experiment. In order to explain these last calculational estimates it is necessary to know how we imagine the dark matter to interact and get slowed down in the air and the earth shielding; also how the dark matter particles get excited and emit 3.5 keV radiation or electrons. About the Xenon1T and DAMA-rates: • Absolute rates very crudely Our estimate of the absolute rates for the two experiments are very very crude, because we assume that the dark matter particles -in our model small macroscopic systems with ten thousands of nuclei inside them -can have an exceedingly smooth distribution of lifetimes on a logarithmic scale. These calculations are discussed in section 19.6. • The ratio of rates The ratio of the rates in the two experiments -Xenon1T electron recoil excess and DAMA -should in principle be very accurately predicted in our model, because they are supposed to see exactly the same effect just in two different detectors in the same underground laboratory below the Gran Sasso mountain! One would therefore expect the rates to be the same, but the Xenon1T rate is 250 times smaller than the DAMA rate.We briefly refer to a possible resolution of this problem, which needs further study, in section 19.6. 19.5 Impact Illustration of Interacting and Excitable Dark Matter Pearls The dark matter pearls come in with high speed (galactic velocity), but get stopped downtomuchlowerspeedbyinteractionwiththeairandthe shielding mountains, whereby they also get excited to emit 3.5 keV X-rays or electrons. Pearls Stopping and getting Excited in Earth Shield What happens when the dark matter pearls in our model of less than atomic size hit the earth shielding above the experimental halls of e.g. DAMA? • StoppingTaking it that the pearls stop in the earth: The pearls are stopped in about 5 * 10-6s from their galactic speed of about 300 km/s down to a speed 49 km/s below which collisions with nuclei can no longer excite the 1 3.5 keV excitations. The stopping length, modulo a logarithmic factor, is m. 4 But takingit that they stopin the air, whichis more likely:They are stopped overa rangeof about7km -as the atmosphere density goesup witha factor e = 2.71.. over such a range in about 2 * 10-2s. • Excitation As long as the velocity is still over the ca. 49 km/s collisions with nuclei in the shielding can excite the electrons inside the pearl by 3.5 kev or moreandmakepairsofquasi electronsandholessay.Weexpectthatoftenthe creation of (as well as the decay of) such excitations require electrons to pass through a (quantum) tunnel and that consequently there will be decay half livesofverydifferent sizes.Wehope evenuptomany hoursor days... 1 • Slowly sinking: After being stopped in of the order of m of the shielding, 4 the pearls continue with a much lower velocity driven by the gravitational attraction of the Earth. After say about 26 hours a pearl reaches the 1400 m down to the laboratories. Most of the pearls have returned to their ground states, but some exceptionally long living excitations survive. Note that the slowly sinking velocity is so low that collisions with nuclei cannot give such nuclei enough speed to excite the scintillation counters neither in DAMA nor in Xenon-experiments. • Electron or . emissionTypicallythedecayofan excitationcouldbethatahole in the Fermi sea of the electron cloud of the pearl gets filled by an outside electron under emission of another electron by an Auger-effect. The electron must tunnelinto the pearl center. This can make the decaylifetime become very long and very different from case to case. Emission as electrons or photons makes Xenon-experiments not see events, except... That the decay energy is released most often as electron energy means that such events are discarded by most of the Xenon-experiments, which only expect the nucleus recoils to be darkmatter events. This would explain the long standing controversy consistingin DAMA seeing dark matter witha much bigger rate than the upper limits from the other experiments. Rather recently though Xenon1T looked for potential excess events among the electron recoil events and found 16 events/year/tonne/keV in the lowest keV-bands over a background of the order of (76 ± 2)events/year/tonne/keV. In our model this rate should be compatible with the DAMA event rate. However they deviatebya factorof250.Ittherefore appearsthatweneedthepearlstorun much faster through the xenon-apparatus than through the DAMA one. 19.6 Numerical Rates for DAMA and Xenon1T-electron-recoil-excess 19.6.1 The Kinetic Energy Flux from Dark Matter The dark matter density Dsol in our partof the MilkyWay and its velocity v are of the orders of magnitude Dsol = 0.3 GeV/cm3 (19.30) = 5.34 * 10-22kg/m3 (19.31) v = 300 km/s (relative to solar system) (19.32) 2 Dkin energy = 1v Dsol (19.33) 2 = 0.5 * (10-3)2 c 2 * 5.34 * 10-22kg/m3 (19.34) = 2.40 * 10-11J/m3 (19.35) meaning an influx of kinetic energy “power per m2” = vDkin energy (19.36) 3 = 1v Dsol (19.37) 2 = 3 * 105m/s * 2.40 * 10-11J/m3 (19.38) = 7.2 * 10-6W/m2 (19.39) Distributing this energy rate over the amount of matter down to the depth 1400 m with density 3000 kg/m3 we obtain the energy rate per kg 7.2 * 10-6W/m2 “ power to deposit” = (19.40) 1400 m * 3000 kg/m3 = 1.7 * 10-12W/kg. (19.41) However, assuming that all the events from the dark matter -as given by the modulated part of the signal found by DAMA-LIBRA -are just due to decays with the decay energy 3.5 keV, the rate of energy deposition per kg observed by DAMA-LIBRA [17] is “deposited rate ” = 0.0412 cpd/kg * 3.5 keV (19.42) 0.0412 cpd/kg * 3.5 * 1.6 * 10-16J = (19.43) 86400 s/day = 2.7 * 10-22W/kg, (19.44) which is 2.7 * 10-22W/m2 = 1.6 * 10-10 times as much. (19.45) 1.7 * 10-12W/m2 Wecanexpressthisbysayingthatthereisaneedforasuppression factorsuppression being 1.6 * 10-10 for the DAMA-LIBRA rate. For the excess of the electronic recoil events found by Xenon1T the corresponding suppression factor must be the 250 times smaller number. This is because the event rate of these excess electron recoil events is 250 times smaller than that of the modulation part of the DAMA rate and the depth of the experiment under the earth is the same 1400 m. Thus we summarize the experimentally determined suppression factors: suppressionDAMA = 1.6 * 10-10 (19.46) 1.6 * 10-10 suppressionXenon1T == 6.4 * 10-13 . (19.47) 250 19.6.2 Estimating “suppression” theoretically The idea for obtaining theoretical estimates of these suppression factors is to say that the observed events comefrom excitationsof our pearls witha lifetimeof the order of the time it takes for the pearl, after its excitation under its stopping in the air or in the stone above the experiments, to reach down to the experimental detectors.Wehere assumethe scatteringcross sectionofdark matteronordinary matter to be similar to that on dark matter. So we estimate the passage time of the pearl down to the detectors as being of the order of 26 hours, by using the low velocity value for the cross section over mass ratio To be used for passage velocity: s = 150 cm2/g (19.48) M Once the pearl has been stopped so much that its velocity is only upheld by the gravitational field with the acceleration g = 9.8 m/s2, the terminal velocity will be obtained formally from the drag-equation2 Drag force D = gM = 0.5 * Cd * s * .v2 . (19.49) Here . is the density of the material passed through and the drag coefficient Cd is of order unity (so the 0.5 is hardly relevant). That is to say the terminal velocity becomes: g vterminal ˜ (19.50) s * . M 9.8 m/s2 ˜ (19.51) 150 cm2/g * 3 g/cm3 = 2.2 cm2/s2 = 1.5 cm/s, (19.52) which allowsapearltopassthough1400min 140000 cm “passage time” = (19.53) 1.5 cm/s = 93000 s = 26 hours. (19.54) 19.6.3 Equally hard to excite and to de-excite In order that there can be any de-excitations of the pearls after such26 hours it is of course needed that an appreciable part of the possible 3.5 keV excitations of our pearls have lifetimesof this orderof magnitude.Apriori these excitations are excitons for which the electron and hole can be close by and decay rapidly or it is possible that one of the partners is outside in the electron cloud and long lived. By arguing that some tunnelling of electrons in or out or around in the pearl may be needed for some (de-)excitations, we can claim that the lifetimes for 2 Strictly speaking this equationis only valid if the pearl velocity is greater than the thermal velocity of the nuclei in the shielding and so needs further study. the various excitation possibilities are smoothly distributed over a wide range in the logarithm of the lifetime; then there will be some pearl-excitations with the appropriate lifetime, although somewhat suppressed by a factor of the order of 1/width where the width here is the width of the logarithmic distribution. We shall take this width to be of order ln suppressionDAMA ~ 23. But more importantly: If a certain excitation is long-lived, it is also hardto produce. So we shall talk about aneffective “ stopping” or “filling time” fora pearl passing into the Earth, and imagine that during this “stopping” or “filling time” the excitations of the pearls have to be created. So the probability for excitation or suppression would be expected to be “filling time” suppression ˜ . (19.55) “lifetime” If the excitation happens to be of sufficiently long lifetime -say of order 26 hours ­then we can expect it to have a sensible chance of de-exciting just in the experi­mental detectors in Gran Sasso, DAMA or Xenon1T say. Butwhatshallwetakeforthis “stopping”or “filling time”?A relativelysimple idea, which is presumably right, is to say that the stopping takes place high in the atmosphere because a pearl entering the Earth’s atmosphere with galactic speed s will be slowed down in the high air with a ~ 2 cm2/g. Now the density of the M atmosphere rises by a factor e = 2.718... per about7km.So as the slowing down beginsit will, becauseof this rising density, essentially stop again after7km. Thus the time during which the pearl is truly slowing down in speed and forming 3.5 keV excitationsisoftheorderofthetimeit takesforittorun7km.Withthepearl velocity of about 300 km/s (essentially the escape velocity for the galaxy) we then have “stopping time” ˜ 7 km 300 km/s (19.56) = 0.023 s (19.57) The crucial factor, which we believe to be most important, is that in order to excite an excitation witha lifetimeof the order 93000sit woulda priori need 93000s so that,ifweonlyhave 0.023s,thentherewillbeasuppression: “stopping time” suppression = (19.58) “lifetime” “stopping time” ˜ (19.59) “passage time” 0.023 s ˜ (19.60) 93000 s = 2.5 * 10-7 . (19.61) given above suppressionDAMA = 1.6 * 10-10 (19.62) suppressiontheory 2.5 * 10-7 . = suppressionDAMA 1.6 * 10-10 (19.63) = 1.6 * 103 (19.64) 1.6 * 10-10 suppressionXenon1T = = 6.4 * 10-13 250 (19.65) suppressiontheory 2.5 * 10-7 . = suppressionXenon1T 6.4 * 10-13 (19.66) = 3.9 * 105 . (19.67) But here can be several corrections to suppressiontheory, at least we should correct by the width in logarithm of the supposed distribution of the lifetimes among the different excitations. Above we suggested a factor 23, which would bring the DAMA rate to only deviate by about a factor 100. Our estimate is of course extremely uncertain. We can never get the DAMA rate and the electron recoil excess rate from Xenon1T agree with the same estimate in as far as they deviate by a factor 250. Our only chance is in a later paper to justify say the story that, because the scintillator in which the Xenon1T events are observed is fluid while the NaI in the Dama experiment is solid, the pearls pass much faster through the Xenon1T apparatus than they pass through the DAMA instrument. Imagine say that the pearls partly hang and get stuck in the DAMA experiment, but that they cannot avoid flowing down all the time while they are in the fluid Xe in the Xenon1T scintillator. 19.7 3.5keV Orderof magnitudewise we see 3.5 keVin 3different places. The energy level difference of about 3.5 keV occurring in3 different places is important evidence motivating our model of dark matter particles being excitable by 3.5 keV: • The line From places in outer space with a lot of dark matter, galaxy clusters, Andromeda and the MilkyWay Center, an unexpected X-ray line with photon energies of 3.5 keV (to be corrected for Hubble expansion...) was seen. • Xenon1T The Dark matter experiment Xenon1T did not find the standard nuclei-recoil dark matter, but found an excess of electron-recoil events with energies concentrated crudely around 3.5 keV. • DAMA The seasonally varying component of their events lie in energy be­tween2keV and6keV, not farfrom centering around 3.5 keV. We take it seriously and not as an accident that both DAMA and Xenon1T see events with energies of the order of the controversial astronomical 3.5 keV X-ray line.We are thereby driven towards the hypothesis that the energies for the events in these underground experiments are determined from a decay of an excited particle, rather thanfroma collision witha particlein the scintillator material.It would namely be a pure accident, if a collision energy should just coincide with the dark matter excitation energy observed astronomically. So we ought to have decays rather than collisions! How then can the dark matter particles get excited? You can think of the dark matter pearls in our model hitting electrons and/or nuclei on their way into the shielding: • Electrons Electrons moving with the speed of the dark matter of the order of 300 km/s towardthe pearls in the pearl frame will have kinetic energy of the order 1 300 km/s Ee ˜* 0.5 MeV * ()2 = 0.25 eV. (19.68) 23 * 105km/s • Nuclei If the nuclei are say Si, the energy in the collision will be 28*1900 times larger ~ 5 * 104 * 0.25 eV ˜ 10 keV. That would allow a 3.5 keV excitation. To deliver such˜ 10 keV energy the nucleus should hit something harder than just an electron inside the pearls. It should preferably hit a nucleus, e.g. C, inside the pearl. 19.8 Conclusion • We have described a seemingly viable model for dark matter consisting of atomic size but macroscopic pearls. These pearls consist of a bubble of a new speculated type of vacuum containing some normal material -presumably carbon -under the high pressure of the skin (surface tension). They each contain about a hundred thousand nucleons in the bubble of radius about 5 * 10-12m. • The electronsina pearl have partly been pushed outof the genuine bubbleof thenew vacuumphase,outtoa distanceofabout 5 * 10-11m. We have compared the model or attempted to fit: • Astronomical suggestions for the self interaction of dark matter in addition to pure gravity. • The astronomical 3.5 keV X-ray emission line foundby satellites, supposedly from dark matter. • The underground dark matter searches. We list below the quantities we have crudely estimated: 1. The low velocity cross section divided by mass. 2. That the signal from Xenon1T and Dama should agree except that the pearls may run with different velocities through the scintillator materials, because the xenon-instruments use fluid xenon, while the DAMA-LIBRA experiment uses the solid NaI. 3. The absolute rate of the two underground experiments. (But unfortunately unless we explain the ratio of the rates for the two experiments as say due tothedifferent velocitiesthroughthe scintillator materials,wecanof course never predict the absolute rate to be better than deviating by about a factor of 250 with at least one of them.) 4. The rate of emission of the 3.5 keV X-ray line from the Tycho supernova remnant due to the excitation of our pearls by cosmic rays [8]. 5. Relation between the frequency 3.5 keV and the overall emission rate of this X-ray line observed from galaxy clusters etc. 6.Wealsopreviouslypredictedtheratioofdark mattertoatomicmatter (=“usual” matter)inthe Universetobeoforder5byconsiderationofthe binding energies per nucleon in helium and heavier nuclei, assuming that the atomic matter at sometime about1 s aftertheBigBangwasspitoutfromthepearls undera fusion explosionfromHefusingintosayC[1]. 19.8.1 Outlook At the end we want to mention a few ideas which we hope will be developed as a continuation of the present model: • Speculative Phase from QCD. QCD and even more QCD with Nambu-Jona-Lasinio type spontaneous symmetry breaking is sufficiently complicated, that possibly a new phase appropriate for our pearls could be hiding there. There is already an extremely interesting observation [35]. • Relative Rates of DAMA and Xenon1T. Acrucial test for our model is to reproduce the relative event rates in DAMA and in the excess of electron recoils in Xenon1T. This requires a careful study of the viscosity of fluid xenon and the properties of our pearls. • Walls in the Cosmos. With the usual expectations for the density per area or equivalently tension S, cosmology would be so severely changed by such domain walls that models with say S1/3 = 10 MeV arephenomenologically not tenable. However, with our fit to a surprisingly small tension with S1/3 ˜ 2.2 MeV it just barely becomes possible to have astronomically extended domain walls, e.g. walls around the large voids between the bands of galaxies; so that these voids could be say formally huge dark matter pearls, though with much smaller density. In fact a series of domain walls with our fitted S = 2.23 MeV3 with distances between them of the order of 13 milliardlight years would have a density not much different for that of the universe we know. • New Experiments? According to our estimates the observed rate of decays of our dark matter pearls should be larger the less shielding they pass through. So an obvious test of our model would be to make a DAMA-like experiment closer to the earth surface where we would expect a larger absolute rate than in DAMA, although there might of course be more background. Actually such an experiment is already being performed by the ANAIS group [18], but they have so far failed to see an annual modulation in their event rate. Acknowledgement HBN thanks the Niels Bohr Institute for his stay there as emeritus. CDF thanks Glasgow University and the Niels Bohr Institute for hospitality and support. Also we want to thank many colleagues for discussions and for organizing conferences, where we have discussed previous versions of the present model, especially the Bled and Corfu meetings. 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. Froggatt and H. B. Nielsen, Mod. Phys. Lett. A30 no.36, 1550195 (2015) [arXiv:1503.01089]. 5. H.B. Nielsen, C.D. Froggatt and D. Jurman, PoS(CORFU2017)075. 6. H.B. Nielsen and C.D. Froggatt, PoS(CORFU2019)049. 7. C. D. Froggatt, H. B. Nielsen, “The 3.5 keV line from non-perturbative StandardModel dark matter balls”, arXiv:2003.05018. 8. H. B. Nielsen (speaker) and C.D. Froggatt, “Dark Matter Macroscopic Pearls, 3.5 keV -ray Line, How Big?”, arXiv:2012.00445. 9. E. Bulbul, M. 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Nielsen, NBI-HE-94-44, GUTPA-94-09-3, Pre­sented at Conference: C94-07-20 (ICHEP 1994), p.0557-560. 22. D. L. Bennett, C. D. Froggatt and H. B. Nielsen, NBI-95-15, GUTPA-95-04-1, Presented at Conference: C94-09-13 (Adriatic Meeting 1994), p.0255-279 [arXiv:hep-ph/9504294]. 23. D.L. Bennett andH.B.Nielsen, Int.J. Mod. Phys. A9 5155 (1994). 24. D. L. Bennett, C. D. Froggatt and H. B. Nielsen, NBI-HE-95-07, Presented at Conference: C94-08-30 (Wendisch-Rietz) p.394-412. 25. C. D. Froggatt and H. B. Nielsen, Phys. Lett. B368 96 (1996) [arXiv:hep-ph/9511371]. 26. H.B. Nielsen (Speaker)and C.D.Froggatt,Presentedat Conference: C95-09-03.1 (Corfu 1995); arXiv:hep-ph/9607375. 27. D.H.Weinbergetal, PNAS 112, 12249 (2015) [arXiv:1306.0913]. 28. SeanTulin and Hai-BoYu, Phys. Rep. 730,1(2018) [arXiv:1705.02358] 29. D. N. Spergel, and P. J. Steinhardt, Phys. Rev. Lett. 84, 3760 (2000) [arXiv:astro­ph/9909386] 30. L.H. Thomas, Proc. Cambridge Philos. Soc. 23, 542 (1927). 31. E. Fermi, Rend. Accad. Naz. Lincei 6, 602 (1927). 32. L.Spruch, Rev. Mod.Phys. 63, 153 (1991). 33.KParandetal Electronic JournalofDifferential Equations,Vol. 2016(2016),No.331,pp. 1-18. 34. T. Jeltema and S. Profumo, MNRAS 450, 2143 (2015) [arXiv:1408.1699] 35. Andrei Kryjevski, David B. Kaplan, Thomas Schaefer, Phys. Rev. D71, 034004 (2005) [arXiv:hep-ph/0404290] Proceedings to the 24th[Virtual] BLED WORKSHOPS Workshop IN PHYSICS What Comes Beyond ... (p. 300) VOL. 22, NO.1 Bled, Slovenia, July 5–11, 2021 20 Galactic model witha phase transition from dark matter to dark energy I. Nikitin email: igor.nikitin@scai.fraunhofer.de Fraunhofer Institute for Algorithms and Scientific Computing, Schloss Birlinghoven, 53757 Sankt Augustin, Germany Abstract. Thiswork continuesthe constructionofarecentlyproposedmodelofdark matter stars. In this model, dark matter quanta are sterile massless particles that are emitted from the central regions of the galaxy in the radial direction. As a result, at large distances r from the center of the galaxy, the mass density of dark matter has the form . ~ r -2, in contrast to the homogeneous model . = Const. In the cosmological context, the homogeneous model with massless particles corresponds to the radiation epoch of the expansion of the universe, while the proposed inhomogeneous model turns out to be equivalent to .CDM. In this paper, scenarios will be considered in which the radial emission of dark matter is brought into hydrostatic equilibrium with a uniform background. It is shown that solutions exist if the uniform background has an equation of state typical for dark energy. Thus, this model describesaphase transitionfromdark matter insidethegalaxytodark energy outsideof it. The specific mechanism for such a transition could be Bose-Einstein condensation. In addition, the question of what happens if dark matter particles are not sterile, for example, are photons of the StandardModel, is considered. Povzetek: To delo je nov korak v predlaganem modelu zvezd iz temne snovi. V tem modeluso kvanti temne snovi sterilnibrezmasni delci,kijih seva center galaksijev radialni smeri. Zato imav velikih oddaljenostih r od sredis.. ca galaksije masna gostota temne snovi obliko . ~ r -2, za razliko od homogenega modela, kjer je . = Const. Homogeni modelzbrezmasnimi delciopi.sirjenja vesolja,vkaterem se temno snovv obdoju .prevladuje sevanje, predlagani nehomogeni model pa se je izkazal enakovreden modelu .CDM.Vtemprispevku obravnava avtor razmere,v katerihje radialna emisija temne snoviv hidrostati.ze,da takare. cnem ravnovesjuz enakomernim ozadjem. Poka .sitev obstaja, .cba stanjaza temno energijo.Tedajopisujeta model fazni ce velja za enakomerno ozadje ena.prehodiz temne snovi znotraj galaksijevtemno energijoizvennje.Sprecifi. cen mehanizem za tak prehod bi lahko bila Bose-Einsteinova kondezacija. Pri tem se pojavi vpra. sanje, kaj se zgodi, . ce delci temne snovi niso sterilni, kot na primer niso sterilni fotoni Standardnega modela. 20.1 Introduction This work continues the construction of the model [1] presented at Bled 2020 Workshop “What Comes Beyond the StandardModels?”. In this model, the sources of dark matter are Planck cores, Planck density objects located inside black holes. These objects are permanently emitting particles of dark matter, of originally Planck energy and Planck flux density. In this work, massless particles will be considered as quanta of dark matter, that are also sterile, which means that they donot enterintoany interactions except gravitational. Radiation occursinaT-symmetric way, into the future and into the past, so no energy is spent during the radiation, and such objects retain their mass. The radiation occurs in the radial direction;therefore,the considered flowshaveno transversepressure.We denote this type of matter as null radial dark matter (NRDM). The solutionof Einstein’sfield equationswithsuch mattertermhasastructuredif­ferent from the Schwarzschild’s one. This explains the designation of such compact massive objects as quasi-black holes, dark stars, Planck stars[2–5]. These solutions do not have an event horizon; instead, a deep gravitational well is formed at the gravitational radius.In our model, calculations forrealistic astrophysical scenarios show the redshift value z ~ 1049, which leads to a shift of the emitted dark mat­ter from the Planck’s .in ~ 10-35m to the ultrahigh wavelengths .out ~ 1014m, respectively, ultralow energies Eout ~ 10-20eV. Such extreme values complicate direct detection of isolated dark matter particles. Nevertheless, the total number of emitted quanta corresponds to the initially high Planck values. The energy densityand radialpressureofsuch radiationcreatesa hidden mass distribution corresponding to the observed rotation curves of galaxies. The total mass con­tained in such radiation, due to its extension, significantly exceeds the mass of the emitting object within itsgravitational radius. The geometric mass density profile . ~ r-2 typical for the radial radiation creates a linearly growing mass 2 profile M ~ r and flat rotation curves v= GM/r = Const.Taking into account the contributions of all black holes in the galaxy, supermassive and stellar mass ones, the distributions are modulated, and the observed nonflat rotation curves of galaxies are reproduced with good accuracy. In addition, consideration of the astrophysical scenario with the fall of an asteroid onto the Planck core leads to electromagnetic radiation with the characteristics of Fast Radio Bursts. In this work, the main attention will be paid to the following question. If we count the massless dark matter as homogeneous (hot dark matter, HDM), then the solution of the Friedmann equations will correspond to the radiation epoch and will not coincide with the current evolution, which in the standardmodel corresponds to a mixture of contributions from uniform cold dark matter (CDM) and dark energy (DE). However,in the model under consideration, the distribution of matter is inhomogeneous, and, as we will see, it allows the construction of models that are in agreement with the experiment. Thus, within the framework of this model, NRDM mimics CDM at the cosmological level. The CDM macro-particles are galaxies with massive halos surrounding them. In more detail, we will consider several scenarios for the connection of galaxies in NRDM configuration with a uniform background. The backgrounds considered are vacuum, CDM and matter with DE equation of state. In the first two cases, totally uniform DE contribution can be also added. The hydrostatic equilibrium of the system and the correspondence of the densities to the observed .-parameters willbe used as selection criteria.Asaresultof the analysis,it turned outthatof the considered scenarios, only NRDM-DE connections meet the selection criteria. Such scenarios can be interpreted as a phase transition of dark matter from the NRDM state inside galaxies to the DE state outside. The specific mechanism for sucha transition canbe Bose-Einstein condensation (BEC). In addition, we will consider the question of what happens if the dark matter particles are not exactly sterile, for example, are photons of the StandardModel. Phase transitions between dark matter and dark energy have been addressed in a number of recent works. In [6], a phase transition in a system of two scalar fields was considered, with a massive phase of dark matter condensing around galaxies, while outside one of the fields was absent, and the other turned into an exponentiallyrolling mode corresponding to dark energy. Conceptually, this model expands the cosmons theory [7], in which there is only one scalar field representing dark energy in the exponentially rolling mode, while its fluctuations representdarkmatter.Intheworks[8,9],aphase transitionsimilartotheIsing model of ferromagnetism was considered, effectively generating two cosmological constants during the evolution of the universe. In the works [10–13] various scenarios of phase transitions at an early stage of the evolution of the universe, with the formation of bubbles of a new vacuum – dark energy were considered. In these scenarios, there wasa transferand filtrationof dark matterthroughthe walls of the bubbles, whichin specific calculationsreproducesitspresent abundance. In earlier works [14,15], bubblesofa different vacuum after the phase transition were stabilized and led to the formation of massive compact objects – dark energy stars. More general scenarios of the interaction of dark energy and dark matter were considered in a number of works [16–19], in the framework of the so-called Q-phenomenology. In this approach, the components of the dark sector are consid­ered as two massive fluids, in which, in the absence of interaction, the energies are conserved separately. When interaction is enabled between components, energy exchange occurs, parameterizedby a single scalar function Q. For this function, one chooses linear dependencies of elementary densities, products of their degrees –byanalogywiththe kineticsof chemicalreactions,andvariousothermodelforms. The calculation results were then compared with the cosmological observables. In works [20–33] the interaction of dark matter and dark energy was considered in relation to cosmological tensions. These are the discrepancies between the Hubble parameter and other cosmological properties, found in different types of observations, in particular, for the early and late stages of the evolution of the universe. The direct relation of the dark matter – dark energy interaction models with cosmological tensions can be explained. In the absence of the interaction, the components of the dark sector evolve independently, being bound only by the common gravitational field.Fromhereitiseasyto obtaintheindividualde­pendences of the component densities on the scale factor of the universe. This makes directly observable variables (such as distribution of CMB inhomogeneities, luminosity-distance-redshift dependence, etc.) related with model parameters (such as Hubble parameter today, linear fluctuation of the matter density field, etc.). When the interaction is turned on, the components begin to pump into each other; as a result, the relationship of the model parameters with the observed variables is modified.A similar approach is used in the models of dynamical dark energy [34–36], where the equation of state or the density of dark energy are modified directly. The resulting changes manifest themselves as tensions between the values of the Hubble parameter, deduced from different types of measure­ments without model modification.Within this framework, withthe right model modification, the cosmological tensions should disappear. The idea that dark matter and/or energy are associated with Bose-Einstein con-densation,arerepresentedby a superfluid liquid, was consideredina numberof works[37–43].In particular,[43] considereda complex scalar field withapotential equivalent to Chaplygin gas. While the specific form of the potential is not impor­tant, thepresenceofa minimuminitis significant.In this model,the dark energy is the stateof Bose-Einstein condensate, asymptotically attainedby the scalar field at this minimum. Dark matter was viewedasan excited state describedbyagasof quasiparticles.In our work,a similar model willbe considered,in which the outer zoneof the galactic halo will alsobe occupiedby Bose-Einstein condensate, while the distribution of dark matter in the inner part of the halo will be associated with the emission of particles from RDM stars. First, in Section 20.2, we will recall the structure of the RDM model, then consider a number of scenarios for its connection with a uniform background. Not all of the scenarios will be successful, but we will describe all in detail to rule out unsuccessful options. Section 20.3 considers separately the photon case. The details of the constructions are given in the Appendix. 20.2 Estimations for various scenarios The model [1] considers three cases: massive, null or tachyon radial dark matter (M/N/T-RDM). The tachyon case is too exotic and will not be considered here. On the other hand, the massive case is similar to the commonly considered uniform cold dark matter (CDM). In this paper, we focus on the intermediate case, null or light-like dark matter. The quanta of such dark matter are massless sterile particles of an unspecified type. The main formulas that determine the distribution of masses and pressures of dark matter in the model under consideration are . = pr = ./(8pr2),pt = 0, .grav = . + pr + 2pt, (20.1) where . is the mass density, pr is the radial pressure, pt is the transverse pressure, .grav is the gravitating mass density, r is the radius, . is constant scaling parameter, in the geometric system of units G = c = 1. Such dependence’s are established at large distances from the center of the galaxy,when all sources of dark matter (RDM stars), distributed over the galaxy in proportion to the density of the luminous matter,canbe consideredasconcentratedinonecenter.The integrated gravitating mass for such a distribution is linear in the radius: Mgrav = .r, and the square of 2 the orbital velocity is constant and equal to v= Mgrav/r = .. Inrelationto the .-parameterfortheMilkyWay(MW)galaxy,[1]provides several estimates. The simplest, if one places a single RDM star in the center of the galaxy and completely neglect the contribution of the luminous matter, leads to a flat rotation curve with an orbital velocity v ~ 200km/s, . =(v/c)2 ~ 4 · 10-7.A more accurate estimate is obtained from the fit of the MW rotation curve, the so-called Grand Rotation Curve (GRC, [44–48]). From this fit it can be seen, [1] Fig.2, that on an approximately flat portion of the rotation curve at the position of the Sun r ~ 8kpc there is a significant contribution of luminous matter, as a resultof whichthe contributionof dark mattertov 2 is less than the trivial estimate. Further, with increasing radius, the contribution of luminous matter decreases, while the contribution of dark matter remains constant up to Rcut ~ 50kpc. This contribution corresponds tothe galactic . = Mdm(Rcut)/Rcut, in geometric units, being averaged over the scenarios considered in [1]: Mdm(Rcut)~ 2.6 · 1011M., . ~ 2.5 · 10-7. In this work, we will carry out estimates in order of magnitude, so itis not so important which definitionofthe galactic . willbe chosen.Weprefer the latter, more precise definition and the corresponding value Mdm(Rcut). When the contribution of individual black holes (identified with RDM stars) is considered, [1] Fig.5 gives estimates for the central supermassive and peripheral stellar black holes: .smbh ~ 10-10 -10-7 , rs,smbh ~ 1.2 · 1010m; .sbh ~ 10-16 - 10-12 , rs,sbh ~ 3 · 104m. This givesa floatingestimate for the external wavelength of DM particles: .out = rs(8p/.)1/2, .out,smbh ~ 1014 -1016m, .out,sbh ~ 1011 - 1013m. This wavelength is highly dependent on the model used to describe the internal structure of RDM stars. In this work, for estimations, we prefer to use the value of the external wavelength .out and the corresponding redshift factor A1/2 = lP/.out as phenomenological parameters. QG The considered scenarios for the connection of galaxies in the NRDM configuration witha uniform background are schematically shownin Fig.20.1. 20.2.1 Rejected scenarios Scenario S0.1: superposition of galactic halos without cutting. In this scenario, the dark matter halo of each galaxy extends to the radius of the visible universe Runi ~ 14Gpc, dark matter from different halos does not interact and gives an additive contribution to the total mass density. If this scenario was valid, then it would be different from the radiation epoch, due to the following reasons. For the radiation epoch,theBigBang(moreprecisely,the momentofrecombination)isthe initial flash, after which the homogeneous photon gas cools down as the universe expands.Theenergyofthephotonschangeswiththescale factoras a -1, and their numerical density as a-3, which gives the dependence a-4 for the mass density. For the RDM model, despite the expansion of the universe and the separation of RDM stars from each other, the energy of DM particles near RDM stars is fixed, related to the above-mentioned parameter .out. It is important that this energy does not fall over time. The number density of RDM stars falls as a -3, and as a result the average mass density also falls as a-3, just like for CDM. The scenario is prohibited due to the following evaluation. According to cal­culations for the Milky Way galaxy [1], the cutoff radius and halo mass are Rcut ~ 50kpc, Mdm(Rcut) ~ 2.6 · 1011M., and the mass of the disk and other emitting structures can be neglected in the order estimate. If one does not use the cutoffand continues the halo to the border of the universe, Mdm(Runi)= Mdm(Rcut)Runi/Rcut ~ 7.3 · 1016M.. If the result is multiplied by the estimated number of galaxies in the universe Ngal ~ 2 · 1012, we get Mdm ~ 1.5 · 1029M.. Compared to the estimated mass of dark matter in the universe Mdm,uni ~ 4.5 · 1023M., this valueis overestimatedbythe factor ~ 3.2 · 105. The mass density averaged over the volume of the universe for the obtained value of Mdm will be .dm ~ 8.6 · 10-22kg/m3, which in ~ 3.2 · 105 more than the estimate from the critical density .dm.crit = 2.7 · 10-27kg/m3 . The following corrections can be made to this calculation. The geometric cutoff by Runi for the galaxies located far from the center (which we locate in the MW) can reduce the halo mass, but the factor is small, at most 2. The mass density is everywhere understood as the gravitating mass density, which also includes the pressure .grav = . + pr. At cosmological distances, the energy of DM particles is redshifted, but considering distances up to 0.03Runi ~ 420Mpc, the mass density decrease factor will not exceed 20% (for the Hubble parameter H0 = 72km/s/Mpc such distances correspond to z ~ 0.1, 10% decrease in energy and 10% slowdown in time, affecting 20% decrease in flux density). This can be used to estimate the massfrom below, asaresult the > 7.6 · 103 discrepancy will remain unexplained. Further, the estimate is based on the assumption that all galaxies have masses of the order of the MW. This, of course, is not the case, there is a distribution of galaxies by masses. An accurate account of the distribution of galaxies will be given in the Appendix, and a similar result will be obtained, within the accuracy of modeling the distribution of galaxies. Now we assume that all galaxies are copies of the MW, the halo of each galaxy is extended to the radius Rgal and the relation NgalMdm(Rcut)/Rcut · Rgal = Mdm,uni holds, at which all the necessary mass relations are joined. Substituting the known data, we get Rgal = 44kpc, which almost coincides with the value of Rcut,MW. An exact match of Rgal = Rcut,MW can be achieved by slightly ' adjusting the estimated number of galaxies to N gal = 1.7 · 1012. Thus, according to this estimate, if we imagine that the universe consists of copies of the MW galaxy, the halo of which is cut offby Rcut ~ 50kpc, then the total mass of dark matter in theuniversewill coincideinorderof magnitudewithits cosmological estimate. With this configuration, dark matter is entirely concentrated in galaxies and is absent in the intergalactic environment. Let’s find out what happens if the Rcut parameter is increased to 1Mpc. This is ' possible when the estimated number of galaxies is reducedto N gal ~ 8.7 · 1010 . It is known that the mass-to-light ratio of galaxies stops changing at distances of this order of magnitude, see [49] Fig.2.5. Moreover, this value is of the order of intergalactic distances. Thus, a scenario is theoretically possible in which galaxies toucheachotherinthe outer zonesoftheirhalos, althoughitmayrequire tensions of the Ngal parameter. Finally, for the original scenario in which the halos overlap and reach the size of ' the universe, it would be N gal ~ 6.2 · 106,atoo strong deviation from the observed value. Therefore, this scenario can be considered as excluded. Scenario S0.2: touching halos in dynamic equilibrium. The above variant with halos touching each otherin the outerregion, with therefinement that galaxies can exchange dark matter. Null dark matter leaking from one galaxy is absorbed by neighboring galaxies, and vice versa. In fact, the world lines of dark matter form a network connectingthe galaxies,andthe conceptofa sphericalhaloisonlyan approximation. The problem here is as follows. As a result of the expansion of galaxies, DM particles coming from neighboring galaxies are subject to a small redshift z and decrease their energy and flux densityby the corresponding factor.We consider RDM starsina stationaryT-symmetric scenario. The parametersof dark matter, in particular, its energy and flux density, coincide for the incoming and outgo­ing flows. Therefore, the exiting particles also have a reduced energy and flux density.With multiplereflections between galaxies, theredshiftofDM particles accumulates, exactly as it would in a homogeneous environment. RDM stars act as spherical mirrors, changing the direction of the DM particles, but not their energy characteristics. Such an environment turns out to be equivalent to HDM, its evolution coincides with the era of radiation, which is different from the observed evolution of the universe today. In fact, the stationary stateof RDM starsrequiresT-symmetry only for the energy flux density ., the individual energies of incoming and outgoing DM particles can be different, compensated by different number flux densities. This will not help, since it is energy density that governs cosmological evolution. Note also that in the equation .P = ./(8pr2 AQG), which determines conditions on the surface sof the Planck core, in the considered scenario the factors . and AQG are scaled equally so that the gravitational radius rs canremain unchanged. Strictly speaking, changing . and AQG in stationary scenarios is also unacceptable, but we consider this change as performed rather slowly, quasi-statically. What happens in fast scenarios, as well as withT-asymmetric . and variable rs, can be found out only after solving the dynamic RDM problem, which goes beyond the scope of this work. Scenario S0.3:ahalo surroundedbyamassive thin shell. Wenowlookatafew scenarios from the termination shock type. This phenomenon occurs at the edge of the solar system when the radially directed solar wind meets the isotropic interstellar medium. Similar phenomena can occur with dark matter at the edge of the galaxy when the radial flowof dark matter meets the intergalactic environment.First, we will consider a scenario in which an NRDM galaxy at radius Rcut is surrounded by a thin CDM layer, and there is a vacuum outside. The CDM layer is held in equilibriumbytheNRDMpressureforceandtheforceofgravity.Ifsuchascenario was possible, the galaxies would be isolated from each other and would be massive balls floating in a vacuum. On a cosmological level, such matter is equivalent to CDM. The equilibrium condition of forces can be written as ./(8pr2)· 4pr2 = ./r · m for r = Rcut, whence the mass of the CDM layer m = Rcut/2, in geometric units. Thisisa huge mass, exceeding the massof the galaxy Mdm(Rcut)= .Rcut, where . « 1, for MW . = 2.5 · 10-7. Formally, with such a mass, the galaxy is covered by its event horizon, becoming a black hole. More precisely, the calculation uses Newtonian equations and only shows that there is no solution in weak fields. The interpretation of this result is that the relativistic pressure at the boundaryof the NRDM galaxy canbe compensated onlybyrelativistic gravitational forces. The distribution of matter in the CDM layer obeys theTolman-Oppenheimer-Volkoff(TOV) equation, the solution of which in weak fields and for a thin layer is described by the one-dimensional hydrostatic equation . = .0 exp(-gh/w), where w -parameter of the equation of state (EOS) p = w., for CDM w = kT/m « 1, all equations are written in geometric units. The pressure equilib­rium at the boundary of the layer leads to ./(8pR2 )= w.0, also g = cut./Rcut, whence . = ./(8pR2 w)· exp(-.h/(wRcut)). Integrating this value, we get R cutm = 4pR2 .dh = Rcut/2. The result is independent of w and coincides with cut the estimate above. Scenario S0.4: halo surrounded by homogeneous dark matter. Avariation of the pre­vious scenario, where instead of vacuum there is a homogeneous dark mat­ter with isotropic EOS outside: pbgr = w.bgr. Here we will consider two op­tions: CDM w « 1, HDM w = 1/3. Pressure equilibrium at the halo boundary: ./(8pR2 )= w.bgr, gravitating masses: Mdm,gal = Ngal.Rcut, Mdm,bgr = cut(1 + 3w).bgr · (4p/3)(Runi3 - NgalRcut3 ), an estimate of the total mass of dark matter in the universe: Mdm,uni = Mdm,gal + Mdm,bgr = Ngal.Rcut + .(1 + 3w)/(6wR2 )(R3 ). Note that, according to earlier calculations, the cutuni -NgalRcut3 first term already corresponds in order to the cosmological estimate for the mass ofdark matter.Onlyifthe secondtermissmall,this correspondencecouldbepre-served. However, if we assume that the intergalactic distances significantly exceed the size of the halo, and the estimate R3 » NgalR3 holds, then the second term uni cut Mdm,uni ~ .R3 /Rcut2 ·(1+3w)/(6w),which for. = 2.5·10-7 ,Runi ~ 14Gpc and uniRcut ~ 50kpc matches Mdm,uni/M. ~ 5.7 · 1027(1 +3w)/(6w). It can be seen that already for w = 1/3 and even more so for w « 1 the result significantly exceeds the value Mdm,uni/M. ~ 4.5 · 1023, obtained from cosmological estimates. Also ' for the above option with tension, Rcut = 1Mpc, N gal = 8.7 · 1010 the resulting formula is Mdm,uni/M. = 4.5 · 1023 + 1.4 · 1025(1 + 3w)/(6w)does not allow CDM/HDM as background matter,for continuous matching with NRDM pressure at halo boundaries. 20.2.2 Accepted scenarios Next, we’ll lookat scenarios involving dark energy.We willrepresent dark energy asa kindof matter, perhapsa kindof dark matter or its other phase state, which has an isotropic EOS pde =-.de, that is, w =-1, with positive .de, constant within each phase. The density of the gravitating mass for such matter is negative and is equal to .de,grav = .de + 3pde =-2.de. The negativity of this density, provided that it prevails over other components, is the driving mechanism for the accelerated expansion of the universe. ScenarioS1.1:jumpinthedark energydensityatthehaloboundary. Let therebe two dif­ferent dark energy densities, outside the halo .de,bgr, inside the halo .de,gal, with cutpde,bgr -pde,gal a jump at Rcut. Equilibrium pressure condition ./(8pR2 )= = .de,gal-.de,bgr,gravitating masses:Mdm,gal = Ngal.Rcut,Mde,gal =-2.de,galNgal · (4p/3)R3 , Mde,bgr =-2.de,bgr · (4p/3)(R3 - NgalR3 ), an estimate of cutuni cutthe total mass of dark matter and dark energy in the universe: Mdm+de,uni = Mdm,gal + Mde,gal + Mde,bgr =(2/3).NgalRcut -(8p/3).de,bgrR3 . The sec- uni ond term here describes the total gravitating mass of dark energy, as if it uniformly filled the entire universe, including galactic halos. The first term is the gravitat­ing mass of the galactic halo reduced by the factor (2/3). In general, the model behaves likea mixtureofuniform cold dark matter and uniform dark energy, like .CDM. In order of magnitude, for Rcut = 50kpc, the CDM mass corresponds to cosmological estimates. Fine tuning is also possible similar to scenario S0.1, the factor (2/3)can be compensated for by a small increase in the estimated number ' of galaxies N gal = 2.6 · 1012 . Let us also analyze the expression for the gravitating mass of one galaxy: M(r)= .r - 2.de,gal(4p/3)r3. In the expression for the internal dark energy density .de,gal = ./(8pRcut2 )+ .de,bgr for the selected value Rcut = 50kpc, after con­version to natural units, the first term is 5.6 · 10-24kg/m3, the second .de,bgr = .de.crit = 6.8·10-27kg/m3,the first term dominates. Thus, continuous matching ofpressuresatthe galactic boundaryrequiresajumpinthedark energy density by a factor of ~ 103. Note that this jump can be reduced by adjusting the Rcut parameter. Further, the expression for the mass function at the selected parameters becomes M(r)/M. = 2.6·1011(r/Rcut)-8.7·1010(r/Rcut)3.Inthe innerpartoftherotation curve, for example, up to the position of the Sun r ~ 8kpc, the first term dominates. Thus,the interioroftherotation curveis unaffectedbythe dark energy introduced into the model. In the outer part of the curve, the contribution of the enhanced internal dark energy density becomes noticeable, finally, it is this contribution that leads to the factor (2/3)in the mass formulas. The term proportional to the external dark energy density for the chosen parameters of the model makes a negligible contribution within the galaxy. It beginsto dominate in the formula M(r>Rcut)=(2/3).Rcut -(8p/3).de,bgrr3 at distances r > 0.6Mpc, at which the effects of cosmological expansion become noticeable. It becomes clear that in the considered scenario the rotation curve undergoes a change only in its outer part, where it decreases by 2/3 factor, about 18%. As we will see later, the measurement errors in this range significantly exceed this variation, which makesit impossibleto distinguishthis solutionfromthereference profile. Thus, we have obtained the first scenario, which connects null matter in galactic halos with a cosmological background of dark energy and turns out to be equiva­lent to the uniform .CDM model.Acalculation basedona simple equilibriumof pressures does not provide any indication for the possible nature of the increased density of dark energy within the galaxy. Phenomenologically, dark energy can be described as a medium in which its constituent particles experience mutual attraction. This attraction corresponds to negative pressure, while the work of external forces -pdV is used to increase the internal energy .dV, in accordance with EOS -p = .. The presence of two phases with different pressures suggests two varietiesforsuchmedia.Ananalogycanbedrawnherewiththestringmodel. The energy of a string is proportional to its length, just like the total mass for dark energy is proportional to its volume. The strings have a fixed tension, which is a constantinthe model.Onecan considerstringswithdifferent tensionsas separate varieties of the same model. The considered scenario demonstrates a fundamental possibility; further possible alternatives will be considered. Scenario S1.2: surface tension at the boundary between the halo and the background from dark energy. Let inside Rcut be NRDM, outside – dark energy with density .de,bgr, and surface tension with coefficient s acts on the boundary. Equilibrium pressure condition ./(8pR2 )= 2s/Rcut + pde,bgr = 2s/Rcut -.de,bgr, gravi- cuttating masses: Mdm,gal = =-Ngals · 4pR2 = Ngal.Rcut, Mde,surf cut, Mde,bgr -2.de,bgr · (4p/3)(R3 -NgalR3 ), an estimate of the total mass of dark matter uni cutand dark energyinthe universe: Mdm+de,uni = Mdm,gal +Mde,surf +Mde,bgr = (3/4).NgalRcut +(2p/3)· NgalR3 .de,bgr -(8p/3).de,bgrR3 . Here the third cutuni term corresponds to the cosmological contribution of dark energy, it grows in the negative direction in proportion to the volume of the expanding universe. The first and second terms are preserved in the expansion and represent CDM. At Rcut = 50kpc, the first term significantly exceeds the second, and, as in the previous scenario, allowsfinetuningofthe parameterstothe cosmological value of CDM density. In the above formulas, the gravitating mass corresponding to the boundary layer is calculated as follows. Surface tension is related to negative transverse pressure and positive energy density as -pt = . = s/dr, where dr is the layer thickness. The gravitating mass of the spherical layer is M =(. + 2pt)Sdr =-s · 4pR2 . cutThere is also a radial pressure pr inside the layer, which continuously interpolates the boundary values, remains bounded, and makes a vanishing contribution at dr . 0. When choosing Rcut = 50kpc, the density jump between external dark matter and NRDM is still ~ 103 times, but here it is compensated by surface tension. As in the previous scenario, the pressure jump can be reduced by adjusting the Rcut parameter. The mass function for rRcut)=(3/4).Rcut +(2p/3)R3 .de,bgr - cut(8p/3).de,bgrr3, which dominates for r > 0.6Mpc. The resulting scenario is very close to the previous one, only a different mech­anism is used to compensate for the pressure jump at the edge of the galaxy. Phenomenologically,if we consider dark energy asa medium consistingof inter-actingparticles,thepresenceofa boundarycanleadtothe appearanceofa surface term in the equations, as for classical media. The jump in the mass function that appearsinthis scenario correspondstoajumpintherotation curvebythe factor 3/4, about 13%. This jump also occurs in the outer region, where the scatter of experimental data is large, so that it can be unnoticed. Also, this jump can be an idealization of a more complex scenario in which the transition layer has a finite thickness. The possibilityofa gradual changeof EOS willbe exploredin the following scenario. Scenario S1.3: phase transition of dark matter to dark energy. In this scenario, we assume that dark energyisa formof dark matter, and with the increasing radius, there is a continuous transition between the corresponding EOS: pr = wr., pt = wt., (wr,wt)change from (1, 0)for r = Rcut1 to (-1, -1)for r = Rcut2 >Rcut1. The result depends on the transition path, which we fix from physical consider­ations as follows. Initially, from r = Rcut1 to the intermediate point r = Rcut1b, only wt changes, from 0 to -1. The inclusion of transverse attraction between flows of dark matter leads to the Joule-Thomson effect known in gas dynamics, cooling of flows, which in our case manifests itselfin a rapid decrease in the mass density .. Further, from r = Rcut1b to r = Rcut2 only wr changes, from 1 to -1. In this range, the contributions of dark matter from different sources are mixed, the matter becomes isotropic. Further, the matter obeys isotropic EOS for dark energy, and its density and pressure become constant. It is convenient to solve the problem in logarithmic variables x = log r, . = log ., with the restriction .>0.To interpolate wt,r in the corresponding intervals, we choose functions linear in x, and the positions of the endpoints {x1,x1b,x2} = log{Rcut1,Rcut1b,Rcut2}will be chosen from the correspondence of the model to the cosmological parameters. The stationary spherically symmetric solutions considered here satisfy the hydro­ ' static equation for anisotropic medium, see Appendix for details: r(pr + .)A + r ' 2A(r(pr)+ 2pr - 2pt)= 0. The first term describes the gravitational interac- r tion, which in our problems can be neglected. The reason for this is that in the ' weak field limit A ~ 1 + 2., A ~ 2g, where . is the gravitational potential, r ' g = . = Mgrav(r)/r2 is gravitational field in the used system of units, |.|« 1, r rg « 1.Inourmodels,thedensityandpressurearecontrolledbyasmall common factor ., and the first term turns out to be of the next smallness order compared to the second one. This property of the weak-field regime can also be verified on the exact solutions of the hydrostatic equation, given in Appendix. ' Thus, we can concentrate on the second term: r(pr)+ 2pr -2pt = 0. Let’s go r '' to logarithmic variables and substitute EOS: wr. +(wr)+ 2(wr - wt)= 0. xx R ' The solution is: . =- dx((wr)+ 2(wr - wt))/wr. In the following, we will x consider regular solutions in which the denominator and the numerator in the ' integrand vanish simultaneously: wr = 0, (wr)= 2wt. Note that, with our choice x of the interpolation order, the condition wr = 0 can be satisfied only at the second stage, in the interval [x1b,x2], while,duetothe linearityoftheinterpolation,the ' condition (wr)= 2wt =-2 holds on this entire interval. x At the first stage [x1,x1b], (wr,wt)=(1, -q), q =(x -x1)/(x1b -x1), calculat­ing the integral, we get .1b -.1 =-3(x1b - x1). At the second stage [x1b,x2], ' (wr,wt)=(1 -2q, -1), q =(x -x1b)/(x2 -x1b), from the condition (wr) x =-2 ' we get q = 1, that is, x2 = x1b +1. Calculating the integral, we get .2 -.1b =-2. x Hence log(.1/.2)= .1 - .2 = x1)+ 2. Choosing .1 = ./(8pR2 ), 3(x1b - cut1Rcut1 = Rcut = 50kpc, . = 2.5·10-7 ,.2 = .de,bgr = .de.crit = 6.8·10-27kg/m3 , and also converting all values into the natural system of units, we get: .1/.2 = 824, Rcut1b = 0.24Mpc, Rcut2 = 0.65Mpc. Thus, the required density variation from NRDM to the background dark energy in the considered scenario fixes the halo cutoffparameters to reasonable values. Next, consider the contribution ofthe galaxy to the cosmological mass density. The gravitating mass density is .grav =(1 + R wr + 2wt)., and the gravitating mass of the spherical layer is .Mgrav = 4p .gravr2dr. After the transition to logarithmic variables, the integrals over two interpolation intervals can be taken analytically. Omitting cumbersome expressions, we will immediately give the numerical answer {M1, .M1, .M2,Mvac}= {2.60, 2.67, -2.60, 2.35}· 1011M.. Here M1 = .Rcut is the mass of the NRDM halo, .M1,2 are the masses of the spherical layers for two interpolation intervals, Mvac =(8p/3).de,bgrR3 is cut2 the compensation mass of the vacuole arising from the rearrangement of the terms Mdm+de,uni = NgalMdm+de,gal -(8p/3).de,bgr(R3 - NgalR3 )= uni cut2Ngal(Mdm+de,gal +Mvac)-(8p/3).de,bgrR3 . The Mvac term should be taken uni into accountin cosmological calculations,whenreducingtothe parametersofa homogeneous medium, while when calculating the rotation curves only the actu­ally present masses should be taken, and Mvac should be omitted. Interestingly, there is an identity M1 +.M2 = 0, which holds exactly, at the analytical level, but is probably a coincidence due to a special choice of interpolating functions. Also of interest is the approximate equality of all mass contributions in their absolute value. The cosmological mass per galaxy is the sum of all these contributions and is equal to Mdm+de,gal + Mvac = 5 · 1011M.. This gives a coincidence with the cosmological CDM mass Mdm,uni = 4.5 ·1023M. in order of magnitude, for exact ' 1011 coincidencethe estimated numberof galaxies shouldbereducedto N gal = 9·, 2.2 times less than the nominal value. One can also adjust the . parameter, but since our estimates of the halo cutoffparameters were tied to MW values, these estimates must be repeated when . changes. The constructed scenario, obviously, contains wide arbitrariness in the choice of interpolating functionsandis ratheraproofofthe existenceofasolution satisfying cosmological estimates. This existence in itself is non-trivial. Recall that in standard cosmology,null,hotdarkmatterleadstoadifferentrateof cosmologicalexpansion today and is forbidden. The possibility of joining hot darkmatter with dark energy within the galactic halo, at a cosmological level equivalent to .CDM, is the main result of this work. The specific way of joining may be different, in the Appendix we will discuss the possibility of narrowing this arbitrariness. For now, note that the interpolation order selected in the model is significant. The reverse order when (wr,wt)changes from (1, 0)to (-1, 0)for r . [Rcut1,Rcut1b] ' leads to the condition (wr)= 2wt = 0, not feasible for linear functions. If x we interpolate both terms at the same time, (wr,wt) =(1 - 2q, -q), q =(x - ' x1)/(x2b - x1), from the conditions wr = 0 , (wr)= 2wt we get q = 1/2, x ' q = 1/2, that is, x2b = x1 + 2. Moreover, .2b = .1 -2, which for Rcut1 = 50kpc x gives Rcut2b = e2Rcut1 = 0.37Mpc, .1/.2b = e2 ~ 7.4, far from the experimental value of .1/.2 ~ 824. The physical rationale with the initial cooling of dark matter due to the Joule-Thomson effect and the subsequent transition to the isotropic phase for the cooled gas was important for obtaining the strong density drop observed in real galaxies. Here are some graphs showing the behavior of the main physical profiles in the considered scenario. Fig.20.2 left shows the dependence of . = log . on x = log r. Initially, the graph contains an NRDM line with a slope of -2, which corresponds to the . ~ r-2 dependence. Further, at point 1, the transverse interaction between the flows turns on, and the Joule-Thomson effect is superimposed on the contin­uing radialdropin density. Here, the slopeof the graph d./dx is continuously changing from -2 to -4. Further, in the interval from 1b to 2, a transition to the isotropic phase follows, the slope in this case being equal to -2. After point 2, there is isotropic dark energy with constant density, slope 0. The resulting density variation between points1 and2 corresponds to the experimentally observed factor of .1/.2 ~ 824. For comparison, the option shown in gray when (wr,wt)are linearly interpolated at the same time. Theslope between points1and2bis -1. After 2b, there is an isotropic phase with a slope of 0. Due to these changes, the graph goes much higher than the previous one, the density variation does not correspond to the observed value. Fig.20.2 right shows the dependence of Mgrav(r). Initially, there is an NRDM part witha characteristic linear dependence,thenatpoint1b,the dependencepasses through a maximum and, after point 2, is described by a negative cubic term corresponding to the contribution of dark energy. Scenario S1.4: Bose-Einstein condensation. In this scenario, two phases are also considered: the internal NRDM phase, described by the classical particle model, and the external phase, describedby a complex scalar field. This field theoryis Fig.20.3: Left:an externalpartoftheMilkyWayrotation curve, accordingto[48]. Avariety of profiles are shown, including the RDMcut scenario from [1]. Right: outer part of the dependence of radial velocity on the distance, according to [46]. The position of the galaxy M31 is marked, the outer part of the graph is fitted with a Hubble-alike dependence. used in phenomenological models of Bose-Einstein condensation, as well as in cosmological models of quintessence and its variants (k-essence, quartessence, Chaplygin gas), see [43] and references therein. Therefore, this scenario assumes that dark matter particles are emitted by RDM stars in the galaxy and undergo Bose-Einstein condensation at large distances. Alternatively, these can be particles of different types that are in contact equilibrium at the edge of the galactic halo. In the field theory under consideration, the Lagrangian, the energy-momentum tensor, and the equations of motion have the form [50] Chap.6.3,7.5: L = -(.µ. * .µ.)/2 -V(|.|2), (20.2) Tµ. =(.µ. * ... + ... * .µ.)/2 + gµ.L, (20.3) (-.2/.t2 + .). = 2V ' (|.|2).. (20.4) Here the equationsof motion are writtenina flat background, and therestof the expressions are valid for an arbitrary metric.We alsoremind that fora scalar field the covariant and coordinate derivatives are equal: .µ. = .µ.. The field equa­tions belong to the well-known nonlinear Klein-Gordon type with the potential. For V(|.|2)= Const + m 2|.|2/2 the equations become linear and describe the behaviorofafree massive scalar field.We neglectthe influenceof gravityonthe scalar field, assuming that the gravitational fields are weak and the corresponding solutions are relativistic. We will usea smooth potentialV(s2), which hasa minimum fora nonzero value 22 of the argument V(s )= Vmin, s 1 >0. For this minimum, the constant function 1 . = s1 isthe exact solutiontotheproblem.Forsucha function,usinga spherical coordinate system and a metric of signature (-+ ++), we write out the mixed components of the energy-momentum tensor: T. = diag(-., pr,pt,pt)=-Vmin · diag(1, 1, 1, 1), (20.5) µ . = Vmin,pr = pt =-Vmin, (20.6) .grav = . + pr + 2pt =-2Vmin. (20.7) The result coincides with the standardEOS of dark energy, which explains the interesttothis modelinthe cosmological context.Wewillfix Vmin >0, and for simplicity we will assume V>0 everywhere. In this paper, we consider stationary spherically symmetric problems for which iEt there are particular solutions of the form . = es(r), with real E, s(r).With this substitution,the dimensionisreduced (E2+.)s = 2V ' (s2)s. Next, we will consider stationary solutions E = 0, . = s(r). The uniqueness of solutions with stationary boundary conditions is demonstrated in the Appendix. Thus, all solutions that can be attached to the constant . = s1 are globally stationary and have the form . = s(r). Calculating EOS for stationary solutions '2'2 T. = diag(0,s ,0,0)-diag(1, 1, 1, 1)· (s /2 + V(s 2)), (20.8) µ '2'2 . =-pt = s /2 + V(s 2)>0, pr = s /2 -V(s 2), (20.9) .grav = . + pr + 2pt =-2V(s 2). (20.10) If the potential is shallow, then .grav ~ -2Vmin, as for DE. This result is quite remarkable. As a consequence, the scenario can be configured in such a way that the gravitating density profile immediately after the NRDM phase .grav = ./(4pr2) >0 drops sharply to the DE phase .grav ~ -2Vmin. This reproduces the phenomenological RDMcut scenario discussed in [1], with a sharp cutoffof the density to almost zero at the Rcut radius. The DE contribution begins to be felt at much larger distances and reproduces the observed effect of accelerated cosmological expansion there. Technically, the condition of connection for the radial pressure component at the boundary between the phases must still be met. This condition can be satisfied if the model has enough degrees of freedom to ensure that in pr, the first term s '2/2 dominates over the second -V(s2). In this case, it is possible to ensure the continuous connection with the positive pr from the NRDM phase, no matter how large this value may be. Physical manifestations are defined only by .grav and do not depend on the details of this connection. We will make such a connection for a particular choice of the potential. First ofall, we write the right-hand side of the equations of motion in the form 2V ' (s2)s = ' V(s2) . Next, using the reparametrization of the argument V(s2)= V1(s), we s choose the potential as given below. Theremarkablepropertiesof sucha potential are the linearity of the equation of motion, the existence of an analytical solution, and also the fact that any potential in the vicinity of the minimum can be written as follows: V1(s)= Vmin + a/2 (s -s1)2, a>0, s1 > 0, (20.11) '' ' s + 2s /r = a(s -s1), (20.12) vv v - arar s = s1 +(eC1)/r +(eC2)/(2 ar). (20.13) Selecting a branch with finite s . s1 at r .8, we get C2 = 0.We also impose the condition C1 >0 in order to ensure s>s1 on the solutions. For s>s1, the ascending branch of V1(s)corresponds to the positive square of the mass, normal particles. At that time, for s