Blejske delavnice iz fizike Bled Workshops in Physics Letnik 21, St. 2 vOL. 21, nO. 2 ISSN 1580-4992 Proceedings to the 23rd Workshop What Comes Beyond the Standard Models Bled, July 4-12, 2020 [Virtual Workshop ] [July 6.-10. 2020] Volume 2: Further Talks And Scientific Debuts Edited by Norma Susana Mankoc Borstnik Holger Bech Nielsen Dragan Lukman dmfa - zaloZniStvo Ljubljana, december 2020 The 23rd Workshop What Comes Beyond the Standard Models, 4.- 12. July 2020, Bled [ Virtual Workshop, 6.-10. July 2020 ] Volume 2: Further Talks And Scientific Debuts 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 (MatjaZ Breskvar) VIA (Virtual Institute of 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 Mankoi BorStnik Holger Bech Nielsen Maxim Yu. Khlopov The Members of the Organizing Committee of the International Workshop "What Comes Beyond the Standard Models", Bled, Slovenia, state that the articles published in the Proceedings to the 23rd Workshop "What Comes Beyond the Standard Models", Bled, Slovenia are refereed at the Workshop in intense in-depth discussions. Workshops organized at Bled > What Comes Beyond the Standard Models (June 29-July 9,1998), Vol. 0 (1999) No. 1 (July 22-31,1999) (july 17-31, 2000) (july 16-28, 2001), Vol. 2 (2001) No. 2 (july 14-25, 2002), Vol. 3 (2002) No. 4 (july 18-28, 2003) Vol. 4 (2003) Nos. 2-3 (July 19-31, 2004), Vol. 5 (2004) No. 2 (july 19-29, 2005), Vol. 6 (2005) No. 2 (September 16-26, 2006), Vol. 7 (2006) No. 2 (july 17-27, 2007), Vol. 8 (2007) No. 2 (july 15-25, 2008), Vol. 9 (2008) No. 2 (july 14-24, 2009), Vol. 10 (2009) No. 2 (july 12-22, 2010), Vol. 11 (2010) No. 2 (july 11-21, 2011), Vol. 12 (2011) No. 2 (july 9-19, 2012), Vol. 13 (2012) No. 2 (july 14-21, 2013), Vol. 14 (2013) No. 2 (july 20-28, 2014), Vol. 15 (2014) No. 2 (july 11-19, 2015), Vol. 16 (2015) No. 2 (july 11-19, 2016), Vol. 17 (2016) No. 2 (July 9-17, 2017), Vol. 18 (2017) No. 2 (june 23-July 1, 2018), Vol. 19 (2018) No. 2 (july 6-14, 2019), Vol. 20 (2019) No. 2 (july 6-10, 2020), Vol. 21 (2020) No. 1 (july 6-10, 2020), Vol. 21 (2020) No. 2 > Hadrons as Solitons (July 6-17,1999) > Few-Quark Problems (July 8-15, 2000), Vol. 1 (2000) No. 1 > Selected Few-Body Problems in Hadronic and Atomic Physics (July 7-14, 2001), Vol. 2 (2001) No. 1 > Quarks and Hadrons (July 6-13, 2002), Vol. 3 (2002) No. 3 > Effective Quark-Quark Interaction (July 7-14, 2003), Vol. 4 (2003) No. 1 > Quark Dynamics (July 12-19, 2004), Vol. 5 (2004) No. 1 > Exciting Hadrons (July 11-18, 2005), Vol. 6 (2005) No. 1 > Progress in Quark Models (July 10-17, 2006), Vol. 7 (2006) No. 1 > Hadron Structure and Lattice QCD (July 9-16, 2007), Vol. 8 (2007) No. 1 > Few-Quark States and the Continuum (September 15-22, 2008), Vol. 9 (2008) No. 1 > Problems in Multi-Quark States (June 29-July 6, 2009), Vol. 10 (2009) No. 1 > Dressing Hadrons (July 4-11, 2010), Vol. 11 (2010) No. 1 > Understanding hadronic spectra (July 3-10, 2011), Vol. 12 (2011) No. 1 > Hadronic Resonances (July 1-8, 2012), Vol. 13 (2012) No. 1 > Looking into Hadrons (July 7-14, 2013), Vol. 14 (2013) No. 1 > Quark Masses and Hadron Spectra (July 6-13, 2014), Vol. 15 (2014) No. 1 > Exploring Hadron Resonances (July 5-11, 2015), Vol. 16 (2015) No. 1 > Quarks, Hadrons, Matter (July 3-10, 2016), Vol. 17 (2016) No. 1 IV > 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 ofHadrons (July 15-19, 2019), Vol. 20 (2019) No. 1 > o Statistical Mechanics of Complex Systems (August 27-September 2, 2000) o Studies of Elementary Steps of Radical Reactions in Atmospheric Chemistry (August 25-28, 2001) Contents Preface to Volume 2 in English and Slovenian Language ................VII Preface in English and Slovenian Language............................XIII Further Talks ....................................................... 1 1 The Concept of Cosmological Inflation and the Origin of 3+1 -dimensional Space-Time in a Universe Consisting of Conserved Vacuum Domains E. Dmitrieff......................................................... 1 2 Neutrino Masses and Mixing Within a SU(3) Family Symmetry Model with Five Sterile Neutrinos A. Hernandez-Galeana............................................... 11 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? H.B. Nielsen and C.D. Froggatt........................................ 28 Scientific Debuts.................................................... 75 4 Problems of the Correspondence Principle for the Recombination Cross Section in Dark Plasma K. Belotsky, E. Esipova, D. Kalashnikov and A. Letunov.................. 77 5 Neutrino Cooling Effect of Primordial Hot Areas in Dependence on its Size K.M. Belotsky, M.M. El Kasmi and S.G. Rubin........................... 84 6 Consideration of a Loop Decay of Dark Matter Particle into Electron-Positron from Point of View of Possible FSR Suppression K. M. Belotsky and A.Kh. Kamaletdinov................................ 90 7 Theoretical Indication of a Possible Asymmetry in Gamma-Radiation Between Andromeda Halo Hemispheres Due to Compton Scattering on Electrons From Their Hypothetical Sources in the Halo K.M. Belotsky, E.S. Shlepkina and M.L. Soloviev......................... 97 8 Numerical Simulation of Dark Atom Interaction With Nuclei T.E. Bikbaev, M.Yu. Khlopov and A.G. Mayorov.........................105 VI Contents 9 Anihelium Flux From Antimatter Globular Cluster M.Yu. Khlopov, A.O. Kirichenko and A.G. Mayorov......................118 10 Domain Walls and Strings Formation in the Early Universe A.A. Kirillov and B. S. Murygin........................................128 11 The Interaction of Domain Walls with Fermions in the Early Universe A.A. Kurakin and S.G. Rubin..........................................134 12 Cosmological Accumulation of Conserved Numbers in Kaluza-Klein Theories W Nikulin.........................................................141 13 Sub-Planckian Evolution of the Universe P.M. Petriakova......................................................149 14 The "Dark Disk" Model in the Light of DAMPE Experiment M.L. Solovyov, M.A. Rakhimova and K.M. Belotsky......................156 15 Dynamical Evolution of a Cluster of Primordial Black Holes V.D. Stasenko and A. A. Kirillov.......................................162 Poem by Astri Kleppe ...............................................169 16 Earth Astri Kleppe........................................................171 Preface to Volume 2 in English and Slovenian Language The series of annual workshops on "What Comes Beyond the Standard Models?" started in 1998 with the idea of Norma and Holger for organizing a real workshop, in which participants would spend most of the time in discussions, confronting different approaches and ideas. Workshops have taken place in the picturesque town of Bled by the lake of the same name, surrounded by beautiful mountains and offering pleasant walks and mountaineering and very fruitful discussions. This year 2020 we still had a workshop in July, but without personal conversations all day and late at night, even between very relaxing walks and mountaineering, due to COVID-19 pandemic. We have, however, a very long tradition of videoconferences (cosmovia), enabling discussions and explanations and exchanging informations and knowledge with laboratories all over the world. This enables us to have this year the total virtual workshop, resembling Bled workshops as much as possible. The "cosmovia" way of workshop enabled more students to participate. Correspondingly the organizers decided to publish two volumes. In this second volume mostly students contributions and those contributions of invited speakers, which arrived the very last moment, are published. We inform the reader that the preface to Volume 1 contains a short information about topics from elementary fermion and boson fields and cosmology, presented and discussed in this year workshop. It appears in this volume after this preface. In up to this year, the long presentations, with breaks and continuations over several days, followed by very detailed discussions, have led to very pedagogical presentations, clarifying the assumptions and the conclusions. Although "cos-movia" way of workshops worked optimally, enabling discussions almost day and night, internet discussions can not at all replace personal discussions. This year we have not succeeded to prepare the discussion section, representing the common work of participants initiated by discussions. The interactions among the participants were not efficient enough. We present in what follows several questions, proposals and doubts, which remain unanswered during the workshop, or having not real response yet. We hope that all the participants — invited speakers and students — will try to participate in looking for the answers, new explanations, new ideas, proofs, doubts, questions, numerical evaluations, by exchanging emails up to the next year. All the participants are welcome to add their own questions, proposals, doubts, etc. We shall publish the results of the common work during the year in the next year proceedings. We shall also take care that all the participants will receive questions, proposals, etc. i. Many a contribution discusses the primordial black holes. This topic was triggered by gravitational wave signal, seemingly merging from black holes with masses of around 150 Solar masses or less. The questions arises: i.a. What is the primordial time? In what position the universe was at the primordial time? With massless ordinary fermions and antifermions interacting with massless ordinary vector gauge fields, or with massive quarks and leptons and with all higher massive families included, as predicted by the spin-charge-family theory after the electroweak phase transition? The higher dimensions, if existing, remained compactified or the theories with higher dimensions, remaining compactified, are not the correct ones? i.b. What causes inflation? i.c. How does the inhomogeneity of the universe with ordinary matter generate the black holes? ii. The spin-charge-family theory starts in d > (13 + 1) with gravity only — with the vielbeins and the two kinds of the spin connecting fields — manifesting in d = (3 + 1) all the known vector and scalar gauge fields and with massless family of fermions manifesting in (3 + 1) the families of quarks and leptons and predicting the fourth family to the existing three, the dark matter, the matter-antimatter asymmetry, the proton decay, new scalar fields, and many other prediction. This theory, starting with very simple starting action, needs no additional assumptions, no additional gauge groups, no additional gauge fields, scalar or vector ones, to explain observed properties of phenomena. Can the predictions be accurate enough to help experimentalists to measure them, like the mass of the fourth family of quarks and leptons, the properties of Yukawa couplings and higgs scalars? Do the proposals with additional groups and additional quarks and leptons relate the spin-charge-family theory? How do the Kaluza-Klein-like theories cause phase transitions if starting in d = oo or any other dimension? iii. What is the dark matter made of? iii.a The lowest of the four families, decoupled from the observed ones, like in the spin-charge-family theory, or they are dark stars with the properties of almost black holes? Or they are a new phase of the vacuum of ordinary quarks and antiquarks captured in bubbles? What interactions do make such bubbles possible? iii.b. What is the decay rate of the dark matter particles in any of the proposed dark matter model? Can any of proposals for dark matter explain the cosmic positron anomaly and all other cosmic measurments? iv. What are indeed black holes? If there are singularities inside a black hole what is the status of fermions and fields inside the black hole? Do they make phase transitions into massless state within the black hole, loosing identity they have in d = (3 + 1)? Do we really understand black holes inside the the horizon? v. If the odd Clifford algebra explains the Dirac's second quantization postulates, what else can we learn out of the talk presented in this workshop? What consequences does the proposal bring for both standard models? vi. Can one relate the model in which universe consists of closed packed vacuum domains and the models with proposed actions, presented in this workshop? Having a poet among us we kindly asked her to offer us a poem. Thanks Astri, we publish for each volume one of your poems. May be our participants can up to next year translate your poems in the language of their countries. Let us conclude this preface by thanking cordially and warmly to all the participants, present through the teleconferences at the Bled workshop, for their excellent presentations and also, in spite of all, for really fruitful discussions. Norma MankoC BorStnik, Holger Bech Nielsen, Maxim Y. Khlopov, (the Organizing comittee) Norma MankoC BorStnik, Holger Bech Nielsen, Dragan Lukman, (the Editors) Ljubljana, December 2020 Predgovor k drugemu zvezku zbornika Vsakoletne delavnice z naslovom ,,Kako preseči oba standardna modela, koz-moloskega in elektrosibkega" ("What Comes Beyond the Standard Models?") sta postavila leta 1998 Norma in Holger z namenom, da bi udeleZenci v izčrpnih diskusijah kritično soočali različne ideje in teorije. Delavnice domujejo v Plemljevi hiši na Bledu ob slikovitem jezeru, kjer prijetni sprehodi in pohodi na čudovite gore, ki kipijo nad mestom, ponujajo priložnosti in vzpodbudo za diskusije. Tudi to leto je bila delavniča v juliju, vendar nam je tokrat čovid-19 onemogočil srečanje v Plemljevi hisi. Tudi diskutirali nismo med hojo okoli jezera ali med hri-bolazenjem. Vendar nam je dolgoletna izkušnja s "čosmovio" —videopovezavami z laboratoriji po svetu — omogočila, da je tudi letos stekla Blejska delavniča, tokrat prek interneta. Uporaba sistema "Cosmovia" je omogočila udelezbo veliko večjemu številu študentov. Organizatorji so se zato odločili da zbornik delavniče izdajo v dveh delih. V tem drugem delu so v glavnem prispevki studentov, pa tudi prispevki, ki sodijo v prvi zbornik, a smo jih dobili v zadnjem trenutku. Predgovor k prvemu zvezku zbornika vsebuje kratek pregled vseh tem, o katerih je tekla beseda na letosnji delavniči, s področij osnovnih fermionskih in bozonskih polj in koz-mologije. V tem zvezku sledi temu predgovoru predgovor prvega zvezka. V dosedanjih delavničah so dolge predstavitve del, s premori in nadaljevanji preko večš dni, ki so jim sledile izčšrpne diskusije, vodile do zelo pedagosških razlag, ki so pomagale razumeti predpostavke in zaključke prispevkov. Cetudi je čosmovia poskrbela, da so diskusije tekle ves čas, tako kot je bilo na vseh delavničah doslej, blejskih diskusij v zivo diskusije po internetu niso mogle nadomestiti. To leto nam ni uspelo pripraviti razdelka z diskusijami, ki bi predstavile skupna dela udelezenčev, ki so se začela z diskusijami. Razprave na daljavo niso bile dovolj učšinkovite. Zato ponujamo vprašanja, predloge in dvome, ki so med delavničo ostali neod-govorjeni, ali pa nanje ni bilo pravih odzivov. Upamo, da bodo vsi udelezšenči — vabljeni predavatelji in študenti — poskusili sodelovati v iskanju odgovorov, novih razlag, idej, dokazov, dvomov, vprasšanj in numeričšnih izračšunov, preko izmenjave elektronskih sporočšil do naslednjega leta. Vsi udelezenči so vabljeni, da dodajo svoja lastna vprasanja, predloge, dvome itd. Orgnizatorji jih bomo posredovali vsem. Nastala skupna dela pa bomo objavili v naslednjem zborniku. i. Veliko je prispevkov na temo "prvotne črne luknje", ki najbi pojasnili izmerjene gravitačijske valove in ki se domnevno sprosčajo pri zlitiju črnih lukenjz masami okrog 150 sončevih mas. Pojavijo se številna vprasanja: i.a. Kaj je "prvotni čas", čas nastanka teh črnih lukenj? V kakšnem stanju je bilo tedajvesolje? Je to čas po elektrošibkem faznem prehodu vesolja, ali je bilo to pred tem faznim prehodom, ko so imeli (običajni) fermioni in antifermioni maso nič in so imela tudi vsa bozonska polja maso nič. Ali pa so pri nastanku teh črnih lukenjbili aktivni tudi stabilni hadroni masivnih kvarkov in leptonov druge gruče stirih druZin, ki jih napove teorija spinov-nabojev-druZin? So dimenzije, večje kot (3 + 1), če obstajajo, ostale kompaktifičirane, in ali so modeli z več kot (3 + 1) dimenzijami sploh ponudijo pravi opis nasega vesolja? i.b. Kajpovzroča inflačijo? i.č. Kako nehomogenost vesolja z masivnimi kvarki in antikvarki ustvari črne luknje? ii. Teorija spinov-nabojev-druzin postavi preprosto akčijo, v kateri fermioni z maso nič interagirajo samo z gravitačijo v d > (13 + 1) — s tetradami in z dvema vrstama polj spinskih povezav — kar sev d = (3 + 1 )-razseznem prostoru času manifestira kot vsa poznana vektorska in skalarna umeritvena polja ter kot brez-masna druzina fermionov, ki jih po elektrosibkem prehodu izmerimo kot masivne druzine kvarkov in leptonov. Teorija napove obstojčetrte druzine k znanim trem, temno snov, asimetrijo snov-antisnov, razpad protona, nova skalarna polja. Teorija, ki začne z zelo preprosto akčijo, ne potrebuje dodatnih prepostavk, dodatnih umeritvenih grup ali polj, skalarnih ali vektorskih za razlago opazenih lastnosti in pojavov. Ali lahko ponudi dovoljnatančne napovedi, da bi eksperimenti potrdili napovedane mase četrte druzine kvarkov in leptonov, lastnosti Yukawinih sklop-itev in higgsovega skalarja? Ali so predlogi z dodatnimi grupami in dodatnimi kvarki in leptoni povezani s teorijo spinov-nabojev-druzin? Kaj sprozi fazne prehode v teorijah Kaluza-Kleinovega tipa z d = oo ali s kakšno drugo dimenzijo, denimo d = (13 + 1)?? iii. Iz česa je temna snov? iii.a Ali je to najnizja od stirih druzin, ki ni sklopljena z opazenimi, kot je to v teoriji spinov-nabojev-druzin, ali pa so to temne zvezde, ki imajo lastnosti skoraj čšrnih lukenj? Ali je morda to nova faza vakuuma z običšajnimi kvarki in antikvarki, ki so ujetih v mehurčških? Katere lastnosti in-terakčij omogočajo obstoj taksnih mehurčkov? iii.b. Kakšna je stopnja razpada delčev temne snovi v predlaganih modelih? Ali lahko predlogi za temno snov pojasnijo anomalijo kozmičšnih pozitronov? iv. Kaj so v resniči črne luknje? Ce so znotraj črnih lukenj singularnosti, kaj se zgodi s fermioni in polji znotraj čšrne luknje? Ali znotraj čšrne luknje preidejo v brezmasno stanje in izgubijo identiteto, ki jo imajo v d = (3 + 1)? Ali v resniči razumemo čšrne luknje znotraj njihovega horizonta? v. Ce lahko Cliffordova algebra pojasni Diračove postulate za drugo kvantizačijo, kajdrugega se se lahko naučimo iz predavanjna tejdelavniči? Kaksna spoznanja prinasša predstavljen predlog druge kvantizačije za oba standardna modela? vi. Kako lahko vzporedimo predlog modela, da vesolje tvorijo vakuumske domene, ki so povsod goste, z modeli, ki gradijo na dinamiki fermionov in bozonov, kijih določa akčija? Prosili smo pesnico med nami, da popestri zbornik s pesmijo. Hvala Astri za obe pesmi, kiju bodo morda udeleZenci prevedli v jezik svoje deZele. Naj zaključimo ta predgovor s prisrčno in toplo zahvalo vsem udeležencem, prisotnim preko videokonference, za njihova predavanja in še posebno za zelo plodne diskusije in kljub vsemu odlicno vzdusje. Norma Mankoč Borštnik, Holger Bech Nielsen, Maxim Y. Khlopov, (Organizacijski odbor) Norma MankoC Borštnik, Holger Bech Nielsen, Dragan Lukman, (uredniki) Ljubljana, grudna (decembra) 2020 Preface The series of annual workshops on "What Comes Beyond the Standard Models?" started in 1998 with the idea of Norma and Holger for organizing a real workshop, in which participants would spend most of the time in discussions, confronting different approaches and ideas. Workshops have taken place in the picturesque town of Bled by the lake of the same name, surrounded by beautiful mountains and offering pleasant walks and mountaineering. This year 2020 we still had a workshop in July, but without personal conversations all day and late at night, even between very relaxing walks and mountaineering due to COVID-19 pandemic. We have, however, a very long tradition of videoconferences (cosmovia), enabling discussions and explanations with laboratories all over the world. This enables us to have this year the total 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 presented and discussed in our Bled workshops concern the proposals how to explain physics beyond the so far accepted and experimentally confirmed both standard models — in elementary particle physics and cosmology — in order to understand the origin of assumptions of both standard models and be consequently able to propose new theories, models and to make predictions for future experiments. Although most of participants are theoretical physicists, many of them with their own suggestions how to make the next step beyond the accepted models and theories, experts from experimental laboratories were and are very appreciated, helping a lot to understand what do measurements really tell and which kinds of predictions can best be tested. The (long) presentations (with breaks and continuations over several days), followed by very detailed discussions, have been extremely useful, at least for the organizers. We hope and believe, however, that this is the case also for most of participants, including students. Many a time, namely, talks turned into very pedagogical presentations in order to clarify the assumptions and the detailed steps, analyzing the ideas, statements, proofs of statements and possible predictions, confronting participants' proposals with the proposals in the literature or with proposals of the other participants, so that all possible weak points of the proposals, those from the literature as well as our own, showed up very clearly. The ideas therefore seem to develop in these years considerably faster than they would without our workshops. This year neither the cosmological nor the particle physics experiments offered much new, as also has not happened in the last two years, which would offer new insight into the elementary particles and fields and also into cosmological events, although a lot of work and effort have been put in, and although there are some indications for the existence of the fourth family to the observed three, due to the fact that the existence of the fourth family might explain the existing experimental data better, what is mentioned in this proceedings, as we did in the last year proceedings. Also the newest analyses of the data from LHC and other experiments has not changed the situation much. Of particular interest is the observed gravitational waves signal triggered by black holes of around 150 solar masses. These measurements are of the central interest of many a contribution in this proceedings. However, there are more and more cosmological evidences, which require the new step beyond the standard model of the elementary fermion and boson fields. Understanding the universe through the cosmological theories and theories of the elementary fermion and boson fields, have, namely, so far never been so dependent on common knowledge and experiments in both fields. We are keeping expecting that new cosmological experiments and new experiments in laboratories together will help to resolve the open questions in both fields. On both fields there appear proposals which should explain assumptions of these models. Most of them offer small steps beyond the existing models. The competition, who will have right, is open. The new data might answer the question, whether laws of nature are elegant (as predicted by the spin-charge-family theory and also — up to the families — other Kaluza-Klein-like theories and the string theories) or "she is just using gauge groups when needed" (what many models assume, also some presented in this proceedings). Can the higgs scalars and the Yukawa couplings be guessed by small steps from the standard model case, or they originate in gravity in higher dimensions as also the vector and scalar gauge fields do? Is there only gravity as the interacting field, which manifests in the low energy regime all the vector gauge fields as well as the scalar fields, those observed so far and those predicted by the spin-charge-family theory, with the scalar colour triplets included ? Should correspondingly gravity be a quantized field like all the vector and the scalar gauge fields — possibly resulting from gravity — are? Is masslessness of all the bosons and fermions, with scalar bosons included, essential, while masses appear at low energy region due to interactions and breaks of symmetries? Do the observed fermion charges indeed origin from spins of fermions in higher dimensions? What is then the dimension of space-time? Infinite, or it emerges from zero? One of contributions discusses also this problem. Does "nature use" odd Clifford algebra to describe fermions, what leads to anticommutation relations for second quantized fermions, explaining the Dirac's postulates, making already the creation operators for single fermion state anticommuting? What "forces" fermions to appear in families? How many families do we have and what is their relation to the observed ones? What are reasons for breaking symmetries — discrete, global and local? Is The Lorentz invariant really violated? Does the symmetry between fermions and antifermions manifest also in the presence of gravity? Do the baryons of the stable family, decoupled from the observed ones, and predicted by the spin-charge-family theory (or can follow from heterotic string model), contribute to the dark matter? Do new stable quarks constitute neutral particles like neutrons, or form negatively charged particles, bound with primordial helium in dark atoms? How close are the additional new fermions, added to quarks and leptons of the standard model "by hand", to the stable fifth family of the spin-charge-family theory? Are also the charged "nucleons" of OHe's atoms explainable with the stable nucleons of the fifth family? Is the dark matter explainable within the standard model? Or does the dark matter manifest in dark stars, which are a kind of black holes? What are indeed the black holes? If they ought to be created in the primordial time during the inflation (early matter stages or phase transitions), what kind of fermions and antifermions should contribute to the creation of black holes, massless (that is before the electroweak transition) or massive? What did cause the inflation? If there are singularities inside a black hole what is the status of fermions and fields inside the black hole? Do they make phase transitions into massless state within the black hole, loosing identity they have in d = (3 + 1)? Do we really understand black holes inside the the horizon? We discussed these and many other open topics during Bled workshop 2020. Like it is the new idea of theory of strings, represented by particle objects, which do not develop in time. The DAMA/LIBRA experiment convinced us again that the group in Gran Sasso do measure the dark matter particles scattering on the nuclei of their measuring apparatus. It is expected that sooner or latter other laboratories will confirm the DAMA/LIBRA results. This has not yet happened and our discussions clarified the reasons for that. Although cosmovia served the discussions all the time (and we are very glad that we did have in spite of pandemia the 23rd 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. As students need personal discussions with a good teacher, Internet discussions can never replace the real one. Let us point out that we still succeeded to discuss the open problems on present understanding of the elementary particle physics and cosmology in the fully online regime, trying to save the most important feature of Bled Workshops - their free streaming discussion resulting in the comprehensive view on the discussed phenomena and ideas. And let us add 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. Their presentations are published in the second part of the proceedings, together with the invited talks, which came at the very last moment. The organizers strongly hope that next year the covid-19 will be defeated, this is the hope for the whole world, for the young generation in particular and for all of us, with the Bled workshop 2021 included. Let us meet at Bled! (This year's experience made us to think on more practical videoconferencing tools, like Zoom to facilitate extension of our discussions online.) 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 not really polish their contributions. Organizers hope that this is well compensated with fresh contents. Questions and answers as well as lectures enabled by M.Yu. Khlopov via Virtual Institute of Astroparticle Physics (viavca.in2p3.fr/site.html) of APC have in ample discussions helped to resolve many dilemmas. Google Analytics, showing more than 242 thousand visits to this site from 154 countries, indicates world wide interest to the problems of physics beyond the Standard models, discussed at Bled Workshop. At XXIII Bled Workshop VIA streaming made possible to webcast practically all the talks. The reader can find the records of all the talks delivered by cosmovia since Bled 2009 on viavca.in2p3.fr/site.html in Previous - Conferences. Most of the talks can be found on the workshop homepage http://bsm.fmf.uni-lj.si/. Having a poet among us, we kindly asked Astri to contribute a poem for our proceedings. It is our pleasure that she did listen us and send two poems. We publish both, in each volume one. Let us conclude this preface by thanking cordially and warmly to all the participants, present through the teleconferences at the Bled workshop, for their excellent presentations and also, in spite of all, for really fruitful discussions. Norma Mankoi BorStnik, Holger Bech Nielsen, Maxim Y. Khlopov, (the Organizing comittee) Norma MankoC BorStnik, Holger Bech Nielsen, Dragan Lukman, (the Editors) Ljubljana, December 2020 Predgovor (Preface in Slovenian Language) Vsakoletne delavnice z naslovom ,,Kako preseči oba standardna modela, koz-moloskega in elektrosibkega" ("What Comes Beyond the Standard Models?") sta postavila leta 1998 Norma in Holger z namenom, da bi udeleZenci v izčrpnih diskusijah kritično soočali različne ideje in teorije. Delavnice domujejo v Plemljevi hisi na Bledu ob slikovitem jezeru, kjer prijetni sprehodi in pohodi na čudovite gore, ki kipijo nad mestom, ponujajo priložnosti in vzpodbudo za diskusije. Tudi to leto je bila delavniča v juliju, vendar nam je tokrat čovid-19 onemogočil srečanje v Plemljevi hisi. Tudi diskutirali nismo med hojo okoli jezera ali med hri-bolazenjem. Vendar nam je dolgoletna iskusnja s "čosmovio" — videopovezavami z laboratoriji po svetu — omogočila, da je tudi letos stekla Blejska delavniča, tokrat prek interneta. K našim zelo odprtim, prijateljskim, dolgim in zahtevnim diskusijam, polnim iskrivega sodelovanja, je prispevalo veliko fizikov in čelo nekaj matematikov. V večini predavanj in razprav so udelelezenči poskusili razumeti in pojasniti predpostavke obeh standadnih modelov, elektrosibkega in barvnega v fiziki osnovnih delčev ter kozmoloskega, predpostavke in napovedi obeh modelov pa vskladiti z meritvami in opazovanji, da bi poiskali model, ki preseze oba standardna modela, kar bi omogočšilo zanesljivejsše napovedi za nove poskuse. Ceprav je večina udelezenčev teoretičnih fizikov, mnogi z lastnimi idejami kako narediti naslednji korak onkrajsprejetih modelov in teorij, so se posebejdobrodosli predstavniki eksperimentalnih laboratorijev, ki nam pomagajo v odprtih diskusijah razjasniti resnično sporočilo meritev in nam pomagajo razumeti kakšne napovedi so potrebne, da jih lahko s poskusi dovoljzanesljivo preverijo. Organizatorji moramo priznati, da smo se na blejskih delavničah v (dolgih) predstavitvah (z odmori in nadaljevanji preko več dni), ki so jim sledile zelo podrobne diskusije, naučili veliko, morda več kot večina udelezenčev. Upamo in verjamemo, da so veliko odnesli tudi študentje in večina udelezenčev. Velikokrat so se predavanja spremenila v zelo pedagoške predstavitve, ki so pojasnile predpostavke in podrobne korake, soočšile predstavljene predloge s predlogi v literaturi ali s predlogi ostalih udelezšenčev ter jasno pokazale, kje utegnejo tičšati sšibke točške predlogov. Zdi se, da so se ideje v teh letih razvijale bistveno hitreje, zahvaljujoč prav tem delavničam. Tako kot v preteklih dveh letih tudi to leto niso eksperimenti v kozmologiji in fiziki osnovih fermionskih in bozonskih polj ponudili rezultatov, ki bi omogočili nov vpogled v fiziko osnovnih delčev in polj, čeprav je bilo vanje vloženega veliko truda in četudi razberemo iz eksperimentov, da četrta druzina k ze izmerjenim trem mora biti, saj lahko s štirimi druzinami lazje pojasnimo izmerjene podatke, kar je omenjeno tudi v tem zborniku. Tudi zadnje analize rezultatov merjenj na LHC in drugih merilnikih niso pripomogle k boljsemu razumevanju naravnih zakonov v fiziki osnovnih delcev in kozmologiji. Posebno pozornost so vzbudile meritve gravitacijskih valov, ki so jih povzrocile crne luknje z masami okoli 150 soncnih mas. Prav te meritve poskušajo razloziti nekateri prispevki v letosnjem zborniku. Vse vec je tudi kozmoloških meritev, za katere se zdi, da jih standardni model osnovnih fermionski in bozonskih poljne more pojasniti. Se nikoli doslejniso bili predlogi za kozmoloske teorije in iskanje nove teorije v fiziki osnovnih polj tako zelo soodvisne od poizkusov in razumevanja predpostavk na obeh podrocjih. Pricakujemo, da bodo kozmološka merjenja in meritve v laboratorijih pomagala razresšiti odprta vprasšanja na obeh podrocšjih. Na obeh podrocšjih je predlogov za novo teorijo cšedalje vecš, vendar velika vecšina teh predlogov ponuja majhna odstopanja od standardnih modelov. Tekma, kdo ima prav, je odprta. Nove meritve bodo morda kmalu ponudile odgovor na vprasanje, ali so naravni zakoni elegantni (kot napoveduje teorija spina-naboja-druzin in tudi druge teorije Kaluze in Kleina, vendar brez druzin in ne tako "udarno") ali pa "narava uporabi grupe, ki in ko jih ravno potrebuje" (kar predlaga velika vecina modelov, tudi nekateri v tem zborniku). Ali je smotrno pojav Higgsovega skalarnega polja in Yukawinih sklopitev dodati k standardnemu modelu osnovnih delcev kot dodatno polje, ki ga zahtevajo poskusi, ali pa je v resnici skalarnih poljvec, njihov izvor pa je gravitacijsko polje v razsesnostih d > (3 + 1)? So vsa osnovna fermionska in bozonska polj, tudi skalarna, brezmasna in je njihova masa, ki jo merimo pri nizkih energijah, posledica sil in zlomitve simetrij? Ali izvirajo naboji fermionov, ki jih izmerimo pri nizkih energijah, v spinih, ki jih ti fermioni nosijo v d > (3 + 1)? Kaj tedaj prostor in cas v resnici pomenita? Sta neskoncna, ali pa se rodita iz nic ? Ali "narava uporabi" liho Clifordovo algebro za opis fermionov, kar zagotovi antikomutacijske relacije med kreacijskimimi in anihilacijskimi operatorji ze med enofermionskimi stanji, kar pojasni Diracove postulate za fermione v drugi kvantizaciji? Kaj"prisili" fermione, da se pojavijo v druzinah? Koliko je druzin kvarkov in leptonov in kako so povezani, ce sploh, z izmerjenimi tremi druzšinami? Kaj povzrocši zlomitev simetrij, diskretnih, globalnih, lokalnih? Ali je Lorentzova simetrija zlomljena in ce je, pod kakšnimi pogoji se zlomi? Ali je simetrija med fermioni in antifermioni v gravitacijskem polju zlomljena? Kajso gradniki temne snovi? Ali so barioni druzin, ki niso sklopljene z izmerjenimi druzšinami kvarkov in leptonov in jih napove teorija spina-nabojev-druzšin, del temne snovi v vesolju? Ali se lahko novi fermioni, ki jih dodajo k kvarkom in leptonom standardnega modela osnovnih fermionskih in bozonskih polj, dajo pojasniti s stabilnimi barioni, ki jih napove teorija spina-nabojev-druzin? So tudi temna jedra atoma O-He-lija clani stabilne druzine? Se da temna snov pojasniti s skupki kvarkov in leptonov standardnega modela? Ali pa k temni snovi prispevajo temne zvezde, ki imajo lastnosti crnih lukenj? Kaj pa so v resnici crne luknje? Ce so nastajale ob inflaciji, kaksni fermioni in antifermioni so sodelovali pri nastanku crnih lukenj, z maso nic (to je pred elek-trosibkim faznim prehodom) ali z nenicelnimi masami? Kajje povzrocilo inflacijo? Ce ima crna luknja singularnost, kako se spremenijo lastnosti fermionov in an- tifermionov znotrajčrne luknje? Ali izgubijo lastnosti, ki so jih imeli v d = (3 + 1 )-razsežnem prostoru? Ali razumemo, kajse dogaja v crni luknji znotrajhorizonta? Ta in še marsikatera druga vprašanja smo naceli v casu Blejske delavnice 2020. Denimo kot to, da v novi teoriji strun, ki jo sestavljajo točkasti delci, cas sploh ne nastopa. Meritve DAMA/LIBRA v Gran Sassu so nas znova prepricale, da so delci, ki se sipljejo na atomskih jedrih merilcev in ki skozi leto periodicno spreminjajo svojo intenzivnost, delci temne snovi. Pricakujemo, da bo laboratorijem po svetu, ki poskusajo potrditi njihove meritve, prej ali slej to tudi uspelo. Vprasanja in odgovori so pomagali razumeti, zakajnobenemu doslejpotrditev se ni uspela. (Četudi je cosmovia poskrbela, da so diskusije tekle ves cas, tako kot je bilo na vseh delavnicah doslej, blejskih diskusijv zivo diskusije po internetu niso mogle nadomestiti. Diskusije so bile ognjevite in ostre, vsajpri nekaterih predavanjih, vendar potrebujejo ucinkovite diskusije osebno prisotnost diskutantov in poslusalcev, ki z vprašanji poskrbijo, da je debata razumljiva vsem. Tudi študentom internet ne more nadomestiti dobrega ucšitelja. Poudariti je potrebno, da nam je kjub temu uspela dokaj plodna diskusija o tem, kako dobro razumemo danes obe podrocji, fiziko osnovnih delcev in polj ter dinamiko nasšega vesolja. In dodajmo, da je delavnica preko interneta omogocšila sudentom aktivno in plodno sodelovanje, ki bi se ga v zivo zaradi stroskov potovanja ne mogli udelezšiti. Sšiudentski prispevki so zbrani v drugem zborniku Blejske delavnice, skupaj s prispevki vabljenih predavateljev, katerih prispevke smo prejeli zadnji trenutek. Organizatorji upamo, da bo naslednje leto virus premagan, naše upanje velja za ves svet, za mlado generacijo pa se posebej, pa tudi za Blejsko delavnico 2021, da bo stekla v zivo na Bledu. Ker je vsako leto le malo casa od delavnice do zakljucka redakcije, manj kot dva meseca, avtorji ne morejo dovolj skrbno pripravti svojih prispevkov, vendar upamo, da to nadomesti svezina prispevkov. Bralec najde zapise vseh predavanj, objavljenih preko "cosmovia" od leta 2009, na viavca.in2p3.fr/site.html v povezavi Previous - Conferences. Vecino predavanj najde bralec na spletni strani delavnice na http://bsm.fmf.uni-lj.si/. Prosili smo Astri, da nam poslje kako od svojih pesmi. Prijazno nam je ugodila in poslala dve. Objavljamo obe, v vsakem zborniku po eno. Naj zakljucimo ta predgovor s prisrcno in toplo zahvalo vsem udeležencem, prisotnim preko videokonference, za njihova predavanja in še posebno za zelo plodne diskusije in kljub vsemu odlicno vzdusje. Norma Mankoč Borštnik, Holger Bech Nielsen, Maxim Y. Khlopov, (Organizacijski odbor) Norma Mankoč Borštnik, Holger Bech Nielsen, Dragan Lukman, (uredniki) Ljubljana, grudna (decembra) 2020 Further Talks All talk contributions are arranged alphabetically with respect to the authors' names. Bled Workshops in Physics Vol. 21, No. 2 A Proceedings to the 23rd [Virtual] Workshop, Volume 2 What Comes Beyond ... (p. 1) Bled, Slovenia, July 4-12, 2020 1 The Concept of Cosmological Inflation and the Origin of 3 + 1 -dimensional Space-Time in a Universe Consisting of Conserved Vacuum Domains E. Dmitrieff * Irkutsk State University Abstract. We start from the assumption that the universe is a dense filling of small (Planck-scale or so) vacuum domains. Also we postulate that the total count of these domains is mostly conserved, but there are no strong restrictions on the coordination number of domains in their mutual packing. The dimensions count of such a universe is not predetermined but is defined by its characteristic coordination numbers. We found out that under these conditions, the universe size, counted by inter-domain hops, falls down exponentially along with growth of dimension count, defined by the coordination number. The universe formed by hypercubic grid with more than 6 dimensions becomes microscopic and therefore causal connected. Therefore the backward process of lowering the dimension count can be considered as a kind of exponential inflation that may start from two domains-sized universe, having hundreds of dimensions, and practically freeze down between 4 and 3 dimensions because of huge emergent sizes and local causality. We explored dense packing in 2, 3 and 4 dimensions and found out that 4-dimensional flat torus with three ordinary and one maximally compactified dimensions pretends to be energetically optimal. So it can correspond to the true vacuum. The discreteness of domains does not allow the compactified dimension to collapse completely, so its length can not be less then two domains. Our results show that such a spontaneous break of dimension symmetry in a tessellation of electrically charged domains slightly violates CP symmetry, leaving CPT symmetry conserved, and could explain the difference in properties of photons and gluons by corresponding symmetry break from 4 to 3+1 colors. Povzetek. Avtor privzame, da vesolje sestavlja gosto pokritje majhnih (reda Planckove skale) vakuumskih območij, da se celotno stevilo domen približno ohranja in da tudi njihova oblika in s tem "koordinacijska stevila", ki dolocajo dimenzijo vesolja, niso strogo predpisani. Avtor ugotovi, da pri teh pogojih velikost vesolja, ki je definirana s stevilom skokov med domenami, eksponentno pada z vecanjem stevila dimenzij. Vesolje, ki ga tvori hiperkubicna mreza z vec kot 6 dimenzijami, postane mikroskopsko in zato vzrocno povezano. Obraten proces, to je znizevanje dimenzij vesolja, vidi avtor kot eksponentno inflacijo vesolja, ki zacne z dvema domenama in z nekaj sto dimenzijami in zamrzne med dimenzijama 3 in 4 kot zelo razsezno in lokalno vzrocno povezano vesolje. * E-mail: elia@sr.isu.ru 2 E. Dmitrieff V prispevku avtor predstavi gosto pakiranje domen v dimenzijah 2, 3 in 4 ter zaključi, da se zdi energijsko optimalen ploski 4-dimenzionalni torus s tremi običajnimi dimenzijami in eno maksimalno kompaktificirano. Zaradi diskretnosti domen se kompaktificirana dimenzija ne more zmanjsati na manj kot dve domeni. Avtor oceni, da taksna spontana zlomitev dimenzij v teselaciji električno nabitih domen vodi k rahli kršitvi simetrije CP, ohranja pa simetrijo CPT. Zakljuci, da njegov pristop morda pojasni zlomitev barvne simetrije na gluonsko in fotonsko. Keywords: tessellation approach, dimension decay, cosmological inflation, causality, flat torus, satori PACS: 02.40.Sf, 02.70.-c, 04.50.Gh, 04.60.Nc, 12.60.-i, 98.80.Bp, 98.80.Jk 1.1 Methodics Speculating about physics and cosmology beyond Standard Models, we follow the concept of 'universe as a tessellation'. The word universe here means an artificial mathematical and computational model. We build it, starting with a set of initial assumptions or axioms. Then, we explore the consequences and compare whether they have suitable correspondences in the real Universe. 1.1.1 Axioms The assumptions that we start with in the tessellation approach, are the following: 1. We treat the physical vacuum, and the universe as whole, as dense and (almost) regular filling of small vacuum domains. Here we neither assume existence of space or time of any kind, nor dimension count; neither of fluctuating fields of different types, nor of gravity, quanta, relativity, nor topology. We just assume that there are domains, they are small enough to be particle's constituents, and they can neither completely overlap each other, nor mutually penetrate, so they form a sort of tessellation or bubble foam. Each particular domain shares some walls with several other neighbor domains. These walls between domains emerge because domains do not overlap. However, domain walls with limited curvature could be postulated, instead of domains, as principal entities. In this case, domains would emerge on sides of walls. 2. Vacuum domains have electrical charge either +| or — |. The magnitude is chosen to be ± 6 e, to be able to reproduce the electrical charges and other quantum numbers of fundamental particles, as combinations of the domains. According to the postulate, each domain carry one bit of information. This assumption can be formulated with the dual way: Each domain carry one bit of information. So there are two kinds of domains. The electrical charge of any sample taken emerges as a numerical difference between counts of domains of one kind and of another kind in this sample. In case of principal walls in the first assumption, the second one would postulate that some of walls are oriented. Positive and negative domains emerge on different opposite sides of oriented wall. 1 The Concept of Cosmological Inflation and the Origin. 3 1.1.2 Higgs field Hereby, the scalar potential of the electric charge density, introduced above, pretends to be the real-valued Higgs-like field. It is non-zero almost everywhere, excepting walls. Inside domains, it keeps near or equal to either positive or negative constant with the same magnitude. Suppose that two domains, either of equal charges, or of opposite charges, get closer to each other. So they would penetrate or overlap. Their electrical charge densities would interfere, and the electrical potential would fall or rise by the magnitude, getting away from the optimal constant. According to the first assumption, these domains would pull back, towards the stable equilibrium point. We can treat it as rising of effective energy on both sides of positive and negative optimal values. So the function of effective energy must be at least fourth-power polynomial, looking like Higgs field V($). Using the Higgs field vacuum expectation value, we estimate the domain size that appears to be about 10-21 m. 1.1.3 Particles We found out that these two assumptions are quite enough to represent all the known fundamental particles with correct quantum numbers [1]. While the pure vacuum is supposed to be a regular alternation of domain of both kinds, the cases of violations of periodicity could be treated as some particles. In other words, particles are anti-structure defects of periodical tessellation, or bound states of them. 1.1.4 Particle formulas Using equivalence between bits and domains, we can write particle formulas as clusters of several bits. Note that these bit clusters are not bit strings that represent integers in computers. The mutual arrangement of bits is significant as well as mutual arrangement of defects among domains. So the bit formulas for particles must be structural, like formulas of organic compounds in chemistry. 1.1.5 Non-particle excitations in tessellation In addition to particle defects, other types of excitation can exist in this model, for example, in the form of compression, displacement, and torsion waves, which could be identified, for instance, with gravitational waves or dark matter. 1.1.6 Dimension count Following the conception defined above, we introduce the space dimension count d as a function of count of walls, shared with domain's neighbors, average or exact. 4 E. Dmitrieff If there are 2 neighbors, then the emergent space is one-dimensional; 3 to 6 neighbors is a characteristic of two-dimensional space. Three dimensions appear when there are 6 to 14 neighbors, four in case of 26 or some more. So the dimension count is emergent, not fixed, determined by the average coordination number, that is a count of the nearest neighbors, that is a characteristic of particular tessellation. In case of domains rearrangement, the coordination number, and therefore, the dimension count can change. Various configurations can have different effective energy, so some of them can be stable, meta-stable, or unstable. Non-stable domain configurations, rearranging into more stable ones with change of the neighbor count can also change the dimension count. It may be called the dimension decay. 1.1.7 Universe size As a distance between two domains we consider the minimal count of inter-domain walls to be crossed on a way from one domain to another. As the universe size L we treat the maximal distance between any pair of domains. 1.1.8 Assumption of domain count conservation In current study, we introduce an additional assumption that the total count of domains in the universe is strictly or mostly fixed, i.e. processes of their formation and destruction are suppressed or absent and we can treat domain count as a constant. We show that this assumption leads to the consequence, that the size of universe depend on the number of dimensions exponentially. We suppose that the maximal number of dimensions can be achieved in the model having the topology of a multidimensional simplex: each domain would be a neighbor of all the others, so universe size would be 1. But already in the case of a hypercube built from the same number of domains as the number of domains (10-21 m in size) in the Universe of about 1010 light years, the maximum number of dimensions is only about 468, while such a 468-dimensional universe has a size of about 10-20 m. Rearrangement of domains with a decrease in dimension count leads to an increase in linear sizes, but only at 6 dimensions the universe becomes macroscopic, of the order of 0.1 mm. A further decrease in dimensions is accompanied by an exponential increase in linear size to about 100 km at d = 5 and to about 1000 astronomical units at d = 4. In this step, there happens a loss of correlation between different parts of the universe, which, apparently, has been kept until this time: we have no reason to believe that the speed of oscillations that determines light speed in multidimensional structures should be significantly lower. The last decay of 4-dimensional space into 3-dimensional, or compactification of one dimension with the formation of a flat torus, leads to the formation of a 1 The Concept of Cosmological Inflation and the Origin. 5 universe of modern size. Further decay into the usual 3-dimensional, or 2- or 1-dimensional space, most likely, is energetically disadvantageous, since it is the four-dimensional space that offers the best saving of cross-domain walls with 26 neighbors. 1.2 Calculations In a tessellation universe with some particular d there will be N repetitions of the elementary translation unit in each of d dimensions. Suppose as first, that the Ns are the same in all d dimensions, so it is a hypercube with the edge of N domains along it. Then, the total amount of domains C = Nd (1.1) In case Ns are different, i.e. there are N0 = Ni, etc, the shape is a hyperrectan-gle, and d-1 C = n Nn (1.2) n=0 But let us examine just the simplest and realistic hyper-cubic case. Following the assumption that C = const, we evaluate this constant using the approximation of the Universe size L (20 billions light year): First, we express universe size as a power of 2 (we use this basis because of simplicity of formulas for hypercubic grids): L = 20 x 109years x 3 • 108 m x 365- x 24 x 60 x 60^^ « 2• 1026m « 287m. (1.3) s 4 year And we use d = 3 as a dimension count of our Universe. The universe volume Vu thus will be of order L3, i.e. Vu « 2261m3. (1.4) Also, we use the inter-domain size I, that is twice of the domain radius. If domain sizes are at the Planck scale, it will be about Wnck = 10-34m « 2-113m. (1.5) If our hypothesis of the Higgs and Coloumb field Unity (HCU) is correct, the domains will be about Ihcu = 10-21 m « 2-70m. (1.6) So the volume of one domain is supposed to be Vd = (10-34)3 = 10-102m3 « 2-339m3(Planck), (1.7) Vd = (10-21)3 = 10-63m3 « 2-209m3(HCU) (1.8) 6 E. Dmitrieff Then, dividing the universe volume by the domain volume, we get the domain count: C - Vu - 2 C Planck — T~r~ — 2 Vd C Vu 2 Chcu — t7" — 2 Vd 261+339 — 2 (Planck), T261 +209 — 2 (HCU). (1.9) (1.10) Since C is supposed to be conserved through d-changing decays, we can get N for any given dimension count d: N = C d Multiplying with the domain diameter l, we get universe size: L = N x I = C d x I For HCU domains with I = 2-70m it gives L = N x I = 2^ x 2-70 = 2470-70(m). (1.13) (1.11) (1.12) Since N is discrete (it is a count of domains in particular direction ), it can not be less than 2: N > 2. Thus, n — C ^ — Cpdmanck > 2 -> 2 ^ >21, (1.14) so dmax < 600 with Planck-size domains, or dmax < 470 with larger HKU domains. In the extreme case, when d = dmax, the universe would consist of only one translation unit, so it would be not periodical, it would not a be a tessellation at all, but it would be just a single polytope. However, it would be wery good connected, all the parts would be correlated, since the sizes are small, there are at most 600 hops from any domain in the universe to anyone else, and the Euclidean distance is not greater than %/600 « 25. All above is correct for hyper-cubic tessellations and polytopes. In case of simplex polytopes (triangle, tetrahedron and so on), the count of dimensions is not so strictly limited, it can rise almost to infinity, up to d = C. 1 The Concept of Cosmological Inflation and the Origin. 7 But we consider only hyper-cubic tessellations, not simplex polytopes, since all known generalized Kelvin's problem's candidate solutions, that are optimal packed tessellations, as we have shown, could be produced from hyper-cubic grids by linear shifts, conserving most of their properties, including the parity. So the estimations above (with minor corrections) can be applied to the bee honeycomb, Kelvin and Weaire-Phelan structures as well as to 4-dimensional Satori structure. Now we can calculate the sizes (in meters) for the hyper-cubic universe of any dimension count. For instance, for d = 6 with HCU domains: . 470 L = N x I = 2tt0-70m = 28m = 256m. (1.15) d = 5 gives L = N x I = 2470-70m = 224m = 16000km. (1.16) 6-dimensional universe with Planck-sized domains will be L = N x I = 2600-113m = 2-13m « 0.1mm, (1.17) while 5-dimensional universe's size L = N x I = 2t0-113m = 27m = 128km. (1.18) 1.3 Clifford flat torus The distribution or alternation of domains is presumably determined by minimizing the energy of their neighboring contact with each other. Therefore, we mainly consider tessellations that are solutions or candidates for solutions to the Kelvin's problem of optimal packing [3]. To work with a four-dimensional space, or with spaces of higher dimensions, as well as to get rid of a pre-selected dimension count, we proposed an approach to measure the economy of filling [4], which is independent of the dimension count and is based on comparison with the corresponding simple hypercubic lattice. Thus, we have made possible the generalized formulation of the Kelvin's problem, in which the search for the optimal filling is not limited to three-dimensional space. Instead, the dimension of space appears along with the solution of the problem, as a characteristic of space, in which this solution can be nested. The four-dimensional structure that we found and named 'Satori', is a candidate solution for the generalized Kelvin's problem. Like the three-dimensional candidate solution of Weair and Phelan (which offers a bit less economy), the Satori structure is chiral, and it offers CPT symmetry. 8 E. Dmitrieff Applying the principle of energy minimization to the Satori structure with anti-structural defects, we found out that the favorable topology for the model is the four-dimensional Clifford torus with a period of one translation unit. The result is 'almost' three-dimensional space, having one additional com-pactified dimension, the radius of which is emergently fixed. Only in this case it 1 The Concept of Cosmological Inflation and the Origin. 9 is possible to have defects whose energy is zero, which corresponds to massless particles like photons and neutrinos. In the absence of compactification or when compactification occures not in one but in two or more periods, such defects would be massive and the space as a whole would decay by strengthening compactification up to a minimum of one period. Since the three-dimensional torus is not flat but curved, the same compactification in more than one dimension is likely energetically prohibited due to the curvature energy. We consider the gluon chain termination with quarks as another way to reduce energy instead of looping the chain. Considering the structure of Satori in the topology of such a torus, we find that it turns out to be oriented along a compactified dimension: all four three-dimensional layers that form the centers of cells or domains are different. Thus, if we consider these layers as phases of the oscillation of a three-dimensional structure, this oscillation will have the appearance of directional rotation, in which four different phases are ordered in turn, and the two possible directions of sequencing are different. In this case, the movement of individual domains occurs in such a way that each domain can either rotate in place or move along the remaining three dimensions. However, there is no difference between the two cases due to the fact that the domains are indistinguishable from each other and it is impossible to say which domain is spinning in place and which one is moving. The foregoing relates to the Satori structure with a compactified dimensioin in the absence of defects. In the case of defects existing in it, a difference in the electric charge appears, and such a defect can either move or spin or alternate one both way of moving. A domain cannot remain in place, since at different phases the same place is occupied by domains of different signs. Thus, when passing from layer to layer, the defect undergoes bifurcation. An exception is when the defect moves along the model with the highest possible speed. In this case, there are no rotational transitions. This behavior of defects allows us to identify it with the motion of particles, for which, with approaching the speed of light, increased half-life is observed, which is usually associated with a slowdown in own time. In our model, the own time of a particle turns out to be a phenomenon associated with branching during the movement of the corresponding defect in a vacuum undergoing directed oscillations: the amount of own time is determined by the fraction of branching at which the choice is not determined. References 1. E.G. Dmitrieff: Experience in modeling properties of fundamental particles using binary codes, in: N.S. Mankoc Borštnik, H.B.F. Nielsen, D. Lukman: Proceedings to the 19th Workshop 'What Comes Beyond the Standard Models', Bled, 11. - 19. July 2016. 2. E.G.Dmitrieff: On triple-periodic electrical charge distribution as a model of physical vacuum and fundamental particles, in: N.S. Mankoc Borstnik, H.B.F. Nielsen, D. Luk- 10 E. Dmitrieff man: Proceedings to the 21th Workshop 'What Comes Beyond the Standard Models', Bled, 23. - 29. June 2018 3. D. Weaire et al., The Kelvin Problem, Taylor & Francis, 1996. 4. E.G.Dmitrieff: Tessellation approach in modeling properties of physical vacuum and fundamental particles, in: N.S. Mankoc Borstnik, H.B.F. Nielsen, D. Lukman: Proceedings to the 22nd Workshop 'What Comes Beyond the Standard Models', Bled, July 6-14, 2019 Bled Workshops in Physics Vol. 21, No. 2 A Proceedings to the 23rd [Virtual] Workshop, Volume 2 What Comes Beyond ... (p. 11) Bled, Slovenia, July 4-12, 2020 2 Neutrino Masses and Mixing Within a SU(3) Family Symmetry Model with Five Sterile Neutrinos A. Hernandez-Galeana * Departamento de Física, ESFM - Instituto Politécnico Nacional. U. P. "Adolfo Lopez Mateos". C. P. 07738, Ciudad de Mexico, Mexico Abstract. Within a broken SU(3) local family symmetry, we study neutrino masses and mixing in a framework with five sterile neutrinos. In this BSM, ordinary heavy fermions, top and bottom quarks and tau lepton, become massive at tree level from Dirac See-saw mechanisms implemented by the introduction of a new set of SU(2) L weak singlet vector-like fermions, U, D, E, N, with N a sterile neutrino. Right handed neutrinos are introduced to cancel anomalies. We provide a parameter space region where this framework can account for the neutrino masses (mi = 0.00584795 , m2 = 0.0104888 , m3 = 0.051461 , m = 1.21534, ms = 2604.12, m6 = 2643.36, m/ = 9.97002 x 106 , ms = 1.00658 x 107) eV, the squared neutrino mass differences m2 - m2 = 7.58162 x 10-s eV2, m2 - m2 = 2.53822 x 10-3 eV2, and m4 — m2 = 1.47441 eV2. We also report the corresponding (Upmns)4x8 lepton mixing matrix. Povzetek. Avtor pojasnjuje obstoj druzin v modelu, v katerem doda grupam standardnega modela grupo SU(3). V prispevku obravnava mase nevtrinov s petimi sterilnimi nevtrini. Avtor doda običajnim kvarkom in leptonom se dva kvarka (U, D) in dva leptona (E, N), vsi so sibki vektorski singleti SU(2)L. Maso kvarkov b in t in leptona tau doloci Diracov "mehanizem see-saw" ze na drevesnem nivoju. N je sterilni nevtrino. Anomalije odpravi tako, da uvede se desnorocne nevtrine. V izbranem obmocju parametrov modela izracuna nevtrinske mase (m1 = 0.00584795, m2 = 0.0104888, ms = 0.051461 , m = 1.21534, ms = 2604.12, m6 = 2643.36, m/ = 9.97002 x 106 , ms = 1.00658 x 107) eV) in kvadrate masnih razlik (m2 — mi = 7.58162 x 10-s eV2, m2 — m2 = 2.53822 x 10-3 eV2 in m4 — mi = 1.47441 eV2). Navede tudi pripadajoco leptonsko mesalno matriko (Upmns )4x8. Keywords: Quark and lepton masses and mixing, Flavor symmetry, AF = 2 Processes. Pacs: 14.60.Pq, 12.15.Ff, 12.60.-i 2.1 Introduction The origen of the hierarchy of fermion masses and mixing continue being one of the most important open problems in particle physics. In this report we address * E-mail: ahernandez@ipn.mx 12 A. Hernandez-Galeana the problem of generating neutrino masses and mixing within the framework of a broken SU(3) gauged family symmetry model [1,2]. This framework introduce a hierarchical mass generation mechanism in which light fermions become massive from radiative corrections, mediated by the massive gauge bosons associated to the SU(3) family symmetry that is spontaneously broken, while the masses of the top and bottom quarks as well as for the tau lepton, are generated at tree level from "Dirac See-saw"mechanisms implemented by the introduction of a new set of SU(2)L weak singlets U, D, E and N vector-like fermions, with N a neutral lepton. In addition this BSM introduce three right handed neutrinos in order to cancel anomalies. Therefore, we have a scenario with five "Standard Model"(SM) singlet "sterile neutrinos" and three active L-handed neutrinos, that is a 3+5 scenario. Previous theories addressing the problem of quark and lepton masses and mixing with spontaneously broken SU(3) gauge symmetry of generations include the ones with chiral SU(3) family symmetry [3]- [6], as well as other SU(3) family symmetry proposals [7]- [10] Neutrinos are one of the most exciting areas of research. Cosmology and Short Baseline Oscillation experiments hint the possible existence of light sterile neutrinos. For recent studies of neutrino masses, including sterile neutrinos, see for instance [11]- [14] 2.2 SU(3) family symmetry model The model is based on the gauge symmetry G = SU(3) SU(3)C <£> SU(2)l U(1 )Y (2.1) where SU(3) is a completely vector-like and universal gauged family symmetry. That is, the corresponding gauge bosons couple equally to Left and Right Handed ordinary Quarks and Leptons, with gH, gs, g and g' the corresponding coupling constants. The content of fermions assumes the standard model quarks and leptons: ¥ = (3,3,2, 3)l , = (3,1,2, —1 )l (2.2) = (3,3,1,4 )R , ¥^(3,3,1, — 2 )r , = (3,1,1, — 2)r (2.3) where the last entry is the hypercharge Y, with the electric charge defined by Q = T3L + 2 Y. The model includes two types of extra fermions: Right Handed Neutrinos: = (3,1,1,0)R, introduced to cancel anomalies [7], and a new family of SU(2)L weak singlet vector-like fermions: Vector like quarks U°,UR = (1,3,1, 3) and DL,DR = (1,3,1, — 2), Vector Like electrons: EL,ER = (1,1,1,—2), and New Sterile Neutrinos: N°,NR = (1,1,1,0). 2 Neutrino Masses and Mixing Within a SU(3) Family... 13 The particle content and gauge symmetry assignments are summarized in Table 2.1. Notice that all SU(3) non-singlet fields transform as the fundamental representation under the SU(3) symmetry. SU(3) SU(3)c SU(2)l U(1)y 3 3 2 1 3 3 3 1 4 3 ^dR 3 3 1 2 - 3 w 3 1 -1 cr 3 1 1 -2 Cr 3 1 1 0 Ul,r 1 3 1 4 3 Dl,r 1 3 1 2 - 3 El,r 1 1 1 -2 NL,R 1 1 1 0 3 1 2 -1 od 3 1 2 +1 m , n)2 3 1 1 0 Table 2.1: Particle content and charges under the gauge symmetry 2.3 SU(3) family symmetry breaking SU(3) family symmetry is broken spontaneously by heavy SM singlet scalars m =(3,1,1,0) and n2 = (3,1,1,0) in the fundamental representation of SU(3), with the "Vacuum ExpectationValues" (VEV's): T = (Ai,0,0) , : ^ (Y+Yr + Y+Y-) + ^ (Z2 + Z2 + 2Zï %) (n2> : ^ (Y+Y- + Y+Y—) + 92hA2z2 The "Spontaneous Symmetry Breaking" (SSB) of SU(3) occurs in two stages SU(3) x Gsm -—— SU(2) ? x GSM —^ GSM FCNC ? (2.6) Notice that the hierarchy of scales A2 > Ai yield an "approximate SU(2) global symmetry" in the spectrum of SU(2) gauge boson masses of order gH A1. Therefore, neglecting tiny contributions from electroweak symmetry breaking, the gauge boson masses read (M2 + M2) Y+Y- + Mf Y+Y- + m2 Y+Y- K a-r2 1 M2 + 4M2 -M2 Z2 + 1 2 2 + -m2 z2 + 1M1 + 4M2 Z2 +1 (m2)7 Zf Z2 (2.7) 3 '73' M2 = ^ M2 = ^ (2.8) Z2 Z2 V3 M2 M Table 2.2: Zf — Z2 mixing mass matrix Diagonalization of the Zi — Z2 squared mass matrix yield the eigenvalues and finally m— = 2 (m1 + m2 — y/(M2 — Mf)2 + m2m2^ M+ = 2 (m2 + M2 + y/(M2 — Mf)2 + Mfm2 (2.9) (2.10) Z2 Z2 (Mf + m2) Y+Y- + Mf Y+Y- + m2 Y+Y- + M- — + M+ (2.11) Z Z M 2 M 2+4M 3 where 2 Neutrino Masses and Mixing Within a SU(3) Family... 15 cos ty sin ty\ /Z sin ty cos ty/ yZ . • , V3_Mi_ cos ty sin ty = —----' - 4 J Mi + M2(M2 - M2) (2.12) (2.13) 2.4 Electroweak symmetry breaking The "Electroweak Symmetry Breaking" (EWSB) is achieved by the Higgs fields and which transform simultaneously as triplets under SU(3) and as Higgs doublets with hypercharges —1 and +1 under the SM, respectively, explicitly: ®u = with the VEV's (®u) = /Yty°V\ ty- ty° ty- ty° (JL NAX 0 ) _L fvu2 V2 I 0 _L (vuA 01/ = / /ty+ ty° ty+ ty° ty+ ty° (®d) = ^ yvdi 0 I vd2 U d3 J The contributions from (®u) and (® d) generate the W and Z° SM gauge boson masses 9S..2 \ (vu + vd) W+W- + ^ ga ^ (vu + vd) Z; (g2 + g' ) ,2 , Z2 d) Z° (2.14) + tiny contribution to the SU (3) gauge boson masses and mixing with Z°, = viu + v2u + v3u , vd = v2d + v2d + v3d. So, if MW = 1 g v, we may write = \JVU + vd « 246 GeV. 2 u 2 3 2 v u 16 A. Hernandez-Galeana 2.5 Fermion masses 2.5.1 Dirac See-saw mechanisms The gauge symmetry G = SU(3) x GSM, the fermion content, and the transformation of the scalar fields, all together, avoid Yukawa couplings between SM fermions. The allowed Yukawa couplings involve terms between the SM fermions and the corresponding vector-like fermions U, D, E and N: The scalars and fermion content allow the gauge invariant Yukawa couplings for quarks and charged leptons Hu f q ur + hi, iUr ni uo + hu IUr n2 ul + Mu u° ur + h.c Hd f q ®d DR + hd, f sr ni dl + hif2 f sr n2 DL + md do dr + h.c He ®d ER + hn, f°R ni EO + h^2 f°R n2 e° + me eo er + h.c MU , Md , Me are free mass parameters and , Hd He , hf, hf2, f = u, d, e are coupling constants. When the involved scalar fields acquire VEV's, we get for charged leptons in the gauge basis f L RT = (e°, |j°,t°,E°)l,r, the mass terms if LM°f R + h.c, where M° = (2.15) / 0 0 0 He vdi\ 0 0 0 Hevd2 0 0 0 He Vd3 VhfAi h|A2 0 ME / It is worth to notice that completed analogous tree level mass matrices are obtained for u and d quarks M° is diagonalized by applying a biunitary transformation f LR = V° R xL,R. V°TM° VR = Diag(0,0,-A3,A4) (2.16) V°TM°M°t V° = V°TM°tM° V° = Diag(0,0,A3,A4) , (2.17) where A3 and A4 are the nonzero eigenvalues, A4 being the fourth heavy fermion mass, and A3 of the order of the top, bottom and tau mass for u, d and e fermions, respectively. We see from Eqs.(2.16,2.17) that from tree level there exist two mass-less eigenvalues associated to the light fermions: 2.6 Neutrino masses Now we describe the procedure to generate neutrino masses 2 Neutrino Masses and Mixing Within a SU(3) Family... 17 2.6.1 Tree level Dirac neutrino masses With the fields of particles introduced in the model, we may write the Dirac type gauge invariant Yukawa couplings ho P0 nr + hi PVni NL + h2PVn2 N + hayvna N + Mo NNL NR + h.c (2.18) ho, h1, h2 and ha are Yukawa couplings, and Mo a Dirac type, invariant neutrino mass for the sterile neutrinos NO R. After electroweak symmetry breaking, we obtain in the interaction basis YVl R = (v0, v0, NO)l,r, the mass terms ho [vi vOl + V2v°l + va vOl] NR + [hi Ai vOr + h2 A2 v°r + ha A3 vOr] NO + Mo NNL NR + h.c. (2.19) 2.6.2 Tree level Majorana masses: Since NL R, Table 1, are sterile neutrinos, we may also write left and right handed Majorana type couplings hL®u(NL)c + mL NNL (N°)c + h.c (2.20) and hiRPPVni (NR)c + h2RPPVn2 (NR)c + haRPVna (NR)c + mR INR (NR)c + h.c , (2.21) respectively. After spontaneous symmetry breaking, we also get the left handed and right handed Majorana mass terms hL [vi v Ol + V2 v °l + va ^Ol] (NL)c + mL IN 0 (N0)c + h.c., (2.22) + [hiR At vOr + h2R A2 v°R + h3R A3 vOR (NR)c + mR INR (NR)c + h.c. (2.23) Thus, in the basis = (vRL, v°L, YRL, NR, (VOR)c, (v°R)c, «R)c, (NR)c ), the Generic 8 x 8 tree level Majorana mass matrix for neutrinos MV, from Table 2.3, YV MV (YV)c,read 18 A. Hernandez-Galeana (Vôl )c (V2l)c (V?l)c (NÔ)c VÔR V2r VÔR NR 0 0 0 hLVi 0 0 0 hoVi V2L 0 0 0 hLV2 0 0 0 ho V 2 0 0 0 hLV3 0 0 0 hoV3 hLVl hLV2 hLV3 mL hi Ai h2A2 0 Mo (vôr)c 0 0 0 hi Ai 0 0 0 hiRAi 0 0 0 h2A2 0 0 0 h2RA2 «r)c 0 0 0 0 0 0 0 0 (NR)c ho vi hoV2 hoV3 Mo hiRAi h2RA2 0 mR Table 2.3: Tree Level Majorana masses MV = 0 0 0 ai 0 0 0 ai ^ 0 0 0 a2 0 0 0 a2 0 0 0 a3 0 0 0 a3 ai a2 a3 mL bi b2 0 mo 0 0 0 bi 0 0 0 ßi 0 0 0 b2 0 0 0 ß2 0 0 0 0 0 0 0 0 Vai a2 a3 mo ßi ß2 0 mR (2.24) Diagonalization of MVo), Eq.(2.24), yields four zero eigenvalues: U°' MV UV = Diagonal(0,0,0,0,,,m°,) (2.25) 2.7 2.7.1 One loop neutrino masses: 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.(2.18-2.23). The vertices read from the SU(3) family symmetry interaction Lagrangian iLint = gH - Zf + ^ - 2V°Y^V° + Z£ + ^ (v"iY2v2 Y+ + v"gY2V? Y2+ + v"2Y2V? Y3+ + h.c.) (2.26) 2 Neutrino Masses and Mixing Within a SU(3) Family... 19 The contribution from these diagrams may be written as aH r*A ^ Cy-m^lMy )ij n aH gH 4n ' mv(My)ij = Y. mO UVik UVjk f(My, mg), k=5,6,7,8 (2.27) (2.28) f(My' mO) = mÄ2 mM^ Inm^T ' My >> m£2 valid for neutrinos. _L \A2 o 2 k < nk > < > Fig. 2.1: Generic one loop diagram contribution to the Dirac mass term m^ v°Lv?R. M = MD,mL,mR vor V°R V?R NR V ol D v 15 Dv 16 0 0 V °°L D v 25 D v 26 0 0 v ol D v 35 D v 36 D v 37 0 NN O 0 0 0 0 Table 2.4: One loop Dirac mass terms Dvij -v°L v?R 2.7.2 One loop L-handed and R-handed Majorana masses Neutrinos also obtain one loop corrections to L-handed and R-handed Majorana masses from the diagrams of Fig. 2 and Fig. 3, respectively. A similar procedure as for Dirac Neutrino masses, leads to the one loop Majorana mass terms Thus, in the basis, we may write the one loop contribution for neutrinos as 20 A. Hernandez-Galeana < > < > Fig. 2.2: Generic one loop diagram contribution to the L-handed Majorana mass term m^ V°L (j ) 'T. M = Md , mL, mR vol V°L NL vol Lvii Lv 12 Lv 13 0L V°L Lv 12 Lv 22 Lv 23 0 Lv 13 Lv 23 Lv 33 0 nl 0 0 0 0 Table 2.5: One loop L-handed Majorana mass terms Lvij v?L (v?L)T Vor V°R NR vor Rv 55 Rv 56 0 0 V°R Rv 56 Rv 66 0 0 0 0 0 0 NR 0 0 0 0 Table 2.6: One loop R-handed Majorana mass terms OH Rvij v0R (y°r)T MOv = Lv11 Lv 12 Lv 13 0 Dv15 Dv16 0 0 Lv 12 Lv 22 Lv 23 0 Dv25 Dv26 0 0 Lv 13 Lv 23 Lv 33 0 Dv35 Dv36 Dv37 0 0 0 0 0 0 0 0 0 D v 15 Dv25 Dv35 0 Rv 55 Rv 56 0 0 D v 16 Dv26 Dv36 0 Rv 56 Rv 66 0 0 0 0 D v 37 0 0 0 0 0 V 0 0 0 0 0 0 0 0' «H n (2.29) 2 Neutrino Masses and Mixing Within a SU(3) Family... 21 Y I I < nk > Fig. 2.3: Generic one loop diagram contribution to the R-handed Majorana mass term m^ V°R ( )T. M = Md , mL, mR where, after using the relationships coming from the zero entries of MV, eq.(2.24); MV = UV Diagonal(0,0,0,0, m°, m°, m°, m°) U°T , (2.30) and in the limit M^ >> m£2, we may write: Lv ij = 3 Fij , i,j = 1,2,3 Dv 15 = 3 F15 + 1 F26 , Dv 16 = — 6 Fi6 , D v 25 =— 6 F25 , D v 26 = 1 F26 + 1 F15 , D v 35 =— 1 F35 , D v 36 = — 6 F36 , Dv37 = 2 (F15 + F26) Rv 55 = 3F55 , Rv 56 = 1 F56 , Rv 66 = 3 F66 where mo 2 mo 2 mo 2 Fij = WVi5 W0j5 ln--0-2 + UVi6 WVj6 IU--0-2 + W0i7 In--^ (2.31) m5 m6 m7 2.7.3 Neutrino mass matrix up to one loop Finally, we obtain the Majorana mass matrix for neutrinos up to one loop Mv = U0T M0V UO + Diag(0,0,0,0,mO,mO,mO,mO), (2.32) 22 A. Hernandez-Galeana 2.7.4 (Vckm)4x4 and (Vpmns)4x8 mixing matrices Within this scenario, the transformation from massless to physical mass fermion eigenfields for quarks and charged leptons is ^L = V° VL1) and ^R = VR vR1 ) ¥R , and for neutrinos YV = U° U. YV; U.T MV U. = Diagonal(A1,A2,A3,A4,A5,A6,A7, A8) (2.33) Recall now that vector like fermions, Table 1, are SU(2)L weak singlets, and hence, they do not couple to W boson in the interaction basis. So, the coupling of L-handed up and down quarks; f£LT = (u0,c0,t0)L and fdLT = (d°,s°,b°)L, to the W charged gauge boson is ^2 f"°uLYfdLW+^ = ^uL [(v0l VU1L))3X41T (v°l Vd[))3x4 Y^dL W+^ , (2.34) with g the SU(2)L gauge coupling. Hence, the non-unitary vckm of dimension 4 x 4 is identified as (VCKM)4X4 = [(v°l VUL))3X4]T (v°l V^^ (2.35) [V°L V(.L)]3X4 = (v°l)3x4 (VuL)4x4 , [V°L VdL)]3x4 = (VdL)3x4 (VdL))4x4 Similar analysis of the coupling between active L-handed neutrinos and L-handed charged leptons to W boson, leads to the lepton mixing matrix (upmns)4X8 = [(v°l V^L))3x4lT ("V "V)3x8 (2.36) [V°L V^L)l3x4 = (v°l)3x4 (V^L))4x4 , ("V "V)3x8 = ("V)3x8 ("V)8x8 2.8 Numerical results for Neutrino masses and mixing in a 3+5 scenario We report here numerical results for lepton masses and mixing, at the MZ scale [15] The input values for the horizontal boson masses, Eq.(8), and the coupling constant of the SU(3) family symmetry are: M. = 5.3 x 103 TeV , M2 = 3.3 x 105 TeV , — = 0.05, (2.37) 2 Neutrino Masses and Mixing Within a SU(3) Family... 23 A1 = 3352.7TeV , A2 = 103 A1 , gH = 2.23561 Horizontal gauge bosons from the SU(3) family symmetry introduce flavor changing couplings, and in particular mediate AF = 2 processes at tree level. The above high scales and heavy boson masses provide the proper suppression of Ko — Ko and Do — Do meson mixing from the tree level exchange diagrams mediated by the SU(2) horizontal gauge bosons Z1 , Y±. 2.8.1 Charged leptons: Tree level: MO = / 0 0 0 2670.25 0 0 0 11902.6 0 0 0 16264.7 \ MeV, \1.21882 x 1010 -2.32202 x 109 0 6.07835 x 1010/ up to one loop corrections: -19.9797 -83.226 -16.9884 \ Me = 0 0.6408 -0.8544 \-2.74 x 10-7 71.9782 293.027 59.814 168.853 -1712.54 480.432 0.000054 0.000755 6.20 x 1010/ MeV the charged lepton masses (me , m^, mT , ME ) = (0.486031 , 102.717, 1746.17 , 6.20 x 1010 ) MeV Mixing matrix: VeL = VOl V(L: 0.986458 0.0744614 0.00276675 -0.898433 -0.163991 0.43275 -0.146138 -0.439101 0.886473 \ 0 5.68933 x 10-8 3.23887 x 10-7 4.30921 x 10-8\ 1.93334 x 10-7 2.62497 x 10-7 1 24 A. Hernandez-Galeana 2.8.2 Neutrino masses and Lepton (UPmNS)4X8 mixing: Tree level MV, eq.(2.24): in eV / 0 0 0 30.9559 0 0 0 13.2472 \ 0 0 0 434.898 0 0 0 62.502 0 0 0 1980.48 0 0 0 76.9286 790642. 114364 0 4000 0 0 0 9.88602 x 106 0 0 0 1.40868 x 106 0 0 0 0 \13.2472 62.502 76.9286 4000. 9.88602 x 106 1.40868 x 106 0 100000 ) M°V, eq.(2.29): in eV 30.9559 434.898 1980.48 40 0 0 0 790642. 0 0 0 114364. 0000 /-0.000216363 -0.00311126 -0.0142886 0 -0.240713 0.00592337 -0.00311126 -0.04474 -0.2054720 -0.0510705 -0.1576590 0 0 -0.0142886 -0.205472 00 -0.240713 -0.0510705 -0.278496 0.00592337 -0.157659 0.367409 00 00 0.943645 0 0.239023 0 0.278496 0.367409 -0.239023 0 0 4136.09 588.853 0 0 0 588.853 -84.537 0 0 0 0 0 0. 0 Mv, eq.(2.32): in eV 0 /4.89118 X 10 -7 0.00534394 0.0268319 -0.0271814 0.0229136 0.0226125 0.16 1679 0.1 60874 0.00534394 -0.00830416 0.0484769 -0.0371462 -0.0211906 -0.0209894 -0.0521349 0.0268319 0.0484769 -0.629274 0.62907 -0.00373038 -0.00353603 -0.209602 -0.0271814 -0.0371462 0.62907 -0.530945 -0.221868 -0.220375 0.228698 0.0229136 0.0226125 0.161679 -0.0211906 -0.00373038 -0.221868 -2604.12 -0.253956 -0.940082 -0.0209894 -0.00353603 -0.220375 -0.253956 2643.36 0.748489 -0.052 1 349 -0.209602 0.228698 -0.940082 0.748489 -9.97002 X 1 06 -0.0518753 -0.208558 0.22756 -0.935401 0.744762 0.0518753 1.00658 X 10' 0.744762 Neutrino masses: (m1 = 0.00584795, m2 = 0.0104888, m3 = 0.051461 ,m4 = 1.21534, m5 = 2604.12,m6 = 2643.36,m7 = 9.97002 x 106 ,m8 = 1.00658 x 107) eV 2 Neutrino Masses and Mixing Within a SU(3) Family... 25 Squared neutrino mass differences: m2 - m1 = 7.58162 x 10-5 eV2 m2 — m = 2.53822 x 10-3 eV2 m2 — m2 = 1.47441 eV2 Neutrino mixing: Uv = UO U\ ( -0.817815 -0.573736 0.398121 -0.525222 0.00987364 0.0271079 0.0000115219 5.86044 x 10 — 0.0139007 -0.0158414 0.0309853 0.0302746 -0.73405 0.0115492 0.142716 -0.645085 -0.0000891429 -0.0000792766 -0.00290156 -0.106192 -0.0975645 -0.40357 0.111201 0.617751 0.0203878 -0.662886 0.74528 0.126871 9.34653 x 10—7 -4.8869 x 10—7 7.12916 x 10—6 6.29816 x 10—6 0.00805693 0.115983 0.532465 -0.00800614 1.12646 x 10 — -0.115003 6.87784 x 10 — 0.528342 0.0000166682 -0.702146 -0.707553 -0.0565233 -0.0644483 0.0634117 0.701853 0.450162 -0.446888 0.100018 0.0000488735 0.0000477752 0 0.0561715 0.056604 -0.702997 -1.08644 x 10—6 \ -6.81358 x 10—6 -0.0000164536 -0.0562542 -0.698211 -0.0994993 0 -0.706708 (Upmns)4x8 lepton mixing matrix : -0.807257 -0.414308 -0.0640545 -0.571865 0.440887 0.29044 0.0560008 0.718949 0.200336 0.0759557 -0.291791 0.571204 \ 4.43209 x 10—8 -1.19151 x 10—7 -1.05789 x 10—7 -1.68405 x 10 -0.0790509 0.0784275 -1.6032 x 10—6 1.60766 x 10—6 \ 0.126822 -0.125914 1.11777 x 10—6 -1.07963 x 10—6 -0.524122 0.520029 -0.0000179606 0.0000177362 1.62541 x 10—7 -1.61267 x 10 — 0 0 / — 7 26 A. Hernandez-Galeana 2.9 Conclusions We have reported an updated numerical analysis for neutrino masses and mixing 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. In section 2.8 we report the fit of parameters, which accommodate the neutrino masses (m, = 0.00584795,m2 = 0.0104888,m3 = 0.051461 ,m4 = 1.21534,m5 = 2604.12, m6 = 2643.36, m7 = 9.97002 x 106, m8 = 1.00658 x 107) eV, the squared neutrino mass differences — m2 = 7.58162 x 10-5 eV2, m3 — = 2.53822 x 10-3 eV2, and m4 — m2 = 1.47441 eV2 as well as the (UPMNS)4x8 lepton mixing matrix. Notice that the majority of the entries in (UPMNS)3x3 submatrix are within the reported limits in [11]- [14]. 2.10 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 stimulating Virtual Bled Workshop 2020. This work was partially supported by the "Instituto Politecnico Nacional", Grant from COFAA. References 1. A. Hernandez-Galeana, Rev. Mex. Fis. Vol. 50(5), (2004) 522. hep-ph/0406315. 2. A. Hernandez-Galeana, Bled Workshops in Physics, (ISSN:1580-4992), Vol. 17, No. 2, (2016) Pag. 36; arXiv:1612.07388[hep-ph]; Vol. 16, No. 2, (2015) Pag. 47; arXiv:1602.08212[hep-ph]; Vol. 15, No. 2, (2014) Pag. 93; arXiv:1412.6708[hep-ph]; Vol. 14, No. 2, (2013) Pag. 82; arXiv:1312.3403[hep-ph]; Vol. 13, No. 2, (2012) Pag. 28; arXiv:1212.4571[hep-ph]; Vol. 12, No. 2, (2011) Pag. 41; arXiv:1111.7286[hep-ph]; Vol. 11, No. 2, (2010) Pag. 60; arXiv:1012.0224[hep-ph]; Bled Workshops in Physics,Vol. 10, No. 2, (2009) Pag. 67; arXiv:0912.4532[hep-ph]; 3. Z. Berezhiani and M. Yu.Khlopov: Theory of broken gauge symmetry of families, Sov.J.Nucl.Phys. 51, 739 (1990). 4. Z. Berezhiani and M. Yu.Khlopov: Physical and astrophysical consequences of family symmetry breaking, Sov.J.Nucl.Phys. 51, 935 (1990). 5. J.L. Chkareuli, C.D. Froggatt, and H.B. Nielsen, Nucl. Phys. B 626, 307 (2002). 6. Z.G. Berezhiani: The weak mixing angles in gauge models with horizontal symmetry: A new approach to quark and lepton masses, Phys. Lett. B 129, 99 (1983). 7. T. Yanagida, Phys. Rev. D 20, 2986 (1979). 8. T. Appelquist, Y. Bai and M. Piai: SU(3) Family Gauge Symmetry and the Axion, Phys. Rev. D 75, 073005 (2007). 9. T. Appelquist, Y. Bai and M. Piai: Neutrinos and SU(3) family gauge symmetry, Phys. Rev. D 74, 076001 (2006). 10. T. Appelquist, Y. Bai and M. Piai: Quark mass ratios and mixing angles from SU(3) family gauge symmetry, Phys. Lett. B 637, 245 (2006). 11. G.H. Collin, C.A. Argiielles, J.M. Conrad, and M.H. Shaevitz, Phys. Rev. Lett. 117, 221801 (2016). 2 Neutrino Masses and Mixing Within a SU(3) Family... 27 12. M.C. Gonzalez-Garcia, Michele Maltoni, Jordi Salvado, and Thomas Schwetz, arXiv:1209.3023[hep-ph]. 13. Ivan Esteban, M.C. Gonzalez-Garcia, Alvaro Hernandez-Cabezudo, Michele Maltoni, and Thomas Schwetz, arXiv:1811.05487[hep-ph]. 14. Suman Bharti, Ushak Rahaman, and S. Uma Sankar, arXiv:2001.08676[hep-ph]. 15. Zhi-zhong Xing, He Zhang and Shun Zhou, Phys. Rev. D 86, 013013 (2012). Bled Workshops in Physics Vol. 21, No. 2 A Proceedings to the 23rd [Virtual] Workshop, Volume 2 What Comes Beyond ... (p. 28) Bled, Slovenia, July 4-12, 2020 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? H.B. Nielsen *1 and C.D. Froggatt2 1 Niels Bohr Institutet 2 Glasgow University Abstract. We study the 3.55 keV X-ray suspected to arise from dark matter in our model of dark matter consisting of a bubble of a new phase of the vacuum, the surface tension of which keeps ordinary matter under high pressure inside the bubble. We consider two versions of the model: • Old large pearls model :We worked for a long time on a pearl picture with pearl / bubbles of cm-size adjusted so that the impacts of them on earth could be identified with events of the mysterious type that happened in Tunguska in 1908. We fit both the very frequency, the 3.55 keV, and the overall intensity of the X-ray line coming from the center of the Milky Way and from galaxy clusters with one parameter in the model in which this radiation comes from collisions of pearls. • New small pearl model: Our latest idea is to let the pearls be smaller than atoms but bigger than nuclei so as to manage to fit the 3.5 keV X-rays coming from the Tycho supernova remnant in which Jeltema and Profumo observed this line. Further we also crudely fit the DAMA-LIBRA observation with the small pearls, and even see a possibility for including the electron-recoil-excess seen by the XenonlT experiment as being due to de-excitation via electron emission of our pearls. The important point of even our small size pearl model is that the cross section of our "macroscopic" pearls is so large that the pearls interact several times in the shielding but, due to their much larger mass than the typical nuclei, are not stopped by only a few interactions. Nevertheless only a minute fraction of the relatively strongly interacting pearls reach the 1400 m down to the DAMA experiment, but due to the higher cross section we can fit the data anyway. Povzetek. Avtorja domnevata, da njun model za temno snov pojasni izvor spektralne crte pri 3.55 keV v rentgenskem območju spektra nase galaksije. Temna snov je po njuno posledica nove faze vakuuma, ki jo tvorijo mehurcki obicajne snovi pri visokem tlaku, ki ga vzdržuje povrsinska napetost. Obravnavata dve razlicici modela: • Stari model velikih biserov: Avtorja sta dolgo razvijala model vakuuma z "biseri" (ali mehurcki) centimetrskih velikosti, ki so povzrocili eksplozjo v Tunguski leta 1908. Da ima rentgensko sevanje, ki nastaja pri trkih takih "biserov" v centru galaksije in v jatah galaksij, frekvenco 3.55 keV in izmerjeno jakost, morata v modelu prilagoditi samo en parameter. * Giving talk 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 29 • Novi model majhnih biserov: Ce pa predpostavita da so "biseri" novega vakuuma manjsi od atomov in vecji od jeder, njun model dobro opise spektralno crto pri 3.5 keV, ki prihaja od ostanka Tychove supernove in sta jo opazila Jeltema in Profumo. Njuni zelo masivni "biseri" doseZejo kljub mocni interakciji z obicajno snovjo v manjsem stevilu experiment DAMA-LIBRA, ki je 1400 m pod povrsjem zemlje in se celo veckrat sipljejo na merilcu ne da bi se ustavili. V grobem opisejo meritve DAMA-LIBRA. Elektroni, ki jih "biseri" sevajo, pa morda pojasnijo presezek elektronov, ki ga izmeri XenonlT. 3.1 Introduction The main purpose of the present article is to put forward the latest developments of our long speculated idea that the so far mysterious dark matter found via its gravitational forces, instead of consisting of particle of atomic masses or an Axion-like condensate, could consist of our proposed type of macroscopic objects with a mass much bigger than that of genuine atoms. We started our speculations already years ago by supposing cm-size pearls make up the dark matter, but they will be developed in the section 3.8 below into the idea that these pearls could indeed be much smaller and of geometrical size even smallish compared to atoms, although the mass should still be appreciably larger than that of atoms. We shall stress small macroscopic pearls. Even such a dramatic change in our old model into a version with much smaller pearls would not be observed via the gravitational effects provided just that the density of mass per unit volume is kept the same. It is also this fact that really only the mass density matters for the gravitational effects, that makes it possible that these effects cannot distinguish our types of heavy or relatively lighter pearls from the more usual assumption of only atomic weight particles, such as supersymmetric partners of Z0 or photon say in superstring theory. However, assuming that indeed the X-ray radiation [1,2] observed by satellites and suspected to come from dark matter does indeed come form dark matter requires more specific models for what the dark matter could be; e.g. it could consist of some new sort of sterile neutrino able to decay although very seldomly into a photon and e.g. an ordinary neutrino. Such a sterile neutrino should then of course have a mass equal to just two times the photon energy number 3.55 keV of the observed X-ray radiation counted in the rest frame of the supposed dark matter in the region observed. • Our Old Model: We develop an alternative version of our model [3-6] in which dark matter consists of cm-size pearls with masses of 108 kg under the attempt to identify the X-ray radiation seen by sattelites [1,2] and supposed to originate from dark matter with the energy per photon 3.55 keV. We shall discuss the possibility that the dark matter pearls be much smaller but still macroscopic. This is our new model with small pearls of a size smaller than atoms but bigger than atomic nuclei. Actually we assume that our pearls have a skin surrounding them keeping some ordinary matter inside the pearls under such an (appropriate) pressure 30 H.B. Nielsen and C.D. Froggatt that, in the electron system of this ordinary matter inside, there appears an energy gap between filled and empty electron states - called the homolumo gap (to be explained later) - of size close to the energy difference just 3.55 keV of the observed radiation. The idea then is that there can be excitations being (loosely) bound states of an electron in one of the lowest empty states and a hole in one of the at first filled states. These excitons should have an energy close to the observed photon energy in the line. Then one could have that the photons observed astronomically by the satellites are photons from the decay of such excitons in the highly compressed ordinary matter material in our model supposed to exist inside pearls making up the dark matter. It is a major part of our work [3] to evaluate the rate of such X-ray radiation that will result under the assumption that the main production of the 3.5 keV radiation comes about when two of our dark-matter-pearls collide with each other. We claim it to be a great success that the magnitude of this rate of radiation can be fit together with the energy per photon, the number 3.5 keV. We shall in the present article have in mind really two models, which are essentially inconsistent with each other, In the first model the mass of one pearl is about 1.4 * 108kg and in the other model the mass is about 104 GeV = 10-23kg. The old value of 1.4 * 108kg was taken as a fit to the famous Tunguska-event in 1908 taken to be due to the impact of one of our pearls. The small mass proposal of about 10-23kg is rather inspired by an attempt to fit to the DAMA (-LIBRA) experiment (by most people presumably believed to be due to something else other than dark matter). (A presentation of the DAMA results is given in the present Bled Workshop proceedings). • Observational Discussion: Our small mass 104 GeV « 10-23kg pearl proposal is filled with ordinary matter with an estimated density of the order of 1014kg/m3 as we fit the size of the pearl, It is clear that the size of such a small pearl will nevertheless be so big - bigger than an atomic nucleus - that the cross section is likely to be so big that it could not possibly pass through about 1400 m into the earth without interacting. So in this sense our dark matter pearls are not WIMPS since the WI in this acronym stands for weakly interacting. It could still be dark in the sense that the interaction with e.g. light per mass unit could be small, but not small per pearl. With such a strong interaction one may worry whether such pearls have any chance of reaching down to give any signal in underground experiments looking for dark matter, because the pearls might be stopped in the shielding above the experimental apparatus; but here the reader should have in mind that a pearl that is heavy compared to atoms or nuclei, when it hits, will not be stopped but just deliver a smaller part of its kinetic energy to the hit particle, so that the latter obtains a speed of the same order as the speed of the incoming pearl. Of course, if one has a hugely heavy pearl as we estimated of cm-size and with the large mass of 1.4 * 108 kg, then it will cause a major catastrophe, like the famous one in Tunguska, and a potential underground laboratory would be destroyed rather than making a proper observation. 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 31 But with the small size pearl having a mass in the 104 GeV range the pearl would still interact a lot with the earth in the shielding, but possibly not enough to be fully stopped before reaching say the DAMA-LIBRA laboratory proper. Actually we shall imagine that a very small fraction of the pearls come though to the laboratory, by accident so to say. If the pearls interact several times passing through the experimental apparatus they will be disqualified as dark matter, which is usually assumed to have so small a cross section that they only interact once in the detector. Even if a dark matter pearl interacts several times in the shielding - but is not observed to interact because a high mass is not stopped but can continue - it may well be observed essentially as a dark matter event anyway. Really we would like to propose a picture for the 104 GeV pearl mass proposal that a major part of the pearls end up getting stopped in the shielding - the earth above the experimental hall underground - but that the pearls with the smallest cross sections come through to the experimental apparatus and is observed there. If the pearls have a much bigger cross section than normal WIMPs they may well produce a non-negligible number of events even if the number reaching through is much lower than the number of WIMPs one would have expected. In other words for the 104 GeV mass pearls we shall speculate that compared to the usual WIMP picture the much higher cross section of our pearls than that of the WIMPs can compensate for the lower number of pearls than of WIMPs reaching to the experimental apparatus for two reasons: - There are fewer pearls than WIMPs if the pearls are as suggested heavier than the WIMPs, because we have to keep the gravitational effects the same to have the same mass density in the universe. - There are few pearls also because some of the pearls get stopped in the shielding due to the bigger cross section in spite of them being heavy and not so easy to stop. Now we should also mention that what is truly measured in the DAMA-LIBRA experiment is not so much the full numbers of presumed dark matter particles interacting with the apparatus, but rather the seasonal variation of the number of events. If indeed what they see in DAMA-LIBRA were due to our rather strongly interacting pearls, then there would be a seasonal effect partly due to the pearls coming in one season with higher speed than in another so they would be able to penetrate deeper. If by chance the depth of the laboratory is close to the average stopping place of the pearls, such an effect of different penetration depths in the different seasons might be delicate to estimate, but could make it possible to get a bigger seasonal effect than estimated in a more simple way. Let us immediately remark, that if indeed such seasonal variation due to relatively small changes with season of the penetration depth of interacting pearls (dark matter particles), then this could mean that the DAMA-LIBRA type of experiment measuring mainly the seasonal effect could be favoured in finding a signal over other experiments not using this technique. This would help solving the main problem or mystery in connection with the DAMA-LIBRA experiment: Why do the other underground experiments looking for 32 H.B. Nielsen and C.D. Froggatt dark matter not see the same amount of it as DAMA-LIBRA ? Now we would answer that DAMA-LIBRA may sit close to the average penetration depth and in one season this penetration depth is a bit deeper and DAMA-LIBRA sees a lot, while in another season the average penetration depth is a bit higher up and one does not see so much. Actually our fit suggests that the average penetration depth is only a small part of the way down the 1400 m but the falling off tail of the distribution which DAMA observes varies exponentially with the variation in average penetration depth and a rather big seasonal effect is indeed expected. We want to conclude that IMPs (= interacting heavy par ticles) as our pearls could be denoted rather that the usual WIMP picture is a possibility for what the underground experiment DAMA-LIBRA could have observed. And our argument about the penetration depth could be used to explain that other experiments did not see the same dark matter. 3.1.1 Plan In the following section 3.2 we present a couple of figures about the dark matter as known already via its gravitational forces, and in the following section we give a couple of figures about impacts of objects like meteors falling on earth with the purpose of comparing the energy delivered with that which dark matter could deliver, if it fell like other objects. Then in section 3.4 we review some of the ideas needed to understand our type of model with pearls consisting of a bubble of a new type of vacuum (this is just our speculation because so far nobody really saw any new vacuum convincingly). In the subsections of this section we present in 3.4.1 our postulated new law of Nature "Multiple Point Principle", which is the main new assumption in our work in as far as, except for this multiple point principle, we only need the Standard Model as the laws of nature. We only make further speculations on the dynamics such as the existence of bound states or in general on results of the too hard to calculate, but by far not excluded possibilities in the Standard Model. In the subsection 3.4.2 we say a few words about the domain walls that will separate such different phases of the vacuum that we speculate exist. In subsection 3.4.3 we mention the effects other than gravitational ones which are probably due to the dark matter. The most important such effect for the present work is the excess X-ray radiation observed as a tiny peak above the best understanding fit to the X-ray spectrum at the photon energy 3.55 keV. Other such likely dark matter effects are an excess of positrons and the associated gamma rays; and then, what we are very keen on, one of the experiments Xenon1T meant to look for dark matter saw a little excess of electrons appearing in the apparatus, at first seemingly not dark matter; but we think it could be our dark matter pearls passing slowly through and delivering electrons with just the energy 3.55 keV. In section 3.6 we mention that the type of dark matter models most popular in the literature, except for black holes making up the dark matter, need to modify the Standard Model by introducing extra particles corresponding to extra fields. Most popular is to use supersymmetry models in which there has to be included as many 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 33 new physics particles as there are particles already. Compared to that one should understand that we only add a new fine-tuning principle the "Multiple Point Principle", which is an extra assumption about the values of coupling constants that can even be checked and at least are close to work, while the usual modified Standard Model has lots of extra particles not yet found. Next in section 3.7 we discuss the fitting with our large pearl model, and in section 3.8 we then consider the model with the "small", meaning little less than atomic size, pearls. Really this "small" size is in fact very large compared to what is considered in more conventional models (such as supersymmetry). In the subsection 3.8.1 we extract the ratio of cross section to mass for the dark matter required from the observation of the 3.55 keV X-rays from the Tycho supernova remnant and compare it to the corresponding ratio for nuclei in subsection 3.8.2. Then in the subsection 3.8.3 we present the fit of the small pearl model, but the fitting is based on the discussion of the DAMA(-LIBRA) experiment that we have first put in the next subsection 3.8.4. In section 3.9 we resume and conclude the article . 3.2 We know something from the gravitational studies As is well-known the dark matter has mainly and in fact possibly only been seen by its gravitational effects - and it could still be a possibility that there is no dark matter, but instead that something is wrong with our understanding of the gravitational force - but even from only observing it via the gravitational force, one can nevertheless derive some understanding of its distribution and velocity. In fact one can already estimate that the solar system as a whole moves relative to the local dark matter average velocity with a speed of 232 km/s according to the figure 3.1. Further the distribution of the dark matter Motion of Dark Matter, stars etc. Numbers for Crude Estimates • Density of Dark Matter in Solar System Neighborhood: D = = 5.35 * 10-22 M (3.1) cm3 m3 • Typical Speed (also relative to each other): v = 200 km/s = 2 * 105 m/s (3.2) • Rate of Impacts on crossing Area, per m2: Rate = vD = 1.07 * 10-i6-k^ (3.3) m2s These numbers may be crudely estimated by looking at the distributions in figure 3.2, which have been gotten from the ERIS simulation of the dark matter. 34 H.B. Nielsen and C.D. Froggatt Fig. 3.1: Motion relative to Dark Matter Here is drawn how the solar system moves along relative to the supposed rest system of the bulk of the dark matter. One shall imagine the earth going around the ellipse drawn which in perspective is an approximate circle representing the orbit of the earth. Note how the speed of the earth w.r.t. the dark matter average will vary with the season. 3.3 Compare to Rates of Impacts on Earth For the dark matter we have thus found the rate Rate = vD = 1.07* 10 -16 kg (3.4) In Table 3.1 we use this vD for dark matter in our neighborhood to derive a few estimates of impact rates for dark matter, if dark matter were indeed macroscopic particles with the masses listed in the first column of this table: Hitting Rates for some Masses: In the first column is given the mass of the dark matter pearl. The second column gives the rate of impacts such a mass would give per m2 and in the third column this rate is translated into the time between the impacts on this square meter. The fourth and fifth column similarly give the rates and the time in between impacts for impacts on the Earth in total instead of just on a square meter. Notice that in the row corresponding to the mass of the dark matter particle being 108kg there is - in the last column - about 100 years between the impacts. Now it was approximately 100 years ago when the famous Tunguska event occurred, meaning that if the Tunguska event should be caused by a dark matter pearl, then the mass would be of the order of 108kg. m2s 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 35 Fig. 3.2: Velocity histogram of different components of the Milky Way, as seen in the ERIS simulation. The black histogram shows the velocity distribution of dark matter. The cyan histogram illustrates the velocity of all stars, and has a much larger central peak than the dark matter distribution. The orange histogram, however, which includes only metal-poor stars, is very similar to the dark matter velocity distribution. (Herzog-Arbeitman et al. [7]) mass m2 rate m2 time earth rate earth time 10-l6kg = 5 * 1010GeV 1s-1 1 s 5 * 10l6s-1 2 * 10-l5s 10-8kg = 10Hg 10-8S-1 108s = 3y. 5 * 108s-1 2 *10-9s 1kg 10-l6s-1 10l6s 5s-1 0.2s 108kg = 105ton 10-24s-1 1024s 5 *10-10s-1 2 *109s ~ 100y Table 3.1: A few rates for hypothetical dark matter pearls Next we now give a similar table for meteor impacts as observed, impacts a priori expected to be made from "ordinary matter"( i.e. atoms). Here it is meant that the impacts are counted for the whole Earth: Compare Impacts of Ordinary Matter 10-2 kg : 105 per year 1 kg : 104 per year. 108 kg : 10-3 per year. You may consider the numbers in this table 3.1 as extracted from the figure 3.3. 36 H.B. Nielsen and C.D. Froggatt Since a year has 3.16 * 107 s this corresponds to a mass density Dmetec the velocity vmetecrs being of the order vmeteorDmetec 104 kg/year/eartharea 3.16 * 107s/year 3 * 10-3 kg/eartharea/s 0.5 * 1015m2/eartharea = 2 * 10-18 kgs-lm times (3.5) (3.6) (3.7) formally a factor 50 smaller than the dark matter. Rather than the mass of the impact object you might use its size and then we get the graph in figure 3.3: Fig. 3.3: Size of Impact goes as square root of "time in between" From this figure 3.3 we can read off an approximate dependence of the size of the impacts on earth and their frequency. Approximately the inverse frequency being the "time between" goes as the square of the size of the impacting object. So a formula easy to remember is: "impact size" in m = \Jav. "time between" in years (3.8) on earth. On the figure 3.4 we see the relation between energy release by the impact and again the frequency measured in impacts per year. Would Macroscopic Dark Matter Dominate Meteors? • Taking very roughly the graph as having the slope -1 in the logarithmic plot we may read off that the energy of impacts per year is of the order of magnitude of 1013J/y to 1014J/y. 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 37 Fig. 3.4: Relation between energy released and impacts per year To compare with that the kinetic energy in the column of dark matter hitting the earth per year is for non-relativistic dark matter particles of the order of "dark matter power on earth" = = 1 * (300km/s)3 * 0.3GeV/cm3 * n * (6.38 * 106m)2 (3.9) (3.10) = 1 (3 * 105m/s)30.3 * 1.78 * 10-21 kg/m3 * n * (6.38 * 106m)2 (3.11) = 1.27 * 1016J/y (3.12) (using 1 year = 31556952 s) So it looks that unless some of the kinetic energy of the dark matter hitting the earth is lost from showing up as observable impacts, there is too much energy in the dark matter to match the impacts as observed. In our old work [6] we took it that because of the smallness of even cm-sized pearls they penetrate so deeply into the earth that it is realistic that an appreciable part possibly 19/20 of the energy is penetrating so deep into the earth, that it does not appear as observed energy on the surface of the earth. Since we could well find it consistent that our big pearl (=cm-size) would go thousands of km into the earth, it would indeed be hard to get all the energy out so quickly as to be identified with the energy of the impact. 38 H.B. Nielsen and C.D. Froggatt 3.4 Requisites for Our Model(s) Before going on to fit our type of model and discussing how well such pearl models for the dark matter matches much of our knowledge about the dark matter, as it actually will, we shall put forward a few prerequisites needed for understanding the speculations making up at least one concrete example of a macroscopic pearl model of the dark matter. As a motivation for just our concrete picture for how the pearls could come about let us stress: Our picture of dark matter pearls can come about in the pure Standard Model, i.e. without any new physics in the sense of new basic particles. We shall rather only speculate about new particles which are bound states of the already known particles, and thus do not require any modification of the Standard Model. We have e.g. no supersymmetric partners, because we do not have supersymmetry at least not in the relevant region of energy for our model. Gia Dvali showed that the existence of several vacua is inconsistent unless they are degenerate in the article "Safety of Minkowski Vacuum" [8]. 3.4.1 Multiple Point (Criticality) Principle The point in our work, which comes closest to assuming new physics, is the principle that the coupling constants of the true model for physics - for our purpose here the Standard Model - are by a "new Law of Nature" tuned in to just arrange that there are a series different phases of the vacuum - different vacua we could say - which all have the same energy density ( = cosmological constant) [9-12]. We call this principle of such fine-tuning of the coupling constants the Multiple Point (Criticality) Principle (MPP) [9-12]. There has been given various arguments for it [8-12], and we can claim that using it we have even made correct predictions, e.g. the number of families, prior to the LEP measurement of the number of light neutrino species. We fitted fine structure constants in a rather complicated model called ANTIGUT and the fitting parameter was indeed the number of families. We predicted that to be 3. Later we obtained a mass prediction [13] for the Higgs of mHiggs = 135 ± 10 GeV before the Higgs was found. For our pearl-models of the dark matter it is important that Nature should have this fine-tuning at least to an appreciable accuracy making the inside and the outside vacua for our pearls of equal energy density. This is because otherwise almost certainly one of the phases would spread out and it would be very hard to get pearls that are stable. Actually even with the degenerate vacua we have in our model the need for getting the pearls filled by ordinary matter under high pressure to withstand the pressure coming from the tension of the surrounding skin or domain wall. Guesses as to the order of magnitude for what the energy density difference should be, if not tuned to be small, would be so high that our model would become unlikely. Though, if e.g. the energy density difference was only of the order corresponding to the observed order of magnitude for the vacuum energy in the universe it would contribute so little over one of our pearls that it would not disturb our calculations taking the difference to be zero. 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 39 3.4.2 Domain walls in general There is also a discussion of walls in another article [14] in these proceedings. We ourselves like to point out, that once we have the "Multiple Point Principle" we have in principle the possibility that some even large regions in space could be filled by one phase while another region could be filled by another phase of the vacuum. Had we had a spontaneously broken discrete symmetry it would induce a case of "Multiple point principle" in as far as two or more phases related by the broken symmetry would of course have for symmetry reasons the same energy density. It is however, not such a case of a spontaneously broken discrete symmetry, which we imagine in our model. We rather speculate that two a priori different, and not connected to each other by symmetry, vacuum phases are to be used. Having the spontaneously broken discrete symmetry is also phenomenologically badly working, in as far it would typically lead to random vacua coming to dominate in various regions outsides the horizons of each other. Such outside each others horizon different dominating vacua would cause domain walls extending over longer distances than the horizon and in turn make up huge amounts of domain walls in cosmology. Unless the wall tension was extremely small such horizon scale walls would get to dominate under all circumstances in the long run; and that would spoil our cosmological models. So we must hope, and we actually do expect, that the domain walls due to the asymmetry between their sides - i.e. due to the fact that the different vacuum phases are not connected by symmetry - will contract a bit more towards diminishing one vacuum than the other one. Thus at an early stage in the history of the universe one of the vacua only survives in small bubbles compared to the universe size. It is such small surviving bubbles that should be the dark matter. Actually even the small bubbles only survive because at a stage they get stopped from contracting by having collected so many nucleons inside that they can provide a sufficient pressure to stop the contraction. For our cm-size pearls we had an estimate that the contraction of the pearls to the stability point where they just have the size given by their content of nucleons, counter acting the pressure, would end about the time in cosmology, when the big bang nuclear synthesis is about to start and temperature is of MeV size. It is very needed for our model that the pearls have become so compact and effectively disconnected from the rest of the plasma before the big bang nuclear synthesis properly begins, because otherwise our model would modify this big bang nuclear synthesis, and it would be an unconvincing refitting even if we managed to fit the abundances of the various light isotopes resulting form the big bang nuclear synthesis. Nevertheless one should of course investigate astronomically if some of the big voids observed in the matter distribution should actually be a result of domain walls. If one had, for some accidental or other reason, an astronomical size region with the same vacuum as inside our pearls, formally an enormously large dark matter pearl, then we would expect there to be the same matter density inside this huge pearl as on the average in the universe. But now there would be no way to have true dark matter in the region, because the whole region is already formally dark matter. Pearls inside it of the present phase vacuum would repel rather than 40 H.B. Nielsen and C.D. Froggatt attract nucleons and would thus totally collapse. Therefore in such regions one would in practice lack the dark matter and have it replaced by a higher density of ordinary matter. The latter would, however, have electrons staying relativistic longer than dark matter would have stayed relativistic. Thus these regions would presumably develop their inhomogeneities later than the regions where the present vacuum dominates. This could then be likely to delay the development of stars and galaxies in such formal huge dark matter bubbles of astronomical size. Such regions might appear as voids? 3.4.3 Non-gravitational Dark Matter Observations We believe it is true to say that all non-gravitational signs from dark matter are somewhat doubtful. Nevertheless our main aim in this article to look especially for whether our model can get support from the observations of one of the presumed non-gravitational observations of dark matter, the 3.55 keV X-ray radiation in outer space, mainly seen [1,2] from our Milky Way Center or from big clusters of galaxies. The 3.55 keV X-rays We have already mentioned this for us so important X-ray observation in a line of frequency 3.55 keV, which seems not to be explained by the atomic ion transitions expected in the plasmas from which the X-rays come. But it is only a tiny little deviation from the main fit of the X-ray spectrum and e.g. an unexpectedly high abundance of potassium in the plasmas could make a line in the region of the 3.55 keV be increased so much as to replace the tiny suspected dark matter line. Using the expectations from the gravitational knowledge about the distribution of the dark matter, fits have been made to the 3.55 keV radiation expected both under the assumption that the emission from a region depends linearly on the density D of dark matter and under the assumption, that the amount of 3.55 keV line radiation is proportional to the square of the dark matter density D2. It is the latter dependence that should come out of our model, because we postulate that the 3.55 keV radiation arises when our pearls collide. Both types of fits are not hopeless, and even the rather well fitting analysis by Cline and Frey [15], which we use in our work, has at least one severe discrepancy: one of the measurements in the outskirts of the Perseus Cluster delivers about 1000 times more 3.55 keV radiation experimentally than one should expect by extrapolating the fits to the other observations. In our use of the analysis of Cline and Frey, we simply had to delete this observation to obtain a meaningful average for the overall scale of the radiation which is then what we ourselves sought to fit. We should investigate, if we could understand this deviating measurement in the Perseus Cluster as due to our pearls getting energy for 3.55 keV radiation in a different way than from the collisions. In fact we have similar problem with the Tycho supernova remnant in which the square of the density D2 over the supernova remnant region is very tiny in comparison to galaxy clusters and the Milky Way Center extensive volumes. The supernova remnant region, even taking 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 41 into account the closeness of the Tycho supernova remnant, is so small that it would not be expected that Jeltema and Profumo should have seen the 3.55 X-ray line from the dark matter there. But in fact Jeltema and Profumo [24] have seen 3.55 keV radiation from the supernova remnant. Our suggestion is that the cosmic rays or X-rays in the Tycho supernova region can excite the pearls, which then whatever the excitation energy - collision or cosmic ray excitation - will emit an appreciable part of the energy as 3.55 keV radiation. One could of course hope - and we hope to find out - that there are some similar cosmic rays or X-rays reaching the outskirts of the Perseus Galaxy Cluster. Of course, if the cosmic ray or X-ray activity is about the same in two neighboring regions in say the Perseus Cluster, then the ratio of the X-ray or cosmic ray feeded radiation relative to the one feeded by the collisions will go in the ratio Df2 = D-1. This is because the rate from cosmic ray feeding goes as D * "density of cosmic rays", while the collision rate goes as D2. In the outskirts of the cluster the density of dark matter D presumably goes down, and thus the cosmic ray feeded radiation becomes relatively more important. Positrons and Other Gamma-rays Also positrons above some 10 GeV in energy have shown an excess suggested to be due to dark matter together, as one could imagine, with gamma-rays not in a line but in a broader spectrum. In this connection there is a little problem: Using usual types of model for dark matter identified with some type of particle simply decaying into among other things the positron to make the excess, it is very hard to avoid that associated with this positron emission one does not also get some gamma-rays. Now, however, the fitting does not go well and it seems that experimentally there are not so many gamma-rays as is almost unavoidably needed for matching the positron excess! This little tension with an elementary particle dark matter interpretation could provide support for our type of model, because at the collision and strong heating up of the uniting pearls a large amount of electrons will be emitted and can easily create electric fields that in a rather low acceleration way can accelerate e.g. positrons. Thus one can get positrons which are not produced at high speed almost abruptly, but which are "slowly" accelerated. The latter gives much less electromagnetic radiation and thus our model has the potential of making positrons with much fewer gamma rays connected with them. This would agree better with the too few observed gamma-rays. Xenon1T Electron Recoil Excess Yet another effect, which we shall count as a non-gravitational effect of dark matter, but which is not obviously dark matter at all: the Xenon1T electron recoil excess. Apart from the DAMA/LIBRA and the DAMA experiment all other experiments seem to find only negative results, when looking for the dark matter in direct searches. There was, however, found one unexpected result [16] although at first not seemingly related to dark matter: 42 H.B. Nielsen and C.D. Froggatt The experiment Xenon1T investigated what they call electron recoil in their Xenon experiment. In the Xenon experiment one has a big tank of liquid xenon with some gaseous xenon above it and photomultipliers looking for the scintillation of this xenon, the philosophy being that a dark matter WIMP e.g. hits a nucleus inside the xenon and the recoil of this creates a scintillation signal S1 and also an electron, which is then driven up the xenon tank by an electric field and at the end by a further electric field made to give a signal at the top S2. By the relative size of the signals S1 and S2 one may classify the events - which are taken to be almost coinciding pairs of these signals S1 and S2 - as being nucleus recoil or electron recoil. One expects to find the dark matter in the nucleus recoils, since a dark matter particle is not expected to make an electron have sufficient energy to make an observable electron recoil event. But now by carefully estimating the expected background, the Xenon1T experimenters found an excess of electron recoil events. Ideas proposed for explaining it include axions from the sun or neutrinos having bigger magnetic moments or perhaps less interestingly that there could be more tritium than expected in the xenon. But here with our model of relatively stronger interacting particles able to radiate the line 3.55 keV when excited we have a possible explanation: Going through the earth above the detector and the rest of the shielding, the pearls or particles get excited so as to emit 3.55 keV X-ray just as they would do in the Tycho supernova remnant, where they also get excited by matter or cosmic rays. But then the particles passing through the deep underground Xenon1T experiment are already excited and prepared for emitting the 3.55 keV radiation. Now they could possibly simply do that in the xenon tank or they might dispose of the energy by a sort of Auger effect by rather sending out an electron with an extra energy of 3.55 keV. Such an electron with an energy of a few keV could be detected and taken for an electron recoil event in the Xenon1T experiment. It is remarkable that the signal of these excess electron recoil events appears as having just an energy of the recoiling electron very close to the value 3.55 keV. Indeed the most important bins for the excess are the bins between 2 and 3 keV and the bin between 3 and 4 keV. So we would claim that there is in our model no need for extra solar axions or a neutrino magnetic moment, nor tritium. But we claim it to be 3.55 keV radiating dark matter one sees in the xenon experiment! The Dark Ages, 21 cm line As a possible place to look for information about dark matter - especially of the pearl type say - is the influence it could have had in the "Dark ages" before the stars lit up the universe, a time that may be investigated through the study of the H1 radio line of 21 cm wavelength. Recent studies [17,18] were pointed out to us by Astri Kleppe. Supernova Introductional Burst As an interesting possibility for studying our dark matter pearls astronomically, we should also mention our older work, in 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 43 which we claim [19] that our dark matter pearls can not only help the supernovae to explode more, which is what is called for, but also to explain a neutrino burst appearing some hours before the genuine explosion, as appears to have been observed by the neutrino experiment LSD [20]. 3.5 Status of Searches Before going on to describe our models for dark matter being pearls of a new phase of vacuum, let us shortly review the status of the searches for dark matter in underground experiments. The plot in figure 3.5 shows the excluded regions in the cross section versus mass plane for dark matter particles in the usual WIMP-theory: It is important to notice for our work below that inside the region excluded 5X10"1 1 2 3 4 56 10 20 30 102 2x10J 103 WIMP Mass [GeV] Figure 27.1; WIMP cross sections (normalized to a single nucleoli) for spin-independent coupling versus mass. The DAM A/LIBRA [72], and CDMS-Si enclosed areas are regions of interest from possible signal events. References to the experimental results fire given in the text. For contcxt, the black contour shows a scan of the parameter space of 4 typical SUSY models, CMSSM, NlIHMl, MUHM2, pMSSMlO [73], Which integrates constraints set by ATLAS Run 1. Fig. 3.5: Areas of the cross section versus mass of WIMP dark matter particles above the curves are excluded. So one sees that regions favoured by DAMA and CDMS-Si are seemingly in disagreement (although not in a theory independent way). See reference [21]. by several experiments there is a spot in which the DAMA-LIBRA experiment -in fact by 9 standard deviations - claim to have found the dark matter (or at least something with very similar properties) by their special technology of looking for 44 H.B. Nielsen and C.D. Froggatt seasonal variations, that should appear because the speed of the Earth relative to the average velocity of the dark matter varies with season (see figure 3.1 above). 3.6 Dark Matter with only the Standard Model (except MPP) Contrary to everybody else, except for the people who take primordial black holes for dark matter, we want to propose a dark matter model inside the Standard Model, only with a certain assumption about the coupling constants in the Standard Model, that there are several vacua fine-tuned to have the same energy density. So we have very little "new physics": • We assume a law of nature - of a somewhat unusual kind - the "Multiple Point Principle" saying: there are several different vacuum phases, and they all have the same energy density (or we can include that they have ~ 0 energy density.) • Apart then from mentioning an attempt mainly with Yasutaka Takanishi to explain the baryon excess, we shall use only the Standard Model, even for dark matter! 3.7 Our Fit We performed a detailed fit with the model [3] in which we first of all looked for the absolute scale of the intensity of our model of dark matter pearls or balls emitting the X-ray line with photon energies of 3.55 keV in the rest system as apparently observed by satellites etc. 3.7.1 The Intensity The intensity we take in our model to be emitted by pearls, that have collided with one another - a rather infrequent event - but when they finally collide it is assumed, that the very strong skin surrounding the pearls can contract and thereby deliver energy, which can be used for the radiation in the 3.5 keV line or for other frequencies. There is in our model so to speak an active "energy production from the contraction". But this we can in fact estimate, if we have the parameters of the model. Of course the fact that we need collisions of a pair of pearls to get the radiation in the 3.5 keV line means, that the intensity resulting in a given region of the space becomes proportional to the square of the density pD of dark matter in that region. A fit to a model of this kind- which would also be applicable for a model in which the dark matter particles annihilate with each other - was performed using the astronomical - mainly satellite - data by Cline and Frey [15]. For the purpose of our model we can interpret it that they measure an intensity proportional parameter, which basically is in our language M, where M is the mass of the typical / average pearl, a the cross section for one such pearl hitting another one, and N the number of 3.5 keV photons emitted when such a collision actually happens. From the results of Cline and Frey we find the number exp (3.13) 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 45 or rather we extract this number from their table: Name Units N < CTCFV > * Í10GeVN2 I M ) 10-22cm3s-1 v km/s boost ( n<^cfv> U v v* boost ! ¡ 10GeVN2 I M ) 10-27cm2 Remark Clusters [1] 480 ± 250 975 30 0.016 ± 0.008 Perseus [1] 1400 - 3400 1280 30 0.037-0.09 Perseus [2] (1 - 2) *105 1280 30 2.7-5.3 ignored Perseus [23] 2600 - 4100 1280 30 0.07-0.11 CCO [1] 1200 - 2000 926 30 0.04-0.07 M31 [2] 10 - 30(NFW) 116 10 0.0086 - 0.026 30 -50 (Burkert) 0.026 -0.043 MW [22] 0.1 -0.7 (NFW) 118 5 0.00017-0,0012 ignored 50 -550 (Burkert) 0.084 - 0.93 in average Average 0.032± 0.006 Table 3.2: This table is based on the table 1 in reference [15]. It should be noticed though that something is not fitting well in the case of the Perseus Cluster in as far as one measurement in the outskirts of this galaxy cluster turns out to give a factor 1000 more radiation in the 3.5 keV line than the one that would have fitted with the proportionality to the squared density estimated from gravitational considerations. In our averaging we left this observation out totally, since it would have led to a very bad fitting for the other observations. But without this badly fitting observation we get the average (3.13). 3.7.2 The Frequency The very frequency or the photon energy 3.5 keV, we sought to fit with the "ho-molumo gap" in the ordinary material under high pressure - comparable to that in white dwarf stars - inside our dark matter pearls. Such a "homolumo gap" is a very general feature for materials containing a degenerate Fermi sea of fermions, say electrons, and in addition has some structure -like a glass or almost all materials - consisting in that the material in detail adjusts so as to partly lower the energy density of the Fermi-sea. It is obvious that the energy of the Fermi sea is lower the lower in energy the filled fermion states, whereas lowering the energy of the empty states does not lower the total energy. The adjustment to a ground state of the material will therefore (almost) unavoidably lead to a lowering of the filled states and thus cause a gap between the filled and the empty states. It is this gap between the filled and the empty single particle states which is called the homolumo-gap. It is namely the gap between highest occupied molecular orbit (the chemist expression for single particle fermion state), HOMO and the lowest unoccupied molecular orbit, LUMO. We estimated in [3] the value in energy of this homolumo gap partly just by a dimensional argument and partly by using a Thomas-Fermi approximation. 46 H.B. Nielsen and C.D. Froggatt The formula for our estimate of the homolumo gap, which also turns out to be the expected frequency or photon energy for the line, was r- i a\ 3/2 Eh = v^ (^J Ef. (3.14) Here a is the fine structure constant considered for the purpose of our dimensional arguments as a velocity (by multiplying it by the velocity of light c) and Ef is the Fermi energy of the electrons in the hard compressed material inside our pearls. 3.7.3 The fitting and theoretical speculations In our model we imagine that there are at least two phases of the vacuum - in addition presumably to several other ones too, but in the work now being reviewed we cared for only two important ones - and that the one in which we do not live, but which is realized inside the dark matter pearls, is distinguished from the present vacuum by there being a (boson) condensate of a speculated bound state of 6 top plus 6 anti-top quarks. In the vacuum phase inside the pearls we would at first have speculated that the expectation value of the Higgs field should go to zero, but that would give us an estimate of the tension of the skin separating interior and the exterior of the pearls, which would not give an acceptable fit. Indeed assuming that the usual Higgs spontaneous breakdown of the weak gauge symmetry in the vacuum inside the pearl is absent would suggest an order of magnitude of the tension in the skin of the pearls of the order of (100 GeV)3, but the fitting we made gives an appreciably smaller tension. Name t*10MeV AV ln ^*10MeV m AV Uncertainty Frequency "3.5keV" 5.0 1.61 100% Intensity 3.8 1.3 90% S1/3 theory 1) 0.28 -1.3 40% S1/3 theory 2) 1 0 40% Combined theory £,, AV 0.18 -1.7 100% Ratio t tspread =1 2.4 0.88 80% l.b. Table 3.3: Table of four theoretical predictions of the parameter on which the quantities happen to mainly depend. The first column denotes the quantities for which we can provide a theoretical or experimental value to be expected for our fit to that quantity. The next column gives what these expected values need the parameter combination ^*10VeV to be. The third column is the natural logarithm of that required value for the ratio , i.e. ln ^oy^. The fourth column contains crudely estimated uncertainties of the parameter thus fitted counted in this natural logarithm. In the last column we just marked the ratio . tspread with ° tradiation l.b. to stress that it is only a lower bound and shall not be considered a great agreement for our theory. 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 47 Fig. 3.6: The values of the ratio as needed for four constraints. There are two experimental constraints from the frequency and intensity of the 3.5 keV radiation respectively and two theoretical constraints in two versions corresponding to taking theory 1 or theory 2 for the tension. We make the simplifying assumption that all energy from the surface contraction in a collision gets emitted as 3.5 keV X-rays. The sixth line "Ratio +tspread = 1 " represents the lower bound ensuring J tradiation 1 ° that all the energy actually goes into 3.5 keV radiation. The essential parameter we used in our fit was defined as £ * 10MeV 10MeV * R/Rcrit AV "potential difference for nucleon in the two vacua" (3.15) In order to reduce the number of parameters in our earlier paper [6] we assumed that the pearls just had such a size that they were on the borderline to collapse and we call the radius of such barely stable pearls Rcrit. We now denote the actual radius of the (typical) pearl by just R and define the parameter £ = rR- . The parameter AV is the binding energy of a nucleon relative to when it is in the vacuum phase in the interior of the pearls. One should imagine that nucleons are attracted by the pearls by having a lower potential by the amount AV inside the pearl. If the pearl gets too small and the pressure from the skin thus too high it will pay energetically for the nucleons inside the pearl to escape and the pearl thus collapses; this is what happens when the radius is smaller than the critical radius Rcrit. The 10 MeV was just a conventional number, we put in to make the parameter dimensionless. It turned out from our calculations that the combined parameter ratio is the main one to fit, because the interesting measurable and theoretically interesting quantities mainly depend on it. We thus used it to make fits especially to the experimentally predictable quantities, the intensity of the 3.5 keV radiation scale and the very frequency 3.5 keV. The fitted values of the combined parameter ^oy^ for these quantities are 48 H.B. Nielsen and C.D. Froggatt presented in Table 3.3 together with those expected for a tension of (100 GeV)3 as obtained from the Higgs field consideration above (theory 1) - even a somewhat smaller value for the tension is speculated about and called theory 2 - and for a theoretical expectation. These predictions are also plotted in Figure 3.6. We see that the theoretical expectations for the tension S tend to fit with too small values of our parameter combination and so does our theoretical estimate of the £ deviation from criticality combined with the expected value for AV represented in the table and the figure below as "combined theory". The last line in the table and the figure represents a parameter value below which it is expected that more and more energy is lost to higher frequency radiation than the 3.5 keV one. This is because the pearl in the collision gets heated up and then the heat spreads out so quickly that only a very little part goes into the line observed as the 3.5 keV line. The point is indeed that we expect the temperature from the contraction of the surface to be much higher than 3.5 keV, but then this heat spreads out of course gradually on a second time scale to the whole pearl. Under this spreading out there is a spreading border at the place to which the heating has reached at any moment. Near that border the temperature is about 3.5 keV and the 3.5 keV radiation is produced and because the pearl material is supposed to be transparent to the 3.5 keV and lower frequency radiation, it is radiated out to outer space. But if the heat reaches all through the pearl the outer surface of the pearl gets appreciably hotter than 3.5 keV; then most radiation comes with higher frequency and is correspondingly lost for radiation in the observed 3.5 keV line. The "time ratio . tspread =1" represents the fitting to the value 2.4 of our ^-radiation. ■*• ^ parameter ^oy^ at which the heat just reaches to the border of the pearl. That is to say for smaller parameter values there is a significant loss in energy to higher frequencies, while for larger values of our parameter we expect that a major part of the energy from the contractions manages to be emitted as the line. 3.8 Latest Idea: Smaller Pearls givng also DAMA observation and Tycho Supernova Remnant Observation of 3.5 keV After the Bled conference we have looked at the idea that we could ignore the connection to the Tunguska event, which was at first so terribly important for our studies and instead seek a combined fitting of not only as just presented the 3.5 keV radiation from the clusters of galaxies and the center of our Milky Way, but also an observation, that would at first look like spoiling the hypothesis that the 3.55 keV line comes from dark matter. In fact this observation was considered by the authors of [24] to be a clear sign that the 3.5 keV line must after all be an effect of some ordinary ions - such as an unexpectedly high abundance of potassium (K) - but not a signal from dark matter. This observation is the observation by Jeltema and Profumo that the 3.5 keV line is indeed also emitted from the Tycho supernova remnant! In almost all usual dark matter models as elementary particles this appearance from the supernova remnant with very little dark matter compared to ordinary matter is rather absurd. It can only come about if the dark matter can somehow absorb the energy present in the remnant region and convert it into the 3.5 keV line. 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 49 We are now working on fitting the requirement to get the sufficient 3.5 keV radiation from the supernova remnant and it certainly points towards smaller values for the tension than even the fit above. In fact we have a crude fit to both the observation by Jeltema et al. and the DAMA and DAMA LIBRA observations, but now with both the cubic root of the tension S1/3 and the potential difference for a nucleon passing through the skin of the pearl AV being of the order of 1 or 2 MeV only. In this picture the pearls are less than atomic size and thus much more like dark matter models with WIMPs. But, especially to cope with the amount of interaction needed for the Tycho supernova remnant observation, they have to interact so strongly that they will interact several times on the way down through the earth to the DAMA-LIBRA observatory. So they should not be called weakly interacting, i.e the W in WIMP should be left out. Because they are, however, still very heavy, say 103 GeV or even heavier, compared to usual WIMP speculations, they are difficult to stop even when they hit matter in the shielding. So they can pass on and penetrate into the apparatus even if they have been somewhat hitting on the way down. Assuming that they as macroscopic objects - they are still pearls although now smaller - have somewhat different cross sections, some pearls may come through. Then even if only a small part comes through the shielding they could cause a number of events, as the observations suggest anyway in experiments like DAMA-LIBRA. Actually such a survival is only expected for some exceptional ones among the dark matter particles, which could easily lead to an enhanced dependence on the season and thus be especially suitable to be detected by DAMA-LIBRA relative to other experiments, that just observe the events independent of their season variation. 3.8.1 MM from Tycho Observation The mysterious 3.55 keV line has been seen, corrected to zero Doppler shift, not only from various galaxy clusters and the Milky Way Center, but also from the remnant of the supernova described by Tycho Brahe after its appearance in 1572. This at first seems to be in contradiction to the hypothesis that the X-rays should come from dark matter at all. The authors Jeltema et al. [24] take it that this Tycho supernova remnant observation means that the 3.55 keV line radiation cannot come from dark matter because basically there would not be dark matter in sufficient amounts in the supernova remnant. It would then have to be an ordinary transition line in excited ions, which must have been underestimated in the theoretical calculation of the other radiation from the supernova remnant say. Actually some underestimate of the abundance of potassium K could deliver a line in the region. But we basically take the point of view, that dark matter consists of some (type of) particles which have the possibility of being excited, and then when excited to send out especially X-rays in the 3.55 keV line. So we have the option of having the activity in the supernova remnant excite the dark matter particles there and thus make them radiate with their characteristic frequency 3.55 keV. (In the galactic clusters etc. we have a model of exciting them by collisions causing 50 H.B. Nielsen and C.D. Froggatt skin contraction and thus extra energy being set free. But the emission is again the characteristic line 3.55 keV.) But of course the absolute imperative for such a model for creating the 3.55 keV line radiation in the supernova remnant is that the dark matter particles (whatever they may be) have sufficiently big cross sections to at least pick up enough energy for the emission of the observed 3.55 keV line radiation. How we got the need for MM > 6 * 10 7m2/kg What observed: Jeltema and Profumo claim [24] that they have observed an X-ray spectral peak - fitted with difficulty, but noneteless fitted to be there - with an intensity of 2.2 * 10-5 photons per cm2 per s. Thus in each cm2 of the sphere around Tycho passing through the earth, there passes 2.2 * 10-5 photons per s per cm2 (or is it 2.2 * 10-6? as there is a discrepancy with the figure in [24], and if the figure is right). The distance to Tycho (SN1572) is about 9000 light-years. In fact, according to Wikipedia: "The distance to the supernova remnant has been estimated to between 2 and 5 kpc (approx. 6,500 and 16,300 light-years), with recent studies suggesting a narrower range of 2.5 and 3 kpc (approx. 8,000 and 9,800 light-years)." Taking 1 light-year = 1016 m, the area of the sphere around Tycho going through the earth is sphere area = 4n * (9000ly * 1016m/ly)2 (3.16) = 1041m2 (3.17) So the number of 3.55 keV photons passing through this surface will be # of photons = (2.2 ± 0.3) * 10-5cm-2s-1 * 1041 * 104cm2 (3.18) = 2 * 1040s-1 (3.19) - ''an energy rate" : 3.5keV * 2 * 1040s-1 (3.20) = 1032 erg/s. (3.21) Rate of Energy Ploughing up The total energy in the remnant region will still in first approximation be equal to the energy ejected from the supernova, if we assume that the energy escaping as light going so far away that we no more can count it as belonging to the remnant is small compared to the part remaining in the remnant region. A major part of this energy is presumably in the form of fast moving particles or even X-rays, so that order of magnitudewise we may count it as cosmic rays moving with the speed of light relative to the dark matter pearls, which of course have a much lower velocity of the order of the escape velocity from the Galaxy. All over the remnant region we assume that the density of dark matter is very similar to that in the neighborhood of our solar system 0.3 GeV ,„ , Dsun = -ev, (3.22) 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 51 so that the number of pearls we have in every cm3 is 0.3MeV. In each second each of these pearls pick up the cosmic rays or whatever material in the remnant in a volume d * v « d * c where d is the cross section for a pearl and v is the average relative velocity of the pearl and the remnant matter or radiation. That is to say, that during a second the fraction of the volume getting ploughed through is "Fraction ploughed through" = * d * v. (3.23) So if one observes a 3.55 keV line with an intensity I = 2.2 * 10-5 photons per s per cm2 we need the total energy rate (power) at a distance d = 9 * 1019m to be W = I * 4n * d2 * 3.55keV * 2.2 * 10-5cm-2s-1 = 1032 —. (3.24) s Then we must have W = Eremnant * vD * M, (3.25) or d M W Tycho Eremnant * Dsunv 1032erg/s (3.26) (3.27) _ 1051 erg * 0.3 GeV/cm3 * 3 * 1010cm/s = 0.56 * 10-2cm2/kg (3.28) = 10-29cm2/GeV (3.29) 1 (3.30) (3.4 GeV )3 3.8.2 Comparing to Nuclear o/M Ratio The material inside our pearls is highly compressed and taken to be mainly carbon (with atomic number A = 12). Then using a crude formula 1.2A1/3fm for the radius of a nucleus and n(1.2A1/3)2fm2 for the cross section for some smaller particle scattering on the nucleus, we get for nucleus scattering: n * 1.22fm2 * A2/3 (3.31) MA nuclear A * 0.94 GeV 123 GeV-3 (3.32) ^2 1 (3.33) (0.265 GeV)3 Combining these numbers for the ratio M needed for the dark matter in the supernova remnant (3.30) with the one for a suitable nucleus (3.31) we see that the needed lower bound is M ITycho _ (0.26 GeV)3 (3 3^ M Inuclear (3.4 GeV) = 0.0763 = 4.5 * 10-4. (3.35) 52 H.B. Nielsen and C.D. Froggatt This means that about 1/2000 of the accessible energy would indeed become 3.5 keV photons, if the cross section for the pearls in Tycho was actually equal to the nuclear cross section. Actually such an efficiency of 4.5 * 10-4 is not at all unlikely. So we could claim that, having in mind that the orders of magnitude could have run out to wildly different values, the rather close agreement could be taken to mean that indeed the true m for the dark matter pearls being excited is indeed equal to the nuclear one (3.31). If indeed the pearls were so small that there was no significant shadowing by one nucleus of another of the nuclei in the pearls, then the cross section to mass ratio would just be the nuclear one. So an order of magnitude agreement with the actual cross section to mass ratio being the nuclear value should be taken almost as successful agreement. Let us say this in other words: If we assume that the tension S and the parameter -f have such values that formally the cross section to mass ratio M would be smaller than the corresponding nuclear ratio (3.31), the actual cross section to mass ratio would only be approximately equal to the nuclear ratio. (Here f is the radius scaling factor for fixed tension S, see section 3.8.3). For so thin pearls a cosmic ray say could with high probability pass through the pearl without hitting any nuclei inside. For such parameters one would obtain for the cross section to mass ratio just the nuclear value, see Figure 3.7. But anyway of course there would be an appreciable loss of energy that would not go to the 3.5 keV radiation, even compared to the amount of 3.5 keV radiation having been corrected with the time ratio for the fact that the emission into 3.5 keV radiation only takes place in a short period of time tspread. Let us say that it is only the fraction 1/1 of the energy available in the period when the surface of the pearl is still cold that really comes out as this radiation. Having in mind instead of the collision events the events in the Tycho supernova remnant this time ratio correction is not present, because the single cosmic ray exciting the pearl is supposed not to heat it up so much that the problem of the pearl being hot comes up. So for the Tycho supernova remnant the emission of the 3.5 keV radiation should be calculated without this time ratio correction. But it should still for "general" inefficiency be reduced by the factor I. For pedagogical reasons we could imagine, that we could estimate the efficiency 1/1 sufficiently accurately that we could say: Fantastic that we just get the radiation as observed by Jeltema et al. from the Tycho Supernova remnant equal to this I divided into the rate expected if all the energy went to 3.5 keV radiation and the cross section to mass ratio was just the nuclear physics one (3.31). In this optimistic thinking we would have an empirically based suggestion saying that the size of the pearls are actually so small that the cross section to mass ratio becomes equal to the nuclear ratio. But for this to happen it would have to be that the formally calculated ratio should be larger than or equal to this nuclear ratio. This in turn will put an upper limit on the tension S depending somewhat on our parameter -f, since the cross section to mass ratio is a decreasing function of the tension S and then of course also as a function of the third root of this tension S1/3 which we mainly use in our text and figures. The upper limit following from this consideration based on claiming the nearness of the ratio actually estimated from 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 53 Jeltema et al. to the nuclear ratio is shown on Figure 3.9 below as the line labeled "nuclear". Fig. 3.7: This figure illustrates that for the density inside a pearl being very high a cosmic ray particle hitting the pearl will sooner or later in the interior hit a nucleus, while for a very little pearl with the same density the thickness of the pearl is insufficient for all cosmic ray particles to hit a nucleus and the cross section will be less than the geometrical one ct = nR2. The ratio M is then rather equal to the nuclear value (3.31). Resume of Comparison with Nuclear Ratio m But let us stress again that, if the loss of energy by the inefficiency of making 3.5 keV radiation from all the energy available could be estimated to be a factor of the order of I = 2000, then we could claim the very value of the Jeltema et al. observation strength as a victory for the picture. 3.8.3 Combined Fitting, Small Pearl Model Formulas for the Critical Case, Pearls Just about to Collapse First let us give a list of the interesting quantities in terms of the cubic root of the tension of the surface S1/3 and the energy difference for the nucleon on passing the domain wall AV in the case of a critical sized pearl. By this we mean the case in which a further parameter has been avoided by adjusting it so that the tension provides a pressure on the material inside the pearl making it just on the border to collapse by spitting out nucleons. In other words providing enough pressure to just barely compensate the potential difference AV per nucleon. So now we should note the various parameters in this borderline/critical situation (see reference [3] for details 54 H.B. Nielsen and C.D. Froggatt on the notation): 3n2S Pearl radius Rcrit = 2(Ay)4 (3.36) Fermi momentum pf crit = 2AV (3.37) Energy release by collision ES crit = S(~ 4n)R;;rit (3.38) = n5 * 9S3/(AV)8 (3.39) Collision cross section Ocrit = n * (2Rcrit)2 = 6 * n3S2/(AV)8 (3.40) ^spread crit = 4— * R |crit (3.41) a55R2T = -25R^ |crit (3.42) ES tradiation crit = 4nR2asT (3.5keV )4 (3.43) 60S n2(3.55keV )4' txcrit 6 * n3S2/(AV)8 M i-r, 24n5S3 Mcrit mN * (AV)9 AV 2mN Ncrit Es crit AV Mcrit Mcrit * 3.55KeV 2mN * 3.55keV Nt M2 —73.5kev , c>it - - - (AV )2 allEs^3.5keV; crit Mcrit Mcrit (3.44) (3.45) (3.46) (3.47) (3.48) (3.49) Ncrit O'crit z0 cr,>, * T~a--(3.50) 4n2SmN Es crit _ S(- 4n)( (fVf)2 Mcrit mN^^Vf AV 8n2SmN * 3.55keV (3.51) tspread Nt ^spread ^radiation AA2 3.5keV frequency _ EH _ 137-3/2V2pf _ 137-3/2V22AV (3.53) 31 9nMcrit =(Rpf)lcrit (3.54) 8mN Mcrit _ 24n5S3 mN _ (AV)9 . (3.55) With Radius Scale up Parameter f The critical case is not realistic except very crudely. The pearls would collapse by the tiniest deformation during the contraction in the early universe situation. We must expect that there must be an appreciable safety margin in the sense, that the number of nucleons inside the 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 55 contracting pearl for the pearl not to collapse immediately must be so large, that the final radius, when the fluctuations from the contraction have died out will be say R = f * Rcrit with f « 5. We estimated in earlier articles this expected ratio of the average radius to the critical or borderline one to be V4n * 24/9 « 5. The dependence of some of the important quantities with this f goes as follows. Here we also include the dependence on AV and on S: Pearl radius R = £,fs Rcrit = f S * 24n2 (2AV )4 Cubic root of tension S3 = S1/3 (fixed) Fermi momentum pf Energy release by collision ES Collision cross section 3.5keV AV 8n2SmN * ( ^Vt } 3.55keV = 1.15 * 10-13GeVAV * AV (3.72) (3.73) 2V2 frequency = Eh = 137-3/^v/2pf = f74 -j^Tl AV (3-74) M mN ^ = £ 2 m. (3.115) So taking the density in this sensitive apparatus to be say papparatus = 3000 kg/m3, we have P apparatus lhit > (31 * 3000kg/m3) * ^m (3 Pwater Istop " 3400 m * 1000 kg/m3 (. ) 9 1 = 1.37 * 10-2 = —. (3.117) This means that there is at most 73 times as much weight in the pearl compared to the important nucleus weight in the shield. If say the important or average nucleus in the shield is silicon with mass 28 GeV, then the pearl's mass is of the order of 73 * 28 GeV = 2000 GeV. Now in order to have a proper macroscopic electron cloud in the pearl that can give the macroscopically estimated homolumo gap, we need that the pearl nuclear charge Z (i.e. the number of protons) is at least large enough that an atom of this atomic number can provide AV order of magnitude binding energies. Taking the binding energy to be of the order of Z Rydberg, it means we need Z > 1Ryd>erg, so that for say AV = 1 MeV we would need Z > 104. This would be a problem for our model if we took the above estimate of 2000 GeV too accurately. But this limit is so close that we shall of course rather take it that now we know the bound must be very close and we shall take the mass ro be M « 2000 to 10000 GeV - see Figure 3.10. An Interesting Coincidence Let us note, that we have got almost coincidence between the mass 104 GeV needed for our macroscopic approximation to be valid and the value obtained above. In other words we can say that the mass needed for keeping a sufficiently high electron density such that e.g. our homolumo-gap calculation is still valid and the mass estimated from DAMA-LIBRA, say 2000 GeV, are essentially the same, which is a funny coincidence! Actually if we begin to fit with a mass a bit smaller than 104 GeV, there will be a correction to the formula for the homolumo gap size and thus for our prediction 68 H.B. Nielsen and C.D. Froggatt Fig. 3.10: How we get the mass suggested by DAMA-LIBRA for our model as a "compromise" (denoted by"compr." on the figure) to be ~ 2000 GeV. Formally, using the "simple formulas", we have an upper bound of 2000 GeV and similar crude upper bounds using the number of events in DAMA-Libra. Using the "figure" or the "text" values in the Jeltema and Profumo paper give upper limits for the pearl mass of 1600 GeV or 160 GeV respectively. But if one corrects for the energy lost from going to the 3.5 keV line, as indicated by "l-impr." in the figure, the for the pearls is increased by a factor of the order 2000 and we get the upper limits indicated with a thick line of about 1 GeV or 0.1 GeV. So formally we have an inconsistency of our requirement but, considering that we only have order of magnitude bounds which should approximately be equalities, we have a good compromise value. of the very frequency 3.55 keV. So the true prediction of this frequency would be a bit lower, if such corrections for the bigger extension of the electron cloud than the size of the skin is corrected for. This actually means that the true homolumo gap has a maximum very near to the values we here use to fit with. This may be of some significance for really getting a peak in the X-ray spectrum (at 3.55 keV), since a priori pearls of a bit different size will give different frequencies for the radiation and thus smear out the peak relative to what would appear, if all the pearls have exactly the same size. It may only go with the fourth root that there is such a dependence but still it is a smearing out. Suppose it happens that the dominant size of the pearls is just around a point where the approximation of the electron cloud keeping inside the skin of the pearls stops being valid. Then there will be a correction that for making the pearl smaller counteracts the increase in frequency that the smaller pearl should cause. The result is a maximum in the frequency spectrum of the X-ray radiation. This means an improvement in the sharpness of the line is predicted. 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 69 If we somehow argue that just such a maximum is favoured it would mean we could consider this coincidence as a success. 3.8.6 XenonlT Electron Recoil Excess An observation that may fit very well into our version of the pearl model for dark matter with the less than atomic size pearls is the XenonlT Electron Recoil Excess [16]. This effect of electrons seemingly appearing with energy close to just 3.5 keV - note the coincidence we want to stress with the 3.5 keV X-ray line photon energy - would independent of the details the dark matter model be very indicative, since we already have a strong suggestion that dark matter tends to emit light with the 3.5 keV frequency. Apart from the DAMA/LIBRA and DAMA experiment the other direct search experiments seem to find only negative results when looking for the dark matter. There was, however, found one unexpected result [16] although at first not seemingly due to dark matter: The experiment XenonlT investigated what they call electron recoil in their Xenon experiment. In the Xenon experiment one has a big tank of liquid Xenon with some gaseous Xenon above it and photomultipliers looking for the scintillation of this xenon. The philosophy behind the experiment that a dark matter WIMP e.g. hits a nucleus inside the xenon and the recoil of this creates a scintillation signal S1 and also an electron which is then driven up the xenon tank by an electric field and at the end by a further electric field made to give a signal at the top S2. By the relative size of the signals S1 and S2 one may classify the events - which are taken to be almost coinciding pairs of these signals S1 and S2 - as being nucleus recoil or electron recoil. One expects to find the dark matter in the nucleus recoils, since a dark matter particle is not expected to make an electron with sufficient energy to make an observable electron recoil event. But now carefully estimating the background expected the Xenon1T experimenters found an excess of electron recoil events. Proposed ideas for explaining it include axions from the sun or neutrinos having bigger magnetic moments or perhaps less interestingly that there could be more tritium than expected in the xenon. But here our model of relatively stronger interacting particles able to radiate the line 3.55 keV when excited provides a possible explanation: Going through the earth and the rest of the shielding the pearls or particles get excited so as to emit 3.55 keV X-ray just as they would do it in the Tycho supernova remnant, where they also get excited by matter or cosmic rays. But then the particles passing through the deep underground Xenon1T experiment are already excited and prepared for sending out the 3.55 keV radiation. Now they could possibly simply do that in the xenon tank or they might dispose of the energy by a sort of Auger effect by rather sending out an electron with an extra energy of 3.55 keV. Such an electron with an energy of a few keV could be detected and taken for an electron recoil event in the Xenon1T experiment. It is remarkable that the signal of these excess electron recoil events appears to have just an energy of the recoiling electron very close to the value 3.55 keV. 70 H.B. Nielsen and C.D. Froggatt Indeed the most important bins for the excess are the bins between 2 and 3 keV and the bin between 3 and 4 keV. So we would claim that there is in our model no need for extra solar axions or neutrino magnetic moment, nor tritium. But we claim it to be 3.55 keV radiating dark matter one sees in the xenon experiment! 3.9 Conclusion We have put up two slightly different models for dark matter being actually pearls which have a new phase or type of vacuum inside, which by our "Multiple Point Principle" is supposed to have the same energy density as the present vacuum. The two models only differ by taking the parameters different, especially the tension of the surface separating the inside with its vacuum from the outside with the present vacuum. The two models are thus given as roughly: • Big pearls, adjusted to the Tunguska event being due to one falling down onto the earth: The cubic root S1/3 of the tension is several GeV, the size of the pearls is cm-size. • Small pearls: The cubic root of the tension S1/3 is of the order of 1 MeV, the size of the pearls a bit bigger than atomic nuclei. Our main result was that we could fit both very frequency 3.5 keV of the X-ray radiation suspected to come from dark matter and the intensity as fitted by Cline and Frey to a series of observations of this line from various galaxy clusters with essentially one parameter, which we wrote as . So two observed quantities by one parameter. Both observations concern the still doubtful 3.5 keV X-ray radiation. We can essentially keep this parameter whether we take the pearls big with a big surface tension or small with a small surface tension. Taking the model with the small pearls, on which we have far from finished everything, we hope that we can further: • Make the DAMA-LIBRA controversial observation of dark matter by the seasonal variation technique compatible with the model. • Fit the a priori very strange observation by Jeltema and Profumo of 3.5 keV radiation coming from the Tycho supernova remnant in the picture with the 3.5 keV radiation coming from dark matter. (Something they take themselves as the sign that this 3.5 keV line is not coming from dark matter but from some ion such as potassium). • We have for our model a very promising coincidence of the electron excess energy from the XenonlT experiment with the number 3.5 keV. The point is that the our pearls - in the small size model - come through the apparatus of the XenonlT experiment and are excited with some extra electrons or simply have some excitons in them - excited during the passage through the shielding 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 71 - which then deliver just the 3.5 keV energy to an electron in the XenonlT experiment. And that is then giving an excess of such events with just an excited electron which was the ununderstood effect seen by XenonlT. 3.9.1 The fitting of the Small Pearl Version We basically make predictions from the small pearl version with the following parameters: • The surface tension represented by its cubic root: §1/3, • Essentially the potential difference AV for a nucleon inside versus outside the rl/4 pearl, represented by the combination fr (where f is the ratio of the radius of the pearl to the "critical" radius at which the nucleons would be just about to be spit out. Presumably even coming in under the fourth root this ratio f is not of much significance and probably is ~ 5. • An efficiency parameter 1 for getting 3.55 KeV radiation compared to what our estimates at first suggest. One gets really 1/1 times the energy available in the time during which the pearl is sufficiently cold for radiating appreciably in the 3.55 keV line. With these parameters we fit 1) the intensities of the Cline-Frey fit, 2) the Supernova remnant intensity, 3) the very frequency 3.55 keV and 4) a crude mass extracted from the observations of DAMA-LIBRA in the way it is interpreted by us, namely with somewhat strongly interacting pearls, only coming through by means of their high mass. So we fit 4 data point with 3 parameters. This is still formally a success, but now we claim that in addition and crudely consistent with the fit we have that the actual cross section to mass ratio for our small pearls coincides with the cross section to mass ratio for e.g. carbon nuclei. This corresponds to the fact that our pearls are so small that cosmic rays in the supernova remnant say passing though the pearls only interact when they hit a nucleus but otherwise can escape through without touching the pearl. The pearls are so to speak so thinly filled that the cosmic rays "see" the single nuclei in the sack making up the pearl. Further it is a coincidence, although not obviously reasonable to understand physically, that the size of the pearls is just such that the electron cloud begins to emerge significantly outside the skin surrounding the pearl. This means that the homolumo gap providing the very frequency 3.55 keV for the radiation has a maximum at just this fitted situation. Thus the 3.55 keV line will be especially sharp compared to the possibility that this coincidence was not realized. If we even counted this last coincidence as understandable as say a stable point more likely than a general point, then we could claim we rather fitted 4 data points with 3 parameters and 2 constraints, meaning really only with 3-2 = 1 parameter. 72 H.B. Nielsen and C.D. Froggatt 3.9.2 Parameters S1/3 and AV Small and Outlook The parameter values we obtained with our "Small Pearls Version" were S1/3 = 3 MeV (3.118) £V4 f- = 0.5 MeV-1, (3.119) AV which with f « 24/9 * V4n « 5 (3.120) gives AV « 1.34 MeV. (3.121) Both these values for the parameters in the notation in which they have dimension of energy are - one would say embarrassingly - small compared to the dimensional argument expectations, if one speculated that Higgs physics and top-quark physics were involved. That would namely instead give e.g. S1/3 ~ 100 GeV. This means that Higgs and/or top-quark physics is not at all a promising possible explanation behind the vacuum-phases. We rather need physics of an energy order of magnitude even under or at least in the very low energy scale end of strong interaction physics, or it should be rather a kind of atomic physics involved. We have ideas under development taking as a starting point the work by Kryjevski Kaplan and Schaefer [26], who calculated the phase diagram for nuclear matter under various high nuclear densities and considered the so called CFL phase. This stands for color flavour locking phase meaning that the SU(3)c color group is broken spontaneously in a direction locked with that of the flavour SU(3)f group. It is remarkable that these authors find a triple point as a function of the light quark masses coinciding with the experimental quark masses. This is, however, not quite what we would need to have a case of MPP degenerate vacuum-phases. Because of the high baryon density used in the study of Kryjevski Kaplan and Schaefer [26] their phases are namely not vacua. Nevertheless we are working on arguing that their phase diagram might be extrapolated down to zero baryon density and thus tell us about vacuum phases. In that case an energy scale for the phase transition physics of the order of the strong interaction scale Aqcd ~ 300MeV could be understandable. Even reaching down to a few MeV is at least closer than if one should begin with the Higgs-mass scale. Such surprisingly low tension domain walls also bring the chances for them to really be acceptable astronomically much closer. The problem with domain walls coming to dominate energetically the whole cosmology and thus being phenomenologically unacceptable is of course weakened the lower the tension and thereby from Lorentz invariance also the energy per unit wall-area is. References 1. E. Bulbul, M. Markevitch, A. Foster et al., ApJ. 789,13 (2014) [arXiv:1402.2301] al., 3 Dark Matter Macroscopic Pearls, 3.55 keV X-Ray Line, How Big ? 73 2. A. Boyarsky, O. Ruchayskiy, D. Iakubovskyi and J. Franse, Phys. Rev. Lett. 113, 251301 (2014) [arXiv:1402.4119] 3. Colin D. Froggatt, Holger B. Nielsen "The 3.5 keV line from non-perturbative Standard Model dark matter balls" arXiv:2003.05018 4. C. D. Froggatt and H. B. Nielsen, Phys. Rev. Lett. 95 231301 (2005) [arXiv:astro-ph/0508513] 5. C.D. Froggatt and H.B. Nielsen, Proceedings of Conference C05-07-19.3 [arXiv:astro-ph/0512454] 6. C. D. 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Murygin, "Domain walls and strings formation in the early Universe", see this Volume, p. 128, arXiv:2011.07041v1 [hep-th] 15. J. M. Cline and A. R. Frey, Phys. Rev. D90 123537 (2014) [arXiv:1410.7766] 16. Xenon-collaboration "Observation of Excess Electronic Recoil Events in XENON1T" arXiv:2006.09721v2 (2020) 17. J. Bowman et al, Nature 555 67 (2018). 1 18. R. Barkana, Nature 555 71 (2018). 19. C. D. Froggatt and H. B. Nielsen, Mod. Phys. Lett. A30 no.36, 1550195 (2015) [arXiv:1503.01089]. 20. M. Aglietta et al., Europhys. Lett. 3 1315 (1987). 21. M. Tanabashi et al (Particle Data Group), Phys. Rev. D98 030001 (2018) 22. A. Boyarsky, J. Franse, D. Iakubovskyi and O. Ruchayskiy, Phys. Rev. Lett. 115,161301 (2015) [arXiv:1408.2503] 23. O. Urban, N. Werner, S. W. Allen, A. Simionescu, J. S. Kaastra and L. E. Strigari, MNRAS 451, 2447 (2015) [arXiv:1411.0050] 24. T. Jeltema and S. Profumo, MNRAS 450, 2143 (2015) [arXiv:1408.1699] 25. R. Bernabei; et al. (2013). "Final model independent result of DAMA/LIBRA-phase1". European Physical Journal C. 73 (12): 2648. arXiv:1308.5109. and contribution in the present volume, and earlier volumes: Workshop 2020 on "Beyond the Standard Models" organized by Norma Mankoc Borstnik, D. Lukman, M. Khlopov and H.B. Nielsen. R. Bernabei, DAMA/LIBRA, LNGS scientific committee meeting, 26-27 March 2018, https://agenda .infn .it /conferenceDisplay.py ?confId =15474, 2018 26. Andrei Kryjevski, David B. Kaplan, Thomas Schaefer "New phases in CFL quark matter" arXiv:hep-ph/0404290v1 Scientific Debuts All contributions are arranged alphabetically with respect to the authors' names. Bled Workshops in Physics Vol. 21, No. 2 A Proceedings to the 23rd [Virtual] Workshop, Volume 2 What Comes Beyond ... (p. 77) Bled, Slovenia, July 4-12, 2020 4 Problems of the Correspondence Principle for the Recombination Cross Section in Dark Plasma K. Belotsky *, E. Esipova **, D. Kalashnikov *** and A. Letunov ^ National Research Nuclear University MEPhI, 115409 Moscow, Russia Abstract. We raise the issues concerning correspondence principle in description of a recombination of oppositely charged particles. These issues have come from cosmological dark matter (DM) problem investigations. Particles possessing Coulomb-like interaction are considered. Such Coulomb-like interaction between DM particles is assumed though the problem seems to be more general. Analysis showed that usage of different semiclassical approaches leads to the apparent discrepancy between numbers of recombination acts. We attempted to find some conditions under which classical cross-section (which relates to multiple soft photon process) reduces to quantum one, which is obtained in semi-classical approximation (Kramers' formula). We just draw attention to this and provide some (not decisive) arguments. Povzetek. Avtorji opozorijo na probleme, ki se pojavijo pri uporabi korespondencnega nacela za opis rekombinacije delcev z nasprotnimi naboji. Na te probleme so naleteli pri raziskavah sipanja delcev temne snovi, ki interagirajo s Coulombovi podobnimi iner-akcijami. Ugotavljajo pa, da se te vrste problemov pojavijo tudi pri drugih interakcijah. Analize so namrec pokaZale, da različni semiklasicni pribliZki ne napovedo enakega števila rekombinacij. Iscejo pogoje, pri katerih se klasicni sipalni preseki za procese, pri katerih se izseva vecje stevilo mehkih fotonov, ujemajo s kvantnimi racuni v semiklasicnem priblizku (Kramersova formula). Keywords: correspondence principle, dark matter, dark plasma, collision theory, semiclassical approach 4.1 Introduction Investigation of dark matter (DM) is one of the most important problems in cosmology and particle physics. Many experiments are being carried out to detect DM particles and explore its properties [1-5]. A part of the models considers self-interacting DM including Coulomb-like interaction [6-13]. Within the framework of such models, several disadvantages of the standard ACDM scenario can be * E-mail: k-belotsky@yandex.ru ** E-mail: esipovaea@gmail.com *** E-mail: impermast@gmail.com t E-mail: letunovandrey11@yandex.ru 78 K. Belotsky, E. Esipova, D. Kalashnikov and A. Letunov avoided. These are cuspy density profile of the halo, number of small halos and similar other. Models with Coulomb-like interaction lead to the differences in cosmological evolution. Dark charged particles can form a bound state. If they are a particle and an antiparticle, they annihilate, what significantly reduces their density. If they are not particle and antiparticle but have opposite "dark" charges (they can be called in this case "dark electron" and "dark proton"), binding processes (recombination) lead to a decrease of free dark charged particles. This significantly changes dynamics of dark matter during the formation of structures in the Universe [8,14] and thermodynamical evolution which we considered previously [15]. Thus, accounting for recombination is an essential part of such models. Theory of atomic kinetics is deeply studied section of plasma physics [16,17]. Density evolution of specific particle's sort is routinely calculated. These methods were applied to the study of dark matter [18]. However, the authors of the present work are deeply convinced that the question of choosing the correct expression for the cross-section of atomic processes in cosmological structures is not clear enough. For example in the famous paper [19] devoted to density of monopoles in the Universe the authors used the classical approach for the recombination calculations. Rate of recombination depends on its cross-section. The Kramers formula (4.1) is typical of kinetic plasma calculations. Its semiclassical expression describes a single-photon recombination for a hydrogen-like atom [20] 32n 3 2 ft—0 _ ^0—^, (4.1) aq (n) = 373a aoe-03, where ft—0 is the energy of ground state, n is the principle number of a bound state, — is the frequency of emitted photon, a is the fine-structure constant and a0 is the Bohr radius. The approximate summing (coming up to integration) of (4.1) leads to the following expression 32n 2 ( c \ 2 Zca aq = 37=ar2Z2 (-) ln — (4.2) 2 where Z is a charge of ion, r0 = me c2 ~ a is the classical electron radius and v is initial velocity. e An another formula for the recombination cross-section was derived by Yelutin [21] 14 ad = n (4n)2 r0Z8 (C) 5 . (4.3) The expression (4.3) was obtained in terms of the classical mechanics. A single electron is considered as moving from infinity losing its energy due to dipole radiation. When electron's energy becomes zero it comes to bound state. Expressions (4.2) and (4.3) have different conditions of applicability. Formula (4.2) is valid when Ze2 > ftv. (4.4) 4 Problems of the Correspondence Principle for the Recombination. 79 That is v C a in natural unites (fi = c = 1 ) and Z = 1. It is a common condition for semiclassical approximation in scattering theory [22]. For the classical cross-section we have i.e. v C a5/2 in natural units with Z = 1. However, if electron's speed satisfies (4.5), one will point out dramatic discrepancies. Firstly, the cross-sections have different initial speed dependencies. Secondly, the expression for the classical cross section is several orders of magnitude larger than the quantum. Finally, expressions have different orders of fine-structure constant. Derivation of an accurate quantum expression for the many-photon recombination is sophisticated problem. The semiclassical consideration of a stimulated bremsstrahlung is presented in [23]. This paper shows that every partial cross-section depends on its own photon frequency. Establishing a relation between the number of photons and their energies is very complex issue. Well-known description of quantum single-photon processes is presented in [24]. Although, cross-sections of considered reactions have an analytical form, its derivation is quite cumbersome. The bremsstrahlung cross-section is expressed in terms of the complex hyper-geometric series. Additional photons make establishing correspondence between quantum expression and (4.3) almost impossible. To sum it up, the investigation of dark matter led us to the problem on scattering theory. How to obtain an expression for recombination cross-section when low energy electron emits infinitely many photons? Unfortunately, classical monographs devoted to atomic physics do not contain the solution [25,26]. Here we list some considerations on this topic which do not give solution of the issue. One of the argument, bases on an action, is taken from our previous work [14]. We just want to collect together something existing for thought just to attract attention to this issue. 4.2 Correspondence between classical expression and Kramers Now we want to find the situation when two expressions would coincide. In order to do this we have to guess energy loss of an electron. One can notice that if w in the denominater of the formula (4.1) will be changed to E it can coincide with (4.3). It will be shown below. We are considering the electron moving from infinity losint it's energy. Just before the last this iteration(the photon emission) it is possible to use Kramers formula for one-photon recombination. In order to achive the coincidence we will establish the following relation (4.5) formula E2 = IE M> 2 (4.6) where E is the total energy of the electron before coming to the bound state and o> is the last emitted photon. 80 K. Belotsky, E. Esipova, D. Kalashnikov and A. Letunov Also the following relations express energy coversation E = hœ - ^ (4.7) E = E - AE (4.8) where AE is the total energy loss of the electron before last emitting of a photon. After substitution of (4.7) and (4.8) in (4.6) one will obtain the quadratic equation for AE. The solution for AE has the following form AE = E + hœ0 ± U*E2 + ( (4.9) 2n2 2\ V n It is necessary to choose the sign — to satisfy the energy conservation law. The next step is summation over new partial cross sections _ 32n 3 2 o-q (n) = a3 a2 0 3v/3 0 E2n3 starting with number k that we assume bigger than unit to be in the framework of semiclassical limit. ^ , , 16nZ4 a3r2 (c)4 „ „ , oq = (n) = iTTV0 (v) (4.10) n=k v After comparsion of (4.3) and (4.10) it is easy to obtain the result for k k = a3 (V) 3 Z6 (4.11) Assume k > 1 and we will obtain 4 ( c „,-5 Z AW > a-5 (4.12) This condition reproduce (4.5). Thus, this approach let one point out the connection between two expression: Kramers single-photon cross-section can be integrated into the process of bremsstrahlung. To clarify the physical situation we will give the following reasoning. Slow electron moving from infinity loses it's energy because of bremsstrahlung. In order to understand the correspondence between classical expression (4.3) and Kramers formula (4.2) it is necessary to do some manipulations. Firstly, one can notice that when electron initial speed satisfies (4.5), energy of a single emitted photon have to be relatively large. Electron's energy must be spent on coming to bound state (it has negative energy). Secondly, we rightly assume that if electron overcomes a great distance is not influenced by any other external factors, it will emit many photons. Finally, we establish total energy loss of electron before coming to bound state. Summing all new partial cross sections and comrasion of obtained formula and (4.3) lets one reproduce original condition (4.5). 4 Problems of the Correspondence Principle for the Recombination. 81 4.3 Estimation of action Evaluating of an electron's action also leads to (4.5). We will consider an electron moving in the external Coulomb-like field. In order to simplify the calculation charge of ion Z is put equal to unit S = rt2 f mv2 e2 tA 2 r + — dt (4.13) where S is the action of the electron. In the region of interest kinetic energy is proportional to potential 22 mv2 e2 2 This immediately implies v ~ J 2m2. (4.14) 1^2 „2 S 2— ~ Vme^Vrl - VTT) (4.15) r, rv Here ri corresponds to radius of coming to bound state and r2 is the same value with adding the distance, which electron needs to cover for losing most of it's it's initial energy (see [14]). Eventually, if one requires S C h and obtains V 5 -< a2. (4.16) c Obviously, this condition is in agreement with (4.5). r 4.4 Conclusion Calculations originally connected with an estimation of dark matter particles in the Universe generated problem on collision theory. Dark matter is considered as self-interacting according to the law of Coulomb. It immediately implies that darkly charged particles will recombine intensively. Rate of recombination is proportional to the cross-section of its process. Dark matter is considered to have very low energy, what is naturally realised in the Universe (CMB has temperature 3 K, non-relativistic DM should have much lower temperature), what possibly accounts for applicability of classical approximation in a recombination process description. This should be understood in the sense of scattering theory. Characteristic speed of particle is lower than atomic speed. This condition is expressed by (4.4). In contrast to the Born approximation, semiclassics is applicable here. The problem of the correct expression for the recombination cross-section was discovered. On the one hand, Kramers formula [20] is widely used in atomic kinetics. On the other hand, classical expression [21] describing the electron capture does not correlate with Kramers cross-section. Of course, this discrepancy is connected with the fact that these sections relate to different reactions. Expression describes a 82 K. Belotsky, E. Esipova, D. Kalashnikov and A. Letunov single-photon recombination. Formula (4.3) relates to the process with emitting of infinitely many photons. An attempt to link these two approaches reproduced initial applicability condition (4.5) for (4.3). Moreover, the estimation of the electron's action also leads to the same inequality (4.16). In order to completely solve this problem, it is necessary to obtain expressions for cross-section of infinitely many photon recombination. This is a rather ambitious task, but without this the question remains open. Acknowledgements The work of K.B. was supported by the Ministry of Science and Higher Education of the Russian Federation by project No 0723-2020-0040 "Fundamental problems of cosmic rays and dark matter". E.E. is also supported by fund Basis, project Na 18-1-5-89-1. Also, we would like to thank A.Barabanov, M.Faifman, V.Lensky, S.Rubin, M.Skorokhvatov for interest to this work and discussions. References 1. R Bernabei, P Belli, F Cappella, R Cerulli, CJ Dai, A d'Angelo, HL He, A Incicchitti, HH Kuang, JM Ma, et al. First results from dama/libra and the combined results with dama/nai. The European Physical Journal C, 56(3):333-355, 2008. 2. CDMS II Collaboration et al. Dark matter search results from the cdms ii experiment. Science, 327(5973):1619-1621, 2010. 3. Elena Aprile, KL Giboni, P Majewski, K Ni, M Yamashita, R Gaitskell, P Sorensen, Luiz DeViveiros, L Baudis, A Bernstein, et al. The xenon dark matter search experiment. New Astronomy Reviews, 49(2-6):289-295, 2005. 4. Ko Abe et al. The xmass experiment. In J. Phys. Conf. Ser, volume 120, 2008. 5. The XENON collaboration et al. Projected wimp sensitivity of the xenonnt dark matter experiment, 2020. 6. Petraki, Kalliopi, Pearce, Lauren, and Kusenko, Alexander. Self-interacting asymmetric dark matter coupled to a light massive dark photon. Journal of Cosmology and Astroparticle Physics, 2014(07):039, 2014. 7. Francis-Yan Cyr-Racine and Kris Sigurdson. Cosmology of atomic dark matter. Physical Review D, 87(10):103515, 2013. 8. Benedict Von Harling and Kalliopi Petraki. Bound-state formation for thermal relic dark matter and unitarity. Journal of Cosmology and Astroparticle Physics, 2014(12):033, 2014. 9. K.M. Belotsky, M. Yu. Khlopov, and K.I. Shibaev. Composite dark matter and its charged constituents. arXiv preprint astro-ph/0604518, 2006. 10. Daniel Feldman, Zuowei Liu, Pran Nath, and Gregory Peim. Multicomponent dark matter in supersymmetric hidden sector extensions. 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I.i. sobelman, e.a. ukov, excitation of atoms and spectral lines broadening, 253p. The Science, 1979. 17. D. R. Bates. Atomic and molecular processes, volume 13. Elsevier, 2012. 18. D. E. Kaplan, G. Z. Krnjaic, K. R. Rehermann, and C. M. Wells. Atomic dark matter. Journal of Cosmology and Astroparticle Physics, 2010(05):021, 2010. 19. Ya. B. Zeldovich and M. Yu. Khlopov. On the concentration of relic magnetic monopoles in the universe. Physics Letters B, 79(3):239-241,1978. 20. HA Kramers. London, edinburgh, dublin philos. mag. J. Sci, 46:836-71,1923. 21. P.V. Yelutin. Classical cross-section for recombination. theoretical and mathematical physics, 34(2):180-184,1978. 22. L. D. Landau and E. M. Lifshitz. Quantum mechanics: non-relativistic theory, volume 3. Elsevier, 2013. 23. I. Ya. BERSON. Semiclassical approximation for stimulated bremsstrahlung. Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki, 80:1727-1736,1981. 24. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii. Quantum Electrodynamics: Volume 4, volume 4. Butterworth-Heinemann, 1982. 25. 1.1. Sobelman. Introduction to the Theory of Atomic Spectra: International Series of Monographs in Natural Philosophy, volume 40. Elsevier, 2016. 26. H Bethe and E Salpeter. Quantum mechanics of atoms with one and two electrons [russian translation]. Pizmatgiz, Moscow I, 960, 1960. Bled Workshops in Physics Vol. 21, No. 2 A Proceedings to the 23rd [Virtual] Workshop, Volume 2 What Comes Beyond ... (p. 84) Bled, Slovenia, July 4-12, 2020 5 Neutrino Cooling Effect of Primordial Hot Areas in Dependence on its Size K.M. Belotsky *l, M.M. El Kasmi **1>2 and S.G. Rubin *** 1 National Research Nuclear University MEPhI, Kashirskoe Shosse 31, Moscow 115409, Russia 2 Physics Department, Faculty of Science, Sohag University,Sohag Center, Sohag 82524, Egypt 3 N. I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal University, Kremlevskaya Street 18, Kazan 420008, Russia Abstract. We consider the temperature dynamics of hypothetical primordial hot areas in the Universe. Such areas can be produced by the primordial density inhomogeneities and can survive to the modern era, in particular due to primordial black hole (PBH) cluster of size R > 1 pc and more. Here we concentrate on the neutrino cooling effect which is realized due to reactions of weak ^u transitions and e± annihilation. The given neutrino cooling mechanism is found to work in a wide range of parameters. For those parameters typical for PBH cluster considered, the cooling mechanism is quite valuable for the temperatures T > 3 MeV. Povzetek. Avtorji obravnavajo temperaturne spremembe v domnevnih prvotnih vročih območjih v vesolju. Taksna območja lahko nastanejo zaradi prvotnih nehomogenosti gostote in lahko preZivijo do danes, če so kopice prvotnih črnih lukenj velikosti R > 1 parsek. Obravnavajo predvsem ohlajanje vesolja z nevtrini, ki se sprosčajo pri sibkih prehodih ^u in z anihilacijo e±. Ugotovijo, da je območje parametrov, ki omogočijo ohlajanje z nevrtini, zelo siroko. Za parametre, ki ustrezajo obravnavanim kopičam črnih lukenj, je ta mehanizem hlajenja ustrezen za temperature T > 3 MeV. Keywords: Primordial hot areas, primordial blačk holes, čosmič neutrinos 5.1 Introduction There are some observations [1] indičating the existenče of ločal heated areas in the early Universe. Hypothetičal nature of ločal heated areas was disčussed earlier [2-4]. Sučh areas čan appear due to large primordial density flučtuations and čan be related to the člusters of Primordial Blačk Holes (PBHs) [1,5,6]. We assume that the baryonič matter has been čaptured by the gravitational forčes of these regions at the early Universe. They would remain hot for a long * E-mail: k-belotsky@yandex.ru ** E-mail: m.elkasemy@sčienče.sohag.edu.eg *** E-mail: sgrubin@mephi.ru 5 Neutrino Cooling Effect of Primordial Hot Areas in Dependence on its Size 85 time. At the same time, many processes can heat or cool the matter inside them during their formation after it. Short list of them is the neutrino cooling [6], nuclear reactions, radiation of the hot plasma and stars formed inside the region [8], gravitational dynamics of the system, shock waves, diffusion of matter, variation of the vacuum state while the region is born [9], energy transfer from collapsing walls [10-13], accretion, the Hawking evaporation. The last mechanisms are relevant in the case of PBHs origin of the regions [5,14-16]. In this proceedings, we continue our consideration of neutrino cooling of such regions. It could be the most important reason for the temperature evolution within initial temperature range - keV< T < 10 MeV. In this research, we follow the initial conditions taken from [5,6], where the mass of trapped matter is in wide range 104-108 M0. The main initial parameters are as follows: the size of the region is about R ~ 1 pc, its mass 104 M.0, initial temperature is in the interval To ~ 1 keV ^ 10 MeV. This temperature of such regions could be reached in several ways. The region can start to be formed at higher temperature and finish to do it having cooled down to T0. Also, the region could be heated up during formation, e.g., in the framework of model with collapsing domain walls [7]. Without specific assumptions, we show that effect of neutrino cooling is wide spread phenomena valid in wide range of parameters. The range of initial parameters is under consideration. Neutrino cooling effect can be suppressed at high temperatures and large sizes when the area becomes opaque to the neutrinos. Neutrinos are produced due to reactions of p n transition and e+e-annihilation. The characteristic time for photons to escape the area is bigger than the modern Universe age, this indicates that the size of cluster is big enough not to lose photons. In the given proceedings we study the impact of the size of the region on the neutrino cooling effect. Mechanism of neutrino cooling rates for the main reactions of the neutrino production is considered in Section 2. The impact of the diffusive character of particle propagation inside the cluster is briefly discussed in Section 3. 5.2 Cooling Rates Let us consider the reactions of the neutrino production: e- + p —> n + Ve, (5.1) e+ + n^p + V e, (5.2) e+ + e- -» Ve,|x,x + Ve,|i,T, (5.3) n^p + e- + V e. (5.4) The produced neutrinos leave the heated area if it is not very big. The energy inside the volume is decreased that leads to the temperature decreasing. The rates 86 K.M. Belotsky, M.M. El Kasmi and S.G. Rubin per unit volume, Yt = rt/V, for reactions listed above are respectively Yep = ne-npCTepV, Yen = nn ffeu V, (5.5) Yee = ne-ne+ CTeeV, Yn = —. (5.6) Tn Here nt is the concentration of the respective species, CTtj is the cross section (see e.g. [6]) of interacting particles i and j and Tn « 1000 s is the neutron lifetime. We consider the relativistic plasma so that the relative velocity v ~ 1. The backward reactions for Eqs.(5.1)-(5.4) are suppressed if neutrinos freely escape the cluster. We consider all densities to be independent of the space coordinate inside the region. The number densities are roughly described by the following formulas, see [6], ne- = ne+ + Ane, ne+ = neq(T) exp (-T^) , (5.7) ne = np + nn = gB nny(To), Ane = ne- - ne+ = np. (5.8) which are slightly corrected for better adjustment to the non-relativistic limit. Here n = nB/ny « 0.6 • 10-9 is the baryon to photon ratio in the modern Universe, gB ~ 1 is the correction factor of that relation due to entropy re-distribution, nY (T) = T3 and neq (T) = ^n3 T3 are the equilibrium photon and relativistic electron number densities respectively. Note that nY(T0) defines baryon density which is supposed to be unchanged starting from initial temperature T0 contrary to that of e± and y. Number of e± (along with y) changes due to e — v-conversion processes (reactions Eqs.(5.1) -(5.4)). The temperature of the system decreases due to neutrino escape. Number densities of the electrons and photons fall down with temperature as ~ T3. 5.3 Escaping Time The escape time of neutrinos from the region of the size R with temperature T can be calculated as: R2 2 tesc ~ D ~ R2 • neCTv (5.9) in diffusion approximation. Here the diffusion coefficient is D = [17], velocity v = 1, the neutrino mean free path is Av = 1/ne ov and e — v interaction cross section was roughly taken as ~ G 2 • T2. The electron number density ne ~ ne- + ne+ ~ ne+ is given by Eq.(5.7) which is ~ T3 at T > me. One can conclude from inequality tesc ~ R2GFT5 me and ne ~ T3 are assumed. 5 Neutrino Cooling Effect of Primordial Hot Areas in Dependence on its Size 87 T (MeV) T (MeV) Fig. 5.1: Left: The relation between escaping time of neutrino and temperature of the area and the blue line is the modern age of the Universe. (Behaviour of the curves at tesc ~ 10-20 Gyr reflects the fact that ne becomes ~ Ane, i.e. constant.) Right: The relation between size and temperature of the area. Neutrino cooling effect plays a prominent role below the thick red dot-dashed curve. The black solid line corresponds to the dependence of the Universe horizon from its temperature (R = 10-7(MeV/T)2 pc). Neutrino cooling effect due to reactions of weak p n transitions and e± annihilation are represented in Figure 5.1 where the escaping time of neutrinos in dependence on temperature is shown. As seen, at the temperature T < 3 MeV the escaping time for the most of considered cluster sizes is less than the modern Universe age, thus neutrino cooling works. Note, that at T C me the curves start to fall until the number density of electrons becomes ne ~ Ane. Dependence R(T) is shown in Figure 5.1, right panel, which follows from Figure 5.1, left panel. The region above the red line relates to the case when neutrino cooling is suppressed (neutrinos do not run away from the region during the Universe age). Black line shows the horizon size of the Universe in dependence on the matter temperature. One can see, horizon size is much smaller than the maximal size of region at the same temperature when neutrino cooling effect is, shown by the red line. Therefore, the neutrino cooling effect holds under usual conditions, and can be suppressed in extreme cases. The region can start its formation at very high temperature so that it could be cooled to the considered temperature during its detachment from Hubble flow and virialization. Also, the region could be heated up additionally during its formation, e.g. due to wall collapsing [7]. During detachment and virialization, the region could expand to some extent and hence, cool down. 5.4 Conclusions In earlier work [6], we have shown that due to neutrino emission (at a fixed size) the primordial hot areas are cooling down to the temperature value ~ 0.01 ^ 0.1 88 K.M. Belotsky, M.M. El Kasmi and S.G. Rubin MeV. Here we just investigated the neutrino cooling mechanism of the heated region and dependence on its size. Considering the result of equation (5.9) for escaping time, we can find the size-changing more effectively with temperature. At the temperature T > 3 MeV, the diffusive character of particle propagation makes the time of escaping or time of cooling more than the modern Universe age. This result is obtained at the definite initial region parameters (size and temperature, relevant for PBH cluster model [5]) that could be slightly varied. It illustrates general property for such possible primordial inhomogeneities. It is seen that neutrino cooling effect should take place for a wide reasonable size/temperature range of parameter. Extreme heating up of the area while it has being formed could change situation. There are a variety of mechanisms that can be responsible for the area heating. Additional heating during their creation is also possible. As was mentioned in the Introduction, the area could be heated by the collapsing walls - the scalar field kinks. The fermion reflection on kinks was studied in [7]. It was shown that the reflection weakly depends on the fermion mass. Therefore the kinks could prevent the neutrinos from escaping. More detailed analysis is necessary to clarify this effect. Acknowledgements The work of K.B. was supported by the Ministry of Science and Higher Education of the Russian Federation by project No 0723-2020-0040 "Fundamental problems of cosmic rays and dark matter" and of S.R. by the project "Fundamental properties of elementary particles and cosmology" No 0723-2020-0041. The work of S.R. is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University and RFBR grant 19-02-00930. References 1. A. Kashlinsky, Y. Ali-Haimoud, S. Clesse, J. Garcia-Bellido, L. Wyrzykowski, A. Achu-carro, L. Amendola, J. Annis, A. Arbey and R. G. Arendt, et al.: Electromagnetic probes of primordial black holes as dark matter, Bull. Am. Astron. Soc.,51, 51 (2019). 2. V. K. Dubrovich and S. I. Glazyrin: Cosmological dinosaurs, [arXiv:1208.3999 [astro-ph.CO]] (2012). 3. S. Kumar, E. Dimastrogiovanni, G. D. Starkman, C. Copi and B. Lynn: CMB Spectral Distortions from Cooling Macroscopic Dark Matter, Phys. Rev. D 99, no.2, 023521 (2019). 4. A. Kogut, M. H. Abitbol, J. Chluba, J. Delabrouille, D. Fixsen, J. C. Hill, S. P. Patil and A. Rotti: CMB Spectral Distortions: Status and Prospects, [arXiv:1907.13195 [astro-ph.CO]], (2019). 5. K. Belotsky, V. Dokuchaev, Y. Eroshenko, E. Esipova, M. Khlopov, L. Khromykh, A. Kirillov, V. Nikulin,; S. Rubin, I. Svadkovsky: Clusters of Primordial Black Holes, Eur. Phys. J. C. 79, 1-20 (2019). 6. K. Belotsky, M. E. Kasmi and S. Rubin: Neutrino Cooling of Primordial Hot Regions, Symmetry 12,1442 (2020). 7. A. A. Kurakin and S. G. Rubin, [arXiv:2011.01757 [physics.gen-ph]]. 5 Neutrino Cooling Effect of Primordial Hot Areas in Dependence on its Size 89 8. A.Dolgov, K.Postnov: Electromagnetic radiation accompanying gravitational waves from black hole binaries, J. Cosmol. Astropart. Phys. 2017,18 (2017). 9. K. M. Belotsky, Y. A. Golikova and S. G. Rubin: Local heating of matter in the early universe owing to the interaction of the Higgs field with a scalar field, Phys. Atom. Nucl. 80, 718-720 (2017). 10. V. Berezin, V. Kuzmin, I. Tkachev: Thin-wall vacuum domain evolution, Phys. Lett. B 120, 91-96 (1983). 11. M. Y. Khlopov, R. V. Konoplich, S. G. Rubin and A. S. Sakharov: Formation of black holes in first order phase transitions, arXiv:hep-ph/9807343 [hep-ph], (1998). 12. S. G. Rubin, M. Y. Khlopov and A. S. Sakharov: Primordial black holes from nonequilib-rium second order phase transition, Grav. Cosmol. 6, 51-58 (2000). 13. H. Deng, A. Vilenkin and M. Yamada: CMB spectral distortions from black holes formed by vacuum bubbles, JCAP 07, 059 (2018). 14. A.Dolgov, J.Silk: Baryon isocurvature fluctuations at small scales and baryonic dark matter, Phys. Rev. D 47, 4244-4255 (1993). 15. A. Dolgov: Massive and supermassive black holes in the contemporary and early Universe and problems in cosmology and astrophysics, Phys.-Uspekhi, 61, 115-132 (2018). 16. J. García-Bellido: Primordial Black Holes, PoS EDSU2018, 042 (2018). 17. E. M. Lifshitz, L. P. Pitaevskii: Physical Kinetics. Course of Theoretical Physics, ButterworthHeinemann, Oxford, 2006. Bled Workshops in Physics Vol. 21, No. 2 A Proceedings to the 23rd [Virtual] Workshop, Volume 2 What Comes Beyond ... (p. 90) Bled, Slovenia, July 4-12, 2020 6 Consideration of a Loop Decay of Dark Matter Particle into Electron-Positron from Point of View of Possible FSR Suppression K. M. Belotsky * and A.Kh. Kamaletdinov ** National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409, Kashirskoe shosse 31, Moscow, Russia Abstract. Cosmic positron anomaly is still not explained. Explanation with dark matter (DM) decay or annihilation is one of the main attempts to do it. But they suffer with shortcoming as overproduction of induced gamma-radiation which contradict to cosmic gamma-background. Final state radiation (FSR) in such processes is supposed under standard conditions (by default) to have the basic contribution in it. Our group elaborates possibility to evade this problem in different ways. Here we continue one of them connected with possibility of suppression of FSR due to specifics of Lagrangian describing DM particle decay. Loop through two new spinors and scalar is considered. Effect of FSR suppression is found to be existing but at the very low level in the considered case. Povzetek. Avtorja isceta model, ki bi pojasnil presežek pozitronov v kozmicnih žarkih. Poskusi, da bi presežek pozitronov razložili z razpadi ali anihilacijami temne snovi, se niso obnesli, ker se pri tem sprosti prevec zarkov gamma. Avtorja in njuna skupina isčejo poti, ki bi razlozile presezek positronov, izsevani zarki gamma pa bi bili v skladu z meritvami. Lagrangeovi gostoti za temno snov sta dodala sklopitev z dvema novima fermionoma in enim skalarnim poljem, vendar nov model količino izsevanih zarkov gama le neznatno zniza. Keywords: dark matter, positron anomaly, cosmic rays, final state radiation, loop decay 6.1 Introduction The problem of Dark Matter (DM) is one of the main long-term problems of fundamental physics. Many direct and indirect searches for DM particles are undertaken. Cosmic rays (CR) relate to the indirect one and the revealed cosmic positron anomaly (PA) [1-5] can be supposed to be a possible indication of DM. But attempts to explain the positron anomaly with DM face a problem of agreement with data on cosmic gamma-rays (see, e.g., our works [6-8] and other * E-mail: k-belotsky@yandex.ru ** E-mail: kamaletdinov.a.h@yandex.ru 6 Consideration of a Loop Decay of Dark Matter Particle. 91 [9-13]) and CMB [14] and some other for specific DM model case. Constraint following from CMB can be more easily avoided (see, e.g., the references in [15-17]) than that from data on cosmic gamma-ray background [19]. This constraint seems to be the least model dependent. When high energy positrons and electrons e± are produced from DM decay or annihilation, it will be accompanied by final state radiation (FSR) and they will scatter on medium photons. Both processes give us gamma of high energy. The most popular alternative approach to the solution of the problem of PA origin is associated with nearby pulsars. But it also strongly constrained (if not excluded) by data on gamma-radiation [20-22]. So the question of PA origin is still open. We consider possibility of PA explanation with the help of DM and elaborate two approaches for it: one is connected with space distribution of DM in Galaxy ('Dark disk' model) [6-9,23,24], other one is connected with possible physics of DM interaction leading to annihilation or decay which can give suppressed FSR [15-17]. The latter was attempted to be considered by other recently [25]. Here we make one more step in this investigation. We study one more decay mode of DM particle which contains a loop from spinor and scalar particle of dark sector. The process is drawn below. The obtained answer is that the effect is negligible in the considered case, though it exists in principle, i.e. relative probability of FSR photon production can be changed. Below we present theoretical initial settings for interaction/decay physics of DM particles and basic calculations, then conclude. 6.2 DM decay model considered and calculation details Let us consider two processes of DM particle (X) decay: X —» e+e- and the same with FSR X —} e+e-y. The goal of the task is to minimize the ratio: r (X^e+e-y) F(X —> e+e~) = min' (6U) where r(X —» e+e-(y)) are the respective decay widths. In order to be able to see the photon suppression at different energies, we study the energy distribution of the photon emission probability in the decays of DM particles (6.2). 3Br(e+e-y) _ 3 (r(X -> e+e-y)\ ) 3w 9w \ r(X —» e+ e-) where w is the energy of the final state photon. As was shown earlier [15], the simplest interaction vertices such as (6.3, 6.4) do not lead to a significant suppression of the photon yield in a such decays. These are Lscaiar = X^(a + by5)^, Lvector = + by5)X^, (6.3) ___ b (Yv d ) Le = X^C(a + by5H, L = ^(a + (Y v) )X^. (6.4) m 92 K. M. Belotsky and A.Kh. Kamaletdinov Also shown that complication of process kinematics does not give an effect [17,18]. Here we consider one of the options for complicating the DM-SM interactions. On the base of previous works we suppose that it is worth to consider other type of the processes. Loop diagrams of DM decays into e+, e- particles can be worth to be studied. We consider here the interaction Lagrangian of the form (6.5): £a = xe(a + iby5)e + n 0(c + i dy5)¥ + V ^(c + i dy5)9, (6.5) where 0 is considered as the fermionic neutral DM component, and X,n - as the scalar DM particles. In this work, to simplify the calculations, the mass of the n particles is assumed to be very large so that the photon emission by the n propagator can be neglected. The leading order of the process X —» e+e- in such case describes by triangle-loop diagram shown in figure 6.1. Fig. 6.1: Feynman diagram of two body decay process. We evaluate the corresponding matrix element (6.6) here through the form-factors Fi and F2 using the Passarino-Veltman (PV) reduction procedure, described in [26,27]. In order to perform calculations with PV-functions the PackageX [29] tool for Wolfram Mathemetica was used. Matrix element is i M = i u(pi) (Fi (Vi) - iF2(VS)Y5)v(p2), (6.6) Fi (Vi) = a(c2 - d2) (B(Vi) + (m2 + mi ma )Co(pi,p2)) + +2bcd(B(Vs) + (m2 - mim3)Co(pi,p2H, (6 7) F2(Vs) = b(c2 - d2)(B(Vi) + (m2 - mima)Co(pi,p2))--2ac^B(Vs) + (m2 + mima)Co(pi ,p2^. We use here and further notation B(Vs) = B0(Vs; mi,m3), Ci(pi,p2) = Ci(pi, p2; mi, m2, m3). In this case, the squared amplitude of the two-body decay averaged over the final state polarizations takes the form: 4 Y_ MM* = (Fi(Vi)2 + F2(Vi)2), (6.8) 6 Consideration of a Loop Decay of Dark Matter Particle. 93 After the same calculations carried out for the three-body decay process (X —» e+e-y) (see figure 6.2) and looking at their ratio one can obtain the expression for final-state photon yield energy distribution (6.2): Fig. 6.2: Feynman diagrams of three body decay process. 1 ic2 + d2)^ / \ MM* = 1 + J (Aii - Ai2 - A2i + A22J, (6.9) 4 A , _ 2 |X++|2 + 2mj(l ■ pi)2(pi ■ p2)|Yi|2 Aii = " ^ + +b2 | X-| 2 + 2m2(l ■ pi)2(pi ■ p2)IKi I 2 + (pi +1)4 ' (6.10) Aij = 2m2(pi ■ p2)(l ■ pi)(l ■ p2)-(p + l)2(p + l)2 a2YiY* - 4a2CiCî + b2KiKJ (pi + l)2(pj + l) / x/ X a2X+X+* + b2X-X-* -(pi ■ (p2 + l)) (p2 ■ (pi + l))-^- (6.11) (pi + l)2(pj + l)2 where Ki = Co(ki,p2), K2 = Co(pi,k2), Ci = Ci (ki,p2), C2 = Ci (pi,k2), X± = 2(l ■ pi)Ci + B(Vi) + Ki(m2 ± mi), X± = 2(l ■ p2)C2 + B(^S) + K2(m2 ± mi), (6.12) Yi = 2Ci + Ki Y2 = 2C2 + K2. The study of the influence of model parameters variation on the photon emission showed that the suppression turns out to be insignificant in order to explain satisfactorily the high energy cosmic positron spectrum not contradicting to gamma-ray background. 94 K. M. Belotsky and A.Kh. Kamaletdinov Fig. 6.3: The comparison of the photon yields in the case of simplest scalar vertice (6.3) (blue) and the loop vertice (red) 6.3 Conclusion In this paper we continue our studying possibility to suppress FSR (gamma emission) in DM explanation of cosmic positron anomaly. Here we consider specific DM-lepton interaction Lagrangian which allows decaying DM particle to e± through the loop of intermediate particles of dark sector. We obtained relative probability of FSR production (branching ratio of the respective mode) in dependence of final photon energy analytically up to the level of the squared matrix element. It is important for understanding whether or not possible FSR suppression looking at dependence on model parameters at high photon energies (most critical) and prospectiveness of possible complication of the model. Now it is obtained that the considered variant of the loop decay is not able to facilitate solution PA origin with DM, though shows (maybe, opens new) principal opportunity of FSR yield changing. Acknowledgements The work was supported by the Ministry of Science and Higher Education of the Russian Federation by project No 0723-2020-0040 "Fundamental problems of cosmic rays and dark matter". Also, we would like to thank M.Solovyov for the help in providing by references. References 1. O. Adriani et al. (PAMELA Collab.): Observation of an anomalous positron abundance in the cosmic radiation, Nature 458, 607-609 (2009). 2. M. Aguilar et al. (AMS Collab.), Phys. Rev. Lett. 110,141102 (2013). 3. L. Accardo et al. (AMS Collab.), Phys. Rev. Lett. 113,121101 (2014). 4. M. 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B. 734, 62—115 (2006). 28. G. Devaraj, R. G. Stuart: Reduction of one loop tensor form-factors to scalar integrals: A General scheme, Nucl. Phys. B. 519, 483— 513 (1998). 29. H.H Patel: Package-X 2.0: A Mathematica package for the analytic calculation of one-loop integrals, Comput. Phys. Commun. 218, 66— 70 (2017). Bled Workshops in Physics Vol. 21, No. 2 A Proceedings to the 23rd [Virtual] Workshop, Volume 2 What Comes Beyond ... (p. 97) Bled, Slovenia, July 4-12, 2020 7 Theoretical Indication of a Possible Asymmetry in Gamma-Radiation Between Andromeda Halo Hemispheres Due to Compton Scattering on Electrons From Their Hypothetical Sources in the Halo K.M. Belotsky *, E.S. Shlepkina ** and M.L. Soloviev *** National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409, Kashirskoe shosse 31, Moscow, Russia Abstract. Dark matter (DM) can give observable effects decaying or annihilating with production of electrons or/and photons. Such probability was widely researched for our Galaxy. Here we consider one aspect of similar effect for Andromeda galaxy. We explicitly estimate the energy of the photon of the medium experiencing Inverse Compton (IC) scattering off electron in halo. These photons can be registered by different experiments. Dark matter annihilation or decay could be the source of high energy electrons in halo, though the source could be of other origins too (e.g. running neutron stars). Because of specifics in space orientation of Andromeda galaxy disk (a little inclined to the line of sight), the difference in energies could arise for the photons from two hemispheres of Andromeda halo. It is obtained that such asymmetry can be at the level of several 10%. Povzetek. Temno snov merilci zaznajo pri različnih procesih, denimo, pri razpadu ali anihilaciji delcev temne snovi, pri cemer se rodijo elektroni in/ali fotoni. Za naso Galaksijo so te procese obravnavali številni avtorji. V tem prispevku obravnavajo avtorji podobne pojave v sosednji galaksiji, v Andromedi. Iz meritev ocenijo energijo fotona, ki se preko inverznega Comptonovega pojava sipa na elektronih v haloju. Izmerjeni visokoenergijski elektroni v haloju galaksije se lahko rodijo pri anihilaciji ali razpadu delcev temne snovi, lahko pa nastanejo tudi v drugih procesih (povzrocijo jih, denimo, gibanja nevtronskih zvezd). Ker Andromedin disk ni pravokoten na smer nasega opazovanja, avtorji ocenjujejo, da se izmerjene energije fotonov, ki prihajajo iz dveh razlicnih koncev Andromedinega haloja, lahko raxlikujejo za nekaj 10%. Keywords: dark matter, gamma-rays, inverse Compton scattering, observational asymmetry effects, Gamma-400 7.1 Introduction Dark matter (DM) can be the source of high energy electrons and photons due to its annihilation or decay. There are many works elaborating possible observational * E-mail: k-belotsky@yandex.ru ** E-mail: shlepkinaes@gmail.com *** E-mail:max07s@mail.ru 98 K.M. Belotsky, E.S. Shlepkina and M.L. Soloviev effects from it in cosmic rays (CR) in our Galaxy. Here we consider issue of possible observational effect from Andromeda galaxy. If the source of high energy electrons or positrons in halo exist, the process of Inverse Compton (IC) scattering can happen for photons of the medium - of star light first of all. It is known that the angle distribution of final photon is anisotropic in this scattering process with respect to momentum of incident photon in the initial electron rest frame. This effect should remain in arbitrary reference frame, and will depend on momenta of electron and photon to be scattered. Since star disk of Andromeda galaxy is little inclined with respect to the line of sight, there will be different predominant scattering angle in 'upper' and 'lower' hemispheres of Andromeda halo. There should be two effects: in energy and in flux. Here we consider effect in energy only. There should be effect in flux also, which consists in the difference of the values of the photon flux. We evaluate a net effect for two fixed points upper and below galaxy disk, what allows doing further predictions of possible effects. We will consider the effects of the Andromeda geometry and the line of sight as well as make calculation of the energy spectrum in our future works. It can wash out effect in part, nonetheless it may remain in to some degree, so a geometry modulation of energetic spectrum can be expected. Considered simple case shows how it works. It gives that three energy intervals exist where effect is different: at very low final photon energy the ratio R of energies from upper and lower hemispheres is about unity, at higher energy upto m2/w ~ 1 TeV R « 0.6, where m is the electron mass, w is the initial photon energy, and at even higher energy R —» 1. Effect should be observed for any photon energy, here we focus on maximum value of final photon energy, though formula obtained is universal. Besides effect in the flux, asymmetry in prompt photons radiation from decay/annihilation process (FSR) [22], comparison with observation sensitivity and background, comparison with other calculation methods [13,16,17,20] are to be taken into account in future. Andromeda is a rare galaxy which has been recently observed in gamma by Fermi-LAT satellite experiment [1,4]. Ground experiments (like HAWC [2], HESS [18], MAGIC [8], LHAASO [6], VERITAS [14]) do not have so high angle resolution (though it depends on energy and they allowed observing several galaxies) and can register only very high energy photons. Effect we are talking about may manifest at any energy including intermediate and low energy ranges. In connection with it, forthcoming satellite gamma-ray telescope project Gamma-400 [15] with especially high angle resolution is of special importance for similar research. There was attempt to connect possible excess in y-rays from Andromeda halo with DM [19]. We suggest general feature of asymmetry related with DM or other sources in halo. 7.2 IC photon energy We assume that there can be sources of high energy electrons or/and positrons in the halo of Andromeda galaxy. Such assumption is based on the attempts to 7 Possible Energy Asymmetry of DM Induced Gamma Radiation from M31 99 explain positron anomaly [3,5,7] in CR with the help of DM annihilation or decay in our Galaxy. These attempts inevitably involve an effect in gamma-radiation, which was investigated, in particular by our group [9-12]. Production of high energy e±, firstly, is accompanied by FSR, and, secondly, gives energetic photons as a result of IC scattering of these e± on photons of the medium. To have IC effect at high energy, photons of star light should be taken since they are most energetic from widespread radiations within galaxy. Also one can note that if dark matter annihilation is indeed the origin of the excess of positrons, then we deal with continuously distributed in space high-energy e± sources, so we can take arbitrary point to consider effect. Let us consider one arbitrary act of electron (or positron, what does not matter in the framework of QED) and photon scattering. The scheme of the process is shown in the Fig. 7.1a and 7.2. k and k' are the initial and final photon 4-momenta, w and w' are their energies, 0 and 0' are the angles between initial momentum of electron and initial and final ones of photons respectively, x is the angle between initial and final momenta of the photon. Index 'lab' relates to the same values in the initial electron rest frame. Fig. 7.1: The scheme for scattering process in upper and lower Andromeda hemispheres with the chosen points. Observer is on the right. Let us take Compton formula (one can refer to any textbook, e.g. [21]) for final photon energy in the 'lab' reference frame ' _ _wlab_ W'ab = 1 + ^ d - cos Xlab) ' (7.1) where m is the electron mass, xlab is the photon scattering angle as shown in the Fig. 7.1b We can easily express wlab from the respective photon energy in the real reference frame (w) through the Lorentz's transformation: wlab = yw(1 — v cos 0), (7.2) 100 K.M. Belotsky, E.S. Shlepkina and M.L. Soloviev Fig. 7.2: The scheme for scattering process in upper and lower Andromeda hemispheres with the chosen points. Observer is on the right. and the same transformation takes place for final photon in the 'lab' frame wiab = yw'(1 - v cos 9') = yw'(1 - v cos(9 + x)). (7.3) We applied relation between angles 9' = 9 + x which is seen from the Fig. 7.1. Here and thereafter we use that the absolute value of photon momentum is equal to its energy w^). Everywhere v and y mean velocity and Y-factor of initial electron. One needs to connect cosx with cosxiab. It can be done through scalar product of initial and final photon momenta written out in different reference frames and using Lorentz transformations for photon energy: (kk') = W w'(1 - cos x) = Wlab diab(1 - cos xlab). (7.4) From where Wlab Wl'ab n ^ 1 - cos xlab = W W (1 - cos x), (7.5) where from Eq. 7.2 and Eq. 7.3 one gets Wlab Wiab 1 ww' y2(1 - v cos 9)(1 - v cos(9 + x))' Substituting in Eq. 7.1 one obtains yw(1 - v cos 9) 1 - cos x (7.6) Wlab = 1 + r^(i-cose) y2(1 -vcos9)(1 -vcos(9 + x)) Y w(1 - v cos 9) 1 i ^ 1 -cos X . \ ' ' + ym 1 —v cos(e+x) 7 Possible Energy Asymmetry of DM Induced Gamma Radiation from M31 101 From Eq. 7.3 one has œ = "M-labm + ^, (7.8) Y(1 — v cos(0 + x)) and, finally, taking into account Eq. 7.7 one gets , w(1 — v cos 0) 1 — v cos(0 + x) + Ym 0 — cos x)' (7.9) where m = Y— ~ (2 + 4) • 10-6. One can see from Eq. 7.9 that there are different situations that can be easily analyzed. Let the velocity be v « 0. Then the small third term in denominator œ/m(1 — cosx) will give a tiny anisotropy between hemispheres (what is not important for us). The energy ratio between upper and lower hemispheres R = œ+/œ- (7.10) will be a little bigger than unity for the chosen two points A+ and A_ in Fig. 7.2, where are the final photon energies from upper and lower hemispheres respectively. Scattering angles in upper and lower hemispheres of Andromeda X± are introduced in the Fig. 7.2. When v ~ 1, third term in denominator of Eq. 7.9 (which is proportional to œ/m) is negligible and œ' comes to maximum at cos(0 + x) = 1 • So 0 = —x, what corresponds to the case when initial electron goes in direction to the observer1. It corresponds to narrow sharp maximum in photon energy which is of bigger interest. Next, if v —» 1 so 1 — v becomes smaller than œ/ym, i.e. when y ^ Ycr, the third term in denominator starts to dominate again and in this limit denominator and numerator are canceled. Finally, we obtain for maximal final photon energy in the real reference frame , œ(1 — v cos x) (1 + v)y2œ(1 — v cos x) V + Ym (1 — cos X) 1 +(1 + v)Y m (— cos x) 1 v = 0 i+m (i-cos x) 1 — cos X 1 C Y C Ycr (7.11) T-oi = 1 Y >> Ycr. So, there exists wide electron energy interval, 1 C y C Ycr (what corresponds to initial electron energy m C E C TeV for w ~ 1 eV), where the effect takes place R = 1 — cosX+ « 0.6 (7.12) 1 — cos X- for the chosen two points A+ and A- in the Fig. 7.2. The ratio R for maximal photon energy as dependent on Y-factor followed from the Eq. 7.11 is illustrated in Fig. 7.3. 1 One notes that consideration of other configuration of momenta (angles) in the Fig.7.1 7.1 can lead to the relation 0' = 0 — x- But it does not change conclusion that cos(0') = cos(0 — x) = 1 and that in the given angle system frame initial electron travels in direction to the observer. 102 K.M. Belotsky, E.S. Shlepkina and M.L. Soloviev gamma-factor Fig. 7.3: Dependence of R showing asymmetry effect for maximal observed photon energy wmax between upper and lower hemispheres of Andromeda galaxy from Y-factor of initial electron. Figure relates to the chosen two points A+ and A_ of Fig. 7.2 Comments on line of sight integration We have shown explicit effect for two fixed points above and below galaxy disk. Certainly, one should take into account the effects caused by our line of sight and distributed photon and electron sources, what could wash out difference between hemispheres. Nonetheless effect should remain in some degree since situation for upper line of sight and lower one is not symmetric (see Fig.7.2) because of dependence of Compton scattering from initial relative angle between momenta of the scattered electron and photon. As was seen from Eq. 7.11, when initial photon and electron move towards each other (9 > 90° in Fig. 7.2) final photon (maximal) energy w' is bigger in wide its value range than when they move co-directionally (9 < 90°). Line of sight Integration can be done taking into account a flux and under extra assumption about e± source distribution. Qualitatively, two factors will make difference between upper and lower lines of sight: it is density of e± sources and density of medium photons. Density of the sources is expected to decrease from distance to the galaxy center, concentration of photons - from distance to the stars in disc. For example, one can consider two nearest to the galactic center points C+ and C_ in Fig. 7.2 of two opposite lines of sight. Source densities in them is expected to be equal and maximal over all given lines of sight. But in point C+ the closest part of galaxy disc will shine co-directionally, while for C_ will do towards. Similarly one can consider other parts of lines of sight and there will be effects of different signs, though their full compensation is hardly expected. 7 Possible Energy Asymmetry of DM Induced Gamma Radiation from M31 103 7.3 Conclusion We considered possible effect of asymmetry of gamma radiation from another galaxy connected with its geometric orientation with respect to line of sight.Here we considered an effect in energy between two hemispheres of galaxy related to IC scattering of medium photons on high energy e± from their hypothetical source in galaxy halo. There also will be an effect in flux. The effect can be achievable for existing or future experiments since it may not seem to be vanishing, it can be at the level 10 (several tens) percents. Our future work will concern the flux, sensitivity and data on cosmic gamma-background, and include the prompt (FSR) photons appearing under the assumption that e± has DM decay/annihilation origin. Acknowledgements The work was supported by the Ministry of Science and Higher Education of the Russian Federation by project No 0723-2020-0040 "Fundamental problems of cosmic rays and dark matter". Also we would like to thank A.Egorov, M.Khlopov, M.Laletin and S.Rubin for interest to the work and useful discussions and also D.Fargion for providing relevant references. References 1. A. A. Abdo, M. Ackermann, M. Ajello, A. Allafort, W. Atwood, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri, K. Bechtol, et al. Fermi large area telescope observations of local group galaxies: detection of m 31 and search for m 33. Astronomy & Astrophysics, 523:L2, 2010. 2. A. Abeysekara, R. Alfaro, C. Alvarez, J. Alvarez, R. Arceo, J. Arteaga-Velazquez, H. Solares, A. Barber, B. Baughman, N. Bautista-Elivar, et al. The hawc gamma-ray observatory: Observations of cosmic rays. preprint arXiv:1310.0072, 2013. 3. L. Accardo et al. 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Mayorovi 1 National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409 Moscow, Russia 2 Institute of Physics, Southern Federal University, Stachki 194 Rostov on Don 344090, Russia 3 Universite de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France Abstract. The old and still not solved problem of dark atom solution for the puzzles of direct dark matter searches is related with rigorous proof of the existence of a low energy bound state in the dark atom interaction with nuclei. Such proof must involve a self-consistent account of the nuclear attraction and Coulomb repulsion in such interaction. In the lack of usual small parameters of atomic physics like smallness of electromagnetic coupling of the electronic shell or smallness of the size of nucleus as compared with the radius of the Bohr orbit the rigorous study of this problem inevitably implies numerical simulation of dark atom interaction with nuclei. Our approach to such simulations of OHe -nucleus interaction involves multi-step approximation to the realistic picture by continuous addition to the initially classical picture of three point-like body problem essential quantum mechanical features. Povzetek. Avtorji obravnavajo atome temne snovi kot rešitev ugank direktnega iskanja temne snovi v povezavi z rigoroznim dokazom obstoja nizko energijskega vezanega stanja interakcije temnega atoma z jedrom. Dokaz mora vključevati usklajen opis jedrskega privlaka in Coulombskega odboja v teh interakcijah. V obravnavanem problemu, za razliko od običajnih atomov, ne vemo vnaprej, kateri parametri so majhni, zato se moramo zateči k numerični resitvi problema. Pristop avtorjev k simulaciji interakcij OHe -jedro poteka tako, da klasicnemu modelu treh tockastih teles postopoma dodajajo kvantne popravke. Keywords: Physics beyond the standard model; stable charged particles; composite dark matter; dark atoms; nuclear interactions; Coulomb interaction; OHe PACS: 02.60.-x; 02.70.-c; 12.60.-i; 36.10.-k; 98.80.-k 8.1 Introduction According to the modern cosmology, dark matter is non-baryonic and is associated with physics that has not yet been sufficiently studied and, in fact, unknown to us. If it consists of particles, then they are predicted beyond the Standard Model. To * E-mail: khlopov@apc.univ-paris.fr 106 T.E. Bikbaev, M.Yu. Khlopov and A.G. Mayorov be considered as candidates for dark matter these particles must satisfy a set of conditions: they must be stable, must explain the measured dark matter density, and decouple from plasma and radiation, at least before the beginning of the matter dominated stage [1,2]. The easiest way to satisfy the above conditions is to assume the existence of neutral, elementary Weakly Interacting Massive Particles (WIMP). However, the results of the WIMP searches are contradictory and the existing uncertainty in the choice of "dark" particles has given rise to many different models suggesting various objects for the role of dark matter candidates [3-6]. In these models, new particles should possess some new fundamental symmetry and the corresponding conserved charge in order to protect their stability [5,7,8]. An important problem for scenarios of hypothetical, stable, electrically charged particles is their absence in the matter around us. If they exist, they should be present in the ordinary matter in the form of anomalous isotopes (with an anomalous Z/A ratio). The main difficulty for these scenarios is the suppression of the abundance of positively charged particles bound with electrons, which behave like anomalous isotopes of hydrogen or helium. Serious experimental restrictions on such isotopes, especially on anomalous hydrogen, very severely limit the possibility of stable positively charged particles [9]. This problem is also unsolvable if the model predicts stable particles with charge —1. Such particles bind with primordial helium in +1 charged ions, which recombine with electrons in atoms of anomalous hydrogen [10]. In this connection, stable negatively charged particles can only have charge —2 - we will denote them by O or in the general case even charge —2n, where n is any natural number. In the present paper, we consider a scenario of composite dark matter, in which hypothetical stable O particles avoid experimental discovery, because they form neutral atom-like states OHe with primordial helium, called "dark" atoms [11]. Since all these models also predict the corresponding +2 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 —2 charged particles O [4] 1. Then their positively charged antiparticles can effectively annihilate in the early universe. There are various models in which such stable —2 charged particles are predicted [12-14]. 8.2 "Dark" atoms OHe "Dark" atom is the bound system of O particle and 4He nucleus. In the approximation of our current numerical model, a-particle is point-like and moves along the Bohr radius. Then the binding energy of OHe for a point charge of 4He is given by: ZO__ZHea2mHe Io = —-y-— « 1.6 MeV, (8.1) where a - is a fine structure constant, ZO— and ZHe - electric charges of O particle and nuclei He respectively, mHe - is the a-particle mass. 1 Electric charge of this excess is compensated by the corresponding excess of positively charged baryons so that the electroneutrality of the Universe is preserved 8 Numerical Simulation of Dark Atom Interaction With Nuclei 107 The Bohr radius of He rotation in "dark" OHe atoms is equal to [15]: Hc Rb = ----« 2 • 10-13 cm (8.2) ZO--¿HemHea In all models of O-helium, O behaves like a lepton or as a specific cluster of heavy quarks of new families with suppressed hadron interaction [16]. Therefore, the strong interaction of OHe with matter is determined by the nuclear interaction of He. The mass O , mO—, is the only free parameter of new physics. The experimental search at the LHC for stable doubly charged particles gives a lower limit for their mass about 1 TeV [17]. The neutral primordial nuclear-interacting objects, that is, "dark" OHe atoms, dominate in the modern density of nonrelativistic matter and play the role of a non-trivial form of strongly interacting dark matter. The active influence of this type of dark matter on nuclear transformations requires special research and development of the nuclear physics of O-helium. This is especially important for a quantitative assessment of the role of "dark" atoms in primordial cosmological nucleosynthesis and in the evolution of stars [15]. The importance of the O-helium hypothesis is that it can explain the conflicting results of a direct search for dark matter, due to the specifics of the interaction of "dark" atoms with the matter of underground detectors [18]. Namely, positive results on the detection of dark matter particles in experiments such as DAMA / Nal and DAMA / LIBRA, which seem to contradict all other experiments, for example, with XENON100, LUX, CDMS, which give a negative result. One of the main problems with the "dark" OHe atoms is that their constituents can interact too strongly with ordinary matter. This is because O-helium, although neutral, initially has an unshielded nuclear attraction to the nuclei of matter. Which can lead to the destruction of the bound OHe system and the formation of anomalous isotopes. In turn, there are very strong experimental limitations on the concentration of these isotopes in the terrestrial soil and sea water [9]. To avoid this problem, it is assumed that the effective potential of OHe-nucleus interaction will have a barrier preventing the merging of He and/or O with nucleus. Under these conditions, "dark" atoms interaction with matter doesn't lead to anomalous isotopes overproduction, which is the key point for the O-helium hypothesis. In this work, a description of the performed numerical simulation of the interaction of "dark" O-helium atoms with the nuclei of baryonic matter is given with the aim to explore the conditions for the existence of their low-energy bound state, which can explain positive results of DAMA/NaI and DAMA/LIBRA experiments. Within the framework of the proposed approach to such modeling, in order to reveal the essence of the processes of nuclear interaction of OHe with nuclei of baryonic matter, the approach is based on the classical model, where the effects of quantum physics are gradually introduced. 108 T.E. Bikbaev, M.Yu. Khlopov and A.G. Mayorov 8.3 Numerical modeling of the interaction of OHe with the nucleus of baryonic matter. 8.3.1 Modeling OHe. To model the "dark" atom of O-helium (the OHe system) was considered, consisting of two point-like, bound particles: the He nucleus and the O particle. A spherical coordinate system was introduced, at the center of which the particle O is meant, and around it along the surface of the sphere, the radius of which is equal to the radius of the atom OHe Rb (see formula (2)) the He nucleus moves stochastically, with a constant Bohr velocity Va. The speed Va is: hc2 . cm Va =-— « 3 • 104— (8.3) mHeRb s The initial task in modeling the interaction of OHe with nuclei was to construct a numerical model of O-helium, which would allow to describe the motion of an a-particle around O . It should be used in the main numerical model, in which the motion and interaction of OHe with the nuclei will be simulated. Let us consider how the OHe system was modeled (see Figure 8.1). Fig. 8.1: Block diagram of the OHe system simulation 8 Numerical Simulation of Dark Atom Interaction With Nuclei 109 An a-particle in the bound OHe system has only two independent degrees of freedom, which are taken as the polar and azimuthal angles. Its Cartesian coordinates are expressed through the projections Rb, that is, the components of the radius vector at each moment of time, in order to use them to construct the trajectory of the a-particle. In Figure 1, the defined matrices mean all the quantities necessary to describe the motion of the a-particle, that is, its polar 9 and azimuthal ^ angles, as well as the change in these angles (d9 and d^) and components of the radius vector r. ^o and 90 in Figure 1 are the initial values of the angles through which the initial components of the radius vector of the a-particle r0 are calculated. Changes in the polar d9 and azimuthal d^ angles are defined as follows: d9 =((2rand - A (8.4) Rb A ^2 v2 d* - Rbcos(9) V 7 (2rand- ]J (8.5) where rand is a random variable with a uniform distribution over the range from 0 to 1. The condition in Figure 8.1 means the following inequality: d9^ + ^cos 9d^ < (VjT") (8.6) The physical meaning of this condition is that the square of the distance traveled by an a-particle in time dt over the surface of a sphere of radius Rb with a constant velocity Va cannot be less than the sum of the squares of the distances covered for that the same time over the surface of a sphere of the same radius with the same velocity in the polar and azimuthal directions. In general, from Figure 8.1 it is clear that in each iteration, changes in the azimuthal and polar angles are determined, which are added to their old values (^i and 9t) and using the new angles obtained ($1+1 and 9i+1) the following components of the radius vector of the a-particle ri+1 are calculated. As a result, according to the obtained data, written in the matrix containing the values of the components of the radius vector of the a-particle at each moment of time r, the program builds its trajectory along the surface of a sphere of the Bohr radius Rb (Figure 8.2). Figure 8.2 shows a sphere of radius Rb, on the surface of which the red dots mark the location of the a-particle between times dt. Filling the sphere with dots depends on the number of loop iterations, that is, if there are too many of them, the sphere will be densely filled with dots. 8.3.2 The coordinate system of the OHe-nucleus system. Before we start modeling the OHe system and the nucleus of baryonic matter, taking into account all the forces acting between particles, that is, modeling the 110 T.E. Bikbaev, M.Yu. Khlopov and A.G. Mayorov Fig. 8.2: The density of the distribution of the coordinates of the a-particle on the surface of the sphere of the Bohr radius Rb interaction of three bodies, let us consider the coordinate system for the OHe-nucleus system. The system OHe-nucleus consists of three charged, pointlike (in this work) particles, in which a linked system of two other particles moves to one particle "fixed" at the center of coordinates. The particle at the origin is target nucleus of the baryonic matter, and the moving particles mean the a-particle and the O .In this case, the a-particle rotates along the Bohr radius Rb around the particle O . In order to describe the trajectories of motion of the a-particle and the O consider a spherical coordinate system with a point target nucleus A at the origin of the coordinate system. It introduces the radius vector (see Figure 8.3) of the O r and the radius vector of the a-particle ra. Wherein: ra = r + Rb (8.7) Accordingly, for the radius vector of the a-particle and the O azimuthal and ^o—) and polar (9a and 0o—) angles. Figure 3 also shows the particle velocity vector O , V, the angle between V and the horizontal line, a, and the initial coordinates of the particle O [X0, Yo, Z0]. Before proceeding to the description of modeling the OHe system and the nucleus of baryonic matter, taking into account interactions between particles, it should be said that it is possible to construct the effective potential between O-helium and the nucleus of baryonic matter (see Figure 8.4). This potential includes electromagnetic and nuclear interactions. And the task of modeling is precisely to introduce these interactions in order to reproduce the effects of this potential numerically. 8 Numerical Simulation of Dark Atom Interaction With Nuclei 111 Fig. 8.3: Coordinate system OHe-core. 8.3.3 Coulomb interaction in the OHe -nucleus system At this stage of modeling, a system of three point systems interacting with each other through the Coulomb forces of charged particles is considered, with the above choice of the coordinate system. A Coulomb interaction acts between the a-particle and the target nucleus in the considered coordinate system, which is determined by the force: FZa — FZa(ra) — ZZffe2r0 (8.8) where Z is the charge of nucleus. Coulomb interaction between the O-- particle and the target nucleus, which is determined by the force: FZO — FZo(r) — ZZoe2r (8.9) r 3 r The task of this stage was to simulate the interaction, by means of Coulomb forces (8.8) and (8.9), in the coordinate system OHe -nucleus, where the motion of the He 112 T.E. Bikbaev, M.Yu. Khlopov and A.G. Mayorov nucleus in the bound state OHe is described according to the algorithm presented in the previous section. The simulation was carried out as follows (see Figure 8.5). We use the following initial conditions: the initial coordinates of O [x0, y o, z0] (or r0) and the initial components of its velocity [VXo, Vy0, Vz0] (or V0). Then the initial values of all previously determined values are calculated. Before condition 1, the algorithm determines the i-th value of the increment of the momentum of a-particle dPai: dP«t = F^. dt (8.10) It corresponds to the termination of the program when the excess of dT kinetic energy transferred to He exceeds the ionization potential of O -helium I0, which results in the destruction of the bound O -helium system: dT < I0 « 1.6MeV (8.11) dP2 dT = —^ (8.12) 2ma Condition 2 is described by formula (6) in the previous section. As you can see from Figure 8.6, at each loop, the program calculates the total force acting on the OHe system: Fsum = F|o + Fa (8.13) 8 Numerical Simulation of Dark Atom Interaction With Nuclei 113 Fig. 8.5: Block diagram for modeling the Coulomb interaction in the OHe -nucleus system With its help, the increment of the momentum dP of OHe system is calculated, which is, in the aggregate, the increment of the momentum of O-- . dp = Fsumdt (8.14) 114 T.E. Bikbaev, M.Yu. Khlopov and A.G. Mayorov Using the momentum increment dP, the O velocity increment dV is calculated for the subsequent finding of the new velocity used in the next iteration: dV = dP mo— + m0 (8.15) The result of the algorithm is the reconstructed trajectories of a-particle and O . One example is shown in Figure 6, where the blue circle shows the location of the target nucleus, the red asterisk and the purple square are the initial locations of the a-particle and the O-- particle, respectively, yellow dots and the green dashed line show the trajectories of the a-particles and particles O respectively. In the figure under consideration, one can observe the deviation of the trajectory O from the initial direction, which is associated with the Coulomb interaction between the He nucleus and the target nucleus. This happens because He is closer to the origin and is repelled from the target nucleus more strongly than the O particle is attracted to it. x 10"' ' * 2.5 3 Fig. 8.6: a-particle and particle O trajectories 8.3.4 Nuclear interaction in the OHe -nucleus system At this stage, the program was supplemented with a nuclear interaction of the Saxon-Woods type, between the He nucleus and the target nucleus, determined 8 Numerical Simulation of Dark Atom Interaction With Nuclei 115 by the force F N. a * Uo (ra - RA ra — exp - — FN =- —-V a Axa2, (8.16) /ra - Rzn 1 + exp where RZ is the radius of the target nucleus, U0 is the depth of the potential well, a is a constant parameter. In this case, the total force acting on the system OHe, FSum, is now calculated as follows: Fsum = F|O + Fa, (8.17) where Fa is the total force acting on the a-particle: Fa = Fa + FN (8.18) Simulation is performed according to the algorithm described in the previous paragraph, where dPa, the increment of the a-particle momentum, is now calculated as follows: dPa = Fadt (8.19) Based on the data obtained, the program builds the trajectories of the a-particle and the O .In Figure 7, which shows the result of the program, the blue circle shows the location of the target nucleus, yellow dots and the green dashed line show the trajectories of the a-particle and the O particle in the XY plane, respectively. Figure 7 shows the effect of adding a nuclear force of interaction between the target nucleus and the a-particle. Which consists in the fact that at small distances between particles, nuclear force can compensate for the effect of electromagnetic interaction. As a result, some beats are observed in the trajectory O . 8.4 Conclusions The advantage of the OHe composite dark matter model is that it includes only one parameter of the "new" physics - the O mass. Atoms OHe - these neutral primary nuclear-interacting objects, provide the modern density of the dark matter and play the role of a non-trivial form of strongly interacting "dark" matter. Also, the OHe hypothesis can explain the conflicting results of a direct search for "dark" matter, due to the specifics of the interaction of O -helium with the substance of underground detectors. However, the correct quantum consideration of this model turns out to be rather difficult. The OHe hypothesis cannot work if no repulsive interaction occurs at some distance between OHe and the nucleus, and the solution of this problem is vital for the further existence of the OHe dark atom model [10]. Nuclear forces fall off exponentially, but they can be quite strong when the OHe system comes close to the outer target nucleus. These are insignificant and insufficient distances for considering the He nucleus as a point object. In this case, the perturbation theory can no longer be applied and it becomes rather 116 T.E. Bikbaev, M.Yu. Khlopov and A.G. Mayorov -13 x 10' yfci \ t O Start coordinates + Start coordinates alpha particle Particle tr^ectory Q (■ari r~ rrl i n ■a fa c fl. . -- — 0" trajectory B__ ■ * - v \ > 10 a 6 4 2 0 -2 ■4 -6 0 3 Xc** x 10' Fig. 8.7: The trajectory of a-particle and the O in the XY plane problematic to solve the Schrodinger equation. Therefore, the purpose of this work was to numerically simulate the interaction of the OHe atom of "dark" matter with the nuclei of baryonic matter in order to reveal the conditions for the existence of their low-energy bound state and to calculate their effective interaction potential by a numerical method. At the current stage, the our model describes a system of three point, interacting with each other through the Coulomb and nuclear forces, charged particles. The results of the work of the numerical model are the trajectories of motion of point particles entering the OHe atom of dark matter, taking into account the electromagnetic and nuclear interactions between O -helium and the target nucleus of baryonic matter, in the coordinate system OHe -nucleus. However, the process of numerical simulation has not yet been fully completed and in the future it is planned to improve it by introducing finite sizes of nuclei, by taking into account the distribution of the density of nucleons and the density of protons, and introducing the quantum-mechanical effect of tunneling He nucleus into the nucleus of baryonic matter. 8 Numerical Simulation of Dark Atom Interaction With Nuclei 117 Acknowledgements The work by MK has been supported by the grant of the Russian Science Foundation (Project No-18-12-00213). The work by AM has been supported by the grant of the Russian Science Foundation, RSF 19-72-10161. References 1. M.Yu. Khlopov in Cosmion-94, Eds. M.Yu.Khlopov et al. (Editions frontieres, 1996) P. 67; M. Y. Khlopov in hep-ph/0612250, p 51. 2. M.Y.Khlopov, Bled Workshops in Physics 8,114 (2007); in arXiv:0711.4681, p. 114; M. Y. Khlopov and N. S. MankocBorstnik, ibid, p. 195. 3. G. Bertone, D. Hooper, J. Silk: Particle dark matter: evidence, candidates and constraints, Physics Reports 405, 279 - 390 (2005) 4. M. Khlopov: Fundamental particle structure in the cosmological dark matter, International Journal of Modern Physics A. 28,1330042 (2013) 5. M. Y. Khlopov: Dark matter reflection of particle symmetry, Modern Physics Letters A. 32,1740001 (2017) 6. P. Scott: Searches for Particle Dark Matter: An Introduction, (2011) 7. M. Fabbrichesi, E. Gabrielli, G. Lanfranchi: The Dark Photon, (2020) 8. M. Khlopov: Cosmoparticle physics of dark matter, EPJ Web of Conferences 222, 01006 (2019) 9. J. R. Cudell, M. Y. Khlopov, Q. Wallemacq: The nuclear physics of OHe, Bled Workshops Physics 13,10 -27 (2012) 10. M. Yu. Khlopov: 10 years of dark atoms of composite dark matter, Bled Workshops Physics 16, 71 -77 (2015) 11. M. Y. Khlopov: Conspiracy of BSM physics and cosmology, (2019) 12. K. M. Belotsky, M. Y. Khlopov, K. I. Shibaev: Composite Dark Matter and its Charged Constituents, (2006) 13. M. Y. Khlopov, C. A. Stephan, D. Fargion: Dark matter with invisible light from heavy double charged leptons of almost-commutative geometry?, Classical and Quantum Gravity 23, 7305 -7354 (2006) 14. M. Y. Khlopov, C. Kouvaris: Strong interactive massive particles from a strong coupled theory, Physical Review D 77, (2008) 15. M. Yu. Khlopov, A. G. Mayorov, E. Yu. Soldatov: The dark atoms of dark matter, Prespace. J. 1,1403 -1417 (2010) 16. M. Y. Khlopov: Composite dark matter from 4th generation, JETP Letters 83,1 -4 (2006) 17. V. Beylin, M. Khlopov, V. Kuksa, N. Volchanskiy: New physics of strong interaction and Dark Universe, Universe 6,196 (2020) 18. R. Bernabei: Dark matter investigation by dama at gran sasso, International Journal of Modern Physics A 28,1330022 (2013) 19. W. M. Seif, H. Mansour: Systematics of nucleon density distributions and neutron skin of nuclei, Int. J. Mod. Phys. E24,1550083 (2015) Bled Workshops in Physics Vol. 21, No. 2 A Proceedings to the 23rd [Virtual] Workshop, Volume 2 What Comes Beyond ... (p. 118) Bled, Slovenia, July 4-12, 2020 9 Anihelium Flux From Antimatter Globular Cluster M.Yu. Khlopov1'2'3, A.O. Kirichenko *2 and A.G. Mayorov2 1 Institute of Physics, Southern Federal University, Stachki 194 Rostov on Don 344090, Russia 2 National Research Nuclear University MEPhI, (Moscow Engineering Physics Institute), 115409 Moscow, Russia 3 Universite de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France Abstract. Macroscopic cosmic antimatter objects are predicted in baryon asymmetrical Universe in the models of strongly nonhomogeneous baryosynthesis. We test the hypothesis of the existence of an old globular cluster of anti-stars in the galactic halo by evaluating the flux of helium anti-nuclei in galactic cosmic rays. Due to the symmetry of matter and antimatter we assume that the antimatter cluster evolves in a similar way as a matter cluster. The energy density of antiparticles in galactic cosmic rays from antimatter globular cluster is estimated. We propose a method for the propagation of a flux of antinuclei in a galactic magnetic field from the globular cluster of antistars in the Galaxy. Povzetek. Modeli za krepko nehomogeno sintezo barionov napovedo, da so v vesolju makroskopski objekti antisnovi. Avtorji preverjajo domnevo o obstoju starih krogelnih kopic antizvezd v haloju galaksije. Ocenjujejo gostoto jeder antihelija v galakticnih kozmicnih Žarkih. Ker so interakcije med antidelci poznane, sklepajo da se kopice antizvezd razvijajo podobno kot kopice običajnih zvezd. Odtod dobijo oceno energijske gostote antidelcev v kozmicnih zarkih, ki prihajajo iz kopic zvezd iz antisnovi. Predlagajo metodo za izracun pritoka antidelcev iz kroglastih kopic antizvezd skozi galakticno magnetno polje nase galaksije. Keywords: Antimatter; cosmic rays; globular clusters of anti-stars; search for antihelium; Baryon asymmetry of the Universe; AMS 02; PACS: 98.80.Bp; 98.70.Sa; 97.60.Bw; 98.35.Eg; 21.90.+f; 9.1 Introduction At the end of the 1920s, Paul Dirac predicted the existence of antiparticles — that is new particles, which are opposite in sign of electric, baryonic, lepton and other charges of already known particles [1]. Antimatter is detected in cosmic rays. According to the modern concepts it has three possible nature of origin: • Primordial antimatter. It could be created in the early Universe as the reflection of nonhomogeneous baryosynthesis [2,3], evolve in antimatter domains and * E-mail: aokirichenko@yandex.ru 9 Anihelium Flux From Antimatter Globular Cluster 119 now it can exist in the form of macroscopic antimatter objects like globular clusters of antimatter stars [2]. • Secondary antimatter. It is formed as a result of the collision of the nuclear component of cosmic rays with interstellar gas or with a supernova shell [4]. • Antimatter from exotic sources like evaporation of primordial black holes or the decay/annihilation of hypothetical particles of dark matter [2]. According to [5], such object can be present in the Galaxy in the form of a globular cluster of antimatter stars. The prediction [5] assumes similarity in the properties of antimatter and matter globular clusters. Based on this similarity we consider here possibilities to test the hypothesis of antimatter globular cluster in searches for antihelium component of cosmic rays. Our approach is aimed to specify the predictions of this hypothesis with the account of realistic description of the production and propagation of cosmic antihelium fluxes in the Galaxy. 9.2 Primordial antimatter The baryon asymmetry of the Universe is the observed predominance of matter over antimatter in the visible part of the Universe. Explaining the origin of the baryon asymmetry of the Universe is one of the key problems of modern cosmology and physics of elementary particles. A. D. Sakharov [6] and V.A. Kuzmin [7] formulated the necessary conditions for bariosynthesis in the early Universe: 1. Asymmetry between particles and antiparticles as a violation of charge C-and combined CP-symmetry. 2. Violation of the law of conservation of baryon charge. 3. Violation of local thermodynamic equilibrium. On the other hand, it was shown in [8] - [11] that almost all existing mechanisms of baryosynthesis allow the existence of domains with an excess of antimatter, if baryosynthesis is strongly nonhomogeneous. The size of domains depends on the details of the considered mechanisms and can be both small and reaching the size of a Metagalaxy. The macroscopic region of antimatter with an excess of antibaryons at the same temperature and density evolves in the same way as ordinary matter of macroscopic size. Experiments on accelerators synthesizing antimatter show that the properties of particles and antiparticles coincide, except of the small effect of CP-violation [12] . An astronomical object smaller than a globular cluster cannot be formed surrounded by matter during cosmological evolution [13]. With smaller sizes, antimatter would annihilate with matter before the Galaxy formation. The larger size of domains is constrained by the observed fluxes of gamma radiation. Globular clusters of antistars could form during the formation of the Galaxy and remain in its halo by now. Cosmic ray fluxes of antinuclei are the profound signature of antimatter stars and provide the probe of their existence. 120 M.Yu. Khlopov, A.O. Kirichenko and A.G. Mayorov 9.3 Secondary antimatter The detected fluxes of cosmic antiparticles are formed as a result of collisions of high-energy nuclear component of cosmic rays with interstellar gas. Study of the processes of antiproton and light antinuclei production at accelerators made it possible to determine the cross section of these processes. The data obtained were used to predict the cross sections for heavier nuclei. This analysis (figure -3 -4 9.1) shows that detection of He , He at level of sensitivity of experiment can not explained by secondary antinuclei. 0.1 0.5 1 5 10 50 100 Ekin (GeV/n) Fig. 9.1: Upper limits for of secondary [14] antihelium, antiproton, antideuteron together with previous results. 9.4 Antimatter from exotic sources Modern cosmology classifies as exotic sources of antimatter annihilation or decay of hypothetical particles of dark matter and evaporation of primordial black holes. 9.4.1 Dark matter Dark matter makes up ~ 85% of all matter in the universe. Its presence is implied in many astrophysical and astrological observations, including gravitational effects and large-scale structure formation. Such effects cannot be explained by the action of baryonic matter. Since dark matter has not yet been observed, if it exists, then it must interact through gravity with baryonic matter and radiation. The decay and annihilation of such particles can lead to the formation of antiparticles [15]. 9 Anihelium Flux From Antimatter Globular Cluster 121 9.4.2 Primordial black hole Primordial black holes are a hypothetical type of black hole formed after the Big Bang. In the early Universe, high densities and inhomogeneous conditions could lead to gravitational collapse in dense regions, forming black holes. Ya. B. Zeldovich and I. D. Novikov in 1966 for the first time suggested the existence of such objects [16]. The theory of their origin was first deeply studied by S. Hawking in 1971 [17]. Hawking showed that, due to quantum effects, black holes radiate like a black body with a temperature inversely proportional to the mass of the black hole. A physical understanding of the process can be obtained by imagining that particle-antiparticle radiation is emitted from beyond the event horizon. According to the modern concept, primordial black holes can also be sources of positrons and antiprotons [18]. 9.5 Globular clusters in the galactic halo A globular star cluster is a collection of stars that forms a spherical cluster rotating around the core of the Galaxy. Globular clusters are very closely connected by gravity, which gives them a spherical shape and a relatively high density of stars towards their centers. The name of this category of star clusters comes from the Latin globulus - a small sphere. Globular clusters are located in the galactic halo and contain more stars and are much older than the less dense open clusters found in the galactic disk. Globular clusters are common, with about 150 globular clusters currently known in the Milky Way [19]. Observations of globular clusters show that these stellar formations originate mainly in regions of effective star formation, where the interstellar medium is denser than normal star-forming regions. Currently, none of the known globular clusters show active star formation, they are free of gas and dust, and it is assumed that all the gas and dust were long ago either turned into stars or blown out of the cluster during the initial explosion of star formation. This is consistent with the opinion that globular clusters are the oldest objects in the Galaxy and were among the first clusters of stars [20]. The trajectories of the globular clusters are eccentric and inclined to the plane of the galaxy. Orbiting the "outskirts" of a galaxy, globular clusters take several hundred million years to complete one orbit. Stars can reach a density of 100 to 1000 stars per cubic parsec in the center of a globular cluster. This is different from the density of stars around our Sun, which is estimated at about 0.14 stars per cubic parsec. Globular clusters are usually made up of stars that have a low proportion of elements other than hydrogen and helium compared to stars like the Sun. The proportion of havier elements may indicate the age of a star, with older stars usually having lower metallicities [21]- [22]. 122 M.Yu. Khlopov, A.O. Kirichenko and A.G. Mayorov 9.6 Discussion about the sourse function The paper considers a typical globular cluster, presumably consisting of anti-stars. As an example, we take one of the closest clusters - M4 in the Messier catalog (fugure 9.1) (NGC 6121 in the new general catalog (NGC)). Age Gy Distance from the Sun, kpc Number of stars 12 1.72 8 • 104 Table 9.1: Parametrs of globular cluster M4 [23] Then we also assume that globular cluster M4 is a source of He in galactic cosmic rays. Three possible mechanisms for the injection of antihelium into cosmic rays from the globular cluster M4: 1. Stationary outflow of matter from the surface of antistars If the diapason of propagation of antimatter from the globular cluster crosses the galactic disk, then the stellar wind will enter the disk, and then into the solar system. A stationary outflow of star matter in a cluster is considered for this. Stars are constantly losing part of their mass, so the concentration of particles from the entire globular cluster can be large. These are very low energies, a process of additional acceleration of particles is required to effectively overcome the solar magnetic field, but this effect is suppressed. In this case, we expect an energy ~ MeV. 2. Flares on antistars It is a known fact that active explosive processes occur on the Sun, accompanied by the acceleration of particles and the appearance of solar cosmic rays. We assume the existence of similar processes on antistars in a globular cluster. Particles from flares on antistars can receive higher energy GeV), forming the antinuclear component of galactic cosmic rays. 3. Explosions of antisupernovae in a globular cluster of antistars Supernova explosions are the result of the evolution of stars, which is accompanied by the release of high energy up to ~ 1051 erg. The shell from the exploded anti-star propagates at high speed. Particles can acquire energy (~ 1015 eV) as a result of various acceleration mechanisms on the supernova shell, accelerate and inject into cosmic rays. By analogy with the fact that stars are the source of particles in cosmic rays, antistars should be the main source of antiparticles in cosmic rays. Supernovae may be the most likely source of antinuclei in galactic cosmic rays. 9 Anihelium Flux From Antimatter Globular Cluster 123 9.7 Results 9.7.1 Supernova explosions The analysis begins with the most probable mechanism - antisupernova explosions, because the magnetic fields of the Galaxy prevent the penetration of low-energy antiparticles into the Galactic disk. But it is also important to note that the frequency of the explosion of such supernovae is low against the background of outbursts of anti-stars and against the background of a constant outflow of stationary matter of anti-stars. The first two cases will be discussed later. Calculation of the energy density of antiparticles in cosmic rays Figure (9.2) shows a graph of the evolution of the population of the M4 cluster [24]. The graph shows the processes occurring in the early stages of the life of the cluster, the results of these processes can be compared with the present time. Let's pay attention to the number of neutron stars on the graph. Their number has not changed over 12 billion years. This means that about 12 billion years ago they could have formed as a result of the explosion of antisupernovae. This fact can be used to calculate the energy density of antiparticles in cosmic rays. -1-1-1-1-1-1— 100000 10000 I 1000 100 1.............—......—...................—..........— 0 3XO 4000 6000 3000 10000 12000 Tine (Myr) Fig. 9.2: Change in the population of M4 in time 124 M.Yu. Khlopov, A.O. Kirichenko and A.G. Mayorov Using the formula for the energy density of cosmic rays of matter EsnN sn^ret in 1 \ Pcr =-Vs--(9.1) Nsn - number of neutron stars in M4, t- cluster age, N sn- average supernova explosion frequency, Esn - energy realized from supernova, tret- cosmic ray lifetime, V- volume of the region of propagation of cosmic rays (to calculate the volume, we considered a model of a cylinder with a height and radius of 30 kpc and 10 kpc, respectively. In order to consider not only the region of the disk, but also the halo of the Galaxy). We present all the numerical values of these quantities in the form of a table figure (9.2). Nsn t, Gy Nsn Esn,erg tret, myr V,kpc3 12 1.72 8 • 104 1051 2•10-5 3 • 103 Table 9.2: Table of numerical characteristics of quantities for calculating the energy density of antiparticles the density using formula (1) and the values in the table: P^ ~ 10-4eV/cm3 (9.2) For comparison, we present the value of the energy density of cosmic rays of matter: P - 1eV/cm3 (9.3) We also pay attention to the fact that the energy density for secondary antiprotons: Pp - 10-5eV/cm3 (9.4) The obtained value does not correspond to the established experimental data for the energy density of antiprotons. But given the fact that particles of cosmic rays pass through the magnetic fields of the Galaxy and lose some of the energy in them, it is necessary to consider in more detail the mechanism of motion of cosmic rays, which will be presented in the following part. Particle motion in a Galaxy's magnetic field In order for us to estimate the real fraction of particles from the initial flux that penetrates into the disk of the Galaxy, it is necessary to simulate the motion of particles in the magnetic field of the Galaxy. Simulation of the magnetic field of the galaxy. Based on the equations according to the data of [25], we have compiled a function program, the input parameters of which are the coordinates in the Galaxy, and the output parameters are the components of the magnetic field in the Cartesian coordinate system. 9 Anihelium Flux From Antimatter Globular Cluster 125 The components of the magnetic field in a cylindrical coordinate system centered at the Galactic center taken from [25]: = - 2R/R0 (VR/R°^/Z0^/Z°)) U z2 Br = i Bi ° tanh(R/Ro) 2 (z + z)2 n 0.IB1Z0 1n z2 /tanh(R/Ro) , sech2(R/R0A Bz = + 2B1 (zrozn—R— + —R0—J Where R0 =5 kpc and z0 = 0.5pc are taken as scale lengths, and the B1 parameter is free in [25] and is determined by calibration, for example, by the magnetic field near the solar system according to the data of [26]. a) b) Fig. 9.3: Topology of the magnetic field of the Galaxy in the projection Z : projection RZ 0 and in the We have constructed the topology of the Magnetic field of the Galaxy in the projection RZ and in the projection Z=0 (R and z are coordinates in a cylindrical coordinate system, a coordinate system centered at the Galactic center). You can see in figure (9.3a), that the magnetic field lines spiral out from the center of the Galaxy, this corresponds to the concept of the global magnetic field of the Galaxy in the plane of the galactic disk. The figure (9.3b) shows the vertical projection of the magnetic field of the Galaxy at y = 0, we see that the lines of force diverge in different directions according to the law determined by the equations from ( [25]). We have reproduced the magnetic field given in ( [25]), and now we will simulate the propagation of particles in this magnetic field and will observe how particles are transported in our Galaxy. 9.8 Conclusion In this work, we considered the typical globular cluster M4, whose observed features can reproduce the expected properties of a globular cluster of anti-stars in 126 M.Yu. Khlopov, A.O. Kirichenko and A.G. Mayorov the Galaxy. Based on the symmetry of the properties of matter and antimatter, we discuss the evolution of this GC and the mechanisms of injection of antimatter in CR. We calculated the energy density of high energy antiparticles ejected by antimatter GC in cosmic rays, and also checked the operation of the program to simulate the propagation of these antiparticles in the magnetic field of the Galaxy. Further work is aimed at modeling the motion of particles in the magnetic field of the Galaxy, in order to estimate the minimum energy that a particle penetrating into a galactic disk should have. Implementation of our research program will help to obtain predictions of the expected flux of antinuclei as the signature of antimatter stars in our Galaxy. Acknowledgements The research of MK was financially supported by Southern Federal University, 2020 Project N VnGr/2020-03-IF. Research of AM 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. P. A. M. Dirac: The quantum theory of the electron, Proc. Roy. Soc. (London) A117, 610—624 (1928). 2. M. Y. Khlopov: Fundamentals of Cosmoparticle Physics CISP-Springer,Cambridge, UK, 2012. 3. A. D. Dolgov: Matter and antimatter in the universe, Nucl. Phys. Proc. Suppl. 113 40 (2002). 4. Nicola Tomassetti, Alberto Oliva: Secondary antinuclei from supernova remnants and background for dark matter searches,35th International Cosmic Ray Conference -ICRC2017, (2017). 5. M.Yu. Khlopov: An antimatter globular cluster in our Galaxy - a probe for the origin of the matter, Gravitation and Cosmology , 4, 69-72 (1998). 6. A.D. Sakharov: Violation of CP-invariance, C-asymmetry and baryon asymmetry of the Universe, JETP Lett, 5, 32 (1967). 7. V.A. Kuzmin: CP violation and baryon asymmetry of the universe, JETP Lett, 12, 228 (1970). 8. V.M. Chechetkin, M.G. Sapozhnikov, M.Yu. Khlopov and Ya.B.Zeldovich: Astrophysical aspects of antiproton interaction with He (Antimatter in the Universe), Phys. Lett. 118B, 359-362 (1982). 9. V.M. Chechetkin, M.Yu. Khlopov and M.G. Sapozhnikov: Antiproton interactions with light elements as a test of GUT cosmologies., Rivista Nuovo Cimento, 5,1-80 (1982). 10. A.D. Dolgov, A.F. Illarionov, N.S. Kardashev, I.D. Novikov, Cosmological model of a baryon island, JETP, 67,1517-1524 (1988). 11. M.Yu. Khlopov, S.G. Rubin, A.S. Sakharov: Possible Origin of Antimatter Regions in the Baryon Dominated Universe., Phys.Rev.D 62, 083505 (2000). 12. M. Charlton, S. Eriksson, G. M. Shore: Fundamental Physics in Antihydrogen Experiments, 97-98 (2020). 9 Anihelium Flux From Antimatter Globular Cluster 127 13. M.Yu. Khlopov, R.V. Konoplich, R. Mignani, et al.: Evolution and observational signature of diffused antiworld., Astroparticle Phys.,12, 367-372 (2000). 14. I. Cholis, T. Linden: Anti-Deuterons and Anti-Helium Nuclei from Annihilating Dark Matter FERMILAB-PUB-20-021-A, (2020). 15. V. Trimble Existence and Nature of Dark Matter in the Universe., Annual Review of Astronomy and Astrophysics., 25, 425-472 (1987). 16. Ya.B. Zeldovich, I.D. Novikov: The Hypothesis of Cores Retarded During Expansion and the Hot Cosmological Model, Soviet Astronomy, 10, 602 (1966). 17. S. Hawking: Gravitational collapsed objects of very low mass. Mon. Not. Roy. astron. Soc.,152 , 75-78 (1971). 18. W. Carroll Bradley, A. Ostlie Dale: An Introduction to Modern Astrophysics, Reading, MA: Addison-Wesley Publishing, 1996. 19. http://gclusters.altervista.org/ 20. M. Paul: Star Clusters. Encyclopedia of Astronomy and Astrophysics, 2014. 21. https://www.astro.keele.ac.uk/workx/globulars/globulars.html 22. J. S. Kalirai, H. B.Richer: Star clusters as laboratories for stellar and dynamical evolution, Royal society publishing, (2009). 23. D. C. Heggie and M. Giersz: Modelling individual globular clusters, Cambridge University Press Access S246,3,121-130 (2007). 24. D. C. Heggie, M. Giersz: Monte Carlo simulations of star clusters - V. The globular cluster M4,Monthly Notices of the Royal Astronomical Society 1, 388, 429-443 (2008). 25. C. J. Nixon, T. O. Hands: The origin of the structure of large-scale magnetic fields in disc galaxies Notices of the Royal Astronomical Society 3, 477, 3539-3551 (2018). 26. M. Opher, F. Alouani Bibi: A strong, highly-tilted interstellar magnetic field near the Solar System, Nature, 462,1036-1038 (2009). Bled Workshops in Physics Vol. 21, No. 2 A Proceedings to the 23rd [Virtual] Workshop, Volume 2 What Comes Beyond ... (p. 128) Bled, Slovenia, July 4-12, 2020 10 Domain Walls and Strings Formation in the Early Universe A.A. Kirillov * and B. S. Murygin ** National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) Abstract. Soliton formation through classical dynamics of two scalar fields with the potential having a saddle point and one minimum in (2+1)-space-time is discussed. We show that under certain conditions in the early Universe both domain walls and strings can be formed even if scalar fields are inflaton ones. Povzetek. Avtorja obravnavata tvorbo solitonov s klasično dinamiko dveh skalarnih polj s potencialom, ki ima sedlo in en minimum v (2+1)-razsežnem prostoru. Pokažeta, da lahko nastanejo zidovi domen in strune, ob določenih pogojih v zgodnjem Vesolju, tudi v primeru skalarnih polj, ki so inflatonska. Keywords: solitons, strings, domain walls PACS: 03.50.-z, 11.27.+d, 98.80.Cq 10.1 Introduction Multi-field inflation models such as the hybrid inflation [8] or the natural inflation [7,10] may contain potentials of non-trivial forms. If potential has at least one saddle point, the field dynamics in such models may lead to formation of topologically non-trivial structures named solitons [9,11,14]. Moreover, under certain conditions, they may produce primordial black holes in the radiation era due to collapse of domain walls [1] or loops of cosmic strings [5] that affects the early Universe [13]. Previously, it was shown solitons may be formed in (1+1) space-time even potential has only one minimum and at least one saddle point [3,4]. In this paper, we continue study of the possibility in (2+1) space-time. 10.2 Model in (2+1) space-time Let us consider the dynamics of two real scalar fields cp and x with the Lagrangian of the system L = + 3^x3vx) — V (V,x), (10.1) E-mail: AAKirillov@mephi.ru E-mail: MuryginBS@gmail.com 10 Domain Walls and Strings Formation in the Early Universe 129 where is the Friedman-Robertson-Walker metric tensor with the cosmic scale factor a(t). Then, the classical motion equations for cp and x in (2+1) space-time take the form dV d? (10.2) Xtt — 3Hxt — Xxx — Xyy = — g^. Here, H = a/a is the Hubble parameter which is Hi ~ 1013 GeV during the inflation and becomes smaller in the radiation era. For equations (10.2), the Hubble parameter plays a role of a friction term, and its time dependence does not affect our conclusions. Thus, we assume it remains constant after the end of the inflation. In addition, the Hubble parameter H gives a natural scale for all units. Therefore, we express all dimension variables in Hi units. To solve the system (10.2), we have to define initial and boundary conditions. We choose the initial conditions in the form ?(x,y,0) = R cos 0 + 91, ?t(x,y,0)= 0; 3) X(x,y,0) = R sin 0 + xi, Xt(x,y,0) = 0, where 1 r0 R(r) = R0 cosh-1 0 = 0. (10.4) It sets correspondence between the fields space (?,x) and the physical plane (x,y). Here, the point (^xO corresponds to the center of the initial fields area in the form of the circular disk with the radius R(r) and the polar angle 0 < 0 < 2n, r = \Jx2 + y2 and 0 are a distance from the coordinate origin and a polar angle in a physical xy-plane, respectively, and R0 and r0 are positive parameters. The boundary conditions are chosen as 9x(±oo ,y,t)= 0, ?y(x, ±00 ,t) = 0; (10.5) Xx(±oo,y,t) = 0, Xy(x,±oo,t) = 0. We study classical evolution of the scalar fields cp and x with the potential used in [3,4]: V = d(?2 + x2) + aexp [ — b(? — 90)2 — c(x — X0)2], (10.6) where a, b, c, d are positive parameters. The parameter a sets a height of a local maximum, b and c set its shape, and d is responsible for a slope of the potential. The described potential has only one saddle point and one minimum, but could be easy modified to obtain any number of saddle points by adding terms like the last one. Additionally, we consider the well-known potential "tilted Mexican hat" [10] V = K*2+x2 — i )2+A4(1 — 7?=?), ^ where A, g, A are positive parameters. The parameter g sets a position of a circle of degenerate minima in the case of the Mexican hat without a tilt, A sets a height 130 A.A. Kirillov and B. S. Murygin of a local maximum at the point (90 ,Xo) = (0,0) and A sets a tilt of the potential. Note, a potential slope makes minima non-degenerate. However, non-degeneracy is not a necessary condition for solitons production. The energy density of the system is given by where partial derivatives are taken over the variables {t, x, y}. 10.3 Results For the potential (10.6), we choose the parameters as follows d = 0.005, a = 2, b = 1, c = 1, <0 = -5, xo = 0 and the parameters of the initial conditions (10.3) R0 = 1, r0 = 1, <1 = — 8 and xi = 0 (all dimensional values are taken in Hi units). The initial fields configuration is separated from the minimum by the peak of the potential, see fig. 10.1a. Note, the potential has the minimum at the point (2 1 National Research Nuclear University MEPHI (Moscow Engineering Physics Institute), 2 N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal University, Kremlevskaya Street 18, 420008 Kazan, Russia Abstract. We consider the scalar field solitons and their interaction with the fermions in the early Universe. The analytical form of the reflection coefficient is obtained. The fermion mass is a function of the distance between the fermion and the soliton (wall). The function was approximated by the Woods-Saxon potential. Povzetek. Avtorja obravnavata solitone skalarnih polj in njihovo interakcijo s fermioni v zgodnjem vesolju. Izpeljeta analiticno formulo za koeficient odboja. Masa fermiona je v njunem modelu funkcija razdalje med fermionom in solitonom (steno), za funkcijo razdalje uporabita Woods-Saxonov potencial. Keywords: domain wall, Dirac equation, PBH, early Universe 11.1 Introduction Primordial black holes (PBHs) have been a source of significant interest for over 50 years. The possibility of the existence of such objects was predicted by Zeldovich and Novikov [14]. Despite the absence of direct evidence of their existence, there is a lot of observational data that can be interpreted in the framework of the hypothesis of the origin of black holes (BH) at the initial stages of the origin of the Universe [3,4,7]. In this paper, we base on the model of PBH formation as a result of the collapse of domain walls [2, 5, 11]. As a result of phase transitions during and after the inflationary stage, closed domain walls are formed. The formed non-spherical wall evolves: when interacting with hot plasma, the kinetic energy of the wall dissipates. As a result, the oscillations of the domain wall decay, the energy is transferred to the surrounding plasma, which leads to its additional heating. Further, the wall spheres and collapses into BH. The rate of energy transfer from the domain wall to the surrounding plasma depends on the wall thickness, the initial plasma temperature and its density. * E-mail: kurakin-1993@mail.ru ** E-mail: sergeirubin@list.ru 11 The Interaction of Domain Walls with Fermions in the Early Universe 135 The wall thickness is characterized by the parameters of the initial Lagrangian and can vary over a wide range. Plasma temperature and density depend on the moment the walls appear. Moreover, the dynamics of plasma parameters depends on whether it participates in cosmological expansion or is separated from it due to the gravitational well created by closed walls. In this paper, we consider the fermion interaction with the scalar field solitons (walls). 11.2 Model of domain wall Consider the domain wall model. We describe We describe the wall by a complex scalar field with a Lagrangian: LwaU = - 7- f2/2)2 - A4(1 - cos 9), (11.1) where ^ - complex scalar field and 9 is its phase. At the end of inflation, the ^ field is captured by the potential minimum for which = f. Then we write the complex field in the form: $ = f eie = f e*/f. V2 V2 (11.2) Substitution of the expression (11.2) into Lagrangian (11.1) gives Lagrangian, that describing the phase of complex scalar field: ¿wall = 1 (3|iX)2 - A4(1 - cos(x/f)). The the phase x is determined as follows [10]: "A2 x(x) = 4f arctan exp f 4f arctan exp 2x T where we introduced the wall thickness parameter d d 2f d = A2 • Let us choose the Lagrangian of fermions in the form: = up + g0 1 + h.c.) — ml 1 = = up Y^S^l + V2g0fil 1 cos(x/f) — mip p. where expression (11.2) is used. The interaction of the fermions with the domain wall is Lint = mo cos(x/f)ff = mJ 1--2 \ cosh2(2x/d) ^ Then Lagrangian of fermions can be rewritten as Lf = ify1 - mo where mf = m - m0 - fermion mass. 2 cosh2(2x/d) f f - mfi|> f, (11.3) (11.4) (11.5) (11.6) ff; mo = V2fgo. (11.7) (11.8) x 136 A.A. Kurakin and S.G. Rubin 11.3 Dirac equation A description of the interaction between fermions and domain wall within the framework of the approach to solving the equation of motion is given in the papers [1,6,12] The result for the interaction of the wall with scalar particles is given in the monograph [13]. In the papers [1,6,12] the description of the domain wall is given by the kink solution: ^ ~ tanh x. In such model, the asymptotic fermion mass takes different values: x —» ±oo. This problem does not arise for the Lagrangian (11.8): the fermion mass is the same on both sides of the wall. The Dirac equation 0 = - g(x)) (11.9) holds for fermion Lagrangian (11.8) where function g(x) =-^-+ mf (11.10) cosh2(2x/d) is effective mass, depending on the coordinate in the coordinate x perpendicular to the wall. Hereinafter, in asymptotics, we have: g(x) mf. The fermion wave function is as follows ^(x) = (U! (x) U2(x) U3(x) U4(x))T e_iEt+ip. (11.11) Here we put pt = 0 for simplicity, i.e. the component of the momentum in the plane of the domain wall is zero and the incident wave is perpendicular to the wall. Then the equation takes the form: 0 = (Ey0 + iY33x - g(x)) ^(x). (11.12) Hereafter, we choose the following representation of gamma matrices: Y0 = (H) ,Y3 = O o) (11.13) As a result of substitution, we obtain a system of equations for the bispinor components: 0 = Eu3(x)+ iu3(x) - g(x)ui (x) (1114) 0 = Eui (x) — iuj (x) — g(x)u3(x). We obtain a similar system for the components u2, u4 if we replace: u —> u4, u3 > u2. Let's consider the following linear combinations of bispinor components: (x) = u (x) + iu3 (x) (11.15) $-(x) = ui (x) - iu3 (x). As a result of such substitution we obtain a system: 0 = iE$_(x) + (x)- g(x)$+(x) (1116) 0 = iE^+(x) + (x) + g(x)$_ (x). ( . ) 11 The Interaction of Domain Walls with Fermions in the Early Universe 137 Excluding the variables, we obtain the equations for the components (x): 0 = (dX2 ^ 9'(x)+ E2 - g2(x^ *±(x). (11.17) Let us carry out an approximation by a function for which the solution can be obtained in an analytical form. Let us choose the Woods-Saxon potential. The scattering problem for the Woods-Saxon potential is considered in detail in the papers [8,9]: After approximation, the function g(x) takes the form A9(x) A9(-x) „„ „ , 9(x) = ^-rr-+ ^-f (r \ u + mf' (11.18) 1 + exp(a(x — xo)) 1 + exp(-a(x + xo)) where parameters: A —> A = 2.392m0, mf = m — m0. Fig. 11.1: Approximation of the function g (x). Blue solid line - g (x), red dotted line - Woods-Saxon potential We solve the equation for two regions: x < 0 and x > 0 Consider the region x < 0. We will solve the equation for (x) (the superscript L denotes the region x < 0). Let's make a replacement: £ = - exp(-a(x + xo)). (11.19) Then the equation (11.17) takes the form: d L d A ( , A V , aA£ , A L 0 = (^^J- lmf + r-^J +E2J (11.20) The solution of the equation (11.20) is as follows: $+(£,) = CiCa(1 - 2Fi(-a - v - p, -a + v - p,l - 2a; £) (1121) + C2£a(1 - 2Fi (a - v - p, a + v - p,1 + 2a; £). . 138 A.A. Kurakin and S.G.Rubin where the parameters a, |3 , v are defined as a = 1 J(mf + A)2 - E2 = ^ a V a | =- A (11.22) a v = 1 Jm2 - E2 = - Je2 - m2 = -. a V 1 a V 1 a Let's consider the limit x —» -œ => £ —> -oo. Then, for the function 9+ we obtain superposition of two waves: incident and reflected waves: 9+ --°°> D1eik(x+x0) + D2e-ik(x+x0). (11.23) The coefficients D1, D2 are determined by the formulas: D = C r (1 - 2a) r (-2v) , 1 1 r(-a - v - |3)r(1 - a - v + |) + D2 = Ci r (1 + 2a) r (-2v) C 2-< 2 r(a - | - v)r(1 + a - v + |) r(1 - 2a) r(2v) e-i r(-a + v - |)r(1 - a + v + |) IP, r (1 + 2a)r (2v) e + 2 r(a - | + v)r(1 + a - v + |) (11.24) where Ci, C2 = const. The asymptotic for is obtained by substituting solution (11.23) into the first equation of system (11.16). As a result, we obtain: -k+imDieik(x+xo) + ^^imD2e-ik(x+xo). (11.25) The region x > 0 can be considered in the similar manner to obtain the function 9+ (x) to the right of the wall. In order to find coupling between coefficients C1 and C2 we match the solutions at x = 0: + | x=0 = 1 x=0 = x=0 (11.26) ($ + )' | x=0 = ($+)' | x=0. The normal component of the fermion current density is written as: j = l|>(x)Y3^(x) = - | ui (x)|2 + |U2(X)|2 + | U3(x)|2 - | U4(x)| 2 = = - ( . ) Substitute the obtained functions As a result, we obtain The final form of the current j = E (ID i |2 - ID2I2) = jinc - jref, (11.28) is obtained by substitution of the explicit form of functions and into the expressions for the current density. Here jinc is the current density of the incident particles, jref - of the reflected ones. 11 The Interaction of Domain Walls with Fermions in the Early Universe 139 1 IS U e a 0.010 o u u a n ■m 0.001 100 200 500 1000 2000 d, 1/GeV Fig. 11.2: Dependence of the reflection coefficient R on thickness d, GeV-1. Blue line - m0 = 10-3 GeV, E — mf = 1 MeV; orange line - m0 = 10-4 GeV, E — mf = 1 MeV 0.001 0.005 0.010 0.050 0.100 0.500 1 E-mf, MeV Fig. 11.3: Dependence of the reflection coefficient R on kinetic energy E — mf MeV. Blue line - m0 = 10-5 GeV, d = 104 GeV-1 ; orange line - m0 = 10-4 GeV, d = 104 GeV-1 The reflection and transmission coefficients are determined through the ratio of the current densities as follows jref = |D2[2 jinc |Dl|2 r = pi = l^il. (11.29) The coefficients D1, D2 are determined by the formulas (11.24). The results of calculating the reflection coefficient for electrons (mf = 0.5 MeV) are shown in Figure 11.2,11.3. 140 A.A. Kurakin and S.G. Rubin 11.4 Conclusion The deceleration of the primordial walls due to the interaction with the surrounding media is the important process that could influence the formation of the black holes clusters. In this paper, we have found the reflection probability of the fermions. This is necessary step for studying the cluster heating by the wall fluctuation. Acknowledgements This research was funded by the Ministry of Science and Higher Education of the Russian Federation, Project "Fundamental properties of elementary particles and cosmology" N 0723-2020-0041 and RFBR grant 19-02-00930. The work of S.R. is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University. References 1. A. Ayala, J. Jalilian-Marian, L. D. McLerran, and A. P. Vischer. Scattering in the presence of electroweak phase transition bubble walls. Phys. Rev. D, 49:5559-5570,1994. 2. K. M. Belotsky, V. I. Dokuchaev, Y. N. Eroshenko, E. A. Esipova, M. Y. Khlopov, L. A. Khromykh, A. A. Kirillov, V. V. Nikulin, S. G. Rubin, and I. V. Svadkovsky. Clusters of primordial black holes. Eur. Phys. J. C, 79(3):246, 2019. 3. B. Carr and F. Kuhnel. Primordial Black Holes as Dark Matter: Recent Developments. 6 2020. 4. S. Clesse and J. García-Bellido. Seven Hints for Primordial Black Hole Dark Matter. Phys. Dark Univ., 22:137-146, 2018. 5. H. Deng, A. Vilenkin, and M. Yamada. CMB spectral distortions from black holes formed by vacuum bubbles. JCAP, 07:059, 2018. 6. G. R. Farrar and J. Mcintosh, John W. Scattering from a domain wall in a spontaneously broken gauge theory. Phys. Rev. D, 51:5889-5904,1995. 7. A. Kashlinsky et al. Electromagnetic probes of primordial black holes as dark matter. 3 2019. 8. P. Kennedy. The Woods-Saxon potential in the Dirac equation. J. Phys. A, 35:689-698, 2002. 9. O. Panella, S. Biondini, and A. Arda. New exact solution of the one dimensional Dirac Equation for the Woods-Saxon potential within the effective mass case. J. Phys. A, 43:325302, 2010. 10. R. Rajaraman. Solitons and instantons. An introduction to solitons and instantons in quantum field theory. 1 1982. 11. S. G. Rubin, A. S. Sakharov, and M. Y. Khlopov. The Formation of primary galactic nuclei during phase transitions in the early universe. J. Exp. Theor. Phys., 91:921-929, 2001. 12. D. A. Steer and T. Vachaspati. Domain walls and fermion scattering in grand unified models. Phys. Rev. D, 73:105021, May 2006. 13. A. Vilenkin and E. S. Shellard. Cosmic Strings and Other Topological Defects. Cambridge University Press, 7 2000. 14. Y. B. Zel'dovich and I. Novikov. The hypothesis of cores retarded during expansion and the hot cosmological model. Soviet Astronomy, 10:602,1967. Bled Workshops in Physics Vol. 21, No. 2 A Proceedings to the 23rd [Virtual] Workshop, Volume 2 What Comes Beyond ... (p. 141) Bled, Slovenia, July 4-12, 2020 12 Cosmological Accumulation of Conserved Numbers in Kaluza-Klein Theories V.V. Nikulin * National Research Nuclear University MEPhI Abstract. We develop a new mechanism for the accumulation of conserved numbers in the early Universe in Kaluza-Klein-like theories. The relaxation of the primordial extra space perturbations existing in the early Universe leads to the establishment of a symmetric final state and the appearance of Killing vectors. As a result, the initial non-zero value of symmetry associated numbers occurs after the inflation. We show this conceptual idea on a toy model of 2-dimensional apple-like extra space with U(1) symmetry. This mechanism naturally arises in the Kaluza-Klein theories and can be used to explain the observed cosmological baryon asymmetry. Povzetek. Avtor opise zgodnje vesolje s teorjo Kaluze in Klaina v vecrazseZnem zgodnjem vesolju. Predlaga nov mehanizem, ki poskrbi za akumulacijo stevil, ki se ohranijo. Zmanjsanje perturbacij, ki jih povzročijo dodatne dimenzije, vodi v njegovem predlogu do simetricnega koncnega stanja in ustreznih Killingovih vektorjev. Posledicno se zacetna nenicelna vrednost stevil, povezanih s simetrijo, pojavi po inflaciji. Avtor idejo pojasni na modelu 2-dimenzionalnega dodatnega prostora v obliki jabolka in s simetrijo U(1). Njegov mehanizem pojasni opažene kozmoloske asimetrije barionov. Keywords: Kaluza-Klein theory, apple-shaped extra space, baryon asymmetry, f (R)-gravity, cosmological inflation. PACS: 04.50.Cd, 04.50.+h, 04.50.-h, 04.50.Kd, 11.30.Fs, 11.30.Ly, 11.30.-j 12.1 Introduction One of the advantages of using Kaluza-Klein compact extra dimensions is that they can explain the origin of internal symmetry in particle physics. The idea of the approach is that the internal symmetry of the gauge theory is considered as a consequence of the geometric properties of compact extra space, characterized by the presence of Killing vector fields [2]. The stability of the compact extra space is the well-known issue of the Kaluza-Klein theory. The stabilization can usually be achieved by introducing external material fields [5] or by modifying action for gravity [4]. The process of stabilization obviously should take place in a very early Universe at the energy scales * E-mail: N-Valer@yandex.ru 142 V.V. Nikulin ~ 1/r0, when r0 is the radius of compact extra space. In our work [13] we show how dramatically the presence of compact extra space can affect the cosmological inflationary process. In this paper, we investigate the process of relaxation of the extra space metric during the cosmological inflation. As a result of symmetrization, Killing vector fields appears at the end of inflation and its Noether-associated numbers is asymptotically conserved. The initial non-zero value of this conserved numbers is caused by the extra metric perturbations that took place during inflation. This mechanism could be an explanation for the observed cosmological baryon asymmetry. 12.2 Theoretical description Today we do not really understand how a compact extra space can be born in higher-dimensional theories. However, we have no reason to believe that its geometry has any symmetry, as this process is clearly random. As a result of further development, the metric of extra space undergoes relaxation and symmetrization. The deep causes for the inevitable appearance of symmetry in this process is related to the establishment of thermodynamic equilibrium and entropy growth [10]. 12.2.1 Conserved numbers in Kaluza-Klein theory We know that according to the Noether theorem symmetries lead to the conservation of associated numbers. In particular, for (extra) spatial symmetries, the conserved numbers can be interpreted as the physical (angular) moments carried by material fields along the corresponding Killing vectors [2,6]. Spatial symmetry (extra spatial in our case) usually characterized by Killing vector field £,a (x). It means that Lie derivative of the extra space metric along the Killing vector field L^gd,mn = 0 and the metric stays invariant under the small shifts xm —» xm + £,m(x). From to the Noether's theorem (see technical details in [2]) we get a conserved current associated with the invariance 9aJa = 0. This current for any material field x is Ja = ^^^dbx — ^aLm(x), (12.1) 3(3ax) where Lm is a matter Lagrangian. The associated conserved number Q = rWv^gd d3*dv (12.2) we can interpret as some component of (angular) momentum. Until the extra metric reaches a symmetrical final configuration, this number will not be conserved (Q = Q (t)). The number will accumulate over time, until the relaxation processes stop. We need to simulate the extra metric and scalar field evolution to the final stable state in order to calculate the value of the accumulated number. 12 Cosmological Accumulation of Conserved Numbers. 143 12.2.2 Gravitational dynamics of compact space Consider as a final result of the stabilization a compact 2-dim apple-like extra space. This configuration is stationary as was shown in the works [3,14]. It has rotational symmetry which we interpret (in 4-dim limit) as U(1) global symmetry with the associated conserved number. In contrast to the one-dimensional circular extra space (which have zero Ricci scalar [15]) our configuration can lose the symmetry in early high-energetic Universe due to the metric perturbations. To stabilize the considered extra space, the modified f(R)-gravity is used. First, the higher-dimensional action is taken in the form S — mD S — 2 dDZvTG| [f(R) + Lm] , f(R) — aR2 + R + c. (12.3) Here D = d + 4, mP is fundamental D-dimensional Planck mass and Lm is a matter Lagrangian. A conserved number is accumulated in material fields during the stabilization of extra space. We will consider the simplest case of matter — massive scalar field: Lm = 1gmn3mx3nX - V(x), V(x) = ^m2X2 . (12.4) Consider a D = d + 4-dimensional manifold with metric ds2 = GMNdZMdZN = g^(x)dx^dxv + gd,mn(x,y)dymdyn , (12.5) here the metrics g^v(x) and gd,mn(x,y) corresponds to the M4, K subspaces respectively. We will consider M4 as a common 4-dim space and K as d-dim compact extra space. The signature of D-dim metric is (+---...) and the Greek indices p, v = 0,1,2,3 refer to common 4-dim coordinates. Latin indices m, n = 4,..., d + 3 refer to the extra coordinates. We will use the following conventions for the Riemann tensor: Rpbc = dCrAB — 9BrpC + rPcrBA — rPBrAC and for the Ricci tensor RMN = R^an. We also use unit system h = c = 1. A time evolution of the metric GMN (x, y) is determined by the f(R) Einstein's equations and depends on initial conditions. The dissipation of energy into the 4-dim part of space M4 leads to the decrease of entropy in the compact part of space K, as was shown in [10]. Ar a result, a friction term appears, which stabilizes the extra metric gd,mn(x,y). In addition, the inflationary expansion strongly smooths inhomogeneity of 4-dim space: 9d, (t,yJ. (12.6) Time evolution of the extra space was discussed within the Einstein's gravity and Kaluza-Klein cosmology framework [1]. If a gravitational action has nonlinear Ricci scalar terms - f(Rj, the extra metric gd,mn have asymptotically stationary configurations [4,10]: gd,mn (t,yH gd,mn(yj. (12.7) See [5,11] for more information. 144 V.V. Nikulin For simplicity, we can assume that 4-dim space has just de-Sitter metric during inflation g^ = diag(1, -e2Ht, -e2Ht, -e2Ht), (12.8) where H is inflationary Hubble parameter. The dynamics of inflaton field is not considered here. To find the stationary configurations of extra space we will use the f(R) Einstein equations: rmNf' — if(R)gd,MN + VmVnf' — gd,MN^f' — —D1—2 tmN. (12.9) Here □ is the d'Alembertian 1 □ —9m(GMNV^G|3N) . (12.10) And the contribution of matter is determined by stress-energy tensor TmN : dL tmn — —2ggmn + GMNLm . (12.11) We assume that postulated 4-dim part of metric (12.8) satisfies the higher-dimensional Einstein equations. Next, we will assume that scalar field only depends only on the extra coordinates. It is a result of smoothing out the inhomo-geneities of the 3-dim space during inflation. Equation of motion for scalar field x(x,y) = x(y) is □dX = -V'(x), (12.12) where is extra dimensional part of d'Alembertian. The very end of the process of forming a compact extra space can be considered as the relaxation of small perturbations of the metric over a stable symmetric vacuum configuration. 12.3 Numerical simulation 12.3.1 Vacuum stationary configuration As a compact extra space (12.6), we take a 2-dimensional sphere-like manifold with the metric _( -rV^,0,^ 0 \ g2,mn o -rVe^) sin2 ej. (12.13) where r is characteristic radius of the compact space and the |3(t, e,$) is the parameterization function for extra geometry. To begin with, we will find a vacuum stationary symmetric configuration, which will be the final stage in the evolution of extra space |(t, e, — |st(e) and for the scalar field x(t, e, — xst(e). The extra metric has rotational U(1) symmetry associated the presence of Killing vector. The Killing vector field is 12 Cosmological Accumulation of Conserved Numbers. 145 directed along the polar coordinate The Noether number associated with this U(1) symmetry can be interpreted as the internal polar angular momentum. A similar configuration is used for example in [3]. Fig. 12.1: A typical result of modeling a stationary configuration satisfying the f(R) Einstein equations (12.9). On the left: plot of the geometry parameterization function pst, the scalar curvature Rst and the material scalar field Xst on the azimuthal angle 0 of compact space. On the right: visualisation of the final "apple-shape" stationary configuration of compact 2-dim manifold with metric (12.13). 12.3.2 Symmetrization process Further, to consider the final stage of the relaxation process, we will simulate small perturbations of the metric parameter, scalar curvature, and material scalar field over the stable symmetric state calculated in the last paragraph: P(t,0,$) = Pst(0) + Sp(t,0,$), Sp(t,0,$) < Pst(0), R(t,0,$) = Rst(0)+ SR(t,0,$), SR(t,0,$) < Rst(0), (12.14) X(t, 0, = Xst(0) + 6x(t, 0, , 6x(t, 0, < Xst(0). By linearizing the Einstein's equations (12.9), and solving it [12] for natural random initial conditions, we obtain damped oscillations, which are shown in Fig.12.2. The dumping occurs for all angles 0 which shows the stability of the resulting configuration. This is due to the friction term commonly generated in the de Sitter space. The latter leads to the final stabilization to the U(1) symmetric extra space configuration. 12.3.3 Initial accumulation of U(1) number After the end of the relaxation processes shown in the previous subsection, a symmetric U(1) configuration is achieved. The U(1)-number associated with the 146 V.V. Nikulin Fig. 12.2: A typical evolution of perturbations Sp, SR, Sx over the stable solution calculated in previous paragraph. As an example, the behavior of the polar mode n = 2 is shown (standing wave along ty coordinate). Oscillations are taken at a point 0 = n/4, at other points damping behaves similarly. Noether theorem (12.2) will now be conserved. But in this section we are interested in how this number Q could have accumulated initially, until the end of the relaxation and symmetrization processes. The perturbed solutions simulated earlier allow us to compute Q (t) number. In the accompanying volume we get (from (12.2),(12.5),(12.14)): Q(t) = 3°x3^xr2e2p sin 0 d0dty = (12.15) 30Sx(t, 0, ty)Mx(t, 0, ty) r2e^pst (e)+6p(t'e'*0 sin 0 d0dty . The end of inflationary process have very rapid transition to the reheating stage via the violation of the slow-roll conditions. Due to this the extra metric is quickly symmetrized (for H < 1/r extra space perturbations are rapidly suppressed), while the scalar field go into the oscillating mode. After the inflation, stationary extra metric p (t, 0, ty) = p st (0) give us the equation of motion for matter (12.12) with nonperturbed symmetrical d'Alambertian. As a result, Noether's theorem starts to be fulfilled and Q ceases to depend on time. Traveling waves of the scalar field, carrying an internal angular momentum is now permanently enclosed inside extra space, since the number Q is now conserved. The initially accumulated Q(t) will now remain unchanged. The Universe enters the hot stage with a nonzero initial value of U(1) global conserved number. 12 Cosmological Accumulation of Conserved Numbers... 147 0.000 -0.002 -0,004 Q(t) -0.006 -0.008 5 10 15 20 25 30 Fig. 12.3: Typical time evolution of the U(1) number Q(t) during the symmetriza-tion of compact extra space. The number calculated numerically from (12.15). 12.4 Conclusion In this research we show how the dynamics of compact extra space leads to a nonzero initial accumulation of some conserved number. Such gravitational dynamics of compact extra metric should naturally occur in the early (H ^ 1/r) higher-dimensional Universe. The stabilization of the extra metric lead to a symmetrical stationary final configuration. We considered the case of a final U(1) rotationally symmetric state with corresponding conserved number. Such an accumulation mechanism arising in Kaluza-Klein theories can be used to explain the origin of the cosmological baryon asymmetry [8,9]. It is known that the baryon number is described by the global U(1 )-symmetry. In Kaluza-Klein theories it could be realized as the rotational symmetry of the 2-dim compact extra space (12.13). However, to transfer the baryon number, additional interaction term between the fermion and the scalar field is required (for details, see work [7]). In future works, we plan to develop a Kaluza-Klein mechanism for transferring asymmetry into the fermions in order to explain specifically the cosmological baryon/lepton asymmetry. Acknowledgements The work was supported by the Ministry of Science and Higher Education of the Russian Federation, Project "Fundamental properties of elementary particles and cosmology" No 0723-2020-0041. References 1. R. B. Abbott, S. M. Barr, and S. D. Ellis. Kaluza-Klein Cosmologies and Inflation. Phys. Rev. D, D30:720,1984. 148 V.V. Nikulin 2. M. Blagojevic, F. W. Hehl, and eds. Gravitation and Gauge Symmetries. Institute of Physics Publishing, Bristol, 2002. 3. K. A. Bronnikov, R. I. Budaev, A. V. Grobov, A. E. Dmitriev, and S. G. Rubin. Inhomoge-neous compact extra dimensions. Journal of Cosmology and Astroparticle Physics, 10:001, 2017. 4. K. A. Bronnikov and S. G. Rubin. Self-stabilization of extra dimensions. Phys. Rev. D, 73(12):124019, 2006. 5. S. M. Carroll, J. Geddes, M. B. Hoffman, and R. M. Wald. Classical stabilization of homogeneous extra dimensions. Phys. Rev. D, 66(2):024036, 2002. 6. F. Cianfrani, A. Marrocco, and G. Montani. Gauge theories as a geometrical issue of a kaluza-klein framework. International Journal of Modern Physics D, 14(07):1195-1231, 2005. 7. A. Dolgov and J. Silk. Baryon isocurvature fluctuations at small scales and baryonic dark matter. Phys. Rev. D, 47:4244-4255,1993. 8. A. D. Dolgov, M. Kawasaki, and N. Kevlishvili. Inhomogeneous baryogenesis, cosmic antimatter, and dark matter. Nucl. Phys. B, 807:229-250, 2009. 9. M. Y. Khlopov. Composite Dark Matter from 4-th Generation. Pis'ma Zh. Ehksp. Teor. Fiz., 83:3-6, 2006. 10. A. A. Kirillov, A. A. Korotkevich, and S. G. Rubin. Emergence of symmetries. Phys. Lett., B718:237-240, 2012. 11. S. Nasri, P. J. Silva, G. D. Starkman, and M. Trodden. Radion stabilization in compact hyperbolic extra dimensions. Phys. Rev. D, 66(4):045029, 2002. 12. V. V. Nikulin, P. M. Petriakova, and S. G. Rubin. Formation of conserved charge at the de sitter space. Particles, 3(2):355-363, 2020. 13. V. V. Nikulin and S. G. Rubin. Inflationary limits on the size of compact extra space. International Journal ofModern Physics D, 28(13):1941004, 2019. 14. S. G. Rubin. Cosmology and matter-induced branes. Symmetry, 12(1):45, 2020. 15. U. Sarkar. Particle and Astroparticle physics. CRC Press, 2007. Bled Workshops in Physics Vol. 21, No. 2 A Proceedings to the 23rd [Virtual] Workshop, Volume 2 What Comes Beyond ... (p. 149) Bled, Slovenia, July 4-12, 2020 13 Sub-Planckian Evolution of the Universe P.M. Petriakova * National Research Nuclear University MEPhI, 115409, Kashirskoe shosse 31, Moscow, Russia Abstract. The dynamics of a space endowed by a metric of a 3-dimensional sphere in the framework of f (R)-gravity acting in D = 4 from the creation at high energies is studied. Povzetek. Avtorica obravnava, v okviru f(R) gravitacije v D = 4, dinamiko prostora z metriko 3-razsezne sfere pri visokih energijah. Keywords: three-dimensional sphere, Starobinsky model, f(R)-gravity, modified gravity. 13.1 Introduction Despite the fact that we live in the era of observational cosmology and have experimental data with very good accuracy, there are a huge number of models that can satisfy the modern data, but use completely different approaches and ideologies. This is especially relevant to issues of the very early Universe: from creation to the end of inflationary stage. The inflationary scenario firstly was detailed by Starobinsky [1], Guth [2] and after by Linde [3] and Albrecht with Steinhardt [4]. The Starobinsky model is based on gravity with added the quadratic term of scalar curvature and it is interesting that the use of such a term may be motivated by conformal anomaly considerations. In this theory the Friedmann equation is modified for large values of the Hubble parameter which leads to a cosmological solution with a scale factor growing exponentially during a certain period of evolution. This model also has a post inflationary heating up mechanism. As result of the evolution the Universe enters a hot stage. The modified gravity description of cosmological evolution of our Universe is one physically appealing theoretical framework. It can explain the various evolution eras due to providing a unified and theoretically consistent description. There exist a huge number of modified gravity models, see for example reviews [5] and [6], that can potentially describe evolution of our Universe. The most important criteria for the viability of a modified gravity theory is the compatibility of the theory with modern observations. The simplest, but being particularly favoured by present observations [7], model of the modified * E-mail: petriakovapolina@gmail.com 150 P.M. Petriakova theory of gravity is the Starobinsky model. This model is quite successful. However, it describes evolution starting from a certain energy scale. In this paper it is proposed to study the dynamics of a three-dimensional sphere at energy scales exceeding the inflationary using f(R)-gravity with R2-term. We should note that many modified theories based on the consideration of a purely gravitational Lagrangian also fit into the available observable data. The tensor-to-scalar ratio in the Planck compatible region and the role of higher order curvature term for stability and the reheating dynamics for the unambiguous prediction for the number of e-foldings up to the R3-term are discussed in [8]. Satisfying observable data, as works [9], [10] and [11] demonstrate, possible using a completely different approach: to study pure multidimensional gravity with higher derivatives. An issues of inflation model in the case of supergravity can also be found in the following works [12], [13] and [14]. 13.2 Basic equations and initial conditions Let us consider the theory described by the action 1 i S[g^] = 2mpi d4x^|detg^|f(R). (13.1) The corresponding equations of motion are as follows fR(R)R^ - 1f(R)g^v + [v,Vv - (R) — 0, □ = g^V^Vv. (13.2) Throughout this paper we use the conventions for the curvature tensor RMNK — 9k rMN — dNrMK + Mm — r/LNrAK and the Rkri tensor is defined as rmn — rk RMKN . Taking into account the choice of the metric of a sphere ds2 — dt2 - e2a(t) (dx2 + sin2 x dy2 + sin2 x sin2 y dz2) (13.3) we obtain the system of equations 6a(t)R(t)fR(R) - 6(a(t) + d2(t)jfR(R) + f(R) = 0, 2R 2(t)fR'(R) + 2(R (t) + 2a (t)R (t)) fR(R)- (13.4) -(2a (t) + 6a 2(t) + 4e-2a(t)) fR (R) + f(R) = 0 where the definition of the Ricci scalar for the used metric (13.3) is R(t) = 12a2 (t) + 6a (t) + 6e-2a(t). (13.5) With the choice of the type of the f(R)-function as f(R) = aR2 + bR + c (13.6) 14 The "Dark Disk" Model in the Light of DAMPE Experiment 151 the definition of the Ricci scalar (13.5) and the second equation of the system (13.4) with assumption (13.6) give us / 1 a(t) =- 2 a2(t) — e-2a(t) + - R(t), R(t) =-2 a(t)R(t)- ^R2(t)+ (a2(t) + e-2a(t) - bW) + (13.7) 12 V 6a/ +2a«2(t)+2ae-2"-t' - ¿a- After replacing a(t) from (13.5) in the first equation (13.4) we get R2(t) - 12 (a2 (t) + e-2a(t))R(t) - 12a(t)R(t)--(d2(t) + e-2a(t)) - C = 0. a a (13.8) This expression for R2(t) and the first equation of the system (13.7) allow us to obtain the equation of damped harmonic oscillations b c R (t)+ 3a (t)R (t) + —R(t) + — = 0. (13.9) 6a 3a As well the same result follows from taking the trace of the system (13.2). The different modes of the solution of this equation are possible for a certain ratio between the values 1.5 a2(t) and b/6a. The solution is damped oscillations under the condition 1.5 a2 (t) < b/6a and when this regime occurs the scalar curvature oscillations begin. Otherwise, the solution is aperiodic. The arising curvature oscillations lead to a slowdown in the growth of the value of the function a(t), i.e. the size of a sphere. We have a system of second-order equations (13.7) in the chosen theory. It is necessary to determine the initial conditions for the values of the unknown functions a(t) and R(t) and their derivatives. Let it be given by constant a(0) = a0 , a(0) = a1 , R(0) = R0 , R(0) = R1 . (13.10) After solving the previously obtained equation (13.8) we find an expression for the function R(t) at the initial time depending on the value of other initial conditions and the parameters of the chosen f(R)-function: R0 = 6(a2 + e-2a0) ± y 36(a2 + e-2a°)2 + ^12a1 R1 + ^ K + e-2ao) + C ^. (13.11) We are interested in the dynamics of a sphere starting from the sub-Planck scale. Therefore, the initial conditions on the function a(t) will be near the value of the Hubble parameter at this moment a0 ~ ln Hsv[b_planck , a1 ~ HSub-Planck , HSub-Planck ~ 0.1. (13.12) Let us discuss the influence of the parameters of the f(R)-function. The value of the coefficient a allows us to adjust the moment of onset of the scalar curvature 152 P.M. Petriakova oscillations due to mentioned 1.5 a2 (t) < b/6a relation. Thus, if we want to significantly increase the size of a sphere we should have a > 2b 27a2 10 (13.13) b = 1 assuming the value of the coefficient b = 1 without loss of generality and defining the value of a as (13.12). The last coefficient c in (13.6) remained undefined and we will look at the asymptotic behavior to restrict it. On the asymptotics the curvature scalar should tend to a constant R —» const = Rc and following the definition (13.5) to the standard relation Rc = 12Hpresent due to modern acceleration of the Universe. Then the equations of motion (13.2) at the constant scalar curvature after taking the trace are reduced to the algebraic equation fR(RC)RC - 2f(Rc) = 0. (13.14) Solving this equation (13.14), we obtain the value of the scalar curvature on the asymptotics Rc = -2c. Therefore, we immediately come to the conditions for the value of the coefficient: c - HpTesent and c < 0. The size of a sphere on the asymptotics will be determined from (13.5) as . e2a(t) _ 6 / | R c| /|R 1 + Ci e » 3 + C2 e-t V 3 Rc _ ra 1 + C2 cos( tJ ^ + C2 sin t\ J-R^ Due to the smallness of the value HpTesent - 10-61 at the present epoch and as consequence the last term in (13.6) we almost obtain the Starobinsky inflation model. Of course, the values of the coefficients can differ significantly at high energies, but we do not know the exact dependence of their values on the energy scale. We are going to study the dynamics of space with a fixed set of parameter values to start calculations from the sub-Planck energy scale for Starobinsky model in next section. R C 13.3 Results Firstly, we will discuss the exact solution of the Starobinsky model. The only parameter of this model a _ 1/6m2 is defined as m/mPl - 10-5 [15] and initial conditions (13.10) in this case are determined by the value of the inflationary Hubble parameter HInfl - 10-6 as ao - ln Hrf , ai - H^n , Ri _ 0 R+ (0) _ 2.9 • 10-11. (13.17) The numerical solution of the system (13.7) with given initial conditions (13.17) leads to the correctly dynamics at the inflationary and post-inflationary stages and 13 Sub-Planckian Evolution of the Universe 153 presented on the Fig.13.1. We see that curvature oscillations lead to a slowdown in the growth of a(t). All relevant quantities, such as: the size of a sphere a(t), the duration of the inflationary stage t ~ 10..107 (the beginning of the curvature R(t) oscillations) the amplitude of this oscillations, the value of the Hubble parameter i.e. a(t) are in accordance with the predictions of inflation theory and experimental data. Fig. 13.1: Solution the system (13.7) with the initial conditions (13.17). Let's continue the construction of the numerical solution up to age of the Universe tuniv ~ 1061. Then we get R (tuniv) ~ 10-122, a(tuniv) ~ 10-61 and ratio of sizes 1Starobinsky e_ 5 "wbT"~ 10 • (13.18) A fundamental observational result of recent years is the fact that the spatial curvature of our Universe is very small. The main source of this fact is the study of the temperature anisotropy of the Cosmic Microwave Background (CMB). It means, at the qualitative level, that the radius of spatial curvature is much greater than the size of the observable part of the Universe, i.e. much greater than H-r1esent. Following the fact that a topology significantly differently than Euclidean is not observed [17] obtained value (13.18) can be insufficient. Let us solve numerically the system of equations (13.7) starting from subPlanck energies with initial conditions (13.11) and (13.12). The result for the case of a three-dimensional sphere is shown on the left side of Fig.13.2. We see that the principal difference with Starobinsky model is the size of the sphere formed by the end of the inflationary stage. Since we significantly (by 5 orders) change the energy scale in the initial conditions Hsub_Planck ~ 105 HInfl, we should check how an insignificant change in the only parameter of the model m will affect to the solution. We get a strong dependence on it, and it is shown on the right side of Fig.13.2. After continuing the construction of the numerical solution up to age of the Universe we get again on the asymptotic behavior correct 154 P.M. Petriakova Fig. 13.2: Left side: solution the (13.7) system with the initial conditions (13.11), (13.12) and a = 1/6m2, m/mPl ~ 10-5, c = —10-122; Right side: dependence on the coefficient a with similar (13.11), (13.12) conditions. values R(tUniv) ~ 10 122, x2 — d-1 L DAMPE (AOeJ2 + L (A®yj2 H (A®yJ Fermi (14.4) Here A®t = ®(th) — ®jobs), are the predicted (th) and measured (obs) fluxes for i = e, y denoting e+e- or gamma points respectively, ot denotes the corresponding experimental errors and d denotes the number of statistical degrees of freedom, which includes all the relevant DAMPE and Fermi-LAT data points. The first sum in Eq. (14.4) goes over the DAMPE data points and the second sum goes over the Fermi-LAT data points. DAMPE points are taken in the range 20 = 1600 GeV. Since we do not try to fit the gamma-ray data, but rather not to go over the experimental limits, the terms in the second sum are non-zero only when > ®y°bs), which is ensured by the Heaviside step function H. We use two different approaches for the minimization procedure. In first, called "combined fit", we just simply minimize expression 14.4. In the second, called "e-fit", we minimize only the first sum in the expression 14.4 and only after that, using the obtained parameters, we calculate total chi-square value. x 0 (j u 14.3 Results Fig. 14.1 illustrates the correlation between x2 values and the disk half-width. In the case of "e-fit" the best results are obtained with zc « 750 pc. However, one can clearly see that the quality of fit is still not satisfactory at all, although still better than one for the thick disks and halo. On the other hand, "combined fit" gives much better results with the minimum of x2 of around 1.6 for the disk half-width in the range of 1500 = 2000 pc. However, in the case of AMS-02 positron fraction best fits were obtained with zc = 400 pc. Unexpectedly, the NFW density profile with cut-off produces better results, than Read's profile, over the whole considered region. We suppose it to be due to higher production of low-energy electrons and positrons for NFW, which helps it to account for the lower energy 14 The "Dark Disk" Model in the Light of DAMPE Experiment 159 region of the spectra. The line in the graphics breaks are mainly caused by the change of degree-of-freedom number (as we dynamically calculate it to include only those Fermi-LAT datapoints, where we have the excess) and interpolation errors. x2 e-fit y2 combined fit 500 1000 1500 2000 2500 3000 500 1000 1500 2000 2500 3000 zc, pc zc, pc (a) (b) Fig. 14.1: Graphs for x2 values in dependence of the disk half-width in case of e-fit 14.1a and combined fit 14.1b. Blue line is used for NFW density profile, the orange one - for Read's density profile. Fit — Model e-fit combined fit Halo 203 (0.53) 3.8 (2.1) Disk 17.85 (0.52) 1.48 (1.20) Table 14.1: The best-fit values of x2 for different DM models and approaches for the minimization procedure. The values in brackets are obtained using only electron-positron part of Eq. (14.4). Table 14.1 contains the best-fit values of chi-square in contrast to the ones, obtained for the halo case. The comparison revealed that the dark disk model allows achieving the same accuracy in positron description, as the halo model, while giving less contradiction with IGRB. In both cases, combined fit improves the fit quality, but still not enough to overcome the discrepancy. 14.4 Conclusion We continue our research of DM explanation of the CR puzzles. In this work, we have applied the "dark disk" model to the case of the wide excess of positrons plus electrons in DAMPE data. We have obtained that it helps to lessen the contradiction with cosmic gamm-ray data. However, it is achieved at the cost of thicker disk, compared to the case of low energy positron anomaly of AMS-02. 160 M.L. Solovyov, M.A. Rakhimova and K.M. Belotsky In our future works we plan to run such analysis for the different masses of initial particle, try different reaction modes and to attempt to describe AMS-02 and DAMPE data simultaneously. Acknowledgments The work was supported by the Ministry of Science and Higher Education of the Russian Federation by project No 0723-2020-0040 "Fundamental problems of cosmic rays and dark matter". Also we would like to thank R.Budaev, A.Kirillov and M.Laletin for their contribution at the early stage of this work. References 1. PAMELA collaboration, O. Adriani et al., An anomalous positron abundance in cosmic rays with energies 1.5-100 GeV, Nature 458 (2009) 607-609, [0810.4995]. 2. AMS Collaboration collaboration, M. Aguilar, G. Alberti, B. Alpat, A. Alvino, G. Ambrosi, K. Andeen et al., First result from the alpha magnetic spectrometer on the international space station: Precision measurement of the positron fraction in primary cosmic rays of 0.5-350 gev, Phys. Rev. Lett. 110 (Apr, 2013) 141102. 3. AMS Collaboration collaboration, L. Accardo, M. Aguilar, D. Aisa, B. Alpat, A. Alvino, G. Ambrosi et al., High statistics measurement of the positron fraction in primary cosmic rays of 0.5-500 gev with the alpha magnetic spectrometer on the international space station, Phys. Rev. Lett. 113 (Sep, 2014) 121101. 4. J. Cao, X. Guo, L. Shang, F. Wang, P. Wu and L. Zu, Scalar dark matter explanation of the DAMPE data in the minimal Left-Right symmetric model, Phys. Rev. D97 (2018) 063016, [1712.05351]. 5. H.-B. Jin, B. Yue, X. Zhang and X. Chen, Cosmic ray e+ e- spectrum excess and peak feature observed by the DAMPE experiment from dark matter, 1712.00362. 6. A. U. Abeysekara, A. Albert, R. Alfaro, C. Alvarez, J. D. Alvarez, R. 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(p. 162) Bled, Slovenia, July 4-12, 2020 15 Dynamical Evolution of a Cluster of Primordial Black Holes V.D. Stasenko * and A. A. Kirillov ** National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) Abstract. Evolution of a cluster of primordial black holes in the two-body relaxation approximation based on the Fokker-Planck equation is discussed. In our calculation, we consider the self-gravitating cluster with a wide range of black holes masses from 1O-4M0 up to 100M© and the total mass 1O5M0. Moreover, we included a massive black hole in the cluster center which determines the evolution rate of the density profile in its vicinity. Povzetek. Avtorja obravnavata razvoj kopice primordialnih crnih lukenj. Uporabita Fokker-Planckove enačbe v približku dvodelcne relaksacije. Obravnavata kopico crnih lukenj z masami od 1O-4M0 do 100Mo in s skupno maso 1O5M0, ki ji dinamiko doloca lastno gravitacijsko polje. V sredisce kopice postavita masivno crno luknjo, ki doloca casovno spremembo profila gostote v njeni okolici. Keywords: primordial black holes, clusters of primordial black holes, the Fokker- Planck equation PACS: 04.25.dg, 05.10.Gg 15.1 Introduction The hypothesis of primordial black holes (PBHs) formation was suggested in [22]. Afterward, a few scenarios of PBHs production have been developed (see reviews [7,13]). In our work, we consider those predicting the formation of PBHs as clusters. This mechanism was proposed in [14,17,18] where a collapse of large closed domain walls was discussed. The produced clusters may have extended mass spectra where masses range from ~ 1017 g [2,3] up to ~ 104MQ [11] or even more [10]. These clusters have essential consequences for shedding light on some cosmological problems. Observational manifestations of the model and smoothing of some constraints (the recent restrictions on PBHs are considered in [7]) are widely discussed in reviews [4, 5] and references within. However, finding of clusters evidences is significantly related to the mass spectrum at a specified moment of the Universe history. Therefore, understanding of cluster dynamic play an essential role and is a main research subject of this paper. * E-mail: StasenkoVD@gmail.com ** E-mail: AAKirillov@mephi.ru 15 Dynamical Evolution of a Cluster of Primordial Black Holes 163 Till now, a comprehensive study of clusters evolution has not been carried out. First efforts to retrace changes of clusters mass spectra were made in [5,9,15,20]. The closest physical model to a PBH cluster is a globular cluster of stars. However, it does not have a wide mass range. Therefore, globular cluster theory could not be directly extrapolated to the PBHs cluster case. Moreover, the PBHs cluster may contain a massive central black hole (CBH). In work [1], stationary distribution of stars around a massive black hole was discussed. It was established that the density obeys the law p