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Axion field and the quark nugget's formation at the QCD phase transition Liang, Xunyu 2016

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Axion Field and the Quark Nugget’sFormation at the QCD PhaseTransitionbyXunyu LiangB.Sc., University of Victoria, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2016c© Xunyu Liang 2016AbstractWe study a testable dark matter (DM) model outside of the standard WIMPparadigm in which the observed ratio Ωdark ' Ωvisible for visible and darkmatter densities finds its natural explanation as a result of their commonQCD origin when both types of matter (DM and visible) are formed atthe QCD phase transition and both are proportional to ΛQCD. Instead ofthe conventional “baryogenesis” mechanism we advocate a paradigm whenthe “baryogenesis” is actually a charge separation process which occur inthe presence of the CP odd axion field a(x). In this scenario the globalbaryon number of the Universe remains zero, while the unobserved an-tibaryon charge is hidden in form of heavy nuggets, similar to Witten’sstrangelets and compromise the DM of the Universe.In the present work we study in great detail a possible formation mecha-nism of such macroscopically large heavy objects. We argue that the nuggetswill be inevitably produced during the QCD phase transition as a result ofKibble-Zurek mechanism on formation of the topological defects during aphase transition. Relevant topological defects in our scenario are the closedbubbles made of the NDW = 1 axion domain walls. These bubbles, in gen-eral, accrete the baryon (or antibaryon) charge, which eventually result information of the nuggets and anti-nuggets carrying a huge baryon (anti-baryon) charge. A typical size and the baryon charge of these macroscopi-cally large objects is mainly determined by the axion mass ma. However, themain consequence of the model, Ωdark ≈ Ωvisible is insensitive to the axionmass which may assume any value within the observationally allowed win-dow 10−6eV . ma . 10−3eV. We also estimate the baryon to entropy ratioη ≡ nB/nγ ∼ 10−10 within this scenario. Finally, we comment on implica-tions of these results to the axion search experiments, including microwavecavity and the Orpheus experiments.iiPrefaceThe results of this thesis has been published on arXiv as Xunyu Liang andAriel Zhitnitsky, arXiv:1606.00435 [hep-ph].Section 2.4 and appendices were my work, but also being improved byAriel Zhitnitsky, especially Appendix A, into a more logical framework. Ad-ditionally, the inclusion of the viscosity term was suggested by me. Therest of new results, chapter 2, 3, and 5, presented in this thesis was doneand written by Ariel Zhitnitsky. Also, the remaining review chapters werewritten by Ariel Zhitnitsky.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Two sides of the same coin . . . . . . . . . . . . . . . . . . . 11.2 Quark (anti) nugget DM confronting the observations . . . . 92 Formation of the Nuggets . . . . . . . . . . . . . . . . . . . . 142.1 The crucial ingredients of the proposal . . . . . . . . . . . . 142.2 Accretion of the baryon charge . . . . . . . . . . . . . . . . . 182.3 Radius versus pressure in time evolution . . . . . . . . . . . 252.4 Qualitative analysis . . . . . . . . . . . . . . . . . . . . . . . 312.4.1 Assumptions, approximations, simplifications . . . . 312.4.2 Time evolution . . . . . . . . . . . . . . . . . . . . . . 353 Baryon Charge Separation. Correlation on CosmologicalScales. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.1 Coherent axion field as the source of CP violation . . . . . . 413.2 Nuggets vs anti-nuggets on the large scale. Generic conse-quences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3 nB/nγ ratio. Model dependent estimates. . . . . . . . . . . 494 Implications for the Axion Search Experiments . . . . . . . 55ivTable of Contents5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62AppendicesA Estimation of Fluxes . . . . . . . . . . . . . . . . . . . . . . . . 68B Formation of the Nuggets: Numerical Analysis . . . . . . . 70C Evaluation of Fermi-Dirac Integrals . . . . . . . . . . . . . . 75vList of TablesB.1 Table for some numerical parameters . . . . . . . . . . . . . . 74viList of Figures1.1 The conjectured phase diagram . . . . . . . . . . . . . . . . . 61.2 Limits on quark nugget mass and fluxes based on current (andfuture) constraints . . . . . . . . . . . . . . . . . . . . . . . . 124.1 Cavity / ADMX experimental constraints on the axion mass 57B.1 Typical underdamped solution of R(t) and µ(t) . . . . . . . . 73B.2 The first few oscillations of an underdamped solution . . . . . 73B.3 Dependence on parameter r0 as defined by eq. (B.1) . . . . . 74C.1 Comparison of I(0)n to In with different values of b. . . . . . . 76viiAcknowledgementsI would like to thank my supervisor, Professor Ariel Zhitnitsky. Startingwith little experience, he guided me into research, and finally allowed thisthesis to be my own work, despite the fact the most difficult problems weresolved by him. Also, he is always pleasant to give support when I hadproblems in physics, conference, and research. All of his effort and patienceare sincerely appreciated.I also want to thank Kyle Lawson, an intelligent postdoctoral fellow inour research group, who is willing to spend a great amount of time to reviewmy thesis and give so many careful and insightful comments.viiiChapter 1IntroductionVery recently, at the time of the thesis is being written, the world’s mostsensitive dark matter (DM) detector – LUX (the Large Underground XenonDM experiment) completes its nearly 2-year search. However, similar toother long-term searches, it hardly provides evidence [1] to support the ex-istence of WIMPs (Weakly Interacting Massive Particles), a prevailing DMmodel. The lack of signals may hint an alternative DM model could befavorable.As one of the motivations, this thesis presents a simple composite object,namely quark nugget, that can be served as an alternative DM candidate.Even better, these little compact nuggets may solve another unsolved fun-damental problem – the baryon asymmetry.1.1 Two sides of the same coinThe origin of the observed asymmetry between matter and antimatter is oneof the largest open questions in cosmology. The nature of the dark matter isanother open question in cosmology. In this thesis we advocate an idea thatthese two, apparently unrelated, problems are in fact two sides of the samecoin. Furthermore, both mysterious effects are originated at one and thesame cosmological epoch from one and the same QCD physics. Normally,it is assumed that the majority of dark matter is represented by a newfundamental field coupled only weakly to the standard model particles, thesemodels may then be tuned to match the observed dark matter properties.We take a different perspective and consider the possibility that the darkmatter is in fact composed of well known quarks and antiquarks but in a newhigh density phase, similar to the Witten’s strangelets, see original work [2]and some related studies [3].There are few new crucial elements in proposal [4, 5], in comparisonwith previous studies [2, 3]. First of all, the nuggets could be made ofmatter as well as antimatter in our framework as a result of separation ofcharges, see few comments below. Secondly, the stability of the DM nuggetsis provided by the axion domain walls with extra pressure, in contrast with11.1. Two sides of the same coinoriginal studies when stability is assumed to be achieved even in vacuum, atzero external pressure. Finally, an overall coherent baryon asymmetry in theentire Universe is a result of the strong CP violation due to the fundamentalθ parameter in QCD which is assumed to be nonzero at the beginning ofthe QCD phase transition. This source of strong CP violation is no longeravailable at the present epoch as a result of the axion dynamics, see originalpapers [6–8] and recent reviews [9–16] on the subject. We highlight the basicsideas of this framework in the present Introduction, while we elaborate onthese new crucial elements in details in section 2.1.It is generally assumed that the universe began in a symmetric statewith zero global baryonic charge and later (through some baryon numberviolating process) evolved into a state with a net positive baryon number. Asan alternative to this scenario we advocate a model in which “baryogenesis”is actually a charge separation process in which the global baryon numberof the universe remains zero. In this model the unobserved antibaryonscome to comprise the dark matter. A connection between dark matter andbaryogenesis is made particularly compelling by the similar energy densitiesof the visible and dark matter with Ωdark ' 5 ·Ωvisible. If these processes arenot fundamentally related the two components could exist at vastly differentscales.In the model [4, 5] baryogenesis occurs at the QCD phase transition.Both quarks and antiquarks are thermally abundant in the primordial plasmabut, in addition to forming conventional baryons, some fraction of them arebound into heavy nuggets of quark matter in a colour superconducting phase(an analogous phase to superconductors in condensed matter). Nuggets ofboth matter and antimatter are formed as a result of the dynamics of theaxion domain walls as originally proposed in refs.[4, 5]. A number of veryhard dynamical questions in strongly coupled QCD which are related to thenuggets’s formation have not been studied in any details in the original pa-pers. The main goal of the present work is to make the first step in thedirection to address these hard questions.If the fundamental θ parameter were identically zero at the QCD phasetransition in early universe, an equal number of nuggets made of matterand antimatter would be formed. It would result in vanishing of the visiblebaryon density at the present epoch. However, the fundamental CP violat-ing processes associated with the θ term in QCD (which is assumed to besmall, but still non-zero at the very beginning of the QCD phase transition)results in the preferential formation of anti-nuggets over the nuggets. Thispreference is essentially determined by the dynamics of coherent axion fieldθ(x) at the initial stage of the nugget’s formation. The resulting asymmetry21.1. Two sides of the same coinis not sensitive to a small magnitude of the axion field θ(x) at the QCDphase transition as long as it remains coherent on the scale of the Universe,see chapter 3 for the details.The remaining antibaryons in the plasma then annihilate away leavingonly the baryons whose antimatter counterparts are bound in the excessof anti-nuggets and thus unavailable to annihilate. All asymmetry effectsare order of one, irrespectively to the magnitude of θ(x) at the moment offormation. This is precisely the main reason of why the visible and darkmatter densities must be the same order of magnitudeΩdark ≈ Ωvisible (1.1)as they both proportional to the same fundamental ΛQCD scale, and theyboth are originated at the same QCD epoch. In particular, if one assumesthat the nuggets and anti-nuggets saturate the dark matter density thanthe observed matter to dark matter ratio Ωdark ' 5 · Ωvisible correspondsto a specific proportion when number of anti-nuggets is larger than numberof nuggets by a factor of ∼ 3/2 at the end of nugget’s formation. Thiswould result in a matter content with baryons, quark nuggets and antiquarknuggets in an approximate ratio|Bvisible| : |Bnuggets| : |Bantinuggets| ' 1 : 2 : 3, (1.2)with no net baryonic charge. If these processes are not fundamentally relatedthe two components Ωdark and Ωvisible could easily exist at vastly differentscales.Though the QCD phase diagram at θ 6= 0 as a function of T and µis basically unknown, it is well understood that θ is in fact the angularvariable, and therefore supports various types of the domain walls, includingthe so-called NDW = 1 domain walls1 when θ interpolates between one andthe same physical vacuum state θ → θ + 2pi. Furthermore, it is expectedthat the closed bubbles made of these NDW = 1 axion domain walls arealso produced during the QCD phase transition with a typical wall tensionσa ∼ m−1a where ma is the axion mass. Precisely this scale determines thesize and the baryon charge of the nuggets, see equations (1.3), (1.4) below.The collapse of these close bubbles is halted due to the Fermi pressureacting inside of the bubbles. The crucial element which stops the collapseof the bubbles from complete annihilation is the presence of the QCD sub-structure inside the axion domain wall. This substructure forms immediately1NDW represents the phase difference between the two side of the end of a domain wallθ → θ + 2piNDW .31.1. Two sides of the same coinafter the QCD phase transition as discussed in [4]. The equilibrium of the ob-tained nugget system has been analyzed in [4] for a specific axion domain walltension within the observationally allowed window 10−6eV ≤ ma ≤ 10−3eVconsistent with the recent constraints [9–16]. It has been also argued in [4]that the equilibrium is typically achieved when the Fermi pressure insidethe nuggets falls into the region when the colour superconductivity (CS)sets in2.The size and the baryon charge of the nuggets scale with the axion massas followsσa ∼ m−1a , R ∼ σa, B ∼ σ3a. (1.3)Therefore, when the axion mass ma varies within the observationally allowedwindow 10−6eV . ma . 10−3eV the nuggets parameters also vary as follows10−6cm . R . 10−3cm, 1023 . B . 1032, (1.4)where the lowest axion mass ma ' 10−6eV approximately3 corresponds tothe largest possible nuggets with 〈B〉 ' 1032. Variation of the axion mass bythree orders of magnitude results in variation of the nugget’s baryon chargeby nine orders of magnitude according to relation (1.3). The correspond-ing allowed region is essentially unconstrained by present experiments, seedetails in section 1.2 below.The fact that the CS may be realized in nature in the cores of neutronstars has been known for sometime [17, 18]. A new element which was advo-cated in proposal [4] is that a similar dense environment can be realized innature due to the axion domain wall pressure playing a role of a “squeezer”,similar to the gravity pressure in the neutron star physics.Another fundamental ratio (along with Ωdark ≈ Ωvisible discussed above)is the baryon to entropy ratio at present timeη ≡ nB − nB¯nγ' nBnγ∼ 10−10. (1.5)2There is no requirement for a first order phase transition (in contrast with originalproposal [2]) for the bubble formation in this framework because the NDW = 1 axiondomain walls are formed irrespectively to the order of the phase transition. Needless tosay that the phase diagram in general and the order of the phase transition in particularat θ 6= 0 are still unknown because of the longstanding “sign problem” in the QCD latticesimulations at θ 6= 0, see few comments and related references in chapter 5.3There is no one-to-one correspondence between the axion mass ma and the baryoncharge of the nuggets B because for each given ma there is an extended window of stablesolutions describing different nugget’s sizes [4].41.1. Two sides of the same coinIf the nuggets were not present after the phase transition the conventionalbaryons and antibaryons would continue to annihilate each other until thetemperature reaches T ' 22 MeV when the baryon density would be 9 ordersof magnitude smaller than observed (1.5). This annihilation catastrophe,normally thought to be resolved as a result of “baryogenesis” was formulatedby Sakharov[19], see also review [20]. In the framework of conventionalbaryogenesis, the ratio (1.5) is highly sensitive to many specific details ofthe models (for example, the general spectrum of the system, the couplingconstants, and the particular strength of CP violation), see e.g. review[20].In our proposal (in contrast with conventional frameworks on baryoge-nesis) this ratio is determined by a single parameter with a typical QCDscale, the formation temperature Tform. This temperature is defined by amoment in evolution of the Universe when the nuggets and anti-nuggetsbasically have completed their formation and not much annihilation wouldoccur at lower temperatures T ≤ Tform. The exact magnitude of tempera-ture Tform ∼ ΛQCD in our proposal is determined by many factors: trans-mission/reflection coefficients, evolution of the nuggets, expansion of theuniverse, cooling rates, evaporation rates, viscosity of the environment, thedynamics of the axion domain wall network, etc. All these effects are, ingeneral, equally contribute to Tform at the QCD scale. Technically, the cor-responding effects are hard to compute from the first principles as even basicproperties of the QCD phase diagram at nonzero θ 6= 0 are still unknown4.We plot three different conjectured cooling paths on Fig. 1.1.However, the estimate of Tform up to factor 2 is quite a simple exerciseas Tform must be proportional to the gap ∆ ∼ 100 MeV when CS phase setsin inside the nuggets. The observed ratio (1.5) corresponds to Tform ' 40MeV, see [5] for the details. This temperature indeed represents a typicalQCD scale, slightly below the critical temperature TCS ' 0.6∆ ' 60 MeV,according to standard estimates on colour superconductivity, see reviews[17, 18].4The basic consequence (1.1) as well as (1.5) of this proposal are largely insensitiveto the absolute value of the initial magnitude of the θ parameter. In other words, a finetuning of the initial θ parameter is not required in this mechanism. The same comment (on“insensitivity” of the initial conditions) also applies to efficiency of the nugget’s formation.This is because the baryon density at the present time is 10 orders of magnitude lowerthan the particle density at the QCD phase transition epoch according to the observations(1.5). Therefore, even a sufficiently low efficiency of the nugget’s formation (still largerthan 10−7, see estimates in section 3.3) cannot drastically modify the generic relations(1.1), (1.5) due to a long evolution which eventually washes out any sensitivity to theinitial conditions. The only crucial parameter which determines the final outcome (1.1),(1.5) is the formation temperature Tform as estimated below.51.1. Two sides of the same coin≈ 170MeVθTµQGPCSHadron1 23Tform ≈ 41MeVTc(Phase Unknown)Figure 1.1: The conjectured phase diagram. Possible cooling paths aredenoted as path 1, 2 or 3. The phase diagram is in fact much more compli-cated as the dependence on the third essential parameter, the θ is not shownas it is largely unknown. Therefore, the paths should be thought as linesin three dimensional parametrical space, not as lines on two-dimensional(µ, T ) slice at θ = 0 as shown on the present plot. It is assumed that thefinal destination after the nuggets are formed is the region with Tform ≈ 41MeV, µ > µc and θ ≈ 0, corresponding to the presently observed ratio (1.5),see text for the details.61.1. Two sides of the same coinUnlike conventional dark matter candidates, such as WIMPs the dark-matter/antimatter nuggets are strongly interacting but macroscopically large.They do not contradict any of the many known observational constraints ondark matter or antimatter for three main reasons [21]:• They carry a huge (anti)baryon charge |B| & 1025, and so have anextremely tiny number density;• The nuggets have nuclear densities, so their effective interaction issmall σ/M ∼ 10−10 cm2/g, well below the typical astrophysical andcosmological limits which are on the order of σ/M < 1 cm2/g;• They have a large binding energy ∼ ∆, such that baryon charge inthe nuggets is not available to participate in big bang nucleosynthesis(bbn) at T ≈ 1 MeV.To reiterate: the weakness of the visible-dark matter interaction is achievedin this model due to the small geometrical parameter σ/M ∼ B−1/3 ratherthan due to a weak coupling of a new fundamental field with standard modelparticles. In other words, this small effective interaction ∼ σ/M ∼ B−1/3replaces a conventional requirement of sufficiently weak interactions of thevisible matter with WIMPs.As we already mentioned, this model when DM is represented by quarkand antiquark nuggets is consistent with fundamental astrophysical con-straints. Furthermore, there are a number of frequency bands where someexcess of emission was observed, but not explained by conventional astro-physical sources. Our comment here is that this model may explain someportion, or even entire excess of the observed radiation in these frequencybands. This phenomenological part of the proposal is the key ingredientin our advocacy of the model, and may play very important role for in-terpretation of the present and future observations. Therefore, we devotenext section 1.2 to review the original results [22–31] where predictions ofthe model have been confronted with the observations in specific frequencybands covering more than eleven orders of magnitude, from radio frequencywith ω ∼ 10−4 eV to γ rays with ω ∼ 10 MeV. We also mention in section1.2 some interesting results [32–36] which are presently perfectly consistentwith the model. However, in future, similar studies with modest improve-ments will provide a powerful test of the viability of the quark nugget darkmatter model.One should emphasize here that the corresponding analysis [22–31] is de-termined by conventional physics, and as such all effects are calculable from71.1. Two sides of the same cointhe first principles. In other words, the model contains no tuneable fun-damental parameters, except for a single mean baryon number of a nugget〈B〉 ∼ 1025 which enters all the computations [22–31] as a single normaliza-tion factor. At the same time, the crucial assumptions of the model, such asspecific mechanisms on the baryon charge separation and dynamics of thenugget formation, etc, have never been explored in our previous studies.We believe that the phenomenological success [22–31] of the model war-rants further theoretical studies of this framework, in spite of its naivelycounter-intuitive nature. Therefore, the present work should be consideredas the first step in this direction where we attempt to develop the theoreticalframework to address (and hopefully answer) some of the hardest questionsabout a possible mechanism for the nuggets’ formation during the QCDphase transition in strongly coupled regime when even the phase diagramat θ 6= 0 as a function of the chemical potential µ and temperature T is stillunknown, see footnote 2.The structure of this work is as follows. In section 1.2 we briefly reviewthe observational constraints on the model. In section 2.1 we highlight thebasic assumptions and ingredients of this framework, while in sections 2.2and 2.3 we present some analytical estimates which strongly substantiatethe idea that such heavy objects indeed can be formed and survive until thepresent epoch during the QCD phase transition in early Universe. Section2.4 as well as Appendices A and B are devoted to a number of technicaldetails which support our basic claim.In chapter 3 we argue that there will be the preferential formation ofone species of nuggets over another. This preference is determined by thedynamics of the axion field θ(x) which itself is correlated on the scales of theUniverse at the beginning of the nuggets’ formation. Finally, in chapter 4 wecomment on implications of our studies to direct axion search experiments.To conclude this long introduction: the nuggets in our framework playa dual role: they serve as the DM candidates and they also explain theobserved asymmetry between matter and antimatter. These two crucial el-ements of the proposal lead to very generic consequence of the entire frame-work expressed by eq. (1.1). This basic generic result is not very sensitiveto any specific details of the model, but rather, entirely determined by twofundamental ingredients of the framework:• the contribution to the present day density of both types of matter (visibleand dark) are proportional to one and the same fundamental scale ∼ ΛQCD;• the preferential formation of one species of nuggets over another is cor-related on huge cosmological scales where CP violating axion phase θ(x)remains coherent just a moment before the QCD phase transition.81.2. Quark (anti) nugget DM confronting the observationsThe readers interested in the cosmological consequences, rather than intechnical computational details may directly jump to section 2.1 where weformulate the basics ingredients of the proposal, to section 3.2 where weexplain the main model-independent consequence (1.1) of this framework,and to chapter 4 where we make few comments on implications to otheraxion search experiments, including microwave cavity [9–11, 14] and theOrpheus experiments [15].1.2 Quark (anti) nugget DM confronting theobservationsWhile the observable consequences of this model are on average stronglysuppressed by the low number density of the quark nuggets ∼ B−1/3 asexplained above, the interaction of these objects with the visible matterof the galaxy will necessarily produce observable effects. Any such conse-quences will be largest where the densities of both visible and dark matterare largest such as in the core of the galaxy or the early universe. In otherwords, the nuggets behave as a conventional cold DM in the environmentwhere density of the visible matter is small, while they become interactingand emitting radiation objects (i.e. effectively become visible matter) whenthey are placed in the environment with sufficiently large density.The relevant phenomenological features of the resulting nuggets are de-termined by properties of the so-called electro-sphere as discussed in originalrefs. [22–31]. These properties are in principle, calculable from first princi-ples using only the well established and known properties of QCD and QED.As such the model contains no tunable fundamental parameters, except fora single mean baryon number 〈B〉 which itself is determined by the axionmass ma as we already mentioned.A comparison between emissions with drastically different frequenciesfrom the centre of galaxy is possible because the rate of annihilation events(between visible matter and antimatter DM nuggets) is proportional to theproduct of the local visible and DM distributions at the annihilation site.The observed fluxes for different emissions thus depend through one and thesame line-of-sight integralΦ ∼ R2∫dΩdl[nvisible(l) · nDM (l)], (1.6)where R ∼ B1/3 is a typical size of the nugget which determines the effectivecross section of interaction between DM and visible matter. As nDM ∼ B−191.2. Quark (anti) nugget DM confronting the observationsthe effective interaction is strongly suppressed ∼ B−1/3 as we already men-tioned in the Introduction. The parameter 〈B〉 ∼ 1025 was fixed in this pro-posal by assuming that this mechanism saturates the observed 511 keV linefrom the galactic centre [22, 23], which resulted from annihilation of the elec-trons from visible matter and positrons from anti-nuggets. It has been alsoassumed that the observed dark matter density is saturated by the nuggetsand anti-nuggets. Such assumptions would correspond to an average baryoncharge 〈B〉 ∼ 1025 for typical density distributions nvisible(r), nDM (r) enter-ing (1.6). Other emissions from different bands are expressed in terms ofthe same integral (1.6), and therefore, the relative intensities are completelydetermined by internal structure of the nuggets which is described by con-ventional nuclear physics and basic QED. We present a short overview ofthese results below.Some galactic electrons are able to penetrate to a sufficiently close tothe surface of the anti-nuggets. These events no longer produce the char-acteristic positronium decay spectrum (511 keV line with a typical widthof order ∼ few keV accompanied by the conventional continuum due to 3γdecay) but a direct non-resonance e−e+ → 2γ emission spectrum. The tran-sition between the resonance positronium decays and non-resonance regimeis determined by conventional QED physics and allows us to compute thestrength and spectrum of the MeV scale emissions relative to that of the511 keV line [24, 25]. Observations by the Comptel satellite indeed showsome excess above the galactic background consistent with our estimates.Galactic protons incident on the anti-nugget will penetrate some distanceinto the quark matter before annihilating into hadronic jets. This processresults in the emission of Bremsstrahlung photons at x-ray energies [26].Observations by the Chandra observatory apparently indicate an excess inx-ray emissions from the galactic centre.Hadronic jets produced deeper in the nugget or emitted in the down-ward direction will be completely absorbed. They eventually emit thermalphotons with radio frequencies [27, 28]. Again the relative scales of theseemissions may be estimated and is found to be in agreement with observa-tions.These apparent excess emission sources have been cited as possible sup-port for a number of dark matter models as well as other exotic astrophysicalphenomenon. At present however they remain open matters for investigationand, given the uncertainties in the galactic spectrum and the wide varietyof proposed explanations are unlikely to provide clear evidence in the nearfuture. Therefore, it would be highly desirable if some direct detection ofsuch objects is found, similar to direct searches of the WIMPs.101.2. Quark (anti) nugget DM confronting the observationsWhile direct searches for WIMPs require large sensitivity, a search forvery massive dark matter nuggets requires large area detectors. If the darkmatter consists of quark nuggets, they will have a flux ofdNdA dt= nv ≈(1025B)km−2yr−1. (1.7)Though this flux is far below the sensitivity of conventional dark mattersearches it is similar to the flux of cosmic rays near the Greisen-Zatsepin-Kuzmin (GZK) limit. As such present and future experiments investigatingultrahigh energy cosmic rays may also serve as search platforms for darkmatter of this type.It has been suggested that large scale cosmic ray detectors may be ca-pable of observing quark (anti-) nuggets passing through the earth’s atmo-sphere either through the extensive air shower such an event would trigger[29] or through the geosynchrotron emission generated by the large numberof secondary particles [30], see also [31] for review.It has also been estimated in [32] that, based on Apollo data, nuggets ofmass from ∼ 10 kg to 1 ton (corresponding to B ∼ 1028-30) must accountfor less than an order of magnitude of the local dark matter. While ourpreferred range of B ∼ 1025 is somewhat smaller and is not excluded by[32], we still believe that B ≥ 1028 is not completely excluded by the Apollodata, as the corresponding constraints are based on specific model dependentassumptions about the nugget mass-distribution.It has also been suggested that the anita experiment may be sensitiveto the radio band thermal emission generated by these objects as they passthrough the antarctic ice [33]. These experiments may thus be capable ofadding direct detection capability to the indirect evidence discussed above,see Fig.1.2 taken from [33] which reviews these constarints.It has been also suggested recently [34] that the interactions of these(anti-) nuggets with normal matter in the Earth and Sun will lead to anni-hilation and an associated neutrino flux. Furthermore, it has been claimed[34] that the antiquark nuggets cannot account for more than 20% of thedark matter flux based on constraints for the neutrino flux in 20-50 MeVrange where sensitivity of the underground neutrino detectors such as Su-perK have their highest signal-to-noise ratio.However, the claim [34] was based on assumption that the annihilationof visible baryons with antiquark nuggets generate the neutrino spectrumsimilar to conventional baryon- antibaryon annihilation spectrum when thelarge number of produced pions eventually decay to muons and consequentlyto highly energetic neutrinos in the 20-50 MeV energy range. Precisely these111.2. Quark (anti) nugget DM confronting the observationsFigure 1.2: Limits on quark nugget mass and fluxes based on current (andfuture) constraints, taken from [33]. Our preferable value 〈B〉 ∼ 1025 istranslated to the axion mass ma ' 10−4 eV according to the scaling rela-tion (1.3). The corresponding constraints expressed in terms of ma haveimportant implication for the direct axion search experiments as discussedin chapter 4. Orpheus experiment “B” is designed to be sensitive exactly tothis value of the axion mass ma ' 10−4 eV, see Fig.4.1.121.2. Quark (anti) nugget DM confronting the observationshighly energetic neutrinos play the crucial role in analysis [34]. However,in most CS phases the lightest pseudo Goldstone mesons (the pions andKaons) have masses in the 5-20 MeV range [17, 18] in huge contrast withhadronic confined phase where mpi ∼ 140 MeV. Therefore, such light pseudoGoldstone mesons in CS phase cannot produce highly energetic neutrinosin the 20-50 MeV energy range and thus are not subject to the SuperKconstraints [36].We conclude this brief overview on observational constraints of the modelwith the following remark. This model which has a single fundamentalparameter (the mean baryon number of a nugget 〈B〉 ∼ 1025, correspondingto the axion mass ma ' 10−4 eV), and which enters all the computations isconsistent with all known astrophysical, cosmological, satellite and groundbased constraints as highlighted above. Furthermore, in a number of casesthe predictions of the model are very close to the presently available limits,and very modest improving of those constraints may lead to a discoveryof the nuggets. Even more than that: there are a number of frequencybands where some excess of emission was observed, and this model mayexplain some portion, or even entire excess of the observed radiation inthese frequency bands.In the light of this (quite optimistic) assessment of the observationalconstraints of this model it is quite obvious that further and deeper studiesof this model are worthwhile to pursue. The relevant developments mayinclude, but are not limited, to such hard problems as formation mechanismsduring the QCD phase transition in early Universe, even though many keyelements for proper addressing those questions at θ 6= 0, µ 6= 0, T 6= 0 arestill largely unknown in strongly coupled QCD as shown on Fig.1.1. Thiswork is the first step in the direction to explore a possible mechanism offormation of the nuggets.13Chapter 2Formation of the NuggetsAs mentioned in the preceding chapter, the observational consequence of thenuggets have been studied for a long time. However, the formation aspect ispreviously just a conjecture. Taken as another motivation, this chapter (andthis thesis) is dedicated to investigating the local formation of a nugget.2.1 The crucial ingredients of the proposal1. First important element of this proposal is the presence5 of the topologicalobjects, the axion domain walls [37]. As usual, it is assumed that the Peccei-Quinn symmetry6 is broken after inflation. As we already mentioned theθ parameter is an angular variable, and therefore supports various typesof domain walls, including the so-called NDW = 1 domain walls when θinterpolates between one and the same physical vacuum state with the sameenergy θ → θ+2pin. The axion domain walls may form at the same momentwhen the axion potential get tilted, i.e. at the moment Ta when the axionfield starts to roll due to the misalignment mechanism. The tilt becomesmuch more pronounced at the phase transition when the chiral condensateforms at Tc. In general one should expect that the NDW = 1 domain wallsform once the axion potential is sufficiently tilted to track (anti)quarks, i.e.anywhere between Ta and Tc. In the conventional models, Ta and Tc areusually considered as about 1 GeV and 170 MeV respectively.Much later it has been realized that the axion (along with η′) domainwalls in general, demonstrate a sandwich-like substructure on the QCD scaleΛ−1QCD ' fm. The arguments supporting the QCD scale substructure inside5To be specific, here mainly refers to the time after the QCD phase transition Tc whenthe domain walls are sufficiently tilted. Also see the the detailed explanation in thisparagraph.6Peccei-Quinn symmetry, is a proposed U(1) symmetry in order to resolve the puz-zling strong CP problem. Such symmetry predicts the existence of axion, a neutral andultralight psuedoscalar particle. Especially, axion may roll to a near potential minimumwhen the symmetry is spontaneously broken. This is also known as the misalignmentmechanism.142.1. The crucial ingredients of the proposalthe axion domain walls are based on analysis [38] of QCD in the large Nlimit with inclusion of the η′ field7 and independent analysis [39] of super-symmetric models where a similar θ vacuum structure occurs.One should remark here that the described structure is a classically stableconfiguration. In particular, the η′ field cannot decay to 2γ simply due tothe kinematical reasons when a single η′ field is off-shell (if not annihilatedwith another off-shell anti-field), and cannot be expressed as a superpositionof on-shell free particles. It can only decay through the tunneling, andtherefore, such NDW = 1 domain walls are formally metastable rather thanabsolutely stable configurations.2. Second important element is that in addition to these known QCDsubstructures [38–40] of the axion domain walls expressed in terms of the η′fields, there is another substructure (the baryonic fields, see below) with asimilar QCD scale which carries the baryon charge. Precisely this novel fea-ture of the domain walls which was not explored previously in the literaturewill play a key role in our proposal because exactly this new effect will beeventually responsible for the accretion of the baryon charge by the nuggets.Depending on the sign of the baryon charge, either quarks or anti-quarksare favoured to accrete on a given closed domain wall making eventuallythe quark nuggets or anti-nuggets. The sign is chosen randomly such thatequal number of quark and antiquark nuggets are formed if the externalenvironment is CP even, which is the case when fundamental θ = 0. Onecan interpret this phenomenon as a local spontaneous symmetry breakingeffect, when on the scales of order the correlation length of the axion fieldξ the nuggets may acquire a positive or negative baryon charge with equalprobability, as discussed in great details in next section 2.2.3. Next important ingredient of the proposal is the Kibble-Zurek mech-anism 8 which gives a generic picture of formation of the topological defectsduring a phase transition, see original papers [41], review [42] and the text-book [43]. In our context the Kibble-Zurek mechanism suggests that oncethe axion potential is sufficiently tilted the NDW = 1 domain walls form.The potential becomes much more pronounced when the chiral condensateforms at Tc. After some time after Ta the system is dominated by a single,7The η′ field has the special property that it enters the effective Lagrangian in uniquecombination [θ− η′(x)] where the θ parameter in the present context plays the role of theaxion dynamical field θ(x). A similar structure is known to occur in CS phase as well.The corresponding domain walls in CS phase have been also constructed [40].8When topological objects, such as domain walls, kinks, and strings, form from sponta-neous symmetry breaking. Kibble-Zurek mechanism describes their dynamical propertieslike correlation length, tension, and energy density.152.1. The crucial ingredients of the proposalpercolated, highly folded and crumpled domain wall of very complicatedtopology of scale of the cosmic event horizon dH . In addition, there will bea finite portion of the closed walls (bubbles) with typical size of order corre-lation length ξ(T ), which is defined 9 as an average distance between foldeddomain walls at temperature T . It is known that the probability of findingclosed walls with very large size R ξ is exponentially small. Furthermore,numerical simulations suggest [43] that approximately 87% of the total wallarea belong to the percolated large cluster, while the rest is represented byrelatively small closed bubbles with sizes R ∼ ξ.The key point for our proposal is the existence of these finite closedbubbles made of the axion domain walls10. One should remark here thatthese closed bubbles had been formed sometime after Ta when original θparameter has not settled yet to its minim value. It implies that the do-main wall evolution starts at the time when θ parameter is not yet zero11.Normally it is assumed that these closed bubbles collapse as a result of thedomain wall pressure, and do not play any significant role in dynamics ofthe system. However, as we already mentioned in Introduction the collapseof these closed bubbles is halted due to the Fermi pressure acting insideof the bubbles. Therefore, they may survive and serve as the dark mattercandidates.The percolated network of the domain walls will decay to the axionsin conventional way as discussed in [44–46]. Those axions (along with theaxions produced by the conventional misalignment mechanism [45, 47]) willcontribute to the dark matter density today. The corresponding contributionto dark matter density is highly sensitive to the axion mass as Ωdark ∼ m−1a .Axion may saturate the observed dark matter density if ma ' 10−6 eV [9–16], while it may contribute very little to Ωdark if the axion mass is slightly9The definition of ξ here refers to the average distance between any two domain walls,disregard closed or open. For axion field, ξ ∼ m−1a is generic from the Kibble-Zurekmechanism and therefore QCD insensitive.10The presence of such closed bubbles in numerical simulations in context of the axiondomain wall has been mentioned in [11], where it was argued that these bubbles wouldoscillate and emit the gravitational waves. However, we could not find any further detailson the fate of these closed bubbles in the literature.11This θ parameter in our work is defined as the value of θ at the moment when thedomain walls form. It is not exactly the same value as the misalignment angle whichnormally enters all the computations due to the conventional misalignment mechanism[45, 47]. This is because the temperature when the domain walls form and the temperatureTa when the axion field starts to roll do not exactly coincide though both effects are dueto the same axion tilted potential. The crucial point is that the θ parameter, as definedabove, could be numerically small, nevertheless, it preserves its coherence over entireUniverse, see chapter 3 for the details.162.1. The crucial ingredients of the proposalheavier than ma ' 10−6 eV. In contrast, in our framework an approximaterelation Ωdark ≈ Ωvisible holds irrespectively to the axion mass ma.We shall not elaborate on the production and spectral properties of theseaxions in the present work. Instead, the focus of the present thesis is thedynamics of the closed bubbles, which is normally ignored in computationsof the axion production. Precisely these closed bubbles, according to thisproposal, will eventually become the stable nuggets and may serve as thedark matter candidates.As we already mentioned the nugget’s contribution to Ωdark is not verysensitive to the axion mass, but rather, is determined by the formationtemperature Tform as explained in Introduction, see also footnote 4 with fewimportant comments on this. The time evolution of these nuggets after theirformation is the subject of section 2.3.4. The existence of CS phase in QCD represents the next crucial elementof our scenario. The CS has been an active area of research for quite sometime, see review papers [17, 18] on the subject. The CS phase is realizedwhen quarks are squeezed to the density which is few times nuclear density.It has been known that this regime may be realized in nature in neutronstars interiors and in the violent events associated with collapse of massivestars or collisions of neutron stars, so it is important for astrophysics.The force which squeezes quarks in neutron stars is gravity; the forcewhich does an analogous job in early universe during the QCD phase transi-tion is a violent collapse of a bubble of size R ∼ ξ(T ) formed from the axiondomain wall as described in item 3 above. If the number density of quarkstrapped inside of the bubble (in the bulk) is sufficiently large, the collapsestops due to the internal Fermi pressure. In this case the system in the bulkmay reach the equilibrium with the ground state being in a CS phase. Aswe advocate in section 2.3 this is very plausible fate of a relatively largesize bubbles of size R ∼ ξ(T ) made of the axion domain walls which wereproduced after the QCD phase transition.5. If θ vanishes, then an equal number of nuggets and anti-nuggetswould form. However, the CP violating θ parameter (the axion field), whichis defined as value of θ at the moment of domain wall formation genericallyis not zero, though it might be numerically quite small. Precisely the dy-namics of the coherent axion field θ(x) leads to preferences in formation ofone species of nuggets, as discussed in chapter 3. This sign-preference iscorrelated on the scales where the axion field θ(x) is coherent, i.e. on thescale of the entire Universe at the moment of the domain wall formation.As we already mentioned, the generic consequence of this framework (1.1)is not very sensitive to an absolute value of θ at the moment of the domain172.2. Accretion of the baryon chargewall formation, see comment in footnote 4 on this matter. One can saythat the coherent axion filed θ(x) 6= 0, being numerically small, plays therole of the CP violating catalyst which determines a preferred direction forseparation of the baryon charges on the Universe scale. This role of CP vio-lation in our proposal is quite different from the role it plays in conventional“baryogenesis” mechanisms.2.2 Accretion of the baryon chargeFrom now on and until chapter 3 we focus on the dynamics of a single closedbubble produced during the domain wall formation as described in item 3in section 2.1. The correlation length ξ(T ) is defined as an average distancebetween folded domain walls at temperature T . We assume12 that initialsize of the bubble ξ(T ) is sufficiently large., few times larger than the axiondomain wall width ∼ m−1a , such that one can locally treat the surface of theclosed bubble being flat.The main goal of this section is to demonstrate that such a bubble willgenerically acquire a baryon (or antibaryon) charge in very much the sameway as the η′ field was dynamically accreted as originally discussed in [38]and briefly explained above as item 2 in section 2.1. In other words, we shallargue in this section that the bubbles with baryon or antibaryon charge willbe copiously produced during the phase transition as they are very genericconfigurations of the system. In both cases the effect emerges as a resultof the nontrivial boundary conditions formulated far away from the domainwall core when the field assumes physically the same but topologically dis-tinct vacuum states on opposite sides of the axion domain wall.The technique we shall adopt in this section has been previously used tostudy the generation of the galactic magnetic field in the domain wall back-ground [48]. This method makes the approximation that the domain wallis flat and that translational and rotational symmetries are preserved in theplane of the wall (which we take to be the x–y plane). These approximationsare marginally justified in our case because the initial curvature R ∼ ξ(T )is assumed to be few times larger that the width of the wall ∼ m−1a .Once this approximation is made, we can reformulate the problem in1 + 1 dimensions (z and t) and calculate the density of the bulk (of baryonicaccumulation) properties along the domain wall. To regain the full four-dimensional bulk properties, we shall estimate the density of the particles12This is a marginal assumption, but can be justified as in the later section 2.4.182.2. Accretion of the baryon chargein the x–y plane to obtain the appropriate density and degeneracy factorsfor the bulk density.We proceed to demonstrate this technique by computing the accumu-lation of baryon charge along the wall. We take the standard form for theinteraction between the pseudo-scalar fields and the fermions (quarks) whichrespect all relevant symmetries:L4 = Ψ¯(i6∂ −mei[θ(z)−φ(z)]γ5 − µγ0)Ψ. (2.1)The subscript of “4” under the Lagrangian stands for dimensions, because itcomes from the standard 4-dimensional Dirac equation after chiral rotationof (θ−φ), where the anomaly term is neglected for the moment, but will bediscussed in the next chapter 3. Here θ(z) and φ(z) are the dimensionlessaxion and η′ domain wall solution. Parameter m is the the typical QCDscale of the problem, while µ is the typical chemical potential at a specifictime in evolution of the system, see below with more precise explanations.We also simplify the problem by ignoring all flavour and colour indices, aswell as an effective 4-fermi interactions, as our main goal is to explain thebasic idea with simplified setting.Parameter m in eq.(2.1) should not be literally identified with the quarkmass, nor with the nucleon mass. Instead, this dimensional parameterm ∼ ΛQCD should be thought as an effective coupling in our model whenparameter m effectively describes the interaction with fermi field Ψ in allphases during the formation time, including the quark gluon plasma as wellas hadronic and CS phases13. The same comment also applies to a numer-ical value of the chemical potential µ: it vanishes during initial time andbecomes very large when CS phase sets in inside the nugget.The strategy is to break (2.1) into two 1 + 1 dimensional componentsby setting ∂x = ∂y = 0 (this is the approximation that the physics in the zdirection decouples from the physics in the x–y plane) and then by manip-ulating the system of equations that result.13In quark gluon phase the colour singlet η′ field does not exist. However, the singletphase which accompanied the quark field is still present in the system. The coefficient m inthis phase can be computed using the instanton liquid model. At very high temperaturethe parameter m is proportional to the quark masses and indeed very small. Whentemperature decreases the instanton contribution grows very fast. At this point parameterm is proportional to the vacuum expectation value of the ’t Hooft determinant. Whentemperature further decreases the parameter m is proportional to the diquark condensatein CS phase or the chiral condensate in the hadronic phase, see Fig.1.1. We shall notelaborate along this line by assuming m ∼ ΛQCD for all our estimates which follow.192.2. Accretion of the baryon chargeFirst, we introduce the following chiral components of the Dirac spinors14:Ψ+ =1√S(χ1χ2), Ψ− =1√S(ξ1ξ2), (2.2)Ψ =1√2Sχ1 + ξ1χ2 + ξ2χ1 − ξ1χ2 − ξ2 = 1√2(Ψ+ + Ψ−Ψ+ −Ψ−), (2.3)where S is the area of the wall. This normalization factor cancels the de-generacy factor proportional to S added in the text below.The associated Dirac equation is( −mei(φ−θ) i(∂t + ∂z)− µi(∂t − ∂z)− µ −me−i(φ−θ))(χ1ξ1)= 0, (2.4a)( −mei(φ−θ) i(∂t − ∂z)− µi(∂t + ∂z)− µ −me−i(φ−θ))(χ2ξ2)= 0. (2.4b)where we decouple the z coordinates from x and y by setting ∂x = ∂y = 0.Remember that we are looking for a two-dimensional Dirac equation, thuswe want the kinetic terms to look the same. For this reason we should flipthe rows and columns of the second equation. Doing this and defining thetwo two-dimensional spinorsΨ(1) =(χ1ξ1), Ψ(2) =(ξ2χ2), (2.5)the equations have the following structure:(iγˆν∂ν −me+i(θ−φ)γˆ5 − µγˆ0)Ψ(1) = 0 (2.6a)(iγˆν∂ν −me−i(θ−φ)γˆ5 − µγˆ0)Ψ(2) = 0 (2.6b)14We are using the standard representation here:γ0 =(I 00 −I), γj =(0 σj−σj 0), γ5 =(0 II 0),σ1 =(0 11 0), σ2 =(0 −ii 0), σ3 =(1 00 −1).202.2. Accretion of the baryon chargewhere the index ν ∈ {t, z}, the Lorentz signature is (1,−1) and we definethe following two-dimensional version of the gamma matrices:γˆt = σ1, γˆz = −iσ2, γˆ5 = σ3.These satisfy the proper two-dimensional relationships γˆ5 = γˆtγˆz and γˆµγˆν =gµν + µν γˆ5. We can reproduce equation (2.6) from the following effectivetwo-dimensional Lagrangian density,L2 =Ψ¯(1)(iγˆµ∂µ −me+i(θ−φ)γˆ5 − µγˆ0)Ψ(1)++Ψ¯(2)(iγˆµ∂µ −me−i(θ−φ)γˆ5 − µγˆ0)Ψ(2), (2.7)where two different species of fermion with opposite chiral charge interactwith the axion domain wall background determined by θ(z) and φ(z) fields.Note that, due to the normalization factor 1/√S we introduced above, thetwo-dimensional fields Ψ(i) have the correct canonical dimension 1/2.We have thus successfully reduced our problem to a two-dimensionalfermionic system. It is known that for systems that are constructed bycomponents of (2.8) in 1 + 1 dimensions, the fermionic representation isequivalent to a 1 + 1 dimensional bosonic system through the followingequivalences[49, 50]:Ψ¯(j)iγˆµ∂µΨ(j) →12(∂µθj)2, (2.8a)Ψ¯(j)γˆµΨ(j) →1√piµν∂νθj , (2.8b)Ψ¯(j)Ψ(j) → −m0 cos(2√piθj), (2.8c)Ψ¯(j)iγˆ5Ψ(j) → −m0 sin(2√piθj). (2.8d)The constant m0 in the last two equations is a dimensional parameter oforder m0 ∼ m ∼ ΛQCD. The exact coefficient of this factor depends onrenormalization procedure and is only known for few exactly solvable sys-tems but in all cases, is of order unity.After making these replacements, we are left with the following two-dimensional bosonic effective Lagrangian density describing the two fieldsθ1 and θ2 in the domain wall background determined by φ(z) and θ(z):L2 = 12(∂µθ1)2 +12(∂µθ2)2 − U(θ1, θ2) + µ√pi∂(θ2 + θ1)∂z(2.9)212.2. Accretion of the baryon chargewhere the effective potential isU(θ1, θ2) = − mm0[cos(2√piθ1 − φ+ θ)]− mm0[cos(2√piθ2 + φ− θ)]. (2.10)The conventional procedure to study the system (2.9) is to add the kineticterms for the axion θ and the η′ field φ into (2.9) and study a resulting solu-tion depending on four dynamical fields by specifying all possible boundaryconditions when the potential energy (2.10) assumes its minimal value15. Inother words, one should take into account the dynamics of the θ and φ fieldstogether with θ1, θ2 because the typical scales for φ, θ1, θ2 are roughly thesame order of magnitude and of order of ΛQCD. Recapitulate it: one cannotstudy the dynamics of θ1, θ2 field by neglecting their back reaction on thebackground axion and φ fields.For our present purposes, however, we do not really need an explicitprofile functions for the large number of different domain walls determinedby various boundary conditions controlled by (2.10). The only importantelement relevant for our future discussions is the observation that some ofthe domain walls may carry the baryon (antibaryon) charge. Indeed, thedomain walls which satisfy the boundary conditions2√piθ1(z = +∞)− 2√piθ1(z = −∞) = 2pin1 (2.11)2√piθ2(z = +∞)− 2√piθ2(z = −∞) = 2pin2carry the following baryon charge N defined for one particle Dirac equationN =∫d3xΨ¯γ0Ψ =∫dz(Ψ¯1γˆ0Ψ1 + Ψ¯2γˆ0Ψ2)= − 1√pi∫ +∞−∞dz∂∂z(θ1 + θ2) = −(n1 + n2), (2.12)where we express the final formula in terms of the auxiliary two-dimensionalfields θ1 and θ2 and corresponding boundary conditions given by eq. (2.11).Factor S also cancels with our normalization for four dimensional Ψ field.To complete the computations for four dimensional baryon charge B ac-cumulated on the domain wall we need to multiply (2.12) by the degeneracyfactor in vicinity of the domain wall which can be estimated as followsB = N · g ·∫d2x⊥d2k⊥(2pi)21exp( −µT ) + 1, (2.13)15In fact it was precisely the procedure which has been adopted in [38] for a similarproblem of computing of the profile functions of the axion, pi- meson and η′-domain walldescribed by θ − pi − η′ fields.222.2. Accretion of the baryon chargewhere g is the appropriate degeneracy factor, e.g. g ' NcNf in CS phase.We note that an additional degeneracy factor 2 due to the spin is alreadyaccounted for by parameter N defined in eq. (2.12). For high chemicalpotential µ T corresponding to CS phase the baryon charge per unit areaaccreted in vicinity of the domain wall can be approximated asBS' N · gµ24pi. (2.14)In the opposite limit of high temperature µ  T which corresponds to thequark gluon plasma phase, the corresponding magnitude can be estimatedas followsBS' N · gpiT224. (2.15)It is instructive to compare the estimate (2.15) with number densityN/Vof all degrees of freedom in vicinity of the domain wall. Assuming that thebaryon charge in the domain wall background is mainly concentrated ondistances of order m−1 from the center of the domain wall we arrive to thefollowing estimate for the ratio of the baryon number density bound to thewall in comparison with the total number density of all degrees of freedomresponsible for the thermodynamical equilibrium in this phaser ∼ (B/S) ·mN/V ∼ N(mT)( pi3g18ξ(3)g∗), (2.16)where the effective degeneracy factor g∗ for a quark gluon plasma is g∗ '[344NcNf + 2(N2c − 1)]and ξ(3) ' 1.2 is the Riemann zeta function. Ratio(2.16) shows that the accreted quark density bounded to the domain wall athigh temperature represents parametrically small contribution to all ther-modynamical observables mainly because of a small parameter m/T  1 inthis phase. The situation drastically changes as we discuss in next section2.3 when the temperature slowly decreases due to expansion of the Universeand the system enters the hadronic or CS phase, as shown on Fig. 1.1. Atthis point the baryon charge accumulation in the domain wall backgroundbecomes the major player of the system, which eventually leads to the forma-tion of the CS nuggets or anti-nuggets when quarks (anti-quarks) fill entirevolume of the nuggets (anti-nuggets).We conclude this section with the following important comments. First,we argued that the domain walls in general accrete the baryon (or an-tibaryon) charge in vicinity of the centre of the domain wall. The effect in232.2. Accretion of the baryon chargemany respects is similar to fractional charge localization on domain walls,while the rest of the charge is de-localized in the rest of volume of the sys-tem as discussed in original paper [51]. The effect is also very similar topreviously discussed phenomenon on dynamical generation of the η′ fieldin the domain wall background. The key point is that at sufficiently hightemperature the NDW = 1 domain walls form by the usual Kibble-Zurekmechanism as explained in section 2.1. The periodic fields θ, φ, θ1, θ2 mayassume physically identical but topologically distinct vacuum values (2.12)on opposite sides of the walls. When the system cools down the correspond-ing fields inevitably form the domain wall structure, similar to analysis inhadronic [38] and CS phases [40].We advocate the picture that the closed bubbles will be also inevitablyformed as discussed in section 2.1. The collapse of these bubbles halts asa result of Fermi pressure due to the quarks accumulated inside the nuggetduring the evolution of the domain wall network. Next section 2.3 is devotedprecisely the question on time evolution of these closed bubbles made ofNDW = 1 domain wall.• The most important lesson of this section is that there is a varietyof acceptable boundary conditions determined by potential (2.10) when theenergy assumes its vacuum values. Some of the domain walls will cary zerobaryon charge when the combination (n1 +n2) vanishes according to (2.12).However, generically the domain walls will acquire the baryon or antibaryoncharge. This is because the domain wall tension is mainly determined bythe axion field while corrections due to QCD substructure will lead to asmall correction of order ∼ m/fa  1, similar to studies of the (axion -η′-pi)domain wall [38]. Therefore, the presence of the QCD substructure with nonvanishing (n1 + n2) 6= 0 increases the domain wall tension only slightly. Inother words, accumulation of the baryon charge in vicinity of the wall doesnot lead to any suppression during the formation stage. Consequently, thisimplies that the domain closed bubbles carrying the baryon or antibaryoncharge will be copiously produced during the phase transition as they arevery generic configurations of the system. Furthermore, the baryon chargecannot leave the system during the evolution as it is strongly bound to thewall due to the topological reasons. The corresponding binding energy perquark is order of µ and increases with time as we discuss in the next section.This phenomenon of “separation of the baryon charge” can be inter-preted as a local version of spontaneous symmetry breaking of the baryoncharge. This symmetry breaking occurs not in the entire volume in theground state determined by the potential (2.10). Instead, the symmetrybreaking occurs on scale ξ(T ) in vicinity of the field configurations which242.3. Radius versus pressure in time evolutiondescribe the interpolation between physically identical but topologically dis-tinct vacuum states (2.11). One should add that a similar phenomenon oc-curs with accumulation of the η′ field in vicinity of the axion domain wall asdescribed in [38]. However, one could not term that effect as a “local spon-taneous violation” of the U(1)A symmetry because the U(1)A symmetry isexplicitly broken by anomaly, in contrast with our present studies when thebaryon charge is an exact symmetry of QCD. Nevertheless, the physics isthe same in a sense that the closed bubble configurations generically acquirethe axial as well as the baryon charge. This phenomenon as generic as for-mation of the topological domain walls themselves when the periodic fieldsφ, θ1, θ2 may randomly assume physically identical but topologically distinctvacuum values on the correlation lengths of order ξ.Finally, one should also mention here that very similar effect of the “localCP violation” can be experimentally tested in heavy ion collisions in event byevent basis where the so-called induced θind- domain with a specific sign ineach given event can be formed. This leads to the “charge separation effect”which can be experimentally observed in relativistic heavy ion collisions[52]. This “charge separation effect” in all respects is very similar to thephenomenon discussed in the present section. As an additional fact, theauthor’s supervisor, Ariel Zhitnitsky, also involves in the studies [52] for apossibility to test the ideas advocated in this work by performing a specificanalysis in the controllable “little Bang” heavy ion collision experiments, incontrast with “Big Bang” which happened billion of years ago. This fieldof research initiated in [52] became a hot topic in recent years as a result ofmany interesting theoretical and experimental advances, see recent reviewpapers [53–55] on the subject.2.3 Radius versus pressure in time evolutionWe assume that a closed NDW = 1 domain wall has been formed as discussedin previous section 2.1. Furthermore, we also assume that this domain wall isclassified by non-vanishing baryon number (n1 + n2) according to eq.(2.12).Our goal now is to study the time evolution of the obtained configuration. Aswe argue below the contraction of the bubbles halts as a result of the Fermipressure due to baryon charge accreted during the evolution. As a result,the system comes to the equilibrium at some temperature Tform when thenuggets complete their formation. We want to see precisely how it happens,and what are the typical time scales relevant for these processes.We start with the following effective Lagrangian describing the time252.3. Radius versus pressure in time evolutionevolution of the closed spatially symmetric domain wall of radius R(t),L =4piσR2(t)2R˙2(t)− 4piσR2(t) (2.17)+4piR3(t)3[Pin(µ)− Pout(t)] + [other terms] ,where the “other terms” represents the effects that are not able to account inour simplified model, but are believed to be only minor corrections 16. Hereσ is the key dimensional parameter, the domain wall tension σ ∼ fpimpifa ∼m−1a as reviewed in Introduction, see eq.(1.3). The tension σ, in principle,is also time-dependent parameter because the axion mass depends on time,but for qualitative analysis of this section we ignore this time dependence fornow. We return to this question later in the text. Parameters Pin[µ(t)] andPout(t) represent the pressure inside and outside the bubble. The outsidepressure in QGP phase at high temperature can be estimated asPout ' pi2gout90T 4out, Tout ' T0(t0t)1/2,gout '(784NcNf + 2(N2c − 1))(2.18)where gout is the degeneracy factor, while T0 ' 100 MeV and t0 ∼ 10−4s rep-resents initial temperature and time determined by the cosmological expan-sion. We also assume that the thermodynamical equilibrium is maintained atall times between inside and outside regions such that the temperature insidethe bubble approximately follows the outside temperature Tout(t) ' Tin(t).Very quick equilibration indeed is known to take place even in much fasterprocesses such as heavy ion collisions. The fast equilibration in our casecan be justified because the heat transport between the phases is mostlydue to the light NG bosons which can easily penetrate the domain wall withlittle on no interaction, in contrast with quarks and baryons discussed in theprevious section. This assumption will be justified a posteriori, see (2.47)on flux exchange rate between interior and exterior regions. Therefore, webelieve our approximation Tout(t) ' Tin(t) is sufficiently good, at least forqualitative estimates which is the main goal of this work.The expression for the pressure inside the bubble Pin(t) depends on anumber of quite nontrivial features of QCD such as the bag vacuum energy,16Some neglected effects could be, for example, the realization of non-constancy of σ,which is nonetheless considered in the numerical computation, see Appendix B.262.3. Radius versus pressure in time evolutioncorrections due to the gap in CS phase and many other phenomena, to bediscussed later in the text.The equation of motion which follows from (2.17) isσR¨(t) = − 2σR(t)− σR˙2(t)R(t)+ ∆P (µ)− 4η R˙(t)R(t), (2.19)where ∆P [µ(t)] ≡ [Pin(µ)− Pout(t)]. We also inserted an additional term(which cannot be expressed in the Lagrangian formulation (2.17)), the shearviscosity η to the right hand side of the equation, which effectively describesthe “friction” of the system when the domain wall bubble moves in “un-friendly” environment17. On the microscopical level this term effectivelyaccounts for a large number of different effects which do occur during thetime evolution. Such processes include, but are not limited to different scat-tering process by quarks, gluons or Nambu Goldstone Bosons in differentphases. All these particles and quasiparticles interact between themselvesand also with a moving domain wall. Furthermore, the annihilation pro-cesses which take place inside the bubble and which result in production ofa large number of strongly interacting quasi-particles also contribute to η.Having discussed an expression for Pout(T ) and viscous term ∼ η we nowwish to discuss the structure of the internal pressure Pin(µ) which enters(2.19). It has a number of contributions which are originated from verydifferent physics. We represent Pin(µ) as as a combination of three terms tobe discussed one by one in order,Pin(µ) ' P (Fermi)in (µ) + P (bag const)in (µ) + P (others)in . (2.20)In this formula P(Fermi)in can be represented as followsP(Fermi)in (µ) =E3V=gin6pi2∫ ∞0k3dk[exp( (k)−µT ) + 1] , (2.21)17We use conventional normalization factor of 4ηR˙(t)/R(t) for the viscous term. Thisnormalization factor is the same which appears in the Rayleigh-Plesset equation in theclassical hydrodynamics when the viscous term, the surface tension term 2σ/R(t) and thepressure term ∆P enter the equation in a specific combination as presented in (2.19).One should emphasize that our equation (2.19) describes the dynamics of the 2d surfacecharacterized by the same surface tension σ in contrast with classical equation of theRayleigh-Plesset equation describing a 3d spherical bubble in a liquid of infinite volume.This difference explains some distinctions between the kinetic terms proportional to factor∼ σ in our case (2.19) in contrast with the classical Rayleigh-Plesset equation.272.3. Radius versus pressure in time evolutionwhere we assume that quarks are massless and the chemical potential µ(t)implicitly depends on time as a result of the bubble’s evolution (shrinking).The degeneracy factor in this formula isgin ' 2NcNf , (2.22)where we keep only the quark contribution by neglecting the antiquarks. Inother words, we simplify the problem by ignoring the time dependence of thedegeneracy factor gin(t) which effectively varies as a result of µ(t) variation.Now we are in position to discuss P(bag constant)in from (2.20) which canbe represented as followsP(bag const)in (µ) ' −EB · θ [µ− µ1][1− µ21µ2], (2.23)where positive parameter EB ∼ (150 MeV)4 is the famous “bag constant”from MIT bag model, see [4] for references and numerical estimates for thisparameter in the given context of the nugget structure. The bag constantcan be expressed in terms of the gluon and quark condensates in QCD. Weshall not elaborate on this problem in the present work by referring to [4]with relevant studies in the given context.The bag “constant” EB describes the differences of vacuum energies(and therefore, vacuum pressure) in the interior and exterior regions of thenuggets. This difference occurs in our context because the phases realizedoutside and inside of the nugget are drastically distinct. For example, at theend of formation the outside region of the nugget is in cold hadronic phase,while inside region is in CS phase. The vacuum energies in these two phasesare known to be drastically different. This term works as a “squeezer”,similar to the role it plays in the MIT bag model, when the vacuum energyoutside of the nugget is lower than the vacuum energy inside the nugget.Therefore it enters with the same sign minus as the domain wall pressure.A specific µ- dependence used in (2.23) is an attempt to model a knownfeature of QCD that the absolute value of the vacuum energy decreaseswhen the chemical potential increases. This feature is well established andtested in conventional nuclear matter physics, and it was analytically de-rived in simplified version of QCD with number of colours Nc = 2, see [4]for references and details. Our parametrization (2.23) corresponds to thebehaviour when P(bag constant)in (µ) = 0 for small chemical potentials µ ≤ µ1when the vacuum energy inside and outside of the nuggets approximatelyequal. This term becomes very important “squeezer” at large chemical po-tential at µ ≥ µ1 when the system outside is in the hadronic vacuum state282.3. Radius versus pressure in time evolutionwhile inside it is in a CS phase. The numerical value for parameter µ1 canbe estimated as µ1 ∼ 330 MeV [4] when the baryon density is close to thenuclear matter density.The last term entering (2.20) and coined as P(others)in (µ) is due to a largenumber of other effects which we ignore in present work. In particular, thereis a conventional contribution due to the boson degrees of freedom whichcancels the corresponding portion of gout from (2.18) at high temperature,T  µ. It does not play any important role in our analysis because we aremainly concerned with analysis of fermion degrees of freedom and buildingthe chemical potential inside the bubble. Another effect which is worth tobe mentioned is the formation of the gap in CS phase due to the quarkpairing, similar to formation of the gap in conventional superconductors.The generation of the gap obviously decreases the energy of the system.There are many other phenomena which are known to occur in CS phase [17].However, we expect that these effects are less important in comparison withthe dominating contributions which are explicitly written down in equations(2.21) and (2.23).The equation (2.19) can be numerically solved for R(t) if time varia-tion of the chemical potential µ(t) entering (2.21) and (2.23) is known. Tostudy the corresponding time evolution for the chemical potential µ(t) weuse expression (2.13) for the baryon charge bounded to the domain wall.We assume that the thermodynamical equilibrium is maintained betweeninternal and external parts of the nugget such that Tin(t) ' Tout(t). Thisassumption will be justified a posteriori, see discussions after eq.(2.47). Atthe same time the chemical potential is quickly increasing with time insidethe nugget due to decreasing of the nugget’s size. We also assume a fast equi-libration for the chemical potential within the nugget in its entire volume.In other words, we describe the system using one and the same chemicalpotential in vicinity of the wall and deep inside the bubble. Justification forthis assumption will be given later in the text.With this picture in mind, we proceed by differentiating eq.(2.13) withrespect to time to arrive to the following implicit equation relating µ(t) andR(t) at fixed temperature T ,B˙ =Ng4pi2S˙(t)∫d2k⊥[exp( −µ(t)T ) + 1] (2.24)+NgS4pi2µ˙(t)T∫ d2k⊥ [exp( −µ(t)T )][exp( −µ(t)T ) + 1]2 + (fluxes) = 0,292.3. Radius versus pressure in time evolutionwhere term “fluxes” in (2.24) describes the loss of baryonic matter due toannihilation and other processes describing incoming and outgoing fluxes,to be discussed later in the text. The relation (2.24) gives an implicit rela-tion between µ(t) and R(t) which can be used for numerical studies of ourequation (2.19) describing the time evolution of the system.We shall discuss the physics related to incoming and outgoing fluxes inAppendix A. If we neglect this term which describes the loss of baryonicmatter we can analytically solve (2.24) for small µ  T when one can usethe Taylor expansion of the integrals entering (2.24). The result is(µ(t)− µ0) ' pi2T6 ln 2ln(R0R(t)), (2.25)where R0 is initial size of the system at t = t0 while µ0 ' 0 is initial chemicalpotential. One can explicitly see that the chemical potential builds in veryfast when the nugget reduces its size only slightly. This formula (2.25) isonly justified for very small µ(t). For larger values of µ one should use exactformula (2.24).Finally, one should note that at the end of formation at time t → ∞when temperature T  µ the evolution stops, in which case all derivativesvanish, R¨form = R˙form = µ˙form = 0. At this point the nugget assumes itsfinal configuration with size R ' Rform, and the equation (2.19) assumes theform2σRform= Pin =ginµ424pi2− EB(1− µ21µ2), µ ≥ µ1. (2.26)This condition is precisely the equilibrium condition studied in [4] with fewneglected contributions (such as the quark-quark interaction leading to thegap). This is of course the expected result as the time evolution, which isthe subject of the present work, must lead to the equilibrium configurationwhen the free energy assumes its minimum determined by (2.26).One should recall that analysis of the equilibrium presented in ref. [4]with typical QCD parameters strongly suggests that the system indeed fallsinto CS phase when the axion domain wall pressure σ assumes its conven-tional value. At the same time, the equilibrium is not likely to emerge withthe same typical QCD parameters without an additional external pressurerelated to the axion domain wall. In this sense the axion domain wall withextra pressure due to σ 6= 0 plays the role of an additional “squeezer” sta-bilizing the nuggets.The key element of this section is equation (2.24) which is the directconsequence of a spontaneous accretion of the baryon (or antibaryon) charge302.4. Qualitative analysisin the domain wall background as discussed in section 2.2. Precisely thisequation explicitly shows that the chemical potential µ(t) grows very fastwhen the domain wall shrinks as a result of the domain wall pressure σ.The presence of a non-vanishing chemical potential in the vicinity of thedomain wall obviously implies the generation of the binding forces betweenthe fermions and the domain wall, such that a typical bound energy of asingle fermion to the domain wall is of order of µ.A generic solution of equations (2.19) and (2.24), as we discuss in the nextsection, shows an oscillatory behaviour of R(t) with a slow damping of theamplitude such that the system eventually settles down at the equilibriumpoint (2.26). However, even the very first oscillation with initial µ0 ≈ 0leads to very fast growth of the chemical potential µ(t) ≈ T as analyticalestimates represented by eq.(2.25) shows. In the next section we develop aquantitative framework which allows us to analyze our basic equation (2.19)for R(t) where time dependence µ(t) is implicitly expressed in terms of thesame variable R(t) as determined by (2.24).2.4 Qualitative analysisOur goal in this section is to solve for R(t) and therefore µ(t) by solving(2.19) and (2.24), which implicitly relate both variables. We shall observethat a nugget experiences a large number of oscillations during its evolutionwith slow damping rate, and eventually settles down at the equilibrium point(2.26). This behaviour of the system will be coined as “underdamped oscil-lations”. In next section 2.4.1 we formulate some assumptions and presentthe technical details, while the interpretation of the obtained results will bepresented in section 2.4.2. We want to make a number of simplificationsin our analysis in the present section to demonstrate the generic featuresof these oscillations. The numerical studies presented in Appendices A, Band C support our basic picture of oscillatory behaviour advocated in thissection.2.4.1 Assumptions, approximations, simplificationsExact analytical analysis of either (2.19) or (2.24) can be obtained onlyduring the first moment of the initial stage of evolution of the system whenµ is sufficiently small (2.25). We need to understand the behaviour of thesystem for a much longer period of time. Thus, we make two importanttechnical simplifications to proceed with our qualitative analysis. The firstone is to neglect the term in (2.24) describing the fluxes. This assumption312.4. Qualitative analysiswill be supported by some estimates presented in Appendix A which showthat incoming and outgoing fluxes cancel each other with very high accuracy,such that net flux is indeed quite small. Hence, (2.24) is now simplified to:B˙ =ddt{Ng4pi2S∫d2k⊥exp( −µT ) + 1}= 0 (2.27)which means in this approximation, the baryonic charge is roughly conservedin the domain wall background at all times during the evolution of thesystem.As our second simplification we neglect the mass of the fermions in com-parison with temperature T and the chemical potential µ, i.e. we use thefollowing dispersion relation  =√k2⊥ +m2 ' k⊥ in vicinity of the domainwall. This approximation is somewhat justified in QGP and CS phases, andtherefore along the path 3 as shown on Fig. 1.1. It is not literally justifiedfor paths 1 and 2 as in the hadronic phase where the quark mass m should beidentified with the so-called “constituent” quark mass which is proportionalto the chiral condensate. Nevertheless, to simplify the problem we neglectthe mass m(T ) for all paths in our qualitative analysis of the time evolutionas we do not expect any drastic changes in our final outcome as a result ofthis technical simplification. With these assumptions we can approximatethe integral entering eq. (2.27) as follows,∫ ∞0dk⊥ · k⊥e(k⊥)−µT + 1= T 2 · I(µT) (2.28)I(µT) ' pi26+12(µT)2 − pi212e−µ/T +O(µTe−µ/T )where the omitted terms ∼ µT e−µ/T will be neglected thereafter, as theywill never dominate in neither small nor large limit of µ. One can numer-ically check that approximation (2.28) describes the relevant integral I( µT )sufficiently well in the entire parametrical space of µ/T , see Appendix Cwith corresponding analysis. As a quick test of this approximation one cancheck that approximate expression (2.28) reproduces an exact (in the smallµ limit) expression (2.25) with accuracy of order 15%, which is more thansufficient for our qualitative studies of this section.As mentioned above, if flux term (2.24) is neglected, the curly-bracketterm in (2.27) is a conserved quantity. Equating it to its initial values whereS(t = 0) = 4piR20, µ(t = 0) = µ0 ' 0 one arrives toT 2R2[pi26+12(µT)2 − pi212e−µ/T]=pi212T 20R20. (2.29)322.4. Qualitative analysisIn what follows we assume that the thermodynamical equilibration is estab-lished very quickly such that one can approximate T ' T0 during the timeevolution as we already discussed in the previous section 2.3. To simplifyfurther the system we wish to represent the equation relating R and µ/T inthe following formf(R) ≡ pi26[12(R0R)2− 1]=12(µT)2 − pi212e−µ/T , (2.30)where we introduced function f(R) for convenience of the analysis whichfollows. Essentially, the idea here is to simplify the basic equation (2.19) asmuch as possible to express the µ(t)− dependent terms entering through thepressure (2.20) in terms of R(t) such that the equation (2.19) would assumea conventional differential equation form for a single variable R(t).Our next step is to simplify the expression for the Fermi pressure (2.21)entering (2.20) using the same procedure we used to approximate formula(2.28), i.e.P(Fermi)in =gin6pi2∫ ∞0k3dkexp((k)−µT)+ 1(2.31)' ginT 46pi2{7pi460+pi22(µT)2 − 7pi4120e−µ/T +14(µT)4+O(µTe−µ/T )}' ginT 46{7pi260+[12(µT)2 − pi212e−µ/T]+14pi2(µT)4}+ginT 46{pi240e−µ/T +O(µTe−µ/T )}.In what follows we neglect the last line in eq. (2.31). The justificationfor this procedure is the same as before: it produces a small contributionin entire region of µ in comparison with accounted terms. The technicaladvantage for this procedure is the possibility to rewrite (2.31) in terms offunction of R(t), rather than µ(t) using our relation (2.30).The formula in the square bracket in (2.31) is just f(R) defined by (2.30).The remaining( µT)4term can be also expressed in terms of R by taking332.4. Qualitative analysissquare of (2.30):f2(R) =[12(µT)2 − pi212e−µ/T]2' 14(µT)4+(pi212)2+O(µTe−µ/T) (2.32)where the correction term ∼ O(µe−µ/T ) will be dropped in what follows, asbefore. Thus, we approximate P(Fermi)in in terms of R(t) as followsP(Fermi)in 'ginT 46[7pi260+ f(R) +f2(R)pi2− pi2144]. (2.33)The expression for the Fermi pressure P(Fermi)in (R) now is expressed in termsof R rather than in terms of µ as in the original expression (2.21).We wish to simplify the expression for P(bag const)in (µ) entering (2.20) ina similar manner to express P(bag const)in in terms of R. This contributionbecomes important as discussed after eq. (2.23) only for sufficiently large µ.In this region f(R) can be well approximated asf(R) ' 12(µT)2, µ T (2.34)so that we haveP bagin ' −EB · θ(√2f(R)− µ1T)(1− µ212T 2f(R)). (2.35)As a result of these simplifications and approximations the pressure termwhich enters the basic equation (2.19), ∆P (µ) ≡ [Pin(µ)−Pout(t)] which wasinitially formulated in terms of the chemical potential µ inside the bubblecan be now written entirely in terms of a single variable, the size of thebubble R(t):∆P [f(R)] ' ginpi26T 4[79720− gout15gin+f(R)pi2+f2(R)pi4]−EB · θ(√2f(R)− µ1T)(1− µ212T 2f(R)), (2.36)where f(R) is defined by eq. (2.30). With these technical simplificationsthe basic equation (2.19) can now be written as a second order differentialequation entirely in terms of R(t) rather than µ:σR¨(t) = −2σR− σR˙2R+ ∆P [f(R)]− 4η R˙R, (2.37)342.4. Qualitative analysiswith ∆P [f(R)] determined by eq. (2.36).This equation can be solved numerically. In fact, it is precisely thesubject of Appendix B. However, the most important quantitative features ofthe obtained solution can be understood without any numerical studies, butrather using a simplified analytical analysis, which is precisely the subjectof the next section.2.4.2 Time evolutionAs we already mentioned a nugget assumes its final form at t → ∞ whenall time derivatives vanish and the equation for the equilibrium is givenby (2.26) at T = 0. In this section we generalize this equation for theequilibrium by defining Rform(T ) as the solution of eq.(2.38), see below, atT 6= 0. In other words, the starting point of the present analysis at T 6= 0 isthe equilibrium condition when the “potential” energy assumes its minimalvalue. The corresponding minimum condition is determined by equation2σRform= ∆P (Rform), (2.38)where ∆P (Rform) is defined by eq.(2.36). This condition obviously reducesto eq. (2.26) at t→∞ when µ T .We follow the conventional technique and expand (2.37) around the equi-librium value Rform(T ) to arrive to an equation for a simple damping oscil-lator:d2(δR)dt2+2τd(δR)dt+ ω2(δR) = 0, (2.39)where δR ≡ [R(t) − Rform] describes the deviation from the equilibriumposition, while new parameters τ and ω describe the effective damping coef-ficient and frequency of the oscillations. Both new coefficients are expressedin terms of the original parameters entering (2.37) and are given byτ =σ2ηRform (2.40a)ω2 = − 1σd∆P (R)dR∣∣∣∣Rform− 2R2form. (2.40b)The expansion (2.39) is justified, of course, only for small oscillations aboutthe minimum determined by eq.(2.38), while the oscillations determinedby original equation (2.37) are obviously not small. However, our simpleanalytical treatment (2.39) is quite instructive and gives a good qualitative352.4. Qualitative analysisunderstanding of the system. Our numerical studies presented in AppendixB fully support the qualitative picture presented below.We start our qualitative analysis with estimates of the parameter ω whichdepends on d∆P (R)dR computed at R = Rform according to (2.40b). First of all,in this qualitative analysis we neglect the bag constant term P(bag constant)inbecause it only starts to play a role for sufficiently large µ ≥ µ1 ∼ 330 MeV,when formation is almost completed. This term obviously cannot change thequalitative behaviour of the system discussed below. Our numerical studiespresented in Appendix B (where the bag constant term ∼ EB is included inthe analysis) support this claim.The key element for our simplified analysis is the observation that theratio (R0/Rform)2 ≥ 14 is expected to be numerically large number. Thisexpectation will be soon confirmed a posteriori. This observation consid-erably simplifies our qualitative analysis because in this case ∆P (Rform)defined by (2.36) can be approximated by a single term ∼ f2(R) in squarebrackets in (2.36) as this term essentially saturates ∆P (Rform). This is be-cause the function f(R)/pi2 becomes numerically large in the relevant regionf(R)/pi2 ∼ (R0/Rform)2 according to (2.30).With these simplifications we can now estimate ω2 as followsω2 ≈(ginpi2216)·(T 4σRform)·(R0Rform)4−(2R2form). (2.41)To simplify analysis further one can represent the last term as(2R2form)=(1Rform)·(∆P (Rform)σ), (2.42)and keep the leading term ∼ f2(R) in expression for ∆P (Rform). One caneasily convince yourself that ω2 > 0 is always positive in this approximationsuch that the condition for a desired underdamped oscillations assumes asimple formf(Rform)pi2& 1 ⇒(R0Rform)&√14 (2.43)when ∆P (Rform) defined by (2.36) is dominated by a single term ∼ ( fpi2 )2,which itself can be approximated by the leading quadratic term ∼ (R0R )2according to (2.30). Our numerical studies presented in Appendix B supportthe numerical estimate (2.43).362.4. Qualitative analysisOne can also check that if condition (2.43) is not satisfied than systemshows an “over-damped” behaviour when very few oscillations occur beforecomplete collapse of the system, in which case the nuggets obviously do notform. These short-lived bubbles will never get to a stage when the tempera-ture drops below the critical value TCS . Therefore, a CS phase cannot formin these “short-lived” bubbles. It should be contrasted with “long-lived”bubbles with much longer formation-time of order τ , see comments below.The condition (2.43) is extremely important for our analysis. It es-sentially states that the initial size of a closed bubble R0 must be suffi-ciently large for a successful formation of a nugget of size Rform. On theother hand, a formation of very large closed bubbles is strongly suppressed∼ exp[−(R0/ξ)2] by the KZ mechanism as reviewed in section 2.1. Thisconstraint will be important in our estimation of a suppression factor insection 3.3 due to necessity to form a sufficiently large bubble (2.43) duringthe initial stage of formation.Assuming that condition (2.43) is satisfied we estimate a typical oscilla-tion frequency as followsω ∼ 1Rform∼ ma, tosc ' ω−1 ' m−1a (2.44)where we used the scaling properties (1.3) to relate the nugget’s size Rformwith the axion mass ma. One should emphasize that the estimate (2.44)is not sensitive to any approximations and simplifications we have madein our qualitative treatment of the time evolution in this section. In fact,all parameters entering relation (2.44) are expressible in terms of the QCDscale ΛQCD and a single “external” parameter, the axion mass ma, whichwe keep unspecified at this point. Of course we always assume that theaxion mass may take any value from the observationally allowed window10−6eV . ma . 10−3eV.We now turn our attention to the damping coefficient defined in termsof the original parameters by eq. (2.40a). It is convenient to estimate thedimensionless combination ωτ as followsωτ ' 1Rform·(σ2ηRform)' σ2η∼ mpima∼ 1011, (2.45)where we substituted ω ∼ R−1form according to (2.44) and assumed that η ∼m3pi has conventional QCD scale of order fm−3 while the wall tension σ canbe approximated with high accuracy as σ ' m4pi/ma. This relation impliesthat the damping is extremely slow on the QCD scales. Therefore, the372.4. Qualitative analysissolution describing the time evolution of a “long-lived” bubble can be wellapproximated as followsR(t) = Rform + (R0 −Rform)e−t/τ cosωt (2.46)which is obviously a solution of the approximate equation (2.39). This so-lution represents an “under-damped” oscillating R(t) with frequency ω ∼1Rformand damping time τ ∼ σ2ηRform. Precisely these “long-lived” bubbleswill eventually form the DM nuggets.The time scale (2.45) is very suggestive and implies that the dampingterm starts to play a role on very large scales when the cosmological expan-sion of the Universe with the typical scale t0 ' 10−4s must be taken intoaccount. We have not included the corresponding temperature variation inour studies because on the QCD scales (which is the subject of the presentstudies) the corresponding variations are negligible. However, the estimate(2.45) shows that for a proper analysis of the time scales τ the expansion ofthe Universe (and related to the expansion the temperature variation) mustbe included. The corresponding studies are beyond the scope of the presentwork. However, the important comment we would like to make here is thatthe emergent large time scale (2.45) is fully consistent with our anticipationthat the temperature of the Universe drops approximately by a factor of ∼ 3or so when a CS phase forms in interior of the nugget during the formationperiod. It is quite obvious that if the time scale (2.45) were considerablyshorter than the cosmological time scale t0 ' 10−4s than the temperatureT ∼ t−1/2 inside the nugget could not drop sufficiently deep into the regionwhere CS sets in as plotted on Fig.1.1. Fortunately, the timescale (2.45) islong enough and automatically satisfies this requirement.Now we want to elaborate on one more element of the dynamics which isalso important for a successful formation of the nuggets. To be more specific,we want to discuss the flux of particle exchange, which was ignored in ourqualitative analysis in this section and which is estimated in Appendix A.This flux describes the rate of number of particle flowing between inside andoutside the system, which can be appreciably large even if the net baryonicflux is negligibly small. To be more precise, there are two kinds of fluxes,both investigated in Appendix A, that we are discussing in this thesis: thenet flux of baryonic charge ∆Φ ≡ Φ⇒−Φ⇐, and the average flux of particlenumber 〈Φ〉 ≡ 12(Φ⇒ + Φ⇐). The first one corresponds to the flux termentering eq. (2.24); while the latter is important in understanding what isthe typical time scale for a complete “refill” of the particles during the timeevolution. The last question is important for understanding of the time scalefor thermal equilibration.382.4. Qualitative analysisWe start our analysis with discussions of an average flux 〈Φ〉 at smallchemical potential. It is estimated to be 〈Φ〉 ' 1 fm−3 according to Ap-pendix A. The magnitude of this flux can be fully appreciated by computingthe total number of particle exchange per one cycle of the oscillation2piω· 4piR2form · 〈Φ〉 ∼ R3formfm−3 ∼ |B| , (2.47)where ω is a typical frequency oscillation estimated in (2.44) while |B| is thetotal number of particles (quarks and antiquark) stored in the nugget. Thephysical meaning of this estimate is that a nugget can in principle entirelyrefill its interior with “fresh” particles within a few cycles of exchange. Sim-ilar estimate for the net baryon flux which includes ∆Φ is suppressed, seeAppendix A.The main reason for emergence of this large scale in expression (2.47)is a long time scale of a single cycle (2.44) which is determined by theaxion mass ma rather than by QCD physics. Nevertheless, estimate (2.47)is quite remarkable and shows that even very low rate of chemical potentialaccretion of (anti)quarks being tracked per oscillation, the high exchangerate (2.47) is still sufficient enough to turn a baryonically neutral nugget intoone completely filled with (anti)quarks. When the quarks become effectivelymassive as it happens in hadronic and CS phases, the flux for the exchangeof the baryon charge is drastically decreased by a factor ∼ exp(−m/T ).The same estimate (2.47) essentially holds for exchange of almost mass-less Nambu-Goldstone bosons for sufficiently high temperature. In fact, thelightest degrees of freedom play the crucial role in cooling processes of theinterior of the nugget as these particles can easily penetrate the sharp do-main wall structure. Therefore, the high exchange rate between exteriorand interior of a nugget essentially implies that the thermal equilibrium ismaintained in our system with very high precision due to a huge rate percycle (2.47) when large number of degrees of freedom ∼ B have a chance oforder one to interact with “fresh” particles from the exterior during a singlecycle. Therefore, our assumption on thermal equilibrium between interiorand exterior is justified a posteriori.We conclude this section with few important comments. The most im-portant result of this section is that the nuggets can be formed during theQCD phase transition provided the initial size of the nuggets is sufficientlylarge as stated in eq.(2.43), in which case they survive the evolution. Thekey role in this successful formation plays, of course, the effect of “localspontaneous violation” of the baryon symmetry as discussed in section 2.2and explicitly expressed by eqs.(2.11) and (2.12). One should emphasize392.4. Qualitative analysisthat our qualitative analyses in this section are fully supported by numeri-cal studies presented in Appendices A and B. Therefore, we do not expectthat any numerical simplifications in our analysis may drastically changethe basic qualitative results presented in this section.Another important point is the observation (2.44) that a typical timescale for the oscillations is of order tosc ' ω−1 ' m−1a . Both these estimateswill be crucial elements in our analysis presented in next chapter 3: equation(2.43) will be important in estimate for efficiency of a bubble formation witha large size ∼ R0, while equation (2.44) will play a key role in our argumentssuggesting a coherent preferential formation of one type of nuggets (baryonicor antibaryonic) on largest possible scale of the visible Universe.40Chapter 3Baryon Charge Separation.Correlation on CosmologicalScales.Until this section we mostly concentrated on the time evolution of a singlenugget (or anti-nugget). The main lesson of our previous discussions is thatsuch nuggets can be formed, remain stable configurations, and therefore,can serve as the dark matter candidates. In other words, the focus of ourprevious studies was a problem of a local separation of charges on small scalesof order nugget’s size. The key element of that separation of charges is eq.(2.11) which can be thought as a local version of spontaneous symmetrybreaking of the baryon charge as explained in section 2.2. However, on alarger scale it is quite obvious that equal number of nuggets and anti-nuggetswill be formed as a result of an exact symmetry as we discuss below.This symmetry, however, does not hold anymore on large scales if the ax-ion CP -odd coupling is included into consideration, which eventually leadsto very generic, essentially insensitive to most parameters, consequence ofthis framework represented by eq.(1.1), which is the subject of next sub-sections 3.1 , 3.2. The subsection 3.3 is devoted to some more specific andmodel-dependent consequences of this framework. In particular, we want toestimate a suppression factor related to the necessary to form a large sizebubble (2.43) in KZ mechanism.3.1 Coherent axion field as the source of CPviolationFirst of all, let us show that the baryon charge hidden in nuggets on averageis equal to the baryon charge hidden in anti-nuggets, of course with signminus. Indeed, the analysis of the anti-nuggets can be achieved by flippingthe sign of the chemical potential in eq. (2.9), i.e. µ → −µ. One canrestore the original form of the µ term in Lagrangian (2.9) by replacing413.1. Coherent axion field as the source of CP violationθ1 → −θ1 and θ2 → −θ2. Finally, one should change the signs for theaxion θ and the pseudo-scalar singlet η′ meson represented by φ field in theinteraction term (2.10) to restore the original form of the Lagrangian. Thesesymmetry arguments imply that as long as the pseudo-scalar axion fieldfluctuates around zero as conventional pseudo-scalar fields (as pi, η′ mesons,for example), the theory remains invariant under P and CP symmetries.Without this symmetry the number density and size distribution of thenuggets and anti-nuggets could be drastically different18.Therefore, the symmetry arguments suggest that on average an equalnumber of nuggets and anti-nuggets would form if the axion field is repre-sented by a conventional quantum fluctuating field oscillating around zeropoint. If it were the case, the baryons and antibaryons would continue to an-nihilate each other as well as annihilate with the nuggets and anti-nuggets inour framework. Eventually it would lead to the Universe with large amountof dark matter in form of nuggets and anti-nuggets (they are far away fromeach other, therefore they do not annihilate each other) and no visible mat-ter. However, the axion dynamics which is determined by the axion fieldcorrelated on the scale of the entire Universe leads to a preferential forma-tion of a specific type of nuggets on the same large scales where the axionfield is correlated as we argue below.First of all we want to argue that the time dependent axion field impliesthat there is an additional coupling to fermions (3.1). Indeed, by makingthe time-dependent U(1)A chiral transformation in the path integral one canalways represent the conventional θ term in the following form∆L4 = µ5(t)Ψ¯γ0γ5Ψ µ5 ≡ θ˙. (3.1)In this formula µ5 ≡ θ˙ can be thought as the chiral chemical potential. Manyinteresting properties emerge in the systems if µ5 is generated. In fact, ithas been an active area of research in recent years, mostly due to veryinteresting experimental data suggesting that the µ5 term can be generatedin heavy ion collisions, see original paper [52] and recent reviews [53–55]for the details. In the present context the µ5 term is generated as a resultof the axion dynamics. As a matter of fact, the original studies [52] weremotivated by the proposal that the separation of the baryon charges whichmay occur in early Universe, as advocated in this thesis, could be tested inlaboratory experiments with heavy ion collisions.18If pi meson condensation were occur in nuclear matter it would unambiguously implythat the CP invariance is broken in such a phase. Some of the phases in CS systems indeedbreak the CP invariance as a result of condensation of a pseudo-scalar Nambu-Goldstonebosons.423.1. Coherent axion field as the source of CP violationNow we are prepared to formulate the main claim of this section whichcan be stated as follows. When interaction (2.10), (3.1) is introduced intothe system there will be a preferential evolution in the system of the nuggetsversus anti-nuggets provided that nuggets and anti-nuggets had been alreadyformed and chemical potential µ had been already generated locally insidethe nuggets as described in the previous section 2.4. As we already explainedearlier, the generation of µ can be interpreted as a “local violation” of Cinvariance in the system.This preferential evolution is correlated with the CP-odd parameter onthe scales where the axion field θ(x) is coherent. In our arguments presentedbelow we make a standard assumption that the initial value of θ(x) and itstime derivative θ˙(x) are correlated on the entire observable Universe, suchthat µ5 ≡ θ˙ is also correlated on the same large scale. This is the standardassumption in most studies on axion physics when one computes the presentdensity of axions due to the misalignment mechanism and/or the domainwall network decay, see recent papers [9–16, 37].For our present studies the key element is that the dynamics of the axionfield until the QCD phase transition is determined by the coherent state ofaxions at rest such that [9–16]:θ(t) ∼ Ct3/4cos∫ tdt′ωa(t′), ω2a(t) = m2a(t) +316t2, (3.2)where C is a constant, and t = 12H is the cosmic time. This formula suggeststhat for ma(t)t  1 when the axion potential is sufficiently strongly tiltedthe chiral chemical potential is essentially determined by the axion mass attime tµ5(t) = θ˙(t) ∼ ωa(t) ' ma(t). (3.3)The crucial point is that θ(t) is one and the same in the entire Universeas it is correlated on the Universe size scale. Another important remark isthat the axion field θ(t) continues to oscillate with frequency (3.3) until theQCD phase transition at Tc, though its absolute value |θ/θ0| ∼ 0.01 mightbe few orders of magnitude lower at Tc ' 170 MeV than its original valueθ0 at T ' 1 GeV when the axion field only started to roll, see e.g. [45]. Aswe discuss below, the relevant physics is not very sensitive to an absolutevalue of |θ(t)| in this regime, and therefore, we do not elaborate further onthis rather technical and computational element of the axion dynamics, seefootnote 19 below with comments on this matter.In the context of the nugget’s evolution (accretion of the baryon charge)this claim implies that on the entire Universe size scale with one and the433.1. Coherent axion field as the source of CP violationsame sign of θ(t) a specific single type of nuggets will prevail in terms ofthe number density and sizes. Indeed, one can present the same arguments(see the beginning of this section) with flipping the sign µ → −µ with theonly difference is that the interaction (2.10) prevents us from making thevariable change θ(i) ↔ −θ(i) for a given θ(t) because it changes its formunder θ(i) ↔ −θ(i). In other words, slow varying (on the QCD scale) CPviolating terms (2.10), (3.1) lead to a preferential evolution of the systemfor a specific species of the nuggets with a given sign of µ.Indeed, it has been known for quite some time, see e.g. [56, 57] that in thepresence of θ 6= 0 a large number of different CP violating effects take place.In particular, the Nambu-Goldstone bosons become a mixture of pseudo-scalar and scalar fields, their masses are drastically different from θ = 0values. Furthermore, the quark chiral 〈ψ¯ψ〉 and the gluon 〈G2〉 condensatesbecome the superposition with their pseudo-scalar counterparts 〈ψ¯γ5ψ〉 and〈GG˜〉 such that entire hadron spectrum and their interactions modify in thepresence of θ 6= 0. All these strong effects, of course, are proportional to θ,and therefore numerically suppressed in case under consideration (3.2) by afactor |θ/θ0| ∼ 10−2 in the vicinity of the QCD phase transition. Naively,this small numerical factor |θ/θ0| ∼ 10−2 may lead only to minor effects∼ 10−2. However, the crucial point is that while the coupling (2.10) of theaxion background field with quarks is indeed relatively small on the QCDscales, it is nevertheless effectively long-ranged and long-lasting in contrastwith conventional QCD interactions. As a result, this coherent CP oddcoupling may produce large effects of order of one as we argue below.Indeed, as we discussed in previous section 2.4 a typical oscillation timetosc when the baryon charge accretes on the wall is of order tosc ∼ m−1aaccording to eq. (2.44). But this time scale tosc ∼ m−1a is precisely thetime scale when θ˙ = ma(t) varies according to (3.3). Therefore, while thedynamical fermi fields θ1, θ2 defined by (2.8) fluctuate with a typical scaleof order ΛQCD  ma, the coherent variation of these fields will occur duringa long (on the QCD scales) coherent process when a nugget makes a singlecycle. These coherent corrections are expected to be different for nuggets(positive µ) and anti-nuggets (negative µ) as a result of many C and CPviolating effects such as scattering, transmission, reflection, annihilation,evaporation, mixing of the scalar and pseudo-scalar condensates, etc whichare all responsible for the accretion of the baryon charge on a nugget duringits long evolution.Important comment here is that each quark experiences a small differencein interacting with the domain wall surrounding nuggets or anti-nuggetsduring every single QCD event (mentioned above) with typical QCD time443.1. Coherent axion field as the source of CP violationscale Λ−1QCD. However, the number of the coherent QCD events ncoherentduring a long single cycle is very largencoherent ∼ ΛQCDtosc ∼ ΛQCDma∼ 1010  1. (3.4)Therefore, a net effect during every single cycle will be order of one, in spiteof the fact that each given QCD event is proportional to the axion field θ(t)and could be quite small.The argument presented above holds as long as the axion field remainscoherent, see also a comment at the very end of this subsection. In otherwords, a small but non vanishing coherent CP violating parameter θ(t) playsthe role of catalyst which determines a preferred direction for separation ofthe baryon charges on the Universe scale. This role of CP violation in ourframework is very different from conventional “baryogenesis” mechanismswhen CP violating parameter explicitly enters the final expression for thebaryon charge production.The corresponding large coherent corrections during a single cycle toscimply that the fast fluctuating fields θ1, θ2 (which effectively describe thedynamics of the fermions living on the wall according to (2.8)) receive largecorrections during every single cycle∆θ1(t) ∼ ∆θ2(t) ∼ 1. (3.5)These changes of order one of the strongly interacting θ1, θ2- fields lead tomodification of the accreted baryon charge per single cycle per single degreeof freedom∆N ∼ (∆θ1 + ∆θ2) ∼ 1 (3.6)on the nuggets according to (2.12). One should emphasize that the correc-tions (3.6) are expected to be different for nuggets and anti-nuggets becausethe interaction (3.1), (2.10) which is responsible for these corrections (3.6)breaks the symmetry between nuggets and anti-nuggets when µ → −µ asdiscussed above.Precise computations of these coherent CP violating effects are hard tocarry out explicitly as it requires a solution of many-body problem of thecoherent wall fermions with surrounding environment in the background ofaxion field (3.2) when a large number of C and CP violating effects take placeand drastically modify evolution of nuggets versus anti-nuggets. A largenumber of cycles of every individual nugget (anti-nugget) also introduces ahuge uncertainty in computations of ∆N during the time evolution when a453.2. Nuggets vs anti-nuggets on the large scale. Generic consequences.single cycle leads to the effect of order one, with possible opposite sign for aconsequent cycle. In other words, it is very hard to predict what would be thefinal outcome of the system after a large number of cycles when each cycleproduces the effect of order 1. We expect that the final result would be againof order one. Such a computation is beyond the scope of the present work.Therefore, in what follows we introduce a phenomenological parameter c(T )of order one to account for these effects. All the observables will be expressedin terms of this single phenomenological parameter c(T ) ∼ 1, see eq. (3.7).Our final comment in this subsection is as follows. The charge separationeffect on largest possible scales is only possible when the axion field (3.2) iscoherent on the scales of the Universe. This coherence is known to occurin conventional studies on the dynamics of the axion field in the vicinityof the QCD phase transition [9–14, 16, 45]. At the same time, soon afterthe QCD phase transition the dominant part of the axion field transfersits energy to the free propagating on-shell axions (which is the subject ofaxion search experiments [9–14, 16]). These randomly distributed free axionsare not in coherent state anymore. Therefore, the coherent accumulationeffect which leads to a preferential formation of one species of nuggets, asdiscussed above, ceases to be operational at the moment of decoherencetdec when the description in terms of the coherent axion field (3.2) breaksdown19. The baryon asymmetry we observe today in this framework is aresult of accumulation of the charge separation effect from the beginning ofthe nugget’s formation until this very last “freeze-out” moment determinedby tdec.3.2 Nuggets vs anti-nuggets on the large scale.Generic consequences.As we already mentioned to make any precise dynamical computations of∆N ∼ 1 due to the coherent axion field (3.2) is a hard problem of stronglycoupled QCD at θ 6= 0. In order to effectively account for these coherenteffects one can introduce an unknown coefficient c(T ) of order one as followsBantinuggets = c(T ) ·Bnuggets, where |c(T )| ∼ 1, (3.7)19The decoherence time tdec is not entirely determined by absolute value of amplitudeof the axion field (3.2). In fact, the amplitude could be quite small, but the field remainscoherent on large scales. The computation of the decoherence time tdec is a hard problemof QFT, similar to a problem in quantum optics when initially coherent light becomesde-coherent superposition of uncorrelated photons.463.2. Nuggets vs anti-nuggets on the large scale. Generic consequences.where c(T ) is obviously a negative constant of order one. We emphasize thatthe main claim of this section represented by eq. (3.7) is not very sensitiveto the axion mass ma(T ) nor to the magnitude of θ(T ) at the QCD phasetransition when the bubbles start to oscillate and slowly accrete the baryoncharge. The only crucial factor in our arguments is that the typical variationof θ(t) is determined by the axion mass (3.3), which is the same order ofmagnitude as t−1osc, and furthermore, this variation is correlated on the scalewhere the axion field (3.2) can be represented by the coherent superpositionof the axions at rest.The key relation of this framework (3.7) unambiguously implies that thebaryon charge in form of the visible matter can be also expressed in termsof the same coefficient c(T ) ∼ 1 as followsBvisible = −Bantinuggets −Bnuggets. (3.8)Using eq. (3.7) it can be rewritten asBvisible ≡ (Bbaryons +Bantibaryons) (3.9)= − [1 + c(T )]Bnuggets = −[1 +1c(T )]Bantinuggets.The same relation can be also represented in terms of the measured ob-servables Ωvisible and Ωdark at later times when only the baryons (and notantibaryons) contribute to the visible component20Ωdark '(1 + |c(T )||1 + c(T )|)· Ωvisible at T ≤ Tform. (3.10)One should emphasize that the relation (3.9) holds as long as the thermalequilibrium is maintained, which we assume to be the case. Another impor-tant comment is that each individual contribution |Bbaryons| ∼ |Bantibaryons|entering (3.9) is many orders of magnitude greater than the baryon chargehidden in the form of the nuggets and anti-nuggets at earlier times whenTc > T > Tform. It is just their total baryon charge which is labeled asBvisible and representing the net baryon charge of the visible matter is the20In eq. (3.10) we neglect the differences (due to different gaps) between the energyper baryon charge in hadronic and CS phases to simplify notations. The correspondingcorrections in energy per baryon charge in hadronic and CS phases, in principle, canbe explicitly computed from the first principles. However, we ignore these modificationsin the present work. This correction obviously does not change the main claim of thisproposal stating that Ωvisible ≈ Ωdark.473.2. Nuggets vs anti-nuggets on the large scale. Generic consequences.same order of magnitude (at all times) as the net baryon charge hidden inthe form of the nuggets and anti-nuggets according to (3.8).The baryons continue to annihilate each other (as well as baryon chargehidden in the nuggets) until the temperature reaches Tform when all vis-ible antibaryons get annihilated, while visible baryons remain in the sys-tem and represent the visible matter we observe today. It corresponds toc(Tform) ' −1.5 as estimated below if one neglects the differences in gapsin CS and hadronic phases, see footnote 20. After this temperature thenuggets essentially assume their final form, and do not loose or gain muchof the baryon charge from outside. The rare events of the annihilationbetween anti-nuggets and visible baryons continue to occur. In fact, theobservational excess of radiation in different frequency bands, reviewed insection 1.2, is a result of these rare annihilation events at present time.The generic consequence of this framework represented by eqs. (3.7),(3.9), (3.10) takes the following form at this time Tform for c(Tform) ' −1.5which corresponds to the case when the nuggets saturate entire dark matterdensity:Bvisible ' 12Bnuggets ' −13Bantinuggets,Ωdark ' 5 · Ωvisible (3.11)which is identically the same relation (1.2) presented in Introduction. Therelation (3.11) emerges due to the fact that all components of matter, visi-ble and dark, are proportional to one and the same dimensional parameterΛQCD, see footnote 20 with a comment on this approximation. In formula(3.11) Bnuggets and Bantinuggets contribute to Ωdark, while Bvisible obviouslycontributes to Ωvisible. The coefficient ∼ 5 in relation Ωdark ' 5 · Ωvisibleis obviously not universal, but relation (1.1) is universal, and very genericconsequence of the entire framework, which was the main motivation for theproposal [4, 5].For example, if c(Tform) ' −2 then the corresponding relation (3.10)between the dark matter and the visible matter would assume the formΩdark ' 3 · Ωvisible. Such a relation implies that there is a plenty of roomfor other types of dark matter to saturate the observed ratio Ωobserveddark '5 · Ωobservedvisible . This comment will be quite important in our discussions inchapter 4 where we comment on implications of this framework for otheraxion search experiments.One should emphasize once again that the generic consequences of theframework represented by (1.1), (3.10) are not sensitive to any specific pa-rameters such as efficiency of the domain wall production or the magnitude483.3. nB/nγ ratio. Model dependent estimates.of θ at the QCD phase transition, which could be quite small, see footnote19 with few comments on that. Nevertheless, precisely the coupling with thecoherent CP odd axion field plays a crucial role in generation of |c(T )| 6= 1,i.e. the axion plays the role of catalyst in the baryon charge separation effecton the largest possible scales. Some other observables which are sensitive tothe dynamical characteristics (e.g. efficiency of the domain wall production)will be discussed below.3.3 nB/nγ ratio. Model dependent estimates.The time evolution of the dark matter within this framework is amazinglysimple. The relations (3.7), (3.8), (3.9) hold at all times. The baryoncharges of the nuggets and anti-nuggets vary until its radiusR(T ) assumes itsequilibrium value as described in sections 2.3, 2.4. It happens approximatelyat time when the CS phase forms in interior of the nuggets, which can beestimated as TCS ' 0.6∆ ' 60 MeV, where ∆ ' 100 MeV is the gap of theCS phase. After this temperature the nuggets essentially assume their finalform, with very little variation in size (and baryon charge). The rare eventsof the annihilation of course continue to occur even for lower tempearures.In fact, the observational consequences reviewed in section 1.2 is a result ofthese annihilation events at present time.The variation of the visible matter Bvisible demonstrates much more dras-tic changes after the QCD phase transition at Tc because the correspondingnumber density is proportional to exp(−mN/T ) such that at the moment offormation Tform ≈ 40 MeV the baryon to entropy ratio assumes its presentvalue (1.5) which we express as followsη ≡ nBnγ' Bvisible/Vnγ∼ 10−10, nB ≡ BvisibleV. (3.12)If the nuggets and anti-nuggets were not present at this temperature theconventional baryons and antibaryons would continue to annihilate eachother until the density would be 9 orders of magnitude smaller than observed(3.12) when the temperature will be around T ' 22 MeV. Conventionalbaryogenesis resolves this “annihilation catastrophe” by producing extrabaryons in early times, see e.g. review [20], while in our framework extrabaryons and antibaryons are hidden in form of the macroscopically largenuggets.In our framework the ratio (3.12) can be rewritten in terms of thenugget’s density as the baryon charge in form of the visible matter and493.3. nB/nγ ratio. Model dependent estimates.in form of the nuggets are related to each other according to (3.9). Thisrelation allows us to infer what efficiency is required for the bubbles to beformed and survive until the present time when the observed ratio is mea-sured (3.12).One should emphasize that any small factors which normally enter thecomputations in conventional baryogenesis (such as C and CP violating pa-rameters) do not enter in the estimates presented below in our frameworkas a result of two effects. First, the C violation enters the computation as aresult of generation of the chemical potential µ as described in section 2.2.It is expressed in terms of spontaneous accretion of the baryon charge on thesurface of the nuggets as given by eq. (2.12) which effectively generates thechemical potential (2.25), which can be thought as the local violation of thesymmetry on the scale of a single nugget. Secondly, the CP violation entersthe computation in form of the coupling with the coherent axion field (3.1).Precisely this coupling as we argued above leads to removing of the degen-eracy between nuggets and anti-nuggets formally expressed as c(T ) ∼ 1 ineq. (3.7). Therefore, the only small parameter we anticipate in our esti-mates below is due to some suppression of the closed bubbles which mustbe formed with sufficiently large sizes during the QCD phase transition.We cannot compute the probability for the bubble formation as it obvi-ously requires the numerical simulations, which is beyond the scope of thepresent work. Instead, we go backward and ask the question: What shouldbe the efficiency of the bubble formation at the QCD phase transition inorder to accommodate the observed ratio (3.12)?With these comments in mind we proceed with our estimates as follows.First, from (3.9), (3.11) we infer that the baryon charge hidden in the nuggetsand anti-nuggets is the same order of magnitude as the baryon charge of thevisible baryons at Tform at the end of the formation, i.e.Bnuggets/Vnγ& 10−10, (3.13)where we use sign & instead of ≈ used in eq (3.12) to emphasize that thereis long time for equilibration between the moment TCS ' 0.6∆ ' 60 MeVwhen CS phase forms in interior of the nuggets and Tform ' 40 MeV whenall antibaryons of the visible matter get annihilated, corresponding to thepresent observed value (3.12). During this period the equilibrium betweenthe visible matter and the baryons from nuggets is maintained, and someportion of the nugget’s baryon charge might be annihilated by the visiblematter. It explains our sign & used in eq. (3.13).503.3. nB/nγ ratio. Model dependent estimates.The relation (3.13) implies that the number density of nuggets and anti-nuggets can be estimated as〈B〉nnuggetsnγ& 10−10, 〈B〉nnuggets ≡ BnuggetsV, (3.14)where 〈B〉 is the average baryon charge of a single nugget at Tform.Now we want to estimate the same ratio (3.14) using the Kibble-Zurek(KZ) mechanism[41–43] reviewed in section 2.1. The basic idea of the KZmechanism is that the total area of the crumpled, twisted and folded domainwall is proportional to the volume of the system, and can be estimated asfollows:S(total DW) =Vξ(T ), (3.15)where ξ(T ) is the correlation length which is defined as an average distancebetween crumpled domain walls at temperature T . Largest part of thewall belongs to the percolated large cluster. It is known that some closedwalls (bubbles) with typical size ξ(T ) will be also formed. These bubbleswith sufficiently large size R ∼ ξ(T ) will eventually become nuggets. Weintroduce parameter γ to account for the suppression related to the closedbubble formation. In other words, we defineSnuggets = γS(total DW) =γVξ(T ), γ  1. (3.16)At the same time total area of the nuggets Snuggets can be estimated asSnuggets = 4piR20(T ) [V · nnuggets] , (3.17)where R0 is the size of a nuggets at initial time, while [V · nnuggets] repre-sents the total number of nuggets in volume V . Comparison of (3.16) with(3.17) gives the following estimate for the nugget’s density when bubblesjust formed,nnuggets ' γ4piR20ξ. (3.18)The last step in our estimates is the computation of the average baryoncharge of a nugget at TCS when CS sets in inside the nugget. The cor-responding estimates have been worked out long ago [4] and reproducedin section 2.3 in the course of the time evolution by taking t → ∞, see513.3. nB/nγ ratio. Model dependent estimates.(2.26). The baryon number density inside the nuggets depends on a modelbeing used [4], but typically it is few times the nuclear saturation densityn0 ' (108 MeV)3 which is consistent with conventional computations forthe baryon density in CS phases. Therefore, we arrive to〈B〉 ' (2− 6)n0 · 4piR3form3, (3.19)where Rform is the final size of the nuggets. By substituting (3.19) and(3.18) to (3.14) we arrive to the following constraint on efficiency of thebubble formation represented by parameter γ(2− 6) · γ3(Rformξ(T ))(RformR0)2(n0nγ)& 10−10, (3.20)where expression for nγ(T ) should be taken at the formation timenγ =2ξ(3)pi2T 3form, ξ(3) ' 1.2, (3.21)while the correlation length ξ(T ) should be evaluated at much earlier times,close to Tc when domain wall network only started to form. Typically bub-bles form with R0 ∼ ξ. However, the bubbles shrink approximately 3-5 timesaccording to (2.43) before they reach equilibrium during the time evolutionas discussed in section 2.4. Therefore, to be on a safe side, we make veryconservative assumption thatRformR0∼ 0.1, R0 ' ξ. (3.22)To proceed with numerical estimates, it is convenient to factor γ on twopieces,γ ≡ γformation · γevolution, γformation ∼ 0.1, (3.23)where the first part, γformation ∼ 0.1 has been estimated using numericalsimulations, see textbook [43] for review. The second suppression factorγevolution is unknown, and includes a large number of different effects. Inparticular, many small closed bubbles with R0 ≤ ξ are very likely to beformed but may not survive the evolution as we discussed in section 2.4.Furthermore, there are many effects such as evaporation, annihilation in-side the nuggets which may also lead to collapse of relatively small nuggets.Furthermore, the formation probability of large closed bubbles with R0  ξ523.3. nB/nγ ratio. Model dependent estimates.(which are most likely to survive) is highly suppressed ∼ exp(−R20/ξ2). Allthese effects are included in the unknown parameter γevolution. Our con-straint (from observations on nB/nγ within our mechanism) can be inferredfrom (3.20)γevolution(Tform) & 10−7. (3.24)One suppression factor which obviously contributes to suppression (3.24) isrelated to necessity to produce a sufficiently large initial bubble for successfulnugget formation as given by eq. (2.43). We do not know a numericalvalue for the correlation length ξ in our system at the initial moment offormation21. However, even in the worst case scenario when R0 ∼ 3ξ insteadof R0 ∼ ξ, which is normally assumed in KZ simulations, the correspondingsuppression factorexp(−R20ξ2)∼ 10−4 (3.25)is still perfectly consistent with the observational constraint (3.24).We emphasize that there is no fine tuning in this estimate and overshoot-ing the estimate (3.24) is perfectly consistent with constraint (3.24). Thisis because the equilibration of the baryon charge from the nuggets with thevisible baryons always lead to the result (1.1), (3.9) when all contributionsare the same order of magnitude. The precise ratio (3.12), as we alreadymentioned, is determined by the moment in evolution of the Universe whenall visible baryons get completely annihilated which is the formation tem-perature Tform ∼ ΛQCD, which is again, perfectly consistent with the mainparadigm of the entire framework that all dimensional parameters are orderof ΛQCD.How one can understand the result (3.24) which essentially states thateven very tiny probability of the formation of the closed bubbles is stillsufficient to saturate the observed ratio (3.12)? The answer lies in the ob-servation that the baryon density nB ' nB¯ was 10 orders of magnitudelarger at the moment of the bubble formation. Therefore, even a tiny prob-ability at the moment of formation of a closed bubble with sufficiently largesize will lead to effects of order one at the moment when the baryon num-ber density drops 10 order in magnitude. Another reason why very tiny21One should emphasize that the correlation length ξ should not be identified withRform when the nugget’s formation is almost completed. Rather the correlation length ξ,defined by (3.15), characterizes the system at the very beginning of the formation whenthe domain wall network starts to emerge. It is normally assumed that ξ ' R0. Mostlikely R0 should be slightly greater than ξ for successful formation.533.3. nB/nγ ratio. Model dependent estimates.probability of the formation of the closed bubbles nevertheless is sufficientto saturate the observed ratio (3.12) is that typical “small factors” whichnormally accompany the conventional baryogenesis mechanisms such as CPand C odd couplings do not appear in estimate (3.24) due to the reasonsalready explained after eq (3.12).• We conclude this section with the following comment: The basic con-sequences of this framework represented by eqs. (1.1), (3.9), (3.10) are verygeneric. These features are not very sensitive to efficiency of the closed do-main wall formation (as long as it is greater than (3.24)) nor to the absolutevalue of θ as long as coherence is maintained, see footnote 19. These genericfeatures hold for arbitrary value of the axion mass 10−6eV ≤ ma ≤ 10−3eV,in contrast with conventional treatment of the axion as the dark matter can-didate, when ΩDM can be saturated by the axions only when the axion massassumes a very specific and definite value ma ' 10−6 eV, see next sectionwith details.The derivation of the observed ratio (3.12) from the first principles(which is entirely determined by Tform) is a hard computational problemof strongly coupled QCD when all elements such as cooling rate, annihila-tion rate, charge separation rate, damping rate, evaporation rate and manyother effects are equally contribute to Tform. However, it is important thatthe “observational” value Tform lies precisely in the region where it should be:Tform < TCS, i.e. slightly below the temperature where CS sets in. There-fore, any fine-tuning procedures have never been required in this framework(in contrast with conventional baryogenesis computations) to accommodatethe observed ratios presented by eqs. (1.1), (3.12).54Chapter 4Implications for the AxionSearch ExperimentsThe goal of this section is to comment on relation of our framework and thedirect axion search experiments [9–16]. We start with the following com-ment we made in section 1.2: this model which has a single fundamentalparameters (a mean baryon number of a nugget 〈B〉 ∼ 1025 entering allthe computations) is consistent with all known astrophysical, cosmological,satellite and ground based constraints as reviewed in section 1.2. For dis-cussions of this section it is convenient to express this single normalizationparameter 〈B〉 ∼ 1025 in terms of the axion mass ma ∼ 10−4 eV as thesetwo parameters are directly related according to the scaling relations (1.3).The corresponding relation between these two parameters occur becausethe axion mass ma determines the wall tension σ ∼ m−1a which itself entersthe expression for the equilibrium value of the size of the nuggets, Rform atthe end of the formation. One should emphasize that it is quite nontrivialthat the cosmological constraints on the nuggets as shown on Fig. 1.2 andformulated in terms of 〈B〉 are compatible with known upper limit on theaxion mass ma < 10−3eV within our framework. One could regard thiscompatibility as a nontrivial consistency check for this proposal.The lower limit on the axion mass, as it is well known, is determinedby the requirement that the axion contribution to the dark matter densitydoes not exceed the observed value Ωdark ≈ 0.23. There is a number of un-certainties in the corresponding estimates. We shall not comment on thesesubtleties by referring to the review papers[9–16]. The corresponding un-certainties are mostly due to the remaining discrepancies between differentgroups on the computations of the axion production rates due to the differ-ent mechanisms such as misalignment mechanism versus domain wall/stringdecays. In what follows to be more concrete in our estimates we shall usethe following expression for the dark matter density in terms of the axion55Chapter 4. Implications for the Axion Search Experimentsmass resulted from the misalignment mechanism [16]:Ω(DM axion) '(6 · 10−6eVma) 76(4.1)This formula essentially states that the axion of mass ma ' 2 · 10−5 eVsaturates the dark matter density observed today, while the axion mass inthe range of ma ≥ 10−4 eV contributes very little to the dark matter density.This claim, of course, is entirely based on estimate (4.1) which accounts onlyfor the axions directly produced by the misalignment mechanism suggestedoriginally in [47].There is another mechanism of the axion production when the Peccei-Quinn symmetry is broken after inflation. In this case the string-domainwall network produces a large number of axions such that the axion massma ' 10−4 eV may saturate the dark matter density, see relatively recentestimates [45, 46] with some comments and references on previous papers.The corresponding formula from refs.[45, 46] is also highly sensitive to theaxion mass with ma- dependence being very similar to eq. (4.1).The main lesson to be learnt from the present work is that in addi-tion to these well established mechanisms previously discussed in the lit-erature there is an additional contribution to the dark matter density alsorelated to the axion field. However, the mechanism which is advocated inthe present work contributes to the dark matter density through formationof the nuggets, rather than through the direct axion production. The cor-responding mechanism as argued in section 3.2 always satisfies the relationΩdark ≈ Ωvisible, and, in principle is capable to saturate the dark matterdensity Ωdark ≈ 5Ωvisible by itself for arbitrary magnitude of the axion massma as the corresponding contribution is not sensitive to the axion mass incontrast with conventional mechanisms mentioned above. A precise coeffi-cient in ratio Ωdark ≈ Ωvisible is determined by a parameter of order one,|c(T )| ∼ 1, which unfortunately is very hard to compute from the first prin-ciples, as discussed in section 3.2.Our choice for ma ' 10−4 eV which corresponds to 〈B〉 ∼ 1025 is entirelymotivated by our previous analysis of astrophysical, cosmological, satelliteand ground based constraints as reviewed in Section 1.2. As we mentionedin Section 1.2 there is a number of frequency bands where some excessof emission was observed, and this model may explain some portion, oreven entire excess of the observed radiation in these frequency bands. Ournormalization 〈B〉 ∼ 1025 was fixed by eq.(1.6) with assumption that theobserved dark matter is saturated by the nuggets. The relaxing of this56Chapter 4. Implications for the Axion Search ExperimentsAxion Coupling |ga | (GeV-1)Axion Mass mA (eV)10-1610-1510-1410-1310-1210-1110-1010-5 10-4 10-3Cavity ExperimentsADMXSolarExpected sensitivityof proposed techniqueA B C DKSVZDFSZAxion Dark MatterFigure 4.1: Cavity / ADMX experimental constraints on the axion massshown in green. The expected sensitivity for the Orpheus axion searchexperiment [15] is shown by blue regions “A”, “B”, “C” and “D”. In partic-ular, experiment “B”, covers the most interesting region of the parametricalspace with ma ' 10−4 eV corresponding to the nuggets with mean baryoncharge 〈B〉 ' 1025 which itself satisfies all known astrophysical, cosmolog-ical, satellite and ground based constraints, see Fig.1.2. The plot is takenfrom [15].57Chapter 4. Implications for the Axion Search Experimentsassumption obviously modifies the coefficient c(T ) as well as 〈B〉.Interestingly enough, this range of the axion mass ma ' 10−4 eV isperfectly consistent with recent claim [58],[59] that the previously observedsmall signal in resonant S/N/S Josephson junction [60] is a result of thedark matter axions with the mass ma ' 1.1 · 10−4 eV. Furthermore, ithas been also claimed that similar anomalies have been observed in otherexperiments [61–63] which all point towards an axion massma ' 1.1·10−4 eVif interpreted within framework [58],[59]. The only comment we would liketo make here is that if the interpretation [58],[59] of the observed anomalies[60–63] is indeed due to the dark matter axions, then the correspondingaxion mass is perfectly consistent with our estimates (based on cosmologicalobservations) of the average baryon charge of the nuggets 〈B〉 ' 1025 asreviewed in section 1.2.We conclude this section on an optimistic note with a remark that themost interesting region of the parametric space corresponding to the nuggetswith mean baryon charge 〈B〉 ' 1025 might be tested by the Orpheus axionsearch experiment [15] as shown on Fig. 4.1.58Chapter 5ConclusionFirst, we want to list the main results of the present studies, while thecomments on possible future developments will be presented at the end ofthis chapter.1. First key element of this proposal is the observation (2.12) thatthe closed axion domain walls are copiously produced and generically willacquire the baryon or antibaryon charge. This phenomenon of “separationof the baryon charge” can be interpreted as a local version of spontaneoussymmetry breaking. This symmetry breaking occurs not in the entire volumeof the system, but on the correlation length ξ(T ) ∼ m−1a which is determinedby the folded and crumpled axion domain wall during the formation stage.Precisely this local charge separation eventually leads to the formation ofthe nuggets and anti-nuggets serving in this framework as the dark mattercomponent Ωdark.2. Number density of nuggets and anti-nuggets will not be identicallythe same as a result of the coherent (on the scale of the Universe) axionCP -odd field. We parameterize the corresponding effects of order one byphenomenological constant c(T ) ∼ 1. It is important to emphasize thatthis parameter of order one is not a fundamental constant of the theory,but, calculable from the first principles. In practice, however, such a com-putation could be quite a challenging problem when even the QCD phasediagram is not known. The fundamental consequence of this framework,Ωdark ≈ Ωvisible, which is given by (1.1) is universal, and not sensitive toany parameters as both components are proportional to ΛQCD. The ob-served ratio (1.2), (3.11) corresponds to a specific value of c(Tform) ' −1.5as discussed in section 3.2.3. Another consequence of the proposal is a natural explanation of theratio (1.5) in terms of the formation temperature Tform ' 40 MeV, ratherthan in terms of specific coupling constants which normally enter conven-tional “baryogenesis” computations. This observed ratio is expressed in ourframework in terms of a single parameter Tform when the nuggets completetheir formation. This parameter is not fundamental constant of the theory,and as such is calculable from the first principles. In practice, however, the59Chapter 5. Conclusioncomputation of Tform is quite a challenging problem as explained in section3.3. Numerically, the observed ratio (1.5) corresponds to Tform ' 40 MeVwhich is indeed slightly below the critical temperature TCS ' 60 MeV wherethe colour superconductivity sets in.The relation Tform . TCS ∼ ΛQCD is universal in this framework asboth parameters are proportional to ΛQCD. As such, the universality ofthis framework is similar to the universality Ωdark ≈ Ωvisible mentioned inprevious item. At the same time, the ratio (1.5) is not universal itself asit is exponentially sensitive to precise value of Tform due to conventionalsuppression factor ∼ exp(−mp/T ).4. The only new fundamental parameter of this framework is the axionmass ma. Most of our computations (related to the cosmological observa-tions, see section 1.2 and Fig. 1.2), however, are expressed in terms of themean baryon number of nuggets 〈B〉 rather than in terms of the axion mass.However, these two parameters are unambiguously related according to thescaling relations (1.3). Our claim is that all universal properties of thisframework listed above still hold for any ma. In other words, there is no anyfine tuning in the entire construction with respect to ma. The constraints(and possible cosmological observations) from section 1.2 strongly suggest〈B〉 ' 1025 which can be translated into preferred value for the axion massma ' 10−4 eV.5. This region of the axion mass ma ' 10−4 eV corresponding to averagesize of the nuggets 〈B〉 ' 1025 can be tested in the Orpheus axion searchexperiment [15] as shown on Fig. 4.1.We conclude with few thoughts on future directions within our frame-work. It is quite obvious that future progress cannot be made without amuch deeper understanding of the QCD phase diagram at θ 6= 0. In otherwords, we need to understand the structure of possible phases along thethird dimension parametrized by θ on Fig 1.1.Presently, very few results are available regarding the phase structure atθ 6= 0. First of all, the phase structure is understood in simplified versionof QCD with two colours, Nc = 2 at T = 0, µ 6= 0, see [64]. In fact, thestudies [64] were mostly motivated by the subject of the present work andrelated to the problem of formation of the quark nuggets during the QCDphase transition in early Universe with non vanishing θ. With few additionalassumptions the phase diagram can be also conjectured for the system withlarge number of colours Nc =∞, at non vanishing T, µ, θ, see [65, 66].Due to the known “sign problem”, see footnote 2, the conventional latticesimulations cannot be used at θ 6= 0. The corresponding analysis of thephase diagram for non vanishing θ started just recently by using some newly60Chapter 5. Conclusioninvented technical tricks [67–70].Another possible development from the “wish list” is a deeper under-standing of the closed bubble formation. Presently, very few results areavailable on this topic. The most relevant for our studies is the observationmade in [11] that a small number of closed bubbles are indeed observed innumerical simulations. However, their detailed properties (their fate, sizedistribution, etc) have not been studied yet. A number of related questionssuch as an estimation of correlation length ξ(T ), the generation of the struc-ture inside the domain walls, the baryon charge accretion on the bubble, etc,hopefully can be also studied in such numerical simulations.One more possible direction for future studies from the “wish list” is adevelopment some QCD-based models where a number of hard questionssuch as: evolution of the nuggets, cooling rates, evaporation rates, annihila-tion rates, viscosity of the environment, transmission/reflection coefficients,etc in unfriendly environment with non-vanishing T, µ, θ can be addressed,and hopefully answered. All these and many other effects are, in general,equally contribute to our parameters Tform and c(T ) at the ΛQCD scale instrongly coupled QCD. Precisely these numerical factors eventually deter-mine the coefficients in the observed relations: Ωdark ≈ Ωvisible given by eq.(3.10) and nB/nγ expressed by eq. 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B 647, 371 (2007).67Appendix AEstimation of FluxesThe main goal of this Appendix is to argue that the approximation in eq.(2.24) which was adopted in the text by neglecting the extra term “fluxes”is justified, at least on qualitative level. In other words, while these “flux”terms are obviously present in the system, they, nevertheless, do not dras-tically change a key technical element (an implicit relation between R(t)and µ(t)) which this equation provides. Precisely this implicit relation be-tween R(t) and µ(t) eventually allows us to express the µ-dependent pres-sure ∆P [µ] in terms of R dependent function ∆P [f(R)] such that the basicequation (2.37) describing the time evolution of the nuggets is reduced to adifferential equation on a single variable R(t).Our starting point is the observation that the relevant flux which entersequation (2.24) is ∆Φ = (Φ⇒−Φ⇐), counting the net baryon charge transferand sensitive to the chemical potential difference, rather than total flux 〈Φ〉which counts the exchange of all the particles, including bosons22. In fact, ifthe average flux 〈Φ〉 were entering equation (2.24) one could explicitly checkthat this term would be the same order of magnitude as two other terms ofthe equation. However, the key point is that the baryon charge transfer ∆Φis numerically suppressed, i.e. ∆Φ  〈Φ〉. In fact, ∆Φ identically vanishesfor µ = 0. Furthermore, one can use the same technique which has been usedin section 2.4.1 to argue that ∆Φ  〈Φ〉 in entire region of µ. Numericalanalysis supports this claim.To reiterate this claim: while a typical flux defined asΦ =gin(2pi)3∫vzd3kexp(k−µT ) + 1+ (bosons) ∼ (fm)−3 (A.1)assumes a conventional QCD value, the net baryonic flux ∆Φ · S throughsurface S is numerically suppressed, and can be neglected in eq. (2.24).22The dominant contribution to the fluxes normally comes from the lightest degrees offreedom which are the Nambu-Goldstone bosons in hadronic and CS phases. These con-tributions are crucial for maintaining the thermodynamical equilibrium between exteriorand interior, but they do not play any role in the baryon fluxes which enter eq.(2.24).68Appendix A. Estimation of FluxesOne can explain this result as follows. Consider a single oscillation of thedomain wall evolution. To be more specific, consider a squeezing portion ofthis evolution when R(t) decreases. During this process the chemical poten-tial (and the baryon charge density) locally grow as we discussed in section2.4.1. The major portion of this growth is resulted from the baryon chargewhich was already bound to the domain wall, rather than from the baryoncharge which enters the system as a result of the baryonic flux transfer.On an intuitive level the dominance of the bound charges (accounted ineq. (2.24)) in comparison with flux-contribution (neglected in eq. (2.24))can be explained using pure geometrical arguments. Indeed, the chemicalpotential increases very fast as a result of rapid shrinking of the bubble withspeed v ' c. The corresponding contraction of a bubble leads to propor-tionally rapid increase of the chemical potential on the domain wall. Thishappens because the baryon charges are strongly bound to the wall, andcannot leave the system due to the topological reasons as the boundary con-ditions effectively lock the charge to the macroscopically large domain wall.As a result of this evolution the binding energy of a quark ∼ µ increaseswhen the bubble contracts. This process represents a highly efficient mech-anism of very rapid growth of the chemical potential due to the domainwall dynamics. It is vey hard to achieve a similar efficiency with the flux-contribution when the probability for a reflection from the domain wall istypically much higher than probability for a transmission. Furthermore, anon-vanishing quark mass make suppression even stronger ∼ exp(−m/T ).To conclude: we do expect that an accounting for the flux- contributionmodifies our equations relating µ(t) and R(t) as expressed by eqs. (2.24),(2.30). However, we do not expect that this modification may drasticallychange the basic qualitative features of eqs. (2.24), (2.30) which have beenheavily employed in this work.69Appendix BFormation of the Nuggets:Numerical AnalysisThis appendix is devoted to exact numerical computation in contrast withanalytical qualitative arguments presented in section 2.4. The basic lessonof this Appendix is that a number of simplifications which have been madein section 2.4 are justified, at least, on a qualitative level.Before we proceed with numerical computations we want to make fewcomments on parameters entering the basic dynamical equation (2.37). Inthe previous sections, σ was treated as a constant in order to simplify theanalysis. This approximation is justified as long as a typical curvature ofthe domain wall is much smaller than the width of the domain wall, i.e.R  m−1a . This condition is only marginally justified as a typical radiusof the bubble is of order m−1a , which is the same order of magnitude as thewidth of the wall. At the same time, the width of the QCD substructure ofthe domain wall (including the η′ substructure and the baryon substructure)is very small in comparison with the curvature, and it does satisfy the criteriaof a thin wall approximation as m−1  R ∼ m−1a . Precisely this QCDsubstructure plays a crucial role in our analysis in section 2.2 where westudied the “local violation” of the baryon charge in the presence of thedomain walls. The broad structure of the domain wall due to the axionfield with the width m−1a does not play any role. However, precisely thisstructure determines the large tension σ ∼ m−1a of the domain wall.We want to effectively account for this physics by assuming that σ(R)effectively depends on the radius of the bubble R. On the physical groundswe expect that σ(R) approaches its asymptotic value at large R when thedomain wall is almost flat, σ(R → ∞) → σ0, while σ reduces its value atsmaller R, and eventually vanishes at some cutoff Rcut. A natural choice isRcut ' 0.24R0 which corresponds to large µcut . 500MeV from (2.30), whenthe chemical potential assumes its typical CS value. To introduce such aninfrared cutoff smoothly, it is convenient to parametrize σ as followsσ(R) = σ0e−r0/2(R−Rcut) (B.1)70Appendix B. Formation of the Nuggets: Numerical Analysiswhere σ0 ' 9f2ama is the conventional domain wall tension, see e.g. [11],while r0 is a free phenomenological parameter, 0 < r0 . R0 as we expectσ(R0) ' σ0. In our numerical studies we verified that the physical results(such as formation size Rform) are not very sensitive to parameter r0.Another parameter which requires some comments is the viscosity η.In the context of the present work, the viscosity accounts for a numberof different QCD effects which lead to dissipation and “friction”. Such ef-fects include, but are not limited to different scattering process by quarks,gluons or Nambu Goldstone Bosons in different phases. Furthermore, theannihilation processes which take place inside the bubble and which resultin production of a large number of strongly interacting quasi-particles alsocontribute to η. The viscosity can be computed from the first principles inweakly coupled quark-gluon phase [71]. However, we are more interestedin behaviour of η below Tc. In this case the computations [72] based onchiral perturbation theory suggest that η ∼ m3pi. This numerical value isquite reasonable in all respects, and consistent with simple dimensional ar-guments. It is also known that η(T ) depends on temperature [72]. However,we neglect this dependence in our estimates which follow.Now we can proceed with our numerical studies. Since σ(R) is a functionof R as explained above, we should start with a modified differential equationfor R(t):σ(R)R¨(t) = −2σ(R)R− σ(R)R˙2R+ ∆P (R) (B.2)− (12R˙2 + 1)dσ(R)dR− 4η R˙R.This equation is not identically the same as equation (2.37) discussed insection 2.4. This is due to the fact that the tension σ(R) is now becomeR dependent function as we discussed above. The equation (B.2) has beensolved numerically using parameters listed in Table B.1. The numericalvalues of these parameters can be obviously somewhat modified. However,the basic qualitative features presented in section 2.4 do not drasticallychange when the QCD parameters are varied within reasonable parametricalregion. Our numerical studies, as we discuss below, do support the analyticalqualitative results presented in section 2.4.We start our short description with Fig.B.1. It shows a typical evolutionof a bubble with time. The frequencies of oscillations are determined by theaxion mass m−1a , while typical damping time is determined by parameter τas discussed in section 2.4. To make the pattern of oscillations visible, the71Appendix B. Formation of the Nuggets: Numerical Analysisviscosity has been rescaled23. At large times, t → ∞, the solution settlesat R0/Rform ' 2.9 and µform ' 330MeV ∼ µ1, consistent with qualitativeanalysis of section 2.4.We now describe Fig. B.2 where we zoom-in first few oscillations of atypical solution shown on previous plot Fig. B.1. We want to emphasize thata seeming cusp singularity is actually a smooth function near Rmin. It looks“cuspy” as a result of a large time scale on Fig. B.1. The “cusp” is relativelynarrow comparing to macroscopic period of oscillation (δtcusp ∼ 10−3R0).Nevertheless it actually lasts much longer in comparison with a typical QCDscale (δtcusp  Λ−1QCD).On Fig. B.3 we demonstrate a (non)sensitivity of the system to pa-rameter r0 introduced in eq. (B.1). One can explicitly see that the initialevolution is indeed quite sensitive to ad hoc parameter r0. However, the fi-nal stage of the evolution is not sensitive to r0. In other words, the physicalparameters Rform and τ are not sensitive to ad hoc parameter r0. Note thatestimation of damping time τ and period of oscillation tosc agree well withqualitative estimations presented in section 2.4.23In this plot we use η ' 108η0, which is eight orders of magnitude larger than η0 ' m3pi.We did it on purpose: First, it simplifies the numerics. Indeed, the η parameter determinesthe dumping time scale (2.45) which is many orders of magnitude longer than any otherscales of the problem. Secondly, we use η ' 108η0 for the demonstration purposes. Indeed,a typical oscillation time ω−1 and the damping time τ are characterized by drasticallydifferent scales. If we take η according to its proper QCD value than the time scale onplots Fig. B.1 would be eight orders of magnitude longer than shown.72Appendix B. Formation of the Nuggets: Numerical Analysis0 20 40 60 800.00.20.40.60.81.001234Figure B.1: Typical underdamped solution of R(t) and µ(t). The oscillationswith frequencies ∼ m−1a shown in orange, the modulation of R(t) is shownin blue. The chemical potential µ(t) shown in red. The initial R0 = 1011fmand r0 = 0.5R0.0.0 0.5 1.00.00.20.40.60.81.00.5086 0.5090 0.50940.2600.2650.2700.2750.280Figure B.2: The first few oscillations of an underdamped solution shown onFig. B.1.73Appendix B. Formation of the Nuggets: Numerical Analysis0 20 40 60 800.00.20.40.60.81.00.51.00.1 0.2 0.3 0.40.3300.3400.3500.360Figure B.3: Dependence on parameter r0 as defined by eq. (B.1). Zoom-inshows small oscillations during the final stage of formation.Table B.1: Table for some numerical parametersQuantity Symbol ValueQCD units(mpi = 1)flavours Nf 2 2colors Nc 3 3degeneracy factor (in) (2.21) gin 12 12degeneracy factor (out) (2.18) gout 37 37baryon charge on DW (2.12) N 2 2axion decay constant fa 1010GeV 7× 1010mass of axion ma 6× 10−4eV 4× 10−12domain wall tension σ0 5× 108GeV3 2× 1011bag constant (2.23) EB (150MeV)4 1.5“squeezer” parameter (2.23) µ1 330 MeV 2.4cosmological time scale t0 10−4s 1019initial µ µ0 1.35 MeV 0.01initial radius R0 10−2cm 1011initial temperature T0 100 MeV 0.74QCD viscosity[72] η0 0.002 GeV3 174Appendix CEvaluation of Fermi-DiracIntegralsThe main goal of this Appendix is to present some supporting argumentssuggesting that the approximation we have used in section 2.4.1 and whichinvolves the Fermi-Dirac integrals are qualitatively justified. Indeed, therelevant integrals which enter eqs. (2.28), (2.31) have the formIn(b) ≡∫ ∞0dx · xn−1ex−b + 1, b =µT> 0, (C.1)where n = 2 appears in integral (2.28), while n = 4 appears in (2.31). Wewill hence focus on evaluation of I2 and I4 in this appendix. They can beexactly evaluated asI2(b) =pi26+12b2 + Li2(−e−b) (C.2a)I4(b) =7pi460+pi22b2 +14b4 + 6 Li4(−e−b) (C.2b)where Li2(−z) and Li4(−z) are the polylogarithm functions of order 2 and4, respectively. Polylogarithm functions are commonly known to representthe Fermi-Dirac and Bose-Eisterin integrals. The Polylogarithm functionsare defined asLin(−z) =∞∑k=1(−1)kknzk. (C.3)Note that |z| = e−b ≤ 1 for any positive b. In this case Lin(−z) is evidentlyfast-converging, so that we can efficiently estimate it by extracting the lead-ing exponent e−b then using the Taylor expansion for the remaining piece:Li2(−e−b) ' e−b[−pi212+ (ln 2− pi212)b+O(b2)]75Appendix C. Evaluation of Fermi-Dirac Integrals0.0 0.5 1.0 1.5 2.0 2.50.50.60.70.80.91.0Figure C.1: Comparison of I(0)n to In with different values of b.Li4(−e−b) ' e−b[− pi4720+ (3ζ(3)4− 7pi4720)b+O(b2)],where ζ(3) ' 1.202 is the Riemann zeta function. Neglecting the terms oforder O(be−b) which are small in both limits, at large and small chemicalpotentials, one can approximate I2 and I4 as followsI(0)2 'pi26+12b2 − pi212e−b +O(be−b) (C.5a)I(0)4 '7pi460+pi22b2 +14b4 − 7pi4120e−b +O(be−b). (C.5b)We test our approximation by comparing our approximate expressions (C.5a),(C.5b) with exact formulae (C.2a), (C.2b). As one can see from Fig. C.1,our approximation shown in blue (I(0)2 /I2) and orange (I(0)4 /I4) is very goodwith errors less than 3% in the entire range of b > 0.On the same plot we also show the approximation I˜(0)4 for approximate76Appendix C. Evaluation of Fermi-Dirac Integralsexpression I(0)4 used in the main text in eq. (2.31)I˜(0)4 '7pi460+pi22b2 +14b4 − pi412e−b. (C.6)The error for I˜(0)4 is quite large for very small chemical potential b 0.5, onthe level of 40%, shown in green. The error becomes much smaller after shortperiod of time when b = µ/T ≥ 0.5 becomes sufficiently large. To conclude:the approximations of the integrals in section 2.4.1 are sufficiently good forqualitative analysis presented in that section.77

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