KKaon oscillations and baryon asymmetry of the universe
Wanpeng Tan ∗ Department of Physics, Institute for Structure and Nuclear Astrophysics (ISNAP),and Joint Institute for Nuclear Astrophysics -Center for the Evolution of Elements (JINA-CEE),University of Notre Dame, Notre Dame, Indiana 46556, USA (Dated: September 19, 2019)
Abstract
Baryon asymmetry of the universe (BAU) can likely be explained with K − K (cid:48) oscillationsof a newly developed mirror-matter model and new understanding of quantum chromodynamics(QCD) phase transitions. A consistent picture for the origin of both BAU and dark matter ispresented with the aid of n − n (cid:48) oscillations of the new model. The global symmetry breakingtransitions in QCD are proposed to be staged depending on condensation temperatures of strange,charm, bottom, and top quarks in the early universe. The long-standing BAU puzzle could thenbe understood with K − K (cid:48) oscillations that occur at the stage of strange quark condensationand baryon number violation via a non-perturbative sphaleron-like (coined “quarkiton”) process.Similar processes at charm, bottom, and top quark condensation stages are also discussed includingan interesting idea for top quark condensation to break both the QCD global U t (1) A symmetry andthe electroweak gauge symmetry at the same time. Meanwhile, the U (1) A or strong CP problemof particle physics is addressed with a possible explanation under the same framework. ∗ [email protected] a r X i v : . [ h e p - ph ] S e p . INTRODUCTION The matter-antimatter imbalance or baryon asymmetry of the universe (BAU) has beena long standing puzzle in the study of cosmology. Such an asymmetry can be quantified invarious ways. The cosmic microwave background (CMB) data by Planck set a very preciseobserved baryon density of the universe at Ω b h = 0 . ± . n B /n γ = 6 . × − . For an adiabaticallyexpanding universe, it would be better to use the baryon-number-to-entropy density ratioof n B /s = 8 . × − to quantify the BAU, which may have to be modified under the newunderstanding of the neutrino history in the early universe (see Sec. IV).From known physics, it is difficult to explain the observed BAU. For example, for aninitially baryon-symmetric universe, the surviving relic baryon density from the annihilationprocess is about nine orders of magnitude lower than the observed one [2]. Therefore, anasymmetry is needed in the early universe and the BAU has to exist before the temperatureof the universe drops below T = 38 MeV [2] to avoid the annihilation catastrophe betweenbaryons and anti-baryons.Sakharov proposed three criteria to generate the initial BAU: (i) baryon number (B-)violation (ii) C and CP violation (iii) departure from thermal equilibrium [3]. The StandardModel (SM) is known to violate both C and CP and it does not conserve baryon numberonly although it does B − L (difference of baryon and lepton numbers). Coupled withpossible non-equilibrium in the thermal history of the early universe, it seems to be easyto solve the BAU problem. Unfortunately, the violations in SM without new physics aretoo small to explain the observed fairly large BAU. The only known B-violation processesin SM are non-perturbative, for example, via the so-called sphaleron [4] which involves ninequarks and three leptons from each of the three generations. It was also found out that thesphaleron process can be much faster around or above the temperature of the electroweaksymmetry breaking or phase transition T EW ∼
100 GeV [5]. This essentially washes outany BAU generated early or around T EW since the electroweak transition is most likelyjust a smooth cross-over instead of “desired” strong first order [6]. It makes the appealingelectroweak baryogenesis models [5, 7] ineffective and new physics often involving the Higgshave to be added in the models [8–10]. Recently lower energy baryogenesis typically usingparticle oscillations stimulated some interesting ideas [11, 12]. Other types of models such2s leptogenesis [13] are typically less testable or have other difficulties.Here we present a simple picture for baryogenesis at energies around quantum chromo-dynamics (QCD) phase transition with K − K (cid:48) oscillations based on a newly developedmirror matter model [14]. K − K (cid:48) oscillations and the new mirror matter model will befirst introduced to demonstrate how to generate the “potential” amount of BAU as observed.Then the QCD phase transition will be reviewed and the sphaleron-like non-perturbativeprocesses are proposed to provide B-violation and realize the “potential” BAU created by K − K (cid:48) oscillations. In the end, the observed BAU is generated right before the n − n (cid:48) oscillations that determine the final mirror(dark)-to-normal matter ratio of the universe [14].Meanwhile, the long-standing U (1) A and strong CP problems in particle physics are alsonaturally resolved under the same framework. II. K − K (cid:48) OSCILLATIONS AND THE NEW MODEL
To understand the observed BAU, we need to apply the newly developed particle-mirrorparticle oscillation model [14]. It is based on the mirror matter theory [15–22], that is, twosectors of particles have identical interactions within their own sector but share the samegravitational force. Such a mirror matter theory has appealing theoretical features. Forexample, it can be embedded in the E ⊗ E (cid:48) superstring theory [17, 23, 24] and it canalso be a natural extension of recently developed twin Higgs models [25, 26] that protectthe Higgs mass from quadratic divergences and hence solve the hierarchy or fine-tuningproblem. The mirror symmetry or twin Higgs mechanism is particularly intriguing as theLarge Hadron Collider has found no evidence of supersymmetry so far and we may notneed supersymmetry, at least not below energies of 10 TeV. Such a mirror matter theorycan explain various observations in the universe including the neutron lifetime puzzle anddark-to-baryon matter ratio [14], evolution and nucleosynthesis in stars [27], ultrahigh energycosmic rays [28], dark energy [29], and a requirement of strongly self-interacting dark matterto address numerous discrepancies on the galactic scale [30].In this new mirror matter model [14], no cross-sector interaction is introduced, unlikeother particle oscillation type models. The critical assumption of this model is that themirror symmetry is spontaneously broken by the uneven Higgs vacuum in the two sectors,i.e., < φ > (cid:54) = < φ (cid:48) > , although very slightly (on a relative breaking scale of ∼ − –10 − )314]. When fermion particles obtain their mass from the Yukawa coupling, it automaticallyleads to the mirror mixing for neutral particles, i.e., the basis of mass eigenstates is not thesame as that of mirror eigenstates, similar to the case of ordinary neutrino oscillations dueto the family or generation mixing. The Higgs mechanism makes the relative mass splittingscale of ∼ − –10 − universal for all the particles that acquired mass from the Higgsvacuum. Further details of the model can be found in Ref. [14].The immediate result of this model for this study is the probability of K − K (cid:48) oscillationsin vacuum [14], P K K (cid:48) ( t ) = sin (2 θ ) sin ( 12 ∆ K K (cid:48) t ) (1)where θ is the K − K (cid:48) mixing angle and sin (2 θ ) denotes the mixing strength of about10 − , t is the propagation time, ∆ K K (cid:48) = m K − m K is the small mass difference of thetwo mass eigenstates of about 10 − eV [14], and natural units ( (cid:126) = c = 1) are used forsimplicity. Note that the equation is valid even for relativistic kaons and in this case t is theproper time in the particle’s rest frame. There are actually two weak eigenstates of K ineach sector, i.e., K S and K L with lifetimes of 9 × − s and 5 × − s, respectively. Theirmass difference is about 3 . × − eV very similar to ∆ K K (cid:48) , which makes one wonder ifthe two mass differences and even the CP violation may originate from the same source.For kaons that travel in the thermal bath of the early universe, each collision or interactionwith another particle will collapse the oscillating wave function into a mirror eigenstate. Inother words, during mean free flight time τ f the K − K (cid:48) transition probability is P K K (cid:48) ( τ f ).The number of such collisions will be 1 /τ f in a unit time. Therefore, the transition rate of K − K (cid:48) with interaction is [14], λ K K (cid:48) = 1 τ f sin (2 θ ) sin ( 12 ∆ K K (cid:48) τ f ) . (2)Note that the Mikheyev-Smirnov-Wolfenstein (MSW) matter effect [31, 32], i.e., coherentforward scattering that could affect the oscillations is negligible as the meson density is verylow when kaons start to condensate from the QCD plasma (see more details for in-mediumparticle oscillations from Ref. [27]), and in particular, the QCD phase transition is mostlikely a smooth crossover [33, 34].It is not very well understood how the QCD symmetry breaking or phase transition occurin the early universe, which will be discussed in detail in the next section. Let us supposethat the temperature of QCD phase transition T c is about 150 MeV and a different value4e.g., 200 MeV) here does not affect the following discussions and results. At this time onlyup, down, and strange quarks are free. It is natural to assume that strange quarks becomeconfined first during the transition, i.e., forming kaon particles first instead of pions andnucleons. A better understanding of this process is shown in the next section. As a matterof fact, even if they all form at the same time, the equilibrium makes the ratio of nucleonnumber to kaon number n N n K (cid:39) ( m N m K ) / exp( − ( m N − m K ) /T c ) ∼ . σ EW ∼ G F T where G F = 1 . × − GeV − is the Fermicoupling constant. Then one can estimate K ’s thermally averaged reaction rate over theBose-Einstein distribution,Γ = g (2 π ) (cid:90) ∞ d pf ( p ) σ EW pm = g π G F T m (cid:90) ∞ dp p exp( (cid:112) p + m /T ) − g = 2 for both K S and K L , m is the mass of kaons, and T is the temperature. Theexpansion rate of the universe at this time can be estimated to be H ∼ T s − where T MeV is the temperature in unit of MeV. The condition for K to decouple from the interaction orfreeze out is Γ /H <
1. It can be easily calculated from Eq. (4) that the freezeout occurs at T fo = 100 MeV. This means that kaon oscillations have to operate between T c = 150 MeVand T fo = 100 MeV. And fortunately the K mesons have long enough lifetime (comparedto the weak interaction rate) for such oscillations and BAU to occur during this temperaturerange.For the standard constraint on the mirror-to-normal matter temperature ratio of x = T (cid:48) /T < / K (cid:48) → K and K → K (cid:48) will be decoupled in a similar way as the n − n (cid:48) oscillationsdiscussed in Ref. [14]. Using a typical weak interaction rate λ EW = 1 /τ f = G F T ∼ T s − and the age of the universe t = 0 . /T s during this period of time, one can get the5nal-to-initial K abundance ratio in the ordinary sector for the second step, X f X i = exp( − (cid:90) P K K (cid:48) ( τ f ) λ EW dt )= exp( − × sin (2 θ )( ∆ K K (cid:48) eV ) (cid:90) T fo T c d ( 1 T ))= 1 − . ≡ − (cid:15) (5)and the first step is negligible due to the much faster expansion rate of the universe [14].However, K S has a lifetime of 9 × − s that is comparable to the weak interactionrate at such temperatures. Owing to this, only one third of K S particles participate in theoscillations while the other two thirds decay to pions. In contrast to K S , K L mesons have amuch larger lifetime (5 × − s) and hence almost all of them take part in the oscillations.Considering the above correction, the final-to-initial K abundance ratio in the normal worlddue to oscillations becomes, X f X i = 1 − (cid:15). (6)The CP violation amplitude in SM is measured as δ = 2 . × − [35] so that δ ∼ × − and the oscillation probability ratio can be estimated as P K K (cid:48) /P ¯ K ¯ K (cid:48) ∼ − δ . Then thenet K fraction can be obtained as follows,∆ X K ¯ K X K ¯ K ≡ X K − X ¯ K X K + X ¯ K = 13 (cid:15)δ ∼ × − (7)If the excess of K ( d ¯ s ) generated above can survive by some B-violation process, i.e.,dumping ¯ s quarks and leaving d quarks to form nucleons in the end, then assuming thathalf of strange quarks condensate into K L,S (with the other half in K ± ) we will end up witha net baryon density of n B /s = 5 . × − that essentially gives the sum of the observedbaryon and dark matter. In the next section, we will demonstrate how such a B-violationprocess could occur during the QCD phase transition.In Eq. (5) the mixing strength sin (2 θ ) ∼ − and the mass splitting parameter∆ K K (cid:48) ∼ − eV are estimated from n − n (cid:48) oscillations in Ref. [14] assuming that thesingle-quark mixing strength is similar and the mass splitting parameter is scaled to theparticle’s mass. Unfortunately, these estimates are still fairly rough as the neutron lifetimemeasurements have not yet constrained the oscillation parameters well [14] resulting in afactor of ∼
10 uncertainty in (cid:15) of Eq. (5). On the other hand, the observed baryon asym-metry can be used to constrain these parameters under the new mechanism, i.e., (cid:15) = 0 .
05 or6in (2 θ )∆ K K (cid:48) = 10 − eV . Remarkably, such parameters are consistent with the neutronlifetime experiments and the origin of dark matter under the new model (see more discus-sions in Sec. IV). More detailed studies of the mirror mixing parameters under the contextof the CKM matrix and proposed laboratory measurements can be found in a separate paper[36]. III. QCD SYMMTRY BREAKING TRANSITION AND OTHER OSCILLATIONS
A massless fermion particle’s chirality or helicity has to be preserved, i.e., its left- andright-handed states do not mix [37]. This is essentially also true for extremely relativisticmassive particles as required by special relativity. Therefore the global flavor chiral symme-try of SU (2) L ⊗ SU (2) R for the family of up and down quarks is very good as their massesare so tiny compared to the QCD confinement energy scale.Under strong interactions like QCD, the non-vanishing vacuum expectation value of quarkcondensates can lead to spontaneous symmetry breaking (SSB) by mixing left- and right-handed quarks in the mass terms. The resulting pseudo-Nambu-Goldstone bosons (pNGB)and Higgs-like field will manifest as light bound states of quark condensates. For example,the approximate SU (2) L ⊗ SU (2) R chiral symmetry is spontaneously broken into SU (2) V ,i.e., the isospin symmetry at low energies in QCD, which can be described under an effectivetheory of the so-called σ -model [37]. In this case, the lightest isoscalar scalar σ or f (500)meson with mass of ∼
450 MeV serves as the quark condensate for SSB [38], a similar role toHiggs in electroweak SSB. The resulting pNGB particles are the three lightest pseudoscalarmesons ( π ± and π ). The Lagrangian for the matter part with omission of gauge fields andHiggs-like parts can be written as, L matter = ¯ q aL ( iγ µ D µ ) q aL + ¯ q aR ( iγ µ D µ ) q aR − m a (¯ q aL q aR + ¯ q aR q aL ) (8)where the left- and right-handed quark fields q L/R are summed over the flavor index a . Thenon-vanishing mass terms can mix left- and right-handed states and hence explicitly breakthe chiral symmetry.There is actually an extra global symmetry of U (1) L ⊗ U (1) R in the above QCD systembefore the SSB, where the U (1) L + R symmetry is conserved and manifests as baryon con-servation in QCD while the axial part U (1) L − R or U (1) A is explicitly broken by the axial7urrent anomaly, resulting in a CP violating term in the Lagrangian involving gauge field G , L θ = θg s π G · ˜ G (9)with θ modified by the Yukawa mass matrices for quarks as the physical strong CP phase¯ θ = θ − arg det( (cid:81) a m a ). This leads to the long-standing so-called U (1) A and strong CP puzzles in particle physics [39] as the ¯ θ parameter has to be fine-tuned to zero or at least ≤ − to be consistent with experimental constraints of the neutron electric dipole moment[40].In the scheme of 1 /N expanded QCD, Witten using a heuristic method [41] discovered aninteresting connection to the η (cid:48) meson as a possible pNGB to solve the U (1) A or strong CP problem although the η (cid:48) mass (958 MeV) seems to be too high for the above chiral SSB. Thegood Witten-Veneziano relation for obtaining the η (cid:48) mass under such an approach [41, 42]indicates some validity of the idea. In addition, it gives the correct QCD transition scaleof about 180 MeV and relates the η (cid:48) mass to the interesting topological properties of QCD[41, 42].At a little earlier time, Peccie and Quinn [43, 44] conjectured a so-called U (1) P Q axialsymmetry to solve the U (1) A problem by dynamically canceling the axial anomaly with animagined “axion” field. Here we could combine the two brilliant ideas and find the clue forsolving the problem as shown below.The key is to realize that the QCD symmetry breaking transition can be staged as shownin Table I. That is, we could have a strange quark condensation first leading to an SSB at ahigher energy scale and then the normal SU (2) chiral SSB at slightly lower energy. At theearly stage, it is the strange U (1) (i.e., U s (1)) symmetry that gets spontaneously broken.The U s (1) L + R is kept as strange number conservation in QCD that will then be broken bythe electroweak force while the other global U s (1) L − R symmetry is broken by mixing left-and right-handed strange quarks in the mass term. At the same time the SU (3) flavorsymmetry of (u,d,s) quarks is broken into SU (2) of (u,d) quarks with five pNGB particlesof K ± , K L,S , and η (more exactly η ). The broken U s (1) L − R or U s (1) A gives another pNGB,i.e., η (cid:48) (more exactly η with quark configuration of u ¯ u + d ¯ d + s ¯ s ), as Witten suspected. TheHiggs-like particle leading to this SSB is the scalar singlet f (980) meson with mass of 990MeV [35] that is perfectly compatible with the seemingly heavy η (cid:48) .The U s (1) A symmetry has all the desired necessary features of the arbitrary U (1) P Q ABLE I. Possible stages of QCD spontaneous symmetry breaking or phase transitions are shown.Candidates of Higgs-like and pNGB particles are taken from the compilation of Particle DataGroup [35]. The major oscillations of neutral condensates and non-perturbative processes at eachstage are listed as well.SSB stages ( u, d ) s ¯ s c ¯ c b ¯ b t ¯ t Higgs-like σ/f (500) f (980) χ c (1 P ) χ b (1 P ) HiggsBroken Symm. chiral SU (2) U s (1) A and SU (3) → SU (2) U c (1) A U b (1) A U t (1) A and EWpNGB π ± , π η ( η (cid:48) ) and K ± , K L,S , η ( η ) η c (1 S ) η b (1 S ) η t (1 S )?Oscillations n − n (cid:48) K − K (cid:48) D − D (cid:48) B − B (cid:48) H − H (cid:48) Non-perturbative s-quarkiton c-quarkiton b-quarkiton t-quarkiton andsphaleron axial symmetry conjectured by Peccie and Quinn [43, 44]. That is, SSB of U s (1) A due tostrange quark condensation provides a Higgs-like field ( f (980)) and a pNGB ( η ) that candynamically drive the U (1) A axial anomaly and the ¯ θ parameter to zero and therefore solvingthe strong CP problem. The imagined “axion” from SSB of the Peccie-Quinn symmetry[45] is not needed and the problem can be solved within the framework of SM without newparticles.However, such a solution does not seem to provide a B-violation mechanism for solvingthe BAU problem as U s (1) L + R or strange number is conserved. Another key insight relatedto the non-perturbative effects and topological structures of QCD and SM will be discussedbelow.The work of ’t Hooft [46, 47] interpreted the U (1) A anomaly in the chiral SSB as the topo-logical effects in QCD and introduced the so-called θ -vacua between which tunneling occursvia instantons non-perturbatively although such quantum tunneling effects are extremelysuppressed. It is actually this kind of non-trivial θ -vacuum structure and instanton-likegauge field solutions leading to the desired B -violation in SM. Below we provide a briefreview of the known electroweak sphaleron under gauge SSB and then propose a new typesphaleron-like process using the above-discussed dynamic SSB on global symmetries.9 saddle-point gauge field solution called “sphaleron” in the electroweak interaction of SU (2) L ⊗ U (1) Y was first discovered in 1984 by Klinkhamer and Manton [4] that inspiredvarious electroweak baryogenesis models later. Finite temperature effects considered by Ref.[5] make the sphaleron-like process rate high enough for B -violation around or above theelectroweak phase transition energy scale. Recently an SU (3) sphaleron has been proposedand calculated [48, 49] and could be related to the non-Abelian chiral anomaly [50].The nontrivial vacuum structure in gauge theories can be characterized by the Chern-Simons integer or the winding number N CS and transitions between topologically inequiv-alent vacuum configurations can then be denoted by the integer Pontryagin index or thetopological charge, Q ≡ ∆ N CS = g π (cid:90) d xG · ˜ G. (10)The electroweak sphaleron is associated with the SSB of the electroweak gauge symmetry SU (2) and the global B and L anomalies of SU (2) U (1) B and SU (2) U (1) L , respectively.The corresponding anomalous baryon and lepton number currents can be written as, ∂ µ J Bµ = ∂ µ J Lµ = g N g π G · ˜ G (11)and therefore the baryon and lepton number conservation is violated for a topological tran-sition as follows, ∆ B = ∆ L = 2 N g ∆ N CS (12)where N g = 3 is the number of generations. The sphaleron sits at the top of the energybarrier between two adjacent vacuum configurations with ∆ N CS = 1 / B and L numbers by three units (i.e., N g ) while conserving B − L at the same time. The sphaleronenergy can be estimated as [4], E s ∼ M W /α ∼
10 TeV (13)which essentially defines the height of the barrier between topologically disconnected vacua.Now the question becomes if there is a similar sphaleron-like process that could occurat the energy scale of the QCD phase transition. The answer is very likely. There couldbe a similar saddle-point solution when the QCD gauge fields are included with a dynamicSSB on global symmetries, we will call it “quarkiton” to distinguish from sphaleron for theelectroweak gauge SSB. 10he quarkiton process is assumed to be associated with the strange quark condensationand the strange chiral U s (1) A SSB as discussed above. As such, it is related to the strangechiral anomaly of SU (3) c U s (1) A under QCD with an anomalous chiral current for strangequarks expressed by its divergence, ∂ µ J sµ = g N c π G · ˜ G (14)and the strange chirality violation can be obtained as follows,∆ S c = 2 N c ∆ N CS (15)where N c = 3 is the quark color degree of freedom. This chirality violation requires threestrange quarks of the same chirality to form the quarkiton at the top of the energy bar-rier between two neighboring QCD vacuum configurations with ∆ N CS = 1 / SU (2) L U (1) A within the 2nd generation of quarks and leptons provide additional selectionrules of ∆ B = ∆ L = 2∆ N CS = 1 for the quarkiton. For the full SM gauge theory, therefore,it is natural to construct the quarkiton as a B and L violating process (by one unit for each)involving three strange quarks and three leptons in the same generation like the following, sss + µ + ν µ ν µ ⇔ Quarkiton ⇔ ¯ s ¯ s ¯ s + µ − ¯ ν µ ¯ ν µ (16)where all of quarks and leptons are left-handed, three strange quarks ensure a color singlet,and the overall B − L is conserved. In particular, a quarkiton is configured to be a neutralsinglet under the SM gauge symmetry of SU (3) c ⊗ SU (2) L ⊗ U (1) Y . A complete topologicaltransition of ∆ N CS = 1 via quarkiton can be described by Eq. (16) as follows: the lhsparticles excited out of one vacuum configuration form the quarkiton over the barrier andthen decay to the corresponding anti-particles on the rhs of Eq. (16) with respect to thenext vacuum configuration.To estimate the quarkiton energy, we apply SSB on the global flavor symmetry SU (3)of (u,d,s) quarks and the chiral strange U s (1) instead of the gauge symmetries as used insphaleron calculations [4, 49]. We can derive a similar saddle-point solution with its energyrelated to the pNGB particles of quark condensates instead of the elementary gauge bosons.In particular, the quarkiton energy can be related to the kaon mass and the kaon-quarkcoupling as follows, E q ∼ m K /α Kqq ∼ . α Kqq = g Kqq / π ∼ g πNN = 13 .
4, the corresponding pion-quark coupling constant g πqq ≈ g πNN / (3 g A ) ≈ .
6, and α πqq = g πqq / π ∼
1. So the quarkiton energy is on the sameorder of the kaon mass m K ∼ . K − K (cid:48) oscillations discussed in the previous section. Like the electroweak transition [6], the QCDphase transition in the early universe is most likely a smooth crossover [33, 34] and theextra K ( d ¯ s ) particles will be deconfined back into free down and anti-strange quarks.That is, the extra down quarks from K can be saved once all the extra anti-strange quarksare converted to strange quarks via the quarkiton process and then condensate again intomesons. Half of the saved down quarks are subsequently transitioned to up quarks by theelectroweak interaction. When the next stage QCD phase transition (i.e., the chiral SU (2)SSB) occurs at possibly around T = 100 −
150 MeV or temperatures mostly overlappedwith the s-quark condensation process for a smooth phase transition crossover, these extraup and down quarks will condensate into protons and neutrons forming the initial baryoncontent of the universe. The net effect after all these processes for one K ( d ¯ s ) excess is, d + ¯ s → p + 16 n + 16 e − + 16 ¯ ν e + 13 ν µ . (18)During the strange quark condensation, kaons are the lightest strange mesons. So it issafe to assume that about half of strange quarks condensate into K while the other halfinto K ± . Before condensation the (anti-)strange quark number to entropy density ratio is n s ¯ s /s = 4 × − owing to an effective number of relativistic degrees of freedom g ∗ = 61 . n B /s = 5 . × − . Considering that mostof the baryon excess generated above will be converted to mirror baryons subsequently via n − n (cid:48) oscillations [14] and today’s observed dark/mirror-to-baryon ratio is 5.4, the eventualleftover baryons in the normal sector will be n B /s = 8 . × − that agrees very well withthe observed value.Note that B − L is conserved at the end of net baryon generation from (18) with extra12mount of ν µ equal to the net baryon number. The fate of these and other neutrinos andtheir effects on the thermal evolution of the universe will be discussed in the next section.Now one may wonder if a similar quarkiton process and SSB could also operate earlierat higher temperatures for charm, bottom, and even top quark condensation. Interestingly,analogous to the strange quarkiton process, the following could be conceived to occur atdifferent condensation stages for c-, b-, and t- quarks, respectively, ccc + µ − µ − ¯ ν µ ⇔ Quarkiton ⇔ ¯ c ¯ c ¯ c + µ + µ + ν µ (19) bbb + τ + ν τ ν τ ⇔ Quarkiton ⇔ ¯ b ¯ b ¯ b + τ − ¯ ν τ ¯ ν τ (20) ttt + τ − τ − ¯ ν τ ⇔ Quarkiton ⇔ ¯ t ¯ t ¯ t + τ + τ + ν τ (21)where the SM gauge singlet configuration is required for all quarkitons. The Higgs-likecandidates could be χ c (1 P ) for c-quark condensation and χ b (1 P ) for b-quark condensationwith the possible pNGB particles of η c (1 S ) and η b (1 S ) for breaking the corresponding U c (1) A and U b (1) A symmetries, respectively, as shown in Table I.Another interesting idea could be conceived from the coincident energy scale of t-quarkcondensation and electroweak phase transition. That is, the actual Higgs could be a boundstate of top quark condensate that breaks both the global QCD top flavor U t (1) A and theelectroweak gauge symmetries at the same time by giving mass to all the fermion particlesand defining the SM vacuum structure. The subsequent b-, c-, s- quark condensation andSSB transitions just modify the QCD vacuum structure further. Together with evidence ofsimilar K mass differences due to CP violation and mirror splitting as discussed earlier,one may wonder if at the scale of T EW the top quark condensation could also break thedegeneracy of normal and mirror worlds and cause the CP violation at the same time.These phase transition processes can lead to more particle oscillations between the normaland mirror sectors from D , B , and Higgs during the c-, b-, and t-quark condensationphases, respectively. For Higgs with ∆ HH (cid:48) ∼ − eV and sin (2 θ ) ∼ − (as a t-quarkcondensate) [14], one can get a small oscillation parameter of (cid:15) ( HH (cid:48) ) ∼ − at T c = 100GeV from Eq. (5). For D with ∆ D D (cid:48) ∼ − eV and sin (2 θ ) ∼ − [14], we can estimate (cid:15) ( D D (cid:48) ) ∼ − at T c = 1 GeV from Eq. (5). Similarly, (cid:15) ( B B (cid:48) ) ∼ − at T c = 10 GeVfor B with ∆ B B (cid:48) ∼ − eV and sin (2 θ ) ∼ − [14]. These oscillations are much weakercompared to the K − K (cid:48) oscillations and therefore they are negligible for the generationof BAU. 13 V. CONSISTENT ORIGIN OF BAU AND DARK MATTER
As demonstrated in the previous sections, K − K (cid:48) oscillations provide an intriguingmechanism for the generation of the matter-antimatter asymmetry. In combination withthe n − n (cid:48) oscillations under the new model [14], a consistent picture for the origin of bothBAU and dark matter will be presented in this section.Besides the two built-in model parameters of the mixing strength sin (2 θ ) and the massdifference ∆, a third cosmological parameter x = T (cid:48) /T has to be constrained for such oscil-lations to work. To be consistent with the results of the standard big bang nucleosynthesis(BBN) model, in particular, the well known primordial helium abundance, a strict require-ment of T (cid:48) /T < / T (cid:48) /T ∼ . n − n (cid:48) oscillations of the new mirror matter model, whichcould potentially solve the primordial Li puzzle [51, 52] as well.As shown in the earlier discussions, the normal and mirror sectors do not exchange muchvia oscillations in the early universe, only on the order of 10 − or less for D , B , andHiggs oscillations. The largest exchange of a few percents comes from K − K (cid:48) oscillations.Although the n − n (cid:48) oscillations [14] are more dramatic, the overall baryon density is toolow at the moment and consequently the n − n (cid:48) induced exchange between the two sectorsis much smaller. Therefore, the entropy of each sector is approximately conserved from theelectroweak phase transition ( T = 100 GeV) until after BBN. Meanwhile, the macroscopicasymmetry on the ratio of T (cid:48) /T is mostly preserved as well.However, neutrino-mirror neutrino oscillations could become significant when the universecools down to T = 0 . νν (cid:48) ∼ − eV [14]. This can significantly change the entropy of each sector and also the temperatures ofnormal and mirror neutrinos. On the other hand, the normal and mirror gamma temper-atures should stay intact since neutrinos have decoupled long before this moment. Whenwe discuss the baryon-to-entropy ratio of n B /s the traditional entropy definition is used byignoring the entropy changes due to possible ν − ν (cid:48) oscillations. Notwithstanding, a new14nderstanding of the nature of neutrinos and mirror neutrinos in the extended StandardModel with Mirror Matter (SM ) predicts that such ν − ν (cid:48) oscillations are not possible [29].Only the criterion of T (cid:48) /T < / T (cid:48) /T = 1 / K (cid:48) and n (cid:48) occur first when the normalsector is still above the QCD phase transition temperature making its effective number ofrelativistic degrees of freedom g ∗ much larger. These early oscillation processes contributelittle as their oscillation parameter of (cid:15) = (cid:82) P ( τ f ) λdt is greatly suppressed by a factor of( T (cid:48) /T ) (cid:112) g ∗ ( T (cid:48) ) /g ∗ ( T ) [14].When the K → K (cid:48) oscillations operate between T = 100 −
150 MeV, initial matter-antimatter asymmetry is generated in the normal sector as discussed earlier while the excessof ¯ K (cid:48) in the mirror sector will quickly decay into mirror pions at much lower mirror tem-peratures. Therefore, nearly all the initial baryon asymmetry originates from the normalsector and the mirror sector contributes little to the net baryon content in the beginning.Possibly slightly after the inception of K → K (cid:48) oscillations, the n → n (cid:48) oscillationsstart to convert the initial net baryons into mirror baryons. For a likely smooth crossoverof QCD phase transition [33, 34], the two oscillation processes probably overlap over a largetemperature range (e.g., between 100 and 150 MeV). The peak of n → n (cid:48) oscillations occursat about 70 MeV well after the end of K → K (cid:48) oscillations. The n − n (cid:48) oscillation ratedrops quickly below T = 60 MeV whereas n − n (cid:48) oscillations still keep a very small exchangerate between the two sectors even at temperatures below 10 MeV. The final mirror-to-normalbaryon ratio eventually becomes about 5.4 as the observed dark-to-baryon ratio.Once the mirror BBN starts, most of mirror neutrons will be fused into mirror helium.Instead of having mirror neutrons depleted as in standard BBN calculations, n − n (cid:48) oscil-lations will keep an appreciable n (cid:48) abundance in the mirror sector. The normal BBN thenfollows and most of normal neutrons are fused into normal helium. At this moment, thereverse n (cid:48) → n oscillations will feed the normal sector with more neutrons at lower energies.These additional low energy neutrons will help destroy the extra Be formed earlier andpotentially solve the primordial Li problem [51, 52]. In particular, lower energy neutronscan make the Be destruction rate much higher than the n+p fusion rate [53] and thereforealleviate the issue of lithium-deuterium anti-correlation [54].Under the above consistent picture of particle oscillations, one can further examine the15
ABLE II. The sequence of events in the early universe is listed during the period of K − K (cid:48) and n − n (cid:48) oscillations using T (cid:48) /T = 1 / T [MeV] T (cid:48) [MeV] events450 150 start of mirror s-quark condensation and suppressed K (cid:48) → K oscilla-tions; start of mirror nucleon formation and suppressed n (cid:48) → n oscilla-tions (possibly slightly later)300 100 end of suppressed K (cid:48) → K oscillations210 70 peak of suppressed n (cid:48) → n oscillations150 50 start of normal s-quark condensation and K → K (cid:48) oscillations; startof normal nucleon formation, n → n (cid:48) oscillations, and generation ofmatter-antimatter asymmetry (possibly slightly later)100 33 end of K → K (cid:48) oscillations70 23 peak of n → n (cid:48) oscillations60 20 major n → n (cid:48) conversion peak done10 3 tailing of n → n (cid:48) oscillations; final dark(mirror)-to-baryon matter ratio1 0.3 normal weak interaction decoupling0.3 0.1 start of mirror BBN0.15 0.05 mirror helium formed; mirror neutrons gained from n → n (cid:48) oscillations0.1 0.03 start of normal BBN0.05 0.017 normal helium formed; low energy normal neutrons gained from n (cid:48) → n oscillations resulting the destruction of Be0.8 keV 0.3 keV start of ν − ν (cid:48) oscillations [14] or no ν − ν (cid:48) oscillations in SM [29] relations between the mixing strength sin (2 θ ), the mass difference ∆ nn (cid:48) , the mirror-to-normal baryon ratio, and the QCD phase transition temperature T c using the frameworkdeveloped in the original work of the new mirror matter model [14]. Figure 1 shows themirror-to-normal baryon ratio as function of the mass difference for varied QCD phasetransition temperatures and sin (2 θ ) = 2 × − . Figure 2 depicts the mixing strengthvs the mass difference for three different QCD phase transition temperatures assuming a16 M i rr o r- t o - N o r m a l B a r y on R a t i o Δ nn' [10 -6 eV]T c = 100 MeV 1 100 10000 1x10 M i rr o r- t o - N o r m a l B a r y on R a t i o Δ nn' [10 -6 eV]T c = 150 MeV 1 100 10000 1x10 M i rr o r- t o - N o r m a l B a r y on R a t i o Δ nn' [10 -6 eV]T c = 200 MeV 1 100 10000 1x10 M i rr o r- t o - N o r m a l B a r y on R a t i o Δ nn' [10 -6 eV]n dark /n B = 5.4 FIG. 1. The mirror-to-normal baryon ratio is shown as function of the mass difference ∆ nn (cid:48) assuming the mixing strength sin (2 θ ) = 2 × − . The QCD phase transition temperature isvaried to be T c = 100 , ,
200 MeV, respectively. S i n ( θ ) [ - ] Δ nn' [10 -6 eV] T c = 100 MeV 0.1 1 10 0.1 1 10 100 1000 S i n ( θ ) [ - ] Δ nn' [10 -6 eV] T c = 150 MeV 0.1 1 10 0.1 1 10 100 1000 S i n ( θ ) [ - ] Δ nn' [10 -6 eV] T c = 200 MeV 0.1 1 10 0.1 1 10 100 1000 S i n ( θ ) [ - ] Δ nn' [10 -6 eV] Sin (2 θ ) = 2x10 -5 FIG. 2. The mixing strength sin (2 θ ) is shown as function of the mass difference ∆ nn (cid:48) assuminga mirror-to-normal baryon ratio of 5.4. The QCD phase transition temperature is varied to be T c = 100 , ,
200 MeV, respectively. mirror-to-normal baryon ratio of 5.4.For the most likely QCD phase transition temperature range (150-200 MeV) [33, 34] andthe well observed dark-to-baryon ratio of 5.4, the n − n (cid:48) mass difference, as shown in Figs.1-2, can be constrained as ∆ nn (cid:48) = 10 − − − eV by the uncertainty of the mixing strength8 × − ≤ sin (2 θ ) ≤ × − inferred from neutron lifetime measurements [14]. The bestvalue of ∆ nn (cid:48) = 3 × − eV corresponds to the mixing strength of sin (2 θ ) = 2 × − .17ote that no upper limit on ∆ nn (cid:48) can be set if the QCD phase transition temperature issomehow much lower (e.g., around 100 MeV). A detailed study of applying the new modelto BBN may help further constrain these parameters. More neutron lifetime measurementswith different magnetic traps can certainly provide a much more accurate value of sin (2 θ )and consequently better ∆ nn (cid:48) . And furthermore, it may also provide a way to pin down theQCD phase transition temperature. V. CONCLUSION
Under the new mirror-matter model [14] and new understanding of possibly staged QCDsymmetry breaking phase transitions, the long-standing BAU puzzle can be naturally ex-plained with K − K (cid:48) oscillations that occur at the stage of strange quark condensation.A consistent picture of particle-mirror particle oscillations throughout the early universe ispresented including a self-consistent origin of both BAU and dark matter. Meanwhile, the U (1) A or strong CP problem in studies of particle physics is understood under the sameframework. The connection between the CP violation in SM and the normal-mirror masssplitting seemingly points to the same mechanism in new physics that needs to be exploredin the future. Non-perturbative processes via quarkitons at different quark condensationstages are proposed for B-violation and could be verified and further understood with cal-culations using the lattice QCD technique. More accurate studies on K L,S at the kaonproduction facilities, in particular, on the branching fractions of their invisible decays [14]that surprisingly are not constrained experimentally [55], will better quantify the generationof baryon matter in the early universe. Future experiments at the Large Hadron Collidermay provide more clues for such topological quarkiton processes and reveal more secrets inthe SM gauge structure, the Higgs mechanism, and the amazing oscillations between thenormal and mirror worlds.
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