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D 100, 063537 (2019)

Kaon oscillations and of the

Wanpeng Tan * Department of , 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

(Received 28 April 2019; revised manuscript received 3 August 2019; published 25 September 2019)

0 of the universe (BAU) can likely be explained with K0 − K0 oscillations of a newly developed mirror- model and new understanding of (QCD) phase transitions. A consistent picture for the origin of both BAU and is presented with the aid of n − n0 oscillations of the new model. The global breaking transitions in QCD are proposed to be staged depending on condensation temperatures of strange, , bottom, and top in the early 0 universe. The long-standing BAU puzzle could then be understood with K0 − K0 oscillations that occur at the stage of strange condensation and violation via a nonperturbative sphaleronlike (coined “quarkiton”) process. Similar processes at charm, bottom, and condensation stages are also discussed including an interesting idea for top quark condensation to break both the QCD global ð1Þ ð1Þ Ut A symmetry and the electroweak gauge symmetry at the same time. Meanwhile, the U A or strong CP problem of physics is addressed with a possible explanation under the same framework.

DOI: 10.1103/PhysRevD.100.063537

I. INTRODUCTION drops below T ¼ 38 MeV [2] to avoid the catastrophe between and antibaryons. The matter- imbalance or baryon asymmetry Sakharov proposed three criteria to generate the initial of the universe (BAU) has been a long standing puzzle BAU: (i) baryon number (B-) violation (ii) C and CP in the study of cosmology. Such an asymmetry can be violation (iii) departure from thermal equilibrium [3]. The quantified in various ways. The cosmic microwave back- (SM) is known to violate both C and CP ground (CMB) data by Planck set a very precise observed 2 and it does not conserve baryon number only although it baryon density of the universe at Ω h ¼ 0.02242 b does B − L (difference of baryon and numbers). 0.00014 [1]. This corresponds to today’s baryon-to- −10 Coupled with possible nonequilibrium in the thermal number density ratio of n =nγ ¼ 6.1 × 10 . For an B history of the early universe, it seems to be easy to solve adiabatically expanding universe, it would be better to ¼ the BAU problem. Unfortunately, the violations in SM use the baryon-number-to- density ratio of nB=s without new physics are too small to explain the observed 8 7 10−11 . × to quantify the BAU, which may have to be fairly large BAU. The only known B-violation processes in modified under the new understanding of the SM are nonperturbative, for example, via the so-called history in the early universe (see Sec. IV). [4] which involves nine quarks and three From known physics, it is difficult to explain the from each of the three generations. It was also found out observed BAU. For example, for an initially baryon- that the sphaleron process can be much faster around symmetric universe, the surviving relic baryon density or above the temperature of the electroweak symmetry from the annihilation process is about nine orders of ∼ 100 breaking or TEW GeV [5]. This magnitude lower than the observed one [2]. Therefore, essentially washes out any BAU generated early or around an asymmetry is needed in the early universe and the TEW since the electroweak transition is most likely just a BAU has to exist before the temperature of the universe smooth cross-over instead of “desired” strong first order [6]. It makes the appealing electroweak *[email protected] models [5,7] ineffective and new physics often involving the Higgs have to be added in the models [8–10]. Recently Published by the American Physical Society under the terms of lower energy baryogenesis typically using particle oscil- the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to lations stimulated some interesting ideas [11,12]. Other the author(s) and the published article’s title, journal citation, types of models such as leptogenesis [13] are typically less and DOI. Funded by SCOAP3. testable or have other difficulties.

2470-0010=2019=100(6)=063537(10) 063537-1 Published by the American Physical Society WANPENG TAN PHYS. REV. D 100, 063537 (2019)

−15 −14 Here we present a simple picture for baryogenesis ∼10 –10 universal for all the that acquired at energies around quantum chromodynamics (QCD) mass from the Higgs vacuum. Further details of the model 0 phase transition with K0 − K0 oscillations based on a can be found in Ref. [14]. 0 K0 − K0 The immediate result of this model for this study is the newly developed model [14]. 0 00 oscillations and the new mirror matter model will be first probability of K − K oscillations in vacuum [14], introduced to demonstrate how to generate the “potential” 1 amount of BAU as observed. Then the QCD phase 2 2 P 0 00 ðtÞ¼sin ð2θÞsin Δ 0 00 t ð1Þ transition will be reviewed and the sphaleronlike non- K K 2 K K perturbative processes are proposed to provide B-violation 0 0 0 0 2 and realize the “potential” BAU created by K0 − K0 where θ is the K − K mixing angle and sin ð2θÞ denotes −4 oscillations. In the end, the observed BAU is generated the mixing strength of about 10 , t is the propagation time, 0 − Δ 0 00 ¼ m 0 − m 0 is the small mass difference of the right before the n n oscillations that determine the final K K K2 K1 mirror(dark)-to-normal matter ratio of the universe [14]. two mass eigenstates of about 10−6 eV [14], and natural ð1Þ Meanwhile, the longstanding U A and strong CP prob- units (ℏ ¼ c ¼ 1) are used for simplicity. Note that the lems in are also naturally resolved under equation is valid even for relativistic and in this case t the same framework. is the proper time in the particle’s rest frame. There are 0 0 actually two weak eigenstates of K in each sector, i.e., KS 0 00 0 9 10−11 5 10−8 II. K − K OSCILLATIONS AND THE and KL with lifetimes of × sand × s, NEW MODEL respectively. Their mass difference is about 3.5 × 10−6 eV very similar to Δ 0 00 , which makes one wonder if the two To understand the observed BAU, we need to apply the K K newly developed particle-mirror particle oscillation model mass differences and even the CP violation may originate [14]. It is based on the mirror matter theory [15–22], that from the same source. is, two sectors of particles have identical For kaons that travel in the thermal bath of the early within their own sector but share the same gravitational universe, each collision or with another particle will collapse the oscillating wave function into a mirror . Such a mirror matter theory has appealing theoretical τ ⊗ 0 eigenstate. In other words, during mean free flight time f features. For example, it can be embedded in the E8 E8 0 00 − 0 ðτ Þ [17,23,24] and it can also be a natural the K K transition probability is PK0K0 f .Thenum- 1 τ extension of recently developed twin Higgs models [25,26] ber of such collisions will be = f in a unit time. Therefore, 0 that protect the Higgs mass from quadratic divergences the transition rate of K0 − K0 with interaction is [14], and hence solve the hierarchy or fine-tuning problem. The mirror symmetry or twin is particularly 1 2 2 1 λ 0 ¼ ð2θÞ Δ 0 τ ð Þ K0K0 sin sin K0K0 f : 2 intriguing as the Large Collider has found no τf 2 evidence of so far and we may not need supersymmetry, at least not below energies of 10 TeV. Such Note that the Mikheyev-Smirnov-Wolfenstein (MSW) a mirror matter theory can explain various observations in matter effect [31,32], i.e., coherent forward scattering that the universe including the lifetime puzzle and dark- could affect the oscillations is negligible as the to-baryon matter ratio [14], evolution and nucleosynthesis density is very low when kaons start to condensate from in stars [27], ultrahigh energy cosmic rays [28], dark energy the QCD plasma (see more details for in-medium particle [29], and a requirement of strongly self-interacting dark oscillations from Ref. [27]), and in particular, the QCD matter to address numerous discrepancies on the galactic phase transition is most likely a smooth crossover [33,34]. scale [30]. It is not very well understood how the QCD symmetry In this new mirror matter model [14], no cross-sector breaking or phase transition occur in the early universe, interaction is introduced, unlike other particle oscillation which will be discussed in detail in the next section. type models. The critical assumption of this model is Let us suppose that the temperature of QCD phase that the mirror symmetry is spontaneously broken by the transition Tc is about 150 MeV and a different value uneven Higgs vacuum in the two sectors, i.e., hϕi ≠ hϕ0i, (e.g., 200 MeV) here does not affect the following although very slightly (on a relative breaking scale of discussions and results. At this time only up, down, ∼10−15–10−14) [14]. When particles obtain their and strange quarks are free. It is natural to assume that mass from the Yukawa coupling, it automatically leads to strange quarks become confined first during the transi- the mirror mixing for neutral particles, i.e., the basis of tion, i.e., forming particles first instead of and mass eigenstates is not the same as that of mirror eigen- . A better understanding of this process is shown states, similar to the case of ordinary neutrino oscillations in the next section. As a matter of fact, even if they all due to the family or generation mixing. The Higgs form at the same time, the equilibrium makes the ratio of mechanism makes the relative mass splitting scale of number to kaon number

063537-2 KAON OSCILLATIONS AND BARYON ASYMMETRY OF THE … PHYS. REV. D 100, 063537 (2019) 3=2 nN mN take part in the oscillations. Considering the above cor- ≃ expð−ðmN − mKÞ=TcÞ ∼ 0.1 ð3Þ 0 nK mK rection, the final-to-initial K abundance ratio in the normal world due to oscillations becomes, very small due to the fact that kaons are much lighter than nucleons. Xf 2 ¼ 1 − ϵ: ð6Þ Once neutral kaons are formed, they start to oscillate by Xi 3 participating in the with cross section of 2 2 −5 −2 δ ¼ σ ∼ G T G ¼ 1 17 10 The CP violation amplitude in SM is measured as EW F where F . × GeV is the Fermi −3 2 −6 0’ 2.228 × 10 [35] so that δ ∼ 5 × 10 and the oscillation . Then one can estimate K s thermally 2 0 0 ∼ 1 − δ averaged reaction rate over the Bose-Einstein distribution, probability ratio can be estimated as PK0K0 =PK¯ 0K¯ 0 . 0 Z Then the net K fraction can be obtained as follows, ∞ Γ ¼ g 3 ð Þσ p d pf p Δ 0 0 0 − 0 1 25 ð2πÞ3 EW XK K¯ XK XK¯ 2 −8 0 m ≡ ¼ ϵδ ∼ × 10 ð7Þ Z 0 0 0 þ 0 3 3 2 2 ∞ 3 XK K¯ XK XK¯ ¼ g GFT pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip ð Þ 2 dp 4 2π m 0 expð p2 þ m2=TÞ − 1 If the excess of K0ðds¯Þ generated above can survive by some B-violation process, i.e., dumping s¯ quarks and ¼ 2 0 0 where g for both KS and KL, m is the mass of kaons, leaving d quarks to form nucleons in the end, then and T is the temperature. The expansion rate of the universe assuming that half of strange quarks condensate into ∼ 2 −1 0 at this time can be estimated to be H TMeV s where KL;S (with the other half in K ) we will end up with a −10 TMeV is the temperature in unit of MeV. The condition for net baryon density of nB=s ¼ 5.6 × 10 that essentially K0 to decouple from the interaction or freeze out is gives the sum of the observed baryon and dark matter. Γ=H < 1. It can be easily calculated from Eq. (4) that In the next section, we will demonstrate how such a the freezeout occurs at Tfo ¼ 100 MeV. This means that B-violation process could occur during the QCD phase kaon oscillations have to operate between Tc ¼ 150 MeV transition. 0 2ð2θÞ ∼ 10−4 and Tfo ¼ 100 MeV. And fortunately the K have In Eq. (5) the mixing strength sin and the −6 Δ 0 ∼ 10 long enough lifetime (compared to the weak interaction mass splitting parameter K0K0 eV are estimated rate) for such oscillations and BAU to occur during this from n − n0 oscillations in Ref. [14] assuming that the temperature range. single-quark mixing strength is similar and the mass For the standard constraint on the mirror-to-normal splitting parameter is scaled to the particle’s mass. matter temperature ratio of x ¼ T0=T < 1=2 [17,19] that Unfortunately, these estimates are still fairly rough as will be discussed further in Sec. IV, the two oscillation steps the neutron lifetime measurements have not yet constrained 0 0 of K0 → K0 and K0 → K0 will be decoupled in a similar the oscillation parameters well [14] resulting in a factor way as the n − n0 oscillations discussed in Ref. [14]. Using of ∼10 uncertainty in ϵ of Eq. (5). On the other hand, a typical weak interaction rate λ ¼ 1=τ ¼ G2 T5 ∼ the observed baryon asymmetry can be used to constrain EW f F ϵ ¼ 0 05 5 −1 ¼ 0 3 2 these parameters under the new mechanism, i.e., . TMeV s and the age of the universe t . =TMeV s 2 2 −16 2 0 or sin ð2θÞΔ 0 ¼ 10 eV . Remarkably, such param- during this period of time, one can get the final-to-initial K K0K0 abundance ratio in the ordinary sector for the second step, eters are consistent with the neutron lifetime experiments and the origin of dark matter under the new model (see Z Xf more discussions in Sec. IV). More detailed studies of ¼ − 0 ðτ Þλ exp P 0 0 f EWdt the mirror mixing parameters under the context of the Xi K K Z CKM matrix and proposed laboratory measurements can be Δ 0 2 T 1 28 2 K0K0 fo found in a separate paper [36]. ¼ exp −5 × 10 sin ð2θÞ d 7 eV Tc TMeV ¼ 1 − 0.05 ≡ 1 − ϵ ð5Þ III. QCD SYMMTRY BREAKING TRANSITION AND OTHER OSCILLATIONS and the first step is negligible due to the much faster A massless fermion particle’s or helicity has to expansion rate of the universe [14]. be preserved, i.e., its left- and right-handed states do not 0 9 10−11 However, KS has a lifetime of × s that is mix [37]. This is essentially also true for extremely comparable to the weak interaction rate at such temper- 0 relativistic massive particles as required by special rela- atures. Owing to this, only one third of KS particles tivity. Therefore the global flavor chiral symmetry of participate in the oscillations while the other two thirds SUð2Þ ⊗ SUð2Þ for the family of up and down quarks 0 0 L R decay to pions. In contrast to KS, KL mesons have a much is very good as their masses are so tiny compared to the larger lifetime (5 × 10−8 s) and hence almost all of them QCD confinement energy scale.

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TABLE 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 [35]. The major oscillations of neutral condensates and nonperturbative processes at each stage are listed as well.

SSB stages ðu; dÞ ssc¯ cb¯ bt¯ t¯

Higgs-like σ=f0ð500Þ f0ð980Þ χc0ð1PÞ χb0ð1PÞ Higgs ð2Þ ð1Þ ð3Þ → ð2Þ ð1Þ ð1Þ ð1Þ Broken Symm. chiral SU Us A and SU SU Uc A Ub A Ut A and EW π π0 η ðη0Þ 0 η ðηÞ η ð1 Þ η ð1 Þ η ð1 Þ pNGB , 1 and K , KL;S; 8 c S b S t S ? 0 0 0 0 0 Oscillations n − n K0 − K0 D0 − D0 B0 − B0 H − H Non-perturbative s-quarkiton c-quarkiton b-quarkiton t-quarkiton and sphaleron

Under strong interactions like QCD, the nonvanishing with experimental constraints of the neutron electric dipole of quark condensates can lead to moment [40]. spontaneous symmetry breaking (SSB) by mixing left- and In the scheme of 1=N expanded QCD, Witten using a right-handed quarks in the mass terms. The resulting heuristic method [41] discovered an interesting connection η0 ð1Þ pseudo-Nambu-Goldstone (pNGB) and Higgs-like to the meson as a possible pNGB to solve the U A or will manifest as bound states of quark con- strong CP problem although the η0 mass (958 MeV) seems ð2Þ ⊗ ð2Þ densates. For example, the approximate SU L SU R to be too high for the above chiral SSB. The good Witten- ð2Þ η0 chiral symmetry is spontaneously broken into SU V, Veneziano relation for obtaining the mass under such i.e., the symmetry at low energies in QCD, which an approach [41,42] indicates some validity of the idea. In can be described under an effective theory of the so-called addition, it gives the correct QCD transition scale of about 0 σ-model [37]. In this case, the lightest isoscalar scalar σ 180 MeV and relates the η mass to the interesting or f0ð500Þ meson with mass of ∼450 MeV serves as the topological properties of QCD [41,42]. quark condensate for SSB [38], a similar role to Higgs At a little earlier time, Peccie and Quinn [43,44] con- ð1Þ in electroweak SSB. The resulting pNGB particles are jectured a so-called U PQ axial symmetry to solve the 0 ð1Þ the three lightest pseudoscalar mesons (π and π ). The U A problem by dynamically canceling the axial Lagrangian for the matter part with omission of gauge with an imagined “” field. Here we could combine fields and Higgs-like parts can be written as, the two brilliant ideas and find the clue for solving the problem as shown below. The key is to realize that the QCD symmetry breaking ¼ ¯ a ð γμ Þ a þ ¯ a ð γμ Þ a − ð¯ a a þ ¯ a a Þ Lmatter qL i Dμ qL qR i Dμ qR ma qLqR qRqL transition can be staged as shown in Table I. That is, we ð8Þ could have a condensation first leading to an SSB at a higher energy scale and then the normal SUð2Þ where the left- and right-handed quark fields q are chiral SSB at slightly lower energy. At the early stage, it is L=R the strange Uð1Þ [i.e., U ð1Þ] symmetry that gets sponta- summed over the flavor index a. The nonvanishing mass s neously broken. The U ð1Þ is kept as strange number terms can mix left- and right-handed states and hence s LþR conservation in QCD that will then be broken by the explicitly break the chiral symmetry. ð1Þ ð1Þ ⊗ electroweak force while the other global Us L−R sym- There is actually an extra global symmetry of U L ð1Þ metry is broken by mixing left- and right-handed strange U R in the above QCD system before the SSB, where the ð3Þ ð1Þ quarks in the mass term. At the same time the SU flavor U LþR symmetry is conserved and manifests as baryon ð2Þ ð1Þ symmetry of (u,d,s) quarks is broken into SU of (u,d) conservation in QCD while the axial part U L−R or 0 η ð1Þ quarks with five pNGB particles of K , KL;S, and (more U A is explicitly broken by the axial current anomaly, η ð1Þ ð1Þ exactly 8). The broken Us L−R or Us A gives another resulting in a CP violating term in the Lagrangian involv- η0 η ing gauge field G, pNGB, i.e., (more exactly 1 with quark configuration of uu¯ þ dd¯ þ ss¯), as Witten suspected. The Higgs-like par- ticle leading to this SSB is the scalar singlet f0ð980Þ meson θ 2 ¼ gs ˜ ð Þ with mass of 990 MeV [35] that is perfectly compatible Lθ 2 G · G 9 32π with the seemingly heavy η0. ð1Þ The Us A symmetry has all the desired necessary θ ð1Þ with modified by the Yukawa mass matrices for quarksQ as features of the arbitrary U PQ axial symmetry conjec- θ¯ ¼ θ − ð Þ ð1Þ the physical strong CP phase arg det a ma . tured by Peccie and Quinn [43,44]. That is, SSB of Us A This leads to the long-standing so-called Uð1Þ and strong due to strange quark condensation provides a Higgs-like ¯ A CP puzzles in particle physics [39] as the θ parameter has field (f0ð980Þ) and a pNGB (η1) that can dynamically drive ≤ 10−9 ð1Þ θ¯ to be fine-tuned to zero or at least to be consistent the U A axial anomaly and the parameter to zero and

063537-4 KAON OSCILLATIONS AND BARYON ASYMMETRY OF THE … PHYS. REV. D 100, 063537 (2019) therefore solving the strong CP problem. The imagined from each generation and violates B and L numbers by “ ” axion from SSB of the Peccie-Quinn symmetry [45] is three units (i.e., Ng) while conserving B − L at the same not needed and the problem can be solved within the time. The sphaleron energy can be estimated as [4], framework of SM without new particles. ∼ α ∼ 10 ð Þ However, such a solution does not seem to provide a Es MW= TeV 13 B-violation mechanism for solving the BAU problem as ð1Þ which essentially defines the height of the barrier between Us LþR or strange number is conserved. Another key insight related to the non-perturbative effects and topo- topologically disconnected vacua. logical structures of QCD and SM will be discussed below. Now the question becomes if there is a similar sphaleron- ’ ð1Þ like process that could occur at the energy scale of the The work of t Hooft [46,47] interpreted the U A anomaly in the chiral SSB as the topological effects in QCD phase transition. The answer is very likely. There QCD and introduced the so-called θ-vacua between which could be a similar saddle-point solution when the QCD gauge fields are included with a dynamic SSB on global tunneling occurs via nonperturbatively although “ ” such quantum tunneling effects are extremely suppressed. , we will call it quarkiton to distinguish from It is actually this kind of non-trivial θ-vacuum structure sphaleron for the electroweak gauge SSB. and instantonlike gauge field solutions leading to the desired The quarkiton process is assumed to be associated with B-violation in SM. Below we provide a brief review of the the strange quark condensation and the strange chiral ð1Þ known electroweak sphaleron under gauge SSB and then Us A SSB as discussed above. As such, it is related to ð3Þ2 ð1Þ propose a new type sphaleronlike process using the above- the strange of SU cUs A under QCD discussed dynamic SSB on global symmetries. with an anomalous chiral current for strange quarks A saddle-point gauge field solution called “sphaleron” in expressed by its divergence, the electroweak interaction of SUð2Þ ⊗ Uð1Þ was first 2 L Y g N discovered in 1984 by Klinkhamer and Manton [4] that ∂μ 5s ¼ c ˜ ð Þ Jμ 16π2 G · G 14 inspired various electroweak baryogenesis models later. Finite temperature effects considered by Ref. [5] make and the strange chirality violation can be obtained as the sphaleronlike process rate high enough for B-violation follows, around or above the electroweak phase transition energy scale. Recently an SUð3Þ sphaleron has been proposed and ΔSc ¼ 2NcΔNCS ð15Þ calculated [48,49] and could be related to the non-Abelian ¼ 3 chiral anomaly [50]. where Nc is the quark color degree of freedom (d.o.f.). The nontrivial vacuum structure in gauge theories can be This chirality violation requires three strange quarks of the characterized by the Chern-Simons integer or the winding same chirality to form the quarkiton at the top of the energy number N and transitions between topologically inequi- barrier between two neighboring QCD vacuum configura- CS Δ ¼ 1 2 valent vacuum configurations can then be denoted by the tions with NCS = like the sphaleron. When the integer Pontryagin index or the topological , electroweak gauge symmetry is also considered, the chiral anomaly of SUð2Þ2 Uð1Þ within the 2nd generation of 2 Z L A ≡ Δ ¼ g 4 ˜ ð Þ quarks and leptons provide additional selection rules of Q NCS 2 d xG · G: 10 32π ΔB ¼ ΔL ¼ 2ΔNCS ¼ 1 for the quarkiton. For the full SM , therefore, it is natural to construct the The electroweak sphaleron is associated with the SSB quarkiton as a B and L violating process (by one unit for ð2Þ of the electroweak gauge symmetry SU and the global each) involving three strange quarks and three leptons in ð2Þ2 ð1Þ ð2Þ2 ð1Þ B and L anomalies of SU U B and SU U L, the same generation like the following, respectively. The corresponding anomalous baryon and þ − lepton number currents can be written as, sss þ μ νμνμ ⇔ Quarkiton ⇔ s¯ s¯ s¯ þμ ν¯μν¯μ ð16Þ 2 μ B μ L g Ng ˜ where all of quarks and leptons are left-handed, three ∂ Jμ ¼ ∂ Jμ ¼ G · G ð11Þ 16π2 strange quarks ensure a color singlet, and the overall B − L is conserved. In particular, a quarkiton is configured to and therefore the baryon and lepton number conservation is be a neutral singlet under the SM gauge symmetry of violated for a topological transition as follows, ð3Þ ⊗ ð2Þ ⊗ ð1Þ SU c SU L U Y. A complete topological tran- sition of ΔN ¼ 1 via quarkiton can be described by ΔB ¼ ΔL ¼ 2NgΔNCS ð12Þ CS Eq. (16) as follows: the left-hand side (lhs) particles excited where Ng ¼ 3 is the number of generations. The sphaleron out of one vacuum configuration form the quarkiton over sits at the top of the energy barrier between two adjacent the barrier and then decay to the corresponding vacuum configurations with ΔNCS ¼ 1=2 and hence it on the right-hand side (rhs) of Eq. (16) with respect to the involves nine quarks and three leptons (all left-handed) next vacuum configuration.

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To estimate the quarkiton energy, we apply SSB on the converted to mirror baryons subsequently via n − n0 global flavor symmetry SUð3Þ of (u,d,s) quarks and the oscillations [14] and today’s observed dark/mirror-to- chiral strange Usð1Þ instead of the gauge symmetries baryon ratio is 5.4, the eventual leftover baryons in the −11 as used in sphaleron calculations [4,49]. We can derive a normal sector will be nB=s ¼ 8.7 × 10 that agrees very similar saddle-point solution with its energy related to the well with the observed value. pNGB particles of quark condensates instead of the Note that B − L is conserved at the end of net baryon elementary gauge bosons. In particular, the quarkiton generation from (18) with extra amount of νμ equal to the energy can be related to the kaon mass and the kaon-quark net baryon number. The fate of these and other coupling as follows, and their effects on the thermal evolution of the universe will be discussed in the next section. ∼ α ∼ 0 5 ð Þ Eq mK= Kqq . GeV 17 Now one may wonder if a similar quarkiton process and SSB could also operate earlier at higher temperatures α ¼ 2 4π ∼ 1 where the kaon-quark coupling Kqq gKqq= is for charm, bottom, and even top quark condensation. inferred from the observed -nucleon coupling constant Interestingly, analogous to the strange quarkiton process, gπNN ¼ 13.4, the corresponding pion-quark coupling con- the following could be conceived to occur at different 2 stant gπqq ≈ gπNN=ð3gAÞ ≈ 3.6, and απqq ¼ gπqq=4π ∼ 1. condensation stages for c-, b-, and t- quarks, respectively, So the quarkiton energy is on the same order of the kaon þ μ−μ−ν¯ ⇔ ⇔ ¯ ¯ ¯ þμþμþν ð Þ mass mK ∼ 0.5 GeV and close to the energy scale of ccc μ Quarkiton c c c μ 19 0.2 GeV for the strange quark condensation or phase þ ¯ ¯ ¯ − transition. Such a low energy barrier ensures that the bbb þ τ ντντ ⇔ Quarkiton ⇔ b b b þτ ν¯τν¯τ ð20Þ quarkiton transition rate is high enough for B-violation − − þ þ around the QCD phase transition energy scale. ttt þ τ τ ν¯τ ⇔ Quarkiton ⇔ t¯ t¯ t¯ þτ τ ντ ð21Þ Such a quarkiton process can help solve the BAU 0 problem under the scenario of K0 − K0 oscillations dis- where the SM gauge singlet configuration is required for cussed in the previous section. Like the electroweak all quarkitons. The Higgs-like candidates could be χc0ð1PÞ transition [6], the QCD phase transition in the early for c-quark condensation and χb0ð1PÞ for b-quark con- universe is most likely a smooth crossover [33,34] and densation with the possible pNGB particles of ηcð1SÞ and 0 ¯ η ð1 Þ ð1Þ ð1Þ the extra K (ds) particles will be deconfined back into free b S for breaking the corresponding Uc A and Ub A down and antistrange quarks. That is, the extra down symmetries, respectively, as shown in Table I. quarks from K0 can be saved once all the extra antistrange Another interesting idea could be conceived from the quarks are converted to strange quarks via the quarkiton coincident energy scale of t-quark condensation and process and then condensate again into mesons. Half of the electroweak phase transition. That is, the actual Higgs saved down quarks are subsequently transitioned to up could be a of top quark condensate that breaks ð1Þ quarks by the electroweak interaction. When the next stage both the global QCD top flavor Ut A and the electroweak QCD phase transition (i.e., the chiral SUð2Þ SSB) occurs at gauge symmetries at the same time by giving mass to all the possibly around T ¼ 100–150 MeV or temperatures fermion particles and defining the SM vacuum structure. mostly overlapped with the s-quark condensation process The subsequent b-, c-, s- quark condensation and SSB for a smooth phase transition crossover, these extra up and transitions just modify the QCD vacuum structure further. down quarks will condensate into and Together with evidence of similar K0 mass differences due to forming the initial baryon content of the universe. The net CP violation and mirror splitting as discussed earlier, one 0 ¯ effect after all these processes for one K (ds) excess is, may wonder if at the scale of TEW the top quark con- densation could also break the degeneracy of normal and 1 1 1 1 1 − mirror worlds and cause the CP violation at the same time. d þ s¯ → p þ n þ e þ ν¯ þ νμ: ð18Þ 6 6 6 6 e 3 These phase transition processes can lead to more particle oscillations between the normal and mirror sectors from D0, During the strange quark condensation, kaons are the B0, and Higgs during the c-, b-, and t-quark condensation lightest strange mesons. So it is safe to assume that about −4 0 phases, respectively. For Higgs with ΔHH0 ∼ 10 eV and half of strange quarks condensate into K while the other 2 −4 sin ð2θÞ ∼ 10 (as a t-quark condensate) [14], one can get a half into K . Before condensation the (anti)strange quark 0 −18 −2 small oscillation parameter of ϵðHH Þ ∼ 10 at Tc ¼ number to entropy density ratio is nss¯=s ¼ 4 × 10 owing 0 −6 100 GeV from Eq. (5).ForD with Δ 0 00 ∼ 10 eV to an effective number of relativistic d.o.f. g ¼ 61.75 D D 2ð2θÞ ∼ 10−4 ϵð 0 00 Þ ∼ 10−8 during this stage. Taking into account the oscillation result and sin [14], we can estimate D D ¼ 1 ϵð 0 00 Þ ∼ 10−13 from Eq. (7) one can obtain a net baryon-number-to- at Tc GeV from Eq. (5). Similarly, B B at 0 −5 2 −10 ¼ 10 Δ 0 ∼ 10 ð2θÞ ∼ entropy density ratio of nB=s ¼ 5.6 × 10 . Considering Tc GeV for B with B0B0 eV and sin that most of the baryon excess generated above will be 10−4 [14]. These oscillations are much weaker compared to

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0 TABLE II. The sequence of events in the early universe is listed during the period of K0 − K0 and n − n0 oscillations using T0=T ¼ 1=3 as an example. The QCD phase transition temperature is assumed to be 150 MeV.

T [MeV] T0 [MeV] Events 0 450 150 Start of mirror s-quark condensation and suppressed K0 → K0 oscillations; start of mirror nucleon formation and suppressed n0 → n oscillations (possibly slightly later) 0 300 100 End of suppressed K0 → K0 oscillations 210 70 Peak of suppressed n0 → n oscillations 0 150 50 Start of normal s-quark condensation and K0 → K0 oscillations; start of normal nucleon formation, n → n0 oscillations, and generation of matter-antimatter asymmetry (possibly slightly later) 0 100 33 End of K0 → K0 oscillations 70 23 Peak of n → n0 oscillations 60 20 Major n → n0 conversion peak done 10 3 Tailing of n → n0 oscillations; final dark(mirror)-to-baryon matter ratio 1 0.3 Normal weak interaction decoupling 0.3 0.1 Start of mirror BBN 0.15 0.05 Mirror helium formed; mirror neutrons gained from n → n0 oscillations 0.1 0.03 Start of normal BBN 0.05 0.017 Normal helium formed; low energy normal neutrons gained from n0 → n oscillations resulting the destruction of 7Be 0.8 keV 0.3 keV Start of ν − ν0 oscillations [14] or no ν − ν0 oscillations in SM3 [29]

0 −8 0 0 the K0 − K0 oscillations and therefore they are negligible early universe, only on the order of 10 or less for D , B , for the generation of BAU. and Higgs oscillations. The largest exchange of a few 0 percents comes from K0 − K0 oscillations. Although the − 0 IV. CONSISTENT ORIGIN OF BAU n n oscillations [14] are more dramatic, the overall AND DARK MATTER baryon density is too low at the moment and consequently the n − n0 induced exchange between the two sectors is 0 As demonstrated in the previous sections, K0 − K0 much smaller. Therefore, the entropy of each sector is oscillations provide an intriguing mechanism for the gen- approximately conserved from the electroweak phase eration of the matter-antimatter asymmetry. In combination transition (T ¼ 100 GeV) until after BBN. Meanwhile, 0 with the n − n0 oscillations under the new model [14],a the macroscopic asymmetry on the ratio of T =T is mostly consistent picture for the origin of both BAU and dark preserved as well. matter will be presented in this section. However, neutrino-mirror neutrino oscillations could Besides the two built-in model parameters of the become significant when the universe cools down to T ¼ mixing strength sin2ð2θÞ and the mass difference Δ,a 0.8 keV or about 20 days after the assuming 0 2 −18 2 third cosmological parameter x ¼ T =T has to be con- Δνν0 ∼ 10 eV [14]. This can significantly change the strained for such oscillations to work. To be consistent entropy of each sector and also the temperatures of normal with the results of the standard big bang nucleosynthesis and mirror neutrinos. On the other hand, the normal and (BBN) model, in particular, the well known primordial mirror gamma temperatures should stay intact since neu- helium abundance, a strict requirement of T0=T < 1=2 at trinos have decoupled long before this moment. When we BBN temperatures [17,19–21] has to be met to ensure a discuss the baryon-to-entropy ratio of nB=s the traditional slow enough expansion of the universe. Such a temper- entropy definition is used by ignoring the entropy changes 0 ature condition can naturally occur after the early due to possible ν − ν oscillations. Notwithstanding, a new and subsequent reheating [17,19]. A typical ratio of understanding of the nature of neutrinos and mirror neutrinos T0=T ∼ 0.3 is evidently supported in the studies of ultra- in the extended Standard Model with Mirror Matter (SM3) high energy cosmic rays under the new mirror matter predicts that such ν − ν0 oscillations are not possible [29]. model [28]. A better constraint on this parameter can Only the criterion of T0=T < 1=2 is needed for the probably be obtained from the thorough BBN simulations studies in this paper. To better illustrate the process, modified with the n − n0 oscillations of the new mirror however, we use T0=T ¼ 1=3 as an example with the matter model, which could potentially solve the primor- sequence of the events listed in Table II. Mirror oscillations 0 dial 7Li puzzle [51,52] as well. for both K0 and n0 occur first when the normal sector is As shown in the earlier discussions, the normal and still above the QCD phase transition temperature making mirror sectors do not exchange much via oscillations in the its effective number of relativistic d.o.f. g much larger.

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8 These early oscillation processesR contribute little as their 1x10 Tc = 100 MeV ϵ ¼ ðτ Þλ oscillation parameter of pPffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif dt is greatly sup- Tc = 150 MeV 0 2 0 1x106 T = 200 MeV pressed by a factor of ðT =TÞ gðT Þ=gðTÞ [14]. c 0 n /n = 5.4 K0 → K0 dark B When the oscillations operate between 10000 T ¼ 100–150 MeV, initial matter-antimatter asymmetry is generated in the normal sector as discussed earlier 100 0 while the excess of K¯ 0 in the mirror sector will quickly 1

decay into mirror pions at much lower mirror temperatures. Baryon Ratio Mirror-to-Normal Therefore, nearly all the initial baryon asymmetry origi- nates from the normal sector and the mirror sector con- 0.1 1 10 100 1000 [10-6 eV] tributes little to the net baryon content in the beginning. nn' 0 → 00 Possibly slightly after the inception of K K oscil- FIG. 1. The mirror-to-normal baryon ratio is shown as function → 0 lations, the n n oscillations start to convert the initial net of the mass difference Δnn0 assuming the mixing strength baryons into mirror baryons. For a likely smooth crossover sin2ð2θÞ¼2 × 10−5. The QCD phase transition temperature is of QCD phase transition [33,34], the two oscillation varied to be Tc ¼ 100, 150, 200 MeV, respectively. processes probably overlap over a large temperature range (e.g., between 100 and 150 MeV). The peak of n → n0 oscillations occurs at about 70 MeV well after the end of Tc = 100 MeV 0 T = 150 MeV 0 → 0 − 0 10 c K K oscillations. The n n oscillation rate drops Tc = 200 MeV 0 ] 2 -5 quickly below T ¼ 60 MeV whereas n − n oscillations -5 Sin (2 ) = 2x10 still keep a very small exchange rate between the two ) [10

sectors even at temperatures below 10 MeV. The final (2

2 1 mirror-to-normal baryon ratio eventually becomes about 5.4 as the observed dark-to-baryon ratio. Sin Once the mirror BBN starts, most of mirror neutrons will be fused into mirror helium. Instead of having mirror 0.1 neutrons depleted as in standard BBN calculations, n − n0 0.1 1 10 100 1000 0 -6 oscillations will keep an appreciable n abundance in the nn' [10 eV] mirror sector. The normal BBN then follows and most of FIG. 2. The mixing strength sin2ð2θÞ is shown as function of normal neutrons are fused into normal helium. At this Δ 0 0 → the mass difference nn assuming a mirror-to-normal baryon moment, the reverse n n oscillations will feed the ratio of 5.4. The QCD phase transition temperature is varied to be normal sector with more neutrons at lower energies. Tc ¼ 100, 150, 200 MeV, respectively. These additional low energy neutrons will help destroy 7 the extra Be formed earlier and potentially solve the −6 7 by the uncertainty of the mixing strength 8 × 10 ≤ primordial Li problem [51,52]. In particular, lower energy 2 −5 7 sin ð2θÞ ≤ 4 × 10 inferred from neutron lifetime mea- neutrons can make the Be destruction rate much higher −6 Δ 0 ¼ 3 10 than the n þ p fusion rate [53] and therefore alleviate the surements [14]. The best value of nn × eV 2ð2θÞ¼2 10−5 issue of lithium-deuterium anticorrelation [54]. corresponds to the mixing strength of sin × . Δ 0 Under the above consistent picture of particle oscilla- Note that no upper limit on nn can be set if the QCD phase tions, one can further examine the relations between transition temperature is somehow much lower (e.g., around 2 100 MeV). A detailed study of applying the new model to the mixing strength sin ð2θÞ, the mass difference Δnn0 , the mirror-to-normal baryon ratio, and the QCD phase BBN may help further constrain these parameters. More transition temperature T using the framework developed in neutron lifetime measurements with different magnetic traps c can certainly provide a much more accurate value of the original work of the new mirror matter model [14]. 2 ð2θÞ Δ 0 Figure 1 shows the mirror-to-normal baryon ratio as sin and consequently better nn . And furthermore, function of the mass difference for varied QCD phase it may also provide a way to pin down the QCD phase transition temperatures and sin2ð2θÞ¼2 × 10−5. Figure 2 transition temperature. depicts the mixing strength vs the mass difference for three V. CONCLUSION different QCD phase transition temperatures assuming a mirror-to-normal baryon ratio of 5.4. Under the new mirror-matter model [14] and new For the most likely QCD phase transition temperature understanding of possibly staged QCD symmetry breaking range (150–200 MeV) [33,34] and the well observed dark- phase transitions, the long-standing BAU puzzle can be 0 0 to-baryon ratio of 5.4, the n − n mass difference, as shown naturally explained with K0 − K0 oscillations that occur −6 −5 in Figs. 1–2, can be constrained as Δnn0 ¼ 10 − 10 eV at the stage of strange quark condensation. A consistent

063537-8 KAON OSCILLATIONS AND BARYON ASYMMETRY OF THE … PHYS. REV. D 100, 063537 (2019) picture of particle-mirror particle oscillations throughout surprisingly are not constrained experimentally [55], will the early universe is presented including a self-consistent better quantify the generation of baryon matter in the ð1Þ origin of both BAU and dark matter. Meanwhile, the U A early universe. Future experiments at the Large Hadron or strong CP problem in studies of particle physics is Collider may provide more clues for such topological understood under the same framework. The connection quarkiton processes and reveal more secrets in the SM between the CP violation in SM and the normal-mirror gauge structure, the Higgs mechanism, and the amazing mass splitting seemingly points to the same mechanism in oscillations between the normal and mirror worlds. new physics that needs to be explored in the future. Nonperturbative processes via quarkitons at different quark ACKNOWLEDGMENTS condensation stages are proposed for B-violation and could be verified and further understood with calculations using This work is supported in part by the National Science the lattice QCD technique. More accurate studies on Foundation under Grant No. PHY-1713857 and the Joint 0 KL;S at the kaon production facilities, in particular, on Institute for Nuclear Astrophysics (JINA-CEE, www. the branching fractions of their invisible decays [14] that jinaweb.org), NSF-PFC under Grant No. PHY-1430152.

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