Baryogenesis and Asymmetric Dark Matter from The Left-Right Mirror Symmetric Model

The paper suggests a left-right mirror symmetric model to account for the baryogenesis and asymmetric dark matter. The model can simultaneously accommodate the standard model, neutrino physics, matter-antimatter asymmetry and dark matter. In particular, it naturally and elegantly explains the origin of the baryon and dark matter asymmetries, and clearly gives the close interrelations of them. In addition, the model predicts a number of interesting results, e.g. the cold dark matter neutrino mass is $3.1$ times the proton mass. It is also feasible and promising to test the model in future experiments.


I. Introduction
The standard model (SM) has been evidenced to be a very successful theory at the electroweak energy scale. The precise tests for the SM physics have established plenty of knowledge about the elementary particles [1,2]. Nevertheless, at present there are a number of the unsolved issues in the particle physics and universe observations [3,4], which are not able to be accounted by the SM. The issues in flavor physics are the two facts, i) the mass spectrum hierarchy of the quarks and charged leptons [5], ii) the distinct difference between the quark flavor mixing pattern and the lepton one [6]. The neutrino physics has to answer such questions as the real origin of the Sub-eV neutrino masses [7]? Whether the nature of the light neutrinos are Dirac or Majorana fermions [8]? Is the CP violation in the lepton mixing vanishing or not [9]? However, the issues in the cosmology are more difficult. What is real mechanism of the genesis of the matter-antimatter asymmetry [10]? What is the nature of the cold dark matter [11]? The current universe observations have given the data of the baryon asymmetry and the relic abundance of the cold dark matter as follows [1], In particular, the two issues about the baryogenesis and dark matter are very significant for both particle physics and cosmology, so they attract great attentions in the experiment and theory fields all the time [12]. The various theoretical suggestions have been proposed to solve the abovementioned problems [13]. The baryogenesis can be achieved by the electroweak baryogenesis [14], the leptogenesis [15], the Dirac leptogenesis [16], and so on. The cold dark matter candidates are possible the scalar boson dark matter [17], the sterile neutrino dark matter [18], the supersymmetry dark matter [19], and so on [20]. The asymmetric dark matter ideas have been discussed in the references [21]. The references [22] has studied the mirror symmetric model. These theories are successful in explaining one specific aspect of the problems, but it seems very difficult for them to solve many aspects of the problems simultaneously. On the basis of the unity of nature, a realistic theory beyond the SM should simultaneously accommodate and account for the neutrino physics, baryon asymmetry and dark matter besides the SM, in other words, it has to integrate the four things completely. It is especially hard for a model construction to keep the principle of the simplicity, feasibility and the fewer number of parameters, otherwise, the theory will be excessive complexity and incredible or infeasibility. However, it is still a large challenge for theoretical particle physicists to realize the purpose [23].
In this work, I construct a simple and feasible particle model. It can simultaneously accommodate the SM, neutrino physics, matter-antimatter asymmetry and dark matter. The model extends the SM to a left-right mirror symmetry theory. It has the local gauge groups SU(3) C ⊗SU(2) L ⊗SU(2) R ⊗U(1) Y and global symmetry U(1) B+ B ⊗ U(1) L ⊗ U(1) L where B and L are respectively the mirror baryon and lepton numbers. In addition, the model has the left-right mirror symmetry and a Z 2 discrete symmetry. Besides the SM particles, the model introduces the corresponding mirror particles. The model symmetries are spontaneously broken step by step at different energy scale as the universe temperature decreasing. In the model, the super-heavy scalar boson can decay into the SM right-handed quarks and the left-handed mirror quarks. The decay processes are out-of-equilibrium and CP violation. The CP -violating source lies in the explicit mirror breaking coupling in the Yukawa sector. This eventually leads to both the baryon asymmetry and the mirror neutrino asymmetry through the two steps of the sphaleron processes [24]. The lightest mirror neutrino, which is a Dirac neutrino with the GeV mass, is exactly the cold dark matter. The model can not only completely accommodate the SM and neutrino physics, but also correctly reproduce the observed value of the baryon asymmetry and the relic abundance of the cold dark matter. In particular, the model predicts some interesting results, for example, the cold dark matter asymmetry is 1.6 times the baryon asymmetry, its mass is 3.1 times the proton mass, and so on. Finally, the model is feasible and promising to be tested in future experiments. I give some methods of searching some of the new particles at the colliders.
The remainder of this paper is organized as follows. In Section II I outline the model. Sec. III I discuss the matter-antimatter asymmetry and dark matter. The numerical results and the experimental searches are given by Sec. IV. Sec. V is devoted to conclusions.

II. Model
The gauge symmetries of the model are characterized by the Local and global gauge groups as follows, Local gauge groups : where B and L respectively indicate the baryon and lepton number of the mirror particles. The gauge groups have evidently a left-right mirror symmetry. The model particle contents and their gauge quantum numbers are in detail listed as follows, , (1,1,1,0), (e R , e L ) (1, 1, 1, −2), These notations are self-explanatory. All kinds of the fermions imply three generations as usual. ν R and ν L are singlet neutrinos. S and S are complex singlet bosons with the lepton number (−2). φ ± and φ 0 are super-heavy bosons, moreover, φ 0 is a real scalar. In addition to the above-mentioned gauge symmetries, the model has a attractive left-right mirror symmetry and a Z 2 discrete symmetry. They are defined by the field transforms as follows, Z 2 parity is"+1" for f, H L , H R , S, S and all the gaugebosons.
Here the called " SM sector" not only contains all the SM particles but also it introduce the non-SM particles as ν R , S. For the four particles, φ ± , φ 0 , G µ , B µ , their mirror particles are exactly themselves, so they are actually shared in the two sectors. The gauge and discrete symmetries will be broken step by step at different energy scales in the evolution of the universe. On the basic of the above symmetries, the model Lagrangian is composed of the following three parts. Firstly, the gauge kinetic energy terms are where f and f denote the SM and mirror fermions in (4), respectively. The covariant derivative D µ is given by where g s , g w , g Y are three gauge coupling constants, λ a and τ i are respectively Gell-Mann and Pauli matrices, and Q Y is the charge operator of U(1) Y . Secondly, the model Yukawa couplings are where H ′ L/R = iτ 2 H * L/R and C is a charge conjugation matrix. Obviously, the parameter freedom is greatly reduced owing to the mirror symmetry. The couplings, Y u , Y d , etc., are 3 × 3 complex matrices but Y 1 , Y 2 are Hermitian matrices. In any case, I can always choose such a set of flavor basic that Y e , Y m , Y 1 , Y 2 are all diagonal matrices and the others are all non-diagonal. The coupling matrices should originate from family symmetry breaking, however, they bring about flavor mixings. In addition, the irremovable complex phases in the couplings become sources of the CP violation. In particular, the irremovable complex phase in Y 0 is also a source of the mirror symmetry breaking, explicitly, the mirror symmetry is broken for Y 0 = Y * 0 . The Yukawa couplings of (7) will lead to reasonable explanations for the neutrino masses, matter-antimatter asymmetry and cold dark matter. Thirdly, the model scalar potential is given by The self-coupling parameters, λ H , λ S , λ φ 0 , λ φ ± , are positive and should be ∼ 0.1. µ 0 is a positive coupling parameter with mass dimension, explicitly, the Z 2 discrete symmetry is broken by the µ 0 term. The other interactive couplings should be sufficient weak. v L , v R , v s , v φ are respectively the VEVs of the corresponding scalar fields, but S and φ ± have vanishing VEVs, see the following equation (9). M φ 0 and M φ ± are respectively the super-heavy masses of φ 0 and φ ± , which are ∼ 10 10 GeV. It can clearly be seen from (8) that v L = v R = 0 will spontaneously break the local gauge symmetries, while v s = 0 will spontaneously break the global gauge symmetry U(1) L , but U(1) L is unbroken due to v s = 0. Obviously, the gauge symmetry breakings simultaneously trigger the spontaneous breaking of the mirror symmetry. After the local gauge symmetry breaking, φ 0 will also be induced to develop a small v φ by the µ 0 term. In short, the model scalar sector undertakes and implements the gauge and discrete symmetry breakings. It is more varied and interesting in comparison with the SM or MSSM Higgs sector [25]. The potential vacuum configurations and the scalar boson masses are derived by the standard program. The detailed results are as follows, where S R and S I are respectively real and imaginary components of S. There are three massive neutral bosons H 0 R , H 0 L , S R , while S I and S become massless Goldstone bosons. Some parameters in (9) 1 GeV is not a independent parameter since M φ 0 is regarded to be independent. This hierarchy of the VEVs clearly shows the symmetry breaking sequence. First of all, SU (2) the electroweak breaking. This is accomplished by the neutral component H 0 L developing v L . Lastly, the real component S R developing v s completes the U(1) L breaking, whereas the U(1) L symmetry is maintained and unbroken all the time. At the present day, M H L has been measured as 125 GeV at the LHC [26]. The model predicts that M H R is ∼ 10 8 GeV, M S R is scores of GeVs, and the Goldstone bosons S I and S are actually a species of hot dark matter. However, the other bosons remain to be searched in future experiments.
In the gauge sector, the local gauge symmetry breakings result in masses and mixings of the gauge fields through the Higgs mechanism. The detailed expressions are as follows, In (10), there are only two independent parameters, i.e. g w and g Y . θ is a mixing angle for the SU(2) R breaking, while θ W is a mixing angle for the SU(2) L breaking.
The two angles are correlated by that equation in (10). The SM hypercharge is . Q w and Q w are two charge operators associated with the two massive neutral gauge fields Z µ and Z µ , respectively. It should be pointed out that the mixing angle between Z µ and Z µ is ∼ it is so small that it can be ignored. For v R ∼ 10 8 GeV, M W R and M Z are ∼ 10 7 GeV. They are too heavy to be detected at the present.
In the Yukawa sector, the fermion masses and mixings are given as follows, By virtue of the small mixing of u R and u R (d R and d R ), the mirror quarks can oscillate into the SM quarks. This plays key roles in the following baryogenesis. As a result of the U(1) L breaking, ν R obtains a Majorana mass about a dozen GeV, it becomes a heavy Majorana neutrino undetected by now. For Y ν ∼ 10 −7 , ν L is generated an effective Majorana mass through the see-saw mechanism [27], it is exactly Sub-eV Majorana neutrino in nature. By contrast, the mirror neutrino ν has only a Dirac mass due to U(1) L being unbroken, so it is actually a Dirac neutrino. The ν masses are several to dozens of GeVs, in particular, the lightest mirror neutrino ν 1 becomes the cold dark matter. This distinction between the L sector and the L one also leads to U P M N S = U P M N S . The flavor mixing matrices U CKM and U P M N S are respectively defined by [28,29]. The mixing angles and CPviolating phases in the two unitary matrices are parameterized by the standard form in particle data group [1].
In conclusion, the above contents form the theoretical framework of the model. In brief, the model extends the SM to the left-right mirror symmetrical theory. The new symmetries and non-SM particles will play key roles in the new physics beyond , which lead to the matter-antimatter asymmetry.
the SM, in particular, in the origin of matter-antimatter asymmetry and cold dark matter.

III. Baryogenesis and Asymmetric Dark Matter
The baryogenesis and asymmetric dark matter have a common origin in the model, so the two things have a close relationship. As the universe expansion and cooling, the model symmetries are spontaneously broken and reduced step by step. In the evolution process, the baryogenesis and asymmetric dark matter will naturally be generated by the following mechanism.
After the universe inflation, the universe reheating temperature is in general ∼ 10 12−13 GeV for most of the inflation models [30]. In the reheated universe, thus there is an immense amount of the super-heavy scalar boson φ 0 which has by nature a mass about 10 10 GeV. In the light of (7) and (8), The decay channels of φ 0 include , but the main decay are actually the former modes and their CP conjugate processes, as shown in Figure  1. For the couplings Y 1,2 ∼ 10 −4 , the decay rates are far smaller than the Hubble expansion rate of the universe, namely where i = 1, 2, M pl = 1.22 × 10 19 GeV, g * is an effective number of relativistic degrees of freedom at T = M φ 0 . At this temperature, the non-relativistic particles are only φ 0 and φ ± in the model, the rest of the model particles are all relativistic states, so one can easy figure out g * = 210. Consequently, the φ 0 decays are actually out-of-equilibrium processes.
The complex phase in the coupling Y 0 is explicitly a source of the CP violation and mirror symmetry breaking. It can surely lead to CP asymmetries of the decays by the interference between the tree diagram and the loop one. The CP asymmetries are defined and calculated as follow, For µ 0 ∼ 10 3 GeV, then µ 0 M φ 0 ≪ Y 1,2 ∼ 10 −4 , so the last two decays can indeed be ignored. The second asymmetry has the same result as the first one. In short, the decays of φ 0 satisfy two items of Sakharov's three conditions [31], namely CP violation and out-of-equilibrium.
In the above decays of φ 0 , the total baryon number B + B is undoubtedly conserved, but the CP violation and out-of-equilibrium surely lead to the respective baryon number asymmetries in the B and B sectors as follow, where N f is fermion generation number, s is entropy density, and κ is a dilution factor. For a very weak decay, it is serious departure from thermal equilibrium, so one can approximate κ ≈ 1. Note that the asymmetries do not depend on the universe temperature in the comoving volume. Obviously, the B and B asymmetries are the same size but opposite sign, so the total baryon number asymmetry of the universe is still vanishing. After φ 0 decaying and decoupling, the B and B sectors are out of connection and separated from each other. The B and B asymmetries will remain in the respective sectors. For |Y 0 | ∼ 10 −3 and its complex phase ∼ 0.1π, the equations of (13) and (14) can correctly give a satisfied baryon asymmetry. Things happened next are the sphaleron processes [32]. In the temperature area of v R < T < M φ 0 , the gauge symmetries are yet unbroken and the mirror symmetry is kept well. In the SM and mirror sector, the baryon and lepton current anomaly and the relevant charge conversion are collected as follows, In view of the left-right mirror symmetry, the sphaleron transitions for SU(2) L and SU(2) R are completed in parallel ways. The only difference is the opposite initial asymmetries in the two sectors which are provided by (14). When the universe temperature decreases to T = v R , SU(2) R is broken and the mirror sphaleron process is stopped. Therefore, the baryon and lepton asymmetries at T = v R are obtained as follows, where c sp is a coefficient of the sphaleron conversion, (16) is in detail derived in appendix A. Below the v R scale, SU(2) L ⊗ SU(2) R ⊗ U(1) Y → SU(2) L ⊗ U(1) Y ′ , and the left-right mirror symmetry is also lost. Accordingly, the right-handed doublets q R and l R are spontaneously decomposed into the uncorrelated states u R , d R and ν R , e R , and the right-handed anomaly C R disappears as well. The B sector and the B one are now connected by virtue of the mixings of u R and u R as well as d R and d R , but the L sector and the L one are still separated, see (11). In the temperature area of v L < T < v R , only the sphaleron transition for SU(2) L goes on. The baryon and lepton current anomaly and the relevant charge conversion are now changed as follows, It can be seen from (17) that the mirror baryons have joined the SM sector, so two new separate sectors are now the B + B − L sector and the L one. When the energy scale decreases to v L , the electroweak breaking occurs and the SU(2) L sphaleron process is stopped. At T = v L , the baryon and lepton asymmetries evolve into the below results, where c ′ sp and c sp are two new coefficients of the sphaleron conversion, a derivation of (18) is in appendix B. Below the v L scale, all kinds of the asymmetries in the B + B and L sectors will be kept at all time, but the L sector is not like that because U(1) L is broken at the energy scale v s ∼ 100 GeV. After this, the heavy Majorana neutrino ν R and the light one ν L are active in the L sector.
In the above evolution, the below reactions play key roles, Firstly, the heavier mirror baryons can oscillate into the lighter SM baryons due to the mixings between the mirror quarks and the SM quarks. This will eventually lead to the B asymmetry disappearing. Secondly, the symmetric parts of e − and Figure. 2. (a) the annihilation way of a pair of ν, (b) the radiative decay of the heavier ν 2,3 into the lightest ν 1 which is namely the cold dark matter.
e + annihilate into photons, while their asymmetric parts convert into the mirror neutrino ν R via the reactions with π + or p + . In particular, the stable mirror atoms can not be formed precisely because of the third reaction in (19). Consequently, all of the mirror baryons, mirror charged leptons and mirror atoms can not survive in the present universe. Thirdly, the symmetric parts of ν and ν annihilate into Goldstone boson pairs of S and S * , and the heavier ν 2,3 are radiative decay into the lightest ν 1 , see Figure 2. Thus only the stable ν 1 survive in the asymmetric parts of ν, it eventually becomes the cold dark matter in the present universe. Lastly, the two special particles ν R and S R in the SM sector (of course both of them are not the well-known SM particles) completely annihilate and decay via the third line processes in (19), so they have no relics in the present universe. Finally, the light Majorana neutrino ν L and the massless Goldstone bosons S I , S are evidently relativistic decoupling, therefore, they become the hot dark matter in the present universe.
There is no doubt that ν 1 is non-relativistic decoupling. Its freeze temperature is determined by the annihilation cross-section σ( ν 1 + ν 1 −→ S + S * ) as follows, where v r is the relative velocity of ν 1 and ν 1 , and v is the ν 1 velocity in the centerof-mass frame. The heat average can be calculated by σv At this freeze temperature, the relativistic particles include the first generation fermions u, d, e, three generations of the light Majorana neutrinos ν L , the Goldstone bosons S I , S, and photon. Therefore, g * in (20) should be input by g * = 34.75. After ν 1 decoupling, the symmetric parts of ν 1 have been exhausted owing to its large annihilation cross-section, only its asymmetric parts survive in the present universe. Finally, it should be noted that T ν 1 is also the decoupling temperature of S, and the S asymmetry does no change before and after it decoupling.
To sum up, the current universe matters consist of the photon, baryon, electron, the light Majorana neutrino ν L and massless Goldstone bosons S I , S which are the hot dark matter, and the mirror Dirac neutrino ν 1 which is the cold dark matter. On account of these matters having a common origin, today their asymmetries and relic abundance have some essential relations as follows, where 7.04 is a ratio of the entropy density s to the photon number density n γ . In addition, Ω S I is smaller than Ω S because the decoupling temperature of S I is higher than one of S. However, Ω S and Ω S I are much smaller in comparison with Ω ν L ≈ 1.7 × 10 −3 and Ω γ ≈ 5 × 10 −5 . In view of their same origin, η B and η ν 1 have a close size, moreover, they are essentially a complementary relationship. At present day, η B has been measured, but η ν 1 hides itself and eludes all kinds of observations. Obviously, all of the results meet BBN constraints [33]. In a word, (21) are interesting and important predictions of the model. They surely provide a clear guide for the future experimental search. Through the above mechanism, the universe has eventually evolved into the final state with both baryon asymmetry and dark matter asymmetry from the initial state with the matter-antimatter symmetry, and it is separated into the visible sector and the dark one. The later numerical results will demonstrate that the model is indeed successful.

IV. Numerical Results
In the section I present the model numerical results. In the light of the foregoing discussions, the model contains a lot of the new parameters besides the SM ones. In principle the SM parameters have been fixed by the experimental data, but the non-SM parameters have yet large freedoms. In fact, there are not many non-SM parameters involved in the numerical calculations. The gauge sector parameters are the two gauge couplings g w and g Y . In view of the relevant relations in (10), I can use the mixing angle θ as a substitute for g Y , furthermore, g w and tan θ are determined by e and sinθ W , which have precisely been measured by the electroweak physics. The scalar sector parameters include the two couplings λ H , λ s , the three VEVs v L , v R , v s , and the three mass parameters M φ 0 , M φ ± , µ 0 . Among which, λ H and v L are determined by the SM physics. A reasonable value of λ s should be around 0.1. v R ∼ 10 8 GeV and v s ∼ 100 GeV are suitable based on the model consistency and experimental limits. M φ 0 , M φ ± ∼ 10 10 GeV can satisfy the outof-equilibrium condition and baryon asymmetry. Finally, µ 0 ∼ 10 3 GeV leads to v φ ∼ 0.1 GeV from (9), this meets multiple requirements of the model. Based on an overall consideration, a set of suitable and typical values of the gauge and scalar parameters are chosen as Now input (22) into the relevant equations in (9) and (10) M W R , M Z and M H R are dominated by v R , while M S R is affected by v s . Although the neutral boson S R is lighter than the SM Higgs boson H L , it will be difficult to detect it because S R has hardly any interactions with the SM particles. The model Yukawa sector contains a great deal of the flavor parameters. However, I can choose such a set of flavor basis that Y e , Y m , Y 1 , Y 2 are all real diagonal. Y e is determined by M e . Since the flavor structures of Y m , Y 1 , Y 2 are as yet unknown, all of them are taken as constant unit matrices for simplicity. For the same reason, the complex coupling Y 0 is set as a complex constant unit matrix. Thus, the non-diagonal coupling Y ν can be given by the lepton mixing matrix U P M N S and three real diagonal parameters, see the below. The Yukawa sector parameters are typically chosen as follows,  (22) and (24).
obtained by fitting η B . Y m ∼ 0.1 is very reasonable in the light of (20). y 1 , y 2 , y 3 are determined by fitting the masses of the cold dark matter ν 1 and the light neutrinos ν L . By use of (11) where △m 2 21 and △m 2 32 are the two mass-squared differences of the light Majorana neutrinos. In addition, one can obtain T ν 1 ≈ 100 MeV from (20). By use of (12), (13), (16), (18) and (21), finally the baryon asymmetry and the relic abundance of the cold dark matter ν 1 are calculated as The above first equation clearly shows that the φ 0 decay is indeed out-of-equilibrium. The results in (26) are precisely the current data of the universe observations [34]. corresponds to the values of |Y 0 | and ϕ in (24). It can be seen from Figure 3 that a reasonable region of |Y 0 | should be ∼ (3 × 10 −3 − 4 × 10 −3 ) for a moderate ϕ.
In brief, all the numerical results are naturally produced without any fine tuning. They have clearly demonstrated the main ideas of the model. In the end, I give a brief discussion about searching the new particles e, ν, ν R , S, S. On the basis of the model interactions, Figure 4 draws some feasible production processes at the LHC [35]. The diagram (a) illustrates the pair production of e or ν by the proton-proton collisions, of course, its cross-section for ν is too tiny to be identified. The diagram (b) can produce a pair of S or S, but these cross-sections are tiny due to S, S having very weak coupling to H L . In addition, H L can decay into ν L and ν R , but it is also very difficult to find ν R on account of Y ν ∼ 10 −7 . The best efficient methods to test the model are of course by the lepton-antilepton collisions at the ILC. The main processes are In particular, the loss of energy in the processes should be regarded as definitive signals of the cold dark matter neutrino ν 1 or the hot dark matter Goldstone bosons S, S I . As long as the luminance and running time are enough large, the non-SM particles e, ν R , S R are possible to be discovered. Although all of the searches are large challenging for the future experiments, the model is feasible and promising to be tested in near future.

V. Conclusions
In the paper, I suggest the left-right mirror symmetric theory to account for the baryogenesis and asymmetric dark matter. The model has the left-right symmetric gauge groups and the global symmetry U(1) B+ B ⊗ U(1) L ⊗ U(1) L , in addition, the mirror and Z 2 discrete symmetries. The decays of the super-heavy scalar φ 0 are the CP violation and out-of-equilibrium. The CP -violating source is the explicit mirror breaking coupling in the Yukawa sector. Through the two steps of the leftright mirror symmetric and asymmetric sphaleron processes, this eventually leads to both the baryon asymmetry and the cold dark matter neutrino asymmetry. The model can not only naturally accommodate the SM and neutrino physics, but also simply and elegantly account for the matter-antimatter asymmetry and dark matter nature, in particular, the both close interrelations are showed. The model gives a number of interesting and important predictions, for instance, the cold dark matter neutrino asymmetry is 1.6 times the baryon asymmetry, its mass is 3.1 times the proton mass, and so on. Finally, the model is feasible and promising to be tested in future experiments. Some non-SM particles in the model will possibly be discovered in near future. However, all these efforts will increase our understanding to the mysteries of the universe. asymmetries in the SM and mirror sectors are listed as follows, where S, S have (−2) lepton numbers. After φ 0 decaying and decoupling, the SM sector and the mirror one are separated from each other. The particles in the SM sector are in the thermal reaction equilibrium via the gauge and Yukawa couplings, and the non-perturbative sphaleron processes. There is complete counterparts in the mirror sector in view of the left-right mirror symmetry. Therefore, there are the below relations between the various chemical potentials, Finally this yields the below relations of the baryon and lepton asymmetries,

Appendix B
After SU(2) L ⊗ SU(2) R ⊗ U(1) Y → SU(2) L ⊗ U(1) Y ′ , the left-right mirror symmetry is lost and the right-handed sphaleron process is vanishing. Now the mixing between the B sector and the B one appears via the v φ terms, but there is no mixing between the L sector and the L one. Therefore, the relevant thermal reaction equilibriums are changed as follows, In terms of µ l L , µ ν R , the chemical potentials are expressed as Thus, the baryon and lepton asymmetries are given by the relations as follows, =⇒ B + B = c ′ sp (B + B − L), L = (c ′ sp − 1)(B + B − L), L e = L e R + L e L = 0, L ν = L ν R + L ν L = c sp L , L S = (1 − c sp ) L ,