Cosmic Evolution of Lepton Flavor Charges

In the early Universe above the weak scale, both baryon $B$ and lepton $L$ numbers are violated by nonperturabive effects in the Standard Model while $B-L$ remains conserved. Introducing new physics which violates perturbatively $L$ and/or $B$, one can generate dynamically a nonzero $B-L$ charge and hence a nonzero $B$ charge. In this work, we focus on the former scenario which is also known as leptogenesis. We show how to describe the evolutions of lepton flavor charges taking into account the complete Standard Model lepton flavor and spectator effects in a unified and lepton flavor basis-independent way. The recipe we develop can be applied to any leptogenesis model with arbitrary number of new scalars carrying nonzero hypercharges and is valid for cosmic temperature ranging from $10^{15}$ GeV down to the weak scale. We demonstrate that in order to describe the physics in a basis-independent manner and to include lepton flavor effect consistently it is necessary to describe both left-handed and right-handed lepton charges in terms of density matrices. This is a crucial point since physics should be basis independent. As examples, we apply the formalism to type-I and type-II leptogenesis models where in the latter case, a flavor-covariant formalism is indispensable.


I. INTRODUCTION
In the early Universe, if the cosmic temperature is above the weak scale, the thermal bath contains all the degrees of freedom of the Standard Model (SM) and perhaps other new physics degrees of freedom as well if they are kinematically accessible. To generate a cosmic baryon asymmetry dynamically (baryogenesis), one needs to violate at least the baryon number B of the SM. Above the weak scale when the SM B-violating process is in thermal equilibrium [1], one needs to identify other charges which are not in thermal equilibrium such that the charge is effectively conserved and can remain nonzero. In the SM, one identifies the baryon minus lepton number with c = 0, implying a nonzero B is generated as well. After baryogenesis is completed, i.e., with c Q = 0. In ref. [3], we have classified all effective charges of the SM and its minimal supersymmetric extension, 16 in the former and 18 in the latter and this opens up a new avenue for baryogenesis.
In this work, we focus on baryogenesis scenario through the violation of B − L which can come from perturbative interaction which violates L and/or B. We consider the former scenario, which is also known as leptogenesis [4]. First of all, we show that in order describe leptogenesis in a basis-independent manner one needs to describe both the number asymmetries in lepton doublet and singlet E in term of matrices of number densities in their respective flavor spaces (we will denote them densities matrices) [5]. It is of fundamental importance since physics should not depend on a particular basis. While the computation of leptogenesis is usually carried out in a charged lepton mass basis, one should be cautious that this description has limited validity, and in particular, if the result is basis dependent, then it is a red flag that something must be wrong. In this flavor-covariant formalism [6,7], the SM lepton flavor effect is consistently taken into account. 1 With the effective charges identified in ref. [3], we are able to include the complete spectator effects due to quark Yukawa and SM sphaleron interactions in a unified manner, which to our knowledge has not been carried out before. (See ref. [9], in which the spectator effects related to tau and bottom-quark Yukawa interactions are investigated.) In ref. [10], asymmetry in E is not taken in account, and as a result, one cannot obtain a fully basis-independent result.
In ref. [6,7], asymmetry in singlet E is considered while other spectator effects [11,12] pertaining to quark Yukawa and SM sphaleron interactions are not considered.
This article is organized as follows. In Section II, we review the effective symmetries and charges of the SM in the early Universe. In Section III, we write down the flavor-covariant Boltzmann equations, taking into account the complete lepton flavor and spectator effects due to quark Yukawa and the SM sphaleron interactions. These results are completely general, and, together with the equations in Appendix C, can be applied to any leptogenesis model (with arbitrary number of new scalars carrying nonzero hypercharges) for cosmic temperature ranging from 10 15 GeV down to the weak scale. In Section IV, we apply our results to type-I and type-II leptogenesis models. Finally, we conclude in Section V. In Appendix A, we discuss how number density asymmetry matrices are related to matrices of chemical potentials; in Appendix B, we show how the flavor-covariant structure can be derived using Sigl-Raffelt formalism [5]; and in Appendix D, we discuss how to determine the transition temperatures related to spectator effects.

II. EFFECTIVE SYMMETRIES AND CHARGES
In the early Universe, due to the additional scale related to cosmic expansion, one should consider effective symmetries and charges. Taking all particles to be massless, the interaction rates among the particles have to scale as T .
At sufficiently high T , all of those interactions will be slower than the Hubble rate. In this case, if one assigns a quantum number or charge to each type of particle, the charge will be effectively conserved since all particle-number-changing processes are out of thermal equilibrium (effectively do not occur within a Hubble time). In the SM, with three families α = 1, 2, 3 of quark Q α and lepton α doublets, charged lepton E α , up-type U α and down-type D α quarks singlets, and a Higgs doublet, one will expect to have up to 16 effective charges or the associated global U (1) symmetries.
One can conveniently choose linear combinations of U (1) charges which are subsequently broken as the cosmic temperature decreases. This choice leads to where we have denoted the charge associated to each type of particle as {U 1 , U 2 , U 3 } = {u, c, t}, In the absence of neutrino mass, the SM Lagrangian contains four accidental U (1) symmetries: the total baryon number U (1) B and three lepton flavors U (1) Lα . There are fewer actual accidental symmetries of the SM due to the Adler-Bell-Jackiw anomaly. We can determine if any of the accidental symmetry U (1) x is preserved from its anomaly coefficient associated with the triangle where the sum is over all fermions i of degeneracy g i , charge q x i under U (1) x , and representation R i under SU (N ≥ 2) gauge group with c(R i ) = 1 2 in the fundamental representation and c(R i ) = N in the adjoint representation. Since the contribution of each fermion i to the SU (N ) sphaleroninduced effective operator is proportional to c(R i ), the effective operator is given by [3] In the SM, we see that U (1) B and U (1) Lα are anomalous [13] with anomaly coefficients Out of four anomalous symmetries, one can form three linear combinations which are anomaly free.
It is convenient to choose the following three anomaly-free symmetries U (1) B/3−Lα we mentioned earlier. Then, the anomalous symmetry U (1) B+L with anomaly coefficient From eq. (5), one obtains the EW sphaleron effective operator The operator above violates only U (1) B , and the interaction due to this operator is in thermal equilibrium [1] from T B ∼ 2 × 10 12 GeV [9] up to T B− ∼ 132 GeV [14].
The SM quark Yukawa terms are given by where the SU (2) L contraction between the left-handed quark Q b and the Higgs H doublets is shown explicitly with the SU (2) antisymmetric tensor 01 = − 10 = 1. If these terms are absent, one has a chiral symmetry U (1) χ where q χ Qa = −q χ Ua = −q χ Da ≡ q. Nevertheless, this chiral symmetry is anomalous with From eq. (5), one can construct the QCD sphaleron effective operator as [15] The operator above violates the chiral symmetry U (1) u , and the interaction due to this operator is in thermal equilibrium for T T u ∼ 2 × 10 13 GeV [9].
The rest of the effective symmetries in eq. (3) are broken when the corresponding Yukawa interactions get into thermal equilibrium, starting from the one involving top Yukawa, tau Yukawa and so on. We can estimate the temperature T x in which U (1) x is broken from the condition when the U (1) x -violating rate is equal to the Hubble rate Γ x (T x ) = H (T x ) and obtain [3] T t ∼ 1 × 10 15 GeV, T u−c ∼ 2 × 10 10 GeV, and we have assumed thermalization at T ∼ 10 15 GeV [16,17]. In principle, one will need to track the evolutions of all the effective charges, starting from some initial condition. For instance, after reheating at the end of inflation with temperature T RH , we can take the initial condition to be when all the effective charges are zero. The charge density associated to each effective charge can be written as where the number density asymmetry of particle i is defined as n ∆i ≡ n i − nī where n i (nī) is the number density of particle i (antiparticleī). In this case, the initial condition will be n ∆x (T RH ) = 0 for all the charges. One should then track the evolutions of all the n ∆x (T ) with the Boltzmann equations including all the SM interactions. To generate some nonzero charges, the three Sakharov conditions should be fulfilled [18]: • C and CP violation corresponding to the process violating U (1) x , • out-of-equilibrium condition for the process violating U (1) x .
If the Sakharov conditions are not met for any of the charges in eq. (3), one will always have n ∆x = 0. If the Sakharov conditions are met for some of the U (1) x (this does not happen in the SM and hence physics beyond the SM is required), one will have n ∆x (T g ) = 0, where T g is the temperature when the charge n ∆x is being generated. If all other U (1) y =x remain effective, we have n ∆y = 0, while for U (1) y =x which are not effective, we will necessarily have n ∆y ∝ n ∆x . This does not necessarily imply that n ∆y = 0 since the constant of proportionality can be zero; i.e., U (1) y and U (1) x are orthogonal to each other. At T B− < T < T e , since U (1) B is not effective, we can construct the baryon charge density from eq. (14) as where we have assumed zero hypercharge density n ∆Y = 0. The coefficient c is not zero since B and B/3 − L α are not orthogonal to each other. With the SM degrees of freedom and assuming that the EW sphaleron interaction freezes out at 132 GeV after the EW symmetry breaking at 160 GeV [14], we obtain where we have excluded the top-quark contribution.
Next, we will review briefly how to relate the number density asymmetries of the SM particles to their corresponding chemical potentials. Since all the SM particles participate in the gauge interactions, they can thermalize at a cosmic temperature T 10 15 GeV [16,17] and follow the equilibrium phase-space distribution where E i is the energy of particle i, µ i is its chemical potential, and ξ i = 1(−1) for i a fermion (boson). For gauge bosons, their numbers are not conserved, and their chemical potentials are zero. For the rest of the SM particles, due to the scatterings with the gauge bosons, the chemical potential of an antiparticle is related to the corresponding particle by a negative sign µī = −µ i .
To take into account flavor correlation of particle i, one can generalize µ i to a matrix in its flavor space. (See Appendix A for details.) In this work, since we are interested in the lepton flavor effect, we will generalize µ and µ E to matrices in their lepton flavor spaces (see the next section).
Integrating the phase space distribution eq. (17) over 3-momentum, at leading order in |µ i | /T 1 (assuming that the number density asymmetries of the SM particles are much smaller than their total number densities in the early Universe in accordance with observation), the number density asymmetries are linearly proportional to their respective chemical potentials where g i is the gauge degrees of freedom and ζ i = 1(2) for i a massless fermion (boson). 3 To scale out the effect of dilution purely due to the Hubble expansion, we will normalize the matrix of number densities Y i ≡ n i /s by the cosmic entropy density s = 2π 2 45 g T 3 with g being the effective relativistic degrees of freedom of the Universe (g = 106.75 for the SM) and we obtain where we have defined Y nor ≡ 15 8π 2 g . Then, one can relate Y ∆i to normalized charge density Y ∆x ≡ n ∆x /s as [2] where 3 For a particle i with mass m i , ζ i = 6 The relation above is completely general (the charges are completely fixed for any given model), and the temperature dependence appears only in ζ i for particles which are not massless and in Y ∆x (T ), which should be solved from the relevant Boltzmann equations. In the next section, we will discuss how to consider lepton flavor charges and their coherences with density matrices while treating the effects of baryons as spectators [11,12].

III. LEPTON FLAVOR EFFECT
In the SM, we have the charged lepton Yukawa term 4 where β and H are, respectively, the left-handed lepton and Higgs SU (2) L doublets while E α is the right-handed charged lepton SU (2) L singlet with family indices α, β = 1, 2, 3. The charged lepton Yukawa coupling can be diagonalized by two unitary matrices U E and V E , GeV the Higgs vacuum expectation value and m e , m µ and m τ are, respectively, the electron, muon, and tau lepton masses (at certain scale). In the charged lepton mass basis, which is also known as the (leptonic) flavor basis, we have E = U E E and = V E where they are labeled as = e , µ , τ and E = E e , E µ , E τ .
In this work, since we are interested in studying the flavor coherence of the lepton charges, from eq. (19), we will consider matrices of number density asymmetries of and E (see Appendix A for details), where Y ∆ , Y ∆E , µ , and µ E are 3×3 Hermitian matrices in the leptonic flavor spaces (one for and the other for E). The diagonal elements denote the number density asymmetries in the "flavors" for any chosen basis (not necessarily the charged lepton mass basis), while the off-diagonal elements encode the correlations between the flavors. As we will see later, this generalization is necessary such that physics is independent of basis. Nevertheless, a convenient basis is usually useful to interpret the physics at hand. Including the EW sphaleron [19] and scatterings due to charged lepton Yukawa, the flavor-covariant Boltzmann equations can be written as [5][6][7]9] In the SM, the Boltzmann equation for the evolution of total baryonic charge Y ∆B is the following The additional factor of 3 comes from the fact that for each scattering, the change of the total baryon number is ∆B = 3 while for the lepton flavors we have ∆L α = 1 for each flavor. In this work, our focus is only on the lepton flavor effect, and hence we have considered Y ∆B as the total baryon charge instead of matrix in the baryon flavor space. The baryon flavor effect will be considered elsewhere. Hence, we will parametrize the transitions across T x due to quark interactions, i.e. with x = {e, µ, τ }, as some exponential functions that we will discuss in the next section. Ignoring baryon flavor effect, let us define the charge matrix which transforms like Y ∆ as in eq. (28) under flavor rotations (27). From eqs. (25) and (29), we obtain the Boltzmann equation for Y ∆ as follows Now we only need to solve (26) and (31) treating Y ∆ and Y ∆E as the only independent variables.
One could have defined the B/3 − L a charge matrix where one would have to keep in mind that Y ∆ and Y ∆E transform differently as in eq. (28).
Clearly, the physics will remain the same but in order to avoid remembering the different transformations within Y ∆ , we will resort to using Y ∆ . Nevertheless, it is instructive to look at the Boltzmann equation for Y ∆ in the flavor basis where we can construct from eqs. (25), (26), and (29) as follows: (33), it is apparent that for a consistent description of evolution of lepton flavor charges which is basis independent both Y ∆ and Y ∆E need to be described by density matrices: if off-diagonal terms of Y ∆ are induced, off-diagonal terms for Y ∆E will be induced as well and vice versa.
In the rest of the work, we will use eqs. (26) and (31), which are valid in any basis. Including new physics interactions that generate either Y ∆E and/or Y ∆ in the two Boltzmann equations, from eq. (16), the final baryon asymmetry will be frozen at T B− to be Next, we will write down the relations between Y ∆ and Y ∆H in terms of Y ∆ and Y ∆E for the SM and the SM augmented with arbitrary scalar fields carrying nonzero hypercharges.

A. Standard Model
With the SM field content, from eq. (20), we obtain 7 where c B and c H are coefficients which vary with temperature. In obtaining the expressions above, we have assumed all effective charges in eq. (3), except (Y ∆ ) αα and (Y ∆E ) αα , to be zero.
where T B = 2.3 × 10 3 GeV. In Appendix D, we discuss how to determine a precise value of T B .
The rest of the spectator effects pertaining to quark sector are encapsulated in the coefficient 7 The number asymmetries of quark fields in term of Y ∆ and Y ∆E are collected in Appendix C.
In the equation above, we can see explicitly that the asymmetry carried by the Higgs is diluted as more charges come into equilibrium. Since the transitions due to the rate Γ ∝ T as compared to the Hubble rate H ∝ T 2 always have an exponential behavior, one can parametrize the transitions with the following function For the purpose of this work, we use the transition temperatures as shown in eq. (13). Precise determination of the transition temperatures can be carried out following the procedure shown in Appendix D.
From the definition of Y ∆ in eq. (30), the off-diagonal terms α = β are Hence we can rewrite the matrix Y ∆ as

B. Standard Model with additional scalar fields
If one introduces additional scalar fields φ i with hypercharge q Y φ i to the system, eq. (35) remains the same, while eq. (36) changes to where Y ∆φ i defined in eq. (19) takes into account additional gauge multiplicity g φ i as well as mass of φ i in ζ φ i (implicitly, we have assumed φ i to be in kinetic equilibrium but not necessarily in chemical equilibrium). The relation above is general, independently of whether φ i are in chemical equilibrium or not. If some of the φ i do not achieve chemical equilibrium, one will have effective Otherwise, the evolution of Y ∆φ i will have to be described by the corresponding Boltzmann equation.
For instance, for type-II seesaw leptogenesis with a heavy triplet Higgs T with hypercharge q Y T = 1, one can apply eq. (42) and obtain

IV. APPLICATIONS
Now, we will apply the flavor-covariant Boltzmann equations (26) and (31) to some wellmotivated leptogenesis scenarios. One just needs the general expressions (41) and (42) to close the equations. Even for leptogenesis models involving quarks, one can use the general relations in Appendix C (ignoring baryon flavor effect). Hence, one no longer needs to solve for flavor matrices for a particular model and which hold only in a particular temperature regime as has been done, for example, in refs. [21] and [22]. In the first example, we will apply the formalism to type-I leptogenesis, while in the second example, we will apply it to type-II leptogenesis where flavor-covariant formalism is indispensable as first pointed out in ref. [22]. In particular, we will demonstrate that the results obtained are independent of basis, showing that it is necessary to take into account flavor correlation in both and E. In other words, it is inconsistent to consider flavor correlation only in or only in E.

A. Type-I leptogenesis
In the type-I seesaw model, the SM is extended by right-handed neutrinos N i as where M i is the Majorana mass of N i and we will work in the arbitrary basis where y E is not necessarily diagonal. While two generations of N i are already sufficient to explain neutrino oscillation data, as an example, we will consider three generations i = 1, 2, 3.
After the EW symmetry breaking with v ≡ H = 174 GeV, the light neutrino mass matrix for where M = diag (M 1 , M 2 , M 3 ). The mass matrix can be diagonalized with U T ν m ν U ν =m ≡ diag (m 1 , m 2 , m 3 ) where U PMNS = V E U ν is identified with the leptonic mixing matrix.
For type-I leptogenesis, an asymmetry is generated through the CP -violating decays N i → α H.

In addition to the Boltzmann equation for
where we have defined z ≡ M 1 /T , we have to append to the right-hand side of eq. (31) a source and washout terms, respectively, given by [10] where to close the equations we apply eqs. (36) and (41). Assuming the Maxwell-Boltzmann with K n (x) the modified Bessel function of the second kind of order n and the decay reaction density γ N i is given by with the total decay width of N i . 9 The matrix of CP -violation parameter i and flavor rotation matrix P i are, respectively, given by [10] ( Under flavor rotations (27) and (28), we have and the whole Boltzmann equation for Y ∆ remains flavor covariant as required.
For illustration, we choose the best-fit point from ref. [24] for the SO(10) model with Higgs 9 Here we consider only decay and inverse decay. We have ignored the helicities of N i and scattering processes which will be relevant for leptogenesis in the weak washout regime Γ Ni /H (T = M i ) 1 since in this case, the physics at T M i will play a relevant role [23]. We have also assumed N i to be well-separated states We will solve the Boltzmann equations in the original basis (as above) and in the flavor basis where y E is diagonalized through a flavor rotation as in eq. (23) assuming zero initial abundance In Figure 1, we show the numerical solutions comparing the results in the nonflavor basis y E (solid curves) and in the flavor basisŷ E (dashed curves). In the top row, we show the diagonal elements of Y ∆ and |Y ∆E |, while in the bottom row, we show their off-diagonal elements (they are Hermitian matrices). Here, we see that independent of basis, once off-diagonal elements of Y ∆ develop from leptogenesis, unavoidably, off-diagonal elements of Y ∆E will be induced as well. In the flavor basisŷ E , the off-diagonal elements start to become suppressed at various temperatures as the charged lepton Yukawa interactions subsequently get into thermal equilibrium and finally at z 100, Y ∆ 12 and (Y ∆E ) 12 start to become suppressed, indicating a transition to the three-flavor regime.
In Figure 2, top row, we plot TrY ∆ and |TrY ∆E | in the two different bases: nonflavor basis y E (red solid curve) and flavor basisŷ E (blue dashed curve). Reassuringly, TrY ∆ and TrY ∆E are basis independent, although the entries of Y ∆ and Y ∆E differ among the two bases by the flavor rotations as in eq. (28) with V = V E and U = V * E since y E is symmetric. Clearly, the physics is invariant under basis transformation, and the benefit of the flavor basis is to help us to interpret the results.
For instance, we can read the diagonal entries of Y ∆ and Y ∆E in theŷ E basis as the flavor charges in the e, µ, τ (red, blue and green dashed curves in the top row of the Figure 1) and also deduce when the system transits to a different flavor regime from the suppression of off-diagonal entries. In the bottom plot of Figure 2, we see that, while Y ∆(B−L) = TrY ∆ − TrY ∆E is conserved at the end of leptogenesis z 10, TrY ∆ and |TrY ∆E |, not being conserved charges, continue to evolve. For a final remark, the final baryon asymmetry produced in this example is Y ∆B (T B− ) = 6.1 × 10 −11 , consistent in sign but smaller than the observed value by about 30%. 10 10 This can be compared with ref. [25], which also obtained a final baryon asymmetry, which is of the right sign but a factor of a few smaller than the observed baryon asymmetry. Besides the improved treatment discussed in the work, we also correct the wrong basis used in ref. [25].

B. Type-II leptogenesis
In the type-II seesaw model, the SM is extended by a massive triplet scalar T under SU (2) L with hypercharge q Y T = 1 as where Since T couples to two lepton doublets which in general do not align in flavor space, one needs to describe them with density matrix as first pointed out in ref. [22].
The CP violation in the decays of T † → α β and T → HH can arise at one-loop level from the contribution of heavier particles of mass scale Λ M T , which generate the Weinberg operator below Λ, After the EW symmetry breaking, the light neutrino mass receives contributions from integrating out the scalar triplet T as well as the Weinberg operator as where In the following, we will utilize the interaction terms derived in ref. [22] but include only decay, inverse decay and gauge scattering processes (other scattering effects are negligible in the parameter space we will consider below). The Boltzmann equations to describe the evolution of where we have defined z ≡ M T /T and to close the equations we apply eqs. (41) and (43). The branching ratios for the decays of T to lepton doublets and Higgses are, respectively, For the generation of Y ∆ , we have to append to the right-hand side of eq. (31) a source and washout terms, respectively, given by [22] where the matrix of the CP violation parameter is Assuming Maxwell-Boltzmann distribution for T , we have Y eq ΣT = Y eq T + Y eq T † = 135 2π 4 g z 2 K 2 (z) and the decay reaction density γ D is given by where the total decay width is Finally, assuming Maxwell-Boltzmann distributions for all the particles, the gauge scattering reaction density for T T † ↔ ψψ, where ψ refers to the SM fields, is where the reduced cross section is given by [26] Taking into account the RGE of the gauge couplings at one loop 11 , we obtain an accurate parametrization within 10% up to z 20, Notice that under flavor rotations in eq. (27), from eqs. (56) and (68), we observe that For the source and washout terms (66) and (67) to transform the same way, one requires 11 The one-loop RGEs of are given by [27,28] where we take µ = 2πT and fix α 2 (m Z ) = 0.0337 and α Y (m Z ) = 0.0169 with m Z = 91.2 GeV.

This can be obtained by a particular choice of ordering of flavor indices as discussed in Appendix
A. Hence, we will take y E → y * E in eqs. (26) and (31) such that the transformation is consistent with the one above. Equivalently, we can take Y ∆ → Y T ∆ in eqs. (26) and (31).
For illustration, we choose a benchmark point from ref. [22], and we fix the neutrino mass matrix to be with m 1 = 10 −3 eV while for the rest of the parameters, we choose the best-fit parameters for normal mass ordering from the global fit [29]. The effect of RGE up to scale around M T is accounted for approximately by taking r = 1.4. We ignore the RGE of charge lepton Yukawa and fix it to be whereŷ E = diag (2.8 × 10 −6 , 5.9 × 10 −4 , 1.0 × 10 −2 ). We will solve the Boltzmann equations in two different bases: nonflavor basis y E with V E = U † PMNS and flavor basisŷ E with V E = I 3×3 . |TrY ∆E | is too small to be shown here but one can easily deduced its value from the plot. See the text for further discussions.
In Figure 3, we show the numerical solutions comparing the results in y E (solid curves) andŷ E (dashed curves) bases. In the top row, we show the diagonal elements of Y ∆ and |Y ∆E |, while in the bottom row, we show their off-diagonal elements. In the flavor basisŷ E , the off-diagonal elements start to become suppressed at various temperatures as the charged lepton Yukawa interactions subsequently get into thermal equilibrium. We see that at z ∼ 1000 Y ∆ 12 and (Y ∆E ) 12 remain large, indicating that one has not entered the three-flavor regime.
In Figure 4, top row, we plot TrY ∆ and |TrY ∆E | in the two bases: y E (red solid curve) and y E (blue dashed curve). As expected, TrY ∆ and TrY ∆E are basis independent, while the entries of The density matrix operator of the SM in thermal equilibrium at temperature T is given bŷ where Z ≡ Tr e − 1 T (ĤSM− i µ iNi) withĤ SM the SM Hamiltonian and µ i andN i the chemical potential and number operator of a SM field i, respectively. If we are interested in the correlation between a particle species of different flavors, we can generalize the chemical potential and number operators to matrix in flavor space as where α and β are flavor indices and we have made use of the fact that in chemical equilibrium with gauge bosons we have µī = −µ i . The operator a † iα,p creates a particle i α of momentum p from the vacuum a † iα,p |0 = |p, i α while b † iα,p creates an antileptonī α of momentum p as b † iα,p |0 = |p,ī α . For fermions (bosons), they fulfill anticommutator (commutator) relations a i β ,p , a † iα,p +(−) Other operator combinations are zero. Sandwiching the operator (A2) between two states with particle of type i of the same momentum p (but their flavors can be different), we have, for example, i β , p µ iNi αβ i α , p = (µ i ) αβ . It also follows that (µ i ) βα = (µ i ) * αβ .
Next, we will define the generalized phase-space distribution f i(ī) for particle i and antiparticlē i, respectively, as [5] δ In what follows, we would like to solve for (f i,p ) αβ and fī ,p αβ . Since the derivation below follows independently of whether µ iNi is a matrix in flavor space or not, we will suppress the flavor indices. Notice that The traces are taken over multiparticle states with energy E i . For fermion f , the occupation number is either 0 or 1, and each of them contributes a factor of 1 + e − 1 T (Ef−µf) , while for boson b, each of them contributes a factor of ∞ n=0 e − 1 For a fermion i, we have (A6) For a boson i, we have (A7) In the second line above, we have used n ne − 1 One can repeat the exercise above for antiparticleī with a i → b i and the only change is µ i → −µ i . Hence, from the definitions (A3) and (A4), we obtain the desired results where ξ i = 1(−1) for i a fermion (boson) and E i = |p| 2 + m 2 i . Integrating the phase space distributions above over 3-momentum, we obtain (matrices) of number densities where we have included g i to take into account additional gauge degrees of freedom.
Expanding to linear order in chemical potential |µ i | /T 1 and integrating over 3-momentum, the difference between the phase-space distributions of i andī, we obtain the (matrix of the) number density asymmetry where we have defined with (f i,p ) βα and fī ,p βα , the transformations will be n ∆ → V * n ∆ V T and n ∆E → U * n ∆E U T .
Normalizing eq. (A10) by the cosmic entropy density s = 2π 2 45 g T 3 with g being the effective relativistic degrees of freedom of the Universe, we have where we have defined Y nor ≡ 15 8π 2 g . The relation above also holds for a particle which does not carry a flavor index, e.g., for the SM Higgs, which is taken to be massless at high temperature, we have Y ∆H = 4Y nor 2µ H T , where g H = 2 for the SU (2) L gauge degrees of freedom and ξ H = 2 for massless boson.
up to linear order for all the particles, we have where in the last step we have used eq. (A12). Similarly, we have the relation for antiparticles by changing the sign of the chemical potentials, The evolution equations of the Heisenberg operators whereĤ =Ĥ 0 +Ĥ int is the Hamiltonian of the system with H 0 denoting the free field Hamiltonian whileĤ int represents all possible interactions among the fields. In the following, we will write down the derivation only for the equation of motion of O i,p since those for Oī ,p will be analogous.
Taking the ensemble average on both sides of eq. (B5), we have Integrating the equation above over momentum p on both sides, we have where we have defined the number density (matrix) as and assume that f i,p goes to zero at large momentum. In the absence of interactionsĤ int = 0, the phase space will evolve purely due to the Hubble expansion. In terms of Y i ≡ n i /s, we can rewrite For massless fields, H 0 does not contribution to the right-hand side of eq. (B8). Next, we would like to write the terms in right-hand side of evolution equation also in terms of number densities.
Doing a perturbative expansion on the Heisenberg operator Ĥ int , (O i,p ) αβ to the first order in H int , and considering that the interaction timescale is much shorter than the evolution timescale, we can take the time integral to infinity and obtain [5] where the subscript 0 denote operators consist of free fields i.e.Ĥ int = 0.
Considering only the SM charged lepton Yukawa interaction term, we havê Since this interaction is linear in the each type of field, it will only contribute to the second term. Considering thermal mass [30], there is a contribution to the first term of eq. (B11), which results in oscillation among flavors. Ref. [6] showed that flavor oscillations are damped by gauge interactions, and hence we will ignore this term.
Expanding the fields in momentum modes, we have ,p e +ip·x , (B13) where p · x = Et − p · x and the sum s is taken over the two spin states. Substituting the fields above into eq. (B11), we obtain the evolution equation of as where δ p ≡ (2π) 4 δ (4) (p) with p a 4-momentum, I is a 3 × 3 identity matrix and we have assumed all external fields to be massless. Clearly, the whole term vanishes since p · p E ∝ p 2 H = 0. For nonvanishing result, one should consider thermal masses and scattering processes involving another external field [20]. For instance, a gauge field can be attached to either , E or H. One can also attach a fermion-antifermion pair to the Higgs fields and the process involving the top-quark Yukawa coupling will be the dominant one. Since the flavor structures involving and E will remain exactly the same, we will not carry out the exercise here. 12 Notice that the last term in the big curly brackets of eq. (B16) will still be zero due to energy-momentum conservation.
Dividing and multiplying the terms in the right-hand side of eq. (B16) by f eq ,p 1 − f eq ,p which follows from eq. (B2) and expanding up to linear term in |µ i | /T 1, the first three terms in the big curly brackets in eq. (B16) all have the same flavor structure, where we have made used of eq. (B3). Similarly, we obtain the evolution equation of¯ by changing the sign of all chemical potentials Hence the evolution equation for Y ∆ = Y − Y¯ has the following form Repeating the exercise above, we obtain the evolution equation for Y ∆E = Y E − YĒ with the following flavor structure: Under rotations in flavor spaces E → U E, → V , y E → U y E V † , the kinetic equations above will have the same form (flavor covariant) since Y ∆ → V Y ∆ V † and Y ∆E → U Y ∆E U † .
As a final remark, in a radiation-dominated Universe and assuming entropy conservation, we can trade the time variable with temperature T using the relation For T u−b < T < T B , we have For T u−c < T < T u−b , we have For T B 3 −B 2 < T < T u−c , we have For T u−s < T < T B 3 −B 2 , we have For T u−d < T < T u−s , we have Finally, for T < T u−d , we have which is valid for T > T u−b . For α 2 , the RGE at one loop is [27,28] α 2 (µ) = 12πα 2 (m Z ) 12π − 19α 2 (m Z ) + 19α 2 (m Z ) ln µ , where we take µ = 2πT and α 2 (m Z ) = 0.0337 with m Z = 91.2 GeV. Here we ignore the milder RGE of the charged lepton Yukawa and fix it in the flavor basis to beŷ E = diag (2.8 × 10 −6 , 5.9 × 10 −4 , 1.0 × 10 −2 ) (the result is independent of basis).
We .
Here Y ∆H should be treated as an independent variable and one will need to construct a Boltzmann equation for Y ∆H taking account all the interactions that change the number of Higgs.