Coherent Flavour Oscillation and CP Violating Parameter in Thermal Resonant Leptogenesis

Solving the Kadanoff-Baym (KB) equations in a different method from our previous analysis, we obtain the CP violating parameter $\varepsilon$ in the thermal resonant leptogenesis without assuming smallness of the off-diagonal Yukawa couplings. For that purpose, we first derive a kinetic equation for density matrix of RH neutrinos with almost degenerate masses $M_i \ (i=1,2) \sim M$. If the deviation from thermal equilibrium is small, the differential equation is reduced to a linear algebraic equation and the density matrix can be solved explicitly in terms of the time variation of (local) equilibrium distribution function. The obtained CP-violating parameter $\varepsilon_i$ is proportional to an enhancement factor $(M^{2}_{i}-M^{2}_{j}) M_i \Gamma_j/ ((M^{2}_{i}-M^{2}_{j})^2 +R_{ij}^2)$ with a regulator $R_{ij}=M (\Gamma_i + \Gamma_j)$, consistent with the previous analysis. The decay width can be determined systematically by the 1PI self-energy of the RH neutrinos in the 2PI formalism.

Solving the Kadanoff-Baym (KB) equations in a different method from our previous analysis [1], we obtain the CP violating parameter ε in the thermal resonant leptogenesis without assuming smallness of the off-diagonal Yukawa couplings. For that purpose, we first derive a kinetic equation for density matrix of RH neutrinos with almost degenerate masses M i (i = 1, 2) ∼ M . If the deviation from thermal equilibrium is small, the differential equation is reduced to a linear algebraic equation and the density matrix can be solved explicitly in terms of the time variation of (local) equilibrium distribution function. The obtained CP-violating parameter ε i is proportional to an enhancement factor (M 2 i − M 2 j )M i Γ j /((M 2 i − M 2 j ) 2 + R 2 ij ) with a regulator R ij = M (Γ i + Γ j ), consistent with the previous analysis [1]. The decay width can be determined systematically by the 1PI self-energy of the RH neutrinos in the 2PI formalism.

Introduction
Leptogenesis is one of very attractive scenario to explain the baryon number asymmetry [2] (For review, see [3]), but if the Majorana masses of the righthanded (RH) neutrinos have a hierarchical structure, the lightest Majorana mass must be heavier than 10 9 GeV [4] in order to produce sufficient amount of lepton number asymmetry. The condition can be evaded when Majorana masses are almost degenerate, which is called the resonant leptogenesis [6][7] [8].
In light of the LHC experiment TeV scale leptogenesis has attracted much attention [10]- [37]. Especially, when we try to solve the naturalness problem via the Coleman Weinberg mechanism in a B−L sector [38] [39], U (1) B−L gauge symmetry must be spontaneously broken around the TeV scale [40] and masses of RH neutrinos are naturally at the same energy scale. The leptogenesis scale can be much lowered by considering neutrino flavour oscillation out-of-equilibrium, which is important in the νMSM scenario [41] [42][43] [44]. Hence it is becoming more and more important to treat coherent flavour oscillation in a systematic way.
In a conventional approach based on the classical Boltzmann equation, the evolution of the phase space distribution functions of on-shell particles is described and the interactions between particles are taken into account through the collision terms that comprise the S-matrix elements calculated separately. So the conventional classical method is not valid when the quantum coherent oscillation becomes important such as the flavour oscillations or the resonant leptogenesis. Density matrix formalism [45] [46] is a multi-flavour generalization of the Boltzmann equation and has been applied to neutrino flavour oscillations [47] [48][49] [50]. Another formulation is to use the Kadanoff-Baym (KB) equation, which is derived from the Schwinger-Dyson equation on closed-time-path. The approach is very systematic but difficult to solve without introducing various approximations. It was first applied to the leptogenesis with a hierarchical structure of the Majorana mass [51], and intensively used in various papers [52]- [63]. KB equation was applied to the resonant leptogenesis and oscillatory behaviour of lepton asymmetry was discussed [64][65] [66]. The quantum oscillations in the flavored leptogenesis are also discussed in [67] [68][69] [70] [71].
In the resonant leptogenesis, CP-asymmetry in the decay of RH neutrinos is generated by an interference of the tree and the self-energy one-loop diagrams. The CP -violating parameter is given by where h is the neutrino Yukawa coupling and Γ i (h † h) ii M i /8π is the decay width of N i . The resonant enhancement of the CP-violating parameter was discussed in [72], and systematically studied in [7][73] [74]. The regulator was given by R ij = M i Γ j . If the mass difference is larger than the decay width, we have |M 2 i − M 2 j | R ij , and ε i is suppressed by Γ i /M ∼ (h † h) ii . However, in the degenerate case, |M i − M j | ∼ Γ and ε can be enhanced to O((h † h) 0 ) ∼ 1.
Hence the determination of the regulator R ij is essential for a precise prediction of the lepton number asymmetry in the resonant leptogenesis. The authors [75] calculated the resummed propagator of the RH neutrinos and obtained a different regulator R ij = |M i Γ i −M j Γ j |. By using their result, the enhancement factor becomes much larger. The origin of the difference of the regulators is discussed in [76] [77].
Recently Garny et.al. [78] systematically investigated generation of the lepton asymmetry in the resonant leptogenesis using the formulas developed in [52] [53]. In the investigation, they considered a non-equilibrium initial condition in a time-independent background and calculated generation of the lepton number asymmetry. Starting from the vacuum initial state for the RH neutrinos, they read the CP-violating parameter from the generated lepton number asymmetry. The effective regulator they derived is R ij = M i Γ i + M j Γ j , which differs from the previous results.
In a previous paper [1], we solved the KB equation in the thermal resonant leptogenesis and obtained the same regulator R ij = M i Γ i +M j Γ j as above. Our derivation is applicable to cases when the background is slowly changing with time but valid only when the off-diagonal component of the Yukawa couplings are small compared to the diagonal ones For practical purposes, this condition is too strong and it is desirable to extend the analysis to more general cases with large off-diagonal Yukawa couplings. The purpose of the paper is to solve the KB equation without assuming smallness of the off-diagonal Yukawa couplings (1.2). In order for it, we first rewrite the KB equation in terms of the density matrix of RH neutrinos. Since Majorana fermions have 2 spinor components, the density matrix is 2N F × 2N F for N F flavours. In deriving the kinetic equation for the density matrix, we assume that deviation of the distribution functions are not very large. If the condition is satisfied, we reproduce the equation [45]. Various terms in the equation can be systematically obtained in the 2PI formalism. The kinetic equation, which is a differential equation, is reduced to a linear equation when an inequality H Γ i in (2.6) between the Hubble parameter H and the decay width Γ i of RH neutrino N i is satisfied. Then it is straightforward to obtain the solution of deviation of the RH neutrino density matrix from the local equilibrium. From the off-diagonal component of density matrix, we can read the CP violating parameter ε. The same CP violating parameter as in [1] with The paper is organized as follows. In section 3, we derive kinetic equations of density matrices starting from the Kadanoff-Baym equations. The derivation is performed under an assumption that distribution functions are not far from the local equilibrium ones. But smallness of flavour mixing interactions is not assumed. Namely, the off-diagonal Yukawa couplings are not necessary small compared to the diagonal ones, and coherent flavour oscillation is fully taken into account. In section 4, we derive kinetic equations of the RH neutrinos and lepton asymmetry in the yield variables. In section 5, we solve the kinetic equations to obtain the RH neutrino density matrix. From the flavour offdiagonal component, we read the CP-violating parameter ε. We summarize in section 6. In Appendix A, we explain derivation of the kinetic term d t f from KB equation. Explicit forms of inverse of matrix C are written in Appendix B.

Comparison of time scales
We introduce multi-flavour right-handed neutrinos ν R,i where i is the flavour index, i = 1 · · · N F . In particular we consider a case that two RH neutrinos have almost degenerate masses. Hence we set N F = 2 in the following. We write N i = ν R,i + ν c R,i . The Lagrangian is given by where α, β = 1, 2, 3 and a, b = 1, 2 are flavor indices of the SM leptons α a and isospin SU (2) L indices respectively. M i is the Majorana mass of N i and h iα is the Yukawa coupling of N i , α a and the Higgs φ a doublet. P R(L) are chiral projections on right(left)-handed fermions. As a concrete model we consider the Lagrangian (2.2) with only the Yukawa couplings, but the following analysis and the results are not restricted to the specific model: we can systematically include other interactions such as the B − L gauge interactions of the RH neutrinos N i .
We compare various time (or inverse mass) scales in the model. First the Hubble parameter H in the radiation dominant universe is given by where T is the temperature of the universe. Thermal masses and decay widths of SM leptons and Higgs φ are given by m ,φ ∼ gT and Γ ,φ ∼ g 2 T where g is the SM gauge coupling. When T is lower than g 2 × 10 18 GeV , Γ ,φ are larger than H. Since we are interested in the TeV scale leptogenesis in the present paper, we have the relation In type I seasaw model, the decay width of the RH neutrino is given by The ratio of Γ i to the Hubble parameter (2.3) at temperature T = M i is rewritten in terms of the "effective neutrino mass"m i as (see e.g. [3]) where v is the scale of the EWSB. Hence if we take the Yukawa coupling so as tom i ∼ 0.1 eV, the ratio becomes K i ∼ 100. This corresponds to the strong washout regime. Hence we have the following inequality among various quantities with mass dimension: The inequality Γ ,φ Γ i is not used in the analysis of the present paper. Hence our results are still valid when the RH neutrinos are charged under B − L gauge interaction and Γ i becomes larger.

From KB to density matrix evolution
In this section, we derive an evolution equation of the multi-flavour density matrix of the RH neutrinos N i [45,49] starting from the Kadanoff-Baym equation (see also [66]).

Green functions
First we define various Green functions. An ij-component of Wightman Green functions is defined by (3.1) The massM and 1PI self-energy function Π are also 2 × 2 matrices (besides the spinor structure) with the flavour indices ij. We also define the spectral function by G ρ = i(G > − G < ) and the statistical propagator by G F = (G > + G < )/2. The retarded (advanced) Green functions are related to the spectral function by the relation For the self-energy function Γ, we can similarly define various types of self-energy functions of R, A, ρ and > <. (See Appendix B of [1].)

Kadanoff-Baym equations
The Kadanoff-Baym (KB) equation of the RH neutrinos in the expanding universe is given by q is the comoving momentum and * is the convolution in the time coordinate. Symbolically we write it as The 1PI self-energy function Π of RH neutrino is obtained by cutting a (full) propagator of 2PI diagrams. In the 2PI formalism, all internal lines represent full propagators while vertices are tree. For more details, see Appendix C, D of [1]. Figure 1 are examples of self-energy diagrams. In deriving the KB equation, Figure 1(a) gives the decay width at tree level while Figure 1(b) gives an interference between the tree and the one-loop vertex diagrams [62]. Hence the direct CP violating parameter is contained in fig. 1 (b). If we include Z gauge boson or a scalar field coupled with the RH neutrinos, other self-energy diagrams in Figure 2 contribute to Π. By taking the Fourier transform with respect to the relative time coordinate (3.5) Here we used the Moyal-Weyl bracket defined by In the expanding universe with the Hubble parameter H, X derivative is often estimated as ∂ X ∼ O(H). On the other hand, derivative with respect to the relative momentum q 0 is estimated as is given by the decay width Γ N of the RH neutrinos. In the strong washout regime, we have an inequality H Γ N . Since the dominant contribution to the self-energy Π comes from the diagram in Figure 1 (a), Γ for Π * is given by the decay widths of the charged lepton and Higgs Γ l,φ propagating in the internal lines. They are much larger than Γ N . An expansion with respect to ♦ is given by H/Γ N, ,φ and hence justified by (2.6).
Taking up to the first order of the derivative expansion of ♦, we have The spectral function G ρ satisfies a similar equation in which > < ( of G and Π) is replaced by ρ.

Green function in the (local) equilibrium
If we drop the derivative term containing ♦, it becomes an equation for the Green function in the local equilibrium at time X; By using the KB equation of the retarded Green fucntion (see eq. (2.11) in [1]), In the thermal equilibrium at temperature T , the Green functions are antiperiodic in the time direction with an imaginary period iβ = i/T . Hence Fourier transform satisfies the Kubo-Martin-Schwinger (KMS) relation where f (eq) is the Fermi distribution function f (eq) (q 0 ) = 1/(e q0/T + 1). Various properties of the equilibrium Green functions are reviewed in section 3.5 in [1]. Especially, as shown in (3.45) in [1], the off-diagonal component of the Wightman functions G (eq) > < (x 0 , y 0 ) vanishes in the limit of x 0 → y 0 . It directly follows from the KMS relation together with the equal-time anti-commutation relation of the fields N i . When the system is out of equilibrium, it deviates from zero whose imaginary part gives the CP violating source for the lepton number asymmetry.
If the system is slightly deviated from the local equilibrium, KMS relation indicates that the deviation is written as We then define which represents a deviation of the distribution function δG ≶ ∼ i(δf )G ρ .

KB equation for small deviation from G (eq) ≶
We now derive the KB equation for a small deviation from the local equilibrium. Taking a variation in (3.7) and picking up to the first order terms of δ, we have We can obtain the same equation for G ρ by replacing ≶ by ρ. By combining these equations and using the KMS relation, some terms are cancelled and we where we defined The deviation from G (eq) ≶ occurs due to the expansion of the universe, and hence δG ≶ is proportional to the Hubble parameter H. Since the derivative expansion of ♦ is an expansion of H, we can drop terms containing more than one δ or ♦ when H Instead of (3.4), we can start from and obtain a similar equation to (3.16), By multiplying a helicity projection operator with h = ±1 , and taking trace of spinors, we get We make the following quasi-particle ansatz for δ G ≶ . In the present paper, we consider a situation that two RH neutrinos have almost degenerate masses. Hence their poles in the Green function can be approximated by a single pole of Breit-Wigner type: where we set the momentum at on-shell q ±µ = (±ω q , −q) µ and is the spectral density of RH neutrino. Two mass eigenstates are summed in the distribution function δf N . As explained in Section 3.3, flavour off-diagonal components of the distribution function is suppressed by a cancellation of two mass eigenstates. But when the system is out-of-equilibrium, off-diagonal component of δf N becomes comparable to its diagonal one. Also note that hermiticity of Wightman function together with spatial homogeneity and isotropy require the relation δf † N,h,q = δf N,h,q . Majorana condition relates the positive and negative frequency parts as in (3.21). We then insert the ansatz of δG ≶ of (3.21) into (3.20) and perform q 0 integration: , and we get an evolution equation for the density matrix; The density matrix f N,h,q contains an equilibrium part f eq N,h,q = f N (ω q )1 2×2 and a deviation from it. The derivation of the l.h.s. (the kinetic term d t f N,h,q ) is given in Appendix A. The first term of the r.h.s. in (3.20) gives an effective Hamiltonian, while the second term gives the collision term, Here we have used smallness of the flavour off-diagonal components of G eq i =j (see discussion after (3.11)), and smallness of flavour dependent thermal corrections to G R .

Kinetic equation for density matrix
In deriving kinetic equations for the density matrix, we need to make quasiparticle ansatz in (3.21). Similar ansatz must be imposed on the internal lines in the self-energy diagrams Π because distribution functions (even when they are matrix-valued) are defined only on mass-shell. This is the most subtle point in the KB approach. In order to take various diagrams contained in each selfenergy diagram in Figure 1, an often-adopted method is to expand the full propagators and cut the self-energy diagram into two. Examples are shown in Figure 3. On the cut-line, on-shell propagatos are used.

Kinetic equation for RH neutrinos
The collision term (3.27) is proportional to (4.1) The first term with G < describes decay (or scattering) of RH neutrino (plus other particles ) into others while the second term with G > is an inverse-decay (or inverse scattering). By expanding the full propagators in the self-energy Π and cutting the diagram into two, we have various diagrams with on-shell external lines. External lines are assigned to either incoming or outgoing particles. If a cut diagram with G < represents a scattering process of N +i+j · · · → a+b+· · · , it can be expressed as where q a and p i are momenta of incoming and outgoing particles. On the other hand, if a diagram with G > represents an inverse scattering process of a + b + · · · → N + i + j + · · · , it can be expressed as

(4.4)
Combining these two contributions, the evolution equation for the density matrix f N,h,q (3.25) is written as In this expression, we combined variations as Π = Π (eq) + δΠ and f N = f (eq) N + δf N for notational simplicity. 0-th order term of the variation δ automatically cancels due to the detailed balance condition in the equilibrium.
Let us now consider a specific diagram of Figure 3 (a). This diagram is reduced to the cut diagram of Figure 4 (a). Figure 4 (b) is its conjugate and N decays into (¯ , φ * ). Other diagrams like Figure 3 (b) are of higher orders in the Yukawa couplings, and we omit them in the following. From Figure 3(a) and its conjugate, we have for (4.2), and for (4.4) where the following relation where we have used the relation (4.10) The first term q · p is even under the helicity flip h → −h, while the second term is odd. The integral vanishes when thermal effects of the SM particles, namely the thermal mass (∼ gT ) and the statistical factor (Pauli blocking) of leptons, are neglected. The kinetic equaction (4.5) describes an evolution of the density matrix f N of the RH neutrinos. Since the equilibrium distribution satisfies the detailed balance condition, the r.h.s. is nonvanishing only when various quantities are out-of-equilibrium. We take a variation of (4.5) around the equilibrium. Here note that the relations δf = −δf δf φ = −δf φ hold since the SM gauge particles are in thermal equilibrium and their chemical potentials are vanishing.
In order to solve the kinetic equations, it is convenient to define helicity even and odd combinations δf even,odd N,q by δf even N,q ≡ δf N,+,q + δf N,−,q , δf odd N,q ≡ δf N,+,q − δf N,−,q . (4.12) Since helicity operator n·σ is parity-odd and RH neutrino is invariant under the charge conjugation, δf even,odd N,q are CP-even and odd components respectively; In terms of these components, eq. (4.5) with the cut-diagram in Figure 3(a) can be rewritten as a set of equations d t (2f eq N,q + δf even N,q ) (4.14) If we can neglect the helicity odd part of the decay width γ φ h as discussed in (4.11) and the backreaction from lepton asymmetry (the last terms) is dropped, these equations for δf even and δf odd are almost decoupled. Note that the helicity dependent mass term (ω + − ω − ) is also negligible if thermal corrections are small.
The dominant source to generate deviations is the time variation of the local equilibrium distribution d t f eq , which is absent in the equation of δf odd . Hence in the decoupling limit, it is sufficient to consider only the equation for for δf even . In section 5.4, we obtain the CP violating parameter under such a condition.

Kinetic equation for lepton number
The evolution equation for the lepton number is similarly obtained from the KB equation. Details of the derivation is given in Sec. 2.4 and 2.5 in [1]. α-th flavour lepton number current is defined by where a is an SU (2) isospin index. Around T eV scale, the charged Yukawa couplings distinguishing the lepton flavours are in equilibrium and the off-diagonal components of lepton flavour density matrix are negligible compared to diagonal ones. In the second equality, we have assumed that SU (2) isospin symmetry is restored.
Since the derivative expansion is an expansion of H/Γ ,φ , higher order terms are highly suppressed and we have Σ is the self-energy of the SM lepton . If we consider, as an example, the Yukawa interaction of ( , φ, N ), the self-energy function for leptons in Figure 5 gives the same cut diagram Figure 4 (a). By using the same γ α φ h in (4.9), the kinetic equation is reduced to the following Boltzmann equation; Here Tr is trace of the RH neutrino flavour.

Kinetic equations in terms of Yield variables
We rewrite the kinetic equations, (4.13), (4.14) and (4.17), in terms of the Yield variables Y defined by Here s is the entropy of the universe. Note that Y N is a flavour matrix while Y α is a c-number (or α-th eigenvalue of a diagonal flavour matrix). In the following, we consider deviations of distribution functions of RH neutrinos N i and charged leptons α , and other SM particles are assumed to be in the equilibrium distributions. We assume N i and are in the kinematical equilibrium. Then we can set δf even Since the equations for δY are approximated by coupled linear differential equations, equations (4.13), (4.14) can be written in a generic form with matrices In the model with only Yukawa interactions, these matrices are given as follows: Similarly the kinetic equation for lepton number (4.17) is also rewritten as

Solution of the kinetic equations
In order to obtain the CP violating parameter, we solve the kinetic equations for δY N . In the derivation of the kinetic equation from the KB equation, we assumed that the system is not far from the local equilibrium at each time of the expanding universe. But smallness of the off-diagonal Yukawa coupling is not assumed, and the coherent flavour oscillation is fully taken into account. Since the deviation from local equilibrium is caused by the Hubble expansion, both of δ and ∂ t are proportional to the Hubble parameter H. Hence we can set

Formal solution of δY N
In the two-flavour case, Y N , H, Γ N etc. are 2 × 2 matrices. We here express a 2 × 2 matrix A as A = The Yield density matrix Y (eq) N in equilibrium has an a = 0 component only 2 In the expanding universe, the deviation of RH neutrino number densities from equilibrium δY N is first generated and then lepton asymmetry Y L is generated by the flavour oscillation and decay. Here we neglect backreaction from Y L and evaluate the deviation of RH neutrino density directly caused by the expansion of universe. Setting µ = 0, δY N is solved as Components in the 0-th column of C −1 are given by where D is the determinant, [ · ] denotes a summation over i = 1, 2, 3.

CP-violation parameter ε
In order to read the effective CP -violating parameter ε, we set Y L = 0 and insert (5.6) into the kinetic equation of the lepton numbers (4.29), The r.h.s. can be rewritten in terms of 2[δY N ] 0 = Tr(δY N ), which is the total RH neutrino number deviated from the local equilibrium. Especially, neglecting the difference of helicity, we can write the r.h.s. of (5.10) in terms of [δY even N ] 0 in (5.6) as Here [Γ N ] 0 is an averaged decay rate of RH neutrinos into charged lepton α . The CP-violating parameter ε α defined by the coefficient is read as The result is valid when it is justified to replace d t Y N by its equilibrium value d t Y eq N . Though our calculation fixes the flavour basis in which the Majorana masses are diagonal, the final form is written in a flavour covariant way. The above definition of ε is appropriate since the numerator of the ordinary definition is replaced by d t Y L /2[δY N ] 0 while the denominator is approximated by Γ N .

Explicit forms of δY N
In this section, we use explicit forms of various quantities to rewrite the formal expression where 1 ω q f eq N q . Γ N comes from the self-energy diagrams of RH neutrinos, and contains information of (inverse) decay or scattering of RH neutrinos. We decompose Γ N into Γ α by fixing the flavour α of lepton α in the final state. Only the real part appears in the KB equation. From (4.23), we can decompose Γ N in the model (2.2) as Γ α is a partial decay width that RH neutrino decays into α . At the leading order, it is given by replacing (h † h) ij in (5.16) by (h † iα h αj ) (no summation over α).
The first term of Γ N is the decay amplitude at the tree level and if we neglect the statistical effects and the thermal mass of the Higgs and lepton, ξ coincides with ξ 0 , and approaches In order to simplify the notation, we write where Γ eff ij and Γ eff ij are effective decay rates including not only thermal effects but also scattering contributions. If interactions do not change the flavour structure, the effective decay matrix is written as [δY even N ] 0 gives an averaged number of the RH neutrinos deviated from the local equilibrium. Equivalently, ii-component of the matrix δY even N is given by where ± represents i = 1, 2 respectively. Off-diagonal components can be similarly obtained. The real part a = 1 and the imaginary part a = 2 of δY even N are given by Here we defined )] = 0, namely when detC = 0. In such a situation, the deviation of RH neutrino number density becomes large and the assumption of our investigation, smallness of the deviation from local equilibrium, becomes invalid.

CP violating parameter when C = 0
Finally we write the formal expression of (5.12) in a more familiar form by introducing further simplifications. We neglect the thermal mass of leptons and drop the Pauli blocking terms. Then the helicity odd part of γ φ h disappears as explained in (4.11) and the off-diagonal components C connecting the CP even and odd parts in δY vanish. Furthermore we use the vacuum value of Γ N (α = α = 0). Then, by using explicit forms of H in (5.14) and Γ N in (5.19) with Γ eff ij = Γ vac ij , the CP-violating parameter ε α is given by This CP violating parameter has the regulator M 2 (Γ 1 +Γ 2 ) 2 which is consistent with our previous result [1]. In the previous analysis we obtained the same result under an assumption that the off-diagonal Yukawa couplings are smaller than the diagonal ones. In the present analysis, we do not use such a condition, and take effects of coherent flavour oscillation fully into account. The decay widths Γ eff i are determined by the effective decay width (5.19), which are obtained from the 1PI self-energy diagrams Π by cutting the diagrams and putting external lines on mass-shell.
Finally we note that we can decompose the r.h.s. of (5.11) into N i (i = 1, 2) as where we define the CP violating parameter of each N i as (5.33) When i = 1, j takes 2, and vice-versa. Such a separation into a different flavour of RH neutrinos is, of course, valid only when the off-diagonal component (h † h) is smaller than the diagonal one. The numerator of the first factor can be rewritten as which gives a consistent result with [5].

Summary
In the paper, we solved the KB equation without assuming that the off-diagonal component of the Yukawa couplings are small compared to the diagonal ones. In order to solve it, we first derive the kinetic equation for the density matrix. The differential equation can be reduced to a linear equation if the background is slowly changing and the deviation of the distribution function from local equilibrium is small. Then the density matrix of RH neutrino can be solved in terms of the time variation of the equilibrium distribution function and the generated lepton asymmetry. Its off-diagonal component determines the CP violating parameter ε. It is resonantly enhanced due to the almost degenerate Majorana masses and the regulator of ε is given by In the 2PI formalism, the decay width Γ i is given by the imaginary part of the self-energy function of the RH neutrinos. In addition to the loop corrections of the vertex functions, scattering effects with particles in medium are contained. The effect of coherent oscillation is fully taken into account by considering the density matrix formalism. The derivation of the kinetic equation of the density matrix from the KB equation is based on an assumption that the distribution function is not far from the local equilibrium. It will be interesting to obtain the kinetic equation when the system is far from equilibrium. We want to come back to this problem in near future.

Note added
During the final stage of writing the manuscript, an interesting paper [79] appeared. In the paper, the authors derived the kinetic equation of density matrix based on the Hamiltonian approach, and solve the equation to obtain δY even N in the flavour covariant way. The result is consistent with ours but the interpretation of the CP violating parameter seems to be different. Also, in [79], the one-loop resummed effective Yukawa coupling is used to define decay and inverse-decay amplitudes (Γ N in our notation), in which the effect of coherent oscillation is included in their analysis. In our approach based on the 2PI formalism, Γ N comes from 1PI self-energies and the effect of coherent oscillation is not contained. The indirect CP violating parameter generated by resummation of RH neutrino propagators is taken into account by considering the multi-flavour formulation of density matrix.
We then take an effect of the remaining terms in (A.1). These terms can be rewritten as In the first equality, we have used the relation ♦{f }{A} = −♦{A}{f } and ♦{f }{A} = A♦{f }{A −1 }A for a given matrix A. In the second line, we have used G −1 R/A = −(/ q −M − Π R/A ) and droped next-to-leading order contributions Π R,A .
In this appendix, we considered a single flavour case in order to see that the distribution function is sharpened as above. The effect of flavour mixing due to the second term in (A.5) may change the flavour structure in the l.h.s. of the kinetic equation. We want to come back to this interesting issue in future. B Appendix B: Explicit forms of C −1 andC −1 where determinant D is given by