because of the partial fraction expansion of the propagator∝(q2−s1)−1(q2− s2)−1= (s1−s2)−1{(q2−s1)−1−(q2−s2)−1}which reflects the superposition of the different mass eigenstates (cf. Eq.(4.27)). To leading order of the coupling constant, the complex symmetric matricesXI are written as XI =iMIeffxIxTI using the complex vectorxL(R)I in the flavor space.
iMNI→ℓαaϕb=−i(h·xI)αuαPRui=Iϵab, (2.149) iMNI→ℓαaϕ∗b =−i(xTI ·h†)αvi=IPLvαϵba (2.150) with the effective couplings (h·xI)α and (xTI ·h†)α. Both of the sets of the effective couplings, (hUT)αI, (V h†)Iα and (h·xI)α, (xTI ·h†)α, are known to give a consistent result. We get theCP-violating parameter
εi = 1 8π
∑
j(̸=i)
ℑ(h†h)2ij (h†h)ii
(Mi2−Mj2)MiMj
(Mi2−Mj2)2+R2ij (2.151)
= 1 8π
∑
j(̸=i)
ℑ(h†h)2ij (h†h)ii(h†h)jj
(Mi2−Mj2)MiΓj
(Mi2−Mj2)2+R2ij
with theregulatorRij =|MiΓi−MjΓj|. We can get the same conclusion from the lepton-number conserving processes (2.131) and (2.132) with the compo-nentsGLR and GRL. In this expression, we used lower-case character in the subscripts and dropped the quantum correction to the Majorana mass. (2.151) should be regarded as theCP-violating parameter of thei-th mass eigenstate decay process.
It’s clear that the Rij = |MiΓi −MjΓj| comes from the factor (2.148).
However, note that this form of regulator becomes zero in the doubly degenerate limitMi =Mj and Γi = Γj, hence it does not seem to be the correct form of the regulator.
Systematic considerations were first performed by Pilaftsis [39], and he found that the regulator in the denominator is given by Rij =MiΓj. Then, in the degenerate case|Mi−Mj| ∼ Γ, ε can be enhanced to O((h†h)0) ∼1. After that, the authors [19, 42] gave the above calculation and obtained a regulator Rij =|MiΓi−MjΓj|. By using their result, the enhancement factor can become much larger.
` φ∗ (a)
`
φ∗ (b)
Figure 8: When we take only the decay and inverse decay processes of RH neutrino in the collision term, then the incoming and outgoing state of RH neutrino must be described as quantum states (a). If we consider only the scattering process, the internal line of RH neutrino mediating the process should have the information of the non-equilibrium quantum state of RH neutrino because this means that the RH neutrinos, which cause the system to be out of equilibrium, appear only as the internal line of scattering diagram (b).
procedure of the RIS-subtraction to get a physically acceptable equation (2.68).
This means that the “(inverse) decay” process of the RH neutrinos are already included in the 2→2 scattering processes as the contributions mediated by the real intermediated states (RIS) of the RH neutrinos. The reason why we need such a separation is that the Boltzmann equations are Markovian evolution equations of on-shell classical particles’ distribution functions: If we consider only the decay and inverse decay processes, we need evolution equations of the off-shell RH neutrinos, that is, time evolution of the “distribution function” of non-equilibrium quantum state must be considered (Fig.8 (a)). And if we con-sider only the scattering processes of the SM lepton and Higgs boson, we have to take into account the time evolution of the quantum state mediating the scattering processes (Fig.8 (b)), then the corresponding collision term is nec-essarily non-local in time. As we see in the later sections, the Kadanoff-Baym (KB) equation describes them in a consistent way. In section 4, solving the KB equation of RH neutrino in advance, we correctly understand the resonant leptogenesis from the view point of (b). On the other hand, in section 5, we rewrite the KB equation into the equation of “distribution function of quantum state” describing the picture (a).
The lower bound of the Majorana mass of the lightest RH neutrino (2.122) is obtained by using the expression (2.71) of theCP-violating parameter in the hierarchical mass spectrum case. Therefore, it can be evaded in the degenerate mass spectrum case. In the conventional calculation, theCP-violating param-eter with the degenerate mass spectrum is read off from the 2→2 scattering processes to do the re-summation of the self-energy diagram of the RH neutrino, as seen in section 2.3. However, note that, in that calculation, the squared am-plitudes of the scattering processes are not considered. One of the motivation for such a calculation could be said to be the cancellation between the “on-shell”
and “off-shell” terms, that occurs inequilibriumcase as a natural consequence of the unitarily andCP T invariance [20] and explicitly checked in [19]. In section 4, it’s shown that,with the non-equilibrium propagator of the RH neutrino, the
contributions of the interferences of the different mass eigenstates, which are dropped in the conventional calculation, become crucial to get the correct form of theCP-violating parameter in the degenerate mass spectrum case.
3 Evolution equations of lepton numbers from non-equilibrium QFT
A systematic method to investigate the evolution of lepton asymmetry is the Kadanoff-Baym (KB) equations. The KB equation corresponds to the time evolution equation offulltwo-point function which does not distinguish on-shell and off-shell states. Accordingly it can take into account quantum coherence of the system. And derived from the first principle of the quantum field the-ory, the KB equation cannot be local (consisting only of physical quantities at the moment) like the Boltzmann equation. The KB equation can be reduced to the classical Boltzmann equation only in special cases where quantum and memory effects can be neglected, and then, the double counting problem can be systematically resolved (see [72] and references therein).
Time-evolution of a quantum system is determined by the Hamiltonian of the system and the initial density operator ˆρat the initial time t =ti. All of the information of the system is encoded in the time-dependent density operator ˆ
ρ(t), or instead, a set of all then-point Green functions. Although the equations for all then-point functions are known as the Schwinger-Dyson equations, it is practically impossible to study the evolution equations containing all then-point functions. We need to select an important set of observables. In the classical approach based on the Boltzmann equation, one-particle distribution function on the phase space is selected to describe the system. In the KB approach, two-point Green functions are selected.
In this section, we summarize notations of various Green functions and their basic properties in the thermal equilibrium. We also summarize the non-equilibrium evolution equation (KB equation) for the Green functions. After brief reviews in section 3.1 and 3.2, we derive the evolution equation of the lepton number in section 3.3 and 3.4.
3.1 Green functions and KMS relations
Various Green functions are introduced in field theories (see also Appendix A).
Consider a fermion field ψ. The statistical propagator GF and the spectral densityGρ are defined as
GF(x, y) =1
2⟨[ ˆψ(x),ψ(y)]ˆ ⟩, (3.1) Gρ(x, y) =i⟨{ψ(x),ˆ ψ(y)ˆ }⟩ (3.2) where⟨· · · ⟩is defined as
⟨Oˆ(x)⟩ ≡Tr{ρ(tˆ i) ˆO(x)}. (3.3) The statistical propagator GF contains information of the particle density of the state on which operators are evaluated. On the other hand, the spectral
density Gρ gives information of spectrum, such as particle’s mass and decay width. Because of the anti-commutator,γ0Gρ(x0, y0) becomes proportional to the spatial delta functionδ3(x−y) at the equal timex0=y0:
γ0Gρ(x, y) =iδ3(x−y)1 (3.4) where1is an identity matrix in the flavor and the spinor indices.
Other useful Green functions are the Wightman functions G>(x, y) =GF(x, y)− i
2Gρ(x, y) =⟨ψ(x) ˆˆ ψ(y)⟩, (3.5) G<(x, y) =GF(x, y) + i
2Gρ(x, y) =−⟨ψ(y) ˆˆ ψ(x)⟩ (3.6) and the retarded and advanced Green functions are given by
GR/A(x, y) =±Θ(±(x0−y0))Gρ(x, y). (3.7) The spectral function can be written asGρ=GR−GA=i(G>−G<).
In this thesis, we assume homogeneity along the spatial directions so that we can always use the Fourier transform in the 3-dimensional space. If the state is described by the thermal equilibrium state, we can further Fourier transform in the time direction.8 In the thermal equilibrium at temperatureT, the Green functionsG(x, y) are anti-periodic in the time direction with an imaginary pe-riod iβ = i/T and their Fourier transforms satisfy the KMS (Kubo Martin Schwinger) relation
G(eq)>< (q) =−i
{1−f(q0)
−f(q0) }
G(eq)ρ (q), G(eq)F (q) =−i (1
2 −f(q0) )
G(eq)ρ (q). (3.8) Here f(q0) is the Fermi-Dirac distribution function f(q0) = 1/(eq0/T + 1). In presence of the chemical potentialµ, q0 is replaced by q0−µ. Since the rela-tion relates the fluctuarela-tion described by the Wightman funcrela-tion to the dissipa-tion described by the retarded Green funcdissipa-tion, it is also called the fluctuadissipa-tion- fluctuation-dissipation relation. By this relation, the spectrum of the system determines all the Green functions. When the system becomes out of equilibrium, the KMS relation is violated. The violation plays an important role in the leptogenesis.
As a final remark in this section, let us recall that the explicit forms of the Wightman functions of free charged fermions (bosons) are given by
Gfree>< (x, y) =
∫ d3q
(2π)3e+iq·(x−y) 1 2ωq
×[
e−iωq(x0−y0)
{1−ηfq
−ηfq
} ˆ
g++e+iωq(x0−y0)
{ −ηf¯−q
1−ηf¯−q
} ˆ g−]
(3.9)
8We often use the Fourier transform in the time direction when the system is in the thermal equilibrium at the local temperatureT(t) at timet. Then the Green functions in the four-momentum representation depends on timetthrough the local temperature.
whereωq is the energy of the on-shell particle, and ˆg± = (±ωqγ0−q·γ+m), η= +1 for Dirac fermions with their mass m and ˆg±= 1, η =−1 for bosons.
fq ≡ ⟨Nˆq⟩ and ¯fq ≡ ⟨Nˆ¯q⟩ are given as the expectation values of the number operator of on-shell particles and anti-particles respectively. In general, they are different from the equilibrium distribution functions.