Leptogenesis via Neutrino Oscillation Magic

The possibility of generating the baryon asymmetry of the Universe via flavor oscillation in the early Universe is discussed. After the inflation, leptons are born in some states, travel in the medium, and are eventually projected onto flavor eigenstates due to the scattering via the Yukawa interactions. By using the Lagrangian of the Standard Model with the Majorana neutrino mass terms, llHH, we follow the time evolution of the density matrices of the leptons in this very first stage of the Universe and show that the CP violation in the flavor oscillation can explain the baryon asymmetry of the Universe. In the scenario where the reheating is caused by the decay of the inflaton into the Higgs bosons, the baryon asymmetry is generated by the CP phases in the Pontecorvo-Maki-Nakagawa-Sakata matrix and thus can be tested by the low energy neutrino experiments.


Introduction
The origin of the baryon asymmetry is a long-standing puzzle in the Standard Model and the Standard Cosmology. The successful generation of the baryon asymmetry requires the deviation from the thermal equilibrium [1]. In the standard inflationary cosmology with the Standard Model particles, there are two processes away from equilibrium: the electroweak phase transition and thermalization. The first process is related to the scenario of the electroweak baryogenesis [2] although some extension of the model is necessary to explain the amount of asymmetry. The thermalization era is also an ideal circumstance for the baryogenesis since by definition the Universe is not in the thermal equilibrium. In general, after the inflation the Universe has experienced a thermalization era called reheating. The baryogenesis at this stage is investigated recently [3,4,5,6,7]. There are also possibilities of having new particles in addition to the ones in the Standard Model, and CP-violating non-equilibrium decays of such particles generate baryon asymmetry such as in the scenario of thermal leptogenesis [8].
The flavor oscillation during the reheating era plays the important role for baryogenesis.
The flavor oscillation of the leptons can happen in the early Universe when the high energy leptons from the decays of the inflatons go through the medium. If the inflaton directly produces the left-handed leptons via its decay, the initial quantum state is some vector in the flavor space, that is generally not an eigenstate of the Hamiltonian in the medium. The lepton flavors will later be "observed" by some interactions. It has been discussed that the CP violation in the quantum oscillation phenomena during this process can explain the baryon asymmetry of the Universe [6].
In this paper, we describe the thermalization process with taking into account the quantum effects of the flavor oscillation. To this end, we should employ the formulation in terms of density matrices rather than the classical Boltzmann equation. By solving the kinetic equation, we show that the baryon asymmetry of the Universe can be created within the Standard Model with the Majorana neutrino mass term, llHH, which explains the neutrino oscillation experiments. (For recent results, see Refs. [9,10,11,12,13].) The baryon asymmetry is generated after a quite non-trivial evolution of the density matrices. The shape of the matrices changes during the travel in the medium and eventually settle into the diagonal form in the flavor basis by the lepton Yukawa interactions at later time. The CP-violating interactions in the neutrino sector create the difference between the lepton and anti-lepton density matrices during this evolution.
We investigate scenarios where the leptons, l, are generated by the direct decays of inflaton, φ, or through the scattering of the Higgs bosons, H. We find that enough amount of asymmetry can be produced when the reheating temperature of the Universe is beyond 10 8 GeV (lepton) or 10 [11][12] GeV (Higgs). In particular, in the case where the inflaton mainly decays into the Higgs boson (for example, through φ|H| 2 interactions, see App. A.), the CP phase stems from the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [14,15] and thus the scenario can be tested in principle by future measurements. The leptons produced by the scattering through the llHH operator undergo the flavor oscillations in the medium, and the flavor dependent lepton asymmetry can be generated by the CP violation in the oscillation. The flavor dependent asymmetries are converted to the net asymmetry via the llHH interactions. The llHH interactions are in fact more important than the gauge interactions at temperatures higher than 10 14 GeV. For such high reheating temperatures, the first thermal bath is formed by the llHH interactions, and many of Standard Model particles are out of thermal equilibrium. As temperature drops, the gauge and Yukawa interactions get gradually important. The evolution of the lepton density matrices during this thermalization era experiences flavor oscillations and multiple scattering processes before they settle into flavor diagonal forms. The CP-violating effects in the evolution explain the baryon asymmetry. In this high-temperature regime, the final baryon asymmetry does not depend on the detail of the inflaton properties, such as mass, branching ratio or the reheating temperature, as they are generated by the history of the medium. The excess of the baryons above anti-baryons indicates that a combination of Dirac and Majorana phases is constrained to be in a certain range depending on the neutrino mass hierarchy and the range of reheating temperature. Interestingly, there exist implications on the effective Majorana neutrino mass, m νee , which determines the rate of the neutrino-less double beta decay. This paper is organized as follows. In Sec. 2, we briefly summarize the quantum equation describing the flavor oscillation. The setup of our scenario is described in Sec. 3. Our kinetic equation describing the time evolution of the density matrix is presented in Sec. 4, and it is solved numerically in Sec. 5. The explanation of the behavior of the solution based on the analytic calculation is given in Sec. 6. The last section is devoted to conclusions.

Quantum equation for lepton flavor
The lepton oscillation and its effect on the lepton asymmetry can be described by the time evolution of the density matrix in the flavor space [16]. By using the free Hamiltonian of the left-handed leptons, l i , the evolution of the density matrices of l i and its anti-particle defined by are given byρ Here V is the volume of the system, V = (2π) 3 δ 3 (0). The expectation values in Eq. (2) are taken by the state to describe the Universe. The flavor indices, i and j, are ordered differently for particles and anti-particles in Eq. (2). In this definition, ρ andρ transform, respectively, as U ρU † and UρU † under a unitary rotation of flavors. The density matrix of the lepton asymmetry is naturally defined by Due to the notation in Eq. (2), CP invariance indicates ρ ij =ρ ji . Even if CP is conserved, the off-diagonal components of ∆ ij can be nonzero while the diagonal entries should vanish.
The trace of the matrix, ∆, describes the total asymmetry for left-handed leptons (righthanded lepton should be added to get total one) which is independent of the basis. Note that ρ andρ evolve differently as in Eq. (3). The difference serves as the "strong phase" in the CP violation in the flavor oscillation. The asymmetry evolves aṡ Even if ∆(p) = 0 at some time, the non-trivial matrix can be generated at a later time if ρ +ρ does not commute with the Hamiltonian, while the trace is kept vanishing.
The effects of the interaction term in the Hamiltonian, H int , have been discussed in Ref. [16]. By using the perturbative expansion and the approximation of the instantaneous collisions, the evolution at a time t = 0 is given bẏ where H 0 int (t) is the interaction Hamiltonian in the interaction picture, e iH 0 t H int (t = 0)e −iH 0 t . One can use this equation for any time t by treating each collision independently.
We use the above formulation for the discussion of the flavor oscillation of the leptons from the inflaton decay.

Basic scenario
As the simplest example for the mechanism, we consider the Standard Model with the Majorana masses for neutrinos: The indices of SU(2) L gauge interaction are implicit. One can obtain the model, for example, by integrating out right-handed neutrinos [17,18,19,20,21]. We assume that right-handed neutrinos (or any other alternative) are sufficiently heavy, and do not show up in the history of the Universe. The Lagrangian of the Standard Model (L SM ) contains the Yukawa interaction of the leptons: where we take the basis where y i is real and positive. In this basis, the symmetric matrix κ is given by where m ν = diag (m ν1 , m ν2 , m ν3 ) is the real, non-negative and diagonal matrix of the neutrino masses and H 174 GeV. There are three CP phases in the PMNS matrix [22]: a Dirac phase δ, and two Majorana phases α M = α 21 and α M 2 = α 31 . If the lightest neutrino is massless, we will take the redundant parameter α M 2 = 0. Throughout this paper, we use the indices (i, j, ...) for the flavor basis while the indices (α, β, ...) for the mass basis. Namely, i, j = e, µ, τ and α, β = 1, 2, 3.
We assume that, after the inflation, the decay of inflaton reheats the Universe and the left-handed leptons are produced as daughter particles. For example, if the production is directly from the inflaton decay, the lepton state is parametrized as The coefficient V i is a normalized vector. In this setup, the inflaton sector is characterized by the reheating temperature T R , the vector V i and the branching fraction to left-handed leptons B. Even if the inflaton does not directly decay into leptons but decays into Higgs bosons, the leptons are, in turn, generated by the scattering of the high-energy Higgs bosons with the medium through the Yukawa or llHH interactions, and hence the parameterization above is still useful. Note that from the constraint on the tensor-to-scalar ratio in the curvature fluctuations, r < r max 0.12 [23], the reheating temperature is bounded from above: We also assume that the time scale for the thermalization is much faster than the decay rate of the inflaton, Γ φ . In that case, Γ φ ∼ T 2 R /M P . (This assumption will be justified later.) The thermalized component of the radiation with temperature T R is quickly produced during the reheating era. Under this assumption, we follow the time evolution of the density matrices while the thermal plasma with temperature T R already exists as the initial condition.
By the interaction with the thermal plasma, the leptons quickly lose their energies by scattering processes, and eventually they are annihilated by hitting their anti-particles. In the course of this non-equilibrium process, the leptons undergo the flavor oscillation since the thermal masses of the leptons are flavor dependent. Working in the "mass" basis, where the neutrino mass matrix is diagonal in the vacuum, the generation (in the mass basis) dependent lepton numbers are produced via the CP violation in the oscillation, while net asymmetry is not created. These flavor dependent lepton asymmetries are partially washed out by the scattering via llHH terms. Since the rate of this process is generation dependent, the net asymmetry is produced. Depending on the decay modes and the reheating temperatures, there are other scenarios which generate flavor or chirality dependent lepton asymmetries.
We will discuss each scenario in detail in Sec. 6.
The ingredients of this baryogenesis are the Yukawa interactions and the llHH interactions, both of which are measurable at low energy. If both of them are in the thermal equilibrium, one cannot obtain the baryon asymmetry. Rather, any baryon or lepton asymmetry would be erased. The important fact is that when llHH interaction is effective at high temperatures, the Yukawa interaction is ineffective due to the cosmic expansion. The opposite is true at low temperatures. Therefore, as we will see later, baryon asymmetry can be generated in a wide range of reheating temperatures.

Kinetic equation
We perform a numerical computation of the lepton asymmetry by solving the kinetic equations for the density matrices of leptons and anti-leptons. The oscillation, decoherence by scattering, annihilation, creation and the lepton number generation are described by the equations.
Following the formalism in Sec. 2, the kinetic equation used in this paper is presented below. The starting point is the master equation obtained from Eq. (6). The kinetic equation is summarized in the form of where flavor indices are implicit [16]. The first term describes the oscillation while the second and third terms correspond to the destruction and production processes, respectively. The Hamiltonian Ω(p) can be parametrized as where the thermal correction δΩ(p) can be written as Here we do not include terms which are proportional to the unit matrix since they do not We focus on the following two components of the density matrices: and those for anti-leptons. Here s is the entropy density. The first component, ρ k , represents the high energy leptons produced by the inflaton decay with initial typical momentum, where is the time at the inflaton decay. g * is the effective degree of freedom for the thermal plasma. The high energy leptons lose their energies by redshift and scattering processes.
The second component, δρ T , represents leptons with the typical momentum |p| ∼ T , with the temperature T ∼ T R (t R /t) 1/2 . Here ρ eq ij = δ ij /(e |p|/T + 1) represents the density matrix in the thermal equilibrium.
In terms of ρ k and δρ T , the kinetic equation becomes where Ω k = Ω(|p| = |k|) and Ω T = Ω(|p| = T ). The destruction and production rates for leptons are given by Here the superscript t denotes the transpose of the matrix. The equations for the anti-leptons can be obtained by exchanging ρ byρ everywhere while changing the sign of Ω's. The solution of the equations at t → ∞ is ρ k =ρ k = δρ T = δρ T = 0 when we ignore the expansion of the Universe as is always the case. The expansion of the Universe makes the llHH interaction ineffective at later time, leaving non-vanishing lepton asymmetry as we discuss below.
In the above equations, we have used the densities of the Higgs boson and the righthanded leptons as the one in the thermal equilibrium. In the actual numerical computation, the kinetic equations of right-handed leptons are taken into account (c.f. Ref. [24] and App.B).
The effects of the U(1) Y gauge interactions are also included.
The first terms in Eqs. (21) and (23) denote the thermalization process through the SU(2) L gauge interactions where the Landau-Pomeranchuk-Migdal (LPM) effects [25,26] are taken into account. The coefficient C = O(1) represents the theoretical uncertainties in the rates and also in the energy distributions of the inflaton decay product. This term converts the high energy leptons into low energy ones while the matrix structure is untouched.
The second terms in Eqs. (21) and (22)  Refs. [27,28,24].) The terms with the coefficient C in Eqs. (22) and (23) represent the pair annihilation and creation of leptons, respectively. This process is important for low energy leptons (p ∼ T ).
For high-energy leptons, the rates are suppressed by the Boltzmann factor or T /m φ . A precise estimation of the rate requires the inclusions of the infrared regularization as well as the LPM effects [29]. We put a parameter C = O(1) which represents the theoretical uncertainty. This term brings the total density matrix ρ +ρ to a one proportional to the unit matrix. By Eq. (5), the flavor oscillation stops when this interaction becomes important as expected.
Finally, the terms with κ are the effects of the scattering via llHH interactions. These terms become unimportant at low temperatures. This interaction brings the density matrices into the diagonal form in the mass basis and lets the asymmetries flow towards zero.
Before ending this section, let us give the kinetic equations for the trace of the asymmetry for later convenience. That is The terms in the first and second lows are essentially different. The first term, i.e. the washout term, decreases or increases the asymmetry for the left-handed leptons, while the terms in the second row transfer the asymmetry into the right-handed leptons without changing the net lepton asymmetry.

Numerical results
The kinetic equations in Eqs. (19) and (20) are solved numerically by setting initial conditions which describe the inflaton decay. For the case where the inflaton decays into a single linear combination of the flavor eigenstates as in Eq. (10), the initial density matrices are given by The normalization factor N is given by where B is the branching fraction of the inflaton into high-energy leptons. For the case of direct decays into leptons, the initial distributions depend on the unspecified main decay mode. Taking the thermal distribution as the initial condition provides a conservative estimate of the baryon asymmetry as we will see later. In general, the decay product can be a weighted sum of different states, i.e., a mixed states, such as a N a (V a i ) * V a j . An interesting possibility is that the inflaton mainly decays into the Higgs bosons. The high energy Higgs bosons, in turn, hit the leptons or Higgs bosons in the medium and produce the high energy leptons through the llHH interactions. In that case, the density matrices are given by For a very high reheating temperature where the llHH interaction is stronger than the gauge interactions, the denominator should be replaced by the trace of the numerator.
Only for the cases of the inflatons decay into the Higgs bosons, the medium leptons are set to be zero initially, rather than assuming the thermal distributions in Eq. (26): This deviation from the thermal distributions plays the relevant role for high reheating temperatures, where the rates of the gauge interactions are slower than the expansion rate of the Universe and thus the thermalization process can be flavor dependent. By the strongest interactions in each temperature regime, the thermal components are created within the time scale t R . The effects of taking Eq. (29) as initial condition are not important for T R 10 13 GeV. The temperature dependences of g * and g * s are not included in the calculation.
They depend on the detail of the thermalization histories. In the numerical calculation, we take g * s = g * = 100.
The kinetic equations are solved numerically and is evaluated at a low enough temperature, where n L /s is already frozen to a constant. Here∆ R is the asymmetry transferred into the right-handed leptons through the Yukawa interaction.
The baryon asymmetry of the Universe is measured to be [23] By assuming that the asymmetry is created above the electroweak scale, the chemical equilibrium of the sphaleron process [30,2] tells us We show in Fig. 1 the absolute value of the lepton asymmetry by varying T R while fixing . The left and right panels, respectively, correspond to the normal and inverted hierarchies of neutrino masses. The lightest neutrino mass, m νlightest , is set to be zero for both cases. The bands represent the uncertainties from the C and C factors in Eqs. (21), (22), and (23). We took C = C and varied the value from 1/3 to 3. The same figures for degenerate neutrino cases are shown in Fig. 2 where the lightest neutrino mass is taken to be m νlightest = 0.07 eV, and α M 2 = 0.
One can see that the baryon asymmetry can be explained for T R 10 8 GeV. * We stress that the contribution discussed here always exists in any models to explain the neutrino masses by the effective llHH terms. Notice that we have assumed the perturbative decay of the inflaton and thus the reheating temperature is taken to be below the mass of the inflaton. However, a non-perturbative reheating allows the temperature to be much higher than the inflaton mass, which may further enhance the asymmetry. (See c.f. Refs. [33,34] for enhancing efficiency for the conversion of the inflaton energy.) We leave the analysis of lepton asymmetry in this case for the future study.
The amounts of the lepton asymmetry in the cases where the inflaton decays into the Higgs bosons are shown in Fig. 3 for the hierarchical neutrino mass cases. The degenerate case is shown in Fig. 4. One can see that the large enough lepton asymmetry is generated * For such low reheating temperatures the Hubble parameter during inflation can be as low as H inf = O(10) MeV. Recently, it was shown that with such a low-scale inflation the QCD axion with decay constant around GUT or string scale can be the dark matter [31,32].  For example, the sign and amount of the measured n B constrain the allowed region of the effective neutrino mass m νee for the neutrino-less double beta decay. An example is shown in the left panel of Fig. 9, where we require that correct sign of the baryon asymmetry is generated with T R 10 13 GeV with a fixed value of δ = −3π/4. The requirement reduces the allowed region to the shaded one between solid lines. The case for T R 10 15 GeV and δ = −π/2 is shown in the right panel. As will be discussed in Sec. 6, the baryon asymmetry in this case has little dependence on the inflation models, and thus it is more predictive. We require the asymmetry to be within the 1/2 − 2 of the measured one, by taking C = C = 1.

Underlying mechanisms for leptogenesis
In this section, we discuss how the lepton asymmetries are generated in each domain of the reheating temperatures and decay modes. We discuss the following situations separately: • inflatons decay into leptons directly and low T R , • inflatons decay into Higgs bosons and low T R , • inflatons decay into leptons directly and high T R , and • inflatons decay into Higgs bosons and high T R , where the separation of high and low T R is around 10 13−14 GeV as will be explained later. The mechanisms are qualitatively different in those four cases. In Sec. 6.1, we clarify the necessary conditions for leptogenesis and how the lepton asymmetry depends on parameters in the Lagrangian from the symmetry perspective. In Secs. 6.2, 6.3, 6.4, and 6.5, we discuss each scenario and provide qualitative and quantitative understandings of the numerical results.

Symmetry argument for CP violation
In order for enough lepton asymmetry to be generated, CP symmetry must be broken physically, i.e., the CP phase should not be rotated away in the interactions that are relevant  in the process. Analogous to the case of the CP violation in the K-meson system, this discussion allows one to find the most relevant parameters for the asymmetry.
We have possible sources for CP violation: Yukawa interaction, llHH interaction, and the initial condition. The initial condition is regarded as the density matrix of inflaton decay product, and the medium around the reheating temperature.
First, let us consider T R 10 14 GeV where the medium is almost flavor blind since the gauge interactions are more important than the lepton Yukawa and llHH interactions.
Suppose the limit of two vanishing neutrino masses. The Yukawa interaction has U(1) 3 symmetry while the llHH interaction has U(2) symmetry. In this limit, one can see that all  the CP phases of the PMNS matrix can be rotated away. Thus, it implies that either • a CP-odd parameter in the inflaton decay product (in the rotated away basis) or • the perturbation of the mass of a lighter neutrino is needed to generate the lepton asymmetry. The lower temperature region of Figs. 1 and 2 correspond to the former case and the lower temperature region of Fig. 3 corresponds to the latter case. In the latter case, y 2 τ m ν2 m * ν3 (y 2 τ m ν1 m * ν2 ) should appear in the lepton asymmetry for normal (inverted) mass ordering at the leading order, which comes from the last two terms of Eq. (23). Thus the asymmetry is suppressed if m ν2 (m ν1 ) is small. Also, we can understand that the Majorana phase is important in this case.  • y 2 τ y 2 µ , or • y 2 τ m ν2 m * ν3 (y 2 τ m ν1 m * ν2 ) for normal (inverted) mass ordering.  When the neutrino masses are degenerate, the llHH interaction preserves an SO (3) symmetry. This plays a relevant role as will be seen in Secs. 6.4 and 6.5.
In the following, we discuss two kinds of scenarios depending on whether the initial condition is in a general matrix (Sec. 6.2 and latter case of Sec. 6.4) or in the diagonal matrix (Secs. 6.3, 6.5 and former case of Sec. 6.4) in the mass basis. In both cases, the flavor dependent asymmetries are first generated and converted into net lepton asymmetry through the lepton-number-violating llHH interactions. We will see that CP and lepton number violations are connected through the "observation" by the medium. Since "observation" in quantum mechanics is a one-way process, it provides the departure from the thermal equilibrium, and hence the Sakharov conditions [1] are satisfied.

Inflatons decay into leptons and T R 10 13-14 GeV
For T R 10 13−14 GeV, the time scale for the thermalization process, is faster than the expansion rate of the Universe, H(T R ) 1/t R . Therefore, the high energy component of leptons is continuously produced by the inflaton decay over the time scale t R , but each lepton loses the energy very quickly by the time scale t th ( t R ). The scattering processes via gauge interactions do not destroy the structure of the density matrices. After losing their energies, the pair annihilation and pair creation processes become important.
The mean free time of the low energy lepton is which is even shorter than t th . Therefore, almost instantaneously after the inflaton decay, the combination of the density matrix ρ T +ρ T flows to ρ T +ρ T ∝ 1 since (δρ T + δρ T ) ∼ e −t/t pair (δρ T + δρ T ) from Eqs. (20) and (22). By Eq. (5), the oscillation is cut-off by the time scale of t th + t pair ∼ t th .
We follow the density matrices of the δρ T component. Even though t th > t pair so that the decoherence is faster for low energy leptons, the oscillation of δρ T is more important than that of ρ k since δΩ is larger for low energy leptons as in Eq. (14). Since the time scale of the Hubble expansion, t R , is longer than t pair or t th , one can ignore the redshift of the momentum in the following discussion. The density matrices in the mass basis of neutrinos, α = 1, 2, 3, evolve as and (δρ mass The thermal corrections δΩ in Eq. (14) are dominated by the ones from the Yukawa interactions in the temperature range T R 10 13-14 GeV. In this case, the differences in the diagonal components appear if there is a phase in U PMNS and/or V . By ignoring the electron Yukawa interaction, one finds ∆ mass αβ := (δρ mass For α = β = 3 and δΩ µµ = 0, we obtaiñ ∆ mass 33 = 2N cos θ 23 sin 2θ 13 Im e −iδ V τ V * e + cos 2 θ 13 sin 2θ 23 At this stage, the asymmetry∆ mass 33 is not physical since it depends on the basis. The trace indeed vanishes. Nonetheless, in the mass basis, the lepton asymmetry is stored in each neutrino-mass eigenstate although the net asymmetry is not created. The finite amount of asymmetry is obtained when we include the effects of the llHH interaction term. Due to the scattering by this term, the "observation" of the neutrino mass basis happens. The 2 to 2 scatterings by this interaction term reduce or increase the lepton number by two. The effects of the llHH interaction can be seen by Eq. (25). The neutrino mass differences imply that the right-hand side is non-vanishing even if the trace of∆ mass vanishes.
The time scale that is important for this ∆L = 2 process is either t R or The former is the time scale where the temperature is kept O(T R ), there the dimension five operators are the most effective, and the latter is the one for the scattering with the top or bottom quarks through the tau (y τ ) and the top (y t ) Yukawa interactions. For T R 10 11 GeV, t Yukawa t R . Beyond t Yukawa , the density matrices get diagonal in the flavor basis by the second term in Eq. (22), which means∆ gets vanishing up to the asymmetry already created.
Therefore, the creation of the net lepton asymmetry happens with the efficiency of for the normal mass ordering. Here, ∆m 2 αβ := m 2 να − m 2 νβ is the neutrino mass square difference. For inverted, ∆m 2 23 is replaced by ∆m 2 12 . Notice that only the difference of the wash-out effects, and thus the neutrino masses, contributes to the net asymmetry (see Eq. (25)). The main contribution is, therefore, from the mass difference for the atmospheric neutrino oscillations, ∆m 2 atm ∼ (0.05 eV) 2 .
The amount of the asymmetry is now estimated as Putting altogether, we find and for normal mass hierarchy. These well fit the numerical results in Fig. 1

Inflatons decay into Higgs bosons and T R 10 13−14 GeV
Even if the initial lepton density matrices are diagonal in the flavor basis or in the mass basis, the off-diagonal components of ∆ are generated through the flavor oscillations (c.f. Eq. (37)).
The off-diagonal elements can become physical later due to multiple scatterings. Depending on the basis, the off-diagonal components are actually parts of diagonal components. One Let us see the above mechanism by explicit calculation. The kinetic equation is given by [Ω mass , ρ mass +ρ mass ] + P (t) ·∆ mass , where we have defined and have neglected several terms which are not important for the discussion below. Since we start from the initial condition∆ mass = 0 at t = t ini , the right-hand side vanishes except for the first term at the early stage. The nonzero value of∆ mass is generated by the first term. When the pair production/annihilation by the gauge interaction becomes effective, ρ mass +ρ mass gets close to the one proportional to the unit matrix and the first term vanishes.
After that, the second term becomes effective. From this observation, the equation we should solve is written as where t cut is the time the oscillation stops, and given by t cut = t ini + t pair for this case.
Eq. (47) is easily solved by neglecting the second term, and one gets Note that we can regard Ω mass and ρ mass +ρ mass as constants for t ini t t ini + t pair , and that only the off-diagonal components are generated here. Up to this stage, no CP violation was necessary. As we discussed before, the off-diagonal component of∆ mass is not a CP-odd quantity. The CP phase in the PMNS matrix can bring this off-diagonal component into the diagonal entries through the Yukawa and llHH interactions. The symmetry argument tells us that the CP phase can be physical when y 2 τ m ν2 m * ν3 appears as a perturbation.
The solution of the Eq. (48) is where T is the time ordered product. The lepton asymmetry can be obtained by taking the trace of the solution. In the case of the inflaton decay into Higgs bosons and low T R , from Eq. (51), the lepton asymmetry is Here t ini corresponds to t ini = min(t R , t Yukawa ), and κ mass := (U PMNS ) t κU PMNS . One can check that O(P 2 ) and O(P ) contributions vanish as indicated from the symmetry argument.
By observing that Γ llHH ∝ T 3 and Γ Yukawa ∝ T , the integration is dominated by the earlier time, and then one gets For normal mass hierarchy,∆ (0) is given bỹ and other components are almost zero, see Eq. (28). The resultant lepton asymmetry is calculated as and where ξ CP ∼ (sin α M + 0.2 sin(α M + δ)) (normal hierarchy).
The calculations for inverted and degenerate cases are straightforward, and the results are

Inflatons decay into leptons and high T R
At a high-temperature range, the lepton asymmetry is generated through multiple "observations" of leptons in the medium as in the previous subsection. Contrary to the previous subsection, the leptons are observed at different temperatures and thus in different basis.
As a result, non-observable off-diagonal components in one basis can later be observed as diagonal components in another basis.
Case with hierarchical neutrino masses Here we consider the high-temperature regime where T R 10 15 GeV for the normal and inverted hierarchy with one massless neutrino. † In this region, the numerical calculation shows an interesting feature that the asymmetry gets almost independent of the reheating temperature.
For simplicity, we consider |k| ∼ T , in which case t th ∼ t pair . It is useful to define the following density matrix, At T R 10 15 GeV, the time scale of t pair , t th and t Yukawa are slower than the expansion rate T R , while the llHH interactions are faster than t R , and these are the interactions to bring the momentum distribution to the thermal one. The discussion is, therefore, qualitatively † Although perturbative unitarity is satisfied in the effective theory at TR 10 16 GeV, the UV physics might contribute to the following mechanism. As we will see, these contributions do not change our prediction much, if we assume that the lepton number for the massless neutrino is not violated in the UV physics at the vanishing limit of the Yukawa couplings. different from the case with lower reheating temperatures. The density matrix is diagonal in the mass basis due to the fast llHH interactions.
Because of the hierarchy, the thermalization is effective only for two of the neutrino generations. The density matrices for the leptons has the following form for T 10 15 GeV: where ρ 1 = ρ 2 ∼ ρ 3 for the normal hierarchy and ρ 1 ∼ ρ 2 = ρ 3 for the inverted one.
The off-diagonal components are highly suppressed due to the decoherence effect via the llHH interaction (see Eqs. (19) and (20)). The density matrices are kept in this form until the llHH interaction for the second heaviest neutrino gets ineffective at T ini ∼ 10 15 GeV (10 13 GeV) for the normal (inverted) hierarchy. Here T ini is the temperature Γ llHH,2 = H(T ) (Γ llHH,1 = H(T )) for normal (inverted) mass hierarchy.
Below the temperature, T ini , the off-diagonal components are started to be generated by flavor oscillation (see Eq. (5)). The differences among ρ 1 , ρ 2 , and ρ 3 are important for this to happen. Since the llHH interaction for the heaviest neutrino is still effective, the oscillation can only generate the∆ mass 12 (∆ mass 13 ) component for the normal (inverted) mass hierarchy. The oscillation continues until the time scale that the pair annihilation by the gauge interactions becomes as fast as the expansion rate. Even after the gauge interaction rate becomes faster than the expansion rate, the generated off-diagonal element is kept unerased in the medium due to the flavor blindness of the gauge interactions. Finally, when the Yukawa interaction becomes effective,t Yukawa ∼ M P /3T 2 τ with T τ ∼ 10 11 GeV which is the time For normal mass hierarchy case, there exist the contributions which depend on y 2 τ y 2 µ and y 2 τ m ν2 m * ν3 , respectively. Flowcharts to describe the dominant processes for leptogenesis can be drawn as∆ for y 2 τ y 2 µ contribution, and for y 2 τ m ν2 m * ν3 contribution.
From the above discussion, we can take t ini ∼ M P /3T 2 ini , t cut =t pair and t end =t Yukawa , wheret pair is the time scale at which the gauge interactions are imporant, α 2 2 T H(T ). The form of the density matrix at t = t ini is for the normal mass hierarchy. Since the strong llHH interactions bring the densities to the thermal ones except for ρ 1 , Then, we obtain by solving Eq. (47) and vanishing other components at t =t pair . Here∆ for y 2 τ y 2 µ contribution, and is Tr ∆ for y 2 τ m ν2 m * ν3 contribution. ... are the subdominant terms for the normal mass hierarchy from the anti-commutation in P . One can see that Eq. (67) is dominated by the large t region while all range of t equally contributes to the integral in Eq. (68). As a result, we obtain for y 2 τ y 2 µ contribution, and n L s y 2 for y 2 τ m * ν2 m ν3 contribution. These formulas well fit the numerical results in the left panel of Fig. 1. Now let us comment on the cases for this mechanism with inverted mass hierarchy.
The same discussion applies by making exchanges between the indices α, β = 1, 2, 3 and α, β = 3, 1, 2 in the previous discussion. However, since two of the llHH interactions are strong,∆ 0 31 = (∆ 0 13 ) * is soon destroyed by the llHH interaction, and the final asymmetry is suppressed. On the other hand, we will see soon that if the reheating temperature is slightly smaller than 10 15 GeV, an approximate symmetry preserves∆ 0 12 and a sufficient amount of the lepton asymmetry can be generated.
One of the essences of this region is the fact, ρ T + ρ T is not proportional to the unit matrix and does not commute with Ω T at T > T th . The mechanism here works in general: e.g. the thermal decoupling of right-handed neutrinos at T > T th would lead to ρ T + ρ T not proportional to 1.
Case with degenerate neutrino masses Let us consider the degenerate case, m 2 να ∼ m 2 ν ∆m 2 αβ , and the time scales of the llHH interaction are faster than t R . Naively, it is expected that the lepton number generated is soon washed out. However, one finds that the imaginary part of the∆ mass is not affected during the scattering via the llHH interaction in Eqs. (22) and (23). This can be understood from an approximate SO(3) lepton flavor symmetry in the llHH interaction. The generators are i αβγ (a † α a β + b † α b β ). Thus the combination of the density matrices, conserves. Since the llHH interaction rate is faster than the expansion rate, a non-vanishing ∆ mass quickly flows to the following form Notice that this symmetry property would be missed in the ordinary Boltzmann equation.
The finite mass differences break the SO(3) symmetry, and from Eq. (48) the decoherence of the imaginary part happens at a rate When the coefficient in the r.h.s becomes faster than the cosmic expansion, the imaginary part becomes almost zero.
The oscillation stops at t llHH because the llHH interaction with degenerate neutrino masses also brings ρ T + ρ T to be proportional to the unit matrix. The relevant component from the oscillation is given bỹ with Notice that we have used the oscillation term at |p| T R which is dominant as in Eq. (14).
The off-diagonal element is produced with a strong phase but the real part quickly approaches to zero due to the decoherence, which results and almost vanishing other components in∆ (0) .∆ mass 12 = (∆ mass 21 ) * is almost frozen until t =t Yukawa due to the symmetry protection.
Therefore the net asymmetry is generated and stored in the right-handed leptons. The net asymmetry can be obtained from the integration, n L s where t end =t llHH is the time Γ llHH,1 Γ llHH,2 H(T ), up to when the net asymmetry is efficiently produced due to the wash-out effect. We obtain The result does not depend much on m ν . A same discussion can be applied to the inverted ordering case at the same range of reheating temperature. In particular, the approximate SO(2) symmetry even works with the lightest neutrino massless. The behavior can be found in Fig. 2 and the right panel of Fig. 1.

Inflatons decay into Higgs bosons and T R 10 14 GeV
When T R 10 14 GeV, the gauge interaction decouples and the scattering and thermalization are made by some/all of the llHH interactions. One can see the asymmetry approaches to UV insensitive values for all the cases. Two kinds of mechanisms are operating for these UV insensitive values depending on the neutrino mass hierarchies. The dominant asymmetry is not from ρ k and ρ k in the kinetic equation. This is because the dominant oscillation frequency is ∝ T R , but it is cutoff by llHH interactions whose time scales are proportional to T −3 R . In total, together with the yield of the high energy leptons, ∝ BT R /m φ , the generated asymmetry is proportional to T −1 R , and thus it is suppressed at large T R . Therefore the dominant asymmetry comes from the thermalization process in the medium.
Normal mass hierarchy For the normal mass hierarchy with one massless neutrino, at T R 10 15 GeV the asymmetry becomes UV insensitive as in Sec. 6.4. Since the leptons in the medium are thermalized through the llHH interaction, the lepton density of the medium has the form Notice that at this reheating temperature, only Higgs boson and two of the left-handed leptons are thermalized. From Eq. (69), one obtains the dominant asymmetry n L s ∼2 × 10 −9 11 g * s (T R ) sin δ T τ 10 11 GeV 2 . (83) The observed asymmetry favors δ 0. This formula fits well the results of normal ordering cases in Fig. 3 and 7. Notice that in the numerical calculation we have conservatively taken g * s = 100. More realistic treatment of g * and g * s may give larger asymmetry than the numerical one.
Inverted mass hierarchy and degenerate masses At T R 10 14 GeV for the degenerate cases or the inverted mass hierarchy case, the UV insensitivity also appears. The key fact is the departure from the thermal equilibrium of the right-handed leptons with T 10 14 GeV, where the pair creation rate of U(1)Y is smaller than the expansion rate of the Universe.
Just above T 10 14 GeV, there are three (two) generations of left-handed leptons, the Higgs bosons and tops are thermalized for degenerate (inverted mass hierarchy) case due to the llHH and top Yukawa interactions. ‡ The Yukawa interaction, whose rate is much slower than the expansion rate of the Universe, tends to thermalize the right-handed leptons through, for example, l τ -top scattering into right-handed tau lepton and top. However, the inverse-process is suppressed due to the absence of thermalized right-handed leptons. In total, the amount of the left-handed leptons are decreased from the thermal equilibrium due to the scattering. This implies that at t < t ini +t Y (t Y is the time at α 2 Y T = H(T )), the deviation from thermal equilibrium, δρ mass + δρ mass , is produced at a rate (See also Eq. (20) and App. B): where ρ th = 0.01 30.25 is the yield of the thermalized left-handed leptons. However, the deviation, δρ mass + δρ mass , approaches to zero at the time scale ∆t llHH (t) Γ −1 llHH,1 (t) Γ −1 llHH,2 (t). Thus the amount of deviation at time t < t ini +t Y can be estimated by the integration of Eq. (84) over the time scale ∆t llHH , Taking t ini = t R , t cut = t ini +t Y and substituting (86), one can solve Eq. (47) and obtains while other components of∆ (0) are nearly zero due to the wash-out effect. Here we have used the fact that only the imaginary part of∆ SO(2) symmetry. Since the second term in Eq. (14) is important for Ω mass T in the commutation relation, the t 1 integration dominates at around t 1 ∼ t cut . For the normal mass ordering, one  73)). § The corresponding asymmetry can be calculated from the same procedure and we do not discuss this further.
Notice that with high reheating temperature, the asymmetry is dominantly generated from the departure of the thermal equilibrium of the left or right-handed leptons, which results from the decoupling of the gauge interactions. This does not depend much on the precise information for the inflaton decay products. In particular, the amount of the asymmetry is independent of the B, m φ and T R . The UV insensitive amount is, interestingly, around the order of the observed one for O(1) CP phases in the PMNS matrix. This indicates, by taking into account the quantum mechanics, a general thermalization process can lead to a good opportunity for baryogenesis.

Summary
The neutrino oscillation has been understood as the macroscopic quantum interference phenomena. The neutrinos traveling in the sun, atmosphere and also terrestrial baselines are superpositions of the waves with different frequencies and thus the probability of observing some flavor becomes dependent on the travel distances. § Although, it is disfavored from the Planck data and baryon acoustic oscillation measurement [23].
In the early Universe, the whole Universe can be thought of as a high-temperature medium. The neutrinos (and also charged leptons) traveling through the medium undergo the flavor oscillation of the cosmic size. Even though the neutrino masses are tiny enough to be ignored in the high-temperature medium, the Universe is in fact opaque and the matter effects are important for leptons/neutrinos due to various interactions such as the gauge interactions, the lepton Yukawa interactions as well as the lepton number violating llHH interaction if the neutrinos are Majorana particles.
At the very first stage of the Universe, the leptons are produced through the decays of inflatons. The quantum states of these leptons can be described by density matrices. The scattering with the medium reduces the matrix into a diagonal form in some basis. For example, the pair annihilation process brings the sum of the density matrices of the leptons and anti-leptons into the one proportional to the unit matrix, which stops the oscillation effects. Also, the scatterings through the lepton Yukawa and the llHH interactions bring the density matrices into diagonal forms in the flavor and the mass basis, respectively. One can think of these scattering processes as "observations." Through these observations, the density matrices evolve non-trivially and settle into a form deviated from the thermal equilibrium due to the cosmic expansion.
We find through the numerical analyses the lepton number is indeed generated by these quantum effects. In particular, if the inflaton decays into the Higgs boson dominantly, the high energy leptons are generated as secondary products via the scattering through the llHH interactions. In this case, the leptons are in the neutrino "mass" eigenstates.
Since the effective Hamiltonian is "flavor" diagonal due to the thermal masses from Yukawa interactions, the oscillation takes place. The net lepton asymmetry is produced by the subsequent scattering processes. The source of the CP violation is the Dirac and Majorana phases in the PMNS matrix, and enough amount of asymmetry can be produced for high enough reheating temperatures.
There is always a contribution to the baryon asymmetry of the Universe from the flavor oscillations of the leptons in the inflationary scenario. Our numerical results have shown that the successful baryogenesis is possible in any models to explain the neutrino masses by the llHH terms at low energy. At least, it works if the UV scale to generate the llHH terms is higher than 10 8 GeV.

A Inflaton decay
If the inflaton φ is gauge singlet, the decay is described by where ... represents other decay channel which is not relevant in the following discussion.
Unless the first term is small, the main decay channel is Higgs boson, and the reheating temperature is given by assuming that c 1,2,3 = O(1).
If the φ has the gauge charge same as Standard Model Higgs boson, we can write the dimension 4 coupling In this case, the reheating temperature is same as Eq. (92) except for the replacement y → y .
Therefore, we can obtain the reheating temperature and branching fraction which realize the successful baryogenesis.

B Kinetic equation for right-handed leptons
For completeness, here the kinetic equation including the right-handed leptons is presented although the numerical impacr is small. The right-handed neutrino gives rise the addition term to δΓ p T ij , which is given by The kinetic equation for the leptons is ( where C Y represents the uncertainty where we have taken to be C Y = C in the numerical calculation. In the inflaton decay to lepton case, is added to Eq. (26) as an initial condition. When the inflaton dominantly decays to Higgs bosons, the initial condition is changed to be δρ T | t=t R = δρ T | t=t R = −0.002 100 g * s (T R ) . (100)
where the right hand side is the experimental value given in [22].