Phenomenology of Baryogenesis from Lepton-Doublet Mixing

Mixing lepton doublets of the Standard Model can lead to lepton flavour asymmetries in the Early Universe. We present a diagrammatic representation of this recently identified source of $CP$ violation and elaborate in detail on the correlations between the lepton flavours at different temperatures. For a model where two sterile right-handed neutrinos generate the light neutrino masses through the see-saw mechanism, the lower bound on reheat temperatures in accordance with the observed baryon asymmetry turns out to be $\gsim 1.2\times 10^9\,{\rm GeV}$. With three right-handed neutrinos, substantially smaller values are viable. This requires however a tuning of the Yukawa couplings, such that there are cancellations between the individual contributions to the masses of the light neutrinos.


Introduction
Observational and theoretical studies of mixing and oscillations are typically concerned with neutral particle states. Important examples are neutral meson mixing, the oscillations of Standard Model (SM) neutrinos [1] and Leptogenesis through the mixing of sterile right-handed neutrinos (RHNs) in the early Universe [2][3][4][5]. In contrast, for charged particles in the SM at vanishing temperature, mass degeneracies between different states are not strong enough to produce observable phenomena of mixing and oscillations. This does however not preclude the fact that these effects are present in principle. Moreover, it has been demonstrated that the mixing of lepton doublets (which are gauged) can be of importance for Leptogenesis [6][7][8][9][10]: At high temperatures, the asymmetries are in general produced as superpositions of the lepton doublet flavour eigenstates of the SM. In the SM flavour basis, this can be described in terms of off-diagonal correlations in the two-point functions, or alternatively in effective density-matrix formulations in terms of correlations of charge densities of different flavours. At smaller temperatures, interactions mediated by SM Yukawa couplings become faster than the Hubble expansion, such that the flavour correlations decohere. In particular, the SM leptons receive thermal mass corrections as well as damping rates that lift the flavour degeneracy. By now, these effects have been investigated in detail. It turns out that due to the interplay with gauge interactions, the flavour oscillations that may be anticipated from the thermal masses are effectively frozen, while the decoherence proceeds mainly through the damping effects, i.e. the production and the decay of leptons in the plasma [9,10]. The appropriate treatment of these flavour correlations turns out to be of leading importance for the washout of the asymmetries from the out-of-equilibrium decays and inverse decays of the RHNs.
The origin of the charge-parity (CP ) asymmetry for Leptogenesis is usually attributed to the RHNs and their couplings [11]. In the standard calculation, when describing the production and the decay of the RHNs through S-matrix elements, one can diagrammatically distinguish between vertex and wave-function terms. The presence of finitetemperature effects as well as the notorious problem of correctly counting real intermediate states in the Boltzmann equations [12] have motivated the use of techniques other than the S-matrix approach: It has been demonstrated that the wave-function contribution can alternatively be calculated by solving kinetic equations (that are Kadanoff-Baym type equations which descend from Schwinger-Dyson equations, see Refs. [13][14][15][16][17] on the underlying formalism) for the RHNs and their correlations, or equivalently, by solving for the evolution of their density matrix [18][19][20][21][22][23][24][25]. The vertex contributions to the decay asymmetry can be obtained within the Kadanoff-Baym framework as well, as it is shown in Refs. [26][27][28][29][30][31][32]. We note at this point that it has more recently been argued that the asymmetry from the wave-function correction and the contribution from the kinetic equation are distinct contributions that should be added together [33,34]. However, it is shown in Refs. [19][20][21] that the kinetic equations derived from the two-particle irreducible effective action capture all contributions of relevance for the CP asymmetry at leading order, which also encompasses the wave-function corrections.
The calculations for Leptogenesis based on Schwinger-Dyson equations on the Closed-Time-Path (CTP) can also be applied to Leptogenesis from oscillations of light (masses much below the temperature) RHNs [35], also known as the ARS scenario after the authors of Ref. [36]. In this approach, we can interpret the CP violation as originating from cuts of the one-loop self energy of the RHNs, that are dominantly thermal. It can be concluded that thermal effects can largely open the phase-space for CP -violating cuts that are strongly suppressed for kinematic reasons at vanishing temperature.
Putting together the elements of flavour correlations for charged particles and of thermal cuts, we can identify new sources for the lepton asymmetry, in addition to the one from cuts in the RHN propagator. In models with multiple Higgs doublets, Higgs bosons may be the mixing particles [37], whereas in minimal type-I see-saw scenarios (with one Higgs doublet), this role can be played by mixing SM lepton doublets [38]. Yet, the RHNs remain of pivotal importance because due to their weak coupling, they provide the deviation from thermal equilibrium that is necessary for any scenario of baryogenesis.
While the set of free parameters of the type-I see-saw model will remain underconstrained by present observations and those of the foreseeable future, the parameter space in that scenario is still much smaller than in models with multiple Higgs doublets. For our phenomenological study, we therefore choose to consider the mixing of lepton doublets in the see-saw scenario, that is given by the Lagrangian In short, the scenario of baryogenesis from mixing lepton doublets can be described as follows [38]: Flavour-off diagonal correlations from the mixing of active leptons ℓ a (where a is the flavour index) can induce the production of lepton flavour asymmetries, corresponding to diagonal entries of a traceless charge density matrix in flavour space. Different washout rates for the particular flavours may then lead to a net asymmetry in total lepton number, i.e, a non-vanishing trace of the charge density matrix. Now, since off-diagonal correlations due to mixing vanish in thermal equilibrium, the mixing of lepton doublets that we aim to describe consequently is an out-of-equilibrium phenomenon. It is thus natural to assume that initially, when the primordial plasma is close to thermal equilibrium, all correlations between the SM lepton flavours vanish. Therefore, we are interested in possibilities of generating these dynamically. Due to gauge interactions, the distribution functions of the SM particles should track their equilibrium forms very closely. Moreover, gauge interactions are flavour-blind, so they can neither generate flavour correlations nor destroy these (up to the indirect effects that we discuss below). Sizeable off-diagonal correlations can however be induced through couplings to the RHNs N, the distributions of which can substantially deviate from equilibrium. The flavour correlations in the doublet leptons ℓ are suppressed however due to the SM Yukawa couplings h with the charged singlets e R and the Higgs field φ, whereφ = (ǫφ) † , and where ǫ is the totally antisymmetric SU(2) tensor. By field redefinitions, we can impose that h and M N are diagonal, which is a common and convenient choice of basis that we adapt throughout this present paper. For simplicity, we therefore write M N i ≡ M N ii .
In this paper, within Section 2, we first review the scenario of Ref. [38]. We improve on the previous discussion by introducing a diagrammatic representation of the mechanism. Moreover, we carefully discuss the generation and the decoherence of lepton flavour correlations at different temperatures, paying particular attention to the fact that both effects take a finite time to fully establish. Section 3 contains a survey of the parameter space of baryogenesis from mixing lepton doublets based on the Lagrangian (1). Under the assumption that only two RHNs are present, we perform a comprehensive scan, given the present best-fit values on the light neutrino mass differences and mixing angles, such that we can identify the point in parameter space that allows for the lowest reheat temperature for which an asymmetry in accordance with observation can result. In addition, we show that for three RHNs, substantially smaller temperatures can be viable, what requires however anomalously large Yukawa couplings of the µ-and the τ -leptons and a cancellation in their contributions to the mass matrix of the light neutrinos. The analysis is however restricted to the strong washout regime, such that it remains an open question of interest whether favourable parametric regions also exist when at least one of the RHNs induces only a weak washout. The concluding remarks are given in Section 4.
2 Generation and Freeze-Out of the Lepton Asymmetry

Diagrammatic Representation of the CP -Violating Source Terms
A detailed derivation of the source term for the asymmetries of the individual lepton flavours is presented in Ref. [38], where the CTP method is employed. Here, we do not reiterate these technical details, but we explain the qualitative form of the main results with the help of a diagrammatic representation of the Kadanoff-Baym equations that arise from the CTP approach. In particular, we express the perturbative approximations to the solutions of these equations diagrammatically. Moreover, we discuss how the mixing of the SM lepton doublets in the CTP formalism can be related to a density matrix formulation of flavour oscillations, that should be familiar e.g. from the problem of oscillations of active neutrinos [39][40][41][42][43].
The CTP formulation of the problem leads to Kadanoff-Baym equations, that we show here in a diagrammatic form in Figure 1(A). One may interpret the Kadanoff-Baym equations as (a subset of) exact Schwinger-Dyson equations, that can only be solved approximately in practice. Since couplings can be assumed to be weak, a perturbative one-loop expansion, that is indicated by Figures 1(B) and (C), amounts to a valid approximation.
We can assume that kinetic equilibrium is established by fast gauge interactions. The distribution functions and the propagators of the SM leptons ℓ are therefore effectively determined by the matrix q ℓab of the charge densities of ℓ and their flavour correlations [9]. The perturbation expansion then explicitly reads q ℓ = q where the superscript (m) stands for mixing and (f) for flavoured asymmetries. The zeroth order term is given by the equilibrium distribution for a vanishing charge density, and therefore q In order to clarify the relation of the present notation with the one used in the derivation of Ref. [38], we make the following remarks: In the present context, the interesting contributions within q (m) ℓ are the off-diagonal correlations of lepton doublets, as these are referred to in Ref. [38]. The leading CP -violating lepton-flavour asymmetries are then contained in q (f) ℓ . Note also that the first term on the right-hand-side of the equation in Figure 1(C) is called the source term in Ref. [38]. . . indicate extra diagrams of different topology than those drawn explicitly and that can be derived from the 2-particle irreducible effective action. Figures (B) and (C), illustrate the scheme that is used in Ref. [38] to obtain approximate solutions, which we also apply in this work. The full propagators for the doublets ℓ are now approximated by the results including flavour correlations. The loops are understood to include gauge-mediated processes and top loops that open up the phase space for the reactions between the particles, that are approximated to be massless, cf. e.g. Ref. [44]. The superscripts (0, m, f) indicate that the charge density matrix that can be computed from the corresponding propagators for ℓ yields the non-zero entries of q (0,m,f) ℓ .

Equations for Mixing Correlations
The first order approximation to the Kadanoff-Baym equations, that is given in Figure 1(B), has the main qualitative features of a density-matrix equation where t denotes time, M is a mass matrix, Γ a matrix that describes relaxation toward equilibrium and Θ a matrix-valued inhomogeneous term. This equation is of a form that is familiar from many applications, among which are neutrino oscillations [39][40][41][42][43] and ARS Leptogenesis [20,36,[45][46][47][48][49]. In particular, it is the commutator term [M, ̺] that induces flavour oscillations among the off-diagonal components of ̺. Now for the present application, we replace the density matrix ρ by the deviations of the lepton and anti-lepton number densities from their equilibrium values δn The off-diagonal components of δn ±(m) ℓ describe the flavour correlations of these nonequilibrium densities. The matrices δn ±(m) ℓ then evolve according to the following equations [38]: Here, η is the conformal time, which is suitable for performing calculations in the background of the expanding Universe. It is determined up to redefinitions of the scale factor, and we make the choice that the physical temperature in the radiation-dominated Universe is T = 1/η, what determines the comoving temperature used in above equations to be where m Pl is the Planck mass and g ⋆ the number of relativistic degrees of freedom. The Eqs.
(3) are derived in Ref. [38] by integrating the corresponding Kadanoff-Baym equations for distribution functions of lepton and antileptons and their flavour correlations over the spatial momentum p, and they are given diagrammatically in Figure 1(B). For the distribution functions, we have assumed that gauge interactions maintain these to be of the Fermi-Dirac form with a matrix-valued chemical potential, as explained in Ref. [9]. This allows to uniquely relate the momentum-space distributions to the number densities δn ℓ in Eq. (3). We now give explaining remarks on the individual terms in Eqs. (3): • The terms ∝ (h 2 aa − h 2 bb ) are present due to thermal masses and correspond to the commutator term in the density-matrix equation (2). Notice the different sign of these terms in the equations for lepton and anti-lepton densities, that was first noted in Ref. [9]. This is what makes it necessary to treat the lepton and antilepton densities differently, rather than considering the matrix of charge densities and their and correlations q • The terms involving B Y i describe the decays and inverse decays of sterile neutrinos into the active leptons. Provided the distribution of the sterile neutrinos N deviates from thermal equilibrium, these processes induce the flavour correlations of active leptons through the off-diagonal components (a = b) of Y † ai Y ib in first place. In this work, we restrict ourselves to the parametric regime where the freeze-out value of the lepton asymmetry is determined at times when the the distribution of the N i is dominated by non-relativistic particles, commonly referred to as strong washout. For this situation, we can assume that M N i ≫ T , and approximate the rate for decays and inverse decays using [38] Here, µ N i denotes a pseudochemical potential that can be employed in order to describe the deviation of f N i , which is the distribution function of the sterile neutrino N i , from its equilibrium form f eq N i . For standard Leptogenesis, results for the lepton asymmetry obtained when using a distribution function numerically derived from the Boltzmann equations before momentum-averaging are compared to the results obtained when using the approximation with a pseudo-chemical potential in Refs. [30,50], and the discrepancy between the two methods is found to be small because for strong washout, the RHNs are non-relativistic, such that they mostly populate modes with momenta that are small compared to the temperature. Within the density-matrix equation (2), the B Y i terms correspond to the inhomogeneous term Θ. In the diagrammatic equation Figure 1(B), this is the first term on the righthand side.
• The terms with B / fl ℓ describe the decay of the flavour correlations due to the SM Yukawa interactions, that discriminate between the different lepton flavours. In the density-matrix equation (2), they correspond to the anticommutator term involving the relaxation rate Γ, and in Figure 1(B) to the second term on the right-hand side. The relevant processes involve the radiation of extra gauge bosons or the decay and inverse decay of a virtual Higgs boson into a pair of top quarks, which are understood to be contained in the loops. (Otherwise, the 1 ↔ 2 processes between approximately massles particles would be strongly suppressed kinematically.) The rates for these processes are calculated to LO in Ref. [44], where it is found that γ fl = 5 × 10 −3 (see also Ref. [51] for an earlier estimate that leads to a similar quantitative conclusion and Refs. [52,53] for a recent LO calculation for the production of massless sterile neutrinos, that is closely related). Taking the momentum average of the kinetic equations of Ref. [38], we find that B / fl what differs from the value used in Ref. [38] due to the updated result of Ref. [44]. We note that the averaging implies the assumption that flavour correlations in all momentum modes decay at the same rate, which is not the case in reality. This procedure should therefore incur an order one inaccuracy that may be removed in the future by extra numerical efforts.
• Finally, the contribution with B g ℓ describes pair creation and annihilation processes that drive δn + ab + δn − ab toward zero. It thus forces an alignment between the correlations among the different flavours of leptons and of anti-leptons, that would otherwise perform oscillations with opposite angular frequency, and as a consequence, the evolution of the off-diagonal correlations is overdamped [9,10]. Gauge interactions thus contribute indirectly to the decay of flavour correlations in addition to the direct damping through the Yukawa interactions. In Ref. [38], the relevant momentum average of the pair creation and annihilation rate is estimated as B g ℓ = 1.7×10 −3 T 2 , based on the thermal rates for s-channel mediated processes, that should yield the dominant contribution due to the large number of degrees of freedom in the SM.
Ignoring the derivative with respect to the conformal time η, the solution to Eqs. (3) is given by where and where Ξ is a matrix specified in Eq. (14) below, that selects only those correlations that have enough time to be built up at the temperatures of interest. The flavour matrix Q ℓ can be viewed as the quantity that multiplies the CP -violation originating from the Yukawa couplings Y of the sterile neutrinos. By comparing the powers of h in the numerator and the denominator, we explicitly see that the q (m) ℓab become suppressed if h aa or h bb become large. This is because processes mediated the SM lepton Yukawa interactions lead to decoherence these off-diagonal correlations, as it is familiar from flavoured Leptogenesis [7][8][9]. Note that these decoherence effects also avoid the resonance catastrophe that would occur for h aa → h bb in their absence.
Neglecting the time derivatives is justified provided that there is enough time for the flavour correlations to adapt to the change in the non-equilibrium density of the sterile neutrinos. From Eqs. (3), it can be seen that rate for the flavour correlations to build up is given by This implies that the correlations do not build up in case the entries of h are so small that Γ − q ℓab < ∼ H when the right-handed neutrinos go out of equilibrium. One should expect this, because in the limit h → 0, the system is flavour blind and there should be no dynamical generation of correlations. For the subsequent discussion, it is useful to compare the rate for the build up of correlations (9) with the more commonly employed rate of flavour equilibration which is valid for |h xx | ≫ |h yy | and for the size of the particular lepton Yukawa couplings as in the SM.

Equations for the Flavoured Asymmetries
In deriving Eq. (8), we have accurately taken account of the impact of the gauge and the Yukawa couplings on generating and also damping off-diagonal correlations in q (m) ℓ . In the equation represented by Figure 1(C), the same couplings enter once more. At this level, we adapt from the usual calculations on flavoured Leptogenesis [7,8] the simplifying approximation that flavour correlations in q (f) ℓab are either unaffected (unflavoured regime) or completely erased (fully flavoured regime). We note that a detailed calculation as for q (m) ℓ should have the result that in the fully flavoured regime, the according components of q (f) ℓab are suppressed rather than fully erased, what would lead to sub-leading corrections to the present calculations. These flavour effects suggest to distinguish between the following regimes: Leptogenesis from mixing lepton doublets should therefore be inefficient at these temperatures.
ℓ , the correlations involving τ are erased due to decohering scattering. In summary, the non-zero entries of the charge-density matrices are given by When above constraints on the rates for the creation and for the decay of flavour correlations are not sufficiently saturated, a treatment of incomplete flavour decoherence is in order. This has been put forward in Ref. [9], but it is yet numerically challenging and requires further developments. For the numerical examples presented in this paper, it turns out however that the approximations as described for regime B1 should be appropriate.
In view of these considerations for the generation of flavour correlations, we can now write down the expressions for the matrix Ξ introduced in Eq. (7), that depend on the temperature regime in which Leptogenesis takes place: Note that the diagonal components are actually irrelevant, because Q ℓaa = 0. From above explicit expressions that indicate the non-zero entries of q (m) ℓ and q (f) ℓ , we see that these charge-density matrices are complementary, what justifies the decomposition of the Kadanoff-Baym equations done in Ref. [38], which is represented here in the Figures 1(B) and (C). We should still make a remark though on the fact why we do not consider terms of order Y 2 that multiply q (m,f) ℓ on the right-hand side of Figure 1(B). The reason is that by virtue of the requirement Γ − qℓab > H for the non-zero correlations q (m) ℓab , the relation (h 2 aa + h 2 bb )γ fl > H should be amply fulfilled as well. Therefore, the flavour damping mediated by the SM Yukawa couplings is more efficient in suppressing the off-diagonal correlations in q (m) ℓab than the damping induced by the couplings Y , that we therefore neglect. Note however, that in the equation represented by Figure 1(C), rates of order Y 2 multiply q (f) ℓ and q (m) ℓ . In particular, the second term on the right hand side is the washout term that suppresses the diagonal charge densities and those of the off-diagonal components, that are unaffected by flavour effects. The second term on the right-hand side is the source term, that we discuss next.
The charge correlations q (m) ℓ , as given by Eq. (7), are of an out-of-equilibrium form, which can simply be inferred from the fact that they are purely off-diagonal. Therefore, they give rise to a non-vanishing source of correlations for entries of the matrix q (f) ℓ , which may be diagonal or non-diagonal. The first term on the right-hand side of Figure 1(C) is the CP violating source that creates flavour asymmetries at the rate Note that the flavour structure of this equation takes the anticommutator form present in Eq. (2). When compared with the corresponding result of Ref. [38], we have generalised this source of lepton flavour asymmetries such that it also includes the off-diagonal correlations that are generated and that should be relevant for the regime A2.

Comparision with the Source Term for Conventional Leptogenesis
It is of course of interest to compare the asymmetry from lepton mixing with the standard asymmetry from the decays and the mixings of sterile neutrinos. For this purpose, we make use of the standard decay asymmetry including flavour correlations, that is given by [2,9,54] where We should compare the function ξ(x) with Q ℓab given in Eq. (8), since both play the role of loop factors, that multiply the CP asymmetry that is present in the Yukawa couplings Y . Inspecting Q ℓab and ignoring first those of the denominator terms that involve B / fl , we observe an enhancement ∼ 1/(h 2 aa − h 2 bb ), which one might naively guess when expecting that the source from lepton mixing is enhanced by the difference of the square of the thermal masses of the leptons. Within ξ(x j ), the corresponding enhancement is explicitly present in the terms ∼ 1/(M 2 N i −M 2 N j ). However, within Q ℓab , the denominator terms that involve B / fl indicate that this enhancement is limited by the damping of the flavour correlations, which induce CP violation from mixing. We observe two different types of damping: First, the terms ∼ B / fl are due to the decoherence of correlations due to scatterings mediated by the SM-lepton Yukawa couplings h. Second, the terms ∼ B / fl B g originate from the effect that leptons and anti-leptons oscillate with opposite frequencies. In conjunction with pair creation and annihilation processes, that mediate between leptons and anti-leptons, this leads to flavour decoherence from overdamped oscillations, as it was first described in Ref. [9] and as it is confirmed in Ref. [10]. An appropriate treatment of the resonant limit x → 1 reveals a similar regulating behaviour also for standard Leptogenesis [20][21][22][23][24].

Flavour Correlations and Spectator Effects
When Leptogenesis occurs at high temperatures, where flavour effects are not important, and when the production and the washout of the asymmetry results from decays and inverse decays of the lightest sterile neutrino, which we call N 1 for now in order to be definite, it is convenient to perform the single-flavour or vanilla approximation. It is based on a unitary flavour transformation (of the left-handed SM leptons) such that N 1 only couples to one linear combination ℓ of the left-handed leptons. On the other hand, when Leptogenesis occurs at temperatures below 1.3 × 10 9 GeV, it is advantageous to remain in the basis where the SM Yukawa-couplings h are diagonal. Interactions mediated by these couplings then rapidly destroy off-diagonal flavour correlations. In a practical calculation, we may then just delete the off-diagonal correlations that are induced by the terms (15) and (16) [7,8]. In the range between 1.3 × 10 9 GeV and 2.7 × 10 11 GeV, it is often convenient to perform a two flavour-approximation, where lepton asymmetries are deposited in the flavour τ and in a linear combination σ of e and µ. Correlations between τ and σ are then erased by h τ τ -mediated interactions. In effect, only diagonal correlations in the flavours τ and σ need to be calculated.
However the reduction to a single flavour at high temperatures or to two uncorrelated flavours between 1.3 × 10 9 GeV and 2.7 × 10 11 GeV only works when N 1 is the only righthanded neutrino that is effectively produced or destroyed in decay and inverse decay processes at times relevant for Leptogenesis, which is typical for hierarchical scenarios where M 1 ≪ M 2,3 , and therefore, the heavier of the N i are strongly Maxwell suppressed. Once more than one of the right-handed neutrinos is effectively produced or destroyed, flavour correlations of the active leptons must be taken into account [10,55], because the combinations ℓ or σ are in general different for the individual N i . Now, because for Leptogenesis from mixing lepton doublets, the relevant CP -violating cut is purely thermal and it involves a right-handed neutrino that must be different from the decaying neutrino, we must require that there are at least two sterile neutrinos N 1,2 with masses M N 1,2 that are not hierarchical. The latter condition is imposed in order to avoid a Maxwell suppression of the CP asymmetry 1 . This implies that for regime A2, we must account for lepton flavour correlations, which in general cannot be avoided by a basis transformation due to the dynamical importance of two sterile neutrinos. In regime B, we may proceed by remaining in the basis where the SM-lepton Yukawa-couplings h are diagonal and simply delete the off-diagonal correlations in the source terms (15) and (16).
In the CTP approach, the collision terms including flavour correlations can be derived in a systematic and straightforward manner. The flavour structure turns out to be in accordance with the anticommutator term in the density matrix equation (2). In order to further improve on the accuracy of the present analysis compared to the one presented in Ref. [38], we include the partial redistribution and equilibration of the flavoured asymmetry within the SM particles present in the plasma, the so-called spectator effects [56][57][58]. The relevant processes are mediated by Yukawa interactions as well as by the strong and weak sphalerons. It is then useful to track within the Boltzmann equations those asymmetries and correlations for which the diagonal parts are only violated by the decays and the inverse decays of the sterile neutrinos. These are given by which we have formulated as a matrix-valued quantity in view of the Boltzmann equations including flavour coherence, that we formulate below. Here, the number density of baryons is given by B, and the diagonal number density of leptons of the flavour a by L a , i.e. it accounts for left-and right-handed SM leptons. In addition to the interactions with sterile neutrinos, the flavour correlations in q (f) ℓ are also altered by processes that are mediated by the SM-lepton Yukawa interactions. However, according to our above discussion, we assume that these are either negligible (unflavoured regime) or lead to a complete decoherence (fully flavoured regime). We also note that the quantities denoted by q X are defined here as the charge densities within a single component of the gauge multiplet X. In contrast, ∆ accounts for a the total charge density that is summed over all gauge multiplicities. This implies that if q ℓaa changes by two units, ∆ aa does so by minus one.
In order to obtain the washout rates, we must reexpress the q ℓaa in terms of the ∆ aa . Moreover, there is also an asymmetry in Higgs bosons that depends on the ∆ aa . We obtain these densities through the relations As explained above, the redistribution of the asymmetries due to the spectators only afflicts the diagonal components of q ℓ . For regime A, we take strong sphalerons, weak sphalerons (that couple to the trace of q ℓ ), interactions mediated by Yukawa couplings of the t, b, c quarks and τ leptons to be in equilibrium. This leads to Note that the factors of 1/2 in Eqs. (19) are due to SU(2) doublet nature of ℓ and φ and to our convention of q ℓaa and q φ to account for one component of the SU(2) doublet only.

Boltzmann Equations
With above explanations and remarks, and with the calculational details given in Ref. [38], we put together Boltzmann equations that describe the freeze-out of the lepton asymmetry. In contrast to the most commonly studied scenarios, we now have more than one sterile neutrino in the game. Therefore, we distinguish the asymmetries that are created through the decays and inverse decays of the individual N i as q (m,f)N i ℓ and ∆ N i , such that Eq. (7) is decomposed as Furthermore, we follow the common procedure of expressing the Boltzmann equations in terms of the ratios It is convenient to parametrise the time evolution through variables z i = M N i /T , in terms of which the equations that describe the freeze-out of the asymmetry can be expressed in the following approximate form: where Y eq N i is the equilibrium value of Y N i . Through Eqs. (5,6,7,15), we can identify the source term for Leptogenesis from mixing leptons In the diagrammatic representation, this source corresponds to the first term on the right-hand side of Figure 1(C). According to what we state above regarding the effect of the SM lepton-Yukawa couplings, we should choose such that correlations that are suppressed by Yukawa-mediated interactions faster than the Hubble rate are deleted from the outset. We re-emphasise that a transformation to an effective two-flavour basis is not possible in Regime A when more than one of the sterile neutrinos is involved in the washout [10,55]. The Boltzmann equations (23) apply to standard Leptogenesis as well. In that case, the source is given byS We have defined here theS ℓab in Eqs. (24,26) such that these comply for a = b with the expressions given in Ref. [38], where these are introduced as a source for q ℓ rather than ∆. The various factors −1/2 and −2 therefore account for the ℓ being SU(2) doublets. The decay rate of the sterile neutrino N k in terms of the variable z i is In the non-relativistic approximation, this agrees with its thermal average up to relative corrections of order T /M N k . In the strong washout regime, a substantial simplification arises from the fact that the deviation of the sterile neutrinos from equilibrium is small, such that we can approximate We use this relation for both scenarios, Leptogenesis from mixing leptons as well as from the decay and mixing of sterile neutrinos. The error incurred through this standard approximation is investigated in Refs. [30,50]. Finally, we need to obtain an expression for the washout rateW [Y N i ℓ ], that is of the anticommutator form indicated in the density matrix equation (2) and that should account for the spectator effects. In terms of Feynman diagrams, the washout term corresponds to the second graph on the right-hand side of Figure 1(C). As explained above, washout affects the flavour-diagonal lepton charges and the off diagonal correlations in a different manner, such that it is useful to define where the superscript t indicates a transposition. The matrix components of the washout rate are given by Putting together the washout terms induced by lepton and by Higgs charge densities, we eventually obtain for the washout term The factor of 1/2 in front of the Higgs-induced term can be understood when noting that q ℓ = µ ℓ T 2 /6, whereas q φ = µ φ T 2 /3, where µ ℓ,φ are chemical potentials and the factor two is due to the difference between Fermi and Bose statistics.

Parametrisation of the Yukawa Couplings
Taking diagonal matrices for the sterile neutrino masses M N , the neutrino sector of the model (1) yet encompasses 18 parameters: 3 sterile neutrino masses and 15 parameters in the Yukawa coupling Y . (While the complex 3 × 3 matrix Y has eighteen degrees of freedom, three of these can be absorbed by phase rotations of the SM leptons ℓ.) This large number of parameters is a typical obstacle to comprehensive studies of the parameter space in type-I seesaw models. The Casas-Ibarra parametrisation [59] facilitates to impose the observational constraints from neutrino oscillations by rearranging the Lagrangian parameters into low and high energy categories. The nine high energy parameters are given by M N 1,2,3 as well as three complex angles ̺ 12 , ̺ 13 , ̺ 23 , in terms of which one defines the complex orthogonal matrix where s ij = sin ̺ ij and c ij = cos ̺ ij . The nine low-energy parameters are given in terms of the diagonal mass-matrix of the active neutrinos m ν = diag(m 1 , m 2 , m 3 ) and the six real angles and phases of the PMNS matrix and U ±δ = diag(e ∓iδ/2 , 1, e ±iδ/2 ). In terms of this parametrisation, the Yukawa couplings of the sterile neutrinos are obtained as A considerable, yet generic simplification occurs when one of the three sterile neutrinos decouples, say N 3 for definiteness. This can happen when the Yukawa couplings Y 3a are very small, when M 3 is very large or when we only assume the existence of two sterile neutrinos to start with. Note that such a configuration requires one of the light neutrinos to be massless. If we therefore take m 1 = 0, we imply that Moreover, the Yukawa couplings as given by Eq. (35) then turn out to be independent of α 1 , as an immediate consequence of m 1 = 0. Altogether, in the decoupling scenario, there are 11 Lagrangian parameters (9 parameters in the Yukawa couplings Y after rephasings and two Majorana masses for the sterile neutrinos). These decompose into 4 high energy parameters (M N 1,2 and ̺ 12 ) and 7 low-energy parameters (m 2 , m 3 , three angles ϑ ij and the two phases δ and α 2 ). Out of the latter, 5 have been measured experimentally (∆m 2 , δm 2 ≈ m 2 2 and the three PMNS mixing angles ϑ ij ). The free parameters of the model are therefore M N 1,2 , ̺, α and δ, while for the PMNS mixing angles in our numerical examples, we choose sin ϑ 12 = 0.55, sin ϑ 23 = 0.63 and sin ϑ 13 = 0.16, which are close to the best-fit values determined by current observations [60,61].

The Parameter Space in the Decoupling Scenario
The production rates of the N i , the washout rates of the asymmetries as well as the CP cuts entering the production rates of the lepton asymmetries that are presented in Section 2 apply all to the non-relativistic regime, i.e. when M N i ≫ T for all N i that are involved in a certain rate. In order to employ these results consistently, we should then avoid situations when at times relevant for the freeze-out value of the asymmetry relativistic N i are present. For the purpose of the present analysis, we therefore choose RHN masses that are not hierarchical, while not necessarily degenerate. Besides, for the source from mixing SM leptons, such parametric configurations are also favoured by the fact that the asymmetry from the decay of the lightest of the N i is exponentially suppressed in the case of hierarchical M i , cf. Eq. (24) above and Figure 4 below. In future work, it may be of interest though to consider relativistic RHNs when the asymmetry does not result from the decay of the lightest RHN, as certain flavour correlations may generically survive the washout from the lighter RHNs [55,62,63]. Now, as we observe below, the dependence of the final asymmetry on the relative size of M 1 and M 2 turns out to be mild (cf. Figure 4). Besides, given the relation (35) and the Boltzmann equations from Section 2.3, we see that the value of the freeze-out asymmetry scales proportional to the M N i when keeping the mass ratios fixed. Therefore, a scan over the four-dimensional parameter space defined by ̺ 12 , δ and α 2 yields comprehensive information on the model in the decoupling scenario [with ̺ 23 and ̺ 13 as in Eqs. (36)], given the constraints mentioned above. For the purpose of the scan, we choose the masses of the two RHNs as these are given in Table 1. The remaining values specified in Table 1 correspond to the point that we find in parameter space for which the maximal asymmetry occurs. We use the flavour approximations as specified for Regime B 1 in Section 2.1.2.  Table 1: Set of parameters that yields the largest asymmetry for Leptogenesis from mixing lepton doublets in the decoupling scenario (effectively two RHNs only).  Table 1.
In Figure 2, we show the freeze-out asymmetry normalised to the observed baryon-minus-lepton asymmetry [64,65] Y obs = 28 79 where the first factor accounts for the conversion to the final baryon asymmetry via sphalerons [66]. We vary parameters in the planes ̺ 12 and δ vs. α 2 , where we fix the remaining parameters as in Table 1. The alignment of some of the contours along δ + α 2 = const. in Figure 1(B) can be attributed to a constant washout of the e-flavour,  Table 1.  Figure 4: Dependence of the freeze-out asymmetry from lepton mixing on the parameter M N 2 , with the remaining parameters as specified in Table 1 (solid blue). For comparison, we show the asymmetry from standard Leptogenesis for the same parameters (dashed red).
since we find for the Yukawa couplings that (cf. Refs. [49,67]) √ m 2 cos ϑ 13 sin ϑ 12 sin ̺ 12 − e −iδ √ m 3 sin ϑ 13 cos ̺ 12 , (39a) √ m 2 cos ϑ 13 sin ϑ 12 cos ̺ 12 − e iδ √ m 3 sin ϑ 13 sin ̺ 12 . (39b) Next, we validate the assumption of strong washout and non-relativistic RHNs by considering the evolution of the individual flavour asymmetries Y ∆i = Y N 1 ∆ii + Y N 2 ∆ii over the parameter z = M N 1 /T , that is commonly used as the time variable when studying Leptogenesis from massive neutrinos. From Figure 3 (A), we observe that, as it is typical for strong washout scenarios, the freeze-out value of the asymmetry settles when z 10. In order to assess further the validity of the non-relativistic approximation for the RHNs as well as in order to determine the minimum value of the required reheat temperature, we start the integration of the Boltzmann equations at some value z = z ini with vanishing asymmetries as boundary conditions (while for all other numerical results, we start the integration at z = z ini = 0). From Figure 3 (B) we see that the result changes by less than 10% as long as z ini < ∼ 3. This independence of the details of the initial evolution of the asymmetries is a typical feature of the strong washout regime. Given the value of M N 1 from Table 1, we may therefore conclude that the minimum required reheat temperature T reh in the decoupling scenario is T reh > ∼ 1.2×10 9 GeV. Due to the order one uncertainties incurred through the estimate of B g ℓ and the momentum averaging leading to B / fl ℓ , this should be considered as coincident with the bound of T reh > ∼ 2 × 10 9 GeV for standard Leptogenesis [58,68,69]. However, the present optimal (by the criterion of minimising the lower bound on T reh ) point given in Table 1 is clearly distinct from the optimal parametric configurations in standard Leptogenesis because here, we are in the strong washout regime, while for the standard source, the lowest viable reheat temperatures occur in between the strong and the weak washout regimes.
The analysis presented in Figure 3 (B) also justifies the use of the flavour approximations for Regime B 1, valid for temperatures roughly below 1.3 × 10 9 GeV. While in fact, for z < ∼ 3, Our scenario falls into Regime A 2, the final prediction for the asymmetry should not be substantially affected by an inaccurate treatment of the flavour effects at early times.
Finally, we vary M N 2 while keeping the remaining parameters fixed as in Table 1. The resulting normalised freeze-out asymmetry is shown in Figure 4. We thus indeed verify that the ratio of M N 1 to M N 2 has no dramatic influence on the freeze-out asymmetry as long it remains of order one. For comparison, we also show in Figure 4 the asymmetry that arises for the same parameters from standard Leptogenesis. While we clearly see the resonance for M N 2 → M N 1 , away from this narrow enhanced region, the result is small compared to the asymmetry arising from lepton mixing. One should note however that there exist parametric configurations that are more favourable for standard Leptogenesis, in particular when saturating the bound on the reheat temperature from Refs. [58,68,69].

Three Sterile Neutrinos
Adding a third RHN N 3 implies that compared to the case with two RHNs only, the resulting asymmetry depends in addition on M N 3 , α 1 , ̺ 23 , ̺ 13 and the absolute mass scale of the light neutrinos, i.e. there are seven extra parameters. This appears to prohibit a comprehensive analysis of the parameter space in practice. Nonetheless, it is interesting to evaluate the asymmetries for an example point, that would be consistent with a smaller reheat temperature. We discuss how an enhanced asymmetry becomes possible even if generated at lower temperatures and how parameters need to be tweaked in order to arrange for such a situation.    Table 2.
In Table 2, we present the point in parameter space for which we determine the freeze-out lepton asymmetry, where it can be seen from Figure 5 that an asymmetry in accordance with the observed value is obtained. Moreover, as exhibited in Figure 5(A), the final asymmetry is dominated by the e-flavour. This can be understood by an inspection of the matrix of Yukawa couplings that satisfies |Y ie | ≪ |Y iµ |, |Y iτ |, such that there is a substantially smaller washout rate for ℓ e than for ℓ µ and ℓ τ . On the other hand, larger Y iµ and Y iτ enhance the asymmetry q ℓee , cf. Eq. (15).
Turning to the parameters in Table 2, we observe the large imaginary part for ̺ 13 , which implies that the Yukawa couplings have a larger magnitude than for configurations with smaller imaginary parts of the ̺ ij . This implies that there is a cancellation in individual terms contributing to the masses of the light neutrinos in the see-saw mechanism, which may be interpreted as parametric tuning. It is noteworthy that situations with large couplings of ℓ µ,τ and relatively small couplings of ℓ e to the RHNs are also favoured in scenarios of Leptogenesis where the CP -violating source arises from the oscillations of relativistic RHNs [35,36,49,63,70], (the so-called ARS scenarios, after the authors of Ref. [36]). We emphasise that however, the source from active lepton mixing is different from the source from RHN oscillations, and while the favoured parametric configurations bear similarities in the pattern of Yukawa couplings, for given masses of the RHNs, the main contributions to the asymmetries are generated in both scenarios at very different temperatures, cf. Ref. [63].

Conclusions
In this work, we have investigated in some detail the possibility of generating the baryon asymmetry of the Universe from the mixing of lepton doublets within the SM extended by two or three RHNs. For this purpose, we have introduced a diagrammatic representation of the underlying mechanism, and we have discussed the dynamics of the flavour correlations of SM leptons at various temperatures. We then have performed a comprehensive parametric study in the setup with two RHNs in the type-I see-saw mechanism. For the case with three RHNs, we have identified a way to achieve lower reheat temperatures that are consistent with the observed baryon asymmetry.
We find that baryogenesis from mixing lepton doublets is a generically viable scenario in the type-I see-saw framework, provided • there are RHNs present in the mass range between 10 9 GeV and 10 11 GeV (cf. the discussion of Section 2.1.1 concerning the upper bound and the numerical findings of Section 3 regarding the lower bound) • and these RHNs are of the same mass-scale, while they do not need to be degenerate.
The lower mass bound on the RHNs and consequently on the reheat temperature can be evaded through a certain alignment of the Yukawa couplings Y , that allows these to be relatively large while the masses of the light active neutrinos remain small, which is possible in the presence of three RHNs, cf. Section 3.3. Methodically, the present calculation draws from formulations of Leptogenesis in the CTP approach that have been applied to the resonant regime [20], to oscillations of relativistic RHNs [35] as well as to the decoherence of active lepton flavours [9]. To this end, we identify the quantities B / fl ℓ and B g ℓ as the main contributors to the theoretical uncertainty. In introducing these, we average over the lepton momentum-modes under the simplifying assumption of identical reaction rates. While such a procedure is common practice in similar calculations for Leptogenesis from oscillations of RHNs (cf. e.g. Ref. [35,45,48]), it would nonetheless be desirable to improve on this approximation in the future by resolving the different reaction rates for each momentum mode.
It is interesting to observe that the minimal reheat temperatures for standard Leptogenesis and for baryogenesis from mixing lepton doublets appear to coincide. For couplings as in the SM, the term (h 2 aa + h 2 bb )B / fl ℓ B g ℓ in the enhancement factor, Eq. (8), numerically dominates in the denominator. Smaller gauge couplings would therefore lead to a larger asymmetry. On the other hand, the size of the gauge couplings does not have a leading influence on the asymmetry for standard Leptogenesis [58,68,69]. Therefore, the similar bound on the reheat temperatures can be attributed to a parametric coincidence.
We also note that the analysis in the present study is valid for the strong washout regime, i.e. the situation where the RHNs can be approximated as non-relativistic during the creation of the asymmetry. It would be interesting to relax this assumption (which should be possible for at least one of the RHNs when three or more RHNs are present altogether) because one may then anticipate substantially larger deviations from equilibrium. In that case, a calculation would however not enjoy the considerable simplifications that arise from treating the RHNs as non-relativistic.
While we have considered here the somewhat minimal framework of the SM augmented by RHNs, our analysis implies that new gauged particles that share the same quantum numbers and that are nearly degenerate or effectively become degenerate at higher temperatures are generic candidates for being involved in creating the matterantimatter asymmetry. This opens new prospects for scenarios of baryogenesis from out-of-equilibrium reactions in the expanding Universe.