Strong Washout Approximation to Resonant Leptogenesis

We show that the effective decay asymmetry for resonant Leptogenesis in the strong washout regime with two sterile neutrinos and a single active flavour can in wide regions of parameter space be approximated by its late-time limit $\varepsilon=X\sin(2\varphi)/(X^2+\sin^2\varphi)$, where $X=8\pi\Delta/(|Y_1|^2+|Y_2|^2)$, $\Delta=4(M_1-M_2)/(M_1+M_2)$, $\varphi=\arg(Y_2/Y_1)$, and $M_{1,2}$, $Y_{1,2}$ are the masses and Yukawa couplings of the sterile neutrinos. This approximation in particular extends to parametric regions where $|Y_{1,2}|^2\gg \Delta$, {\it i.e.} where the width dominates the mass splitting. We generalise the formula for the effective decay asymmetry to the case of several flavours of active leptons and demonstrate how this quantity can be used to calculate the lepton asymmetry for phenomenological scenarios that are in agreement with the observed neutrino oscillations. We establish analytic criteria for the validity of the late-time approximation for the decay asymmetry and compare these with numerical results that are obtained by solving for the mixing and the oscillations of the sterile neutrinos. For phenomenologically viable models with two sterile neutrinos, we find that the flavoured effective late-time decay asymmetry can be applied throughout parameter space.


Introduction
Resonant enhancement from mass degeneracies is a way of obtaining sizeable chargeparity (CP ) violating effects, that would be strongly suppressed by powers of small couplings otherwise. Depending on the ratio of the mass splitting to the decay rate in a system of mixing particles, it may either be more advantageous to describe the CPviolating effects as a time-dependent phenomenon due to mixing and oscillations of the almost mass-degenerate states, or, further away from the mass degeneracy, in terms of a time-independent effective decay asymmetry [1]. The important role that resonant CP -violation assumes in many systems that can be tested in the laboratory has lead • We show how the non-relativistic approximations and simplifications, that are of relevance in the strong washout regime, follow from the general treatment of Refs. [24,25].
• We define the effective decay asymmetry ε as the lepton asymmetry that results on average from the decay of one out-of-equilibrium sterile neutrino. When compared to the decay asymmetry introduced in Ref. [28], this definition resembles more closely the expressions that are usually employed in Leptogenesis calculations, such that it leads to a simple and straightforward way of obtaining the lepton asymmetry. We present the relevant equations that determine the freeze-out asymmetry as well as example solutions.
• We give an expression for the decay asymmetry taking account of active lepton flavours and their possible correlations. We emphasise that flavour effects should be phenomenologically relevant throughout the parameter space. Again, we illustrate the use of this effective asymmetry with numerical examples.
• Since it is crucial for resonant Leptogenesis to treat the decay rate Γ of the sterile neutrinos as matrix-valued, the criterion H/Γ ≪ 1 for the applicability of the approximation in terms of an effective decay asymmetry can only be of schematic meaning. For a simplified scenario with one active lepton flavour only, we determine the smallest eigenvalue associated with the linear differential equation that governs the evolution of the sterile neutrino densities and their flavour-off-diagonal correlations. By comparison with the Hubble rate, this eigenvalue can be used in order to assess whether the approximation in terms of the effective decay asymmetry ε is applicable.
• For a phenomenological scenario with two sterile neutrinos, that explains the observed oscillations of active neutrinos, we find that the use of the effective late-time decay-asymmetry can be justified for all regions of parameter space. This conclusion is also based on comparing the eigenvalues of the equations that govern the mixing and the oscillations of the sterile neutrinos with the Hubble expansion rate prior to the freeze out of the lepton asymmetry.

Relativistic Resonant Leptogenesis
We consider the usual see-saw model for neutrino masses that is given by the Lagrangian Here, the N i are the sterile neutrinos, that observe the Majorana condition N c i = N i , where the superscript c stands for charge conjugation. The Higgs doublet is given by φ and ǫ SU (2) is the antisymmetric, SU(2)-invariant tensor with ǫ 12 SU(2) = 1. The Standard Model (SM) lepton doublets are given by ℓ a , where a = e, µ, τ . When considering the single-flavour model, we drop the index a on the fields ℓ as well as the on Yukawa couplings Y . We make use of the freedom of field redefinitions in order to choose the symmetric matrix M to be real and diagonal, and we refer to the diagonal elements as We describe the generation of the comoving lepton charge density q ℓab in terms of a source term S ab and a washout term W as [24,25] The charge density accounts for the gauge multiplicity, hence we include here the factor g w = 2. Moreover, as mentioned in the Introduction, we allow for the possibility of correlations of the SM lepton flavours. The expansion of the Universe is accounted for through the metric in conformal coordinates g µν = a(η)η µν , where η µν is the Minkowski metric, a(η) is the scale factor and η is conformal time. A prime denotes a derivative with respect to η. In Ref. [24], it is shown that the source term for resonant Leptogenesis through the lepton-number violating Majorana mass can be computed by first solving for the flavour correlations of the oscillating sterile neutrinos, similar to the standard calculations for CP -violation in mixing meson systems [1] or to the lepton-number conserving source in the scenarios that are usually referred to as Leptogenesis from neutrino oscillations [25,[30][31][32][33][34][35][36][37]. The result of Ref. [24] is generalised to include flavour correlations in Ref. [25] and then reads where/ Σ A N (k) is the reduced spectral self-energy of the sterile neutrinos as defined in Ref. [25]. The correlations of the sterile neutrinos are described by iδS N ij (k). Besides the indices i, j for the sterile neutrino flavours, this function corresponds to a rank two tensor in terms of Dirac spinors. It satisfies Kadanoff-Baym equations and the solutions can be decomposed as where σ and ρ are Pauli matrices. In the resonant regime |M i − M j | ≪M, the different components may be written as [24] g ahij (k) = 2πδ(k 2 − a 2M 2 )2k 0 δf ahij , whereM = (M i + M j )/2. Moreover, the Kadanoff-Baym equations also imply the relations [24] δf 1hij (k) = δf 3hij (k)a In view of the non-relativistic approximation below, the a = 0 component is of particular interest. The function δf 0hij may be interpreted as the distribution function of the sterile neutrinos and of their flavour correlations. Using the decomposition (4) and the relations (6), the source term (3) can be expressed as S ab ≡ +hk , ω(k) = √ k 2 + aM 2 ,k = (|k|, k 0 k/|k|) and δf * 0hij (k 0 ) = δf 0hij (−k 0 ). The Kadanoff-Baym equations imply that the sterile neutrino distributions and their correlations satisfy [24,28] where f eq is the equilibrium Fermi-Dirac distribution of the sterile neutrinos. One may alternatively derive this equation using a more heuristic approach in terms of a density matrix instead of the two-point function of the sterile neutrinos. The solution may be substituted back into the source term (3) and eventually into the equation for generating the lepton charge-density (2) in order to obtain predictions for the freeze-out asymmetry.

Non-Relativistic Approximations
Now, we consider a situation, whereM ≫ T (and all sterile neutrinos are assumed to be close together in mass, |M i − M j | ≪M ), as it is of relevance in strong washout scenarios around the time of freeze out. The main simplification arises here due to the fact that modes that do not satisfy |k| ≪ aM are strongly Maxwell suppressed, such that we may approximate the four momenta as Due to the same reason, we can neglect the thermal contributions to the spectral selfenergy of the sterile neutrinos, such that it takes its vacuum form For the terms involvingΣ A N that appear in Eq. (8), this implies that we can take the approximate forms Then, we integrate that equation with the result where we have defined This is the comoving non-equilibrium number density of sterile neutrinos, δn ± 0hij = δn ± * 0hji , which is of the form of a Hermitian matrix. The comoving equilibrium number density is denoted by n eq . The Majorana nature of the sterile neutrinos implies that δn + 0hij = δn − * 0hij , a property that is directly inherited from the distribution δf 0h (±ω, k) and that is derived in Ref. [24]. Note that in the non-relativistic limit, the solutions for the sterile neutrino densities are helicity independent. The relativistic generalisation that accounts for helicity is worked out in Ref. [24].
In order to substitute these results into the source term (3), we use the relations (6) that imply a vanishing axial density δf 3hij in the non-relativistic limit. Note moreover that the Dirac trace in Eq. (3) selects then contributions from δf 0h only. The result for the flavoured source term in the non-relativistic approximation then is Note that we do not sum over h here and make use of the fact that in the non-relativistic limit, we can approximate n ± 0+ij = n ± 0−ij .

Strong Washout Regime
In the radiation-dominated Universe, a(η) = a R η. A particularly convenient choice is η = 1/T , what requires a R = m Pl 45/(4g ⋆ π 3 ) ≡ T 2 /H. Moreover, one can then easily define the parameter z =M /T =M η, that is often used in Leptogenesis calculations.
We investigate under which circumstances the maximal enhancement of the decay asymmetry can be attained. For this purpose, we solve the Eq. (12) in the form that is obtained when using above parametrisation in terms of z where andΓ = 1/(8π). Since larger entries of Y correspond to larger washout, it is proposed in Ref. [28] to obtain a simplified approximation in the strong washout regime by neglecting the first term of Eq. (15). To put this more precisely, note that out of the first three terms of Eq. (15), which are the homogeneous terms, the second and the third grow with z. Therefore, neglecting the first term corresponds to taking the late-time limit of the solution. If the late time-limit applies before the freeze-out of the lepton asymmetry, that occurs for z = z f , it leads to a valid approximation of the freeze-out asymmetry. The evolution of the lepton asymmetry is governed by the equation where the last equality defines the time-dependent effective decay asymmetry ε ab (z), in consistency with Eq. (21) below. In view of flavour effects, we have written this in terms of the asymmetries ∆ ℓaa = B/3 − q ℓaa that are conserved by SM interactions and where B is the baryon number density. Off-diagonal flavour-correlations can be accounted for by ∆ ℓab = −q ℓab for a = b, if necessary. Moreover, q φ stands for the charge density in Higgs bosons, that is present in general. We have also expressed Eq. (17) in a way that defines the decay asymmetry ε as the the lepton asymmetry that results from one sterile neutrino that initially drops out of equilibrium as a mass eigenstate. Note that the factor of four in front of ε ab arises because of the two helicity eigenstates of to the two sterile neutrinos. In addition, this equation includes the crucial washout term W in its flavoured variant, that is derived in Ref. [21] 1 , see also Refs. [38,39]. In the present context, we are interested in the situation where the sterile neutrinos are non-relativistic, such that the washout matrix can be approximated by Lepton-flavour violating interactions mediated through SM Yukawa-couplings are described by the term Γ fl ℓab , that is defined and explained in Ref. [21]. In the fully flavoured approximation, one assumes that these interaction delete the off-diagonal correlations in q ℓ and ∆. Effectively, one may then just set the off-diagonal elements to zero and ignore Γ fl ℓab . Solving Eq. (15) when neglecting the derivatives acting on δn ± 0h yields for the offdiagonal correlations (i = j) of the sterile neutrinos Comparing with Eqs. (14) and (17), we identify the time-dependent effective decayasymmetry It can be straightforwardly interpreted as the asymmetry yield per sterile neutrino that drops out of equilibrium. This quantity differs from the CP -violating parameter defined in Ref. [28], that quantifies the yield in terms of the out-of-equilibrium neutrinos that are present at a given point in time. The discrepancy is due to the time delay in the transition from diagonal out-of equilibrium densities to off-diagonal correlations due to oscillations. We write the late-time limit of the decay asymmetry (21) by dropping the argument z, . e.g. ε ≡ ε(∞), for which we find when using Eq. (19) Provided the strong washout approximation holds, it is then easy to solve Eq. (15) numerically. In the fully flavoured regime, q ℓab can be reduced to its diagonal components and the flavoured asymmetry can be calculated in straightforward generalisation (see e.g. Refs. [40,41]) of the methods for the single-flavour case [9,42]. The flavoured expression (22) for the decay asymmetry in resonant Leptogenesis is of importance throughout the parameter space. If the sterile neutrino mass is below 10 9 GeV, the usual treatment of flavoured Leptogenesis should apply, i.e. ε ab can be reduced to its diagonal components, because interactions mediated by SM-lepton Yukawa-couplings effectively erase all coherence [43,44]. (See however Ref. [38] for a counterexample, where even Yukawa-suppressed correlations at low temperature are of importance, due to a special flavour alignment.) At higher temperatures, when the asymmetry results from the decay of one sterile neutrino only, it is sufficient to either deal with two (a linear combination of e and µ) or one single flavour (a linear combination of e, µ and τ ) only. Once the decay of more than one neutrino contributes, as it is the case for resonant Leptogenesis, there will be decay asymmetries in different linear combinations [39,45] that in general cannot be aligned simultaneously. It then appears simplest to take the full expression for ε ab , including the off-diagonal correlations, and compute their evolution following Ref. [21] (see also Ref. [38]).

Applicability of Approximations
The effective decay asymmetry (22) and the equation for the evolution of the lepton asymmetry (17) offer a simple way of accurately calculating the freeze-out asymmetry even in the resonant regime, where approximations based on the mass splitting of the sterile neutrinos being larger than their width are not applicable. In order to describe the parametric range of validity of neglecting derivatives acting on δn ± 0h in Eq. (15) more precisely, we first take the simplifying assumption of a single lepton flavour only. The effective decay asymmetry can then be expressed in the simple form where X is a dimensionless parameter defined as and where ∆ = is the normalised mass difference, y 1,2 = |Y 1,2 | and ϕ is the relative phase of the Yukawa couplings, ϕ = arg(Y 2 /Y 1 ). Note that the solutions to Eq. (15) remain unaltered as a function of z, provided we leave the ratiosM : ∆ : Y 2 invariant. Therefore, such a rescaling leaves ε(z) and the late-time solutions unchanged as well. This invariance can also be explicitly observed in the late-time asymmetry (23).
The late-time asymmetry (23) can also be constructed from the solutions given in Ref. [28], such that we note agreement with the results of that work. However, our definition for ε differs from the CP -violating parameter proposed in Ref. [28]. Our choice is motivated by the fact that the result (23) quantifies the yield of lepton asymmetry in a transparent manner and that it allows for a straightforward calculation of the final asymmetry, provided the late-time limit is a good approximation at the time of freeze out, what we illustrate in the remainder of this Section.
The expression for the late-time decay asymmetry (23) only leads to an accurate approximation for the process of Leptogenesis, provided the solutions to Eq. (15) reach their late-time form, where the derivatives acting on δn ± 0h may be neglected, prior to the freeze-out of the asymmetry. Based on this requirement, we derive a more precise analytical condition that allows to identify the parametric regions where neglecting the derivatives of δn ± 0h is indeed justified. Since δn ± 0h are Hermitian two by two matrices and moreover, n + 0h = n −t 0h , Eq. (15) corresponds to a coupled set of four real differential equations. The smallest eigenvalue 2 in vicinity of the parametric points where ε is close to unity [cf. Eq. (29)] is given by ǫ = ǫ R2 , which is presented explicitly by Eq. (B1), or alternatively by , whereȳ 2 = (y 2 1 + y 2 2 )/2. Notice also that ǫ is invariant when keeping the ratioM : ∆ : Y 2 fixed. This is more easily seen in the democratic case y 1 = y 2 , where the smallest eigenvalue is given by where ϑ is the Heaviside step function and where we have definedǭ = (a R z/M )ȳ 2Γ . Since (dn eq /dz)/n eq = O(1) around freeze out, one should require ǫ ≫ 1 in order to neglect derivatives acting on δn ± 0h . [A condition that amounts to requiring that the slowest eigenmode of Eq. (15) is faster than the Hubble expansion rate.] This also implies that ǫ ≫ǭ/ǫ. The quantityǭ/ǫ therefore is of phenomenological interest, because it indicates how strong the washout must at least be such that we can justify the neglect of the derivatives of δn ± 0h . In order to relate to the parameters that are typically employed in calculations on Leptogenesis, note thatǭ/z =K = (K 1 + K 2 )/2, where the K i = y 2 iMΓ /H| T =M are the usual washout parameters [9]. In order to satisfy ǫ ≫ 1 at the time of freeze-out, that occurs for z = z f = O(10), it follows that we must requirē We can therefore use the ratioǭ/ǫ in order to infer the minimal washout strength that is necessary for consistently neglecting the derivatives of δn ± 0h . Note that the washout strengthK can also be employed as an expansion parameter for a series approximation that generalises the truncation of the derivative of δn ± 0h in Eq. (15) in a systematic manner. Details of this are worked out in Appendix A.
It is interesting to consider the situation where, for a given value of X, the phase ϕ maximises the decay asymmetry (23). This occurs for ϕ = ϕ M , where and where the asymmetry is then given by For X → 0, the decay asymmetry attains its maximum value ε → 1. Curiously, in this case the CP -violating phase tends to be vanishing, ϕ M → 0. The exact limit can however Ε Ε Figure 1: The ratioǭ/ǫ of the diagonal relaxation rate of the sterile neutrinos to the smallest eigenvalue, with ϕ given by Eq. (28). In order for the derivatives of δn 0h to be negligible, the washout strength should satisfy relation (27).
not be reached because for such an alignment scenario, it takes infinitely long for the off-diagonal correlations in δn ± 0h to build up. In particular, this does not occur before freeze-out. In the examples below, we observe however that it is possible in practice to obtain asymmetries that are at least close to maximal.
For comparison, we also comment the opposite regime, where X ≫ 1 (which may still allow for ∆ ≪ 1). In that case the asymmetry is maximal when ϕ M (X ≫ 1) = π/4. Substituting ϕ = ϕ M and the value of X 2 in terms of ε from relation (29) into Eq. (26), we find This ratio vanishes as the asymmetry ε goes to 1, which reflects the fact that for large asymmetries, it takes a longer time to build the off-diagonal correlations in δn ± 0h , and the washout should be sufficiently strong in order for the late-time decay asymmetry ε to be a good approximation. The ratioǭ/ǫ is presented in Figure 1.
As an illustration for how to interpret the quantityǭ/ǫ, in Figure 2, we show how the parameter ε(z) [as defined in Eq. (21)] evolves in the case where it approaches the late-time value ε = 0.98. We choose two washout strengths, where the weaker one violates the criterion (27) while the stronger one marginally complies with it. In order to obtain these results, we assume thermal initial distributions for the sterile neutrinos and begin to integrate at z = 0. We observe indeed that when relation (27) holds, where z f = O(10), a stationary form for ε(z) corresponds to a good approximation. To see the effect on the freeze-out lepton asymmetry, we take both, the late-time value ε and the time-dependent solution ε(z), and solve Eq. (17), where we assume one single flavour (and consequently suppress the flavour indices), set q φ = 0 for simplicity and take q ℓ = −∆ ℓ . We express the result in terms of the ratio of the lepton-number to the entropy density s, Y ℓ = −∆ ℓ /s and use the value for s with 106.75 relativistic degrees of freedom. For both washout strengths, we observe that initially, there is a substantial deviation between the solutions for Y ℓ that are based on the time dependent ε(z) and its late-time limit. While for the larger washout strength, the freeze-out asymmetries agree eventually up to about 40% accuracy, there is a discrepancy of about a factor of five for the smaller washout strength, that does clearly not satisfy relation (27). Next, we again take y 1 = y 2 but impose fixed values of ϕ, in order to allow for a deviation from the relation (28). In Figure 3, the ratiosǭ/ǫ are presented as functions of ϕ for various values of ε. The curves exhibit two branches, because for a given asymmetry ε and phase ϕ, Eq. (23) has two solutions for X. The two branches join at the point where there is only one root. It is easy to show, using Eq. (23), that the condition for a unique root is ε = cos(ϕ), for which X = sin(ϕ). There are two more curves that we display in Figure 3. First, we show the ratios of the eigenvalues when identifying ϕ = ϕ M , what fixes X through Eqs. (28), and with Eq. (30), we obtain Second, we determine the value of ϕ that minimises the eigenvalue ratio, what defines the graph From Figure 3, we observe asymptotic proximity between these two curves (31) and (32), and moreover, one can check that the junction points for the two solutions for X are close to these curves as well. This implies that ϕ = ϕ M corresponds to a preferable choice for obtaining large asymmetries not only because it maximises ε but also because at the same time, it minimisesǭ/ǫ and therefore the required washout strength. Again, we present in Figure 4 the evolution of the parameter ε(z) and the leptonnumber to entropy ratio Y ℓ for two different washout strengths, what exemplifies the use of the criterion (27) for approximating the freeze-out asymmetry using the late-time decay asymmetry ε. We now move from the simplifying single-flavour model to a more realistic scenario, where several flavours are present and where we take account of constraints from neutrino oscillation data. In order to avoid a proliferation of free parameters, we consider the case where there are only two sterile neutrinos or, alternatively, where a third sterile neutrino decouples. It follows that one of the masses m 1,2,3 of the observed light neutrino states vanishes, i.e. m 1 = 0 for a normal mass hierarchy, which is what we assume here. This leads to a simplified form of the Casas-Ibarra parametrisation of the Yukawa couplings [46] where U ν is the PMNS matrix and v = 246 GeV is the vacuum expectation value of the Higgs field. Note that here, Y is a 2 × 3 matrix. For the PMNS matrix and for the light neutrino masses, we take the best-fit parameters from the global analysis of Ref. [47], and for simplicity, we fix the Dirac and the Majorana phase therein to be zero. The parameter ̺ is a complex angle, and its imaginary part acts here in absence of the PMNS phases as the only source of CP -violation. Moreover, this imaginary part largely controls the absolute value of cos ̺ and sin ̺, i.e. large imaginary parts imply a large washout strength.
For definiteness, we are considering this setup at temperatures of about 10 8 GeV, where all second-generation but none of the first-generation Yukawa couplings are in equilibrium. The qualitative picture does not change when going to different temperatures, where other spectator fields give rise to O(10%) corrections to the freeze-out asymmetries [48][49][50]. We can then relate Moreover, at temperatures below 10 9 GeV, the off-diagonal correlations of the left-handed leptons are strongly suppressed due to the SM Yukawa interactions, such that we can neglect the off-diagonal elements of Eq. (17) (see however Ref. [38], where due to alignments of the Yukawa couplings Y the off-diagonal correlations remain non-negligible at even smaller temperatures). The eigenvalues of the equation for mixing and oscillating sterile neutrinos (15) in terms of the Casas-Ibarra parametrisation are given in Eq. (B3). As the oscillatory contributions due to the mass splitting enter as an imaginary part and the damping contributions due to the Yukawa couplings as a real part, we can find a lower bound on the magnitude of these eigenvalues by setting ∆ = 0, what leads to a considerable simplification of the expressions: We also present the individual flavoured baryon-minus lepton asymmetries |Y ℓaa | = |∆ ℓaa |/s obtained from Eq. (17), using the time-dependent decay asymmetry (solid) and the late-time limit (dashed). The quantities ∆ ℓaa , q ℓaa and q φ are related through Eqs. (34).
Since the smallest ratio is ǫ CI R2 /ǭ CI > ∼ 1/6 for normal hierarchy, neglecting the derivatives on δn ± 0h in Eq. (15) is by the criterion (27) (assuming z f = O(10)) a good approximation everywhere in the strong washout regime of resonant Leptogenesis for the phenomeno-logical model with two sterile neutrinos. Moreover, as washout is always strong in that scenario, what we show in Appendix C, we can conclude that using the late-time asymmetry (22) is a valid approximation for any point in parameter space. For the phenomenological model specified above, we solve Eq. (17) with the effective decay asymmetry (21) based on the full numerical solution to Eqs. (15). This, we compare with the solution obtained when using the late-time limit for the decay asymmetry (22) for all times prior to freeze-out. Since by above arguments, there should be no points where the freeze-out asymmetries obtained by the two methods differ by substantial amounts, we show in Figure 5 the evolutions of ε aa (z) from Eq. (21) and the values of ε aa from Eq. (22), along with the asymmetries |Y ℓaa | = |∆ ℓaa |/s obtained using the time-dependent and the effective late-time decay asymmetries for a typical point in parameter space, for which the width dominates the mass splitting, ∆ ≪ (tr[Y Y † ]Γ) 2 . As anticipated from the analysis of the eigenvalues, albeit the different time evolution at early stages, the freeze-out asymmetries agree very well.

Conclusions
We have studied the applicability of the late-time decay asymmetries ε for sterile neutrinos in their multi-flavoured and single-flavoured forms (22) and (23) to computations of the freeze-out asymmetry in resonant Leptogenesis. This has been done by comparison with the results obtained from the time-dependent decay asymmetry (21) that is based on the solution to the evolution equation (15) for the mixing and oscillating sterile neutrinos. The evolution equation can be straightforwardly derived from its relativistic generalisation, that was first presented in Ref. [24]. Following Ref. [28], the approximations (22) and (23) are obtained by neglecting the time derivative acting on the non-equilibrium number densities and correlations in Eq. (15).
In addition to the numerical comparisons, to gain analytical insight, we have derived expressions for the eigenvalues of the equation that governs the mixing of the sterile neutrinos and their deviation from equilibrium. This analysis reveals that ε can reach its maximum value one provided ∆ → 0 and ϕ → 0 simultaneously. In that case however, also the smallest eigenvalue of the equation describing mixing and oscillations tends to zero, indicating that the approximation in terms of the late-time decay asymmetry is not valid in that limit. Nonetheless, the quantitative analysis (by studying the smallest eigenvalue as well as the numerical solution) reveals that the late-time asymmetry can be a good approximation already for moderately strong washout, even when ε is close to one. To quantify this, cf. Figures 1 and 3 in conjunction with the criterion (27). An increase of the washout strength generally leads to a better approximation.
While the derivation of the single-flavour decay asymmetry (23) makes use of the approximation proposed in Ref. [28], its definition is different from the CP -violating parameter introduced in that work. We find the form that is suggested here somewhat more transparent, as it corresponds to the asymmetry yield per sterile neutrino that initially drops out of equilibrium through the Hubble expansion. Moreover, with its definition as in the present work, the parameter ε can be employed in the same way the usual vacuum decay asymmetry is used in standard calculations on Leptogenesis [8,9,50]. We have exemplified this point by explicitly calculating the freeze-out lepton asymmetry in a phenomenological see-saw model that is consistent with the neutrino mixing and oscillation data.
We can draw the conclusion that the approximation proposed in Ref. [28], which leads to the late-time asymmetries that we derive and study here, is applicable for Leptogenesis calculations in the strong washout regime of the single-flavour model, unless the CP asymmetry and the mass splitting are very small simultaneously, cf. Eqs. (25,26,B1) and relation (27). For the phenomenological model with two sterile neutrinos that is consistent with the oscillations of active neutrinos, we find that the late-time asymmetries always lead to a good approximation for the freeze-out values of the lepton number densities. One potential caveat is that the early-time evolution of ε(z) may strongly affect the asymmetry present within spectator fields, that in turn can have a substantial impact on the freeze-out lepton asymmetry [52]. It should also be noted, while the strong washout approximation always applies for resonant Leptogenesis with two sterile neutrinos, this does not need not to be the case when more of these are present. When the use of the late-time decay asymmetry cannot be justified, one should simply replace it with the time-dependent decay asymmetry (21) that is based on numerical solutions for the mixing and the oscillations of the sterile neutrinos. Methods for obtaining accurate quantitative results for Leptogenesis in the strong washout regime are therefore available throughout parameter space. and Γ by whereK = (K 1 + K 2 )/2 and X can in the single flavour case be identified with the parameter defined in Eq. (A1).
To obtain the solutions, a matrix Ξ is defined, similarly to the one in Ref. [24]: The solution (A1) can now be rewritten as: where the matrices U and V diagonalise Ξ and Ξ * . The corresponding eigenvalues Ξ D are: where ∆ K = (K 1 − K 2 )/(2K), which is zero in the democratic case. We define γ and ω as the real and imaginary parts of the above root.
The transformation matrix U is then given by: In the case of a symmetric matrix Ξ, if c is chosen such that det(U) = 1, the matrix inverse can be calculated as U −1 = U T , and there is also the relation V = U * . Rewriting Eq. (15) in terms of Ξ and Ξ * , we can easily obtain the eigenmatrices: where D = (y 2 1 − y 2 2 ) 2Γ2 + 4y 2 1 y 2 2Γ 2 cos 2 ϕ + 2i∆(y 2 1 − y 2 2 )Γ − ∆ 2 . In order to compare the magnitude of the individual eigenvalues, we define in addition and in analogy with the single-flavour model the parameter

C Washout Strength in Resonant Leptogenesis with Two Sterile Neutrinos
As for the equilibration of the sterile neutrinos, we note that which can be inferred by substituting the observed neutrino masses (with m 1 = 0) and mixing angles [47] into Eq. (B5). Using the relations (36)  which takes for Im[̺] = 0.87 its minimum value 0.31. Therefore, the e flavour will always equilibrate sufficiently long before freeze out at z f = O(10).