{'fraction_non_alphanumeric': 0.04280776389258176, 'fraction_numerical': 0.02242311441992378, 'mean_word_length': 4.234632335884945, 'pattern_counts': {'":': 0, '<': 14, '<?xml version=': 0, '>': 9, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Lepton Flavour Violation and theta(13) in Minimal Resonant Leptogenesis

We study the impact of minimal non-supersymmetric models of resonant leptogenesis on charged lepton flavour violation and the neutrino mixing angle theta(13). Possible low-scale flavour realisations of resonant tau-, mu- and e-leptogenesis provide very distinct and predictive frameworks to explain the observed baryon asymmetry in the Universe by sphaleron conversion of an individual tau-, mu- and e-lepton-number asymmetry which gets resonantly enhanced via out-of-equilibrium decays of nearly degenerate heavy Majorana neutrinos. Based on approximate flavour symmetries, we construct viable scenarios of resonant tau-, mu- and e-leptogenesis compatible with universal right-handed neutrino masses at the GUT scale, where the required heavy-neutrino mass splittings are generated radiatively. The heavy Majorana neutrinos in such scenarios can be as light as 100 GeV and their couplings to two of the charged leptons may be large. In particular, we explicitly demonstrate the compelling role that the three heavy Majorana neutrinos play, in order to obtain successful leptogenesis and experimentally testable rates for lepton flavour violating processes, such as mu -->e gamma and mu -->e conversion in nuclei.


Introduction
The observed baryon asymmetry in the Universe (BAU), which amounts to a baryon-tophoton ratio of number densities η B ≈ 6.2 ×10 −10 [1,2], provides one of the strongest pieces of evidence for physics beyond the Standard Model (SM) [3]. One interesting scenario for explaining the BAU is leptogenesis [4]. Leptogenesis does have a profound link to neutrinos and the origin of their extraordinary small masses. In particular, the famous seesaw mechanism [5] can give rise to their small observed masses through the presence of superheavy can decay out of equilibrium and produce a net leptonic asymmetry. The so-produced leptonic asymmetry gets rapidly reprocessed into the observed BAU [4] by equilibrated (B + L)-violating sphaleron interactions [6].
A potentially interesting alternative to GUT-scale leptogenesis is the framework of low-scale resonant leptogenesis (RL) [7,8]. Within this framework, the lowering of the scale may rely on a dynamical mechanism, in which heavy-neutrino self-energy effects [9] on the leptonic asymmetry become dominant [10] and get resonantly enhanced [7], when a pair of heavy Majorana neutrinos has a mass difference comparable to the heavy neutrino decay widths. As a consequence of thermal RL, the heavy Majorana mass scale can be as low as ∼ 100 GeV, while maintaining agreement with the solar and atmospheric neutrino data [8].
One of the advantages of low-scale RL is that the reheating temperature T reh resulting from inflaton decays does not need to be very high, e.g. T reh ∼ 1-10 TeV [11], thereby avoiding comfortably the overproduction of gravitinos in supersymmetric models whose late decays may cause dissociation of the light elements during the nucleosynthesis era [12].
Flavour effects due to heavy-neutrino Yukawa couplings play an important role in models of RL [13,14]. In particular, in [13] a scenario was put forward, called resonant τ -leptogenesis (Rτ L), in which the BAU originates from a τ -lepton asymmetry, resonantly produced by quasi-in-equilibrium decays of heavy Majorana neutrinos. This mechanism makes use of the property that sphalerons preserve the individual quantum numbers 1 3 B − L e,µ,τ [15][16][17][18]. In a Rτ L model, the generated excess in the L τ number will be converted into the observed BAU, provided the L τ -violating reactions are not strong enough to wash out such an excess. In such a scenario, the heavy Majorana neutrinos can be as light as 100 GeV and have sizeable couplings to two of the charged leptons, specifically to electrons and muons. Consequently, depending on the flavour dynamics of heavy neutrino Yukawa coupling effects, phenomenologically testable models of RL can be built that could be probed at the LHC or in low energy experiments of lepton number violation (LNV) and lepton flavour violation (LFV). For instance, observables of particular interest are the neutrinoless double-beta (0νββ) decay of heavy nuclei [19], the photonic decay µ → eγ analyzed by the MEG experiment [20] and the coherent µ → e conversion in nuclei to be looked for in the planned COMET/PRISM experiment [21].
In this paper we study all possible alternatives to Rτ L, including the minimal models of resonant µ-leptogenesis (RµL) and resonant e-leptogenesis (ReL). Collectively, we refer to these three different lepton-flavour realisations of RL as RℓL. We assume that the RℓL models have an SO(3)-symmetric heavy neutrino mass spectrum at the GUT scale, with all heavy Majorana neutrinos being exactly degenerate at this scale. We consider a minimal non-supersymmetric framework, in which the heavy-neutrino mass splittings required for successful RL are generated radiatively [22] and can therefore be naturally comparable to the decay widths of the heavy neutrinos. Since all charged-lepton Yukawa couplings are in thermal equilibrium at temperatures T < ∼ 10 TeV [23], we consider a flavour diagonal basis for these couplings, while setting up the Boltzmann equations (BEs). In addition, we include the flavour effects due to individual heavy-neutrino Yukawa couplings [13,24], which can have a dramatic impact on the predictions for the BAU in RL models [13,14].
The layout of the paper is as follows. Section 2 describes the basic structure of the minimal SM with three heavy Majorana neutrinos and introduces the flavour symmetries needed to realize the different lepton-flavour scenarios associated with RℓL. As mentioned above, we assume a SO(3) symmetric heavy-neutrino sector at the GUT scale and calculate the renormalization-group (RG) effects on the mass spectrum of the electroweak-scale heavy Majorana neutrinos and their Yukawa couplings to charged leptons. Taking the lightneutrino oscillation data into account, we are able to determine most of the theoretical parameters of the RℓL models. In Section 3, we present analytic results and predictions for LFV observables in the three different RℓL models. Section 4 briefly reviews the basic framework of RL and presents the BEs, upon which our numerical analysis is based. We also clarify the necessity of having at least three heavy Majorana neutrinos in RL models in order to obtain experimentally testable LFV. Section 5 presents numerical estimates of representative RℓL models and their impact on the neutrino mixing angle θ 13 . Finally, Section 6 summarizes our conclusions.

Flavour Models of Minimal Resonant Leptogenesis
In this section, we describe the basic theoretical framework underlying the different flavour models of minimal RL. In particular, our interest is in scenarios in which the BAU is generated by the production of an individual lepton number [13]. For definiteness, we first consider a minimal model for Rτ L in Section 2.1, and then generalize to the other two cases RµL and ReL in Sections 2.2 and 2.3, respectively.
The leptonic Yukawa and Majorana sectors of the SM symmetrically extended with one singlet right-handed neutrino ν iR per i family (with i = 1, 2, 3 = e, µ, τ ) are given by the Lagrangian where Φ is the SM Higgs doublet andΦ = iτ 2 Φ * its isospin conjugate. Moreover, we have suppressed the generation index i from the left-handed doublets L i = (ν iL , l iL ) T , the right-handed charged leptons l iR and the right-handed neutrinos ν iR , while ordinary multiplication between vectors and matrices is implied * .
To obtain a phenomenologically relevant model in this minimal setup, at least 3 singlet heavy Majorana neutrinos ν 1,2,3 R are needed and these have to be nearly degenerate in mass [13]. To ensure the latter, we assume that to leading order, the singlet Majorana where 1 3 is the 3 ×3 identity matrix and ∆m M is a general SO(3)-breaking matrix induced by RG effects. As we will discuss below, compatibility with the observed light neutrino masses and mixings requires that (∆m M ) ij /m N < ∼ 10 −7 , for electroweak-mass Majorana neutrinos, i.e. for m N ≈ 0.1-1 TeV. We will explicitly demonstrate, how such an SO(3)breaking matrix ∆m M , of the required order, can be generated radiatively via the RG evolution of the right-handed neutrino mass matrix m M from the GUT scale M X ≈ 2 × 10 16 GeV to the mass scale of the right-handed neutrinos m N .
To one-loop order, the RG equations governing the 3 × 3 matrices of the neutrino Yukawa couplings h ν , the charged-lepton Yukawa couplings h ℓ and the singlet Majorana * Occasionally, we will also denote the individual lepton numbers with L e,µ,τ , but hopefully the precise meaning of L e,µ,τ can be easily inferred from the context, without causing confusion.
neutrino masses m M are given by [25] dh ν dt = 1 16π 2 where t = ln(M X /µ) and µ is the RG evolution scale. Moreover, g 1 and g 2 are the gauge couplings of the U(1) Y and SU(2) L gauge groups, respectively, and T is the shorthand notation for the trace Here, h u and h d are the 3×3 matrices for the up-and down-type Yukawa couplings, respectively. At the GUT scale M X , we impose the universal boundary condition: m M (M X ) = m N 1 3 . The corresponding boundary values for the neutrino Yukawa couplings h ν depend on the particular model of RℓL, which we now discuss in detail.

Resonant τ -Leptogenesis
In the physical charged-lepton mass basis, the SO(3) symmetry imposed on the singlet Majorana sector at the GUT scale M X gets explicitly broken by a set of neutrino Yukawa couplings to the subgroup of lepton symmetries: U(1) Le+Lµ × U(1) Lτ . The flavour charge assignments that give rise to such a breaking are presented in Table 1.
As a consequence of the U(1) Le+Lµ × U(1) Lτ symmetry, the neutrino Yukawa coupling matrix takes on the general form: and δh ν vanishes, if the symmetry is exact. In this symmetric limit, the light neutrinos remain massless to all orders in perturbation theory, whilst a and b are free unconstrained complex parameters. The phases accompanying these parameters in (2.7) are simply chosen for convenience to maximize the lepton asymmetry in leptogenesis (see Section 4.1) when a and b are real. In order to give masses to the light neutrinos, the following minimal departure δh ν from the flavour symmetric limit is considered: where |ǫ e,µ,τ |, κ 1,2 ≪ |a|, |b| and the phases γ 1,2 are unrestricted. A more precise determination of the range of parameter values can be obtained from the low-energy neutrino data and successful leptogenesis.
It is important to notice that the flavour structure of the neutrino Yukawa couplings h ν is preserved through the RG evolution, as long as δh ν remains a small perturbation. In detail, RG effects violate the SO(3)-invariant form of the Majorana mass matrix m M (M X ) = m N 1 3 by the 3-by-3 matrix ∆m M . In the leading-log approximation, the SO(3)-breaking matrix ∆m M reads: withγ 1,2 = γ 1,2 + π 4 . Correspondingly, the neutrino and charged-lepton Yukawa couplings modify via RG running from M X to m N as follows: where U and U ′ stand for the shorthand expressions We may now determine the Yukawa parameters (a, b, ǫ e , ǫ µ , ǫ τ ), in terms of the lightneutrino mass matrix m ν in the positive and diagonal charged lepton Yukawa basis. To do so, we first notice that the chosen symmetry U(1) Le+Lµ × U(1) Lτ is sufficient to ensure the vanishing of the light neutrino mass matrix m ν . In fact, if it is an exact symmetry of the theory, the light neutrino mass matrix will vanish to all orders in perturbation theory [26].
To leading order in the symmetry-breaking parameters ∆m M and δh ν , the tree-level light neutrino mass matrix m ν is given by where v = 2M W /g w = 245 GeV is the vacuum expectation value of the SM Higgs field Φ.
In deriving the last equation in (2.13), we have also assumed that ∆m N m N κ 1,2 ≪ ǫ e,µ,τ , where ∆m N stands for the expression As a consequence of the flavour symmetry U(1) Le+Lµ × U(1) Lτ , the symmetry-violating parameters ǫ e,µ,τ and κ 1,2 enter the tree-level light neutrino mass matrix m ν in (2.13) quadratically. This in turn implies that for electroweak-scale heavy neutrinos m N ∼ v, the symmetry-breaking Yukawa couplings δh ν in (2.8) need not be much smaller than the electron Yukawa coupling h e ∼ 10 −6 . Moreover, one should observe that only a particular combination of SO(3)-violating terms (∆m M ) ij appears in m ν through ∆m N . Nevertheless, for electroweak-scale heavy neutrinos with mass differences |∆m N |/m N < ∼ 10 −7 , one should have |a|, |b| < ∼ 10 −2 to avoid getting too large light neutrino masses much above 0.5 eV.
Given the analytic form (2.13) of the light neutrino mass matrix, we may directly compute the neutrino Yukawa couplings (a, b, ǫ e , ǫ µ , ǫ τ ), as functions of m N , the phenomenologically constrained neutrino mass matrix m ν and the symmetry-breaking parameters κ 1,2 and γ 1,2 : Since the approximate light-neutrino mass matrix m ν in (2.13) has rank 2, the lightest neutrino mass eigenstate ν 1 will be massless in this approximation. The relations given in (2.15) will be used to obtain numerical estimates of the BAU, in terms of m N and the symmetry-breaking parameters κ 1,2 and γ 1,2 , for both normal and inverted hierarchy scenarios of light neutrinos. In the following, we discuss the two remaining flavour variants of RL: RµL and ReL.

Resonant e-Leptogenesis
A third possible flavour scenario pertinent to ReL is realized by the symmetry-breaking where the flavour charge assignments are given in Table 3. In ReL, the neutrino Yukawa coupling h ν has the structure: and the breaking terms δh ν of the U(1) Lµ+Lτ × U(1) Le flavour symmetry are In close analogy with the previous two scenarios of Rτ L and RµL, the light neutrino mass matrix in ReL is given by where ∆m N retains its analytic form of (2.14). By analogy, we may derive in ReL the relations of the neutrino Yukawa coupling parameters (a, b, ǫ e , ǫ µ , ǫ τ ) to m ν , m N , κ 1,2 and γ 1,2 .
These are given by

Low Energy Observables
Low-energy neutrino data provide indisputable evidence for neutrino oscillations. We use these data to determine and constrain the fundamental parameters of the theory. We also present the full range of predictions for the half life of neutrinoless double beta decay of a heavy nucleus, within the different flavour scenarios of RL. In the same context, we present analytic results and estimates for LFV observables, such as µ → eγ and µ → e conversion in nuclei.

Light Neutrino Oscillation Data
The interpretation of the experimental results on solar neutrinos suggests that ν e → ν µ,τ oscillations are mainly driven by the mass squared difference ∆m 2 12 = m 2 ν 2 − m 2 ν 1 , while the corresponding experimental data on atmospheric neutrinos are interpreted by ν µ → ν τ oscillations driven by ∆m 2 23 = m 2 ν 3 −m 2 ν 2 , within a minimal scheme of three active neutrinos. For the present analysis, we use the global fits performed in [27], even though global fits of other groups give compatible results [28]. The best fit values for the light neutrino masses and mixings, including their uncertainties at the 2σ level, are given by where the sign of ∆m 2 13 ≡ ∆m 2 12 + ∆m 2 23 remains still undetermined, corresponding to the cases of the so-called normal (∆m 2 13 > 0) and inverted (∆m 2 13 < 0) neutrino mass hierarchy, respectively. As outlined in Section 2, we use the oscillation parameters as input to reduce the number of free parameters in the neutrino Yukawa coupling matrix h ν .

0νββ Decay
Models that include Majorana neutrinos violate the L-number and so can give rise to neutrinoless double beta decay (0νββ) of a heavy nucleus A Z X, where two single β decays [29][30][31] can occur simultaneously in one nucleus, A Z X → A Z+2 X + 2e − . The measurement of the half life of this decay provides further information on the structure of the light neutrino mass matrix m ν . The half life T 0νββ 1/2 for a 0νββ decay mediated by light Majorana neutrinos is given by where m denotes the effective Majorana neutrino mass, m e is the electron mass and M 0νββ and G 01 are the nuclear matrix element and the phase space factor of the decay, respectively. The effective neutrino mass m is given by the entry {11} ≡ {ee} of the light neutrino mass matrix m ν , which can be expressed as where U ν is the PMNS neutrino-mixing matrix [32]. As described in Section 2, RℓL models realise a light neutrino mass spectrum with a vanishing lightest neutrino mass, m ν 1 = 0, and either a normal or inverted mass hierarchy. The prediction for the effective Majorana neutrino mass for these two scenarios is given by The above prediction takes into account the uncertainty of the observed oscillation parameters at the 2σ level and the variation of the unknown Dirac and Majorana phases. These predictions are to be compared to the experimental bound T 0νββ 1/2 > 1.9 × 10 25 years in the isotope 76 Ge reported by the Heidelberg-Moscow collaboration [33], implying an upper limit on m in the range: Here, the main uncertainty is due to the choice for the nuclear matrix element that occurs in (3.2). Future 0νββ-decay experiments are expected to probe m to sensitivities of order 10 −2 [19] and so fall within the range given in (3.4) to validate the mass scenario of inverted hierarchy.
For example, in the Rτ L model, the parameters |Ω l 1 l 2 | 2 are, to a good approximation, given by (3.14) Because of the relations (2.15), a and b are approximately inversely proportional to κ 1,2 , a, b ≈ 3 × 10 −7 κ −1 1,2 (in the case of a normal light neutrino mass hierarchy), and for a typical value of κ 1,2 ≈ 10 −4 , a and b are of the order a, b ≈ 3 × 10 −3 . Thus, in the Rτ L scenario, the following typical values for the branching ratios of the photonic LFV µ and τ decays are predicted: Consequently, only B(µ → eγ) is expected to be within reach of future experiments, whereas the LFV τ decays are far beyond the realm of detection. By analogy, the predictions in the Re(µ)L model are found to be Again, B(τ → µγ) and B(τ → eγ) are expected to be below current and future experimental sensitivities for parameter choices compatible with successful leptogenesis. While the prediction for B(µ → eγ) is far below the expected MEG sensitivity, the LFV µ → e transition rate might still be high enough to be testable at future experiments measuring µ → e conversion in nuclei (see next section). Because of the inverse proportionality between a, b and κ 1,2 the prediction for B(µ → eγ) is essentially independent of the choice for κ 1,2 . In Section 5, we present detailed numerical results for LFV µ-and τ -decays that confirm the validity of these simple estimates.

Coherent µ → e Conversion in Nuclei
One of the most sensitive probes of LFV is the coherent conversion of µ → e in nuclei [37,38].
The µ → e conversion rate in a nucleus with nucleon numbers (N, Z) is given by [37][38][39] where α em = 1/137 is the electromagnetic fine structure constant, Z eff is the effective atomic number and Γ capt is the muon nuclear capture rate. For 48 22 Ti, experimental measurements give Z eff ≈ 17.6 [40] and Γ capt ≈ 1.705 × 10 −18 GeV [41]. Moreover, |F (−m 2 µ )| ≈ 0.54 is the nuclear form factor [42] and is the weak matrix element, where In the case of 48 22 Ti, B µe (26,22) is then approximately related to B(µ → eγ) through for a right-handed neutrino mass scale m N ≈ 100 GeV. On the experimental side, the strongest upper bound is obtained from data on µ → e conversion in 48 22 Ti [43], Using the relation (3.21), we observe that this sensitivity is comparable to the current bound on B(µ → eγ). However, the proposed COMET and mu2e experiments, measuring µ → e conversion in 27 13 Al, are expected to be sensitive to conversion rates of order 10 −16 [44,45], thereby improving the sensitivity compared to the current limit by four orders of magnitude.

Leptogenesis
In this section, we briefly review the central results of the field-theoretic formalism for RL developed in [7,8] which will be used in our analysis. We then set up the BEs and present approximate solutions for the kinematic regime of interest to us. Finally, we clarify the necessity of introducing at least three right-handed neutrinos into the theory, in order to explain the BAU and obtain testable rates of LFV, such as B(µ → eγ) ∼ 10 −12 -10 −13 .

Leptonic Asymmetries
Within the framework of leptogenesis, a net non-zero leptonic asymmetry results from the CP-violating decays of the heavy Majorana neutrinos N α into the left-handed charged leptons l − L and light neutrinos ν lL . Consequently, we have to calculate the partial decay width of the heavy Majorana neutrino N α into a particular lepton flavour l, For temperatures above the electroweak phase transition, the SM VEV vanishes and only the would-be Goldstone and Higgs modes will predominantly contribute to Γ αl .
In RL models, resumming the absorptive parts of the heavy Majorana-neutrino selfenergy transitions N β → N α plays an important role in the computation of Γ αl [7,8]. In order to take this resummation consistently into account, we first introduce the leptonflavour dependent absorptive transition amplitude [46] A which represents the contribution of a single charged lepton and light neutrino flavour l running in the loop. Summing over all flavours l, we then get the total transition amplitude Note that the diagonal transition amplitude A αα is related to the tree-level decay width of the heavy Majorana neutrino N α through: Γ . Since all charged-lepton Yukawa couplings will be in thermal equilibrium in the lowscale leptogenesis scenarios of our interest, we consider the weak basis in which the matrices h ℓ and m M are both diagonal and positive. To account for unstable-particle mixing effects between the 3 heavy Majorana neutrinos, we follow [7,8] and define the resummed effective Yukawa couplings h ν lα and their CP-conjugate ones h νC lα related to the vertices LΦN α and L CΦ * N α , respectively. The resummed neutrino Yukawa couplings h ν lα are given by [8,46] where |ε αβγ | is the modulus of the usual Levi-Civita anti-symmetric tensor (ε 123 = 1), The respective CP-conjugate effective Yukawa couplings h νC lα (h ν ) are obtained from (4.4) by replacing the tree-level couplings h ν lα with their complex conjugates h ν * lα : In our calculations, we neglect the 1-loop corrections to the proper vertices LΦN α , whose absorptive parts are numerically insignificant in RL. In terms of the absorptive transition amplitudes A l αβ given in (4.2), the partial decay widths Γ αl and their CP-conjugates Γ C αl may now be expressed in the compact form: where we have explicitly indicated the dependence of the absorptive transition amplitudes on h ν and h νC . Given the analytic expressions (4.7), it is straightforward to compute the leptonic asymmetries for each individual lepton flavour: The analytic results for the leptonic asymmetries δ αl simplify considerably in the 2-heavy neutrino mixing limit, in which R αβ defined in (4.5) is set to zero. In this limit, δ αl are given by [7,8] where α, β = 1, 2 and Γ (0) Na is the tree-level decay width of the heavy Majorana neutrino N α , given after (4.3). Based on the simplified expression (4.9), the following two conditions for having resonantly enhanced leptonic asymmetries δ αl ∼ O(1) may be derived [7]: Note that the first resonant condition (i) is exactly met, when the unitarity limit on the resummed heavy-neutrino propagator gets saturated [47], i.e. when the regulating expression in (4.9), takes its maximal possible value: f reg = 1. Within our RL scenarios, the first condition in (4.10) is naturally fulfilled as the heavy-neutrino mass splittings are generated via RG effects and are of the required order. The second condition is crucial as well and controls the size of the leptonic asymmetries. As we will see below, the condition (ii) in (4.10) has a non-trivial impact on approximate L-conserving RL models.

Comparison with Other Methods
It is now worth commenting on some of the attempts made in the literature [48][49][50] to calculate the resonant part of the leptonic asymmetries δ αl . Their results differ by the way in which the singularity m N 2 → m N 1 occurring in the denominator of the second fraction in (4.9) gets regulated, when heavy-neutrino width effects are taken into consideration.
Specifically, the various approaches differ in their derivations for the analytic form of f reg given in (4.12). For instance, the authors of [48] use a perturbative quantum-mechanical approach to obtain a regulator of the form: 13) where m N = 1 2 (m N 1 +m N 2 ) and ∆m N = |m N 1 −m N 2 |. It is easy to observe that for scenarios, for which Re(h ν † h ν ) 12 = 0, but Re(h ν † l1 h ν l2 ) = 0, the unitarity upper bound given in (4.12) gets violated, in the degenerate heavy-neutrino mass limit m N 2 → m N 1 . In particular, in the same limit, the individual lepton-flavour asymmetries δ 1l (with l = e, µ, τ ) become singular.
Although this singularity disappears when the lepton-flavour sum δ N 1,2 = l=e,µ,τ δ 1,2l is taken, the regulator (4.13) will still be inapplicable to lepton-flavour RL scenarios, for which Based on a modified version of the field-theoretic approach introduced in [7,8], the authors of [49,50] obtain a different regulating expression for the leptonic asymmetry δ 1l : (4.14) It is not difficult to observe that f reg diverges as (m N 1 − m N 2 ) −1 in RL scenarios, for which (h ν † h ν ) 11 = (h ν † h ν ) 22 , even though one could still have (h ν † l1 h ν l1 ) = (h ν † l2 h ν l2 ) for each single lepton flavour l. For instance, such a situation can naturally occur in approximate leptonnumber conserving models of RL discussed recently in [51,52]. Evidently, the prediction for the leptonic asymmetries δ N 1,2 in such scenarios may get overestimated by many orders of magnitude.
In order to illustrate this last point, let us consider a one-generation inverse seesaw model [53] with two singlet neutrinos of opposite lepton number, ν R and S L . The neutrino sector of this model is described by the lepton-number-conserving Lagrangian approach. This gives rise to dismally small lepton asymmetries: Notice that the lepton asymmetries δ 1,2l are proportional to µ 2 in agreement with the NW theorem and, as they should, both vanish identically in the L-conserving limit µ → 0.
As we will see in the next section, lepton asymmetries of order 10 −15 will fall short by at least 7 orders of magnitude to explain the BAU. Instead, had one used the regulator f reg in (4.14), one would have obtained the enormous enhancement: f reg ≈ Γ N 1,2 /µ ∼ 10 5 ≫ 1, leading to lepton asymmetries of order [51]: δ 1,2l ∼ Γ N 1,2 /m N ∼ 10 −5 . However, this result is independent of the L-conserving parameter µ = µ L = −iµ R , and taking the Lconserving limit µ → 0, one obtains a non-zero lepton asymmetry, which clearly signifies an erroneous result. The above exercise shows that one has to go beyond the two-heavy neutrino mixing framework and consider non-trivial lepton flavour effects [13], in order to obtain a phenomenologically relevant model that predicts testable rates for LFV.

Baryon Asymmetry
In this section we present the relevant Boltzmann equations (BEs) which will be used to evaluate the heavy-Majorana-neutrino-, the lepton-and the baryon-number densities, η N 1,2,3 = n N 1,2,3 /n γ , η Le,µ,τ = n Le,µ,τ /n γ and η B = n B /n γ , normalised to the photon number density n γ . In our computations, we include the dominant collision terms related to the 1 → 2 decays of the right-handed neutrinos and to resonant 2 → 2 lepton scatterings that describe ∆L = 2 and ∆L = 0 transitions. We neglect chemical potential contributions from the right-handed charged leptons and quarks.
In order to appropriately take into account the flavour effects on the BEs for η N 1,2,3 and η Le,µ,τ , we closely follow the approach and the notation established in [14]. More explicitly, the BEs are found to be in terms of which the photon number density is given by where K n (x) is the nth-order modified Bessel function [55].
The BEs (4.17) and (4.18) include the collision terms for the decays N α → L l Φ, as well as the ∆L = 0, 2 resonant scatterings: L k Φ ↔ L l Φ and L k Φ ↔ L C l Φ † , which are defined as [14] γ Nα LΦ ≡ k=e,µ,τ Since we are only interested in the resonant part of the above 2 → 2 scatterings, we make use of the narrow width approximation (NWA) for the resummed heavy-neutrino propagators.
Thus, at the amplitude squared level, we employ the NWA for the complex-conjugate product of Breit-Wigner propagators [56]: .
The collision term pertinent to the heavy-Majorana-neutrino decays γ Nα LΦ is given by whilst the corresponding collision terms for the ∆L = 0, 2 lepton-flavour transitions are calculated by means of the NWA (4.23) to be Separating the diagonal α = β RIS contributions from the off-diagonal α = β terms in the sum, we may rewrite the BE (4.18) in the form [14] dη RIS denote the RIS-subtracted collision terms and B αl are the branching ratios (4.28) The collision terms proportional to η L k and to δ αl δ αk on the RHS of (4.27) turn out to be numerically negligible for the RℓL scenarios under consideration. Instead, the RIS-subtracted collision terms proportional to η L l in (4.27) become significant. Their importance in RL models was originally raised in [8,14] and confirmed most recently in [51].
The next step is to include the effect of the (B + L)-violating sphaleron processes [6].
For temperatures T larger than the critical temperature T c ≈ 135 GeV of the electroweak phase transition, the conversion of the total lepton-to-photon ratio l=e,µ,τ η L l to the baryon-to-photon ratio η c B at T c is given by the relation: For T < T c , the so-generated baryon asymmetry η c B gets diluted by standard photon interactions until the recombination epoch, leading to the BAU It is instructive to derive approximate numerical solutions to the BEs (4.17) and (4.27).
initially ignore the RIS subtracted collision terms. But, as we will see below, their inclusion is straightforward. With these approximations and simplifications, the BEs may be recast into the compact form: independently of the initial conditions (see Figure 1). In this regime, the BE (4.33) becomes with δ l = 3 α=1 δ αl and K l = 3 α=1 K α B αl . We may include the numerically significant RIS-subtracted collision terms proportional to η L l in (4.27), by rescaling K l → κ l K l ≡ K eff l , where In determining the scaling factor κ l , we have assumed that η L l ≫ η L k =l in (4.18), which is a valid approximation within a given RℓL scenario under study. Note that if only the diagonal α = β terms representing the RIS contributions are considered in the sum, κ l reaches its maximum value, i.e. κ l = 1 + O(δ 2 l ). We also have checked that in the L l -conserving limit of the theory, the parameter κ l vanishes, as it should.
As is illustrated in Figure 1, the solution η L l to (4.35) exhibits different behaviour in the three kinematic regimes, characterized by the specific values of the parameter z = m N /T : (4.37) For z values in the range z l 2 < z < z l 3 , the solution η L l may well be approximated by For z > z l 3 , the lepton number density η L l freezes out and approaches the constant value η L l = (3δ l )/(2K eff l z l 3 ) ‡ . The general behaviour of η L l in the different regimes is displayed in Figure 1.
In this paper we only consider RℓL scenarios, for which the washout is strong enough, such that the critical temperature z c = m N /T c where the baryon asymmetry η B decouples ‡ The onset of the freeze-out is defined as the position z l 3 where the relative slope of (4.35) drops below 1, i.e. when |η ′ L l /η L l | = 2/3(z l 3 ) 3 K 1 (z l 3 )K eff l = 1. The solution to this equation can be analytically expressed in terms of the Lambert W function, to which z l 3 in (4.37) proves to be an excellent interpolating approximation over the wide range of values K eff l ≈ 2 − 10 10 .
from the lepton asymmetries η L l is situated in the linear drop-off or constant regime.
Specifically, we require that for all heavy neutrino species N α = N 1,2,3 and lepton flavours l = e, µ, τ . As a consequence, the baryon asymmetry η B becomes relatively independent of the initial values of η L l and η Nα . In this case, taking into account all factors in (4.29), (4.30) and (4.38), the resulting BAU is estimated to be .

(4.40)
We note that the formula (4.40) provides a fairly good estimate of the BAU η B to less than 20%, in the applicable regime of approximations given by (4.39) for K eff l > ∼ 5 for a right-handed neutrino mass scale of the order of the EW scale. Hence, to account for the observed BAU η obs B given in (4.31), lepton asymmetries δ l > ∼ 10 −7 are required. In the next section, we present numerical estimates of the BAU for RℓL scenarios, based on the simplified BEs (4.32) and (4.35), with K l replaced by K eff l as defined by means of (4.36).

Numerical Results
In this section we present numerical estimates of the BAU η B and the low-energy LFV observables for µ → e and τ → (e, µ) transitions, based on the analytic results derived in Sections 3 and 4. Our aim is to delineate the viable parameter space of the RℓL models: Rτ L, RµL and ReL, by considering both cases of a normal and inverted hierarchy for the light neutrino mass spectrum. In all the RℓL scenarios under study, the lightest neutrino is massless. As described in Section 2, we use the neutrino oscillation data to determine the theoretical parameters of the light neutrino mass matrix m ν . Specifically, we invert the seesaw formula and solve for the neutrino Yukawa couplings a, b and ǫ e,µ,τ . In addition,  of Re(a) were known. There is also a similar freedom for an overall sign in the ǫ e,µ,τ parameters, but this turns out to be irrelevant, since it applies to a complete column in the neutrino Yukawa matrix h ν and can be rotated away. In addition to the light neutrino masses and mixings, the light neutrino mass matrix may also contain the CP-violating Dirac phase δ and the Majorana phases φ 1,2 . As our ansatz for the neutrino Yukawa matrix h ν uses maximal CP phases for a and b, the inclusion of the light neutrino CP phases is not expected to increase or change significantly our predictions for the baryon asymmetry η B . We therefore consider only the extreme cases of CP parities, δ, φ 1,2 = 0, π, as natural choices which can help us to reduce the dimensionality of the parameter space.
To start with, we show in Figure 2 numerical estimates of the baryon asymmetry η B and the LFV observables B(µ → eγ) and B µe ( 48 22 Ti), as functions of the Yukawa-coupling parameters κ 1 and κ 2 , in a Rτ L model with m N = 120 GeV and a normal hierarchical light neutrino mass spectrum. The neutrino mixing angle sin 2 θ 13 is set to its upper experimental 2σ limit [27]: sin 2 θ 13 = 0.033. The remaining theoretical parameters are chosen to maximise the overlap between successful generation of the required baryon asymmetry and high LFV rates. For definiteness, we set γ 1 = 3π/8, γ 2 = 1/2π (left panel) and assumed approximation κ 1,2 ≪ a, b, required to invert the seesaw formula. The degree of violation of this approximation is indicated by the green shaded areas in Figure 2. These green shaded areas denote the parameter space, in which the quantity is bigger than 0.25 (dark green) and 0.10 (light green), respectively. The quantity δm κ ν is a measure that quantifies the accuracy of our analytic approximation for neglecting the contribution of the neutrino Yukawa couplings κ 1,2 in the light neutrino mass matrix m ν [cf. (2.13)]. Hence, the values of δm κ ν = 0.10 and 0.25 mean that the inversion of the light neutrino mass matrix is accurate at the 10% and 25% level, respectively, due to the assumed absence of the κ 1,2 terms. It should be stressed that the corresponding parameter space is not ruled out; larger values of κ 1,2 5 × 10 −4 would require a numerical approach to invert the light neutrino mass matrix m ν , beyond our analytic approximation. In Figure 3 we display numerical estimates of η B and the LFV observables B(µ → eγ) and B µe ( 48 22 Ti), as functions of κ 1 and κ 2 , in a Rτ L model with m N = 120 GeV and an inverted hierarchical light neutrino mass spectrum, characterised by ∆m 2 13 < 0. As before, we set the neutrino mixing angle sin 2 θ 13 to its upper experimental 2σ limit [27]: sin 2 θ 13 = 0.033. As can be seen from Figure 3, the regions of the parameter space that yield successful baryon asymmetry η B are smaller than those found in the normal hierarchical case for the light neutrino mass spectrum. If such a scenario gets realized in nature, then successful leptogenesis implies rates for B µe ( 48 22 Ti) < ∼ 10 −16 , which can still be within reach of the projected PRISM experiment [21].
The higher rates for the LFV µ → e transitions in a Rτ L model with an inverted hierarchical light-neutrino mass spectrum may be attributed to the fact that the squared  Figure 7: The same as in Figure 6, but for an inverted light neutrino mass spectrum and the following choice of parameters: parameter a 2 is proportional to the not yet well determined neutrino-mixing angle sin 2 θ 13 .
In detail, the branching ratio B(µ → eγ) is enhanced in the case of an inverted hierarchical light neutrino mass spectrum (∆m 2 13 < 0). Instead, in the normal hierarchical light-neutrino scenario with ∆m 2 13 > 0, B(µ → eγ) is essentially independent of sin θ 13 . This is demonstrated in Figure 4, showing B(µ → eγ) as a function of sin 2 θ 13 , where κ 1 = 10 −5 and κ 2 = 10 −4 . In the same figure, the vertical solid and dashed lines denote the current best fit value and the expected sensitivity of future experiments for measuring sin 2 θ 13 , respectively. The branching ratio B(µ → eγ) depends linearly on sin 2 θ 13 in the inverted hierarchical light-neutrino scenario where ∆m 2 13 < 0. Unlike in the normal lightneutrino scenario, the predicted value for η B in the Rτ L model with an inverted hierarchical light neutrino mass spectrum falls short of explaining the BAU by two orders of magnitude, for a potentially observable branching ratio of B(µ → eγ) ∼ 10 −13 .
In Figure 5 we illustrate the importance of including the full three-heavy-neutrino mixing in the calculation of the baryon asymmetry. We present a comparison between the full calculation based on using the effective Yukawa couplings h ν lα given in (4.4) (solid line) and its two-heavy-neutrino mixing approximation (dashed line) where we set R αβ = 0.
Note that the two-heavy-neutrino mixing approximation can differ from the full calculation by up to one order of magnitude. In addition, Figure 5 shows the baryon asymmetry η B calculated by omitting the RIS-subtracted collision terms in the BE (4.27) (dotted line), or equivalently by taking the parameter κ τ defined in (4.36) equal to 1 in the BE (4.33).
Such a simplification may reduce the predicted values for η B by as much as 60%. Since the current experimental sensitivity to this process is B exp (τ → eγ) ≈ 10 −7 which is  Figure 9: The same as in Figure 8, but for an inverted light neutrino mass spectrum and the following choice of parameters: not expected to increase by more than one order of magnitude in the foreseeable future, it would be difficult to probe the parameter space of the RµL scenario compatible with observable BAU. On the other hand, the µ → e transitions in the RµL model, although being proportional to max(κ 1 , κ 2 ) 2 a 2 and so smaller than the predictions obtained in Rτ L scenario, are still sizeable enough to produce a µ → e conversion rate of B µe ( 48 22 Ti) ≈ 2 × 10 −17 (for ∆m 2 13 > 0) and 5 × 10 −16 (for ∆m 2 13 < 0). As the parameters a and b are approximately inversely proportional to κ 1,2 , these values are largely independent of κ 1,2 and apply to the whole (κ 1 , κ 2 ) parameter plane, as depicted in Figures 6 and 7. Consequently, this scenario could be probed at a future µ → e conversion experiment with a sensitivity of ∼ 10 −16 (COMET, mu2e), or ∼ 10 −18 (PRISM) [21].
Finally, Figures 8 and 9 display numerical estimates, for a ReL scenario with normal and inverted light neutrino mass spectra, respectively. Our results are quite analogous to the RµL case. Correspondingly, the largest LFV rate is obtained for the process τ → µγ.

Conclusions
We have analyzed minimal low-scale seesaw scenarios of resonant leptogenesis and studied their potential implications for observables of charged LFV, such as µ → eγ and µ → e conversion in nuclei. We have considered three physically interesting flavour realisations of resonant leptogenesis, where the observed BAU originates from an individual τ -, µor e-lepton-number asymmetry which gets resonantly enhanced via the out-of-equilibrium decays of nearly degenerate heavy Majorana neutrinos.
By means of approximate lepton-flavour symmetries, we have been able to construct viable and natural models of RℓL compatible with universal right-handed neutrino masses at the GUT scale, where the required heavy-neutrino mass splittings are generated via RG effects. Particular attention has been paid that the effective resummation method introduced in [7] and used in our study to compute the resonantly enhanced lepton asymmetries respects the Nanopoulos-Weinberg no-go theorem [54] in the L-conserving limit of the theory. Specifically, we have checked that the leptonic asymmetries δ αl given in (4.8) and (4.9) vanish in all parametrically possible L-conserving limits of the RℓL scenarios. In agreement with earlier studies [13,14], we find that at least three heavy Majorana neutrinos are required, in order to potentially have both successful leptogenesis and experimentally testable rates for LFV processes, such as µ → eγ and µ → e conversion in nuclei.
We have found that the heavy Majorana neutrinos in RℓL scenarios can be as light as 100 GeV, whilst their couplings to two of the charged leptons may be large so as to lead to LFV effects that could be tested by the MEG and the COMET/PRISM experiments.
Specifically, in the Rτ L model with a normal light neutrino mass hierarchy, there is a sizeable model parameter space with successful leptogenesis and large LFV process rates, with B(µ → eγ) ≈ 10 −12 . This prediction is largely independent of sin 2 θ 13 and the other light neutrino oscillation parameters. On the other hand, in the Rτ L model with inversely hierarchical light neutrinos, B(µ → eγ) is linearly proportional to sin 2 θ 13 , and can be enhanced by more than one order of magnitude compared to the normal hierarchy case for sin 2 θ 13 close to its upper experimental 2σ limit. Unfortunately, the generated baryon asymmetry η B is suppressed in this scenario, and to test the viable parameter space for successful leptogenesis would require an experiment for µ → e conversion in nuclei which is sensitive to B µe ≈ 10 −17 -10 −16 . This feature is also quite generic for the case of the RµL and ReL models, where µ → e flavour transitions are suppressed by the small τ -Yukawa couplings κ 1,2 . In all RℓL models, charged LFV in the τ -lepton sector turns out to be at least 6 orders of magnitude beyond the current experimental sensitivity, as the predicted branching ratios are B(τ → eγ, µγ) < ∼ 10 −14 in parameter regions required for successful leptogenesis.
Further studies will be needed to analyze the full range of theoretical, phenomenological and cosmological implications of the three different universal models of ReL, RµL and Rτ L. For instance, the consideration of thermal effects [58] may provide a significant improvement on the standard framework of classical BEs adopted in the present analysis.
An equally significant issue is whether our minimal RL models can account for the wellknown problem of cold Dark Matter (CDM) in the Universe. An obvious solution would be to consider supersymmetric versions of RℓL scenarios [59,60] and study the relic abundance of the lightest stable SUSY particle, which could be a thermal right-handed sneutrino [61].
Alternatively, one may consider scale-invariant extensions of the SM with right-handed neutrinos [62,63], which are minimally augmented with one complex singlet scalar field whose one-loop induced VEV can naturally explain the origin of the electroweak-scale mass of the heavy Majorana neutrinos. It has been shown recently [63] that a minimal Z 4 -symmetric variant of these models can stay perturbative up to the Planck scale, as well as provide a CDM candidate through the so-called Higgs-portal mechanism [64]. It is therefore rather motivating to perform a dedicated analysis of observing electroweak-scale heavy Majorana neutrinos within the specific context of the RℓL scenarios studied here, through their possible like-sign dilepton [26,65] and/or trilepton [66] signatures at the LHC or at other future high-energy colliders.