Leptogenesis in $\Delta(27)$ with a Universal Texture Zero

We investigate the possibility of viable leptogenesis in an appealing $\Delta(27)$ model with a universal texture zero in the (1,1) entry. The model accommodates the mass spectrum, mixing and CP phases for both quarks and leptons and allows for grand unification. Flavoured Boltzmann equations for the lepton asymmetries are solved numerically, taking into account both $N_1$ and $N_2$ right-handed neutrino decays. The $N_1$-dominated scenario is successful and the most natural option for the model, with $M_1 \in [10^9, 10^{12}]$ GeV, and $M_1/M_2 \in [0.002, 0.1]$, which constrains the parameter space of the underlying model and yields lower bounds on the respective Yukawa couplings. Viable leptogenesis is also possible in the $N_2$-dominated scenario, with the asymmetry in the electron flavour protected from $N_1$ washout by the texture zero. However, this occurs in a region of parameter space which has a stronger mass hierarchy $M_1/M_2<0.002 $, and $M_2$ relatively close to $M_3$, which is not a natural expectation of the $\Delta(27)$ model.


Introduction
The Standard Model (SM) has been experimentally confirmed as the correct description of Nature, with excellent precision, up to scales of O(TeV). Nevertheless we know that the SM is not a complete theory. It includes a host of free parameters, the majority of which relate to the Yukawa sector, into whose origin and nature the SM offers no insight. This is despite obvious indications of internal structure, such as large mass hierarchies between generations of fermions, and small CKM mixing. It is also unclear as to how the SM should be extended to account for massive neutrinos and lepton mixing. The combined questions of charged fermion hierarchies and the CKM and PMNS mixing patterns is typically referred to as the flavour puzzle. Moreover, the SM fails to accommodate several observational facts in cosmology. It lacks dark matter and inflaton candidates, has no explanation for dark energy, and does not account for the baryon asymmetry of the Universe (BAU).
Among these cosmological issues, perhaps the BAU is the most distressing one. The SM is (nearly) symmetric in particles and anti-particles; despite this, no antiparticles are observed in Nature. The BAU, that is, the difference between the baryon n B and antibaryon n B number densities, is measured with respect to the entropy density s to be Although the SM includes all the necessary ingredients to generate this BAU dynamically [1], namely, CP violation in the CKM matrix, B violation through sphaleron interactions, and outof-equilibrium processes in the electroweak phase transition, the asymmetry obtained in the SM is too small by orders of magnitude [2].
It is well-known that extending the SM by several heavy right-handed (RH) neutrinos can yield a BAU via leptogenesis [3]. Lepton number-violating decays of the RH neutrinos, some portion of which occur out of equilibrium, produce a lepton asymmetry. This is partially converted into a baryon asymmetry by sphaleron interactions, which are efficient above the electroweak scale. Heavy RH neutrinos simultaneously provide a natural answer to the smallness of left-handed (LH) neutrino masses via the seesaw mechanism.
It is interesting to note that since RH neutrinos are SM singlets, leptogenesis links the resolution of the BAU with their Yukawa couplings, and thus connects with the flavour puzzle. If seesaw is indeed the origin of light neutrino masses, then qualitatively leptogenesis is unavoidable. Whether it accurately reproduces the observed BAU becomes a quantitative question for a given spectrum of RH neutrinos and their interactions with SM particles. Remarkably, the original (and arguably simplest) model of leptogenesis requires a RH neutrino scale M 10 9 GeV, which closely corresponds to the "natural" seesaw scale.
The flavour sector of the SM, including massive neutrinos, comprises 20 arbitrary parameters. A popular approach to relate these parameters, and reduce the number of degrees of freedom in the effective SM, is that of spontaneously broken flavour (or family) symmetries. Non-Abelian discrete symmetries have been especially successful, able to simultaneously describe charged lepton and neutrino parameters, and in several cases, also the quark sector [4][5][6][7][8][9][10][11][12]. A very appealing ∆(27) model was introduced in [13], consistent with an underlying SO(10) grand unified theory (GUT). The family symmetry leads to a predictive structure with a universal texture zero (UTZ) for all fermion mass structures, including the effective neutrinos after seesaw. The family symmetry is also responsible for controlling flavour-violating processes, which are sufficiently suppressed for certain regions of the model parameter space, as shown in [14]. A complete model ought to account for the observed BAU, which provides an additional constraints on its parameters. In particular, as we shall see in this analysis, matching to the observed BAU allows us to constrain the otherwise unknown parameters of the RH neutrino sector.
The paper is organized as follows. Section 2 summarizes the main features of the model, originally presented in [13]. The seesaw implementation is described in Section 3. In Section 4 we write down the Boltzmann equations describing the evolution of the neutrino and asymmetry densities; this is supplemented by Appendix A. Section 5 presents the results and our analysis. We conclude in Section 6. Appendices B and C provide additional insight into the model and leptogenesis within it.

Overview of the ∆(27) Model
In this section we review the model introduced in [13]. Given that we are interested in leptogenesis, we focus on the lepton sector, where SM fermions are contained in superfields L (lepton SU (2) doublets) and e c , N c (following conventional notation, conjugates of the RH charged leptons and neutrinos, respectively). The field content and their transformation properties under the G f = ∆ (27) × Z N flavour group are given in Table 1. The superpotential that generates the Yukawa structures at leading order for Dirac fermions is where i, j = (1, 2, 3) lower (upper) indices denote the ∆(27) triplets (anti-triplets). Nonrenormalizable terms are suppressed by messenger masses, which are in general different [13]; they are denoted here by a common scale Λ, with variations in messenger masses contained in an arbitrary coupling g for each term. The superpotential responsible for RH neutrino Majorana masses is The φ fields are flavons that break ∆ (27) and provide the structure of the mass matrices, with the vacuum alignment The core prediction of the model is universal complex-symmetric mass matrices with the UTZ in the (1,1) entry, of the form for some complex a, b, c. Assuming a strong hierarchy a < b < c, the eigenvalues are approximately given by |a 2 /b|, |b| and |c|. This applies in particular to the Dirac and Majorana mass matrices. Up to O(1) coefficients, they yield the following hierarchies between families: ν  3  e,ν  3  e,ν  2  e,ν  2  e,ν  3  e,ν with y e,ν c and M c the dominant contributions to the third and heaviest generation. Masses and mixing are compatible with the expansion parameters e 0.15 and N ∼ 3 ν . The expansion parameter for Dirac neutrinos, ν , is not constrained by phenomenology, but internal consistency of the model requires that it remains perturbative, i.e. ν 0.5. We shall see that numerically viable regions in parameter space correspond to ν ∈ [0.05, 0.5]. Note that the large hierarchy between the first two RH neutrinos N 1,2 and the heaviest one N 3 is characteristic of this kind of model [15][16][17][18][19], where rather different mixing patterns in the quark and lepton sectors are obtained from the same universal Yukawa structures, on the condition that φ c is dominant in the quark and charged lepton sectors and irrelevant for the neutrino mass matrix.
The structure in Eq. (5) can written as where The set of parameters (y e,ν a , y e,ν b , y e,ν c ) and (M a , M b , M c ) in Eq. (7) are generally complex 6 with phases coming either from the VEVs or the coefficients, y e,ν a ≡ |y e,ν a |e iγe,ν , y e,ν b ≡ |y e,ν b |e iδe,ν , y e,ν c ≡ |y e,ν c |, Given that phenomenology depends only on two independent combinations of the phases, we follow [13] in taking just δ and γ as independent phases (see also Table 2). In terms of the fundamental parameters of the superpotential in Eqs. (2)-(3), they are The superfield S is a gauge singlet, while Σ is not [13] and introduces Clebsch-Gordan (CG) coefficients, although for our purposes here it is sufficient to consider their respective VEVs S and Σ as real numbers, and absorb the different CG contributions to charged leptons and neutrino into g e,ν b . The expansion parameters of the model in Eq. (6) are recovered from the parameters in Eq. (7) as e,ν ∼ y e,ν a /y e,ν b ∼ y e,ν b /y e,ν The lepton asymmetries are obtained in the flavour basis, wherein the charged lepton Yukawa matrix Y e and RH neutrino mass matrix M N are diagonal. They are diagonalized by unitary matrices, such thatŶ where hats (ˆ) denote diagonal matrices of positive eigenvalues and, Y e being complex symmetric, we have V † eR = V T eL . In the flavour basis, in the LR phase convention (where the Yukawa couplings are given by L ∼ LH d e R + LH u ν R + h.c.), the neutrino Yukawa matrix is given by λ ν , where where the additional conjugation on λ ν appears due to the change from the supersymmetry basis to the seesaw basis [21].

The UTZ seesaw mechanism
As the Dirac and Majorana matrices are expressed in terms of the same rank-one matrices, the application of the usual seesaw formula, provides a light neutrino mass matrix m ν which can be expanded in the same fashion, i.e.
Notably, the UTZ is preserved. A detailed discussion of this elegant property can be found in Appendix B. The parameters m a,b,c entangle the combinations of Dirac and Majorana neutrino couplings as hierarchy between them is obtained by the relative size of the coefficients g N a , g N b . For those values of the Dirac neutrino expansion parameter ν preferred by the model, we thus expect a hierarchical spectrum for the Majorana neutrino masses in which M 1 < M 2 M 3 . 7

Boltzmann equations
The generation of a BAU through N i -leptogenesis is a non-equilibrium process which is generally treated by means of Boltzmann equations for the number densities of RH (s)neutrinos, Y N i and Y N i (for an N i neutrino with mass M i ), and leptons, Y Lα . It is useful to consider the quantities is conserved by sphalerons and other SM interactions. L α and ∆ α asymmetries are related by a flavour coupling matrix A, i.e. Y Lα = α A αα Y ∆α . The form of A depends on which interactions are in thermal equilibrium during leptogenesis; it is defined explicitly in Appendix A. The produced lepton asymmetries are partially converted into a baryon asymmetry Y B by the sphalerons, given in the MSSM by with Y ∆α computed at a temperature T M i , where the densities Y N i , Y N i are effectively zero. In the fully flavoured regime, M i 10 9 (1 + tan 2 β) GeV, all lepton flavours are to be treated separately, i.e. α = e, µ, τ . In the two-flavour regime, 10 9 (1 + tan 2 β) GeV M i 10 12 (1 + tan 2 β) GeV, only the interaction mediated by the τ Yukawa coupling is in equilibrium and the asymmetries in the e and µ flavours can be treated with a combined density Y ∆eµ = Y ∆e+∆µ .
In the MSSM, assuming hierarchical RH neutrinos, the Boltzmann equations take the form [24] are the equilibrium densities of (s)neutrinos N i , N i , respectively. The factors D and W govern the decay and washout behaviour, respectively, and contain information about decays, inverse decays, and scattering processes. [25][26][27][28]. The expressions used in our calculation are collected in Appendix A, where we follow in particular the notation and method of [24]. The decay factors K α N i and CP asymmetries ε α N i , arising from the interference between tree-level and loop diagrams of the RH neutrino decay, are determined by the flavour parameters of the model, to which we now turn our attention.

Decay factors and CP asymmetries
The lepton asymmetry in each flavour is governed by two sets of parameters which can be computed within a given neutrino model: the decay factors K α N i and CP asymmetries ε α N i , for a neutrino N i decaying into a Higgs H u and lepton doublet L α (or their conjugates). The Majorana nature of the RH neutrino masses implies the decays N i → L α H u and N i → L α H * u violate lepton number by one unit (∆L = 1). The decay factors are defined as where H(T ) is the Hubble parameter at the temperature T , and H(M i ) 1.66 √ g * M 2 i /M Pl . The CP asymmetries are defined as .
The decay factors are dominated by the single tree-level diagram, while the CP asymmetries arise only at one-loop level from the self-energy plus vertex diagrams. In the two-flavour regime, with the corresponding decay asymmetry ε eµ N i = ε e N i + ε µ N i . Explicitly in terms of the neutrino Yukawa matrix in the flavour basis, λ ν , the decay factors are given by where m * (1.58 × 10 −3 eV) sin 2 β. For N i decay, the relative phase between the tree diagram and the loop diagram with an intermediate N j will be the phase of (λ † ν λ ν ) ij . Then the CPasymmetries for the two lightest RH-neutrinos is expressed as where g(x) is a loop function given by the sum of the vertex and the self energy contributions [24,29]; in the MSSM, An exploration of the CP asymmetries and decay factors -responsible for the production and washout of a lepton asymmetry, respectively -provides some insight into how leptogenesis proceeds in this model. The decay factors appear in the arguments of exponential damping terms, and a large K α N i is associated with strong washout. As it is inversely proportional to the RH neutrino mass, in the "vanilla" picture of flavour-independent N 1 leptogenesis, this yields a lower bound on the N 1 mass, M 1 10 9 GeV [25]. When considering asymmetry generation from next-to-lightest RH neutrinos (N 2 leptogenesis), typically a crucial requirement is that K α N 1 1 in some lepton flavour, to not completely wash out a previously generated asymmetry from N 2 decays [27]. This depends in particular on the Yukawa structures that give λ ν ; N 2 leptogenesis and its compatibility with low-scale neutrino phenomenology has been studied in [30][31][32][33].
As we are considering the case in which M 1 M 3 then we can also neglect the i = 3 contribution to ε α N 1 . In Appendix C we show that λ ν (in the flavour basis) maintains the hierarchical structure suggested by the model, and a rough estimate for the leptogenesis parameters gives From this we can make some a priori considerations: (i) due to the UTZ in the electron coupling to N 1 , lepton asymmetries from N 1 decays are dominated by the µ and τ flavours, while ε e N 1 is generally too small to contribute significantly to asymmetry production, (ii) we similarly expect a strong washout in the µ and τ flavours for both N 1 and N 2 leptogenesis, and comparatively weak washout for the electron, and (iii) despite the large hierarchy between M b and M c , the ε α N 2 are typically dominated by the first term, which generates non-negligible asymmetry only if M b and M c are not too separated.

Numerical results
In this section we present the numerical solutions to the fully flavoured Boltzmann equations in the MSSM as given in Section 4, following a similar procedure to the one already adopted in [34]. The analysis has been performed under the assumption that the spectrum of the heavy neutrinos in the model is hierarchical, • a possible asymmetry generated by the heaviest RH neutrino N 3 is always washed out and assumed to be negligible, • the generation of the asymmetry and the washout from decays and inverse decays of the N 1 neutrinos starts only after the end of the analogous processes from the N 2 . The two lightest RH neutrinos do not interfere with each other, such that the generation of the asymmetry from N 1 decays and from N 2 decays proceed independently.
Consequently, the Boltzmann equations in Eq. (20) are solved twice for each point in the model parameter space. In the first step, we solve for Y ∆α arising from N i=2 decays, assuming thermal initial conditions (zero neutrino and asymmetry densities). The solutions for Y ∆α are then used as initial conditions for the N i=1 calculation. The final asymmetry is obtained from the sum over Y ∆α after N 1 leptogenesis.
The input parameters are comprised of those not already fixed by the fit to low-scale neutrino phenomenology. In particular, we use the fit to quark and lepton masses and mixing for our flavour model performed in [13], with relevant best fit values for the lepton sector given in Table 2. We fix tan β = 10 in this analysis. Note also that, as the model does not determine the absolute mass scale for fermions, the fit only provides estimates for the parameters in Eq. (9) up to an overall scale, which is set by the third generation, i.e. by M c and y e,ν c . With tan β fixed, we can infer the charged lepton scale y e c , while the neutrino scales y ν c and M c remain unfixed.

Neutrinos
Charged leptons Table 2: Fitted values for the low-scale model parameters, extracted from the computation in [13].
The fit fixes the values of the neutrino mass parameters m a,b,c and charged lepton parameters y e a,b,c . The seesaw relation in Eq. (16) entangles the three Dirac and three Majorana neutrino couplings (y ν a,b,c and M a,b,c , respectively), constrained only by the three fitted values of m a,b,c ,  Figure 1 shows the regions that reproduce the experimental value of Y B to within 20%, in terms of the RH neutrino mass eigenvalues M 1,2 , with M 3 5 × 10 14 GeV. In Figure 1a we see the successful leptogenesis regions taking into account only N 1 decays, while in Figure 1b we consider both N 1 and N 2 decays. The comparison between plots allows us to conclude that, over most of the parameter space of the model, the BAU is consistent with leptogenesis proceeding entirely from N 1 decays, assuming thermal initial conditions. In other words, the asymmetries generated by N 2 decays are efficiently washed out in all flavours. In the N 1 case, the dominant contributions to the viable regions are from the µ and τ asymmetries, while the electron asymmetry is completely negligible. This agrees well with the expectations from the analytical approximations in Section 4. Nevertheless, in Figure 1b we find a small region where the N 2 contribution to the BAU dominates. We can see that this scenario requires a small splitting between the heavy RH neutrinos, M 2 /M 3 0.1. The N 2 case is discussed further in Section 5.2, where we show that this region of parameter space is not natural in the UTZ model. . Therefore, we conclude that the correct BAU is found for RH neutrino masses above M 1 4 × 10 9 GeV and M 2 2 × 10 11 GeV. In this regime it is relevant to discuss the issues related to the potential overproduction of gravitinos [35]. There are several ways around it [36,37], one of which is to keep the reheating temperature low and to produce the RH neutrinos non-thermally (e.g. produced in decays of the inflaton). Nonetheless, even for thermal production scenarios, if the gravitino is unstable with mass m 3/2 10 TeV, these relatively high reheating temperatures around 10 9 or 10 10 GeV remain borderline viable.
Finally, it is interesting to analyse the restrictions that a requirement of successful leptogenesis set on the flavour model. As we have seen in Figures 1 and 2, the best possibility, if we demand a relatively low reheating temperature, would correspond to RH eigenvalues M 1 4 × 10 9 GeV and M 2 2 × 10 11 GeV, with M 3 ≥ 10 14 GeV. In terms of the model parameters these points correspond roughly to M a 3 × 10 10 GeV, M b 10 11 GeV, y ν a 0.003 and y ν b 0.009. We emphasize again that independent information on the neutrino Yukawa couplings and RH neutrino masses is not available from oscillation experiments, but when the BAU is accounted for we can obtain several unknown parameters. With the above values we obtain the expansion parameter ν 0.3. The heaviest RH neutrino is then M c × g ν b /g ν GeV. However, these restrictions depend strongly on the details of the flavour model and may change with small variations [14]. If supersymmetry is found in the neighbourhood of the electroweak scale, we would obtain additional information on the flavour symmetry that could help restrict these possibilities [14,[38][39][40].

N 2 leptogenesis and comparison with other models
As we have seen in the comparison of Figures 1a and 1b, there is a small region where the BAU is generated mainly by N 2 leptogenesis. This region corresponds to M 1 ≤ 0.002M 2 , with M 2 ≥ 10 13 GeV and M 3 = 5 × 10 14 GeV. These points correspond to relationships between model parameters, M a < 0.002M b and 0.01 < ν < 0.1.
Here, the mechanism of asymmetry generation and washout is slightly more involved [28,[41][42][43]. At temperatures T ∼ M 2 ∼ 10 13 GeV, a comparatively large asymmetry is generated in each of the two active lepton flavours α = (eµ), τ from N 2 decays. These serve as initial conditions of the subsequent N 1 system, which occurs at much lower temperatures T ∼ M 1 ∼ 10 9 GeV. This lies in the fully flavoured regime, wherein the active flavours are α = e, µ, τ . The asymmetry Y ∆eµ , initially generated in the combined eµ flavour, is split into the e and µ flavours proportionally to K e N 2 ∝ |λ e2 | 2 and K µ N 2 ∝ |λ µ2 | 2 , respectively [28,42,43]. Assuming the number density of e and µ asymmetries are equal at the moment where µ couplings reach equilibrium, the initial conditions for the N 1 decays are thus As the CP asymmetries ε α N 1 are sensitive to the ratio M 1 /M 2 1, no significant additional contribution to the BAU is generated by N 1 decays in this regime. However, if a large asymmetry is generated by N 2 decays, even a small portion stored in the electron flavour can survive washout and reproduce the observed asymmetry. To understand this, we make two observations: 1) the decay factors K α N 1 are approximately proportional to 1/M 1 , and 2) in each lepton flavour, they go like 1, 1). In other words, the flavour structure of the model implies the washout in the electron flavour is generally weaker than other flavours. Indeed, we observe that the Y ∆µ and Y ∆τ asymmetries are efficiently washed out, while some portion of Y ∆e remains.
So, as we can see, N 2 leptogenesis is possible (in part) due to the texture zero, which is enforced by symmetry. However, from the perspective of the UTZ model based on ∆(27), described above, this N 2 -dominated scenario is not "natural", while N 1 leptogenesis is still viable and natural in large parts of the parameter space. This unnaturalness is a direct consequence of the structure of the neutrino matrices (see Eq. (7)) and can be understood by looking at Eqs. (16)- (18). Using Eq. (18) with the measured value for sin θ 13 0.15, we obtain Barring accidental cancellations, this expression fixes As a consequence, the structure of Yukawa matrices enhances the leptogenesis effects from N 1 , proportional to M 1 /M 2 , and suppresses N 2 effects, proportional to M 2 /M 3 (see Eq. (26)). N 2 leptogenesis can be important in situations where M 2 /M 1 1, as seen in Figure 1, but this requires a strong cancellation of several orders of magnitude in Eq. (27). Moreover, the structure of the neutrino Yukawa matrices in the UTZ model is not hierarchical in this region, as we have 0.1 ≤ y ν with y ν c ∝ m t , y ν b ∝ m c and y ν a ∝ m u . In models with special flavon directions like the so-called Constrained Sequential Dominance 3 alignment [34], we have simply sin θ 13 m 1 /(m 1 + m 2 ), which does not constrain the ratio M 2 /M 1 . The only constraint on RH neutrino masses comes from Eq. (28). Setting m 1 = m sol and m 2 = m atm implies M 2 10 11 GeV and M 2 /M 1 m 2 c /(6m 2 u ) 5 × 10 4 . Under these conditions N 1 contributions to the BAU are far too small, but N 2 can still successfully contribute as shown explicitly in [46][47][48].
Unlike the traditional case for N 2 leptogenesis, which is typically aimed at resolving the problem of having a lightest neutrino with too small a mass (M 1 10 9 GeV), in our case even the N 2 region requires M 1 10 9 GeV, to avoid too-large washout. In some sense, separate to the above discussion on naturalness, some balancing is also required to ensure the initial N 2 asymmetry, which may be one or two orders of magnitude larger than anticipated by the observed BAU, is washed out just the right amount by N 1 interactions to yield the correct value of Y B .

Conclusions
We have studied the generation of the baryon asymmetry of the Universe through leptogenesis in the Universal Texture Zero SO(10) × ∆(27) × Z N flavoured GUT model [13]. Here, leptogenesis yields the observed BAU for a considerable region of the parameter space. When expressed in terms of the RH neutrino masses M 1 and M 2 , which are functions of the model parameters M a and M b . The viable ranges for the mass of the lightest RH neutrino eigenstate have a lower bound of M 1 4 × 10 9 GeV, which is still barely compatible with a gravitino mass m 3/2 10 TeV, provided the gravitino is unstable [35].
We specifically considered the effect of N 2 leptogenesis, which we conclude to be disfavored: although there exists a region of parameter space where N 2 leptogenesis provides the dominant contribution to the final asymmetry, this corresponds to a scenario with both a very strong hierarchy between the two lightest RH neutrinos, i.e. M 1 M 2 , and comparatively small hierarchy between M 2 and M 3 . This is not a natural expectation in the model, which predicts a strong hierarchy between the heaviest neutrino and the two lighter ones, i.e. M 1 < M 2 M 3 . Lepton asymmetries generated by decays of the heaviest neutrino N 3 are therefore also negligible.
The preferred mechanism is thus N 1 leptogenesis. The requirement that it accounts for the entire baryon asymmetry allows us to restrict the parameters governing the neutrino Yukawa matrix and RH neutrino mass matrix. These are otherwise only partially constrained by the observed neutrino masses and mixing, namely those combinations of parameters which appear in the neutrino matrix after seesaw. We find that viable N 1 leptogenesis requires M 1 4 × 10 9 GeV, with M 2 2 × 10 11 GeV, while 0.002 M 1 /M 2 0.1. By consistency with low energy observables, we can similarly constrain the neutrino Yukawa couplings, which are bounded from below, y ν a 0.003, y ν b 0.008.
In conclusion, flavoured leptogenesis is viable for the UTZ model in the standard N 1 regime. Through this we are able to place further constraints on the parameter space of the UTZ model, leading to direct constraints on the scale of the parameters M a , M b governing the RH neutrino masses. Given that in the model the active neutrino masses originate from type-I seesaw leading to normal ordering with a strong hierarchy, the leptogenesis constraint on M a , M b can then be combined with the observed mass-squared differences to indirectly constrain the Dirac neutrino couplings y ν a , y ν b . These constraints are complementary to those provided by the study of flavourchanging processes [14] in the UTZ model. are partially funded through POCTI (FEDER), COMPETE, QREN and EU. AM and OV were supported under MICIU Grant FPA2017-84543-P and by the "Centro de Excelencia Severo Ochoa" programme under grant SEV-2014-0398. OV acknowledges partial support from the "Generalitat Valenciana" grant PROMETEO2017-033. AM acknowledges support from La-Caixa-Severo Ochoa scholarship.

A Boltzmann equations
Recall that the baryon asymmetry Y B can be expressed as where Y ∆α are the B/3 − L α asymmetries for each active lepton species α. In the fully-flavoured scenario, these are simply the usual three lepton flavours, α = e, µ, τ . Assuming hierarchical RH neutrinos and thermal leptogenesis, the lepton asymmetries are obtained by solving the Boltzmann equations where z = M i /T . As noted in Section 4, the factors D and W govern the decay and washout behaviour. In this appendix we make these explicit, noting how information about decays and scattering are incorporated. In particular, we follow [24].
The equilibrium number density for a given field f is denoted Y eq f , and are functions of z. The RH (s)neutrino densities are given by where g * = 228.75 is the effective number of degrees of freedom in the MSSM and K 2 (z) the modified Bessel functions of the second kind. The (s)lepton distributions are given by We now turn to the decay and washout factors, D and W . There are three classes of processes that contribute to the Boltzmann equations: (1) decays and inverse decays (N ↔ L α H u ), (2) ∆L = 1 scatterings (L α H u ↔ L α H u ) and (3) ∆L = 2 processes (L α L α ↔ H u H u ). Following [27] and [24], in this analysis we include ∆L = 1 scatterings involving neutrino and top Yukawa

C Analytic expression for the leptogenesis parameters
In this appendix we give the analytic expression for λ ν , i.e. for the Dirac neutrino matrix in the basis where Y e and M N are diagonal, as well as for the decay factors K α N i and the CP asymmetries ε α N i entering in our numerical calculation. The Yukawa and Majorana mass matrices share a unified symmetric and hierarchical texture zero, where (a, b, c) stand for either y ν,e a,b,c or M a,b,c . Given this complex symmetric structure with the hierarchy a < b < c, the diagonalizing matrices satisfy the relation They can be approximated by where P = diag(e −i(2γ−δ+π)/2 , e −iδ/2 , 1) is a matrix of phases ensuring the eigenvalues inM are real and positive. With a ∼ 3 ν , b ∼ 2 ν , and c ∼ 1, Eq. (53) is accurate (and unitary) to O( 3 ν ), and preserves the (1,1) texture zero to O( 4 ν ). Expanding V as V = (1 + ∆V )P , we split λ ν into two parts, one which preserves Y ν (up to rephasing) and a correction term ∆λ * ν , i.e. λ * ν ≡ V e Y ν V T N = P e (1 + ∆V e )Y ν (1 + ∆V T N )P N = P e (Y ν + ∆λ * ν )P N having discarded the term ∆V e Y ν ∆V T N . Notice that the corrections in ∆λ * ν are typically of the order of Y ν except in the 22,32 and 33 elements where they are smaller. They are never larger; in this sense λ * ν preserves the same hierarchy between the elements as Y ν . Considering only the first term in Eq. (54), λ * ν ∼ P e Y ν P N , we can deduce for the decay factors: