Flavor and CP symmetries for leptogenesis and 0nubb decay

We perform a comprehensive analysis of the phenomenology of leptonic low and high energy CP phases in a scenario with three heavy right-handed neutrinos in which a flavor and a CP symmetry are non-trivially broken. All CP phases as well as lepton mixing angles are determined by the properties of the flavor and CP symmetry and one free real parameter. We focus on the generation of the baryon asymmetry Y_B of the Universe via unflavored leptogenesis and the predictions of m_ee, the quantity measurable in neutrinoless double beta decay. We show that the sign of Y_B can be fixed and the allowed parameter range of m_ee can be strongly constrained. We argue on general grounds that the CP asymmetries epsilon_i are dominated by the contribution associated with one Majorana phase and that in cases in which only the Dirac phase is non-trivial the sign of Y_B depends on further parameters. In addition, we comment on the case of flavored leptogenesis where in general the knowledge of the CP phases and light neutrino mass spectrum is also not sufficient in order to fix the sign of the CP asymmetries. As examples we discuss the series of flavor groups Delta (3 n^2) and Delta (6 n^2), n>= 2 integer, and several classes of CP transformations.


Introduction
The baryon asymmetry Y B of the Universe has been precisely measured [1] Y B = n B − n B s 0 = (8.65 ± 0.09) × 10 −11 . (1) The error is given at the 1 σ level and the subscript "0" refers to the present epoch. The generation of Y B requires the fulfillment of the three Sakharov conditions [2]: C and CP violation, departure from thermal equilibrium and baryon number violation. All three of them are met by the mechanism of unflavored and flavored (thermal) leptogenesis [3]. In fact the decay of right-handed (RH) neutrinos N i (we always assume the existence of three such states and thus i = 1, 2, 3) leads to a lepton asymmetry that is partially converted into a baryonic one via sphaleron processes [4]. Departure from thermal equilibrium arises, since the rate of the Yukawa interactions of RH neutrinos is small compared to the Hubble rate, while complex Yukawa couplings Y D are responsible for C and CP violation. The relevant quantities for computing Y B are: the yield of RH neutrinos at high temperatures and the sphaleron conversion factor giving together rise to 10 −3 , the efficiency factors, taking into account washout and decoherence effects and usually of order of 10 −3 ÷ 10 −1 , and the CP asymmetries (α) i , generated in the decays of the i th RH neutrino N i (and to the flavor α = e, µ, τ , if flavored leptogenesis is studied). As has been shown in [5], even in a scenario in which light neutrinos acquire masses via the ordinary type 1 seesaw mechanism [6] the CP phases present in the Yukawa couplings are in general unrelated to the CP phases potentially measurable at low energies: the Dirac phase δ in neutrino oscillation experiments [7][8][9][10][11] and a combination of the two Majorana phases α and β in neutrinoless double beta (0νββ) decay [12]. However, it is also well-known that low energy CP phases may be crucial for having successful leptogenesis in case flavor effects are relevant [13].
In theories with flavor G f and CP symmetries all CP phases can be related. Such symmetries are usually introduced in order to explain the mixing pattern(s) observed in the lepton (and quark) sector. Since two of the lepton mixing angles (and thus most of the entries of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix U P M N S ) are large [14] 0.270 ≤ sin 2 θ 12 ≤ 0.344 , 0.382 [9] ≤ sin 2 θ 23 ≤ 0.643 [4] and 0.0186 [8] ≤ sin 2 θ 13 ≤ 0.0250 [1] (2) (3 σ ranges for light neutrino masses following normal ordering (NO) and in square brackets for inverted ordering (IO), see also [15,16]), it is natural to assign the three generations of left-handed (LH) lepton doublets l to an irreducible three-dimensional representation 3 of the flavor symmetry. If the setup contains RH neutrinos, it is reasonable to also assign these to such a representation. Indeed, it is convenient to use the same representation 3 as for l in a non-supersymmetric (non-SUSY) context, while in SUSY models ν c are usually put in the complex conjugated representation with respect to l. The most promising choice of G f turns out to be a discrete, finite, non-abelian group [17,18] that is broken to different residual symmetries G e and G ν in the charged lepton and neutrino sectors, respectively [19]. These symmetries are in general abelian subgroups of G f with three different elements at least and, in the particular case of three Majorana neutrinos, that we consider, G ν = Z 2 × Z 2 . As is known, with this approach only the Dirac phase can be predicted and the analyses in [20] have shown that it always turns out to be trivial, if mixing angles are required to be in good agreement with experimental data. 1 Thus, in order to fix the Majorana phases via symmetries as well as to achieve δ different from 0 or π, this approach has to be modified. An extension that has been proven very powerful [21][22][23][24][25][26][27] is to amend the flavor with a CP symmetry [21] (see also [28]). The latter acts in general also in a non-trivial way on the flavor space [29], requiring certain conditions to be fulfilled for having a consistent theory [21,30,31]. The choice of G e remains the same, whereas G ν is taken to be the direct product of a Z 2 symmetry, contained in G f , and the CP symmetry: G ν = Z 2 × CP [21]. The main feature of such a setting is predicting the mixing angles as well as all three CP phases in terms of only one free real parameter θ that can be chosen without loss of generality to lie in the interval 0 ≤ θ < π. This parameter is present and not fixed in this approach, simply because the residual flavor symmetry in the neutrino sector is only a Z 2 group. On the other hand, involving CP leads to the clear advantage to determine all three CP phases and not only the Dirac phase. Thus, such a scenario is in particular suitable for studying the phenomenology related to CP phases, like leptogenesis and 0νββ decay.
Leptogenesis, unflavored as well as flavored, has already been studied in scenarios with the flavor symmetries A 4 and S 4 only [32][33][34][35][36][37][38]. One striking feature of these scenarios is the fact that CP asymmetries vanish in case one only considers the leading order (LO) terms in the theory, i.e. those which preserve G e = Z 3 and G ν = Z 2 × Z 2 in the charged lepton and neutrino sectors, respectively. Thus, taking into account next-to-leading order (NLO) corrections (in the neutrino sector, in particular, to Y D ) is mandatory. These are in general proportional to the (small) symmetry breaking parameter κ ∼ 10 −(2÷3) . As shown in [32,33], the CP asymmetries i are proportional to κ 2 for unflavored leptogenesis, whereas in the case of flavored leptogenesis the suppression is less and α i are proportional to κ only. 0νββ decay has already been discussed in contexts with a flavor and a CP symmetry, see first reference in [23], last reference in [22], [39], first reference in [27] and [25]. In [39] the authors have considered the series of groups ∆(6 n 2 ), but they have assumed that the residual symmetry in the neutrino sector is larger G ν = Z 2 × Z 2 × CP . Thus, the Dirac phase is fixed to be trivial as well as one of the Majorana phases which clearly affects the predictions for m ee . In [25] the authors instead study G ν = Z 2 × CP as residual group like in our analysis. However, the presented results are mostly in the limit in which the group theoretical parameter n is taken to be very large and no particular choice of the CP symmetry is made.
In the present paper we study leptogenesis and 0νββ decay in a scenario with the flavor symmetry ∆(3 n 2 ) or ∆(6 n 2 ) and a CP symmetry that are broken in a non-trivial way. As discussed in [25,26], for several choices of flavor groups ∆(3 n 2 ) or ∆(6 n 2 ), CP symmetries, residual groups G e and G ν and the free parameter θ it is possible to obtain lepton mixing angles in agreement with the experimental data and in turn to predict the CP phases δ and α, β. We base our study on these results and we assume throughout that the charged lepton mass matrix is governed by the residual symmetry G e = Z 3 , while the RH neutrino mass matrix is taken to be of the most general form compatible with G ν = Z 2 × CP . Since LH lepton doublets and RH neutrinos transform according to the same three-dimensional representation 3 of the flavor group in a non-SUSY framework (or in a SUSY context l and ν c as3 and 3), the Yukawa coupling Y D of neutrinos with trivial flavor structure is invariant under the entire flavor group and CP. We note that RH neutrinos are not strongly hierarchical in our scenario and hence all three of them are expected to be relevant for leptogenesis. We mainly focus on the case of unflavored leptogenesis and only highlight the main differences occurring, if the baryon asymmetry of the Universe is instead generated via flavored leptogenesis. Similarly, most of our results for leptogenesis are obtained in a non-SUSY framework, however, we emphasize the changes that have to be implemented in order to apply these to a SUSY model. We find that the CP asymmetries i vanish for unflavored leptogenesis, if we only consider LO terms. Thus, we have to rely on NLO terms. In a generic model these can be arbitrary, but frequently it turns out that one type dominates and that the latter is still invariant under one of the residual symmetries. Here we consider a case in which the dominant NLO contribution arises in the neutrino sector and only corrects Y D . Furthermore, we assume that this correction δY D stems from the charged lepton sector and is thus constrained by the residual group G e . As expansion parameter we use κ and hence δY D ∝ κ. It induces an appropriate suppression of the CP asymmetries i ∝ κ 2 , similar to what has been already observed in scenarios with a flavor symmetry only. We show that the sign of i depends on the low energy CP phases and on the loop function, whose sign can be traced back to the light neutrino mass spectrum in our scenario. Light neutrino masses can be hierarchical with NO, IO or quasi-degenerate (QD). In particular, the phases, contained in δY D , are irrelevant for the LO result of i . The sign of Y B is then determined as well and it is given by i . We do not only perform a comprehensive analytical study in which we consider the different choices of residual symmetries, found in [26], but we also take in account the case in which the mixing in the RH neutrino sector is given by the general form of the PMNS mixing matrix. The latter study reveals three interesting features: the CP asymmetries i can be expressed in a certain limit in terms of the CP invariants I i that are proportional to sin α, sin β and sin(α−β) (for definition see appendix A.1). The dominant contribution is expected to arise from terms proportional to sin α, as long as this Majorana phase does not take a value close to 0 or π or the loop function suppresses these terms for particular values of the light neutrino masses. In case sin δ dominates the CP asymmetries i their sign cannot be predicted and can depend on e.g. the relative size of the parameters appearing in δY D as well as on the octant of the atmospheric mixing angle θ 23 . We exemplify these findings with several cases and detail instances in which the sign of Y B is correctly determined. Furthermore, we comment on the case of flavored leptogenesis. The CP asymmetries α i vanish as well, if we only take into account the LO terms, since the Yukawa couplings of neutrinos are taken to be invariant under the flavor symmetry G f and CP in our analysis. Also in this case corrections δY D ∝ κ induce non-zero α i . However, the sign of the latter depends in general on the parameters contained in δY D . A way to generate α i = 0 without corrections δY D is to assume the Yukawa couplings of neutrinos to be of the most general form compatible with the residual symmetry G ν = Z 2 × CP . Again, the sign of α i depends on parameters that are not directly related to the low energy CP phases (and the light neutrino mass spectrum). So, we arrive at the conclusion that the sign of the CP asymmetries cannot be univocally predicted in the flavored regime. Regarding 0νββ decay we carefully study it analytically for different choices of symmetries G f , CP, G e and G ν and present several numerical examples, pointing out the following interesting features: for light neutrino masses following IO it is possible to achieve only values close to the very upper limit (m ee 0.05 eV) of the parameter space, generally compatible with experimental data, and thus enhancing the prospects for a discovery of 0νββ decay in the not-too-far future; for NO light neutrino masses we observe that the well-known cancellation of the different terms contributing to m ee can be avoided and thus a lower limit of m ee 10 −3 eV can be set or at least the interval of the lightest neutrino mass m 0 for which a noticeable cancellation in m ee occurs to be considerably shrunk.
The paper is structured as follows: in section 2 we describe our scenario in which a flavor group G f = ∆(3 n 2 ) or G f = ∆(6 n 2 ) combined with a CP symmetry is broken to G e = Z 3 and G ν = Z 2 × CP and corrections from the charged lepton sector to the neutrino one are crucial for the generation of non-vanishing CP asymmetries. We also briefly repeat the results obtained in [26] for lepton mixing, in particular for the CP invariants, in section 2. Section 3 contains the analysis of unflavored leptogenesis in our scenario, first mentioning some general properties of the underlying leptogenesis framework, then studying the dependence of Y B on the low energy CP phases, discussing the case in which the mixing matrix U R , diagonalizing the RH neutrino mass matrix, is taken to be of the general form of the PMNS mixing matrix, and afterwards studying analytically the properties of the scenarios in which U R is determined by the breaking of G f and the CP symmetry (distinguishing the different cases case 1) through case 3 b.1) and case 3 a) that give rise to different results for lepton mixing) and eventually also numerically. We emphasize the cases in which the sign of the CP asymmetries (and thus of the baryon asymmetry of the Universe) that is intimately related with the CP phases δ, α and β and the light neutrino mass spectrum can be predicted and also point out in which cases this is in general impossible. In section 4 we discuss flavored leptogenesis and show with several examples that the sign of the CP asymmetries α i can in general not be predicted just from the knowledge of the low energy CP phases and the light neutrino mass spectrum. We also demonstrate that for Y D invariant under G ν (and not G f and CP) non-zero CP asymmetries can be achieved without corrections δY D . We comment on the changes occurring, if our scenario is realized in a SUSY context, in section 5 and point out the similarities and crucial differences. Section 6 is dedicated to an analytical and numerical study of 0νββ decay in which the effects of constraining the lepton mixing parameters, in particular the two Majorana phases α and β, become apparent. The summary of our results can be found in section 7. Our conventions for mixing angles θ ij and the CP phases δ, α and β together with the results of the global fit [14] are given in appendix A. The choice of generators of the flavor groups ∆(3 n 2 ) and ∆(6 n 2 ) is presented in appendix B. For completeness, in appendix C some results for i are shown for the case in which U R is identified with the general form of the PMNS mixing matrix. Appendix D contains sketches of explicit models which realize the envisaged scenario in a non-SUSY as well as a SUSY context, choosing G f = ∆(6 n 2 ) with n = 4 and for the parameter s characterizing the CP symmetry s = 1.
2 Approach and lepton mixing resulting from G f = ∆(3 n 2 ) and G f = ∆(6 n 2 ) and CP Here we present the approach we follow and the results obtained for lepton mixing from the groups G f = ∆(3 n 2 ) and ∆(6 n 2 ) discussed in [26].

Approach
We consider in the following a scenario with three RH Majorana neutrinos N i . These form a faithful and irreducible representation 3 of the flavor group G f . To the very same three-dimensional representation we also assign the three generations of LH leptons l. 2 The RH charged leptons α R , α = e, µ, τ are all assumed to transform trivially under G f , i.e. α R ∼ 1 under G f . In order to distinguish them we employ an auxiliary symmetry Z (aux) 3 under which they carry the charges 1, ω and ω 2 . The phase ω is defined as the third root of unity: ω = e 2 π i/3 . LH leptons and RH neutrinos are instead neutral under Z (aux) 3 . In addition, the theory is invariant under a CP symmetry whose action is represented by the CP transformation X r in the different representations r of G f . In general, this CP symmetry acts also non-trivially on the flavor space [29] (see also [30]). The matrix X r is unitary and symmetric in flavor space (for details see [21]) As shown in [21], the consistent combination of a flavor group G f and a CP symmetry, represented by X r , requires that the condition (X −1 r g r X r ) = g r is fulfilled for all elements g of G f with g being also an element of G f that is in general different from g. Here g r and g r indicate that both elements, g and g , are given in the representation r. Since we only make explicit use of the trivial representation and the representation 3 (and its complex conjugate), we only need to specify the form of the CP transformation in these two representations. Without loss of generality we can choose X 1 = 1, while the form of X ≡ X 3 is in general nontrivial. Its particular form is given explicitly in the different cases, see (28). Similarly, we only need to consider the representation matrices g r of the abstract elements g of the group G f in the trivial representation and in 3. Those belonging to the former representation always equal the identity, g 1 = 1, while we denote those of the latter, for simplicity, with the same symbol as the abstract elements themselves, i.e. g 3 ≡ g. 3 We focus on a non-SUSY framework (and comment on the implementation in a SUSY context in section 5 and appendix D) in the present paper and thus the form of the relevant Lagrangian is with H being the Higgs SU (2) L doublet and H c = H . Note that the Higgs field does neither transform under the flavor nor the CP symmetry of the theory.
The residual symmetry in the charged lepton sector is chosen as where Z (D) 3 is the diagonal subgroup of the Z 3 symmetry contained in G f and the additional Z 3 group Z (aux) 3 . The requirement to preserve G e in the charged lepton sector is equivalent to requiring that the Yukawa matrix Y l is invariant under this symmetry. For convenience, we choose a basis for all groups G f we discuss in such a way that the generator Q of the subgroup Z 3 contained in G f is diagonal. Thus, the charged lepton mass matrix m l is diagonal and it contains three independent parameters that can be identified with the charged lepton masses The value of the Higgs vacuum expectation value (VEV) is fixed to v EW = H ≈ 174 GeV in our convention. If the ordering of m e , m µ and m τ is not the standard one, we apply a permutation matrix P . Thus, the contribution of the charged leptons to the PMNS mixing matrix is in the case at hand given by U l = P (up to unphysical phases). In an explicit model the matrix m l can be generated when appropriate flavor symmetry breaking fields take a VEV leaving invariant Z 3 , see appendix D. In such a realization also the mass hierarchy of the charged leptons is usually explained either with the help of an additional Froggatt-Nielsen symmetry U (1) FN [40] under which RH charged leptons carry different charges, see e.g. [41], or it is generated through operators with multiple insertions of flavor symmetry breaking fields, see e.g. [42]. Notice the additional symmetry Z (aux) 3 , not present in the model-independent approach [26], does not have a direct impact on our results on lepton mixing and thus the results in subsection 2.2 hold without loss of generality.
Since LH leptons and RH neutrinos transform in the same way under all flavor and CP symmetries, the second term in (5) is automatically invariant under these symmetries and the Yukawa coupling Y D is proportional to the identity matrix in flavor space. The Dirac mass matrix of the neutrinos hence reads It is clear that the form of m D preserves G f and the phase of the parameter y 0 can always be chosen in such a way that the imposed CP symmetry remains intact. Throughout this paper we take y 0 to be positive, since the phase of y 0 can be absorbed in a re-definition of fields and is thus unphysical. Clearly, m D is, as a consequence, also invariant under all subgroups of G f and CP. The presence of the non-trivial residual symmetry that is the direct product of a Z 2 subgroup of the flavor symmetry G f and the CP symmetry instead manifests itself in the form of the Majorana mass matrix M R of RH neutrinos. Before presenting its explicit form we mention that the generator of the Z 2 group, denoted by Z in 3, and the CP transformation X, have to fulfill since the product of these two symmetries is required to be a direct one. Furthermore, we note that also ZX is a CP symmetry present in the neutrino sector, satisfying the condition in (10) [21]. The Majorana mass matrix M R of RH neutrinos is assumed to be invariant under the residual group with M i being the three RH neutrino masses and the unitary matrix U R is of the form The matrix Ω is determined by X and Z, R ij (θ) is a rotation matrix in the (ij)-plane through the real parameter θ, 0 ≤ θ < π, and K ν is a diagonal matrix with elements ±1 and ±i. The latter is necessary for making the masses of RH neutrinos real and positive and we parametrize it as Thus, the matrix M R contains in general four independent real parameters: the RH neutrino masses M i and θ. In an explicit model M R is generated, like the charged lepton mass matrix m l , via the couplings of RH neutrinos to flavor symmetry breaking fields. Obviously, the VEVs of the latter should leave the residual symmetry G ν invariant. The light neutrino masses originate from the type 1 seesaw mechanism [6] This matrix is diagonalized by U ν The light neutrino masses m i are inversely proportional to the RH neutrino masses M i Note that we omit the minus sign appearing in (16) in the following. The matrix U ν appears in (15), since the light neutrino mass matrix is given in the basis ν L m ν ν c L and we identify U ν with the matrix that brings the fields ν L into their mass basis so that the lepton mixing matrix U P M N S is given by In accordance with what has been shown in [21], in such an approach the form of the PMNS mixing matrix is fixed by the symmetries G f , CP, G e and G ν up to the free real parameter θ, possible permutations of rows and columns, since the masses of charged leptons and neutrinos are not predicted, and possible, but unphysical, phases. Consequently, all mixing angles and CP phases are strongly correlated, because they all only depend on θ. The latter has to be fixed to a particular value in order to accommodate the data on lepton mixing angles well and thus it has to be explained by some mechanism in an explicit model, see e.g. [22]. Since all lepton mixing angles θ ij have been measured with a certain accuracy by now [14], the admitted values of θ are usually constrained to a rather narrow range, even if the experimentally preferred 3 σ ranges for sin 2 θ ij , given in (2), are taken into account. In addition, we note that due to symmetries of the formulae for sin 2 θ ij two intervals of values of θ lead to good agreement with experimental data in several occasions. As mentioned, masses are generally not predicted in this approach, unless a particular model is considered, see e.g. second reference in [22]. Thus, we parametrize the three light neutrino masses in terms of the two measured mass squared differences for IO (18) and the lightest neutrino mass m 0 . For NO the light neutrino mass spectrum is thus of the form while we find for IO If m 0 0.1 eV > |∆m 2 atm |, the light neutrino mass spectrum is QD and m 1 ≈ m 2 ≈ m 3 . The best fit values obtained from the global fit analysis in [14] are ∆m 2 atm = 2.457 × 10 −3 eV 2 (NO) and ∆m 2 atm = −2.449 × 10 −3 eV 2 (IO) . For the 3 σ intervals of the mass squared differences see appendix A.2. The sum of the three light neutrino masses is constrained by cosmology and an upper bound is given by the Planck Collaboration (using TT, TE, EE and lowP constraints) [1] 3 k=1 m k < 0.492 eV at 95% C.L. .
As a consequence the lightest neutrino mass m 0 has to be smaller than Using (16) we see that the masses of the RH neutrinos are determined by the light neutrino mass ordering and by the lightest neutrino mass m 0 and thus fixed once the latter are fixed. The LO results are in general perturbed in an explicit model, e.g. the Dirac mass matrix of the neutrinos can receive contributions from flavor symmetry breaking fields dominantly coupling to charged leptons. Thus, we have in general (7), (8) and (11). The corrections δY l , δY D and δM R are suppressed with respect to the LO results by (powers of) the small (real, positive) parameter κ, κ 1. For simplicity, we include this suppression factor in the definition of the corrections. The latter are responsible for changes in the matrices that diagonalize m l , M R and m ν , i.e. the mixing matrices read leading to a PMNS mixing matrix of the form In the discussion of leptogenesis in our scenario we are particularly interested in the corrections δY D to the Yukawa coupling of the neutrinos. There are three principle possibilities: either the dominant correction δY D arises from fields coupling dominantly to RH neutrinos, then δY D is also invariant under G ν , or δY D arises from fields coupling dominantly to charged leptons, then δY D is expected to possess as residual symmetry G e = Z (D) 3 or δY D respects none of these residual symmetries, since combinations of both types of flavor symmetry breaking fields give the dominant correction. A particularly interesting case is the second one. We can parametrize the correction δY D in this case as with z 1,2 being complex numbers with absolute values of order one. These parameters are in general complex, because we do not require a residual CP symmetry to be preserved in the charged lepton sector. Note that the trace of δY D vanishes, since the correction proportional to the trace can be absorbed in the LO term. Note further that the corrections are normalized in such a way that the trace of the square of the matrices that accompany z 1 and z 2 , respectively, is the same, while the trace of the product of these two matrices vanishes. In section 3 we focus on leptogenesis in a scenario with δY D as in (27). We present sketches of a non-SUSY and a SUSY model in appendix D, in particular, in order to motivate the size of the parameter κ that we assume later in our phenomenological study, see (79).

2.2
Lepton mixing from G f = ∆(3 n 2 ) and G f = ∆(6 n 2 ) and CP In [26] the mixing patterns that can -for certain values of the parameters of the theory -fit the experimental data on the lepton mixing angles well [14] have been found in a scenario with G f = ∆(3 n 2 ) or G f = ∆(6 n 2 ), n being an integer, and CP. Like in [26], we require in the following that three does not divide n as well as for case 1) and case 2) also that n is even. The residual symmetries in the charged lepton and neutrino sectors are G e = Z 3 and G ν = Z 2 × CP . As regards the mixing there is no difference in considering as residual symmetry G e = Z 3 generated by Q = a (see definition of a in appendix B) or the residual symmetry is the diagonal subgroup of the Z 3 symmetry generated by Q = a and the additional Z 3 group Z (aux) 3 , since the relevant property, namely the fact that the charged lepton mass matrix is diagonal, is not altered. In [26] all possible choices of Z 3 and Z 2 groups and a certain set of CP transformations X have been studied. 5 In particular, the following representative cases that lead to different mixing patterns have been singled out ∆(3 n 2 ) , ∆(6 n 2 ) case 1) with s, t, m taking integer values in the interval 0 ≤ s, t, m ≤ n − 1, a, c and d (and b) being the generators of the group ∆(3 n 2 ) (and ∆(6 n 2 )), see appendix B and [26], and the matrix P 23 reading We note that the CP transformation X = P 23 in the representation 3 corresponds to the following automorphism acting on the generators a, b, c and d as Since in our scenario only the Majorana mass matrix of the RH neutrinos is non-diagonal, mixing solely originates from this sector. As shown in the preceding subsection, the diagonalization matrix U R and thus also the lepton mixing matrix U P M N S are given at LO by the matrix in (12), up to permutations of rows and columns.
The case X = P 23 in the charged lepton mass basis is well-known in the literature [28] and is called µτ reflection symmetry. Its predictions are maximal atmospheric mixing and maximal Dirac phase as well as trivial Majorana phases, while the reactor and the solar mixing angle remain in general unconstrained without additional symmetries.
In the following we repeat the form of the CP invariants J CP , I 1 and I 2 for the cases in (28), as these are relevant for our analysis of leptogenesis. We note that we add the results of a third Majorana CP invariant, called I 3 and defined in (181) in appendix A.1. Clearly, this third invariant is not independent of the two other ones I 1 and I 2 . However, it proves useful to employ I 3 in the discussion of leptogenesis, see e.g. formulae in (75)(76)(77). For the formulae of the mixing angles we refer the reader to [26].
In case 1) the PMNS mixing matrix is of the form where 5 These results have been confirmed in [25] and there also extended to the case in which Ge is not a Z3 symmetry.
In addition, we note that by permuting the rows of the mixing matrix U P M N S,2 = U R,2 , defined in (36), i.e. we consider either using the permutation matrix P 1 two slightly different mixing patterns can be obtained whose results for the CP invariants can be deduced from those given in (39)(40), if we apply either the transformations For details see [26]. We only notice here that applying any of the two sets of transformations also changes the sign in the approximate formula for the sine of the Majorana phase α given in (41). In case 2) replacing θ with π − θ does not affect the mixing angles, whereas the sign of J CP and I 2 changes and the CP invariants I 1 and I 3 do not transform in a definite way under this replacement. Nevertheless, using the approximate relation in (41) for sin α, we see that it does not change, if we only replace θ with π − θ. The two CP invariants I 1 and I 3 can be made transforming in a definite way by additionally sending k 1 into k 1 + 1 and v into k n − v (with k being an odd integer chosen in such a way that k n − v is still in the admitted interval for the parameter v). Then, both, I 1 and I 3 , change sign so that all CP invariants change sign. This statement is in agreement with the observation that sin α in (41) is not affected by the replacement of θ by π − θ alone, but it changes sign, if we change k 1 into k 1 + 1.
The last approximation holds, because θ ≈ π/2 (see [26] for details), and the plus (minus) sign refers to θ smaller (larger) than π/2. For s = n/2 in addition, sin δ is actually independent of θ [26] and the approximation in (51) is exact. So, we find in this case sin δ = ∓1, i.e. the Dirac phase is maximal, while the Majorana phases α and β are trivial for s = n/2. For case 3 a) we consider instead of U P M N S,3b1 = U R,3b1 in (46) the matrix without the permu- We find the same result for the Jarlskog invariant derived from U P M N S,3b1 and U P M N S,3a , while the CP invariants I i are permuted among each other J CP,3a = J CP,3b1 , I 1,3a = I 3,3b1 , I 2,3a = −I 1,3b1 and I 3,3a = −I 2,3b1 .
Obviously, the results for the mixing angles in case 3 a) are different from those obtained in case 3 b.1). Thus, the sets of parameters n, m, s and θ that lead to lepton mixing angles in accordance with the experimental data are different in the two cases, see the extensive analysis in [26] for details. We remark that the formulae of mixing angles and CP invariants in case 3 b.1) and case 3 a) have certain symmetry properties. Applying the set of transformations the mixing angles remain invariant, whereas the CP invariants change sign and thus also the sines of all three CP phases. For s = n/2, thus, sending θ into π − θ leaves the mixing angles untouched, while all CP phases change their sign. Using instead the transformations the solar and reactor mixing angle as well as the CP invariants I i remain unchanged, while sin 2 θ 23 becomes cos 2 θ 23 and J CP changes sign. In the particular case m = n/2, we can conclude that the replacement of θ with π − θ does not affect the solar and reactor mixing angle as well as the CP invariants I i , while the atmospheric mixing angle changes its octant and J CP its sign. If we set m = n/2 and s = n/2 and apply the transformation θ → π − θ, we see from (54) and (55) that the solar and reactor mixing angle are unchanged, the atmospheric mixing angle must be maximal, the CP invariants I i must vanish and J CP changes sign and is in general non-zero. For more symmetry transformations of this type see table 6 in [26]. As in case 2), mixing matrices arising from permutations of the rows of the PMNS mixing matrices derived from U R,3b1 and U R,3a do also admit good agreement with the experimental data on lepton mixing angles for certain choices of the parameters. The formulae for mixing angles and CP invariants can be obtained from those found for the PMNS mixing matrices with non-permuted rows by taking into account shifts in the parameters m and θ. Again, details can be found in [26].

Leptogenesis
We first collect several pieces of information regarding leptogenesis in general. We continue with the discussion of leptogenesis in our framework. In particular, we determine conditions on the form of Y D under which the CP asymmetries i can be directly related to the low energy CP phases. We also study the results for i , assuming a generic form of the PMNS mixing matrix. Subsequently, we turn to the detailed analysis of leptogenesis in the different scenarios of mixing, case 1) through case 3 b.1) and case 3 a). In doing so, we separate our discussion in an analytical and a numerical part. In order to not expand the latter too much we concentrate on case 3 b.1) when studying the third class of mixing.

Preliminaries
As already mentioned, we focus on unflavored leptogenesis as mechanism to generate correctly the measured value of the baryon asymmetry of the Universe Y B = (8.65 ± 0.09) × 10 −11 [1]. Thus, we assume RH neutrino masses to be larger than 10 12 GeV [43]. Since the RH neutrino mass spectrum is not expected to be strongly hierarchical, we consider the contributions of all three RH neutrinos to the generation of the baryon asymmetry of the Universe The quantities Y Bi correspond to the part of the baryon asymmetry produced by the i th RH neutrino. The latter can be expressed in the following way [44] Y Bi ≈ 1.38 × 10 −3 The numerical value in (57) depends on the yield of RH neutrinos at high temperatures and on the sphaleron conversion factor, while i is the CP asymmetry, generated in decays of the RH neutrino N i . The efficiency factor η ij parametrizes the washout and decoherence effects of the lepton charge asymmetry generated in N i decays due to lepton number violating interactions which involve the states N j present in the thermal bath at temperatures T ∼ M i . The CP asymmetry i , arising from the decay of the RH neutrino N i with the mass M i , is defined as and the loop function f (x) in the Standard Model (SM) reads [45] f (x) = x Notice that the quantities i are specified as the CP asymmetries for anti-leptons such that the sign of i is the same as for Y Bi in (57). The matrixŶ D is given by the neutrino Yukawa coupling Y D in the RH neutrino mass basis and thus reads with our conventions, see (5) and (11), The efficiency factors η ij are expected to lie in the interval 10 −3 η ij 1, see comments at the end of subsection 3.3 and figure 2. Their computation requires in general the numerical solution of Boltzmann equations. However, in our scenario several instances allow for a simplified treatment. The three different RH neutrinos N i couple at LO to orthogonal linear combinations of the lepton flavors, since we find that the Yukawa couplingŶ D is proportional to the unitary matrix U R , if we take into account that Y D is proportional to the identity matrix at LO, see (8). The RH neutrino masses are taken to be larger than 10 12 GeV so that lepton flavor dynamics is not resolved at the temperatures relevant for leptogenesis and the Boltzmann equations of the three orthogonal lepton charge asymmetries generated by the decays of each RH neutrino are almost independent [44]. In the RH neutrino mass basis the efficiency factors η ij thus reduce to Furthermore, we constrain RH neutrino masses to be smaller than 10 14 GeV. Thus, possible washout effects due to scattering processes which violate lepton number by two units are out of equilibrium and the efficiency factors η ii in (61) can be approximated well as [46] withm i being defined asm Using that the RH neutrino masses M i are directly related to the light neutrino masses m i at LO, see (16), and are, in particular, not degenerate, we can show that the relative mass splitting of any pair N i and N j (i = j) satisfies the bound for y 0 given in (69) and typical values of the expansion parameter κ, see (79). Thus, the perturbative expansion of the CP asymmetries in Y D in (58) is still valid, see e.g. [47,48]. Similarly, thanks to the relation in (16) we can express the argument of the loop function f (x) in (58) in terms of m i In addition, using that Y D is proportional to the identity matrix at LO, see (8), and that (16) holds, we findm Thus, also the efficiency factors η ii are functions of the light neutrino masses We can distinguish two regimes: form i ≈ m i 1.1 × 10 −3 eV we are in the weak washout regime, while for larger values of m i in the strong washout regime, see [43]. In the first case the neutrino Yukawa interactions are so small that the number density of RH neutrinos does not reach thermal equilibrium. Consequently, there is a partial cancellation between the anti-lepton asymmetry generated during the production of the RH neutrinos and the lepton asymmetry produced by their subsequent decays. In the second case the abundance of RH neutrinos matches the equilibrium density and a sufficiently large lepton asymmetry can be entirely realized from the out-of-equilibrium decays of RH neutrinos. Given the strong correlation between light and RH neutrino masses, focussing on the range 10 12 GeV M i 10 14 GeV also constrains the masses m i . In particular, we use as range for the lightest neutrino mass m 0 in our numerical analysis This corresponds, if we fix the mass of the heaviest RH neutrino to 10 14 GeV, to the following interval of the Yukawa coupling y 0 0.04 y 0 0.6 .
In this case the masses of the two lighter RH neutrinos automatically lie in the interval 10 12 GeV M i 10 14 GeV. As one can see, the coupling y 0 is (slightly) smaller than an order one number. 6 As has been noticed in [32,33], if RH neutrinos transform in an irreducible three-dimensional representation of a flavor group G f , the CP asymmetries i vanish in the limit of exact symmetry in the neutrino sector, since the combinationŶ † DŶ D is always proportional to the identity matrix. This also happens in our scenario. For this reason, corrections δY D to the Yukawa coupling Y D of the neutrinos play a crucial role for having successful leptogenesis. We can expand the matrix Y † DŶ D which enters in the expression of the CP asymmetries (58) aŝ at lowest orders in the correction δY D . The first term in this expansion is proportional to the identity matrix and therefore does not contribute to the numerator of i . However, the other two terms lead to off-diagonal entries in the matrix combinationŶ † DŶ D . This expansion in δY D also corresponds to an expansion in the parameter κ, simply because any correction δU R to the diagonalization matrix of the RH neutrino mass matrix can only be effective, if at the same time also the correction δY D is present. Thus, we expect from (58) and (70) This observation is in accordance with the results obtained in models with a flavor symmetry only [32][33][34]. Furthermore, using (58) and (70) we see that the leading term in the CP asymmetries is independent of the parameter y 0 . Since this statement does not hold for higher order terms in κ, we study this issue in our numerical analysis (see discussion of case 1) in subsection 3.5). We show that for y 0 in the range given in (69) and κ,κ, see definition in (74), chosen as in (79), the relative difference between the exact expression of the CP asymmetries defined in (58) and their LO expansion in κ is typically less than 10%.

Dependence of i on the low energy CP phases
We determine the conditions under which the dominant source of CP violation in leptogenesis is given by the low energy CP phases, contained in the PMNS mixing matrix. We thus study under which conditions the non-vanishing off-diagonal elements of the matrix combinationŶ † DŶ D depend only on the CP phases encoded in the mixing matrix U R = U P M N S . We perform this study at LO in the expansion parameter κ.
From (70) we see that CP violation relevant for leptogenesis is related to the Dirac phase δ and the Majorana phases α and β provided that the matrix combination fulfils one of the following conditions: is complex and antisymmetric. The first two possibilities could be ensured with the help of a CP symmetry. 7 Notice that, if conditions i) and iv) or conditions ii) and iii) are simultaneously realized, the CP asymmetries i become more suppressed, i ∝ κ 4 instead of proportional to κ 2 , see (71). It is thus not possible to reproduce the measured value of Y B for the expected size of κ, see (79) for an estimate.
In our scenario Y 0 D is real and proportional to the identity matrix, see (8), and the LO correction δY D has the form of a complex diagonal matrix, see (27), which is the most general form under the assumption that the dominant corrections to the neutrino Yukawa matrix arise from the charged lepton sector. The matrix combination (Y 0 D ) † δY D in (72) thus satisfies condition iii). Consequently, all our results for the CP asymmetries are independent of the phases of the parameters z 1,2 at lowest order. In our numerical analysis we also study the effect of the latter phases, entering in the subdominant terms, on the results for the CP asymmetries and find it to be typically less than 10%, if we compute the relative difference between the results using the expression of the CP asymmetries in (58), which include phases of the complex parameters z 1,2 in (27), and the corresponding LO expressions in κ (see discussion of case 1) in subsection 3.5).
Finally, we note that also in several models with the flavor symmetry A 4 only [32][33][34]36] the LO results of the CP asymmetries i turn out to be independent of the phases of the correction δY D and to only depend on the phases appearing in the diagonalization matrix U R that are identified with the Majorana phases at low energies. This result can be traced back to the fact that the matrix combination (Y 0 D ) † δY D in (72) fulfils condition iii) also in these models.

General results in our framework
Before discussing explicitly the results for the different cases, case 1) through case 3 b.1) and case 3 a), we would like to work out the form of the CP asymmetries obtained for U R = U P M N S . In this case mixing angles and CP phases are not assumed to be predicted by any flavor or CP symmetry, but only to have values compatible with the experimental data. We thus parametrize U R = U P M N S using the convention given in (176) in appendix A.1. The form of the neutrino Yukawa coupling Y D is taken to be the sum of the LO term Y 0 D in (8) and the correction δY D in (27). As argued in the preceding subsection, at the dominant order in κ only the real parts of the parameters z 1,2 , see (27), enter the expressions of the CP asymmetries. Thus, we define Re(z 1 ) = z cos ζ and Re(z 2 ) = z sin ζ .
We assume z ≥ 0 and ζ to lie between 0 and 2 π. The vanishing of one of the parameters z 1 and z 2 refers to particular choices of ζ, ζ = π/2, 3π/2 and ζ = 0, π, respectively. As explained in appendix D, in an explicit model such special values can be achieved, for example, with a particular alignment of the VEV of a flavor symmetry breaking field. Since κ is always accompanied by z, we furthermore define as (effective) expansion parameterκ The analytic expressions of the CP asymmetries at LO inκ in terms of the mixing angles, CP phases and ζ are in general rather lengthy. However, they considerably simplify, if we choose z 2 = 0 or equivalently ζ = 0, π, The first line of these equations tells us that the sign of all CP asymmetries depends on the sines of the Majorana phases α and β as well as on a possible sign coming from the loop function f (m i /m j ), i = j. Especially, the lepton mixing angles θ 13 and θ 12 do not have any influence on the sign. As we see, the CP asymmetries i do neither depend on the Dirac phase δ nor on the atmospheric mixing angle θ 23 . In the second line of these equations we have used the CP invariants I 1,2,3 given in appendix A.1. This establishes the connection between low and high energy CP violation. In order to understand the sign of the CP asymmetries and to estimate the size of the expansion parameterκ, necessary for achieving i that permits a sufficiently large baryon asymmetry, we first briefly analyze the behavior of the loop function. The light neutrino mass spectrum, whether it follows NO or IO, affects the value of the loop function f (x). In particular, it affects also the sign of f (x) as shown in figure 1. We display the behavior of the loop function f (x) for the six different arguments m i /m j , i = j, i, j = 1, 2, 3, with respect to the lightest neutrino mass m 0 . In the panels on the left we assume light neutrino masses with NO, while in the right ones they are i, j = 1, 2, 3, with respect to the lightest neutrino mass m 0 . Plots on the left (right) side correspond to a light neutrino mass spectrum with NO (IO). Note that m 0 is given by m 1 in the former and by m 3 in the latter case. The mass squared differences ∆m 2 sol and ∆m 2 atm are taken to be in their experimentally preferred 3 σ range given in [14] and reported in (184) and (187) in appendix A.2, respectively. This variation explains the width of the red and blue bands.
supposed to follow IO. The solar and atmospheric mass squared differences, ∆m 2 sol and ∆m 2 atm , take values in their experimentally preferred 3 σ ranges given in [14] and reported in appendix A.2, see (184) and (187). These ranges determine the width of the blue and red bands in the six plots of figure 1. This is particularly relevant for a light neutrino mass spectrum with IO, since m 1 and m 2 are almost degenerate in this case. If we assume a hierarchical light neutrino mass spectrum, namely for m 0 4 × 10 −3 eV, We also estimate the size of the CP invariants using the experimentally preferred 3 σ intervals of the reactor and the solar mixing angles shown in appendix A.2 Thus, if the Majorana phase α is not small and the value of f (m 1 /m 2 ) and f (m 2 /m 1 ) is not close to zero for a particular choice of the lightest neutrino mass m 0 (only occurring for NO and for m 0 ≈ 4 × 10 −3 eV), we expect the contributions involving I 1 (sin α) to dominate over the others. As a consequence, the absolute values of the CP asymmetries 1 and 2 should be (much) larger than | 3 |. This suppression of 3 (in addition to the one coming from the loop function f (x)) can also be understood by noticing that 3 is proportional to sin 2 2θ 13 , whereas 1,2 are proportional to cos 2 θ 13 . With (75) the expected size of the CP asymmetry is found to be: for z being of order one and no particular enhancement or suppression of the loop function, i.e. f (x) is also of order one, the expansion parameter has to fulfill κ ,κ 10 −3 in order to achieve | 1,2 | 10 −6 which is the typical size of the CP asymmetry necessary to ensure successful leptogenesis [43]. Our estimate in (79) for κ is naturally reproduced in typical non-SUSY as well as SUSY models, see our sketches of models in appendix D.
In the other limiting case, z 1 = 0 and thus ζ = π/2, 3 π/2, one can show that all three CP phases contained in the PMNS mixing matrix enter the expressions of the CP asymmetries. In order to simplify the form of the latter, we thus consider cases in which two of the three CP phases are trivial. We focus here on the case in which the Majorana phase β and the Dirac phase δ are trivial, since this also happens in case 1), see CP invariants in (34). We find compact formulae for 1,3 , if the conditions in (80) are imposed, 1 ≈ −κ 2 2 π sin α 1 2 1 + sin 2 θ 13 sin 2θ 12 cos 2θ 23 + (−1) k δ sin θ 13 cos 2θ 12 sin 2θ 23 while 2 can be expressed in terms of the other two CP asymmetries This occurs, because Im (Ŷ † DŶ D ) 2 ij vanishes for ij = 13, 31 (not only at LO inκ, but exactly) in this case. At this order in the expansion parameterκ the CP asymmetries 1 and 3 depend on three different pieces: the sine of the non-trivial Majorana phase α, the loop function f (x) and a combination of trigonometric functions of the mixing angles that forms a square. Thus, the sign of 1 and 3 depends on the sign of sin α (and whether β = 0 or π) and the sign of the loop function. Like in the case above, we reach the conclusion that the sign of the CP asymmetries can be fixed with the knowledge of the Majorana phase(s). Results very similar to those in (81-82) are obtained, if the Majorana phase α is trivial instead of β. For completeness, we mention the formulae belonging to this choice of CP phases in appendix C. Moreover, we can analyze the case in which both Majorana phases are trivial, see also appendix C. All three CP asymmetries i are still non-zero in the latter case and all are proportional to sin δ. However, the sign of the CP asymmetries i does not only depend on the sign of sin δ (and the loop function f (x)) in this case, but also, for example, on the octant of the atmospheric mixing angle (cos 2θ 23 ≶ 0), see (195).
Before closing this general part we would like to also comment in a quantitative way on the efficiency factors η ii ≈ η(m i ), given in (62), that are necessary for the computation of the baryon Note that m 0 is given by m 1 in the former and by m 3 in the latter case. The mass squared differences ∆m 2 sol and ∆m 2 atm are taken to be in their experimentally preferred 3 σ ranges given in [14] and reported in (184) and (187)  In particular, for m i . In the case of NO light neutrino mass spectrum, the maximum efficiency is obtained in the production of Y B1 , since η(m 1 ) is the largest, unless the CP asymmetry 1 is suppressed. The strongest washout effects are expected for Y B3 , which is controlled by the heaviest neutrino mass m 3 . The opposite happens in the case of a light neutrino mass spectrum with IO, where the lightest neutrino mass is m 3 , while m 2 ≈ m 1 |∆m 2 atm |. This does not necessarily mean that 1,2 become much suppressed for IO, since the strong suppression of the baryon asymmetry Y Bi due to washout effects can be easily compensated by the large enhancement of the loop function f (m 1 /m 2 ) and f (m 2 /m 1 ) in the expression of the CP asymmetries, as shown in figure 1. The latter argument has also to be taken into account in the case of a QD light neutrino mass spectrum.

Analytical discussion
After the general analysis we discuss the features of case 1) through case 3 b.1) and case 3 a) in an analytical way first. Then, we also show a numerical study for examples of each of the cases.

Leptogenesis in case 1)
We turn now to the predictions of the baryon asymmetry of the Universe for case 1). Accordingly, we express the CP asymmetries, defined in (58), in terms of the relevant parameters which characterize the lepton mixing angles and CP phases of the PMNS mixing matrix in this case, namely θ, the group theoretical quantities n and s (in the combination φ s ) as well as k 1,2 -see (31)(32)(33)(34)(35). The expressions at LO in the expansion parameterκ, given in (74), are Here I 1,3 (θ → θ + ζ) have to be read as meaning that the free parameter θ is shifted by ζ, which characterizes the correction δY D , see (27) and (73). Thus, the CP asymmetries i can be formally written in terms of the CP invariants I 1 and I 3 given in (34) (I 2 is zero anyway). For ζ = 0, π the expressions in (86) coincide with the CP invariants I 1,3 of case 1) and thus the formulae in (83-85) represent a special case of the general ones found in (75)(76)(77). We can also compare the formulae in (83-85) with the ones in (81-82) that have been obtained for ζ = π/2, 3π/2 and a non-zero Majorana phase α only. In doing so, we only have to remember that sin 6 φ s can be identified with sin α, see (35). In particular, we see that case 1) offers an example in which 1 and 3 only receive one contribution and 2 can be written in terms of the former two ones as in (82). Furthermore, with this explicit case we can also confirm several of the observations made in subsection 3.1, e.g. no explicit dependence of the CP asymmetries on the Yukawa coupling y 0 at LO, i are proportional toκ 2 and the phases of the parameters z 1,2 only enter at higher orders inκ, i.e. these terms are suppressed by (κ/y 0 ) σ with σ ≥ 1 with the respect to the LO terms shown in (83-85). The sign of the CP asymmetries is only determined by φ s and by the one of the loop function f (x) for a given value of m 0 and, in particular, is independent of the values of the parameters θ and ζ. The identification of sin 6 φ s with sin α also shows that in the case of no low energy CP violation, s = 0 or s = n/2, also no CP violation occurs at high energies. We note that replacing s by n − s reverses the sign of all three CP asymmetries i This equality holds exactly. Furthermore, we note that at LO inκ the CP asymmetries are periodic functions in ζ with the period π i (ζ) = i (ζ + k π) , for k = 0, 1, 2, . . .
As can be checked by explicit computation, this does not hold, if we include the higher order terms inκ in (83-85). Similarly, at LO inκ the CP asymmetries i remain invariant, if we replace θ by π − θ and ζ by k π − ζ where k is an integer chosen in such a way that k π − ζ lies in the fundamental interval [0, 2 π). Thus, we expect to obtain very similar results (for a different value of the parameter ζ however) for θ in the two different admitted intervals, 0.169 θ 0.195 and 2.95 θ 2.97, see also table 1. Differences, indeed, only arise at orderκ 3 at most.

Leptogenesis in case 2)
In this case the parameters which determine the lepton mixing angles and CP phases are φ u and φ v that characterize the chosen CP transformation X (remember u and v are related to s and t, see (38)), the free parameter θ and k 1,2 . Computing the CP asymmetries at the lowest order inκ we obtain Similar to case 1), we can express the LO results for the CP asymmetries in terms of the quantities I i , if we shift the group theoretical parameter φ u by appropriate multiples of ζ. Again, only if the latter is taken to be 0 or π, we re-cover expressions that depend on the CP invariants I i , as defined in (39)(40), and thus on the sines of the Majorana phases α and β, while the Dirac phase δ does not appear. In this special case we can also match to the formulae found in (75)(76)(77) where the PMNS mixing matrix is of a general form. In addition, we find that i in (89-91) fulfill at LO the equality in (88), i.e. the expressions are invariant under the shift of ζ in ζ +k π where k is an integer. In contrast to case 1), it is not straightforward to determine the sign of i just by looking at (89-91). However, given that θ is close to 0, π/2 or π, the terms proportional to sin 2 θ are suppressed compared to those proportional to cos (φ u + 2ζ) ± cos 2θ ≈ cos (φ u + 2ζ) ± 1, unless sin φ v is small. An example for the latter case is v = 0 (if this choice of v is admitted) and we see that with c 1,2 being expressions of the loop function f (m i /m j ) and k 1,2 . This shows that the terms in i become proportional to sin (φ u + 2ζ) or sin 2 (φ u − ζ) and thus for fixed u (φ u ) crucially depend on the choice of ζ; for example, changing the latter into ζ ± π/2 changes the overall sign of the CP asymmetries. This eventually explains why in our numerical analysis for v = 0 positive and negative values of the baryon asymmetry of the Universe are equally likely, see light-blue areas in the upper-left panel of figure 6. As a consequence, a prediction of the sign of Y B , depending on the low energy CP phases, is not possible in this case. The expression in (92) also shows that the choice of the parameter θ becomes relevant, in particular whether θ π/2 or θ π/2, and thus changing the admitted interval of θ inevitably changes the sign of the CP asymmetries. 9 If v = 0, however, we expect that our predictions of the CP asymmetries only slightly differ for θ being replaced by π − θ, since the dominant terms in i are then proportional to sin φ v , i.e. sin α see (41), and in turn depend on cos 2 θ. We can also relate the expressions found in case 2) to those in case 1). As has been shown in [26], this is possible, if we set θ = 0, identify the parameters of case 2) φ u and v with 2 θ 1 and 6 s 1 of case 1), respectively, assuming that the group index n is the same in both cases, as well as replace k 2 by k 2 + 1. Then, i of case 2) coincide with those of case 1), in particular I 2 (for any argument) vanishes. Another way to relate case 2) and case 1) that has been mentioned in [26] is to set u = 0 (φ u = 0) and to identify v with 6 s 1 . Albeit the solar and reactor mixing angles and the two Majorana phases (β is trivial) are the same in both cases, 10 the CP asymmetries are different for case 1) and case 2), since, in particular, i in case 2) include additional terms proportional to . Only for ζ = 0, π the CP asymmetries i of case 1) and case 2) do coincide.
In addition, we can consider leptogenesis in the case in which the PMNS mixing matrix/the matrix U R is given by one of the two matrices in (42). As explained in subsection 2.2, the formulae for mixing angles and CP invariants can be derived from those given for case 2), i.e. for U P M N S /U R as in (36), using the transformations displayed in (44) and (45), respectively. In the same vein, we can obtain the formulae for the CP asymmetries i in these cases. This is relevant for our numerical discussion, since we study an example in which the permutation matrix P 1 is applied from the left to the matrix in (36). In doing so, we assume that the additional permutation originates from the Majorana mass matrix of the RH neutrinos and for this reason we have included it in the matrix U R . However, in principle we can also consider a different situation in which U R is still given by (36), whereas the PMNS mixing matrix is given by one of the matrices in (42). If so, the permutation has to arise from the charged lepton sector due to non-canonically ordered charged lepton masses. One may wonder whether this difference can lead to new results. Indeed, this is not the case and it is no restriction of our discussion to focus only on the scenario with the permutation originating from the RH neutrino sector. The other case can be simply obtained from the latter one by a permutation P of the elements of the (diagonal) correction δY D , i.e.
Thus, we only need to re-define the parameters z 1 and z 2 appropriately. Clearly, this affects (mainly) the explicit value of the parameter ζ corresponding to a certain choice of the ratio of the real parts of z 1 and z 2 , see (73), but not the general conclusion on the size and the sign of the baryon asymmetry of the Universe, obtained in the different cases.

Leptogenesis in case 3)
We finally consider the predictions of Y B for the third type of mixing patterns, which are classified as case 3 b.1) and case 3 a). As we show, the CP asymmetries predicted in the two cases are closely related.
The analytic approximations of the CP asymmetries in case 3 b.1) are expressed at LO inκ as a function of the group theoretical quantities φ m , φ s , the free parameter θ, k 1,2 as well as ζ, that is Like in the other cases, we have arranged the LO expressions of i in terms of the CP invariants I 1 , I 2 and I 3 , defined in (49)(50). This requires the group theoretical parameter φ m to be shifted into φ m + ζ as well as to add a further piece We also confirm in this particular case all statements made in the general part, regarding the dependence on the Yukawa coupling y 0 , the expansion inκ and the appearing of the imaginary parts of the parameters z 1,2 in the subdominant terms only. We can furthermore check that the LO expressions of the CP asymmetries in (94-96) are periodical in ζ with periodicity π. As in the other cases, this does not hold at higher orders inκ. In addition, we find, like in the general case (see (75)(76)(77)), that for ζ = 0, π the CP asymmetries can be written in terms of the CP invariants I i in (49)(50) which shows that there is no explicit dependence on the Dirac phase. As discussed in [26], the replacement of θ with π − θ does not give rise to a symmetry transformation in case 3 b.1), unless we also replace m (φ m ) with n − m (π − φ m ) or s (φ s ) with n − s (π − φ s ), see (54)(55). For this reason and because R ± (ζ) also change sign for θ → π − θ and s → n − s, we can make the following observation This equality holds for all three asymmetries, at all orders inκ and for all choices of the parameters z 1,2 . We use this fact in our numerical analysis and only display results for s ≤ n/2. If m = n/2, like in the example studied in subsection 3.5, we see that changing θ to π−θ becomes a symmetry transformation, as defined in [26]. Thus, two admitted intervals of θ are expected. We, however, only discuss results for one of them in subsection 3.5, since the CP asymmetries i are likely to be very similar for both intervals θ. The reasoning is as follows: replacing θ with π − θ does not affect the CP invariants I i and thus the Majorana phases, while it changes the sign of J CP ; at the same time, we know that the CP asymmetries i mostly depend on sin α; in the special case in which only δ is non-trivial we have already argued in subsection 3.3 (see also appendix C) that fixing the sign of Y B becomes impossible. We can confirm the latter statement in the case at hand by studying the expressions in (94-96) for s = n/2 in addition to m = n/2 (assuming that n is even) Here we clearly see that we cannot predict the sign of the CP asymmetries i , since it crucially depends on the choice of the free parameter ζ. It also crucially depends on whether θ is smaller or larger than π/2. Thus, changing the admitted interval of θ leads to a change in the sign of i . Notice the similarity between the formulae in (99-101) and those in (92) valid for case 2). In addition, we find that the CP asymmetries are zero at LO inκ for ζ = 0, π/2, π, 3π/2, while they take maximal positive (negative) values for ζ = π/4, 5π/4 (ζ = 3π/4, 7π/4). If we set either the imaginary part of z 1 or z 2 to zero, it turns out that i vanish at all orders inκ for ζ = 0, π and not only at LO. The vanishing of i (at LO) for ζ = 0, π is consistent with the formulae in (75-77) obtained in the general case, since case 3 b.1) entails trivial Majorana phases for m = n/2 and s = n/2. In the same vein, i = 0 at LO for ζ = π/2, 3π/2 can be matched to the formulae in (193)(194)(195) in appendix C, if we take into account that case 3 b.1) predicts for m = n/2 and s = n/2 maximal θ 23 and δ. Moreover, we comment on the results for the CP asymmetries i in case 3 a). These can be easily obtained from (94-96), making the following replacements and using the relations among the CP invariants of the two cases, see (53). Finally, if we consider U P M N S /U R of case 3 b.1) and case 3 a) with rows permuted (similar to what is shown in (42) for case 2)), the given formulae for the CP asymmetries i in (94-96) and (102-104) can still be applied, as long as we use the same replacements of the parameters m and θ that need to be taken into account when computing the lepton mixing angles and CP invariants, see discussion at the end of subsection 2.2.

Numerical discussion
We first summarize our specific choices of several of the involved parameters that we use throughout the numerical discussion of the different cases and then show results for examples of case 1), case 2) as well as case 3 b.1). We also comment on the results expected for case 3 a). We note that we always use the exact expressions for the CP asymmetries i in our numerical analysis, i.e. those that are not expanded in the parameterκ and are instead computed using (58) and the corresponding form of the mixing matrix U R,i , see (31,36,42,46) and (52).

Preliminaries
The mixing matrix U R contains in all cases the two parameters k 1 and k 2 encoded in the matrix K ν , see (13). In the following, we set, if not stated otherwise, Furthermore, we fix the heaviest RH neutrino mass to M 1(3) = 10 14 GeV for NO (IO) .
Using (16) we can then express y 0 as The lightest neutrino mass m 0 is chosen in the interval in (68) and y 0 , consequently, falls into the range in (69). The other neutrino masses (light and heavy ones) are then determined by the two experimentally measured mass squared differences, see (19)(20)(21). In particular, the masses of all three RH neutrinos vary between 10 12 GeV and 10 14 GeV in this setting.
The expansion parameterκ is taken as constant and fixed tõ Thus, when z varies, also the expansion parameter κ has to slightly vary. The value ofκ in (108) is appropriate in order to achieve large enough CP asymmetries, since it satisfies the bound in (79). The parameter ζ that defines the relative size among the real parts of the two parameters z 1 and z 2 , see (73), is either fixed to specific values ζ = 0, π equivalent to Re(z 2 ) = 0 or ζ = π 2 , 3 π 2 equivalent to Re(z 1 ) = 0 or is taken to lie in the intervals I i which is equivalent to constraining the real parts of In our numerical discussion these different choices of the parameter ζ are indicated in different colors in the figures ζ = 0, π in red , ζ = π 2 , 3 π 2 in green and ζ ∈ I i in the light-blue areas.
If not otherwise stated, the imaginary parts Im(z 1,2 ) of z 1 and z 2 are set to zero in the following, since we have argued below (72) that these do not enter the expressions of the CP asymmetries i at the dominant order in κ.
In the discussion of the numerical example for case 1) we not only show figures of the baryon asymmetry Y B of the Universe, but also of the CP asymmetries i . Moreover, we present for this example results for both, NO as well as IO, light neutrino mass spectra. We also study the effect of choosing the two parameters k 1 and k 2 differently from our standard choice in (105). In addition, we analyze the impact and form of subleading terms in the expansion inκ as well as the effect of non-vanishing imaginary parts of the parameters z 1,2 . In the subsequent discussion of an example for case 2) and one for case 3 b.1) we focus on a light neutrino mass spectrum with NO and choose k 1 = k 2 = 0, since the relations to and changes coming from the other possible choices become apparent from the study for case 1). We also set the imaginary parts of z 1,2 to zero for case 2) and case 3 b.1) after having shown for case 1) that their impact on the final result for Y B is small.

Leptogenesis in case 1)
For all (even) n and all CP transformations with s = 0, . . . , n − 1, the values of the free parameter θ, which allow to reproduce the lepton mixing angles in accordance with the experimental data at  (35). In case s is even we find no CP violation at all and consequently vanishing CP asymmetries i . This information together with the results for the other lepton mixing parameters is gathered in table 1.
For n = 4 and s = 1 (meaning α = π/2) we show the predictions of the CP asymmetries i in figure 3 as a function of the lightest neutrino mass m 0 for a light neutrino mass spectrum with NO (left panel) and IO (right panel). We vary the parameter θ in the interval 0.169 θ 0.195 and the solar and atmospheric mass squared differences within their experimentally preferred 3 σ ranges, determined by the global fit analysis [14], see appendix A.2. First of all, we notice that the CP asymmetry 1 , generated in N 1 decays, is suppressed for ζ = π/2 and ζ = 3π/2, see green area in the upper panels of figure 3, since 1 is proportional to cos 2 (θ + ζ) at LO inκ, see (83), and θ is small. Conversely, the absolute value of 1 is maximized for ζ ≈ 0 and ζ ≈ π, as can be read off from the red area in the upper panels of figure 3. Similarly, 3 is suppressed for ζ ≈ 0 and ζ ≈ π, compare red area in the lower panels of figure 3, because it is proportional to sin 2 (θ + ζ) at LO iñ κ, see (85), while this proportionality leads to an enhancement of the CP asymmetry 3 for ζ ≈ π/2 and 3π/2. The behavior of 2 is more complex, if light neutrino masses follow NO, since there are two different contributions that can be of similar size, see (84). If m 0 is small, i.e. m 0 0.02 eV, one can see that for the special choices of ζ, see (109), only one of these dominates. For IO light neutrino masses instead 2 behaves very similar to 1 , because one of the two contributions clearly dominates. Regarding the width of the red and green areas in the different plots of figure 3 we see that these are mostly determined by the variation of the solar and atmospheric mass squared differences in their experimentally preferred 3 σ ranges, like in the case of the loop function f (x), compare to figure 1. In the particular case of the CP asymmetry 2 for NO light neutrino masses the reason for the broadness of the green band for larger values of m 0 , see left plot in the middle of figure 3, is not this variation, but the fact that there are two different contributions to 2 , see (84), that are of similar size in this parameter space. In the case of IO this does not happen, since the  In contrast, 3 is in this case positive for small m 0 3 × 10 −2 eV and negative for larger values of m 0 . It is, however, always strongly suppressed with respect to the other two CP asymmetries and thus irrelevant for the computation of Y B (unless ζ ≈ π/2, 3π/2), since such a strong suppression cannot be compensated by the efficiency factor η(m i ), see plot on the right of figure 2. The described behavior of the CP asymmetries i for NO and IO, respectively, explains the results obtained for the baryon asymmetry Y B , shown in the upper panels of toκ 2 , an increase ofκ by less than a factor of two is in this case sufficient. The situation strongly differs in the case of a light neutrino mass spectrum with IO, since there Y B is mostly driven by the CP asymmetries 1 and 2 that are both negative for s = 1, see panels on the right of figure  3. Only for m 0 ≈ 3 × 10 −3 eV also positive values can be achieved. However, this occurs only in a small portion of the parameter space for ζ (less than 10%). Moreover, the size of Y B is below the experimentally preferred lower 3 σ limit (the maximum value achieved is about 6.5 × 10 −11 ). Thus, we conclude that for s = 1 and k 1 = k 2 = 0 only for NO light neutrino masses and m 0 ≈ 10 −3 eV we can consider the sign as well as the size of Y B to be correctly explained.
The situation is reverse in the case s = 3, since in this case the sign of sin α is opposite. This is expected, since s = 1 is related to the choice s = 3 via the transformation where s is replaced by n−s, which changes the sign of all CP asymmetries, see (87). Thus, for m 0 3×10 −3 eV and NO of the light neutrino masses the sign of Y B can be correctly achieved for most of the choices of ζ. Even more pronounced is the situation for IO, because for s = 3 the sign of Y B is for (almost) all values of ζ correctly accommodated in the whole range of m 0 displayed, 5 × 10 −4 eV m 0 0.1 eV. Also the size of Y B can be correctly reproduced for some choice of ζ for both neutrino mass orderings.
In figure 5, we also consider different choices of the parameters k 1 and k 2 . In particular, we show in the plots on the left the effect of k 2 = 1. Most importantly, we see that for small values of m 0 , m 0 2.8 × 10 −3 eV and NO light neutrino masses Y B is positive for all admitted values of ζ, see (109-110). This leads to a unique prediction of its sign compared to the case in which k 2 = 0 and thus the sign is correctly determined for all values of ζ. It can be traced back to the change in the CP asymmetry 2 , compare left panel in the middle of figure 3 with the lower plot on the left of figure 5. At the same time, Y B is no longer positive for m 0 0.014 eV. On the right of figure 5 we see that for k 1 = k 2 = 1 and IO light neutrino masses Y B behaves very similar for s = 1 as for s = 3 and k 1 = k 2 = 0, see figure 4, showing clearly the importance of the choice of k 1,2 . In particular, the choice k 1 = 1 is needed for achieving the correct sign of the baryon asymmetry. This value of k 1 and s = 1 entail α = 3π/2 which is also obtained for k 1 = 0 and s = 3, compare (35).
The approximate formulae given in subsection 3.4 describe in most of the parameter space the results obtained with the formulae not expanded inκ quite well. To quantify this statement better we first derive the NLO terms inκ contributing to the three CP asymmetries i . The latter, expanded up to NLO inκ, can be written in the following compact form 12 where we have defined Im(z 1 ) = w cos ψ and Im(z 2 ) = w sin ψ with w > 0 being of order one and 0 ≤ ψ < 2 π, since also the imaginary parts of the two complex parameters z 1,2 enter at this level. We have, furthermore, made use of the LO expressions LO i of the CP asymmetries that can be found in (83-85). As one can see, the NLO terms are suppressed byκ/y 0 relative to the LO term. Leaving aside for the moment the effect of the imaginary parts of z 1,2 , by setting w = 0, we can check that the NLO terms are small compared to the LO term, if assuming as allowed range of y 0 the one given in (69). With our choice ofκ in (108) this bound is clearly fulfilled. Consequently, we find in our numerical analysis that for all admitted values of ζ the relative difference between the LO approximations in (83-85) and the results obtained with the formulae not expanded inκ is less than 10%. If we consider the impact of the imaginary parts of z 1,2 , we see that for w/z 1 the LO approximations still describe the unexpanded result very well. Indeed, in more than 90% of the parameter space of ζ, ψ and w with w/z 1 the relative difference between the two is less than 10%. Only if w is taken to be larger than z, we find a considerable decrease in the goodness of the LO approximations, i.e. a relative deviation of the latter from the unexpanded results less than 10% is only given in about 60% ÷ 70% of the parameter space. Similar results are also obtained in the numerical analyses of case 2) and case 3 b.1) [as well as case 3 a)]. This shows that the simple LO expressions are rather powerful in describing the results for the CP asymmetries i adequately.

Leptogenesis in case 2)
As example of case 2), we perform a numerical calculation of Y B for the group theoretical parameters n = 10 and u = 4 .
It is important to note that in this case the matrix U R (and thus also the PMNS mixing matrix) is given by the first matrix in (42), i.e. in order to achieve compatibility with the measured values of the lepton mixing angles we have to apply a cyclic permutation to the rows of the original matrix in (36). As a consequence, when using the formulae for the mixing angles, found in [26], the CP invariants given in (39)(40) and the LO approximations for the CP asymmetries i in (89-91), we have to apply the set of transformations in (44). For completeness, we mention the results of the  table 2 refer to k 1 = 0, compare (41). In the same vein, the sign of sin β given in this table holds for k 2 = 0 and changes for k 2 = 1, see (39). In figure 6 the variety of results for Y B , its sign and its size, are shown for three different choices of the group theoretical parameter v. As mentioned, in case 2) we only display plots for a light neutrino mass spectrum with NO and always set k 1 = k 2 = 0. The smallness of sin α for v = 0 entails a strong suppression of the otherwise dominant contribution to the CP asymmetries. A consequence of this suppression is that the sign of Y B cannot be predicted in this case, but depends crucially on ζ, see (92) with φ u being appropriately replaced according to (44). In this case, as can be checked by explicit computation, the Dirac phase δ provides the dominant source of CP violation in leptogenesis, yielding |Y B | 5 × 10 −11 for m 0 0.02 eV (see the light-blue area in the upper-left panel of figure 6). As in the numerical example of case 1) discussed before, the red and green bands represent the special choices ζ = 0, π and ζ = π/2, 3π/2, respectively. These confirm our analytic findings, see (92), that a shift in ζ by π/2 changes the sign of all the CP asymmetries and therefore of Y B . Notice that the red and green bands do not need to overlap with the light-blue area, since the intervals of ζ in (110) used to obtain the latter area do not contain the special values ζ = 0, π/2, π, 3π/2.  can be obtained, we do not consider this case as a successful one, since the value of Y B is equally likely positive and negative. In contrast, for v = 6 the sign of Y B can be univocally predicted for m 0 3 × 10 −3 eV and is in accordance with the experimental observations. The dominant contribution to the baryon asymmetry (with the correct sign) for NO is produced by the decays of the heaviest RH neutrino, N 1 , as expected from the general results given in subsection 3.3. For our particular choice ofκ in (108) we see that in the interval 7 × 10 −4 eV m 0 2 × 10 −3 eV the size of Y B can be correctly achieved. We also see that for m 0 3 × 10 −3 eV the sign of Y B is almost always negative. If we consider another choice of the parameters k 1,2 as in (105), we can make Y B positive for m 0 3 × 10 −3 eV. The situation is reverse for the choice v = 12, simply because the sign of sin α is opposite compared to the one in the case v = 6. So, for v = 12 the baryon asymmetry Y B is always negative for m 0 2 × 10 −3 eV and almost always positive for m 0 2 × 10 −3 eV. Consequently, only for values of m 0 larger than 6 × 10 −3 eV also the correct size of Y B can be obtained for a certain choice of ζ, if we keepκ fixed to the value in (108). In the range 6 × 10 −3 eV m 0 0.1 eV positive Y B is obtained for more than 90% of the choices of ζ. Again, like in the case v = 6 the sign of Y B can be changed by changing the values of the parameters k 1,2 . Comparing the results for v = 6 and v = 12 we note in addition that the size of |Y B | is a bit smaller for v = 12 than for v = 6, because | sin α| is smaller for v = 12 than for v = 6, see table 2.
Concerning the IO light neutrino mass spectrum, the results are very similar to the ones discussed in case 1), that is the dominant contribution to Y B comes from N 1 and N 2 decays, which can generate large CP asymmetries of order few times 10 −5 and with the correct sign. For this reason we do not show the corresponding plots of Y B . We only remark that, as for the NO light neutrino mass spectrum, in the case v = 0, the sign of Y B is not fixed, see (92), while for other choices of v the sign can be determined.

Leptogenesis in case 3)
A representative example for case 3) is given by n = 8 and m = 4 (119) which leads to good agreement with the experimental data on lepton mixing angles, if we consider case 3 b.1), as has been shown in [26]. The discussion of this case has two advantages: the value of the index n of the group ∆(6 n 2 ) is still moderately small and thus the group itself is not too large (it has 384 elements) and, at the same time, several values of the group theoretical parameter s, characterizing the CP transformation X, are admitted by the requirement to accommodate all three lepton mixing angles at the 3 σ level or better, so that different types of results for leptogenesis are achieved. Concretely, the viable choices of s are [26] s = 1, 2, 4 (120) as well as the values of s with s > n/2 = 4 that are related to the mentioned ones via n − s = 8 − s, i.e. s = 6 and s = 7. These are not discussed independently, since results for these cases can be deduced from those for s ≤ n/2 = 4, see table 6 and explanations in [26] regarding the mixing parameters and (98) for the CP asymmetries i . The results of lepton mixing angles and CP phases for the different choices of s are summarized in table 3. As one can see, all values of s give rise to a similar fit to the solar and the reactor mixing angles, while the results obtained for θ 23 differ. The value of the latter also depends on the interval chosen for the parameter θ, since replacing θ by π − θ changes sin 2 θ 23 into cos 2 θ 23 . Consequently, for most of the choices of s two intervals of θ are admitted that lead to different results not only for the atmospheric mixing angle, but also to a different sign in sin δ. In our numerical analysis of leptogenesis, we always stick to choices of θ belonging to the intervals displayed in table 3 and only briefly comment on results originating from other choices of θ. A feature of this example is m = n/2 = 4 which leads to a considerable simplification of the formulae for the CP phases, see (51). From there we see immediately that sin α and sin β must be equal up to a sign (for the particular choice k 1 = k 2 = 0 they are equal) for all choices of s and, moreover, that for s = n/2 = 4 both Majorana phases are trivial, whereas the Dirac phase is maximal. As shown in table 3 in our particular example the Dirac phase is large in all cases and also the Majorana phases are sizable, if they are not trivial. Furthermore, we note that sin α and sin β turn out to be negative for s = 1, while they are positive for s = 2. All these observations are relevant for the prediction of the sign of the baryon asymmetry Y B . We display the results for Y B depending on the light neutrino mass m 0 and for the different choices s = 1, s = 2 and s = 4 in figure 7. Like for case 2), we focus on a light neutrino mass spectrum with NO. Again, the areas in different colors represent different choices of the parameter ζ, see (109-111). As can be seen, the correct sign and size can be achieved for all three values of s. For s = 1 the sign of Y B is mostly negative for small m 0 , m 0 3 × 10 −3 eV. In fact, for such values of the lightest neutrino mass we have 1 < 0 and 2 > 0. Then, the dominant contribution to the baryon asymmetry (with a negative sign) comes from the heaviest RH neutrino, N 1 , provided | 1 | is not strongly suppressed (this suppression happens for ζ ≈ π/2, 3π/2). In the interval 3 × 10 −3 eV m 0 10 −2 eV both 1 and 2 are positive and thus the overall sign of Y B . This is also true for m 0 10 −2 eV as long as the contribution coming from N 1 is not suppressed due to the particular choice of ζ. Thus, we correctly predict the sign of Y B in more than 90% of the parameter space for 3 × 10 −3 eV m 0 0.1 eV. Regarding the size of Y B , this is correctly achieved forκ = 4 × 10 −3 , as long as m 0 5 × 10 −3 eV. The results for s = 2 are similar to those for s = 1, with the sign of Y B reversed, because the dominant contribution to the CP asymmetries that is proportional to sin α changes sign, as sin α is negative for s = 1 and positive for s = 2, see table 3. Consequently, only for small values of m 0 the sign is correctly reproduced. Y B compatible with its measured value at the 3 σ level or better prefers m 0 in the   The group theoretical parameter s, characterizing the CP transformation X, can take different values. Here we display those leading to good agreement (at the 3 σ level or better) with the experimental results on the lepton mixing angles and that fulfill s ≤ n/2 = 4. Results for s > n/2 = 4 can be easily deduced from the ones shown in this table using the transformation in (54). The equality of the absolute values of sin α and sin β originates from m = n/2 = 4, see (51). Setting k 1 = k 2 = 0 leads to sin α = sin β. Using (51) we can also check that s = n/2 = 4 entails trivial Majorana phases and a maximal Dirac phase. We display only one interval for θ. However, in most cases θ can also be in a second admitted interval. This is related to the one shown via the transformation of θ in π − θ. In this case, sin 2 θ 23 becomes cos 2 θ 23 and sin δ changes sign.
For further details see [26]. interval 6.8 × 10 −4 eV m 0 1.7 × 10 −3 eV. In this particular interval, the sign of Y B is positive in more than 80% of the parameter space. The size of the maximally achievable value of |Y B | is slightly smaller for s = 1 than for s = 2, simply because the dominant term is proportional to sin α that is smaller by a factor 1/ √ 2 ≈ 0.71 for s = 1 than for s = 2, compare also table 3. The remaining choice s = 4 allows to achieve the correct sign and size of Y B in some parameter space for larger m 0 , for which light neutrino masses become more and more degenerate. However, like for v = 0 in case 2), we cannot call this a prediction of the sign of Y B , since there is no preference for a certain sign observable in figure 7. This is explained by the absence of the leading term(s) in the CP asymmetries that are sourced by non-trivial Majorana phases. A way to see this is to use the formulae in (99-101) that are derived under the assumption that m = n/2 and s = n/2 in case 3 b.1). As one can see, the resulting CP asymmetries are proportional to sin 2ζ as well as to sin 2θ. Thus, considering ζ in the intervals in (110) leads equally likely to sin 2ζ > 0 as to sin 2ζ < 0. Consequently, the sign of Y B cannot be predicted in this case, only by fixing the group theoretical parameters, k 1 , k 2 and the interval of θ. In addition, we note that also a change in the interval of θ gives rise to a change in the sign of the CP asymmetries, since both intervals are connected by replacing θ with π − θ. According to the approximations in (99-101) the CP asymmetries are very small for the special choices ζ = 0, π and ζ = π/2, 3π/2, represented by the red and green bands in the figures. As already mentioned in subsection 3.4, for the first choice the CP asymmetries vanish exactly to all orders inκ, whereas for the second choice this statement is only true at LO. Nevertheless, also in the latter case the CP asymmetries become way too small for explaining the experimentally observed value of Y B . The two light-blue areas are delimited by the curves defined by ζ = π/4 (ζ = 3π/4) and ζ ≈ 0.24 (ζ ≈ 1.82). The former two choices refer to the maximally achievable values of Y B (with positive and negative sign, respectively). This can be understood by using the formulae in (99-101) which show that the LO expressions of all CP asymmetries i are maximized for maximal | sin 2ζ| which occurs, for example, for ζ = π/4 and ζ = 3π/4. The latter two choices describe the boundaries of the light-blue areas that lead to the smallest absolute values for Y B for a certain value of m 0 . They simply correspond to two limiting values of the intervals I i in which ζ can vary, if it does not take special values, compare (110). It is interesting to compare in more detail the results for s = 4 in this case with the ones obtained for v = 0 in case 2), see figure 6. As said, they share the feature that the sine of the Majorana phase α is small (or even exactly zero), see table 2 and 3, and thus the otherwise dominant contribution to the CP asymmetries i is absent. Furthermore, the Dirac phase is in both cases large, | sin δ| 0.8. However, there is also a crucial difference between them, namely the fact that for v = 0 in case 2) the Majorana phase β is large, whereas it is trivial for s = 4 in this case. Thus, comparing these two cases allows us to disentangle the effects of the CP phases β and δ on the resulting baryon asymmetry. Indeed, we see that the two figures, upper-left plot of figure 6 and lower-left plot of figure 7, are quite similar, with the striking difference that for m 0 2 × 10 −3 eV for s = 4, sin β = 0, Y B is very small, while Y B can reach values up to ±0.3 × 10 −11 for v = 0 in case 2), i.e. 0.83 | sin β| 0.94.
Finally, we comment on some peculiarities of examples belonging to case 3 a): clearly, for the (always admitted) choice s = 0 no non-vanishing CP asymmetry can be obtained, because in this case an accidental CP symmetry, common to charged lepton and neutrino sectors, exists, for details see [26]. Furthermore, we note that in some cases such an accidental CP symmetry arises from a particular choice of the parameter θ, see e.g. n = 16, m = 1 and s = 8 in which the best fitting point of θ is at θ bf = 0. Considering the whole experimentally preferred 3 σ ranges of the lepton mixing angles also non-zero values of θ are admitted. However, we expect then that CP phases are still small and thus the size of Y B is suppressed. Eventually, we notice that there are cases in which sin α = 0 is obtained at the best fitting point θ bf (in these cases the best fit value of the solar mixing angle cannot be accommodated). These are cases in which a prediction of the sign of Y B is impossible, since we expect a similar behavior as for s = 4 for case 3 b.1), discussed here, or for v = 0 in case 2).

Comments on flavored leptogenesis
In this section we briefly comment on the case of flavored leptogenesis which is realized if RH neutrino masses are smaller than 10 12 GeV [49][50][51] (and larger than 10 8 GeV in order to correctly reproduce the light neutrino mass scale [52] for y 0 10 −3 ). In particular, for 10 9 GeV M i 10 12 GeV the Yukawa interactions of RH charged leptons of τ -flavor are in thermal equilibrium and hence leptogenesis occurs in the so-called two-flavor regime. For even lower RH neutrino masses, M i 10 9 GeV, also the µ-flavor becomes dynamical.
The formula for the flavored CP asymmetry α i (α = e, µ, τ ) reads [43] α i where f (x) in the SM is given in (59) and Notice that the term in (121) with g(x) does not depend on Majorana phases, because it originates from the lepton number conserving, but lepton flavor violating one-loop contribution to the CP asymmetries. The analytic expression of the unflavored CP asymmetries i in (58) is recovered, if we sum over the flavor index α The contribution to Y B from the decay of each RH neutrino has a form similar to the expression in (57), but with α i instead of i and efficiency factors with an additional index, depending on the dynamical flavor.
In analogy to the study of leptogenesis performed in the unflavored regime, we consider a neutrino Yukawa matrix Y D of the form Y D = y 0 1 + δY D , with the correction δY D defined as in (27) and proportional to the small expansion parameter κ (κ). We can first determine the allowed interval of the parameter y 0 using (16) and taking into account 10 9 GeV M i 10 12 GeV 0.001 y 0 0.04 .
Applying (121) we can check that for our form of Y D the flavored CP asymmetries α i are proportional to the product of y 0 and κ (κ) at LO. Thus, corrections δY D are crucial also in this case in order to achieve non-vanishing CP asymmetries. However, the dependence on the parameters y 0 and κ is different than in the unflavored case in which i are independent of y 0 and scale as κ 2 . Since (123) holds, we see that the sum of α i over α must add up to zero at LO. Hence, we can always express one of the flavored CP asymmetries as the negative of the sum of the other two at this order. For y 0 varying in the interval in (124) the absolute value of the CP asymmetries α i is larger than 10 −6 , if we chooseκ 10 −3 . This lower bound coincides with the one necessary to obtain successful leptogenesis in the unflavored regime, see (79). As explained, this is also the expectation of the natural size of κ from model building. Another important feature of the LO result of the flavored CP asymmetries, common to the case of unflavored CP asymmetries, is the fact that the only source of CP violation relevant for leptogenesis is provided by the phases in the matrix U R , coinciding with the CP phases of the PMNS mixing matrix. The reason for this to happen is the symmetric form of δY D in flavor space so that the quantity (Ŷ † DŶ D ) ij , i = j, in (70) only depends on the real part of δY D at LO.
We first derive the predictions for the flavored CP asymmetries, assuming that U R = U P M N S with U P M N S being of its general form as in (176) in appendix A.1. In order to simplify the resulting formulae we assume in the following ζ = k ζ π with k ζ = 0, 1 (equivalent to vanishing real part of z 2 ) and the low energy CP violation to be encoded only in the Dirac phase δ, i.e. α = k α π and β + 2 δ = k β π, with k α,β = 0, 1 . (125) At LO inκ, the flavored CP asymmetries α 1 due to N 1 decays then read where the CP invariant J CP ∝ sin δ is defined in (178) and we have used relation (16). First of all, this example confirms that α i are proportional to the product y 0κ . It also reflects the property that the contribution containing g(x) must be independent of the Majorana phases, in particular of the low energy ones α and β, appearing in the PMNS mixing matrix. Furthermore, this example clearly shows that, unlike in the case of unflavored CP asymmetries, the sign of the CP asymmetries α i (and Y B ) cannot be predicted just from the knowledge of the sign of the sine of the CP phases, because it explicitly depends on k ζ which encodes the sign of the real part of the correction z 1 . Finally, we note that in this particular case e 1 ≈ 0 and τ 1 ≈ − µ 1 ∝ J CP , if the CP phase δ is (close to) maximal, δ ≈ π/2, 3π/2. The form of α 2 and α 3 is very similar to the one of α 1 , in particular, regarding relations between the CP asymmetries of the different flavors α as well as the dependence on the CP phase δ.
Concerning the different scenarios with a flavor and a CP symmetry we focus on the flavored CP asymmetries α i for case 1), since simple expressions can be derived. We obtain for α 1,3 at LO with ρ e = 3, ρ µ = 1 and ρ τ = −1. The formulae for α 2 are given by the linear combination in (82), replacing 1,3 with their flavored counterparts α 1,3 . Notice that the contribution to the CP asymmetries proportional to g(x) is absent in this case. This is due to the fact that the Dirac phase is trivial, see (34). Most importantly, the sign of α i (and Y B ) depends on the particular choice of the parameter ζ (the ratio between the real parts of z 1 and z 2 that parametrize the correction δY D , see (73) and (27)). This is in contrast to the unflavored case where ζ appears only as argument of positive semi-definite functions, cf. (83-85).
If Y D is only invariant under the residual symmetry G ν instead of the full flavor group G f and the CP symmetry, α i are in general non-zero even for vanishing corrections δY D . For Z and X representing the residual Z 2 and the CP symmetry in 3, respectively, Y D is in this case constrained by the conditions such that it can be in general written as with R ij (θ L,R ) being rotations in the (ij)-plane around θ L,R , lying in the interval [0, π), and y i real parameters. All these five are not further restricted by G ν . As a consequence, the light neutrino masses as well as the lepton mixing angles and low energy CP phases are less correlated with the parameters appearing in M R in (11), i.e. U ν (and thus U P M N S ) does not only depend on θ and the relation between the light and heavy neutrino masses in (16) does not hold except for the light neutrino mass m k with k = i, j. In particular, only for this generation the CP parity of the light neutrino is given by the one of the RH one N k , encoded in the matrix K ν . Transforming Y D in (129) to the mass basis of RH neutrinos using U R in (12), we find As can be checked, the flavored CP asymmetries, computed with (121), are proportional to sin 2(θ − θ R ) (but not θ L ), the product y i y j as well as the difference y 2 i − y 2 j . Thus, if any of these vanishes, i.e. θ = θ R + p π/2, p integer, or y i = 0 or y j = 0 or y i = ±y j , the CP asymmetries all must vanish. Furthermore, α k with k = i, j is always zero. The correct size | α i | 10 −6 can be achieved for y j 10 −3 . This constraint on the neutrino Yukawa couplings is consistent with the requirement to reproduce light neutrino masses of order 0.1 eV. Interestingly enough, computing the unflavored CP asymmetries i we see that they vanish exactly in this scenario, unless a small correction δY D , which is not invariant under G ν , is introduced.
In addition to this general discussion, we have explicitly studied the predictions for α i for all cases 1) to 3 b.1). In case 1) not only the unflavored CP asymmetries i , but also the flavored ones vanish identically. For this reason, we present the exact expressions of α i for case 2) with ρ α defined as below (127). All characteristics mentioned above can be verified: α 2 = 0, since the rotation R 13 (θ) acts in the (13)-plane; furthermore, α 1,3 are proportional to the product y 1 y 3 , to the difference y 2 1 − y 2 3 as well as to sin 2 (θ − θ R ); and the sum over all flavors α, see (123), vanishes. In addition, we observe that α 1,3 are not sensitive to the parameter φ v , which determines the Majorana phase α, see (41). 13 This result is in stark contrast with the ones of the unflavored CP asymmetries in (89-91) where the terms with φ v usually dominate, if v does not vanish. Again, the sign of Y B is not determined, since y 1,3 can be both positive and negative. This observation is contrary to the findings in the case of unflavored leptogenesis, see figure 6. The formulae for α i for case 3 a) and case 3 b.1) show a similar structure as those in (131) and are thus not explicitly discussed here.
Finally, we remark that for the particular choice φ u = 0, π (u = 0, n), the flavored CP asymmetries in case 2) in (131) fulfill This result is related to the presence of the µτ reflection symmetry [28], since X = P 23 or Z X = P 23 , if we set in addition φ v = 0 (v = 0), compare (28). A similar result can also be obtained for case 3 b.1): if we set m = n/2, we find e 2 (3) = 0 and µ 2 (3) = − τ 2 (3) (the CP asymmetries α 1 vanish in all flavors due to the form of the rotation R ij (θ)). The µτ reflection symmetry is then recovered for s = n/2 [26] and we find explicitly ZX = P 23 . 14 Flavored leptogenesis in a scenario with µτ reflection symmetry has also been discussed in [53]. In their analysis the authors only impose this symmetry on the neutrino mass matrix. A diagonal mass matrix for charged leptons is ensured by gauging L µ − L τ symmetry. The authors stick to the two-flavor regime and discuss the case in which the RH neutrino mass spectrum is strongly hierarchical. Thus, only the out-of-equilibrium decays of the lightest RH neutrino are relevant for generating the baryon asymmetry of the Universe. In [54] the authors focus on the flavor symmetries belonging to the series ∆(3 n 2 ) and ∆(6 n 2 ) combined with a CP symmetry, as in our analysis. They, however, concentrate on the case of flavored leptogenesis and a strongly hierarchical RH neutrino mass spectrum, which is similar to the study performed in [53]. Consequently, their results in general differ from ours.

Comments on SUSY framework
Here we comment on the implementation of our scenario and results for leptogenesis in a SUSY framework, since most of the concrete models with a flavor (and a CP) symmetry are formulated in a SUSY context, see e.g. [22,41]. The relevant superpotential of such a scenario reads with h u,d denoting the two Higgs multiplets of the Minimal SUSY SM (MSSM). The latter are taken to transform trivially under all non-gauge symmetries. We assign the three generations of RH neutrinos ν c to 3 under G f , while the fields l now transform as3. In this way, the term l h u ν c is invariant under G f and thus arises at the renormalizable level. In addition, the Yukawa coupling Y D is proportional to the identity matrix. The RH charged leptons α c are in the trivial representation of G f . Like in the non-SUSY framework, also here we assume the existence of an auxiliary symmetry Z (aux) 3 . Only the two fields µ c and τ c carry non-trivial phases under the latter symmetry: µ c ∼ ω and τ c ∼ ω 2 .
We note a few crucial differences relevant for our results of leptogenesis. First of all, the range of RH neutrino masses in which unflavored leptogenesis is the dominant generation mechanism of the lepton asymmetry is rescaled by the factor 1 + tan 2 β with tan β = h u / h d so that we have to require RH neutrino masses M i to lie in the interval 10 12 (1 + tan 2 β) GeV M i 10 14 (1 + tan 2 β) GeV. We can estimate the allowed values of tan β in our scenario by considering the operator that gives rise to the tau lepton mass. This operator requires (at least) one insertion of a flavor symmetry breaking field, since LH leptons transform as3 under G f , whereas τ c and h d are trivial singlets. Assuming that the size of the suppression due to the necessary insertion of one flavor symmetry breaking field is ε ≈ (0.01 ÷ 0.1), see (202) in appendix D, and that the tau Yukawa coupling varies between 1/3 and 3, we get as range of tan β the following 15 ε = 0.01 : tan β ≈ 3 , ε = 0.05 : 3 tan β 15 , ε = 0.1 : 3 tan β 30 .
Thus, RH neutrino masses are expected to be larger than in the SM case. In order to obtain the correct values for the light neutrino masses we have to rescale the coupling y 0 accordingly, see (16).
On the other hand, the masses of RH neutrinos cannot be too large either in a SUSY framework, since larger values give rise to larger contributions to flavor non-universal soft terms of sleptons through renormalization group running effects [55]. These flavor non-universal soft terms play a crucial role in charged lepton flavor violating processes [56] such as µ → eγ and µ − e conversion that are strongly constrained experimentally [57,58] (for a review see [59]).
Furthermore, large RH neutrino masses also require a large reheating temperature T R 10 12 GeV. This in turn gives rise to the well-known gravitino problem [60], unless e.g. the gravitino production is suppressed and/or it decays [61] and/or there is an additional substantial contribution to the total energy density of the Universe that dilutes the gravitino abundance, see discussion in [46].
In a SUSY framework the CP asymmetries (α) i not only arise from decays of the RH neutrinos ν c i to SM particles, but also from decays of their SUSY partners and to SUSY particles, see e.g. [46]. Thus, also the form of the loop functions f (x) and g(x) in the MSSM is different. In fact, the former now reads [45] f (x) = −x 2 Comparing its behavior to the loop function in (59), relevant in the SM, we notice the following: both of them can be zero, however, the exact position in x is slightly different (x ≈ 0.42 for the SM loop function and x ≈ 0.34 for f (x) in (134)); otherwise, their values are of the same order of magnitude for x 1; for x 1 it holds to good approximation that (f (x)) MSSM ≈ 2 (f (x)) SM . They have a divergence at x = 1 in common. Concerning the loop function g(x) which enters in the flavored CP asymmetries, we have the exact relation (g(x)) MSSM = 2 (g(x)) SM , where (g(x)) SM is given in (122). At the same time, additional washout effects are induced by the SUSY particles. The numerical factor in (57) also slightly changes and reads 1.48 × 10 −3 due to the additional particles. Moreover, the sphaleron conversion factor is modified due to the presence of a second Higgs doublet.
Summarizing all the effects and assuming leptogenesis to be the dominant mechanism for generating a lepton (and thus the baryon) asymmetry, the results for the baryon asymmetry of the Universe Y B,MSSM in the MSSM framework are obtained by rescaling Y B,SM , computed in an SM context, in the following way [43]

0νββ decay
In this section we study 0νββ decay of even-even nuclei. This process, unobserved so far, is important for testing the Majorana nature of neutrinos and it explicitly depends on the values of the Majorana phases α and β (see e.g. [12] for a recent review). Therefore, it is relevant in order to put constraints on the scenarios introduced in subsection 2.2. Earlier studies of 0νββ decay in the context of models with combined flavor and CP symmetries can be found in the first reference in [23], last reference in [22], in [39], first reference in [27] as well as in [25]. After a short subsection containing general information about the quantity measurable in 0νββ decay we discuss its predictions in different scenarios of lepton mixing separately, scrutinizing, in particular, the examples for which leptogenesis has been analyzed.

Preliminaries and general results
The half-life T 0νββ 1/2 of a decaying nuclear isotope via this process is where G 0ν denotes the phase space factor, M 0ν the nuclear matrix element (NME) for a lepton number violating transition, m e the electron mass and m ee the effective Majorana neutrino mass. The values of G 0ν and M 0ν can be computed and depend on the nuclear isotope, whereas m ee is expressed only in terms of neutrino masses and lepton mixing parameters, m ee = U 2 P M N S,11 m 1 + U 2 P M N S,12 m 2 + U 2 P M N S,13 m 3 , that, according to the parametrization of U P M N S , given in appendix A.1, reads m ee = cos 2 θ 12 cos 2 θ 13 m 1 + sin 2 θ 12 cos 2 θ 13 e iα m 2 + sin 2 θ 13 e iβ m 3 .
with the largest uncertainty arising from the one of the associated NME. For a hierarchical light neutrino mass spectrum, i.e. for m 0 ≈ 0 in (19) and (20), the expected value of m ee strongly depends on the ordering of the neutrino masses. In fact, in this limit we derive from (138), for NO m NO ee ≈ sin 2 θ 12 cos 2 θ 13 e iα ∆m 2 sol + sin 2 θ 13 e iβ ∆m 2 atm (140) and for IO m IO ee ≈ cos 2 θ 12 + sin 2 θ 12 e iα cos 2 θ 13 |∆m 2 atm | .
In the last expression we have neglected the subdominant term proportional to ∆m 2 sol . For a given value of θ 12 and θ 13 , the maximum and minimum of m ee are obtained for trivial Majorana phases, see e.g. the lower-left panel of figure 9. In particular, for IO, using that sin 2 θ 12 ≈ 1/3, we get In the case of a QD light neutrino mass spectrum, we expect from (138) m ee to be proportional to m 0 , for both NO and IO. Indeed, neglecting sin θ 13 , we have which, for sin 2 θ 12 ≈ 1/3 and a trivial Majorana phase α, takes values in the interval As is well-known, for NO m ee can be strongly suppressed due to cancellations for m 0 between 10 −3 eV and 0.01 eV. This can occur in principle, if both Majorana phases are trivial or both are nontrivial. In our numerical analysis we find examples for both situations, see e.g. case 1) for the former one (compare (153)) and case 3 a) for the latter one, see figure 10.
In figures 8-10 we show in light-blue (orange) the most general region in the m 0 − m ee plane for NO (IO), obtained by varying the lepton mixing angles within their experimentally preferred 3 σ ranges, see appendix A.2, and all CP phases between 0 and 2 π. The boundaries of the areas are associated with trivial Majorana phases α and β and do depend on the lower and upper 3 σ limits of the solar mixing angle θ 12 . The experimental constraints in the m 0 − m ee plane are set by the various 0νββ decay experiments, see (139), as well as by the Planck Collaboration that puts an upper limit on the lightest neutrino mass m 0 , see (23). The former is displayed as horizontal dashed line in figures 8-10, while the latter as vertical line.

Predictions of m ee in case 1)
In this scenario the Majorana phase β (as well as the Dirac phase δ) is always trivial, for any choice of θ and the group theoretical parameters n and s (or their combination φ s ), while α can take non-trivial values, see (35). Then, for a hierarchical neutrino mass spectrum (m 0 ≈ 0), m ee reads m NO ee ≈ 1 3 ∆m 2 sol + 2 (−1) k 1 +k 2 sin 2 θ e 6 i φs ∆m 2 atm , Remembering that θ is close to 0 or π, m IO ee can be simplified to with the plus (minus) sign corresponding to even (odd) k 1 . Similarly, we obtain for a QD light neutrino mass spectrum As one can see, for even k 2 the effective Majorana neutrino mass is independent of θ m QD ee ≈ with the plus (minus) sign valid for even (odd) k 1 . Since θ ≈ 0, π, (149) is a good approximation also in the case of k 2 being odd. This result is consistent with the one found in (143) for the generic case.
In the case n = 4, that is the smallest value of the group theoretical parameter n allowing for non-trivial α (see table 1), we obtain for k 2 even and s = 0, k 1 even (odd) or s = 2, k 1 odd (even) Here we used the best fit values of the neutrino mass squared differences, see (21), m 0 = 10 −4 eV for NO and IO as well as m 0 = 0.1 eV for QD, and θ ≈ 0.18 or θ ≈ 2.96 that lead to the best fitting of the experimental data on lepton mixing angles [26]. If we choose k 2 even (odd) and instead s = 1 or s = 3 as well as any value of k 1 , we find with the range of the intervals coming from the variation of the neutrino mass squared differences in their experimentally preferred 3 σ range and 0.169 θ 0.195 or 2.95 θ 2.97. For m 0 being in the first interval a cancellation requires k 1 odd (even) and k 2 even, while for m 0 in the second interval k 1 and k 2 are required to be odd (k 1 is required to be even and k 2 odd) in order to find m NO ee 10 −4 eV. We note that due to the constraints on lepton mixing angles and CP phases the range of m 0 in which m ee can be very small is considerably reduced with respect to the generic case. Especially, the smallest value of m 0 for which a cancellation can occur is larger than in the generic case because of the lower bound on the solar mixing angle, sin 2 θ 12 1/3. For s = 1, 3 such a suppression is not possible and we, indeed, find a lower limit on m NO ee that is m NO ee 0.0029 eV . (154)

Predictions of m ee in case 2)
In this case both Majorana phases α and β can have in general non-trivial values, depending on θ and the group theoretical parameters n, u and v (or their combinations φ u and φ v ). Taking the lepton mixing matrix U P M N S,2 as defined in (36), the general expression (137) for a hierarchical neutrino mass spectrum can be approximated as For a QD light neutrino mass spectrum and k 2 even, as in case 1), the resulting m ee is actually independent of θ m QD ee ≈ with the plus (minus) sign referring to even (odd) k 1 . For k 2 odd instead one can show that m ee is independent of φ u , if θ ≈ 0, π or π/2 (these values are typically required for reproducing the observed lepton mixing angles see [26]), with again the plus (minus) sign referring to even (odd) k 1 . This expression coincides with the one derived in the generic case in (143), if we use that in case 2) holds cos α ≈ (−1) k 1 cos φ v (a relation similar to the approximate relation in (41)).
Expressions like in (155-158) are also obtained, if we perform a permutation of the rows of U P M N S,2 , see (42). In this case m ee can be computed by applying the transformations given in (44)(45) to the approximations derived.
We note that the quantity m ee is invariant under the set of transformations (see below (45)) which shows that results for different values of v are related to each other, if we also take into account that the interval of θ has to be changed.
In the explicit example, we choose for the flavor group the index n = 10 and for the parameter characterizing the CP symmetry u = 4, like in the numerical study of leptogenesis in subsection 3.5. We thus discuss a case in which we can use the formulae shown in (155-157), only after having applied the transformations in (44). This choice of parameters predicts | sin β| ≈ 1, while | sin α| can take three different values, depending on the parameter v (or φ v ), i.e. v = 0, 6, 12, 18 and 24, see table 2. In figure 8 we display the predictions of m ee as function of the lightest neutrino mass m 0 for two different choices of v: v = 0 in the left and v = 6 in the right panel. These values lead either to α ≈ 0, π (v = 0) or to almost maximal α (v = 6). The blue and red regions in each plot correspond to the predictions for NO and IO, respectively, in which we allow for all four possible combinations of k 1 and k 2 . Among these, we highlight the choice k 1 = k 2 = 0 with the dark-grey area. Moreover, we vary θ in the range 1.40 θ 1.44 and the neutrino mass squared differences within their experimentally preferred 3 σ intervals, see appendix A.2. Using (159) we see that for v = 0 values of θ in the second interval 1.70 θ 1.74 lead to the same allowed areas, up to the exchange of k 1 = 0 with k 1 = 1. For v = 0 instead, applying (159) shows, for example, that the plot on the right panel in figure 8 for v = 6 and 1.40 θ 1.44 is the same as the plot for v = 24 and θ in the second interval 1.70 θ 1.74. The only difference is that the dark-grey area corresponds for v = 24 to the choice k 1 = 1 and k 2 = 0 instead of k 1 = k 2 = 0. The predictions of m ee for v = 12 and v = 18 are related to each other in a similar way. Figures for these two values of v resemble the ones displayed in figure 8.
The most important feature of this case is the fact that there is no cancellation in m ee for any values of v in the case of NO. Furthermore, we note that for v = 0 the predictions for IO coincide with the boundaries of the area allowed in the generic case. This happens, since the Majorana phase α is nearly trivial, see table 2, and the effect of β is suppressed by the reactor mixing angle as well as (the small mass) m 3 , compare approximation in (141).
Using (158) with v = 0 shows that m QD ee either equals m QD ee ≈ m 0 for k 1 odd or m QD ee ≈ m 0 /3 for k 1 even (remember k 1 has to be replaced by k 1 + 1 in (158) when applied to the case at hand).
In contrast, for v = 6 we see two different regimes realized m NO ee ≈ 0.0018 eV for k 1 + k 2 odd and m NO ee ≈ 0.0039 eV for k 1 + k 2 even, respectively. Since the Majorana phase α is non-trivial and not small for v = 6, we find a non-trivial lower bound on m ee in the case of IO that is by a factor of two larger than the generic lower bound, m IO ee ≈ 0.031 eV. The other value of m ee is m IO ee ≈ 0.039 eV, arising, if k 1 is even. Two different values are also obtained in the regime of QD light neutrino masses in which we predict for m 0 = 0.1 eV, according to (158), m QD ee ≈ 0.065 eV or m QD ee ≈ 0.083 eV depending on the value of k 1 . Future experiments searching for 0νββ decay [67][68][69] can probe almost the whole region for IO, down to m ee ≈ 0.02 eV, thus allowing for the possibility to distinguish between the different choices of the CP transformation X.

Predictions of m ee in case 3)
We also discuss the effective Majorana neutrino mass given in the last case introduced in subsection 2.2. We first consider case 3 b.1) that is characterized by the lepton mixing matrix in (46). We focus on the choice m = n/2, as done in our numerical analysis of leptogenesis. In this case, it follows directly from (55) that m ee must be invariant under the replacement of θ with π − θ. Again, we can derive simple approximations that work well for hierarchical and QD light neutrino mass spectra. For vanishing m 0 we find Interestingly enough, m NO ee is independent of φ s (and thus of the chosen CP transformation) and it takes two distinct values for k 1 even and odd, respectively. Using for the neutrino mass squared differences the best fit values and choosing θ ≈ 1.31 (which sets the reactor mixing angle to its best fit value [26]), we get m NO ee ≈ 0.0038 (0.0016) eV for even (odd) k 1 , m IO ee ≈ 0.015 e 6 i φs + (−1) k 2 0.033 eV .
In figure 9 we show the quantity m ee versus m 0 for the group theoretical parameters n = 8, m = 4 and s = 1, 2, 4 with θ chosen as in table 3. We, thus, consider the same example like in the numerical analysis of leptogenesis in subsection 3.5. The blue (red) areas correspond to NO (IO), for all combinations of k 1,2 , with the particular case k 1 = k 2 = 0 shown in dark-grey.
For s = 1 and s = 2, the Majorana phases are indeed non-trivial (see table 3) and m ee has a lower bound for a NO light neutrino mass spectrum. This result is similar to what we found in the numerical example of case 2), see figure 8. Taking In both cases the numerical values are in agreement with the analytic estimates for m NO ee given in (164). In the case of IO we find that for s = 2 (φ s = π/4), m IO ee is actually independent of k 1,2 , see (163), and thus only one narrow dark-grey shaded area exists in the right panel. It corresponds to m IO ee ≈ 0.036 eV for small values of m 0 . Instead, for s = 1 m ee is given by m IO ee ≈ 0.024 (0.045) eV for even (odd) k 2 in the hierarchical regime. Again, these numerical results coincide with the analytic estimate in (165). Similarly, for a QD light neutrino mass spectrum two values are possible for m ee , if s = 1, namely m ee ≈ 0.49 m 0 (k 2 = 0 and k 1 = 0, 1) and m ee ≈ 0.93 m 0 (k 2 = 1 and k 1 = 0, 1), whereas we only find one value for s = 2, i.e. m ee ≈ 0.75 m 0 .
On the contrary, in the case s = 4 both Majorana phases α and β are trivial and m ee can be strongly suppressed for a hierarchical NO light neutrino mass spectrum, as shown in the bottom-left panel of figure 9. Since our approach constrains not only the CP phases, but also the lepton mixing angles, see table 3, the suppression of m ee occurs only in two small intervals of m 0 , which depend on the integers k 1,2 , i.e. m NO ee 10 −4 eV is achieved for 0.0019 eV m 0 0.0033 eV (0.0059 eV m 0 0.0074 eV) and k 1 = k 2 = 1 (k 1 = k 2 = 0) .
(169) In the strongly hierarchical regime, that is for m 0 10 −4 eV, we find, instead, in agreement with the analytic estimate, shown in (164). For light neutrino masses with IO or being QD, m ee is either close to its lower or its upper limit, as expected for trivial Majorana phases, compare figure 9. Finally, we remark that for case 3 a), which requires the group index n to be larger than 10, several choices of the CP transformation X (or, equivalently, the parameter s) are admitted, see [26].
as one can see in figure 10. Such strong cancellations can also be achieved for s = 0 and s = 8. In these cases an accidental CP symmetry, common in the charged lepton and neutrino sectors, is present [26]. For other values of s with s ≤ 8, not shown here, the resulting m ee has a lower bound, typically m ee 10 −3 eV, within the range 10 −4 eV m 0 0.1 eV. In summary, we have shown that in the case of NO three different situations can be realized: no cancellations in m ee e.g. for case 2), n = 10 and u = 4, strong suppression of m ee and trivial Majorana phases e.g. for case 3 b.1), n = 8, m = 4 and s = 4, as well as cancellations in the presence of two non-trivial Majorana phases e.g. for case 3 a), n = 16, m = 1 and s = 2 or s = 3. For light neutrino masses following IO or being QD we observe that m ee can only obtain values in a very limited range for all possible m 0 . These values mainly depend on the chosen CP transformation and the parameters k 1 and k 2 . For this reason, at least part of these scenarios can be tested in future 0νββ decay experiments [67][68][69].

Summary
We have studied leptogenesis and 0νββ decay in a scenario with a flavor G f and a CP symmetry that are broken non-trivially in the charged lepton and neutrino sectors. We have chosen as G f groups belonging to the series ∆(3 n 2 ) and ∆(6 n 2 ), n ≥ 2, while the CP symmetry is represented by a CP transformation X that is a unitary and symmetric matrix, acting non-trivially on the flavor space. The residual symmetry G e in the charged lepton sector is fixed to a Z 3 subgroup of G f and G ν , the symmetry preserved in the neutrino sector, is given by the direct product Z 2 × CP with Z 2 ⊂ G f . In this way, the charged lepton mass matrix can always be constrained to be diagonal, while the light neutrino mass matrix contains four independent parameters, corresponding to the three neutrino masses and θ ∈ [0, π), on which lepton mixing angles and CP phases in general depend.
Under the assumption of three RH neutrinos and a Dirac Yukawa coupling Y D that is invariant under G f and CP the CP asymmetries (α) i vanish in the case of flavored as well as unflavored leptogenesis. If Y D is taken to be invariant under the residual symmetry G ν = Z 2 × CP only, α i become non-vanishing and a non-zero value of Y B can be achieved via flavored leptogenesis. Still, in the case of unflavored leptogenesis this is not sufficient and i turn out to vanish.
In our study of unflavored leptogenesis we introduce corrections δY D to Y D in order to obtain i non-zero. These corrections are expected to be in general proportional to the symmetry breaking parameter κ of our scenario. A particularly interesting case is to assume that δY D is induced by the breaking of G f to G e and thus is invariant under G e , the residual symmetry of the charged lepton sector. The two main consequences are: the suppression of the CP asymmetries i ∝ κ 2 and the fact that the Dirac and Majorana phases, potentially measurable in terrestrial experiments, determine the sign of Y B . The first observation has already been made in approaches with a flavor symmetry only, whereas the second one, namely the prediction of the sign of Y B , is only possible thanks to the presence of a CP symmetry that controls the CP phases. Especially, we have found the following: phases that are present in the correction δY D are irrelevant for the sign of Y B at LO; if not suppressed (due to a special choice of the neutrino masses), we expect the terms involving the sine of the Majorana phase α to dominate the CP asymmetries and thus the sign of Y B (see figures 4 and 5, the upper-right and lower panel in figure 6 and the upper two panels in figure 7); in case the dominant contribution to the CP asymmetries arises from terms involving the Dirac phase the sign of Y B cannot be predicted and positive and negative values are equally possible (see the upper-left panel of figure 6 and the lower panel of figure 7). These features are also confirmed by our analysis of the form of Y D , needed for the CP asymmetries to only depend on the low energy CP phases, see subsection 3.2, and by our study that assumes lepton mixing to be of a general form (and not constrained by symmetries), see subsection 3.3.
For flavored leptogenesis we have demonstrated with several examples that the sign of the CP asymmetries α i depends on additional input, e.g. the relative size of the parameters of the correction δY D , and thus fixing the sign of α i is in general impossible, having only information about the CP phases and the light neutrino mass ordering.
We have argued that the symmetry breaking scenario considered by us is well-motivated and have presented examples of non-as well as SUSY realizations of this scenario in appendix D where we show that small corrections δY D of the advocated form are naturally obtained.
We have also studied in detail the predictions for m ee , accessible in experiments searching for 0νββ decay. The following interesting properties are found: the constraints on CP phases and lepton mixing angles allow for cases in which no cancellation in m ee occurs even for NO light neutrino masses and thus m NO ee 0.002 eV can be achieved; in the case of an IO light neutrino mass spectrum such constraints can lead to values of m IO ee 0.05 eV thus increasing the chances to measure this quantity in the not-too-far future.
In summary, we have thoroughly analyzed a scenario in which flavor and CP symmetries can determine low and high energy CP phases together with the lepton mixing angles. Thus, phenomena requiring CP violation can be related. In particular, we have shown that the knowledge of low energy CP phases (and the light neutrino mass spectrum) can be sufficient for fixing the sign of the baryon asymmetry Y B of the Universe, if the latter is generated via unflavored leptogenesis. here in radian), the 1 σ errors as well as 3 σ ranges are 2.317 × 10 −3 eV 2 ≤ ∆m 2 atm ≤ 2.607 × 10 −3 eV 2 .
C Results for CP asymmetries i in the limit z 1 = 0 For completeness, we report here the formulae for the CP asymmetries i that are obtained in the limit z 1 = 0 (equivalent to ζ = π/2, 3π/2) and for only one non-vanishing CP phase. First, we consider the case in which the Majorana phase α and the phase combination β + 2 δ are trivial, i.e. α = k α π , β + 2δ = k β π with k α,β = 0, 1 .
This choice takes our non-standard definition of the second Majorana phase β into account, see (176) in appendix A.1 and compare to the convention used by the Particle Data Group Collaboration [70].
In this case the only source of low energy CP violation (and thus in our scenario also of CP violation at high energies) is the Dirac phase δ. We find Assuming all VEVs φ α to be of that order, the correct hierarchy among the charged lepton masses can be achieved with an additional Froggatt-Nielsen symmetry [40] under which RH charged leptons carry different charges, see e.g. [41]. Since RH neutrinos transform like LH leptons under the flavor and CP symmetry, the neutrino Yukawa coupling Y D arises from a renormalizable operator. Its flavor structure is trivial and the coupling is real. The Majorana mass term of RH neutrinos originates from couplings to fields in two different triplets of ∆(96), one equivalent to 3 and another one to 3 which is a real and unfaithful representation of ∆(96), ϕ ν ∼ (3, 1, ω 4 12 ) and ψ ν ∼ (3 , 1, ω 4 12 ) .
As indicated, these fields are neutral under Z (aux) 3 , but carry the charge ω 4 12 under Z 12 , so that the Lagrangian contains the following terms with f 1 and f 2 being real couplings. The VEVs ϕ ν and ψ ν are aligned as follows with v 1,2,3 and w real parameters of order ε 2 Λ. Thus, RH neutrino masses between 10 12 GeV and 10 14 GeV are achieved for Λ close to the scale of grand unification. Notice that we have chosen the VEVs of ϕ ν and ψ ν to be smaller than those of the fields φ α . In this way the dominant correction to the Dirac neutrino mass matrix arises from the fields φ α only, see (208). As one can check, ϕ ν and ψ ν leave G ν , generated by Z = c 2 and the CP transformation X, invariant. This breaking pattern hence allows us to obtain the PMNS mixing matrix of case 1), see (31), for n = 4 and s = 1. This is a choice of parameters also employed in our numerical discussion of unflavored leptogenesis in case 1), see subsection 3.5. The free parameter θ depends on the VEV of the field ψ ν Its particular value, necessary for describing correctly the lepton mixing angles, should be explained in a more complete model. The three RH neutrino masses M i read 17 and thus are functions of both couplings f 1,2 as well as all parameters of the VEVs of the fields ϕ ν and ψ ν . Using these we can also compute the masses of the light neutrinos. As one can see, we can accommodate in this way both mass orderings as well as a QD light neutrino mass spectrum. The discussed operators necessary at LO in our scenario contain either no or one flavor symmetry breaking field. Each mass matrix, m l , m D and M R , receives corrections of relative order ε 2 with respect to the corresponding LO result. Those to m l arise from insertions of three fields φ α and are of the generic form φ α φ β φ † γ with α, β, γ being e, µ or τ . Clearly, these do not change the form of m l and thus the charged lepton mass matrix is still diagonal. The dominant corrections to m D instead change the form of the latter and hence constitute the leading form of δY D . They stem from the terms − α=e,µ,τ r=1,2,6 y ν α,r The index α indicates which field φ α is coupled, while the index r takes into account the different possible contractions via a one-, two-or six-dimensional representation of ∆(96). The contribution arising from the contraction to a singlet can be absorbed into the LO term, since it is always real and proportional to the identity matrix in flavor space. The one coming from the contraction to a doublet, indeed, is not there, since the residual symmetry G e that is left invariant by the VEVs of the fields in (201) forces it to vanish. Consequently, the correction δY D is generated via the terms with y ν α,6 in (208). Matching the form of δY D given in (27) the two couplings z 1 and z 2 turn out to be z 1 = √ 3 2 2 y ν e,6 − y ν µ,6 − y ν τ,6 and z 2 = 3 2 y ν τ,6 − y ν µ,6 , if we set φ † α φ α to ε 2 Λ 2 for α = e, µ, τ for simplicity. The parameter κ is thus of the order κ ≈ ε 2 meaning 10 −4 κ 10 −2 .
We note that in this particular case z 1,2 in (209) turn out to be real. Subleading corrections to the Dirac neutrino mass matrix arise from two types of terms: terms with two flavor symmetry breaking fields (ϕ ν and ψ ν and the conjugate fields) and terms with four flavor symmetry breaking fields of the type φ † α φ † β φ γ φ δ with α, β, γ, δ = e, µ, τ . Both lead to corrections relatively suppressed by ε 4 with respect to the LO term. 18 The RH neutrino mass matrix, being at LO of order ε 2 Λ, also receives corrections. The dominant ones are of order ε 4 Λ and arise from three types of terms: terms with two conjugate fields ϕ † ν and ψ † ν , terms with three fields of the form φ † α φ α ϕ ν or φ † α φ α ψ ν for α = e, µ, τ as well as terms with four fields of the type φ α that are in general different. Clearly, the latter two types of terms break the residual symmetry G ν , if the VEVs of the flavor symmetry breaking fields are inserted. These together with the correction δY D that is also of relative order ε 2 with respect to the LO term induce small corrections to the LO results for the lepton mixing parameters. However, these are expected to be suppressed by ε 2 and thus are at most at the percent level.

D.2 SUSY setup
If we consider instead a SUSY framework, we can also construct a model of this type. Apart from the fact that l and ν c transform in complex conjugated three-dimensional representations, l ∼3 and ν c ∼ 3, see section 5, the three main differences are: a) we slightly change the additional symmetry and we use a Z 5 instead of a Z 12 group. The transformation properties of the fields are l ∼ ω 3 5 , α c ∼ ω 2 5 , ν c ∼ ω 2 5 , ϕ ν , ψ ν ∼ ω 5 ( ω 5 = e 2 π i/5 ) (211) and the Higgs multiplets h u and h d are neutral; b) we use less fields in the charged lepton sector φ τ ∼ (3, ω, 1) and χ ∼ (2, ω, 1) .
The terms in the superpotential contributing at lowest orders to the charged lepton mass matrix are y e Λ 3 l h d φ τ χ 2 e c + y µ Λ 2 l h d φ τ χ µ c + y τ Λ l h d φ τ τ c .
In this way, we can generate the mass hierarchy among the charged leptons with the help of insertions of several flavor symmetry breaking fields. Furthermore, the correct ratio between muon and tau lepton masses and electron and tau lepton masses is achieved, if we assume that the field χ acquires a VEV of the order 19 | χ | /Λ ≈ λ 2 ≈ 0.04 , while the VEV of φ τ is chosen like in (202) and its actual value depends on the size of tan β, see details in section 5. We note also that the VEVs of ϕ ν and ψ ν still have the form as in (205), but we now choose their size to be ε Λ; c) the lowest order correction to the Dirac mass matrix of the neutrinos and thus the source of δY D are two operators with three insertions of the field φ τ 20 i=1,2 with y ν τ,i being both real. Thus, the size of the small parameter κ is estimated as κ ≈ ε 3 meaning 10 −6 κ 10 −3 .
In both cases y ν Φ is a real coupling. The most general form of the VEV of Φ that leaves the residual group G e = Z and assuming κ like requested in (79). Thus, z 1 and z 2 , parametrizing the correction δY D in (27), read This shows that the special cases, z 1 = 0 or z 2 = 0, discussed in section 3, can be achieved with a particular form of the VEV of the field Φ. Clearly, the latter can also arise from some combination of flavor symmetry breaking fields.