Texture Zero Neutrino Models and Their Connection with Resonant Leptogenesis

Within the low scale resonant leptogenesis scenario, the cosmological CP asymmetry may arise by radiative corrections through the charged lepton Yukawa couplings. While in some cases, as one expects, decisive role is played by the $\lambda_{\tau }$ coupling, we show that in specific neutrino textures only by inclusion of the $\lambda_{\mu }$ the cosmological CP violation is generated at 1-loop level. With the purpose to relate the cosmological CP violation to the leptonic CP phase $\delta $, we consider an extension of MSSM with two right handed neutrinos (RHN), which are degenerate in mass at high scales. Together with this, we first consider two texture zero $3\times 2$ Dirac Yukawa matrices of neutrinos. These via see-saw generated neutrino mass matrices augmented by single $\Delta L=2$ dimension five ($\rm d=5$) operator give predictive neutrino sectors with calculable CP asymmetries. The latter is generated through $\lambda_{\mu , \tau }$ coupling(s) at 1-loop level. Detailed analysis of the leptogenesis is performed. We also revise some one texture zero Dirac Yukawa matrices, considered earlier, and show that addition of a single $\Delta L=2$, $\rm d=5$ entry in the neutrino mass matrices, together with newly computed 1-loop corrections to the CP asymmetries, give nice accommodation of the neutrino sector and desirable amount of the baryon asymmetry via the resonant leptogenesis even for rather low RHN masses($\sim $few TeV -- $10^7$~GeV).

1 Introduction them we pick up those which involve complexities and have potential for the CP asymmetry. With the updated neutrino data, we give updated results of the corresponding neutrino models which are highly predictive and determine cosmological CP violating phases in term of the δ phase. In section 4, applying results of the previous sections we determine cosmological CP violation for each considered model and use them for calculating of the baryon asymmetry. The latter is generated via resonant leptogenesis. We demonstrate that successful scenarios are possible for the low RHN masses (in a range few TeV -10 7 GeV). In section 5 we revise textures of Ref. [17] and make model improvements of the obtained neutrino mass matrices by adding the single ∆L = 2, d = 5 mass terms to certain non zero entries (in a spirit of Sect. 3). This makes the neutrino scenarios compatible with the best fit values of the neutrino data [3] and also proves to blend well with the leptogenesis scenarios. We stress that in the P 4 neutrino texture scenario (discussed in Sect. 3) and also in the texture B 2 ′ (considered in Sect. 5), for successful leptogenesis to take place crucial role is played by the λ µ Yukawa coupling which via 1-loop correction generates sufficient amount of the cosmological CP asymmetry. Such possibility has not been considered in the literature before.
(The general expressions for the corresponding corrections are presented in Sect. 2). Sect. 6 includes discussion and outlook where we also summarize our results and highlight some prospects for a future work. Appendix A includes some expressions, details related to the renormalization group (RG) studies and description of calculation procedures we are using. In appendix B the contribution to the net baryon asymmetry from the decays of the scalar components (RHS) of the RHN superfields is considered in detail. These analyses also include new corrections due to λ µ and corresponding soft SUSY breaking trilinear A µ coupling (besides λ τ , A τ and other relevant couplings).

Loop Induced Calculable Cosmological CP Violation
Before going to the calculations we first describe our setup. The framework is the MSSM augmented with two right-handed neutrinos N 1 and N 2 . This extension is enough to build consistent neutrino sector accommodating the neutrino data [3] and also to have a successful leptogenesis scenario. The relevant lepton superpotential couplings are given by: where h d and h u are down and up type MSSM Higgs doublet superfields respectively and l T = (l 1 , l 2 , l 3 ), e cT = (e c 1 , e c 2 , e c 3 ), N T = (N 1 , N 2 ). We work in a basis in which the charged lepton Yukawa matrix is diagonal and real: Y diag e = Diag(λ e , λ µ , λ τ ). (2.2) Moreover, we assume that the RHN mass matrix M N is strictly degenerate at the GUT scale, which will be taken to be M G ≃ 2 · 10 16 GeV. 6 Therefore, we assume: This form of M N is crucial for our studies. Although it is interesting and worth to study, we do not attempt here to justify the form of M N (and of the textures considered below) by symmetries. Our approach here is rather phenomenological aiming to investigate possibilities, outcomes and implications of the textures we consider. Since (2.3) at a tree level leads to the mass degeneracy of the RHN's, it has interesting implications for resonant leptogenesis [16,17,22] and also, as we will see below, for building predictive neutrino scenarios [17], [18]. For the leptogenesis scenario two necessary conditions need to be satisfied. First of all, at the scale µ = M N 1,2 the degeneracy between the masses of N 1 and N 2 has to be lifted. And, at the same scale, the neutrino Yukawa matrixŶ ν -written in the mass eigenstate basis of M N , must be such that Im[(Ŷ † νŶ ν ) 12 ] 2 = 0. [These can be seen from Eq. (4.1) with a demand ǫ 1,2 = 0.] Below we show that both of them are realized by radiative corrections and needed effect already arises at 1-loop level, with a dominant contribution due to the Y e Yukawa couplings (in particular from λ τ and in some cases from λ µ ) in the RG.
As it was shown [17], [15], within considered setup, radiative corrections are crucial for generating cosmological CP violation. In [15] it was shown that needed asymmetry is generated at 1-loop level due to λ τ Yukawa coupling provided that the condition (Y ν ) 31 (Y ν ) 32 = 0 is satisfied. Here, to be more generic and to not limit the class of the models, we also include the effects of the λ µ Yukawa coupling in the calculation. 7 Thus, in this section we present details of these calculations. We will start with radiative corrections to the M N matrix. RG effects cause lifting of the mass degeneracy and, as we will see, are important also for the phase misalignment (explained below).
At the GUT scale, the M N has off-diagonal form with (M N ) 11   N (obeying the RG equations investigated below). That's why M N was parametrized in a form given in Eq. (2.4). With |δ (1,2) N | ≪ 1, the M (at scale µ = M) will determine the masses of RHNs M 1 and M 2 , while δ (1,2) N will be responsible for their splitting and for complexity in M N (the phase of the overall factor M do not contribute to the physical CP). As will be shown below:  (2. 7) In the N's mass eigenstate basis, the Dirac type neutrino Yukawa matrix will beŶ ν = Y ν U N . In the CP asymmetries, the components (Ŷ † νŶ ν ) 21 Therefore, the CP violation should come from P * N Y † ν Y ν P N , which in a matrix form is: We see that η ′ − η difference (mismatch) will govern the CP asymmetric decays of the RHNs. Without including the charged lepton Yukawa couplings in the RG effects we will have η ′ ≃ η with a high accuracy. It was shown in Ref. [13] that by ignoring Y e Yukawas no CP asymmetry emerges at O(Y 4 ν ) order and non zero contributions start only from O(Y 6 ν ) terms [14]. Such corrections are extremely suppressed for Y ν < ∼ 1/50. Since in our consideration we are interested in cases with M 1,2 < ∼ 10 7 GeV leading to |(Y ν ) ij | < 7 · 10 −4 (well fixed from the neutrino sector and the desired value of the baryon asymmetry), these effects (i.e. order ∼ Y 6 ν corrections) will not have any relevance. In Ref. [17] in the RG of M N the effect of Y e , coming from 2-loop corrections, was taken into account and it was shown that sufficient CP violation can emerge. Below we show that including Y e in the Y ν 's 1-loop RG, will induce sufficient amount of CP violation. This mainly happens via λ τ and in particular cases (which are considered below) from λ µ Yukawa couplings. Thus, below we give detailed investigation of λ τ,µ 's effects. Using were in second lines of (2.10) and (2.11)

21
(2.12) 8 Omitted terms are either strongly suppressed or do not give any significant contribution to either the CP violation or the RHN mass splittings.
where t = ln µ, t G = ln M G and we have used the boundary conditions at the GUT scale δ N (t G ) = 0. For evaluation of the integral in (2.12) we need to know the scale dependence of Y ν and Y e . This is found in Appendix A.1 by solving the RG equations for Y ν and Y e . Using Eqs. (A.5) and (A.6), the integral of the matrix appearing in (2.12) can be written as: and we have ignored λ e Yukawa couplings. For the definition of η-factors see Eq. (A.6). The Y νG denotes corresponding Yukawa matrix at scale µ = M G . On the other hand, we have: (Derivations are given in Appendix A.1.) Comparing (2.13) with (2.16) we see that difference in these matrix structures (besides overall flavor universal RG factors) are in the RG factors r τ,µ (M) andr τ,µ (M). Without the λ τ,µ Yukawa couplings these factors are equal and there is no mismatch between the phases η and η ′ [defined in Eqs. (2.7) and (2.9)] of these matrices. Non zero η ′ − η will be due to the deviations, which we parameterize as The values of ξ µ and ξ τ can be computed numerically by evaluation of the appropriate RG factors. Approximate expressions can be derived for ξ τ,µ , which are given by: (2.20) shows well that in the limit ξ τ,µ → 0, we have η = η ′ , while the mismatch between these two phases is due to ξ τ,µ = 0. With ξ τ,µ ≪ 1, from (2.20) we derive: We stress, that the 1-loop renormalization of the Y ν matrix plays the leading role in generation of ξ τ,µ , i.e. in the CP violation. 9 [This is also demonstrated by Eq. (2.18).] When the product (Y ν ) 31 (Y ν ) 32 is non zero, the leading role for the mismatch between η and η ′ is played by ξ τ . However, for the Yukawa texture, having this product zero, important will be contribution from ξ µ . [As we will see on working examples, this will happen for T 9 of Eq. (3.1) and texture B 2 of Eq. (5. 2)]. The value of |δ N (M)|, which characterizes the mass splitting between the RHN's, can be computed by taking the absolute values of both sides of (2.20): (2.22) These expressions can be used upon the calculation of the leptogenesis, which we will do in sections 4 and 5 for concrete models of the neutrino mass matrices.
3 See-Saw via Two Texture Zero 3 × 2 Dirac Yukawas Augmented by Single d=5 Operator. Predicting CP Violation Within the setup with two RHNs, having at the GUT scale mass matrix of the form (2.3), we consider all two texture zero 3 × 2 Yukawa matrices. As given in [18], there are nine such different matrices: Note that since RG equations for M N and Y ν in non-SUSY case have similar structures (besides some grouptheoretical factors) the ξ τ,µ would be generated also within non-SUSY setup.
where "×"s stand for non-zero entries. From these textures one can factor out phases in such a way as to make maximal number of entries be real. As it was shown in [18], phases can be removed from all textures besides T 4 , T 7 and T 9 . Thus, here we pick up only T 4,7,9 textures, which lead to cosmological CP violation and have potential to realize resonant leptogenesis [16], [17] (due to quasi-degenerate N 1 and N 2 states). Therefore, we can parametrize these three textures as: with with with The phases x, y and z can be eliminated by proper redefinition of the states l and e c . As far as the phases ω and ρ are concerned, because of the form of the M N matrix (2.3), they too will turn out to be non-physical. As we see, in textures T 4 , T 7 and T 9 there remains one unremovable phase φ (i.e. in the second matrices of the r.h.s of Eqs. (3.2) (3.4) and (3.6) respectively). This physical phase φ is relevant to the leptogenesis [17] and also, as it was shown in [18], it can be related to phase δ, determined from the neutrino sector. Integrating the RHN's, from the superpotential couplings of Eq. (2.1), using the see-saw formula, we get the following contribution to the light neutrino mass matrix: For Y ν in (3.8) the textures T 4,7,9 should be used in turn. All obtained matrices M ss ν , if identified with light neutrino mass matrices, will give experimentally unacceptable results. The reason is the number of texture zeros which we have in T i and M N matrices. In order to overcome this difficulty, in Ref. [18], the following single d = 5 operator was included for each case: whered 5 , x 5 and M * are real parameters. (3.9), together with (3.8) will contribute to the neutrino mass matrix. This will allow to have viable models and, at the same time because of the minimal number of the additions, we will still have predictive scenarios. The operators (3.9) can be obtained by another sector in such a way as to not affect the forms of T 4,7,9 and M N matrices (one detailed example was presented in [15]). See Sect. 6 for more discussion on a possible origin of the (3.9) type operators. Above we have written the Yukawa textures in the form: where P 1 , P 2 are diagonal phase matrices and Y R ν contains only one phase. Making the field phase redefinitions: the superpotential coupling will become: with: Now, for simplification of the notations, we will get rid of the primes (i.e. perform l ′ → l, e c′ → e c ,...) and in Eq. (3.8) using Y R ν instead of Y ν , from different T 4,7,9 textures we get corresponding M ss ν , and then adding the single operator (3.9) terms to zero entries of (3.8), one per M ss ν , obtain the final neutrino mass matrices. Doing so, one obtains the neutrino mass matrices [18]: where each type of texture originate as: where subscript for M indicates which Yukawa texture the see-saw part [of Eq. (3.8)] came from, while superscript denotes the non zero mass matrix element arising from the addition of the d =5 operator of type (3.9). Since within our setup we are deriving neutrino mass matrices, we are able to renormilize them from high scales down to M Z . With details given in the Appendix A of Ref. [15], we here write down P 1,2,3,4 textures at scale M Z and give results already obtained in [18]. Before doing this, we set up conventions, which are used below. Since we work in the basis in which charged lepton Yukawa matrix is diagonal and real, the lepton mixing matrix U is related to the neutrino mass matrix as: are light neutrino masses) and the phase matrices and U are: where s ij ≡ sin θ ij and c ij ≡ cos θ ij . For normal and inverted neutrino mass orderings (denoted respectively by NH and IH) we will use notations: As far as the numerical values of the oscillation parameters are concerned, since the bfv's of the works of Ref. [3] differ from each other by few %'s, we will use their mean values: sin 2 θ 12 = 0.308, sin 2 θ 23 = 0.432 for NH 0.591 for IH , sin 2 θ 13 = 0.02157 for NH 0.0216 for IH , In models, which allow to do so, we use the best fit values (bfv) given in (3.20). However, in some cases we also apply the value(s) of some oscillation parameter(s) which deviate from the bfv's by several σ.
This texture, within our scenario, can be parametrized as: NH, sin 2 θ 23 = 0.451, sin 2 θ 12 = 0.323 and best fit values for remaining oscillation parameters, (m 1 , m 2 , m 3 ) = (0.00694406, 0.0110914, 0.0509217), m ββ = 0 Table 1: Results from P 1 type texture. Masses are given in eVs. and RG factors rm and r ν3 are given in Eqs. (A.17) and (A.18) of Ref. [15]. (For notations and definitions see also Appendix A.2 of the present paper.) Due to the texture zeros, it is possible to predict the phases and values of the neutrino masses in terms of the measured oscillation parameters. Referring to [18] for the details, in Table 1 we summarize the results. [Only normal hierarchical (NH) neutrino mass ordering scenario works for the P 1 type texture.] Results from this texture are given in Table 2. The best fit values (bfv) of the oscillation parameters are taken from Eq. (3.20). For the details of the analysis of this model we refer the reader to [18].
The results of this texture for NH and inverted hierarchical (IH) neutrino mass orderings are summarized in Table 3.  Table 3: Results from P 3 type texture. Masses are given in eVs.
The results obtained from this texture P 4 for NH and IH cases are presented in Table 4. The value of s 2 23 we are using is deviated from the bfv, because the conditions (M ν ) 1,3 = (M ν ) 3,3 = 0 do not allow to use bfv's. Note that in NH, case 2 and for IH the values of s 2 23 are less deviated from bfv, but the NH's case 1, as it turns out, is preferred for obtaining needed amount of the baryon asymmetry. Without the latter constraint, just for satisfying the neutrino data, we could have used smaller values of s 2 23 , but this would give higher values of neutrino masses which would not satisfy the current cosmological constraint i m i < 0.23 eV (the limit set by the Planck observations [26]). Upon leptogenesis investigation we will use NH, case 1 given in Tab.4.

Resonant Leptogenesis
Expression for δ N (M) with effects of λ µ,τ and ignoring λ e , is given by Eq. (2.20). The CP asymmetries ǫ 1 and ǫ 2 generated by out-of-equilibrium decays of the quasi-degenerate fermionic components of N 1 and N 2 states respectively are given by [9], [10]: 10 Here M 1 , M 2 (with M 2 > M 1 ) are the mass eigenvalues of the RHN mass matrix. These masses, within our scenario, are given in (2.6) with the splitting parameter given in Eq. (2.22). For the decay widths, here we will use more accurate expressions [5]: where M S is the SUSY scale and we assume that all SUSY states have the common mass equal to this scale. s β and c β are short hand notations for sin β and cos β respectively. N i decays proceed via N i → h u l i and N i →h uli channels. Upon derivation of (4.2) we took into account that h u is a linear combination of the SM Higgs doublet h SM and the heavy Higgs doublet H: Mass of the h SM has been ignored, while the mass of the H has been taken≃ M S . Moreover, the imaginary part of [(Ŷ † νŶ ν ) 21 ] 2 will be computed with help of (2.8) and (2.9) with the relevant phase given in Eq. (2.21). Using general expressions (2.21) and (2.22) for the given neutrino model we will compute η − η ′ and |δ N (M)|. With these, since we know the possible values of the phase φ [see Eqs. (4.6),(4.8),(4.10),(4.12)], and with the help of the relations (4.7), (4.9), (4.11), (4.13) we can compute ǫ 1,2 in terms of |M| and a 2 or a 1 (depending on the texture we are dealing with). Recalling that the lepton asymmetry is converted to the baryon asymmetry via sphaleron processes [27], with the relation we can compute the baryon asymmetry. The notion n f b is used for the baryon asymmetry created through the decays of the fermionic components of N 1,2 superfields. The net baryon asymmetry n b receives the contribution from the decays of the scalar componentsÑ 1,2 . The latter contribution we denote byñ b . The computation of it (being suppressed in comparison with n f b ) will be discussed in appendix B. For the efficiency factors κ f (1,2) we will use the extrapolating expressions [5] (see Eq. (40) in Ref. [5]), with κ f (1) and κ f depending on the mass scalesm 1 = v 2 Within our studies we will consider the RHN masses ≃ |M| < ∼ 10 7 GeV. With this, we will not have the relic gravitino problem [28], [29]. For simplicity, we consider all SUSY particle masses to be equal to M S < |M|, with M S identified with the SUSY scale, below which we have just SM. As it turns out, via the RG factors, the asymmetry also depends on the top quark mass.
It is remarkable that within some models the observed baryon asymmetry (the recent value reported by WMAP and Planck [26]), can be obtained even for low values of the Below, we perform analysis for each of these P 1,2,3,4 cases (and for revised models of Ref. [17] discussed in Sect.5) in turn and present our results. As an input for the top's running mass we will use the central value, while for the SUSY scale M S we will consider two cases: Procedure of our RG calculation and used schemes are described in Appendix A.3. As it was shown in [18], for neutrino mass matrix textures P 1,2,3,4 , we will be able to relate the cosmological phase φ to the CP violating phase δ. We will introduce the notation: which will be convenient for writing down expressions for the φ and for expressing neutrino Dirac type Yukawa couplings in terms of one independent coupling element. (The latter will be selected by the convenience.) For P 1 Texture For this case, using the form of the M ν [given by Eq. (3.21) and derived within our setup] in the relation (3.15) and equating appropriate matrix elements of the both sides, we will be able to calculate the phase φ [18], [15]: Moreover, expressing a 3 , b 2,3 in terms of a 2 (taking a 2 to be an independent variable) and other known and/or predicted parameters, we will have: As we see from Eqs. (4.6) and (4.7), there is a pair of solutions. When for the a 3 in (4.7) we are taking the " + " sign, in (4.6) we should take the sign " − ", and vice versa. (The same applies to the cases of textures P 2,3,4 .) For this case, the baryon asymmetry via the resonant leptogenesis has been investigated in Ref. [15]. In this work, for the decay widths we use more refined expressions of Eq. (4.2). Because of this, the values of tan β (given in Table 5) are slightly different. Since in this model (Y ν ) 31 and (Y ν ) 32 are non zero, according to Eq. (2.20) the mismatch η − η ′ (e.g. CP asymmetry) is mainly arising due to ξ τ . However, in numerical calculations we have also taken into account the contribution of ξ µ . The results are given in Table 5 (for more explanations see  also caption of this table). While in the table we vary the values of M and tan β, the cases with I and II correspond respectively to the cases (I) and (II) of Eq. (4.4) (i.e. M S = 1 and 2 TeV resp.). For the definition of the RG factors given in this table see Appendix A.2 of Ref. [15]. For finding maximal values of the Baryon asymmetries (given in Tab.5) we have varied the parameter a 2 . As we see, the value of the net baryon asymmetry n b slightly differs from n f b . This is due to the contribution fromñ b [coming from the right handed sneutrino (RHS) decays], which is small (less than 3.4% of n f b ). Details ofñ b 's calculations are discussed in Appendix B.   For P 2 Texture With a pretty similar procedure, for this case we get: Expressing a 3 , b 2,3 in terms of a 2 and other parameters (yet known or predicted in this scenario), we will have: Results for this case are presented in Table 6.      Table 3 and computed from Eq. (4.10) (for IH case) φ = ±3.124. For all cases r ν3 ≃ 1.
(For notations and definitions see also Appendix A.2 of the present paper.) Expressing a 3 , b 1,3 in terms of a 1 and other fixed parameters, we will have: Results for this texture for cases of NH and IH neutrinos are presented in Tables 7 and 8 respectively.
For P 4 Texture For this case cosmological phase is given by: Expressing a 1 , b 1,2 in terms of a 2 and other known and/or predicted parameters, we will have: In this scenario, since (Y ν ) 31 and (Y ν ) 32 are zero, according to Eq. (2.20) the mismatch η − η ′ (e.g. CP asymmetry) is arising due to ξ µ . Since the latter is suppressed by λ 2 µ , as it turns out large values of the tan β are required and only in NH case needed amount of the Baryon asymmetry can be generated. Results are given in Table 9.

Revising Textures of Ref. [17] and Improved Versions
In this section we revise the textures considered in the work [17]. Since some of them are excluded by the current neutrino data [3](see also Eq. (3.20)), we apply d = 5 contributions (in a spirit of section 3) and achieve their compatibility with the best fit values. Together with this, we investigate resonant leptogenesis and show that one loop corrections via λ τ and/or λ µ are crucial. In [17], while ignoring λ µ the two loop correction to λ τ was taken into account and this suggested for textures A and B 1 specific low bounds on the values of tan β. As demonstrated below, one loop effects of λ τ (giving dominant contribution for textures A and B 1 ) and λ µ (for the texture B 2 ) significantly change results.
In the setup of two degenerate RHNs, in Ref. [17] the following three possible one texture zero neutrino Dirac Yukawa couplings have been considered : where for notational consistency with the whole paper, we have shown phases α i , β j , while assuming that the couplings a i , b j are real. 11 Below we will (re)investigate these textures in turn.

Texture A
The A Yukawa texture can be written as: As we see, besides the phase φ all phases are factored out and have no physical relevance. With the RHN mass matrix of Eq.(3.13), via the see-saw[see expression in Eq.(3.8)] we will get the light neutrino mass matrix:  [15], and comments therein.] This texture has only two non-zero mass eigenvalues.
As it was shown in [17], this for NH (m 1 = 0) and IH (m 3 = 0) neutrino mass patterns, gives respectively the predictive relations tan θ 13 = m 2 m 3 s 12 and tan θ 12 = m 1 m 2 . Both of them are in a gross conflict with the current neutrino data, which excludes this scenario.

A ′ Neutrino Texture: Improved Version
The drawbacks coming from the A neutrino mass matrix can be avoided by adding d 5 terms to the one of the entries. Here we consider this addition to the (2, 3) and (3, 2) elements of the light neutrino mass matrix, which would make the model viable. (We refer to this improved version as the A ′ neutrino texture.) After this, the M ν will have the form: With this modification, all masses are non zero. One can check out, that with the fixed phase redefinitions [given in Eq. (5. 3)], in general d 5 is a complex parameter. Thus, together with additional mass, we will have one more independent phase. As it turns out, only NH scenario is possible to realize. Therefore as additional independent parameters we take one of the mass and ∆ρ = ρ 1 − ρ 2 . From the condition M (Here and below we use short-handed notations t ij ≡ tan θ ij .) From the first relation of (5.6) one can check that IH scenario can not be realized. As far as the NH scenario is concerned, it will work with low bound on the lightest neutrino mass m 1 . In fact, the first relation of (5.6) gives the allowed range for m 1 . For example, with bfv's of the oscillation parameters (3.20) we have: Thus, as independent parameters we will take m 1 and ∆ρ. We will select them in such a way as to get desirable baryon asymmetry. For example, with the choice As far as the baryon asymmetry is concerned, using (5.5) in (3.15) for the CP pase φ and expressing couplings a 1,3 , b 2,3 in terms of a 2 we get For the values of (5.8), (5.9) and bfv's of s 2 12,23,13 we get φ = −2.9297 .   The B 1 Yukawa texture can be written as: With the RHN mass matrix of Eq. (3.13), via the see-saw we will get the light neutrino mass matrix: This texture works only for inverted neutrino mass ordering [17] (with m 3 = 0) and has two predictive relations. In particular, in terms of measured oscillation parameters we can calculate the phases δ and ρ 1 . The exact expressions are: Although the first expression in (5.14) excludes the possibility of using the best fit values for all oscillation parameters, it allows for keeping values of s 2 23 and s 2 13 within 1σ, while confining s 2 12 to 2σ. Remarkably, needed baryon asymmetry can be achieved with relatively low values of tan β. By addition of the d 5 term to (1,3) and (3,1) entries of the B 1 neutrino texture, the light neutrino mass matrix becomes: which gives all neutrinos massive and opens up a possibility of choosing two variables such as m 3 and ∆ρ ≡ ρ 1 − ρ 2 as independent ones to operate with. From the condition M (2,2) ν = 0 we have: As far as the baryon asymmetry is concerned, using (5.17) in (3.15), we get: Using all these, we can calculate the baryon asymmetry. The results are given in Tab    This texture is interesting because, due to specific form of Y ν , the radiative corrections through the λ τ coupling do not generate cosmological CP asymmetry. Thus λ µ may be important, which we investigate below. Thus, this model (and its slight modification discussed below) serves as a good demonstration of the role of ξ µ correction in emergence of needed Baryon asymmetry.
The B 2 Yukawa texture can be written as: Via the see-saw we will get the light neutrino mass matrix: This texture works only for inverted neutrino mass ordering [17] (with m 3 = 0) and has two predictive relations. In particular, in terms of measured oscillation parameters we can calculate the phases δ and ρ 1 . The exact expressions are:  In order to avoid difficulties with the texture B 2 we add d 5 term to the (1, 2) and (2, 1) elements of the light neutrino mass matrix. After this, the M ν will have the form: With this modification, all masses are non zero, and therefore two additional parameters m 3 = 0 and ρ 2 enter. Thus our relations will involve two more independent quantities. For convenience we take m 3 and ∆ρ = ρ 1 − ρ 2 as such. From the condition M   From these relations the phases δ and ρ 1 can be calculated in terms of m 3 and ∆ρ.
As it turns out, in this improved version the IH case works well for both neutrino sector and for the baryon asymmetry as well. So, we will start with discussing the IH case. For measured oscillation parameters we take the best fit values given in (3.20) and select pairs (m 3 , ∆ρ) in such a way as to get needed baryon asymmetry. One such choice is: These for the observable ν02β-decay give m ββ ≃ 0.0193 eV. As far as the baryon asymmetry is concerned, using (5.29) in (3.15) for the CP pase φ and expressing couplings a 2,3 , b 1,2 in terms of a 1 we get For the values of (5.31), (5.32) and bfv's for the θ ij angles we get φ = 2.2301 . With these, and for given values of M and tan β by varying a 1 we can investigate the baryon asymmetry. Results are given Tab. 13.
As far as the NH case is concerned, the neutrino sector can work well by certain selection of (m 3 , ∆ρ). However, in order to generate needed baryon asymmetry we need to take values of sin 2 θ ij deviated from the bfv's by the ( Note that the B 2 ′ neutrino texture coincides with the texture P 7 of Ref. [18] if all entries in (5.29) are taken to be real. As was shown in [18] the real neutrino texture with M (3,3) ν = 0 will work for both NH and IH neutrinos (see Tab. 6 of Ref. [18]). Advantage of complex d = 5 entry [like in texture (5.29)] is that it gives good possibility for generation of the baryon asymmetry with the λ µ 's radiative correction playing the decisive role. Similar possibility has not been considered in the literature before.
Concluding, note also that the neutrino textures A ′ and B 1 ′ are generalizations of the textures P 5 and P 6 (respectively), considered in [18]. The latter two had no complex phases, while A ′ and B 1 ′ scenarios besides good neutrino fits give possibility for the generation of the baryon asymmetry.

Discussion and Outlook
In this work we have investigated the resonant leptogenesis within the extension of the MSSM by two right handed neutrino superfields with quasi-degenerate masses < ∼ 10 7 GeV. It was shown that in this regime the cosmological CP asymmetry arises at one loop level due to charged lepton Yukawa couplings. In particular, needed corrections may come from either of the λ τ and λ µ couplings. Which one is relevant from these two couplings depends on the structure of the 3 × 2 Dirac type Yukawa matrix Y ν . Aiming to make close connection with the neutrino sector, we first examined all viable neutrino models (considered earlier in Ref. [18]) based on two texture zero Y ν 's augmented by single ∆L = 2, d = 5 operators. This setup is predictive and allows to relate leptonic CP violating phase δ with the cosmological CP violation. In one of such scenarios the role of the λ µ coupling in CP asymmetry generated at quantum level has been demonstrated. We have also revised the models of Ref. [17] and considered their improved versions by including proper ∆L = 2, d = 5 operators. This allowed to have good fit with the neutrino data and generate needed amount of the baryon asymmetry.
Without specifying their origin, in our considerations we have extensively applied the ∆L = 2, d = 5 operators, of the form given in Eq. (3.9). Such d = 5 couplings can be generated from a different sector via renormalizable interactions. For instance, introducing the pair of MSSM singlet states N , N and the superpotential couplings it is easy to verify that integration of the heavy N , N multiplets leads to the operator in Eq. (3.9) withd Important ingredient here is to maintain forms of the matrices Y ν , M N . In [15] considering one such fully consistent extension, it was demonstrated that all obtained results (e.g. neutrino masses and mixings, and baryon asymmetry as well) can remain intact. Although the way demonstrated above is rather simple, there can be considered also alternative ways for generating those ∆L = 2 effective couplings. These could be done either in a spirit of type II [30], or type III [31] see-saw mechanisms, or even exploiting alternative possibilities [32], [33] through the introduction of appropriate extra states. Details of such scenarios should be pursued elsewhere. Throughout our studies we have studied texture zero coupling matrices, but did not attempt to explain and justify considered structures by symmetries. Our approach, being rather phenomenological, was to consider such textures which give predictive and/or consistent scenarios allowing for transparent demonstrations of the suggested mechanism of the loop induced cosmological CP violation. It is desirable to have explanation of texture zeros at more fundamental level, and exploiting flavor symmetries seems to be a good framework. We are planning to pursue this approach in a future work [34].
Since the supersymmetry is a well motivated construction, we have performed our investigations within its framework. However, it would be interesting to examine the considered models also within the non-SUSY setup. For the latter, the scenarios with low tan β look encouraging to start with.
Finally, it would be challenging to embed considered models in Grand Unification (GUT) such as SU (5) and SO(10) GUTs. Due to the high GUT symmetries, additional relations and constraints would emerge making models more predictive. These and related issues will be addressed elsewhere.
Acknowledgments Z.T. thanks CERN theory division for warm hospitality and partial support during his visit there.

A Renormalization Group Studies
A.1 Running of Y ν , Y e and M N Matrices RG equations for the charged lepton and neutrino Dirac Yukawa matrices, appearing in the superpotential of Eq. (2.1), at 1-loop order have the forms [35], [36]: c a e = ( 9 5 , 3, 0), (A.1) The RG for the RHN mass matrix at 2-loop level has the form [36]: Let's start with renormalization of the Y ν 's matrix elements. Ignoring in Eq. (A.2) the O(Y 3 ν ) order entries (which are very small because within our studies |(Y ν ) ij | < ∼ 10 −4 ), and from charged fermion Yukawas keeping λ τ , λ µ , λ t and λ b , we will have: This gives the solution where Y νG denotes Yukawa matrix at scale M G and the scale dependent RG factors are given by:  [15]. At scale M, after decoupling of the RHN states, the neutrino mass matrix is generated and has the form: where '×' stand for entries depending on Yukawa couplings. After renormalization, keeping λ τ , λ t , λ b and g a in the RGs, the neutrino mass matrix at scale M Z has the form:  [15]. We will also need the RG factor relating the VEV v u (M) to the v(M Z ). Thus we define: Analytic expression for r vu derived from appropriate RGs is given by Eq. (A.20) of Ref. [15].
The factor p t is p t ≃ 1/1.0603 [41], while the recent measured value of the top's pole mass is [42]: We take the values of (A.13) as boundary conditions for solving 2-loop RG equations [43], [38] for λ t,b,τ,µ and λ from the M Z scale up to the scale M S . Above the M S scale, we have MSSM states including two doublets h u and h d , which couple with up type quarks and down type quarks/charged leptons respectively. Thus, Yukawa couplings we are considering at M S are ≈ λ t (M S )/s β , λ b (M S )/c β and λ τ,µ (M S )/c β , with s β ≡ sin β, c β ≡ cos β. Above the scale M S we apply 2-loop SUSY RG equations in DR scheme [35]. Thus, at µ = M S we use the matching conditions between DR − MS couplings: Throughout the paper, above the mass scale M S without using the superscript DR we assume the couplings determined in this scheme.

B Baryon Asymmetry from RHS Decays
In this appendix we give details of the contribution to the net baryon asymmetry from the right handed sneutrinos (RHS) -the scalar partners of the RHNs. Estimation of this contribution for specific textures was given in [17], while more detailed investigation was given in [15] (from the lepton couplings taking into account only λ τ and A τ in the proper RGs). Since we have seen that for some cases for the cosmological CP asymmetry decisive is the RG correction via the λ µ Yukawa coupling, here we extend its calculation by taking into account also effects from λ µ and A µ into the asymmetry generated by the RHS decays. We will consider soft SUSY breaking scalar potential which will be relevant for deriving RHS masses and their couplings to the components of the l and h u superfields. Using general expressions of Ref. [35] we write down 1-loop RGs for A ν and B N , which have the forms: We parameterize the matrices B N and A ν as: where entries (M N ) 12 , m B , δ (1,2) BN and elements of the matrix a ν run (their RGs can be derived from the RG equations given above), while m A is a constant. The matrixÂ e (similar to the structure of Y e Yukawa matrix) isÂ e = Diag (A e , A µ , A τ ) . (B.5) Assuming proportionality / alignment of the soft SUSY breaking terms and corresponding superpotential couplings, we will use the following boundary conditions: Using (B.3) for B N 's entries in (B.4) we have: For the elements of a ν we have which show the violation of the alignment between a ν and Y ν due to RG effects. At r.h.s. of (B.8) we kept λ µ,τ,t , A µ,τ,t , gauge couplings and gaugino masses. From this we derive Keeping in mind that the powers of the Y ν couplings can be ignored due to their smallness, the m B can be treated as a constant, and from (B.9), (B.7), (B.4) we obtain: and The form of B N given in Eq. (B.10) will be used to construct the RHS mass matrix. Before doing this, using Eq. (A.5) and ignoring the coupling λ e (as it turns out from the lepton Yukawa couplings all relevant effects are due to λ µ,τ ), forǭ 1,2 at scale µ = M we can get expressions: With the transformation of the N superfields N = U N N ′ (according to Eq. (2.6), the U N diagonalizes the fermionic RHN mass matrix), we obtain: With phase redefinitioñ and by going to the real scalar components and using (B.10), we will have: From (B.14) and (B.17) we obtain the mass 2 terms: and 20) The coupling ofñ 0 states with the fermions emerges from the F -term of the superpotential l T Y ν Nh u . Following the transformations, indicated above, we will have:

B.1 Calculatingñ b s -Asymmetry Viañ Decays
Due to the SUSY breaking terms, the masses of RHS's differ from their fermionic partners' masses. For each mass-eigenstate RHS'sñ i=1,2,3,4 we have one of the massesM i=1,2,3,4 respectively. With the SUSY M S scale M S M < ∼ 1/3, the statesñ i remain nearly degenerate and for the resonantñ-decays the resummed effective amplitude technique [9] will be applied. Effective amplitudes for the real n i decay, say into the lepton l α (α = 1, 2, 3) and antilepton l α respectively are given by [9] where S αi is a tree level amplitude and Π ij is a two point Green function's (polarization operator ofñ i −ñ j ) absorptive part. The CP asymmetry is then given by With Y F and Y B given by Eqs. (B.23) and (B.24) we can calculate polarization diagram's (with external legsñ i andñ j ) absorptive part Π ij . These at 1-loop level are given by: where p denotes external momentum in the diagram and upon evaluation of (B.26), for Π one should use (B.27) with p =M i . In (B.27), taking into account the SUSY masses M S of all non SM states, we are using the refined expression for the Π ij .
In an unbroken SUSY limit, neglecting finite temperature effects (T → 0), theÑ decay does not produce lepton asymmetry due to the following reason. The decays ofÑ in the fermion and scalar channels are respectivellyÑ → lh u andÑ →l * h * u . Since the rates of these processes are the same due to SUSY (at T = 0), the lepton asymmetries created from these decays cancel each other. With T = 0, the cancelation does not take place and one has with a temperature dependent factor ∆ BF given in [44]. 12 Therefore, we just need to compute ǫ i (ñ i → lh u ), which is the asymmetry created byñ i decays in two fermions. Thus, in (B.25) we take S αi = (Y F ) αi and calculate ǫ i (ñ i → lh u ) with (B.26). The baryon asymmetry created from the lepton asymmetry due toñ decays is given by: where an effective number of degrees of freedom (including two RHN superfields) g * = 228.75 was used. η i are efficiency factors which depend onm i ≃ (v sin β) 2 M 2(Y † F Y F ) ii , and account for temperature effects once integration of the Boltzmann equations is performed [44].
Calculating the contribution ∆n b s =ñ b s to the baryon asymmetry from the RHS decays, we have examined various values of pairs (m A , m B ) in the range of 100 GeV -few TeV. As it turned out, the ratioñ b n f b is always suppressed(< 3.4 · 10 −2 ). The results for each neutrino scenario, we have considered in this paper, for one specific choice of (m A , m B ), are given in Table 14 (see its caption for more information). The ranges forñ b s are due to the fact that for each scenario we have considered different values of tan β, M and M S . Upon the calculations, with obtained values ofm i , according to Ref. [44] we picked up the corresponding values of η i and used them in (B.29). While giving the results of the net baryon asymmetry, for each case (see sections 4 and 5), we have included corresponding contributions fromñ b s as well. As we see from the results of Tab. 14, theñ b s is suppressed/subleading for all cases. We have also witnessed (by varying the phases of m A,B ) that the complexities of m A and m B practically do not change the results. This happens because the m A in the Y B coupling matrix appears in front of the Y ν [see Eq. (B.24)], which is strongly suppressed. Irrelevance of the m B 's phase can be seen from the structure of (B.19). Suppression ofñ b s will always happen for the value of |m B | in the range of 100 GeV -few TeV, because the mass degeneracy of n i states is lifted in such a way that resonant enhancement ofñ b s is not realized. (Unlike the case of soft leptogenesis [44]