Leptogenesis in the presence of exact flavor symmetries

arXiv:1110.3781v1 [hep-ph] 17 Oct 2011 Leptogenesis in the presence of exact flavor symmetries D. Aristizabal Sierraa,1 , Federica Bazzocchib,2 a IFPA, Dep. AGO, Universite de Liege, Bat B5, Sart Tilman B-4000 Liege 1, Belgium. b SISSA and INFN, Sezione di Trieste, Via Bonomea 265, 34136 Trieste, Italy. Abstract In models with flavor symmetries in the leptonic sector leptogenesis can take place in a very different way compared to the standard leptogenesis scenario. We study the generation of a B − L asymmetry in these kind of models in the flavor symmetric phase pointing out that successful leptogenesis requires (i) the righthanded neutrinos to lie in different representations of the flavor group; (ii) the flavons to be lighter at least that one of the right-handed neutrino representations. When these conditions are satisfied leptogenesis proceeds due to new contributions to the CP violating asymmetry and -depending on the specific model- in several stages. We demonstrate the validity of these arguments by studying in detail the generation of the B − L asymmetry in a scenario of a concrete A4 flavor model realization. 1 2 e-mail address: [email protected] e-mail address: [email protected] 1 Motivation Observational data from the abundances of light elements (D, 3 He, 4 He and Li) in addition to precision observations of the cosmic microwave background (CMB) temperature fluctuations allow the determination of the cosmic baryon asymmetry, Y∆B = (8.75±0.23)×10−11 [1]. Though the conditions for dynamically generating this quantity are well known and established [2] the cosmic baryon asymmetry poses a puzzle in particle physics: the standard model (SM) fails to explain such a large asymmetry, thus implying the presence of new physics accounting for Y∆B . Leptogenesis is a scenario in which a lepton asymmetry Y∆L is generated in the lepton sector and partially reprocessed into Y∆B by SM electroweak sphaleron processes (for a comprehensive review see [3]). The generation of Y∆L requires, in addition to CP violation and departure from thermodynamical equilibrium, lepton number breaking. Accordingly, in these class of scenarios two in principle unrelated puzzles are linked, the origin of neutrino masses and the baryon asymmetry. Among all the possible neutrino mass models present in the literature the standard seesaw (type I seesaw) [4] provides the framework for standard leptogenesis, in which the lepton asymmetry proceeds via the out-of-equilibrium and CP violating decays of the lightest right-handed (RH) electroweak singlet neutrino. Most of the studies of leptogenesis are based on the assumption that there is no new physics between the lepton number breaking scale and the electroweak scale that can sizable affect the way in which leptogenesis takes place. Though some analysis in scenarios including flavor symmetries above the electroweak scale have been done, and have proved that the presence of new energy scales and new degress of freedom may have an impact on the way leptogenesis proceeds [5–8], all of them are based on the same assumption, namely the lepton number breaking scale is below the scale at which the flavor symmetry is broken 1 . The idea to ascribe to a flavor symmetry to explain particle masses and mixings dates backs to the late 1970’s [11]. Originally flavor symmetries were introduced to explain quark structures and only after neutrino oscillation data the use of horizontal symmetries in the lepton sector has become more challenging and interesting. In particular, in the 1 The exceptions being references [9, 10]. 1 last years it has been shown that lepton mixing may be well described by discrete non abelian symmetries (see [12] and references therein for further details). Given that in these kind of models both, the lepton number and flavor breaking scales are free parameters the question about how does leptogenesis proceeds in the flavor symmetric phase proves to be quite reasonable. In more detail, let us suppose to have a non abelian flavor symmetry group GF under which the RH neutrinos and left-handed SM leptons have definitive transformations. We introduce a cutoff scale Λ since we will deal with non-renomalizable operators. The scenario in which we are interested in is the following: the lepton number breaking scale, characterised by the RH neutrino mass, MN , is larger than the scale at which GF is broken, vF . This means that the Yukawa mass matrices above and below vF are different: in particular the Yukawa Dirac matrix below vF is proportional to the Dirac mass matrix– the proportionality factor represented by the SM Higgs vacuum expectation value (vev). Flavon masses, Mφ , are taken as free parameters, clearly not too far from vF but above it. Clearly it holds Λ > MN , Mφ > vF . Our discussion is based on the class of symmetries that explain neutrino masses and lepton mixings and may be be generalized to any flavor symmetry, abelian or not, discrete or continuos. We will exemplify our arguments by doing a full analysis of the generation of the B −L in a concrete model that at low scale exhibits exact TriBiMaximal (TBM) mixing at leading order. The reason for this choice is very simple: it has been shown that when exact TBM mixing is induced by type I seesaw the CP violating asymmetry is zero and acquires a non-vanishing value only when lepton mixing deviates from TBM [7, 13, 14]. It may be proved that even in the context of type II seesaw with exact TBM the CP asymmetry at leading order is zero. Moreover the interplay between type I and type II seesaw in the class of models with form-diagonalizable mixing matrix is more constrained than expected [15]. The most recent analysis indicates that TBM is not anymore in perfect agreement with neutrino experimental data since the reactor angle predicted by TBM is zero, while this value is now excluded at 3σ level [16]. However TBM remains a good approximation for the lepton mixing matrix and we will consider a model that predicts exact TBM for its simplicity in showing the feasibility of leptogenesis in the regime Λ > MN , Mφ > vF . 2 The paper is organized in the following way: next section is general and we enumerate the general conditions necessary to obtain a CP asymmetry ǫN in the flavor symmetric regimen. Section 3 shows how the proposed conditions work by mean of a complete analysis of a specific model based on the flavor symmetry A4 . The model main features and neutrino phenomenology are briefly discussed and the generation of the B − L asymmetry is explained in detail. Section 4 is devoted to our conclusions. The calculation of the reaction densities necessary for the analysis of the washout processes studied in sec. 3 are given in appendix A. 2 Leptogenesis in the flavor symmetric phase We have already anticipated in the introduction that in models for leptonic flavor mixing four energy scales can be distinguished, namely a cutoff scale Λ-or in general a scale of heavy matter-, the lepton number breaking scale-determined by the RH neutrino massesMN , the flavons scale-determined by the masses of the scalars that trigger the flavor symmetry breaking-Mφ and the scale at which the flavor symmetry is broken, vF . Though Λ > MN , Mφ , vF the scales MN , Mφ and vF , being free parameters, can follow any hierarchy. Since we are concerned about leptogenesis in the flavor symmetric phase it is clear vF < MN , Mφ . This constraint in turn has an implication: if the flavor symmetry enforces the RH neutrinos to belong to the same flavor group GF representation R leptogenesis will not be achievable: in the flavor symmetric phase the RH neutrinos have a common universal mass and therefore the CP violating asymmetry ǫN vanishes [17]. As a consequence, in these kind of models viable leptogenesis requires RH neutrinos to belong to different GF representations Ri , so a mass splitting among the masses of the different representations can be accomodated. Let us assume the existence of k electroweak lepton doublets placed in r representations Lr , m RH neutrinos lying in p representations Np and n electroweak singlet scalars arranged in q representations Sq . Assuming the SM Higgs SU(2) doublet H to be a GF singlet the i-th RH neutrino representation can only decay to final states containing Li . Accordingly, three type of models can be distinguished: 1. For any RH neutrino representation there is a lepton doublet representation with 3 which a gauge flavor invariant renormalizable operator L̄i Ni H can be built 2 . 2. Only for a set of the RH neutrino representations a gauge flavor invariant renormalizable operator L̄i Ni H exist 3. For non of the RH neutrino representations the operator L̄i Ni H can be built. In cases 1 and 2 the standard one-loop vertex and wave-function corrections to the treelevel decay exist however the CP violating asymmetry derived from their interferences and the corresponding tree-level process vanishes. The proof of this statement is easy. Suppose Ni corresponds to the representations Ri . To recover the correct kinetic term we know that it holds Ri∗ Ri = δαi βi where αi , βi are the indeces of the representation Ri . Now, the standard contribution to the CP asymmetry when the flavor symmetry is exact is proportional to the contraction (Ri∗ Ri )αi βi , with βi 6= αi and is therefore zero. Thus, viable leptogenesis is possible only if new contributions to ǫN are present, and this is possible only if at least for one of the RH neutrino representations the condition MN > Mφ is satisfied. In case 3 it is possible that Ni may decay through a n > 2 body decay by means of non renormalizable operators. In this scenario one should modify the standard case in order to include n-body decays. However, the CP asymmetry generated in this case is expected to be small. For this reason in what follows we do not consider case 3. Assuming the flavons are lepton conserving states no B−L asymmetry can be generated via φ decays. However, once the condition MN > Mφ they can play an essential role in the generation of the B − L asymmetry not only because they lead to novel contributions to ǫN , but because in some cases they can even allow some RH neutrino representations to have new decay modes that can change the way in which leptogenesis takes place. Once the conditions discussed above are satisfied not much more, from a general perspective, can be said and the way in which the B − L asymmetry is generated depends upon the particular flavor model. Hereafter are discussion will rely on a particular A4 flavor model realization. 2 Here i labels the index representation, not the flavor index. 4 3 Setup We consider the non supersymmetric version of a model inspired by the Altarelli-Feruglio model discussed in [18] of which the type-I seesaw formulation has been analyzed in [19]. In the original model supersymmetry is introduced to induce the correct spontaneous breaking of the flavor symmetry. Here we assume that by adding additional discrete abelian symmetries or ad hoc soft terms the scalar potential may be arranged in such a way that the desired breaking is realized. At this level the model presented is still a toy model, however our findings will hold even in its supersymmetric version. 3.1 The A4 group Before entering into the details of the model, for completeness, we will briefly discuss the basic ingredients of the A4 discrete group in which the model presented here is based. A4 is the group of even permutations of 4 objects. It has 4 irreducible representations: one triplet and three singlets 1, 1′ , 1′′ . A4 may be thought as generated by two elements S, T satisfying S 2 = T 3 = (ST )3 = 1 . (1) In what follows we will work in the A4 basis in which the triplet representation of T is diagonal, namely S=  −1 1  2 3 2 2 −1 2 2    2  , −1 1 0 0     T =  0 ω 0  0 0 ω2 (2) with ω 3 = 1. The multiplication rules in this basis are given by (ab)1 = (a1 b1 + a2 b3 + a3 b2 ) , (ab)1′ = (a3 b3 + a1 b2 + a2 b1 ) , (ab)1′′ = (a2 b2 + a1 b3 + a3 b1 ) , (ab)3s = (2a1 b1 − a2 b3 − a3 b2 , 2a3 b3 − a2 b1 − a1 b2 , 2a2 b2 − a3 b1 − a1 b3 ) , (ab)3a = (a2 b3 − a3 b2 , a1 b2 − a2 b1 , a3 b1 − a1 b3 ) , (3) 5 where a and b are triplets of A4 , namely a ∼ (a1 , a2 , a3 ), b ∼ (b1 , b2 , b3 ) and for the singlet representations the multiplication rules are trivial 1′ ⊗ 1′′ = 1 , 3.2 1′ ⊗ 1 = 1′ , 1′′ ⊗ 1 = 1′′ . (4) The model In our model four RH neutrinos are added at the SM field content. Three of them, νT , form an A4 triplet while the fourth, ν4 , is an A4 singlet. The lepton doublets, l1 , l2 , l3 , transform as an A4 triplet. For simplicity we assign flavor charges using the Weyl spinor notation. The RH 4-dim Majorana fermion will be therefore defined as   C ν . N = ν (5) It is important to notice that if l ∼ (l1 , l2 , l3 ) transforms as a triplet the requirement of recovering the correct kinetic term according to the group multiplication rules imposes that l† transforms as a triplet but ordered as l† ∼ (l1† , l3† , l2† ). On the contrary we order νT† as (ν1† , ν2† , ν3† ) thus νT ∼ (ν1 , ν3 , ν2 ). RH charged leptons transform as the 3 one dimensional representation of A4 , namely 1, 1′ , 1′′ . Two A4 scalar triplets, φT and φS , our flavons, are added. Once the flavor symmetry is broken they will give rise to the correct mass matrices. 3.2.1 Neutrino mass matrices Given the field content previously described the Lagrangian for the lepton sector reads as Mν4 c† MνT † c (νT νT + νTc† νT )1 + (ν4 ν4 + ν4† ν4c )1 2 2 + λ[νT† νTc φS ]1 + λ∗ [νTc† νT φ∗S ]1 −L = + ξ[νT† φS ]1 ν4c + ξ ∗ ν4c† [νT φ∗S ]1 + y1 ǫαβ (νT† lα )1 H β + y1∗ǫαβ (l†α νT )1 H β 1 1 + y2 ǫαβ [νT† lα φS ]1 H β + y2∗ ǫαβ [l†α νT φ∗S ]1 H ∗β Λ Λ 1 1 + y3 ǫαβ ν4† [lα φS ]1 H β + y3∗ ǫαβ [l†α φ∗S ]1 ν4 H β Λ Λ 1 + yei eci [φT lα ]i H̃ α + H.c. . Λ 6 (6) In eq. (6) we have assumed the presence of an abelian ZN with N > 2 that forbids φS,T and φ∗S,T to couple in the same way and we are assuming the flavons to be complex fields. Note that in this class of models this kind of Abelian symmetries are always present to prevent interference between φT and φS . Note also that if we take MνT and Mν4 to be real we have the freedom to put other two phases to zero, absorbed by redefinitions of φS and l respectively. We choose λ and y3 to be real. In eq. (6) the last row describes charged leptons: i stays for 1, 1′, 1′′ and the [...]i stays for the triplet contractions in the one dimensional representations. In the other rows [...]1 stays for 2 or 3 triplets contracted in a singlet. α and β are SU(2)L indices, H the SM Higgs doublet, and as usual H̃ = iσ2 H. We assume that the additional ZN symmetry forbids φS coupling to neutrinos and φT coupling to the charged sector. The A4 basis chosen is useful because when φT develops a vev according to hφT i ∼ vT (1, 0, 0) the charged lepton Yukawa mass matrix is diagonal. On the other hand when φS acquires the vev hφS i ∼ vS (1, 1, 1) the Dirac Yukawa matrix, Yν , and the RH neutrino mass matrix, MN are diagonalized by the so-called TBM mixing matrix. Thus after electroweak symmetry breaking the light neutrino mass matrix, mν ∼ −mTD · M−1 N · mD (7) is diagonalized by the TBM mixing matrix as well. Clearly mD = Yν vH , with hHi = vH . Both MN and Yν get a contribution above (>) and below (<) the scale vF ∼ vT ∼ vS , so we may write < MN = M> N + MN , Yν = Yν> + Yν< , 7 (8) with M> N M< N  a 0 0   0 0 a  ∼   0 a 0  0 0 0  2b −b   −b 2b  ∼   −b −b  c c 0   0   , 0   d −b c Yν>   −b c   , 2b c   c 0  a 0   0 0  ∼  0 a  0 0  Yν< 0   a    0   0 2b −b −b     −b 2b −b    ∼ .  −b −b 2b    c c c (9) with a ∼ MνT (y1 ), b ∼ λvS /Λ(y2vS /Λ), c ∼ ξvS /Λ(y3 vS /Λ) for the RH neutrino (Yukawa Dirac) mass matrix and d = Mν4 . Without loss of generality vS may be taken real. Defining ǫS = vS /Λ, clearly in the limit ǫS → 0 the symmetry is restored and the light neutrinos are degenerate. Thus we expect in the majority of the cases a quasi-degenerate (QD) neutrino mass spectrum in which |y1 |2 /MνT controls the absolute scalar mass while y2 , λ, ǫS = vS /Λ parametrize the neutrino atmospheric mass splitting . We may find an approximate analytical solution for the spectrum expanding in ǫS . We have 2 vH , MνT (|y1 |2 ξ 2 − 2|y1|y3 ξ cos φy1 + y32 cos 2φy1 ) 1 ǫS + (ǫX + ǫ2X ) ∼ 2 2 |y1 |(|y1|λ − 2|y2 | cos ∆φ12 ) m0 ∼ |y1|2 ∆m2sol ∆m2atm (10) with ǫX = (MνT − Mν4 )/MνT , φy1,2 = Arg(y1,2) and ∆φ12 = φy1 − φy2 . Equation (10) holds only in the regime ǫS < ǫX . When ǫX < ǫS the analytical ex- pressions become more cumbersome because the QD scenario is broken and it is possible to recover both the normal hierarchical (NH) and inverse hierarchical (IH) neutrino mass spectrum. The spectrum predicted by our model is shown in fig.1 by means of mββ , the parameter relevant for neutrinoless double beta decay defined as mββ = [UTBM diag(m1 , m2 , m3 ) UTBM ]11 . It has been obtained by numerically diagonalizing the neutrino mass matrix below the scale vS . All the Yukawa parameters, λ, ξ, yi are taken of order ∼ O(1) and for completeness in the figure we have reported the future experimental bounds on |mββ | and m1 . 8 1 present 0Ν2Β bounds + + + Èm ΒΒÈ HeVL 10 + + + + ææ æ+ + æ + + ææ + + ++ æ æ + æ æ ++ æ æ æ + æ + + + æ + æ æ+ + + æ + + æ + æ+ + + + + æ + æ æ æ æ æ+ + + æ +æ+ æ +æ+æ + + + + æ+ æ+ + + ++ æ + ++ æ æ ææ ææ + + ææ+ + æ æ + æ++ æ æ + ææ + + æ+ ++ æ +æ+ + + æ æ + æ + + + + + + æ + æ + æ + æ æ ææ + æææ æ + + + æ + + ++ + æ æ + æ + æ æ + + ++æ+ ææ+ æ+ æ +æ++ + æ+ + + + + æ æ + + æ+ ++ æ æ ++ + +++ + +æ++ + + æææ +æ+ æ + æ+ æ + + ++ æ ++ æ+ + + + æææ + + ææ+ ææ+ + ++ ææ ++ æ++ ++ + + + æ+ æ æ+ +++ + + +ææ+ + +æ+ +æ+ + ++ + æ + + + æ æ + + æ+ +++ æ + æ + + + æ+ æ ++ æ+ + + æ++ ++ + + +æ++ ææ+ + + + æ æ+ æ + + æ æ+ ++ ææ + + + + æ+ æ+ + ++ + æ æ æ+ + +++ + æ æææ + + + + +++ ææ GERDA II ++ ++ ++ + + + + + + +ææ + æ+ + + -1 + æ ++++ ææ+ + + æ æ æ æ++ + + + æ æ + + æ + + + ææ + æ+ + + æ + æ+ æ+ + æ++æ+++ + æ æ + + ++ + ++ æ + ++ æ+ æ+ ++ + ++ +ææ+æ+ æ+ ++ æ + +ææ+++ +æ++æ++æ + ++ ++ + æ++ æ+ +++ æ æ æ ææ +æ+æ+ ++ ++ +ææ +æ++ æ ææ + ++ æ+ ææææ æ æ ++ æ+ ææ æ + ++++ + æ+ æ æ + ææ+ æ+ +æææææ++æ++ ++ +æ+ + æ + ææ + ++ +++++ ++ æ +æ + æ+++ææ+++æ+ +++ +++ ++++++ ++ æ+æ ++æ+ +++ +++ ææ +æ+ + æ+ ++ ææ+++ + ++ æ + æ ++æ æ+ + +æ+æ+ ++ + ++++ æ+æ+ +++++ + æ+ + ææ æ+ + ++ + +æ+ +æ ææ+ + æ ++++ +++ ++ æ+ + ++ + + æ+ + + + + + + æ ææ ++ ææ+ + + + + æ + + + + ++ + + æææ ++ æ +++ + + ++ + + + + ææ++ + ++ + + + + + ++ +++++ ++++ æ ++ ++ + æ ++ææ++++ +++ + +æ+++ + + + ++ ++ ++++æææ++ææ++++æ+ + + ææ ++++ +ææ+ææ++ æ+ + + + + +æ+ + + + + + æ + ++ ++ æ æ + æ + æ+ + + + + æ æ æ ++ +æ++ + III + ++æ æ æ ++++ æ+++ +ææ æ MajoranaGERDA æ ++ æ æ + + + ææ + + æ + + æ+ + æ+ + + + ++ + + + + +æ æ+ æ æ + + + +++ + æ ææ + + + æ+æ+æ ææ++ ææ ++ æ æ ++ + æ + + + +++ æ æ æ æ æ æ æ æ ææ æ æ æ æ æ æ æ ææ æ KATRIN CUORE 10-2 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ 10-3 -3 10 10-2 10-1 m1HeVL Figure 1: The predictions for |mββ | as a function of the lightest neutrino mass. The natural spectrum predicted by the model is QD, with both normal ordering (NO) and inverse ordering (IO) as indicated by the analytical approximations (see the text). Both QD-NO (QD-IO) and NH (IH) spectrum are indicated with green points (blue crosses). For what concerns the flavon sector the ZN symmetries allows only the mass term Mφ2 (φS φ∗S )1 . (11) Recalling now that φ∗S ∼ (φ∗S1 , φ∗S3, φ∗S2 ) we have that the flavon mass matrix is diagonal and CP even and odd states are degenerate. 3.2.2 Flavon Interaction Matrices < Above vF instead of M< N , Yν we have flavon-neutrino and flavon-Higgs-neutrino interac- tion matrices IN,ν . Starting from the interactions 1 1 † c λ[νT νT φS ]1 + λ∗ [νTc† νT φ∗S ]1 + ξ[νT† φS ]1 ν4c + ξ ∗ν4c† [νT φ∗S ]1 2 2 1 1 y2 ǫαβ [νT† lα φS ]1 H β + y2∗ ǫαβ [l†α νT φ∗S ]1 H ∗β , + Λ Λ − LI = (12) we may write them using the 4-component spinors N and L as 1 1 k INR (φ∗Sk )ij N̄i PR Nj + INk L (φSk )ij N̄i PL Nj 2 2 k α β k +IDL (φSk )ij ǫαβ N̄i PL Lj H + ID (φ∗Sk )ij ǫαβ L̄αi PR Nj H̃ β , R 9 (13) k† k and k labels the φS , φ∗S flavons. Notice that in eq. (13) and = ID where INk L = INk†R , ID R L for the rest of the paper we will indicate with Ni the four RH neutrinos of the model under study, while in sec. 2 Ni was referred to the group representations. First of all we change basis going in the basis in which M> N is diagonal. Thus we have T > M̂> N = UR · MN · UR = Diag(MνT , MνT , MνT , Mν4 ) = Diag (MNi(i=1,2,3) , MN4 ) , UR  √ 2 0  1   0 = √  2  0 0 ŶD> = UR† · YD> 0 1 −i 0   0   , 0   √ 2 1 i 0 0  √ 2 0 0   1 1 y1  0 = √  2 2  0 −i i  0 0 0     .   (14) It proves useful to write the interaction matrices as INk R (φ∗Sk )ij = (IN>R )kij φk , k k ID (φ∗Sk )ij = (I > DR )ij φk R (15) k k and similarly for INk L (φSk )ij , ID (φSk )ij . The ID are 3 × 4 matrices, being 4 the total L R number of RH neutrinos. Equation (15) may appear as redundant since in our model φS couples always linearly. However the advantage of our notation is that it holds even when operators of dimension higher than 5 are included. 3.3 CP asymmetries In the standard leptogenesis scenario the lightest RH neutrino CP asymmetry, ǫN , arises from the interference between the tree-level decay Feynman diagram and the one-loop vertex and wave function corrections [20]. Since N4 does not have renormalizable couplings to the lepton doublets such diagrams do not exist in the case under consideration, regardless 10 φi H Ni Ni H N4 Li Li (b) (a) Figure 2: Tree-level and one-loop correction diagrams accounting for ǫNi . of the RH neutrino mass spectrum. New contributions due to the presence of the flavons degrees of freedom exist and depend upon the RH neutrino spectrum: 1. The MNi > MN4 case: Ni has standard Li H tree-level decays and, given the interactions in the Lagrangian (6), the only possible correction to this process arise from the one-loop correction to the effective vertex Ni φi Li H, as shown in fig. 2. Thus, the CP asymmetry in this case is obtained from the interference between diagrams 2(a) and 2(b). 2. The MN4 > MNi case: Since N4 couples to Ni φi at the renormalizable level a CP asymmetry for the two body decay process N4 → Ni φi can be calculated from the interference between the corresponding tree-level diagram and the two-loop level diagram involving both effective couplings N4 Li φi H and Ni Li φi H (since N4 does not couples to lepton doublets at the renormalizable level the one-loop correction to the process N4 → Ni φi does not exist). There is another option involving N4 three-body decays induced by the effective coupling N4 Li φi H. In this scenario a one-loop correction to the effective process does not exist either and the calculation of the CP asymmetry relies again on the two-loop level correction of the previous case. In case 1 the CP asymmetry arises in a different way compared to the standard case but in what regards the generation of the B − L asymmetry there is no difference. In contrast, the cases in 2 are quite different: for the three-body decay scenario the differences are obvious, for the other scenario leptogenesis will take place in two stages, a first stage in which an asymmetry in Ni is generated via the decays N4 → Ni φi and a second stage in 11 which the asymmetry in Ni is partially transfered to the lepton doublets via Ni decays and scatterings (a scenario of this kind has been discussed in [9, 21]). Note that in this case the CP asymmetry, being a two-loop order effect, would most likely yield a very tiny B − L asymmetry. All these scenarios however exhibit a common feature, the generation of a B − L asymmetry takes place in the flavor symmetric phase. So from now on we will focus on case 1, that as was already pointed out resembles standard leptogenesis. The CP asymmetry in the decay of Ni is defined according to ǫNi = X k=e,µ,τ ǫLNki X ΓkN − Γ̄kN i i , = k k Γ + Γ̄ Ni k=e,µ,τ Ni (16) where ΓLNki (Γ̄LNki ) denotes the Ni partial decay width for final states of flavor k and carrying +1 (−1) unit of lepton number, and ǫLNki are the flavored CP asymmetries. However, here we are working in the context of an exact flavor symmetry. Since flavor is unbroken only flavor conserving processes may happen, that in our framework means k = i. Moreover flavor invariance and the representation used for our right and left handed neutrinos imply that the three RH neutrinos produce the same amount of CP asymmetry and have exactly the same dynamics. Due to the Majorana nature of the RH neutrino states there are two possible one-loop diagrams of the type 2(b) (as in the standard leptogenesis case for the wave-function correction), so the interference between the tree and one-loop level amplitudes (M0 and M1 ) involve two terms. For two-body decays this interference is phase-space independent and consequently the calculation of ǫNi can be simply done in terms of the products of M0 and M1 and approximating the denominator of (16) with |M0 |2 [9]. In the limit MNi >> MN4 , Mφ the CP asymmetry can be written as ǫℓNi i = −  > k  1 MNi 1 Im (Y I ) . i4 D R 8π |Y > Y >† |2ii Λ (17) In the general case, without assuming any large hierarchy among the heaviest RH neutrino representation, the lightest one and the flavons, the expression for ǫNi is far more involved. Fig. 3, obtained from the exact expression, shows the possible values of the CP asymmetry as a function of the effective cut-off scale Λ. 12 1 æ æ æ æ æ æ -1 10 + æ + + æ æ ++ + æ æ + æ + æ + æ + + + + æ + + ++æ+ æ æ + + æ + + æ+ æ æ + + + æ æ+ æ + + + + + + + + æ ++ æ + + æ æ + + + + + ++ +æ ææ + + æ æ æ +++ + + + æ +++æ+++ + ææ + + + ++ æ+ +++ æ ++ + + + ++ æ + æ + ææ æ +æ + + + +æ + æ +ææ æ+ ++ + + æ + æ æ + æ æ ++ + æ + ++ æ æ + ++ æ ææ + æ æææ + + + æ+ + æ ++ + æ + + + ++ ++++ æ + + æ æ ++ + + + + æ æ + ++ æ+ +æ æ æ æ + + + +æ ++ + + + æ æ æ + + + + + + + + æ + + æ + + æ + æ + æ ++ æ æ + +æ + + + + + + ++æ æ + æ + ææ æ æ + æ + æ æ+ + + + + æ æ æ+ + æ æ æ + æ+ + æ æ + +æ + +æ +++ æ + æ æ æ æ æ+ æ æ + ++ + + + + æ + æ+ æ æ æ æ + æ + æ æ + ææ æ æ + + + æ æ æ æ æ+ æ æ ++ æ +æ + æ æ + æ æ æ æ æ +æ æ æ ++ æ æ æ æ æ æ æ + ææ æ + æ ææ + Ε Ni 10-3 10-4 æ æ æ æ æ æ æ æ æ æ + æ+ æ+ ++ + ææ æ æ æ æ æ æ æ æ æ æ æ + æ 10 æ æ æ -6 æ æ æ æ ææ + æ 10-5 æ æ ææ æ æ æ æ æ + + æ æ 10-2 æ æ + + æ 10-7 10 10 1011 1012 1013 1014 1015 L HGeVL Figure 3: ǫNi as a function of the cutoff scale Λ for the spectrum shown in fig. 1. Green points (blue crosses) corresponds to QD-NO and NH (QD-IO and IH) neutrino mass spectrum. 3.4 Generation of the B − L asymmetry The generation of the B −L asymmetry is entirely determined by Ni dynamics but its final value depends on the washout induced by the A4 flavor singlet, N4 . Thus, leptogenesis in the case we are interested in is a two-step process: generation of the B − L asymmetry and its subsequent washout via N4 interactions (such scenario has been analysed in the context of type-III seesaw in [22]). In what follows we will analyze both stages in the unflavored regimen. 3.4.1 Ni dynamics The determination of the B − L asymmetry relies on the solution of the kinetic equations for the Ni abundance and the B − L asymmetry itself. At leading order in the coupling y1 , that is to say including only Ni → LH decays and inverse decays and ∆L = 2 scatterings 13 (LH † ↔ LH † and L̄H ↔ L̄H) 3 , they can be written according to ! 1 YNi (zi ) dYNi − 1 γDi (zi ) , =− dzi s(zi )H(zi )zi YNEq (zi ) # ! " i T dY∆B−L Y∆B−L 1 Y Ni γDi (zi ) , − 1 ǫNi + =− dzi s(zi )H(zi )zi YNEq (zi ) 2YℓEq (18) (19) i where zi = MNi /T , s is the entropy density, YX = nX /s (with nX the X number density), H(z) is the expansion rate of the Universe and the reaction density γDi (zi ) is given by γDi (zi ) = 1 MN5 i K1 (zi ) m̃T , 8π 3 v 2 zi (20) with v ≃ 174 GeV, K1 (zi ) the modified Bessel function of first-type and the parameter m̃T = v 2 |y1 |2 /MNi . An exact solution of the kinetic equations in (18) and (19) can only be done numerically, however a reliable approximate solution can be found [25], which we now discuss in turn. Equations (18) and (19) can be recasted according to h i dYNi Eq = −DT (zi ) YNi (zi ) − YNi (zi ) , dzi h i dY∆TB−L T = −ǫNi DT (zi ) YNi (zi ) − YNEq (z ) − WID (zi )Y∆B−L , i i dzi (21) where the new decay and inverse-decay functions read DT (zi ) = KT zi K1 (zi ) K2 (zi ) 1 T and WID (zi ) = KT zi3 K1 (zi ) , 4 (22) with KT = m̃T /m⋆ (m⋆ = 8πv 2 H(zi = 1)/MN2 i = 1.08 × 10−3 eV) and K2 (zi ) is the modified Bessel function of the second-type. In terms of KT the strong (weak) washout regimen is defined as KT ≫ 1 (KT ≪ 1). The B − L asymmetry is obtained from the formal integration of eqs. (21) by means of the integrating factor technique: Y∆TB−L (zi ) = −3 × ǫNi YNEq (zi → 0) η(zi ) . i Here η(zi ) is the efficiency function defined as [25] Z zi ′ Rz 1 ′ dYNi (z ) − z ′i dz η(zi ) = − Eq e dz ′ YNi (zi → 0) z0 3 T (z ′′ ) dz ′′ WID (23) . (24) The inclusion of these processes is mandatory to obtain kinetic equations with the correct thermody- namical behavior [23, 24]. 14 10-5 10-6 ǫT = 10−1 Y∆TB−L (KT ) 10-7 10-8 10-9 10-10 ǫT = 10−6 10-11 10-12 102 10 103 KT Figure 4: B − L asymmetry produced by the A4 flavor triplet (Ni ) dynamics. The values ǫT = 10−6 − 10−1 correspond to extreme cases. Note that we have included a factor of 3 in (23) to account for the Ni flavor degrees of freedom. The final B −L asymmetry is therefore obtained for zi → ∞ once the parameters KT and ǫNi are specified. The problem of determining the final Y∆TB−L analytically is thus reduced to find an approximate expression for the efficiency function at zi → ∞ (efficiency factor). Such an expression can be derived in the strong washout regimen by: (i) noting that at low temperatures YNi (zi ) follows closely the equilibrium distribution, so the replacement dYNi (zi )/dzi → dYNEq (zi )/dzi in eq. (24) can be done; (ii) replacing the washout function i T T (zi )/z, where zB is the minimum of the function WID (zi ) by W ID (zi ) = zB WID  T ′  Z zi WID (z ) ′ T ψ(z , zi ) = − ln + dz ′′ WID (z ′′ ) . z′ z′ (25) Following this procedure the efficiency factor can be derived [25]: η= with
2 1 − e−KT zB (KT )/2 , KT zB (KT ) 1 zB (KT ) = ln 2 (   5 ) 3125 π KT2 π KT2 ln . 1024 1024 (26) (27) With eqs. (26) and (27) at hand we can determine the maximum and minimum (still consistent with the measured baryon asymmetry) B − L asymmetry one can get through 15 φ N4 N4 φ N4 L N4 φ L H φ H H L L H Figure 5: Relevant 1 ↔ 3 and 2 ↔ 2 s, t and u scattering processes accounting for the A4 singlet washouts. Ni dynamics. The results are displayed in figure 4. Particularly relevant is the maximum value Y∆TB−L ≃ 10−5 as it allows to derive an upper bound on the washout induced by N4 dynamics. 3.4.2 N 4 washout The B − L asymmetry produced at zi ∼ 1 remains frozen up to the temperature at which N4 washouts become effective, z4 = MN4 /T ∼ 1. Since N4 couples to lepton doublets via an effective five-dimensional operator the dynamics of N4 washouts, at leading order in the couplings, involves not only the processes N4 ↔ LφH but the 2 ↔ 2 s, t and u channel scatterings (see figure 5), in contrast to the standard leptogenesis scenario. The derivation of the corresponding kinetic equations in this case is tricky and requires -even at leading order in the couplings- the inclusion of 3 ↔ 3 and 2 ↔ 4 scattering processes (see ref. [10] for more details). Since ǫN4 = 0 the kinetic equations accounting for the N4 washouts can be written according to ! 1 YN4 (z4 ) dYN4 − 1 γtot (z4 ) , =− dz4 s(z4 )H(z4 )z4 YNEq (z ) 4 4 " ! # S S dY∆B−L Y∆B−L YN4 (z4 ) s γtot (z4 ) + − 1 γ2→2 (z4 ) . =− dz4 2YℓEq YNEq (z ) 4 4 (28) (29) s where γ2→2 (z4 ) is the reaction density for the 2 ↔ 2 s-channel scattering process and γtot (z4 ) involves the reaction densities for the full set of processes shown in figure 5, namely γtot (z4 ) = γ1→3 (z4 ) + X C=s,t,u 16 C γ2→2 (z4 ) . (30) 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-1 s (zi ) γ2→2 102 10 1 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-1 KS = 10 t,u (zi) γ2→2 γtot (zi ) s(zi )H(zi )zi γ(zi ) s(zi )H(zi )zi 102 10 1 γ1→3 (zi ) 1 10 KS = 10 KS = 10−2 1 10 zi zi Figure 6: Reaction densities as a function of zi for the different processes present in NS washout (left panel) and total reaction densities for different values of the decay parameter KS (right panel). As in the standard case the strong washout (weak washout) regimen is defined according to KS ≫ 1 (KS ≪ 1). As explained in appendix A all the reaction densities can be written in terms of the total decay width Γ(N4 → LHφ), which we have calculated to be 1 MN3 4 |y3|2 . Γ(N4 → LHφ) = 3 2 192π Λ (31) In terms of the reaction densities given in (43) the kinetic equations in (28) and (28) can be rewritten in such a way they resemble eqs. (21): dYN4 = −DS (z4 ) dz4 dY∆SB−L dz4 YN4 (z4 ) −1 YNEq (z ) 4 4 ! , S = −WID (z4 )Y∆SB−L , S where now the functions DS and WID are given by   3 1 3 DS (z4 ) = KS z4 K1 (z4 ) + (Ss (z4 ) + St (z4 )) 4g⋆ 2 !# " 1 3 Y N4 S Ss (z4 ) + St (z4 ) . WID (z4 ) = KS z43 K1 (z4 ) + 4 2 YNEq 4 (32) (33) Some words are in order regarding these equations. The relativistic degrees of freedom are g⋆ = 118 as in our calculations we use Maxwell-Boltzmann distributions, and the functions 17 1016 KS = 4.6 × 10−2 10-6 10-7 1014 Λ [GeV] Y∆SB−L (zi) 10-5 10-8 1012 1010 10-9 108 10 -10 10-1 1 ëëëë ëëëëëëëëëëëë ëëëëëëëëëëëëëëëëëëëëëëëëëëëë ë ë ëë ëëëë ◦ KS < 10−4 ëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë ëëëëëëëëëëëëëëëë ëëëëëëëëëëëëëëëëëëëëëëë ◦ KS > 10−4 ëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë ëëë ë ëë ëëë ë ë ëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë ë ëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë ë ë ë ë ë ë ë ëëëëëëë ë ë ë ë ëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë ëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë ë ë ë ë ëëëëëë ë ëëëëëë ë ëë ëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë ëëëëëëëëëëëë ë ëëëëë ëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë ëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë ë ë ëë ëë ëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë ëëëëëëëëëëëëë ëëëëëëëëëëëëëëëëëëëëëëëëëëëë ëëëëëëëëëëëëëëëëë ëë ëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë ëëë ë ëëëë ëë ëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë ëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëëë ë ë ë ë ë ë ë ë ë ë ë ë ë ëëëëëëë ëëëëëëëëëëëëëëëëëëëëëëëëëëëë ëëëëëëëë ë ëëëëëëëëëëëëë ëëëëëëëëëëëëëëëëëëëëëëëëëëëëë ë ë ë ë ë ë ë ëëëëëëëëëë ëëë ë 104 10 106 108 1010 1012 MN4 [GeV] zi Figure 7: Washout induced by NS on the maximum Y∆TB−L (see fig. 4) generated in NT dynamics (left panel). The largest allowed KS for which the resulting Y∆B still fits the measured value is KSmax = 4.6 × 10−2 , any value for which KS > KSmax is excluded. On the right panel allowed regions of Λ − MNS as required by the condition KS < KSmax . Ss,t are given in eqs. (44) in the appendix. The decay parameter KS is defined in the same way it is defined in the case of Ni dynamics, KS = m̃S /m⋆ but with m̃S = 8πv 2 Γ(N4 → LHφ) . MN2 4 (34) The presence of the 2 → 2 scattering processes may drive the system to the strong washout regimen even when the 1 → 3 process is slow. Thus, the appropiate definition of the strong (weak) washout regimen in this case reads: γtot (zi ) s(zi )H(zi )zi > 1 (< 1 for weak washout) . (35) zi ∼1 Figure 6 (left panel) shows an example in which though γ1→3 (zi )/s(zi )H(zi )zi |zi∼1 ≪ 1 the system is driven to the strong washout regimen by scattering processes. Note however that the condition KS ≫ 1 (KS ≪ 1) still determines the regimen in which the washout dynamics of N4 takes place, as can be seen in figure 6 (right panel). From the integration of eqs. (32) an upper bound on KS for a given Y∆TB−L can be determined by the condition of not erasing this asymmetry below ∼ 2.6 × 10−10 . The maximum value KSmax is found for the largest possible B − L asymmetry generated in Ni dynamics that as has been argued in sec. 3.4.1 we have found to be ∼ 10−5 . Figure 7 (left 18 panel) shows the final Y∆SB−L matches the required value ∼ 2.6 × 10−10 (for Y∆TB−L = 10−5 ) when KSmax ≃ 4.6 × 10−2, any value KS > KSmax will induce a washout that will damp the B − L asymmetry below the allowed value. Taking |y3 | = 10−2 the decay parameter KS becomes  2  GeV MN4 9 , KS = 12 × 10 GeV Λ (36) with the purpose of placing the more stringent bounds on the Λ − MN4 plane we fix 102 < Λ/MN4 < 104 and take into account the restriction KS < KSmax . The results are displayed in figure 7 (right panel) where the allowed Λ − MN4 region can be seen. Any discussion of leptogenesis in the scenario we have considered here should be done at least within that region. 4 Conclusions In this paper we have study the necessary conditions that have to be satisfied whenever the generation of the cosmic baryon asymmetry of the Universe via leptogenesis takes place in the presence of a lepton flavor symmetry accounting for lepton mixing. In the scenario we have discussed, leptogenesis occurs in the flavor symmetric regime thus before the flavons (that trigger the breaking of the flavor symmetry) acquire vevs, accordingly the decays responsible for generating a net B − L asymmetry are liable of selection rules dictated by the flavor symmetry. In the core of the paper we exemplify how the general conditions for the generation of the baryon asymmetry, in the flavor symmetric phase, work by analysing a specific model based on the flavor symmetry A4 . We briefly discussed the low energy phenomenology of the model and studied in detail, by using the corresponding kinetics equations, the generation of the baryon asymmetry. In the model considered, due to the constraints imposed by A4 , the asymmetry proceeds through the CP violating and out-of-equilibrium decays of the heaviest RH neutrino A4 representation. Subsequent washouts induced by the lightest A4 representation, being potentially dangerous, were properly taken into account. Our onset shows these washouts can always be circumvented and the correct amount of baryon asymmetry can be produced. 19 In conclusion we have shown that under certain conditions, in models containing flavor symmetries in the lepton sector, leptogenesis can occur even in the flavor symmetric phase. The conditions we have enumerated can be regarded as a general recipe for constructing lepton flavor models in which the lepton number violating scale is above the flavor symmetry breaking scale and the generation of the baryon asymmetry proceeds via leptogenesis. 5 Acknowledgments DAS would like to thank Luis Alfredo Munoz for discussions. DAS is supported by a FNRS belgian postdoctoral fellowship. A Appendix: Reaction densities for 1 → 3 and 2 → 2 processes In this appendix we present the relevant equations used in the calculations discussed in section 3.4.2. The thermally averaged reaction densities for 1 → 3 and 2 → 2 processes are given by [24] K1 (z4 ) Γ1→3 , K2 (z4 ) Z ∞ √ √ MN4 4 x K x) σ bC (x) , dx = (z 1 4 64 π 5 z4 1 γ1→3 = nEq N4 (37) C γ2→2 (38) where x = s/MN2 4 (with s the center of mass energy) C = s, t, u, Γ1→3 ≡ Γ(N4 → LHφ) and σ b(x), the reduced cross section, defined as σ b(x) = 2 MN2 4 x λ(1, x−1 , 0) σ(x) with λ(a, b, c) = (a − b − c)2 − 4bc . 20 (39) Neglecting the lepton doublets, Higgs and flavones masses we have found for the differential cross sections the following results: 1 |y3 |2 1 1 dσ s = , 2 2 dt 16π Λ MN4 1 − x   1 |y3 |2 1 t 1 dσ t , = 1− 2 dt 16π Λ2 MN2 4 (1 − x)2 MN4   dσ u 1 |y3 |2 1 t 1 = x+ 2 . dt 16π Λ2 MN2 4 (1 − x)2 MN4 (40) Integrating over t in the range t− = MN2 4 (1 − x) and t+ = 0 and using the definition for the reduced cross section, eq. (39), we get |y3 |2 σ b (x) = 8π s |y3 |2 σ b (x) = 16 π t,u   MN4 Λ MN4 Λ 2 2 (x − 1)2 , x (41) (x2 − 1) . x (42) With these results at hand and taking into account the expression for Γ1→3 given in eq. (31) the different reaction densities become MN3 4 K1 (z) , Γ1→3 , π2 z 3MN3 4 Ss (z) Γ1→3 , = 2π 2 z 3MN3 4 St,u (z) Γ1→3 , = 4π 2 z γ1→3 = s γ2→2 t,u γ2→2 where the functions Ss,t,u (z) are given by Z ∞ Z ∞ √ √ (x − 1)2 x2 − 1 Ss (z) = dx √ K1 (z x) and St,u (z) = dx √ K1 (z x) . x x 1 1 (43) (44) References [1] G. Hinshaw et al. [ WMAP Collaboration ], Astrophys. J. Suppl. 180, 225-245 (2009). [arXiv:0803.0732 [astro-ph]]. [2] A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5, 32-35 (1967). [3] S. Davidson, E. Nardi, Y. Nir, Phys. Rept. 466, 105-177 (2008). [arXiv:0802.2962 [hep-ph]]. 21 [4] P. Minkowski, Phys. Lett. B 67 421 (1977); T. Yanagida, in Proc. of Workshop on Unified Theory and Baryon number in the Universe, eds. O. Sawada and A. Sugamoto, KEK, Tsukuba, (1979) p.95; M. Gell-Mann, P. Ramond and R. Slansky, in Supergravity, eds P. van Niewenhuizen and D. Z. Freedman (North Holland, Amsterdam 1980) p.315; P. Ramond, Sanibel talk, retroprinted as hep-ph/9809459; S. L. Glashow, inQuarks and Leptons, Cargèse lectures, eds M. Lévy, (Plenum, 1980, New York) p. 707; R. N. Mohapatra and G. Senjanović, Phys. Rev. Lett. 44, 912 (1980); J. Schechter and J. W. F. Valle, Phys. Rev. D 22 (1980) 2227; Phys. Rev. D 25 (1982) 774. [5] C. Hagedorn, E. Molinaro, S. T. Petcov, JHEP 0909, 115 (2009). [arXiv:0908.0240 [hep-ph]]. [6] E. Bertuzzo, P. Di Bari, F. Feruglio, E. Nardi, JHEP 0911, 036 (2009). [arXiv:0908.0161 [hep-ph]]. [7] D. Aristizabal Sierra, F. Bazzocchi, I. de Medeiros Varzielas, L. Merlo, S. Morisi, Nucl. Phys. B827, 34-58 (2010). [arXiv:0908.0907 [hep-ph]]. [8] R. G. Felipe, H. Serodio, Phys. Rev. D81, 053008 (2010). [arXiv:0908.2947 [hep-ph]]. [9] D. Aristizabal Sierra, M. Losada, E. Nardi, Phys. Lett. B659, 328-335 (2008). [arXiv:0705.1489 [hep-ph]]. [10] D. Aristizabal Sierra, L. A. Munoz, E. Nardi, Phys. Rev. D80, 016007 (2009). [arXiv:0904.3043 [hep-ph]]. [11] C. D. Froggatt, H. B. Nielsen, Nucl. Phys. B147, 277 (1979). [12] G. Altarelli, F. Feruglio, Rev. Mod. Phys. 82 (2010) 2701-2729. [arXiv:1002.0211 [hep-ph]]. [13] E. E. Jenkins, A. V. Manohar, Phys. Lett. B668, 210-215 (2008). [arXiv:0807.4176 [hep-ph]]. [14] Riazuddin, [arXiv:0809.3648 [hep-ph]]. 22 [15] D. Aristizabal Sierra, Federica Bazzocchi, Ivo de Medeiros Varzielas, in preparation. [16] T. Schwetz, M. Tortola, J. W. F. Valle, [arXiv:1108.1376 [hep-ph]]. [17] G. C. Branco, A. J. Buras, S. Jager, S. Uhlig, A. Weiler, JHEP 0709, 004 (2007). [hep-ph/0609067]. [18] G. Altarelli, F. Feruglio, Y. Lin, Nucl. Phys. B775 (2007) 31-44. [hep-ph/0610165]. [19] F. Bazzocchi, L. Merlo, S. Morisi, Phys. Rev. D80 (2009) 053003. [arXiv:0902.2849 [hep-ph]]. [20] L. Covi, E. Roulet, F. Vissani, Phys. Lett. B384, 169-174 (1996). [hep-ph/9605319]. [21] F. -X. Josse-Michaux, E. Molinaro, [arXiv:1108.0482 [hep-ph]]. [22] D. Aristizabal Sierra, J. F. Kamenik, M. Nemevsek, JHEP 1010, 036 (2010). [arXiv:1007.1907 [hep-ph]]. [23] E. Nardi, J. Racker, E. Roulet, JHEP 0709, 090 (2007). [arXiv:0707.0378 [hep-ph]]. [24] G. F. Giudice, A. Notari, M. Raidal, A. Riotto, A. Strumia, Nucl. Phys. B685, 89-149 (2004). [hep-ph/0310123]. [25] W. Buchmuller, P. Di Bari, M. Plumacher, Annals Phys. 315, 305-351 (2005). [hep-ph/0401240]. 23 View publication stats