Predictions for"mu ->e gamma"in SUSY from non trivial Quark-Lepton complementarity

We compute the effect of non diagonal neutrino mass in l_i ->l_j gamma in SUSY theories with non trivial Quark-Lepton complementarity and a flavor symmetry. The correlation matrix V_M=U_{CKM}U_{PMNS} is such that its (1,3) entry, as preferred by the present experimental data, is zero. We do not assume that $V_M$ is bimaximal. Quark-Lepton complementarity and the flavor symmetry strongly constrain the theory and we obtain a clear prediction for the contribution to mu ->e gamma and the tau decays tau ->e gamma and tau ->mu gamma. If the Dirac neutrino Yukawa couplings are degenerate but the low energy neutrino masses are not degenerate, then the lepton decays are related among them by the V_M entries. On the other hand, if the Dirac neutrino Yukawa couplings are hierarchical or the low energy neutrino masses are degenerate, then the prediction for the lepton decays comes from the U_{CKM} hierarchy.


Introduction
The present experimental situation is such that we are very close to obtain a theory of flavor that is able to explain in a clear way all the Standard Model masses and mixing. The last but not least experimental ingredient has been the neutrino data and the determination of ∆m 2 12 , |∆m 2 23 |, θ 12 and θ 23 . From all these results we are able to extract strong constraints on the flavor structure of the SM. In particular the neutrino data were determinant to clarify the role of the discrete symmetry in flavor physics.
The disparity that nature indicates between quark and lepton mixing angles has been viewed in terms of a 'Quark-Lepton complementarity' (QLC) [1,2] which can be expressed in the relations Despite the naive relations between the PMNS and CKM angles, a detailed analysis shows that the correlation matrix V M = U CKM U P M N S is phenomenologically compatible with a tribimaximal pattern, and only marginally with a bimaximal pattern. Future experiments on neutrino physics, and in particular in the determination of θ 23 and the CP violating parameter J, will be able to better clarify if a trivial Quark-Lepton complementarity, i.e. V M bimaximal, is ruled out in favor of a non trivial Quark-Lepton complementarity, i.e. V M tribimaximal or even more structured [3]. From present experimental evidences a non trivial Quark-Lepton complementarity arises [4]. Moreover the clear non trivial structure of V M and the strong indication of gauge coupling unification allow us to obtain in a straightforward way constraints on the high energy spectrum too. Within this framework we get information about flavor physics from the correlation matrix V M too. It is very impressive that for some discrete flavor symmetries such as A 4 dynamically broken into Z 3 [5,6,7] or S 3 softly broken into S 2 [8,9,10] the tribimaximal structure appears in a natural way.
In supergravity theories if the effective Lagrangian is defined at a scale higher than the Grand Unification scale, the matter fields have to respect the underlying gauge and flavor symmetry. Hence, we expect quark-lepton correlations among entries of the sfermion mass matrices. In other words, the quark-lepton unification seeps also into the SUSY breaking soft sector [12]. In general we do not get strongly renormalization effects on flavor violating quantities from the heavy neutrino scale to the electroweak scale because of the absence of flavor violation. In fact the remaining flavor violation related to the low energy neutrino sector gives a negligible contribution with the exception of the case with highly degenerate neutrinos and tan β > 40 [13,14].
In this work we compute the effect of non diagonal neutrino mass in l i → l j γ in SUSY theories with non trivial Quark-Lepton complementarity and flavor symmetry. In comparison with previous works (i.e. [15,16]), where a bimaximal V M matrix is assumed, in the present work the correlation matrix V M = U CKM U P M N S is such that its (1, 3) entry, as preferred by experimental data, is zero. All the other entries are assumed to vary as allowed by the experimental data [3,4]. Nevertheless We obtain a clear prediction for the contribution to l i → l j γ. By using the non trivial Quark-Lepton complementarity, flavor symmetry, and the see-saw mechanism we will compute the explicit spectrum of the heavy neutrinos. This will allow us to investigate the relevance of the form of V M in l i → l j γ. There are three cases. They depend on the spectrum of the Dirac neutrino mass matrix and the low energy neutrinos. We may have: 1) hierarchical Dirac neutrino eigenvalues (in this case we have very hierarchical right-handed neutrino masses); 2) degenerate Dirac neutrino eigenvalues, with non degenerate low energy neutrino masses (in this case the hierarchy of the right-handed neutrino masses is close to the one of the low energy spectrum); 3) degenerate Dirac neutrino eigenvalues and low energy neutrino spectrum (that implies right-handed neutrinos close to degenerate). For each of these cases we have different contributions to l i → l j γ.
We will show that only when Dirac neutrino eigenvalues are degenerate and low energy neutrino masses are not degenerate, the explicit form of V M plays an important role.
The plan of the work is the following. In Sec. 2 we explain our notations and clarify the meaning of the correlation matrix V M in flavor theories. In Sec. 3 we introduce the relation between l i → l j γ and the Dirac neutrino matrix. In Sec.
Let us introduce the following symmetric complex matrix C where V 0 is the mixing matrix that diagonalizes on the right the Dirac neutrino mass matrix. From eqs. (4)(5) we see that the inverse of V M diagonalizes the symmetric matrix C, in fact we have

Flavor symmetry implies V M as correlation matrix
In the quark sector we introduce Y u and Y d to be the Yukawa matrices for up and down sectors. They can be diagonalized by where the Y ∆ are diagonal and the Us and V s are unitary matrices. Then the quark mixing matrix is given by To relate the U CKM with the U P M N S normally one makes use of GUT models, such us generic SO(10) or E 6 , where there are some natural Yukawa unifications. In fact these cases give an interesting relation between the U CKM quark mixing matrix, the U P M N S lepton mixing matrix and V M obtained from eq. (7). The mixing matrix V M turns out to be the correlation matrix defined in eq. (12). The reason for it is that in SO(10) or E 6 one has intriguing relations between the Yukawa couplings of the quark sector and that of the lepton sector. For instance, in minimal renormalizable SO (10) with Higgs in the 10, 126, and 120, we can have Y l ≈ Y T d . However this feature is much more general and may depend on the flavor symmetry instead of the gauge grand unification. The presence of a flavor symmetry usually implies the structure of the Yukawa matrices and the equivalent entries of Y l and Y d are of the same order of magnitude. We conclude that, as long as the flavor symmetry fully constraints the mixing matrices that diagonalize the Yukawa matrices, we have U l ≃ V ⋆ d . Notice that if there is a flavor symmetry that constrains the Yukawa couplings in such a way that the diagonalizing unitary matrices are fixed, then the entries of Y l can still be very different from the entries of Y T d . However both Yukawa matrices are diagonalized by the same mixing matrices. This is exactly the case in the presence of an A 4 discrete flavor symmetry dinamically broken into Z 3 [5,6,7] and can be partially true in the case of S 3 softly broken into S 2 [8,10]. From eq. (6) we get If we denote by Y ν the Yukawa coupling that generates the Dirac neutrino mass matrix M D , we have also the relation This relation, together with the previous one, implies If the Yukawa matrices are diagonalized by a similar matrix on the left and on the right, for example in minimal renormalizable SO(10) with only small contributions from the antisymmetric representations such as 120 or more important in models where the diagonalization is strongly constrained by the flavor symmetry, the previous relationship translates into a relation between U P M N S , U CKM and V M . In fact we have

The first relation tells us that
Finally, using the second relation in eq. (12) and the definition of the CKM mixing matrix of eq. (10) we get where we introduced the matrix to allow us to write the CKM and P MNS matrices in their standard form (i.e. three rotation angles and one phase for the CKM and the equivalent for the P MNS) and to take into account the phase mismatching between quarks and leptons. The form of V M can be obtained under some assumptions about the flavor structure of the theory. Some flavor models give for example a correlation V M with (V M ) 13 = 0. As a consequence of the from of the non trivial Quark-Lepton complementarity there are some predictions from the model, such as for θ P M N S 13 from [4] and the correlations between CP violating phases and the mixing angle θ 12 of [3].

The observables
As explained in the introduction, in this work we are interested in extracting informations from non trivial quark-lepton complementarity and flavor symmetry about the l i → l j γ decays. We report here the usual formula obtained in the literature on these processes. It is obtained in the weak eigenstate neutrino base, where charged lepton and Majorana right-handed neutrino mass matrices and weak interactions are diagonal. These processes depends onM D , the Dirac neutrino mass in the weak base.

l i → l j γ
The contribution at first order approximation to the process l i → l j γ in SUSY models is given by where m 0 is the universal scalar mass, A 0 is the universal trilinear coupling parameter, tan β is the ratio of the vacuum expectation values of the up and down Higgs doublets, and m s is a typical mass of superparticles with [17] 14) is valid in the base where right-handed Majorana neutrino mass matrix, charged lepton mass matrix and weak gauge interactions are diagonal. The experimental limit for the branching ratio of µ → eγ is 1.2 × 10 −11 at 90% of confidence level [18] and it could go down to 10 −14 as proposed by MEG collaboration.
We want to redefine the fields in such a way that the only source of flavor violation is in the Dirac neutrino Yukawa coupling. We introduce the following definitions where the unitary matrices V l , U l are defined in eqs. (2).
Consequently we have In this primed base we get and we defineM We want now to relate thisM D matrix to the CKM mixing matrix by using the non trivial Quark-Lepton complementarity and flavor symmetry. First of all we rewrite this matrix asM Then we notice that the matrix V † 0 V ⋆ R is related via the C matrix to the diagonal low energy neutrino mass matrix m ∆ low and to V M . In fact we have where we used the inverse of eq. (17) We multiply on the left and on the right both sides of eq. (21) by 1/M ∆ D and we get If one uses the method of [19] one can extract the matrix V † 0 V ⋆ R by making the square root of the matrices in eq. (23). One has where R is a complex orthogonal matrix such that R T R = 1, and one obtains Finally one concludes that Notice that in eq. (27) does not appear the matrix V M , and any information from V M is hidden into the R matrix. In our discussion however eq. (23) unequivocally fixes V † 0 V ⋆ R and the R matrix, once we know the eigenvalues of the Dirac neutrino mass matrix and the low energy neutrino spectrum. In fact the V M matrix is assumed to be known because of the non trivial Quark-Lepton complementarity. Once we computed the V † 0 V ⋆ R matrix form eq. (23), by using eq. (20), we get where in the last line we used the relations in eq. (6) and (12).
Eq. (28) is the equivalent of the general eq. (27) in presence of non trivial Quark-Lepton complementarity and flavor symmetry. We observe that the main modification is the presence of U † CKM instead of U P M N S thanks to the fact the these matrices are related to each other through V M as shown in eq. (12). Moreover the R is absent and is substantially substituted by the known V † 0 V ⋆ R matrix, computed with eq. (23). Let us now compute the V † 0 V ⋆ R matrix in a general scenario. In the following we use the experimental constraint from [4] that says that (V M ) 13 is zero. With this single constraint on V M we write Eq. (32) is general and must be specified depending on the explicit form of V M . For example for V M tribimaximal we get where we remind the reader that m i are complex numbers, and their sign is not defined.

Hierarchical
The numbers α, β, γ are of order 1 but the corresponding angles must be computed up to order λ 6n to obtain the right heavy neutrino masses. The parameters m α , m β , m γ are of order of the low energy neutrino masses. Notice that the rotation angles (1, 2) and (2, 3) in V † 0 V ⋆ R are of order λ n while the (1, 3) angle is of order λ 2n . We observe that in this scenario, with hierarchical Dirac neutrino eigenvalues, the result depends on the explicit value of the angle θ V M 12 and θ V M 23 only at higher order in λ and via the value of m α , m β , m γ . For example, if the (2, 3) angle of V M is π/4, i.e. for V M maximal, we obtain and for V M tribimaximal we get For any V M , the heavy neutrino spectrum is hierarchical with ratios given mainly by In fact on one hand we have that, for normal low energy neutrino hierarchy m α is of order m 2 , m β is of order m 3 , and m γ is of order m 1 . Then we obtain On the other hand, for inverted low energy neutrino hierarchy m α is of order m 2 , m β is of order m 1 (≈ m 2 ), and m γ is of order m 3 (< m 1 , m 2 ) and then Moreover the mixing matrix V † 0 V ⋆ R is close to the identity. Notice that the lightest right-handed neutrino has a mass smaller than M P lanck (M 1 /M 3 ) 2 if we want the mass of the heaviest right-handed neutrino to be smaller than M P lanck .

Degenerate M D
We remind the reader that the fact that the non trivial quark-lepton complementarity can come from a flavor symmetry implies that the Dirac neutrino may have a different hierarchical structure than the up sector, as clarified in sec. 2.2. For example the same argument applies to the charged lepton and down sectors, where we know that the hierarchical structure differs from each other. The idea beyond this fact, as explained in Sec. 2, is that the quark-lepton complementarity comes both from an unified gauge theory and from a flavor theory. It is supposed that, as the recent progresses show us [3,4,5,6,7,8,9,10,11], the nature of the mixing angles and that of the mass come from different type of flavor symmetries. For this reason, the non trivial quark-lepton complementarity can survive even if there is no Yukawa matrices unification. The important point is that the mixing in the Yukawa are related among them. In Sec. 2 we assumed these relations, but from recent literature about flavor physics we know that this is the case.

Non degenerate m low
If the Dirac neutrino mass eigenvalues are degenerate then, from eq. (23), we obtain In this case, if the low energy neutrino masses are not degenerate, Let us define δM i = M 3 − M i . By performing the full computation up to orders (δM i /M 3 ) 2 , we get where The parameters m α , m β , m γ are of order of the low energy neutrino masses. The angles α, β, γ are of order δM i /M 3 with the exception of degenerate low energy neutrino masses. In this case α is enhanced by a factor m 2 /δm 2 12 , while the other two angles β and γ have a factor m 2 /δm 2 13 , and our approach here is not valid any more because the three angles can be small only if the degeneracy of the Dirac neutrino eigenvalues is such that δM i /M < 10 −5 . We notice that there is not any substantial difference for normal (m 1 < m 2 < m 3 ) or inverted hierarchy (m 3 < m 1 < m 2 ) of the low energy neutrino masses, and the only effect is to change the sign of β and γ angles. From eq. (28) we getM andM D can be computed using the expressions in eq. (45) and U CKM . Notice that in this case the resultingM D strongly depends on the V M matrix. For any V M , the heavy neutrino spectrum is degenerate. However the mixing

Degenerate m low
If the low energy neutrino masses m i and the Dirac neutrino eigenvalues are degenerate then we get In this case the value of V M plays a marginal role. The mixing matrix V † 0 V ⋆ R is close to a small rotation in the (1, 3) plane and the heavy neutrino spectrum is degenerate too: For any V M compatible with the experiments, the heavy neutrino spectrum is almost degenerate. Moreover the mixing matrix V † 0 V ⋆ R is close to the identity matrix.

Contribution to l i → l j γ
Using the result in eq. (28) and the general eq. (14), we get where V = V † 0 V ⋆ R is the mixing matrix computed with eq. (23). Notice that the Ω phase differences exp i(φ i −φ j ) cancel because we take the absolute value. We want to stress here that the result in eq. (49) depends on the Quark-Lepton complementarity (and the underlying flavor symmetry) assumption only, and not on the explicit form of the correlation matrix V M .
At zero approximation we neglect the different normalizations for different right-handed neutrinos. We assume L =L = 1 log M X /M R where M R is the common heavy neutrino mass. The BR(µ → eγ) can be rewritten as where λ is the sine of the Cabibbo angle, and A, ρ and η are the other parameters of the unitary CKM matrix. For each Dirac neutrino mass we introduced, its first contribution. Similarly to the process µ → eγ we can compute the contribution to the τ decays. For τ → eγ we get The other τ decay process that violates the individual lepton number is such that To understand the main contribution we must make some assumptions about the hierarchy of the Dirac neutrino masses M i . Moreover to include the effect of non degeneration for heavy neutrino masses we must include V , whose form depends also on the hierarchy of the low energy neutrino masses. (57)

Hierarchical
(58) The ratios among them become of order one and We notice that in this case, with respect to the one considered in the previous section, the value of the branching ratio of µ → eγ is bigger by a factor λ 6 . So we obtain that, despite the fact that this case is the most promising to extract information on the structure of V M , degenerate M D and non degenerate m low is excluded by the experimental data for most of the SUSY parameters. Naturally one can fine-tuning the SUSY parameter and/or the neutrino mass parameters in such a way to escape from our general analysis.

Degenerate m low
If the spectrum of the low energy neutrino is degenerate, then the mixing matrix V † 0 V ⋆ R becomes close to the identity. In this case the branching ratios depend on the common M D mass and the Cabibbo parameter. By assuming 3  To compare this case with the case of hierarchical M D of sec 4.2, we observe that here BR(µ → eγ) is the largest one, while in the other case it is the smallest one. Moreover the value of the branching ratios here depends on the differences In this case, not only we cannot extract information on the V M structure, but also we have no hope to observe these branching ratios because they are too small even with respect to the future experimental sensitivities.

Conclusions
We analized the consequences of a non trivial Quark-Lepton complementarity and a flavor symmetry on BR(l i → l j γ). The non trivial Quark-Lepton complementarity, together with the flavor symmetry, states that the correlation matrix V M , product of the CKM and the P MNS mixing matrix, is related to the diagonalization of the Majorana right-handed and Dirac neutrino mass matrices.
In this framework we obtained that BR(l i → l j γ) is related to the CKM mixing matrix and the Dirac neutrino masses. We have three cases: 1. Hierarchical Dirac neutrino eigenvalues (very hierarchical right-handed neutrino masses, V † 0 V ⋆ R ≃ I) where we get the usual ratios BR(µ → eγ) : BR(τ → eγ) : BR(τ → µγ) = λ 6 : λ 4 : 1 ∝ M 4 3 λ 4L . This case is the most promising one for a future observation of the branching ratios. However it will not give us any information about the structure of the V M matrix. In this case the branching ratios are too small even with respect to the future experimental sensitivities.