Lepton Flavor Violation and Neutrino Masses from $A_5$ and CP in the Non-Universal MSSM

We analyze the phenomenological consequences of embedding a flavor symmetry based on the groups $A_5$ and CP in a supersymmetric framework. We concentrate on the leptonic sector, where two different residual symmetries are assumed to be conserved at LO for charged and neutral leptons. All possible realizations to generate neutrino masses at tree level are investigated. Sizable flavor violating effects in the charged lepton sector are unavoidable due to the non-universality of soft-breaking terms determined by the symmetry. We derive testable predictions for the neutrino spectrum, lepton mixing and flavor changing processes with non-trivial relations among observables.


Introduction
In recent years, different experiments have accumulated a wealth of experimental data on neutrino parameters that have allowed us to extract with reasonable precision the PMNS mixing matrix and the neutrino mass differences.Still, the determination of the absolute neutrino mass scale and the Dirac CP phase remain to be completed.Future experiments like DUNE [1-4], T2HK [5], T2HKK [6] and NOνA [7] will shed light on these quantities; however, even a full determination of the neutrino mass matrix m ν will not be enough to fix the mechanism responsible for it and uncover the origin of the observed flavor patterns.Although this is also true in the case of the quark and charged lepton Yukawa couplings, the smallness of neutrino masses and its favorite explanation through a seesaw mechanism makes this problem specially critical.If neutrinos are Majorana particles, their masses are well described through a d = 5 Weinberg operator [8] in the Standard Model.But there exist different possibilities to generate this effective operator from a more fundamental theory at higher energies, like type I, type II or type III seesaw, with X a unitary and symmetric matrix [21] and A, A ∈ A 5 .In particular for A 5 , Eq.( 2) is fulfilled with A = A .A 5 as a family symmetry for leptons leads to the Golden Ratio (GR) mixing, which predicts a vanishing reactor angle θ 13 .A consequence of introducing CP as a symmetry is that the pure GR mixing is modified and a continuous parameter θ that quantifies this departure is introduced.In fact, the small value of the reactor angle can be reproduced in terms of this variable.The set of combinations of residual symmetries for A 5 and CP that accommodates well the observed mixing in the leptonic sector has been discussed in previous works [14][15][16].Assuming that the lepton SU (2) L -doublet, , transforms like a triplet representation of A 5 , the authors in [15] conclude that only four possibilities are allowed: two for G e = Z 5 , one for G e = Z 3 and another for G e = Z 2 ×Z 2 ; for neutrinos, G ν = Z 2 ×CP has been always considered.Here we are interested in the phenomenology of Case II of [15], corresponding with G e = Z 5 .The neutrino spectrum for this scenario has been fully analyzed in [18].The tuple of generators, Q of G e = Z 5 and (Z, X) of G ν = Z 2 ×CP, characterizing this realization is: where X 0 expressed in the 3 representation is Its form in the rest of irreps of A 5 can be found in Appendix A.

Charged-lepton masses
Residual symmetries constrain the form of the flavon vevs that break the invariance under the family symmetry G .Assuming that charged leptons are symmetric under the subgroup G e = Z 5 , with the generators Q r and the flavon fields φ e r in the r representation, must be satisfied.The condition in Eq. (5) implies that non-zero vevs are possible only for the triplet and pentaplet representations and their form is forced to be where ω 3 , ω 3 and ω 5 are real parameters.The flavons in Eq. ( 6) generate non-renormalizable operators that enter the superpotential and give rise to the charged lepton masses.They may be schematically written as where we have not assumed any specific representation for the RH fields.The residual symmetry also imposes an invariance requisite under the (3 × 3) effective mass matrix: where the generators are given in the triplet representation.A straightforward consequence of Eq.( 8) is that m † e m e must be diagonal in the basis where the set of generators of the group {Q r } are diagonal.In our case, the generator of G e = Z 5 in the tuple of Eq.( 3) is Q = T , which is diagonal according to Eq. (50).Therefore, the operators in Eq.( 7) must produce a diagonal mass matrix at LO (higher order corrections are discussed at section 2.4).Its elements should exhibit the correct hierarchy between generations: where λ c = 0.2257 stands for the Cabbibo angle.In order to keep our discussion as modelindependent as possible, we do not suppose any specific mechanism responsible for this pattern.However, for completeness, we mention some possibilities that have been already proposed in the literature.
One way to generate the observed structure of masses for charged leptons is through processes like those depicted in Figure 1, where the effective mass of each generation involves a different number of flavon insertions.Each of them is proportional to the expansion parameter ε ≡ φ e /M χ 1 that, in this specific case, might be ε ∝ λ 2 C .This type of diagrams can be easily arranged with an Abelian symmetry (continuous or discrete) assigning adequate quantum numbers to the lepton and flavon fields.Another strategy could be having a symmetry that is broken at separated scales.In this scenario, a natural hierarchy between the flavon vevs is expected.Notice that all these constructions involve heavy mediators χ which may be generally LH or RH under SU (2) L .In the absence of further hypothesis, we naturally expect all of them to be present and have masses of the same order.

Neutrino masses
The residual symmetry in the neutrino sector, G ν = Z 2 ×CP, delimits the form of the light neutrino mass matrix m ν through the following invariance conditions: where Z 3 and X 3 are the generators of Z 2 and CP in the triplet representation.The texture of the matrix that satisfies Eq.( 10) is where {s, x, y, z} are dimensionless and real numbers, ϕ = (1 + this sector must satisfy: Mechanism Table 1: Minimal particle content for the different mechanisms and flavor symmetry assignment under A 5 and CP.
where φ ν r and the generators of Z 2 and CP, Z r and X r , are expressed in the r representation.The vacuum alignment is then subject to verify the general structure , where all the coefficients are real.
Majorana neutrino masses can be generated through the so-called dimension 5 Weinberg operator, which is produced at tree level by type I, II or III see-saw mechanisms.Each case is related to the addition of one extra particle to the SM spectrum: RH neutrinos, scalar triplet or fermion triplet, respectively.In the effective low-energy theory some of these constructions are equivalent to others (a detailed discussion about this can be found in Appendix C of [18]) so that we can reduce the discussion to just two cases: Mechanism I, consisting of the Weinberg operator and type II see-saw, and Mechanism II, which includes type I and III see-saw realizations.In the following, we specify the operators generating the neutrino masses for each mechanism.The quantum numbers of the fields for each scenario are displayed in Table 1, where the nomenclature of [18] has been conserved.

Mechanism I
Here we consider neutrino masses generated by the Weinberg operator.In this case, a lepton doublet transforming as the 3 or the 3 representation produces the same phenomenological results under a redefinition of the parameters as indicated in [18].Therefore, without loss of generality, one may simply consider ∼ 3. The LO contributions to the effective superpotential responsible for the neutrino masses are: where y ν i are dimensionless parameters and brackets mean that different contractions are possible.In the low-energy theory all of them are equivalent, since they give rise to the same predictions.However, as can be seen in Figure 3, the mediator sector involved in each process is very different and thereby their UV origin: while in the first diagram the heavy messengers are both LH and RH, the second process considers RH fields and the last one scalar triplets.In the absence of additional hypothesis about the high-energy theory, all these mediators are present and have similar masses, M χ .Once the flavor symmetry is broken, the mass matrix in Eq.( 11) is generated with where υ 1 and {x i , x r , x r,2 } are the vevs of the singlet and pentaplet flavons respectively, see Eq. (13).

Mechanism II
The type I see-saw formulation is investigated in Mechanism II.Two different situations are analyzed in this framework: both and ν c transform in the same triplet representation (II a-2) or in opposite ones (II c-2).The specific choice, ∼ 3 or ∼ 3 , does not affect the resulting phenomenology.Therefore, ∼ 3 is always assumed.Here we consider a trivial Majorana matrix, namely M M = M P 23 , with P 23 defined in Eq.( 4), while the Dirac mass matrix M D is more complicated.The opposite option, non-trivial Majorana mass and trivial Dirac matrix, does not introduce observable effects in the flavor violating observables examined in the following section.Thus, we do not discuss this possibility further.The effective operators entering the superpotential that produces the Dirac mass are: for Mechanism II a-2, and for Mechanism II c-2.Again the brackets indicate that different contractions are possible.Figure 4 shows the ultraviolet (UV) origin of each of them involving left-handed (LH) mediators, right-handed (RH) mediators and triplets under SU (2) L .Once the flavor symmetry is broken, the neutrino mass matrix is generated through the usual type I see-saw process, The coefficients {s, x, y, z} entering m ν as in Eq.( 11) are intricate linear combinations of the dimensionless parameters: for Mechanism II a-2, and for Mechanism II c-2.

Lepton Mixing
The U PMNS is the matrix that measures the misalignment between the rotations of LH charged leptons and neutrinos to the mass basis.It is defined as The conservation of CP and residual transformations determine the form of this matrix up to permutations of rows and columns.In our case, the residual symmetry in the charged sector indicates that for G e = Z 5 generated by Q = T diagonal, the unitary rotation to the mass basis for LH charged leptons U e is the identity.Therefore the U PMNS will be determined only by the neutrino mixing at LO.The unitary transformation that diagonalizes the neutrino mass matrix in Eq.( 11) is [21] where Ω ν defines a change of basis that block-diagonalizes the initial matrix in Eq.( 11) and it is directly related to the GR mixing, R 13 θ is a rotation of an angle θ in the 1 − 3 plane and K ν is a diagonal matrix with entries {±1, ±i} needed to have positive eigenvalues.The explicit form of the matrix Ω ν is The rotation is given by where the size of the angle θ is totally fixed by the entries of the block-diagonalized mass matrix.
In our case: Mixing angles and complex phases can be directly extracted from Eqs.( 22)−(25) using the standard U PMNS parametrization that we detail in Appendix B. The reactor angle is proportional to the angle θ as: Eq.( 26) implies that a small value of θ is required in order to reproduce θ 13 ∼ 9 • .Indeed, θ bf = 0.175 has been found in [15] as the best fit value for this realization.Inspecting Eq.( 25), one may see that such a tiny value for θ can only be obtained considering the following hierarchy among vevs6 : y x, z, s.The atmospheric angle is predicted to be maximal, sin 2 θ 23 = 1/2, and the solar angle is related to the reactor angle through the sum rule: CP invariants and complex phases are also predicted in this framework.The Jarlskog invariant [28] is The Dirac phase δ can be inferred from Eq.( 28) and it is maximal, | sin δ| = 1.The other CP invariants, defined as vanish exactly.Hence Majorana phases, α and β, must be 0 or π.

Next-to-Leading Order Corrections
Notice that Eqs.( 6) and (13) indicate that the LO masses of the neutral and charged leptons must be induced by two separate set of flavons.In practice, this division can be always ensured introducing an additional Z N symmetry that distinguishes among them at LO 7 .However, at higher orders, flavons belonging to the opposite sector (the wrong flavons) are allowed to enter the LO diagrams and may introduce sizable corrections.In this section we comment how these effects can be adequately taken into account.
The NLO corrections to the leading predictions of masses and mixing usually come from higherorder operators that enter the scalar potential and the superpotential.The former induce a shift in the flavon vevs while the latter is usually generated by extra flavon insertions to the LO operators.
As commented before, some of these insertions may be due to the wrong-flavons.For neutrinos, one may check that these contributions are usually subleading.Assuming that the heavy fields χ mediating the diagrams in both sectors have masses of the same order, corrections to the neutrino mass matrix are of the form where m ν is a 3 × 3 matrix with elements given in terms of the {s, x, y, z} parameters defined in Eq. (11).Therefore, barring accidental cancellations in the LO elements which may make the NLO terms dominant, we expect these corrections to be mostly subleading.
Similarly, a suitable choice of charge assignments under a Z N symmetry can guarantee that insertions of the neutrino flavons to the charged sector are also subleading.For instance, if the following set of charges are considered 7 For instance, considering the diagrams in Figure 2 and the first diagram in Figure 4 as the responsible mechanisms for lepton masses, one may see that the charges discriminating both sectors would be: with l ≡ {e, µ, τ, ν}.For the remaining cases, similar assignations can be done.
such that Q Z N (φ) = e i 2πN φ /N , it is easy to see that one charged-lepton flavon could only be substituted by k neutrino-flavon insertions 8 .Then, in its most general form, corrections to the charged-lepton mass matrix will be given by Adjusting the value of k, the required suppression can always be obtained.

Kähler Potential and Soft Terms
The phenomenological consequences of embedding a flavor symmetry in supersymmetric theories where Λ SUSY Λ f have been discussed in previous works [9][10][11][12][13].Even in the most conservative case where SUSY breaking is parametrized by a single universal spurion field, tree-level flavor violating effects generally arise from the mismatch between the order one coefficients in the soft-breaking structures and the Yukawa and kinetic terms due to the different equivalent options of inserting the SUSY-breaking spurion field [9].Moreover, we observe that, in the absence of further hypothesis over the UV spectrum, the common origin of the flavor structures for charged sleptons and neutrinos may induce testable relations between observables belonging to these two different sectors.Finally, we notice that, for some configurations, significant corrections from the Kähler potential emerge that break the residual symmetries and could modify the LO predictions.In fact, this is independent of Supersymmetry and it corresponds to the usual wave-function renormalization that should always be taken into account [13,[29][30][31][32].

Kähler corrections
The effective Kähler potential for a multiplet Ψ of A 5 and CP can be schematically written as with l = e, ν and M χ the generic mass of the χ heavy mediators.The specific terms entering the sum depend on the symmetries of the model.We are interested in the Kähler potential for lepton For other cases, one may proceed in a similar way.
doublets and singlets.Under no further assumptions, the former is given by: where dots stand for higher order terms and the sum in r accounts for the flavons at work depending on the mechanism, see Table 1.The first line is associated with singlet contractions that contribute to every element of the Kähler matrix, so they can be factorized as a global constant.The second line corresponds to non-trivial contractions and generates off-diagonal entries in the Kähler metric.Note that, being φφ † combinations, all terms in Eq.( 35) are neutral under any possible charge.Hence, they cannot be avoided and must be adequately taken into account.
The Kähler function for RH charged-leptons can be written as: In contrast to the previous case, here only the flavons associated with charged leptons contribute.
An analogous expression can be written for RH neutrinos replacing φ e → φ ν and e c R → ν c .The Kähler structures in Eqs.( 35) and ( 36) can be explicitely computed simply inserting the flavon vev in Eqs.(6) and (13).The results are rather intricate, so we omit them here.We find that the charged-lepton flavons only contribute to the diagonal of the Kähler matrix, while neutrino flavons also induce off-diagonal entries.Because of that, a redefinition of the fields is required in order to go to the physical basis where the kinetic terms are canonical, that is K c Ψ = .This can be always achieved making use of an upper triangular matrix that decomposes the Hermitian Kähler metric as K Ψ = U † c U c [30]: The upper triangular matrix defined by Eq. ( 37) has the schematic form where ε 2 ij are complex entries linearly dependent on the flavon vevs squared.In our case, the size of these entries may vary from 10 −7 up to 10 −2 , depending on the mechanism 9 .The field redefinition to the canonical basis affects the lepton masses in Eqs.( 9) and (11), which should be rotated as: Initially, the charged leptons in the non-canonically normalized flavor basis satisfy that m † e m e is diagonal and hierarchical.The canonical rotation introduces corrections to the product in Eq. (9) as: It enters the new U PMNS but does not introduce sizable modifications to the initial angles.This is in agreement with [30], where the authors analyzed the effects of the canonical rotation for hierarchical Yukawa matrices.
The situation for neutrinos is slightly more involved because their mass matrix is not completely hierarchical and analytical expressions accounting for these effects are difficult to obtain.We expect that corrections from the canonical rotation are important when affecting small quantities, such as the reactor angle or the lightest mass for a very hierarchical spectrum, and in those cases where the neutrino spectrum has two or more quasi-degenerated masses, since small contributions can modify the ordering.Numerically, this has been confirmed: while the correlations among vevs obtained in [18] approximately remain after the canonical rotation, the range of possible values for the free parameters is significantly reduced because of these effects10 .

Soft-breaking Masses
Once we have the Kähler potential, the soft-breaking masses before canonical normalization can be obtained just examining the effective operators in K Ψ .At LO, the soft-mass matrices are proportional, element by element, to the Kähler metric.The proportionality factor, f ij , accounts for the different ways in which the spurion F-term F X can be inserted in the representative Kähler diagram, which can be characterized by the number of flavon fields entering, n in , and the number of flavon fields leaving, n out [9,10]: and PLANCK .In our case: Going to the canonical basis, and then to the mass basis, one observes that some off-diagonal terms in the LH soft mass matrix of Eq.( 47) survive the rotations.They can manifest in flavor-violating (FV) processes such as l j → l i γ, l j → 3 l i or µ − e conversion.Moreover, since the FV contributions for LH charged-sleptons and the neutrino flavor structure arise from the same flavons, testable relations between observables from each sector can be inferred.On the other hand, RH charged leptons do not receive observable corrections to their soft masses since they are generated by the flavons in Eq.( 6), which only contribute to the diagonal.As previously discussed for the Kähler potential, they can be simply reabsorbed through a redefinition of the fields.

Predictions on flavor observables
Here we perform a combined analysis of the phenomenology of charged leptons and neutrinos for the two mechanisms exposed in Section 2.2.Following the strategy of [18], each case is divided in several subcases, where one (Mechanism I) of two (Mechanism II) vevs of the active neutrino flavons are set to zero 11 .This allows to reduce the number of independent parameters and to inspect relations among the vevs that correctly reproduce the neutrino properties showed in Table 2.We make predictions for the total sum of the light neutrino masses, the effective mass m ββ and the flavor changing processes collected in Table 4.Our numerical scan is realized as follows: • First, we randomly generate the independent parameters corresponding to each mechanism in a range [−1, +1], Eqs.( 15), (19) and (20).Then, we compute the light neutrino mass matrix m ν in Eq. ( 11), the Kähler metric in Eq.( 35) and the soft masses in Eq. (44).The mass matrix for charged leptons is assumed to be like in Eq.( 9).
• The structures obtained before have to be rotated to the canonical basis as indicated in Eqs.(39), (40) and (46).After that, the lepton mass matrices are diagonalized and re-phased to obtain real and positive eigenvalues.
At this stage, the U PMNS defined as in Eq.( 21), with U l → U c l , and the mass splittings, j with j = 1 for Normal Ordering (NO) and j = 2 for Inverted Ordering (IO), are checked to be in the 3σ-allowed region 12 for both orderings, see Table 2.
• For those points which correctly reproduce lepton masses and mixing in the 3σ region, we fix the value of m ν 0 comparing with the best fit result for ∆m 2  21 .The value of m ν 0 is expected to be some eV at most, so we discard points that correspond to m ν 0 > 5 eV.
• The soft mass matrices are evolved from the GUT to the EW scale by means of the MSSM renormalization group equations and the branching ratios for the lepton flavor violating observables in Table 3 are computed 13 .We also provide estimations for the total sum of light neutrino masses and the effective mass m ββ .

Neutrinos
For the neutrino sector, we follow the analysis performed in [18].The results are displayed in Figure 5, where the effective neutrino mass m ββ is plotted against the minimum neutrino eigenvalue, that is m min = m 1 for NO and m min = m 3 for IO.The upper panel reproduces the estimations presented in [18] whilst the lower panel shows the effect of rotating to the canonical basis where the Kähler is the identity.Blue points are related to Mechanism I, where neutrino masses are produced by the Weinberg operator or type II see-saw.Red points to Mechanism II a-2, corresponding to type I and III see-saw with trivial Majorana matrix and left-and right-handed neutrinos transforming in the same triplet representation.Green points are associated with Mechanism II c-2, which considers the same framework as Mechanism II a-2 but with the lepton doublet and the neutrino singlet transforming in different triplet representations.According to [18], each case is divided in several subcases where one (Mechanism I) or two (Mechanism II) flavon vevs are set to zero.Each subcase allows for NO, IO or both of them.For Mechanism I, we have three subcases corresponding to z = 0, x = 0 and s = 0.For Mechanism II a-2, a total of six subcases are studied: f = 0, h r = 0 and h r,2 = 0 with either h i = 0 or g = 0. Finally, Mechanism II c-2 consist of six more cases: f i = 0, h r = 0 and h r,2 = 0 with either h i = 0 or f r = 0.For conciseness we identify the different cases with the first vev and specify the second vev only when necessary.In addition, we use the subsidences "N" and "I" as shorthand notation for NO and IO.
From Figure 5 we conclude that the scenarios z = 0 N of Mechanism I and h r = 0 N of Mechanism II a-2 are incompatible with the latest data from PLANCK + BAO, while other cases like Mechanism I with s = 0 N and x = 0 I or Mechanism II with h r = 0 N and h r,2 = 0 N could be tested in the future with further cosmological data.As shown in Table 3, future sensitivity in neutrinoless double beta decay experiments could probe those realizations that predict IO and the region for quasi-degenerate masses with a sensitivity m ββ = (0.01 ÷ 0.05) eV (for reviews see [36,37]).
Comparing upper and lower panels of Figure 5, it is evident that the predictions on m min for some subcases change significantly and result in being extended by one or two orders of magnitude.This occurs for those hierarchical subcases where the smallness of m min make it more susceptible to higher order corrections from the canonical rotation.Significant cancellations between leading and higher order terms happen in Mechanism I with s = 0 I , Mechanism II a-2 with f = 0 I , and Mechanism II c-2 with f i = 0 N , see the second plot of Figure 5.However, note that a direct measurement of m min is of no feasible pursue in the near future.From this point of view, m min can be considered as much as an additional model parameter so we may conclude that the canonical rotation has no substantial consequences on the testable predictions of the model in [18].The two exceptions to 10 -6 10 -5 10 -4 0.001 0.010 0.100 1 10 -4 0.001 0.010 0.100 this are the subcases Mechanism II a-2 with f, h i = 0 N and Mechanism II c-2 f i = 0 N where, with enough resolution, a small discrepancy with the prediction in [18] for m ββ may be experimentally found.

Charged Leptons
It has been pointed out in previous works the importance of correctly account for the flavor effects that emerge from the embedding of a flavor symmetry in Supersymmetry [9][10][11].On the one hand, it allows for a characterization of the flavor models of interest through their contributions to flavor-changing observables while, on the other hand, the search for experimental signals on those processes will help us to explore the supersymmetric spectrum at energies that go far beyond the LHC high-luminosity upgrade.In this section we follow the strategy in [10] and we study the flavor contributions arising from the breaking of A 5 and CP in the lepton flavor violating observables collected in Table 4.The structure of the soft mass matrix for LH fields is given in Eq.( 44).We compute its numerical value for each of the points that reproduce the neutrino experimental parameters in the 3σ region, Table 2, plus the PLANCK bound on the total sum of light neutrino masses, Table 3.The range of possible values for the off-diagonal entries results in being constrained by the phenomenology (upper bound) and by the absolute mass scale m ν 0 ∼ 5 eV (lower bound).The interval corresponding to each case is reported in Table 5.The minimum value of the off-diagonal elements in Table 5 can be used to constrain the MSSM parameter space as it has been done in [9][10][11].Concretely, we compute the BR for the LFV processes displayed in Table 4 considering an hypothetical soft mass matrix for LH sleptons whose off-diagonal elements correspond to those minimum values.We choose a representative value of tan β = 10 and calculate our predictions for different values of {m 0 , M 1/2 }.Comparing with current and future experimental bounds in Table 4, we are able to set excluded regions in the {m 0 , M 1/2 } plane.Barring accidental cancellations, we expect that bounds for specific models based on the realizations studied throughout this work will be like those presented in Figure 6 in the most optimistic scenario.The most interesting results are shown in Figure 6, where the upper panel has been obtained from the most restrictive process nowadays, BR(µ → eγ) ≤ 4.3 × 10 −13 , and the lower panel corresponds to the expected future sensitivity on CR(µ − e) Al ≤ 10 −17 .Shaded green areas correspond to the stau as the lightest supersymmetric particle (m 0 M 1/2 ) and no correct electroweak symmetric breaking (M 1/2 m 0 ).The ATLAS mSUGRA limit covers up to M 1/2 = (500 ÷ 800) GeV for m 0 ≤ 6 TeV.We note that the excluded regions, the remaining areas below the corresponding lines, have the typical shape due to LH insertions, see [10].Current bounds are already competitive with the experimental limit in Mechanism I, for which the excluded slepton and gluino masses in the less constrained case reach l 800 GeV and g 1.5 TeV14 .These bounds could be increased up to l 3 TeV and g 5 TeV with future sensitivity from µ − e conversion, (see lower panel in Figure 6).More severe are the constraints imposed by Mechanism II.In particular, for current bounds, we can infer the following global lower limits: l 1.5 TeV and g ≥ 2.8 TeV.They become much stronger considering the future reach of µ − e conversion which translates into l ≥ 5 TeV and g ≥ 8 TeV.It means that precision experiments will allow us to put constraints over the supersymmetric spectrum for masses that are beyond the LHC sensitivity in a factor of 3 ÷ 5. Conversely, the discovery of SUSY partners in the TeV range will put significant constraints on these simple realizations with  LH mediators, which will have to be reformulated in more elaborated scenarios: for instance, suppressing the LH contributions in the Kähler function or increasing the degrees of freedom in the neutrino sector by considering non-trivial Majorana and Dirac structures simultaneously.

Relations among observables
In this framework it is possible to predict testable relations among the LFV observables in the charged sector, Table 4, and the neutrino mass observables m ββ and m min discussed in Section 4.1.This allows to disentangle cases that are not distinguishable only through the analysis in [15].To this end we fix the mSUGRA parameters to m 0 = M 1/2 = 1 TeV, which would correspond to sleptons and gluinos masses around 1.5 TeV and 2.5 TeV respectively, and a very conservative value tan β = 2, which ensures that the tan β-enhanced contributions to the flavor-changing processes are minimized.Then, we study the dependence of both m ββ and m min on BR(µ → e γ) and BR(τ → e (µ) γ).Our results for each process are shown in Figures 7 and 8.It is worth emphasizing the interplay of the experimental bounds coming from the two sectors, which acts in a complementary way in constraining the different realizations of the neutrino masses.In fact, current bounds on LFV observables allow to constrain some regions that otherwise were not testable with present neutrino data, since they predict too low values for m ββ and m min (bottom-right).In particular, the cases h r = 0 and h r,2 = 0 of Mechanism II (both a-2 and c-2) are severely constrained, while the cases s = 0 and x = 0 for Mechanism I, f = 0 for Mechanism II a-2 and f i = 0 for Mechanism II c-2 result partially excluded.Moreover, with the expected sensitivity of MEG II [38] in BR(µ → eγ) all the scenarios detailed before will be completely (dis)proved.Vice versa, those realizations out of the scope of MEG II, namely z = 0 in Mechanism I and h r = 0 in Mechanism II a-2, happen to be incompatible with the recent cosmological data presented by the PLANCK collaboration for the neutrino masses [39].

Conclusions
In this work we have analyzed the phenomenological consequences of combining A 5 and CP as a flavor symmetry in Supersymmetry.We have focused on the leptonic sector where two residual symmetries, Z 5 and Z 2 ×CP, remain conserved at LO for charged leptons and neutrinos.The main features of A 5 and CP as a group have been reviewed in Section 2, where the structure for the lepton mass matrices, the vev of the flavon fields and the leptonic mixing have been derived.The mass matrix for charged leptons results in being diagonal at LO while, for neutrinos, all possible realizations to generate their masses at tree level have been investigated.We noticed that two different set of flavons are required to generate the neutrino and charged lepton masses at LO.Each of them can induce corrections in the opposite sector at NLO, as discussed in Section 2.4.Section 3 has been devoted to compute the minimal set of effective operators entering the Kähler potential and soft masses for LH and RH fields.Under no additional assumptions about the UV theory, the proposed operators are always present and cannot be avoided through the introduction of additional symmetries.For the residual symmetries considered here, we have found that the presence of LH mediators is specially relevant and that the flavons associated with neutrino masses also induce flavor violation in the charged-lepton sector.This allows for a combined analysis of neutrino observables and LFV processes.
10 -15 10 -13 10 -11 0.005 0.010 0.050 0.100 10 -18 10 -16 10 -14 10 -12 10 -10 10 -8 0.005 0.050 0.100  In Section 4.1, we have computed our predictions for the neutrino effective mass m ββ versus m min .We observed that the canonical rotation may have some effect over the model parameter m min .Although these variations seem to be difficult to measure, we think that this information could be useful when (re)constructing theoretical models from experimental data.Regarding charged leptons, in Section 4.2, we have interpreted our predictions for the LFV processes µ → eγ and µ − e conv. in terms of exclusion limits on the plane {m 0 , M 1/2 }.The results depend on the mechanism responsible for the neutrino masses; however, even in the less restricted realizations, we obtain limits that are competitive with those coming from direct searches of ATLAS on mSUGRA scenarios.This type of analysis is very useful to indirectly explore the supersymmetric spectrum in concrete models: in the absence of experimental signals of new physics, stringent limits can be set over the superpartner masses for each specific setup; if SUSY is discovered at the TeV scale, the simplified constructions analyzed throughout this article will have difficulties to accommodate it so that more refined scenarios must be considered.Finally, the common origin of neutrino masses and flavor violation for charged leptons induce testable relations between neutrino and charged-lepton observables.This has been expounded in Section 4.3, where a nice complementarity between both sectors has been found: those realizations difficult to test with neutrino data will be totally probed with the expected sensitivity of MEGII on µ → eγ; conversely, the two scenarios that remain out of the scope of this experiment should be already discarded if the latest limits from PLANCK+BAO on the total sum of the neutrino masses are imposed.

Acknowledgments
OV thanks Universita Roma III for its hospitality.This work has been partially supported under MICINN Grant FPA2017-84543-P and by the "Centro de Excelencia Severo Ochoa" programme under grant SEV-2014-0398.OV acknowledges partial support from the "Generalitat Valenciana" grant PROMETEO2017-033 and AM acknowledges support from La-Caixa-Severo Ochoa scholarship.All Feynman diagrams have been drawn using Jaxodraw [40,41].

A Group Theory
Here we present the matrix form of the A 5 and CP generators, S r and T r , for each of the irreducible representations of the group: r = 1, 3, 3 , 4 and 5.

B U PMNS parametrization
We use the following convention for the PMNS matrix: U PMNS = Ũ diag{1, e iα/2 , e i(β/2+δ) }, where α and β are the Majorana phases and Ũ the CKM-like parametrization of the mixing matrix, ( The Majorana phases can be derived from the numerical PMNS mixing matrix taking into account the unphysical phases described by the diagonal matrix that multiplies the U PMNS from the left, Diag{e iδe , e iδµ , e iδτ }.Those can be eliminated with a redefinition of the charged lepton fields.We can obtain the Majorana phases as For sake of completeness we report the values of the unphysical phases A similar procedure is discussed in [33] using a slightly different parametrization for the PMNS matrix.
Table 4: Relevant LFV processes considered in our analysis.

D Bounds on Mass Insertions
Table 5 collects the obtained intervals for the LL mass insertions after imposing all the constraints regarding lepton mixing and masses.Specifically, we report the off-diagonal elements of the soft mass matrix for LH sleptons divided by the mSUGRA parameter m 2 0 , i.e. the value of the MI at the scale of flavor breaking, before RGE evolution.The allowed ranges are specified for each realization of the neutrino mass mechanisms analyzed in this work.In Section 4.2, the minimum of those intervals has been employed to constrain the mSUGRA plane m 0 − M 1/2 for a representative value of tan β = 10.

√ 5 )Figure 1 :Figure 2 :
Figure 1: Example where the hierarchy in the charged-lepton sector is due to the number of flavon insertions: tau, muon and electron masses are generated by diagrams involving one, two and three flavons, respectively.In these examples the heavy mediators are doublets under SU (2) L .

Figure 3 :
Figure 3: Different contractions generating the Weinberg Operator of Mechanism I in Section 2.2.The first diagram involves SU (2) L doublets and singlets as mediators whereas the second and third only require singlets or triplets.The phenomenological implications in each case are different.

Figure 4 :
Figure 4: Different contractions generating the Dirac mass matrix of Mechanism II a-2 and c-2 in Section 2.2.The first and third diagrams involve SU (2) L doublets as mediators whereas the second requires singlets.The phenomenological implications in each case are different.

Figure 5 :
Figure 5: The observable m ββ against m min .The (bottom) top panel corresponds to the assumption of UV LH-mediators originating (non-)negligible corrections to the Kähler metric.Blue points correspond to Mech.I, red points to Mech.II a-2 and green points are due to Mech.II c-2.The green and blue shadows are the ones allowed for NO and IO respectively.Red shaded regions are already excluded by experimental observations.

Figure 6 :
Figure 6: Excluded regions of the MSSM parameter space due to LFV constraints for tan β = 10.The upper panel is obtained imposing the current bound for BR(µ → eγ), the bottom panel show the regions expected to be ruled out if the future sensitivity for CR(µ − e) Al is reached with no discovery.The different mechanisms are distinguished through different colors: I (blue), II a-2 (red) and II c-2 (green).The black line is the ATLAS bound for mSUGRA models.

Figure 7 :
Figure 7: The quantities m ββ and m min versus the muon flavor violating decay BR(µ → eγ).The mSUGRA input parameters are fixed to the conservative values m 0 = M 1/2 = 1 TeV, tan β = 2.The shaded regions are already excluded by current bounds.

Table 5 :
Obtained intervals for the LL mass insertions: δ 12 , δ 13 , δ 23 .The allowed ranges are specified for each realization of the neutrino mass mechanisms analyzed in this work.