LFV and (g-2) in non-universal SUSY models with light higgsinos

We consider a supersymmetric type-I seesaw framework with non-universal scalar masses at the GUT scale to explain the long-standing discrepancy of the anomalous magnetic moment of the muon. We find that it is difficult to accommodate the muon g-2 while keeping charged-lepton flavor violating processes under control for the conventional SO(10)-based relation between the up sector and neutrino sector. However, such tension can be relaxed by adding a Georgi-Jarlskog factor for the Yukawa matrices, which requires a non-trivial GUT-based model. In this model, we find that both observables are compatible for small mixings, CKM-like, in the neutrino Dirac Yukawa matrix.


Introduction
After the discovery of neutrino oscillations in 1998 by Superkamiokande [1,2], the Standard Model (SM) was forced to include massive neutrinos. Yet, the smallness of their masses seems to require a new framework different from the SM Yukawa couplings in the charged lepton or quark sectors. Simultaneously, it was confirmed experimentally that neutrino mass eigenstates are a non-trivial combination of the flavour states. Consequently, we know that the family lepton numbers, L e , L µ and L τ , are violated in Nature. This necessarily implies some degree of violation in the charged-lepton sector, although it has not yet been observed.
Several mechanisms have been concocted to explain the extreme smallness of neutrino masses and all of them require the existence of new physics (NP). Unfortunately, the available experimental information on the mass splittings and mixing is still insufficient to JHEP05(2020)102 disclose the physics behind their true origin. New observables are required to improve our understanding and charged-lepton-flavour violation (CLFV) is one of the best options at our reach. Nevertheless, the absence of any signal of CLFV may indicate that the associated NP is considerably heavy. On the other hand, the persistent discrepancy between the experimental measurement of the anomalous magnetic moment of the muon [3] and its SM prediction [4][5][6][7][8], ∆a µ = a exp µ − a SM µ = (2.7 ± 0.7) × 10 −9 , (1.1) remains an interesting motivation to explore new models [9][10][11][12][13][14], for a review see [15]. 1 With respect to neutrino masses, the type-I seesaw mechanism [19][20][21][22][23][24] seems to be the most natural extension of the SM to generate them. It assumes the existence of right-handed neutrinos (RHν), which are singlets of the SM and, therefore, are allowed to have both Dirac and Majorana mass terms by the gauge symmetries. Mediated by them, an effective dimension-5 Weinberg operator [25] would be induced, producing the light neutrino masses after the electroweak (EW) symmetry breaking. Although RHν may live at any scale between the EW and the GUT scale, a naïve dimensional analysis of the Weinberg operator signals a Majorana mass around the usual GUT scale ∼ 10 15−16 GeV. That means that the SM supplemented with RHν at high energies suffers from a serious hierarchy problem [26,27].
Despite the fact that no signal of superparticles has been found at LHC, supersymmetric models may still be the appropriate candidate to alleviate this problem and may also answer other questions, such as the nature of dark matter [28,29] and the exact unification of the gauge couplings at the GUT scale [30][31][32][33]. In addition, the presence of right-handed neutrinos induces slepton mixing through the renormalization group equation (RGE), which may produce visible CLFV effects [34][35][36][37]. In particular, SUSY models with non-universal scalar masses at the GUT scale seem to be favoured from naturalness considerations [38][39][40], allowing for a Higgs boson mass at 125 GeV and low electroweak fine-tuning [41,42]. Here we will analyse supersymmetric non-universal Higgs models with an additional parameter for the third generation of scalar superpartners (NUHM3) [43][44][45]. We focus on the so-called light higgsino-world scenario [46][47][48][49][50][51][52][53][54][55] in which the SUSY matter scalars are pushed into the multi-TeV scale while µ 1 TeV, as natural SUSY requires. We observe that a characteristic spectrum where the lightest neutralinos and chargino are higgsino-like is automatically selected when the correct (g −2) µ phenomenology is required through a general scan that includes solutions with the usual gaugino-higgsino hierarchy. This light-higgsino spectrum is favoured by naturalness and also by dark matter searches, as higgsino is efficiently annihilated, unlike the typical bino-like dark matter that is usually over-abundant.
In the literature, refs. [56,57] present related analysis, although following a different approach. In [56], the authors studied the preferred regions of the supersymmetric parameter space according to the muon g-2, implementing the LHC bounds from the Run-I 1 A recent lattice calculation [16] renders a (g − 2)µ result that, if confirmed, will lead to a disappearance of the discrepancy and the role of new physics in it. Yet, other works [17,18] argument that this shift in the value of the anomalous magnetic moment has important consequences in the EW fit so that other observables will inherit the need for new physics with a similar significance. We will discuss the potential impact of this result on our analysis in section 5.1.

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but imposing no low-energy constraints from CLFV. In contrast, the second article analysed the CLFV processes for NUHM3 models but did not comment on the possibility of reproducing (g − 2) µ .
The paper is organised as follows: in section 2, the main ingredients of a supersymmetric type-I seesaw model are presented. We also comment on some specific features common to GUT-motivated scenarios. In section 3, the effect of the right-handed neutrinos on the running of the slepton soft masses is described. In section 4, we discuss the main observables of our analysis providing some useful analytic relations between CLFV processes and (g − 2) µ . Section 5 is devoted to the results of our numerical scan. We summarize our conclusions in section 6.

Supersymmetric type-I seesaw
The supersymmetric type-I seesaw considers the MSSM particle content augmented with three chiral superfields, one for each generation of right-handed neutrinos (RHν). The interactions among chiral supermultiplets are determined by the superpotential which, in this framework, contains new interactions involving RHν: 3) The SUSY soft-breaking interactions introduce mass terms for the superpartners as well as trilinear couplings between the new sfermions and the Higgs. The relevant terms in our case, not including the quark sector, are: At an energy scale above the heavy RHν mass, µ ≥ m ν c 3 , an effective operator with the left-handed neutrinos and Higgs fields is generated by the process in figure 1. Integrating out the heavy neutrinos and replacing the Higgs by its vev at low energies, the following mass matrix is produced:  [58]. Similar results have been found in [59,60].
Extensions of the SM may provide additional information through related observables, like CLFV processes. One of the best examples is supersymmetric extensions of the SM supplemented with a type-I seesaw, where slepton soft-breaking masses are related to the neutrino Yukawa couplings. However, additional assumptions are usually made to simplify the analysis of the phenomenology of these models. The minimal strategy consists in presuming universal soft-masses and a concrete structure for the neutrino Yukawa couplings at the GUT scale. Then non-universal entries are generated in the soft masses through the renormalization group evolution (RGE), proportional to the hypothesised Yukawa couplings. It is important to remark that these RGE contributions are always present in SUSY models irrespective of the presence of additional non-universal entries at the GUT scale. Therefore, barring accidental cancellations, these effects are the minimal outcome of supersymmetric seesaw models.
In this framework, we perform the analysis in two limit cases: one in which the rotation angles in V ν L are small, or CKM-like, and another where the mixing is large, or PMNS-like. These two scenarios should span any intermediate possibility so that general conclusions can be fairly derived. For instance, based on an underlying SO(10) gauge symmetry, one may consider that Y ν and Y u , in the basis of diagonal Yukawas for charged leptons and down-type JHEP05(2020)102 quarks, are deeply connected at the unification scale through relations such as [61][62][63][64]: where, in these equations, we have introduced a Georgi-Jarlskog (GJ) factor, k GJ , which may arise in some GUT realisations due to the different representations of the unified group that mix to generate the SM Higgs doublet. Within a SO(10) model, if the dominant contribution to the Yukawa interactions is due to a Higgs field transforming as a 10, a perfect unification between neutrinos (charged leptons) and up-type (down-type) quarks is expected, so k GJ = 1. Conversely, if the dominant contribution comes from a 126 representation, a factor k GJ = 3 appears between leptons and quarks. Another possibility is having an effective Higgs field transforming as a 120, which may be the product of a 45 and 10 representations. The 45 can acquire a nonzero vev in the flat direction B − L + κT 3 , which preserves the SM and distinguishes between RH fermions 2 [65,66]. In fact, complete flavour models usually require the combined effect of more than one representation to generate dissimilar hierarchies among generations [67][68][69][70][71][72]. For instance, unification in the down sector as in the classical Georgi-Jarslkog scheme demands |y τ /y b | GUT = 1 and |y µ /y s | GUT = k GJ = 3. However, nowadays, these relations are no longer favoured phenomenologically [73][74][75][76] but the updated range y µ /y s GUT = [ 2.5, 6.5 ], (2.11) obtained in [75]. This is the reason we take to generalize our k GJ factor to k GJ = B−L+κT 3 . In the case of neutrino Yukawa couplings, we are allowed to consider that the dominant contribution to the up and neutrino Yukawas comes from the representation giving rise to this k GJ factor. Then, varying κ in the interval where eq. (2.11) is satisfied, one observes that This is the interval taken for the k GJ factor in our numerical analysis. We explore the parameter space of seesaw NUHM3 models in which some of the stringent conditions of the typical mSUGRA models are relaxed. We introduce three additional degrees of freedom in the scalar soft-breaking sector: instead of one common scalar mass, we will consider the following four where we have included a non-universal, but diagonal, charged-slepton mass matrix in the basis of diagonal charged-lepton Yukawa couplings at the GUT scale. The supersymmetric sector of the models is determined by five parameters at the GUT scale, 14) and two more at the EW scale, µ, which can be taken in exchange of m Hu and m H d .

JHEP05(2020)102 3 RGEs and lepton flavour violation
The introduction of RH neutrinos makes the effect of the RGEs specially relevant for sleptons [34][35][36][37]. During the running, the heavy neutrinos induce off-diagonal entries in the slepton soft terms through radiative corrections. Those flavour-violating interactions allow for CLFV processes that otherwise, within the SM, would be greatly suppressed by the neutrino masses. The main effect occurs for the soft-mass matrices of the LH sleptons and can be worked out by solving the RGEs where the first term denotes the MSSM contribution in the absence of RHν, Hu . In the basis of diagonal RHν and charged-lepton Yukawas, the leading log approximation is proportional to the square of the neutrino Yukawas as: where we take the limit m m 0 and approximate m GUT to be of the order of the scale at which the soft terms appear in the Lagrangian (the typical scale of SUSYbreaking transmission). The main effect, with hierarchical Yukawas, is due to the heaviest Majorana neutrino and happens before its decoupling at µ > m ν c 3 . Trilinear couplings receive similar corrections, although they have a smaller impact on the CLFV observables studied here. In contrast, no flavour violation is produced in the RH charged-slepton sector at one-loop, since the RGEs only depend on Y and on the gauge couplings, hence they are diagonal in the basis where Y is diagonal. The off-diagonal elements produced radiatively enter the total 6 × 6 slepton mass matrix as small insertions (compared to the diagonal terms) in the LL and LR/RL sector: JHEP05(2020)102

Observables
Up to now, we have defined the supersymmetric model that we analyse in this project and its RGE evolution to the electroweak scale. The next step will be to compare its predictions with the low-energy observables, to constrain the allowed parameter space or to find possible discrepancies from the SM predictions.
The first observable we have to reproduce is the recently measured value of the Higgs mass, which is a strong constraint on any supersymmetric extension of the SM. Then, as we are mainly interested in the leptonic sector, we concentrate on two main observables: the anomalous magnetic moment of the muon and the radiative process µ → eγ.

Higgs mass
Previous works have extensively discussed how to accommodate the 125 GeV observed Higgs boson [77,78] within a minimal supersymmetric framework [79][80][81][82][83][84]. In the MSSM, it is known that the tree-level value of the lightest Higgs mass is bounded from above by M Z whilst radiative corrections, coming from the fermion-sfermion loops, may increase it up to 135 GeV [85][86][87][88][89]. As those corrections are proportional to the corresponding fermion Yukawa couplings, the dominant contribution is due to the top-stop diagram and can be written as where α is the mixing angle between the scalar components of H 0 u and H 0 d after EWSB and ∆ thr stands for the threshold corrections dependent on the stop mixig [90]. While constrained versions of the MSSM, such as mSUGRA, GMSB or AMSB, usually have difficulties to generate the observed mass, scenarios with non-universal conditions at the GUT scale are able to improve their predictions and provide realisations with a low amount of fine-tuning [41,42,80].
In the models considered here, the stop mass is determined by the scalar mass parameter m (3) 0 at the GUT scale. As we will see below, in order to obtain the adequate mass, large values for m with stops masses in the few-TeV regime, Note that even the most conservative measures of EW fine tuning, ∆ EW , establish that mt 1 3 TeV for ∆ EW < 30 [42,91] so that the result in eq. (4.3), confirmed by the numerical analysis in section 5, might be in the limit of natural radiative SUSY.

Anomalous magnetic moment of the muon
In the MSSM, leptons receive supersymmetric corrections to their anomalous magnetic moment due to neutralino and chargino loops that effectively generate the dipole operators, JHEP05(2020)102 defined in eq. (B.10) in appendix B.2, where we can find the full expressions, [36,92], The supersymmetric amplitude is usually dominated by the processes where the chirality flip of the fermion occurs at the vertex, which is proportional to the Yukawa coupling and therefore tan β-enhanced. The mass insertion approximation (MIA) [93][94][95][96][97][98] allows us to see this explicitly by means of expanding the full amplitude, extracting the relevant diagrams and identifying the main parameters. It has been implemented in appendix B.2.
The diagramatic interpretation of the dominant processes is depicted in figure 2 (left).
In the light higgsino-world scenario, the LSP is the neutralino, which is mainly higgsino and quasi-degenerate in mass with the second-lightest neutralino (NLSP) and the lightest chargino. We observe that the region where a susy µ is within the 3σ range exhibits the following hierarchies between masses: µ M 2 m˜ L . Then, the process is expected to be dominated by the chargino loop, 3 since its loop function for x = µ 2 /m 2 L 1 is the largest one (see figure 9). Therefore, From eq. (4.5), another phenomenological consequence can be inferred: a SUSY contribution that accounts for the current discrepancy between the experimental and the SM theoretical value of the muon anomalous magnetic moment will require light sneutrinos in the second generation. In the models analysed here, the masses for sfermions of the first JHEP05(2020)102

LFV Process
Current Limit Future Limit at the GUT scale. Therefore, contrary to m In practice, eq. (4.6) favours a scalar spectrum where the third generation is heavier than the first two ones. The phenomenology of these models has been previously explored in [101].

CLFV observables
CLFV transitions such as j → i γ, j → 3 i and µ → e in Nuclei are extremely rare in the SM. Actually, we do not expect observing them if the SM is only theory at low energies. The experimental limits on these processes are collected in table 2.
In the MSSM, the j → i transitions are dominated by the dipole operators arising at one-loop level by the exchange of charginos-sneutrinos and neutralinos-charged sleptons, JHEP05(2020)102 eq. (B.10). The flavour switch is feasible due to the flavour-violating interactions generally present in the soft-breaking terms (trilinears and soft masses). Actually, the diagrams for j → i and a susy are essentially identical, except for the flavour transition happening in the former which is absent in the latter. Motivated by this, several articles have evaluated the correlations between µ → eγ and a susy µ in the MSSM [15,[110][111][112][113][114][115], although a dedicated discussion in the context of the light higgsino-world scenario is still lacking.
As discussed in section 4.2, in the models under consideration, the lightest neutralinos and chargino are higgsino-like and quasi-degenerate in mass. Assuming that the offdiagonal entries are much smaller than the diagonal ones and applying the MIA for sneutrino and chargino propagators, we find that the branching fraction given by is well reproduced by the following amplitudes 3 (x µ ) and f Finally, it is worth commenting the processes µ → 3e and µ−e conversion. In this case, both receive contributions from penguin diagrams (with Z-boson and photon exchanges), as well as, from box-diagrams. However, again, the γ-penguin transition (directly connected to µ → eγ), being tan β-enhanced and not suppressed by M Z , dominates and the following simple relations hold: The parallelism between the amplitude in eq. (4.5) and the one in eq. (4.8) is evident. Taking the ratio between them, it is obtained .
(4.12) The branching ratio in eq. (4.7) can therefore be rewritten as: where the interval is due to the variation of the loop functions in the obtained ranges x µν ∈ [0.01, 5] and x 2ν ∈ [0.1, 100], see figure 9 (right panel). From eqs. (4.13)-(4.14), one may see that, for those values of M 2 , µ and mν µ where a susy µ is within the 3σ range, an off-diagonal element in the slepton soft-mass matrix around 10 −4 is required to fulfill current and future limits on BR(µ → eγ). Additionally, eq. (3.5) reveals that the induced off-diagonal term is proportional to the square of the GJ factor introduced in eqs. (2.9)-(2.10). Therefore, the absence of any signal in the FC process becomes an indirect upper bound on k GJ for our benchmark scenarios. We have explicitly checked the relation between BR(µ → eγ) and k GJ for some representative points. The result is displayed in figure 4. We select some points that, among other constraints, fulfill ∆a µ at different levels of accuracy when k GJ = 1 and evaluate how BR(µ → eγ) is modified when k GJ decreases. As expected, BR(µ → eγ) is reduced. Consequently, a factor would be needed to conciliate both observables, considering future sensibility from MEG-II. Notice that eq.  Table 3. Parameter space of the numerical scan.

Analysis
We perform the numerical scan in the ranges shown in Regarding the neutrino sector, once the neutrino Yukawas are known, a phenomenologically viable seesaw mechanism can always be realised by a suitable choice of the righthanded neutrino Majorana mass matrix [61], see appendix C. The running of the neutrinos is performed as detailed in [119]. As a result of the RGE, the neutrino masses and mixing can be slightly modified. To optimize the scan, we feed the program with Y ν at the GUT scale and the effective mass matrix for the light neutrinos at low energies as inputs.
• Tau rare decays below the current limits in table 2.
We also check the thermal relic density of the neutralinos. For higgsino-like LSP, the dark matter candidate is a WIMP and the predicted relic abundance is usually below the WMAP measurement [121], Ω DM h 2 = 0.113 ± 0.0035 at 68%. We accept those solutions that predict a lower density and assume that additional effects due to exotic matter, for instance gravitinos, axions, moduli fields or others, can account for the rest [48,55,122,123].

Lepton observables
First, we analyse the dependency of the two transitions, BR(µ → e) and a susy µ , on the mass of the sparticle that enter the loop. As discussed in section 4.2 and 4.3, both amplitudes are inversely proportional to the muonic sneutrino mass, see eqs. , the parameter that controls the mass of the first two generations of sfermions. The first (second) row corresponds to the CKM-like (PMNS-like) case. The left column is for BR(µ → eγ) and the right column for a susy µ . As expected, the largest branching fraction and magnetic moment are observed for m (1,2) 0 2 TeV. The CKM-like case exhibits, in general, smaller values than the PMNSlike models. This is totally reasonable since the off-diagonal elements of the former are produced by the CKM matrix (almost diagonal) whilst the seconds are dictated by the JHEP05(2020)102  PMNS matrix with larger mixing angles, see eqs. (2.9)-(2.10). The colours red/light and green/dark refer to k GJ = 1 and k GJ = 1, respectively. We observe that the amplitude for the FC process decrease when the k GJ factor is reduced while, the same variation in k GJ , has little effect on a susy µ . This is in agreement with eqs. (4.5), (4.8) and (3.5). In figure 6, the results for BR(µ → eγ) and a susy µ are compared. Again, the colour legend is associated with k GJ = 1 (red/light) and k GJ = 1 (green/dark). The first row shows the distribution of points for the CKM-like case. In the left column, the full range of points is represented whereas, in the right column, a magnification of the region of interest can be found. The second row is for the PMNS-like case and shows similar plots. In the case where k GJ = 1, both scenarios are ruled out by the MEG limit, since they predict a larger branching fraction for the process µ → eγ. As discussed in section 4.4, a k GJ factor that introduces a splitting between the up-type quarks and neutrinos may reduce the discrepancy and make possible to conciliate all the experimental limits.
For the CKM-like case, the value k GJ = 1/3 is enough to fulfill the current bound from MEG and the expected limit from MEG-II. This is in agreement with what was observed in figure 6. In contrast, the PMNS-like case requires a significantly smaller value, k GJ ∼ 1/75. This is consistent with eq. (3.5), which indicates that the generated off-diagonal elements should be larger in the PMNS case, although we obtain a slightly smaller k GJ than our estimate in section 4.4. These tiny values of k GJ would allow a supersymmetric explanation of the discrepancy with PMNS-like mixings, but they are not natural in a GUT scenario. In eq. (2.10), we are assuming that the representation with a Georgi-Jarlskwithog factor gives the dominant contribution to the neutrino Yukawa couplings. However, a k GJ ∼ 1/75 would require all other contributions to be absent or more than two orders of magnitude smaller.
On the other hand, we would like to emphasize that a smaller discrepancy of the experimental results from the SM predictions would enlarge the portion of the parameter space where both observables are consistent. In particular, some points in the CKMlike case with k GJ = 1 may be permitted. This is the case if the discrepancy on the anomalous magnetic moment is reduced due to a slight decrease of the experimental central value [124,125] or because of an improvement in the SM calculation in the direction of the measured quantity [16]. 4 The last row of figure 6 is dedicated to the CLFV decay µ → 3e. Although the present limit for the process (black, continuous line) is less restrictive that the correspondent to µ → eγ, the expected sensitivity (black, dashed line) will become more limiting. Specifically, the region of points where k GJ = 1 and a susy µ is compatible with the experimental measurement at 3σ or more, will be totally scrutinised. In other words, if the discrepancy in the anomalous magnetic moment of the muon is due to supersymmetric corrections like the ones analysed here, a signal in the µ → 3e channel is expected within the ∼ 10 −12 − 10 −16 range. Similar conclusions are extracted from the inspection of the observable CR(µN → eN).

Benchmark point
In this section we discuss the main phenomenological features of a representative benchmark point that satisfies all the constraints detailed before, reproduces ∆a µ at 2.5σ and is in agreement with MEG limits. The input parameters are presented in table 4, which belong to the CKM-like case. The produced spectrum is depicted in figure 7.
As commented throughout the text, the models analysed here with µ M 1/2 predict lightest neutralinos and chargino which are higgsino-like and, therefore, almost degenerate in mass. Because of that, the two-body decay ofχ 0 2 andχ ± 1 toχ 0 1 is forbidden and LHC limits for this compressed spectrum are rather weak. The third neutralino is mostly bino whilst the forth neutralino and second chargino are wino-like. As expected from models with universal gaugino masses at the GUT scale, the gluino is the heaviest superpartner in the gauge sector and, in our case, a mass of ∼ 3.5 TeV is predicted.
The sfermion sector displays the following hierarchy among generations: the lightest states for both sleptons and squarks belong to the first two generations whereas the third generation is associated with the heaviest states. This is consistent with m The main effect of the RGE on the sfermion masses is due to the gauge interactions, which increase the mass as they are evolved down to the EW scale, and, mostly for the third generation, the Yukawa couplings and trilinears, which decrease their masses. Thus, in general, heavier left-handed states are expected for the first two generations. For up-type JHEP05(2020)102  squarks, this is also the case in the third generation since the right-handed stop contribution that lowers its mass is proportional to 2y 2 t while the term associated with the left-handed superpartner goes as y 2 t + y 2 b . These two effects make the right-handed stop the lightest squark. For down-type squarks, the situation is reversed since the left-handed sbottom receives corrections proportional to y 2 t + y 2 b while the right-handed sbottom contribution is ∝ 2y 2 b . This effect is, therefore, opposite to the gauge contribution and, in this particular case, makes the left-handed sbottom lighter than the right-handed one.
The situation is slightly more involved in the slepton sector, where the combined effect of m (1,2) 0 m (3) 0 and up-like neutrino Yukawas interfere in the evolution of the left-handed sleptons and make them lighter than the right-handed superpartners, also for the first two generations. This hierarchy is indeed congruent with the explanation of ∆a µ , since they are the ones mediating the transitions µ → eγ and µ → µγ. Left-handed sneutrinos follow the same pattern than their charged counterpart and satisfy mν m˜ . We check that the predicted spectrum is compatible with LHC direct searches. Regarding the Higgs sector, we obtain a decoupled spectrum where the lightest Higgs is SM-like and the other CP-even, CP-odd and charged states are much heavier and quasidegenerate in mass. We would like to emphasize that it is possible to reproduce the observed Higgs boson mass and ∆a µ simultaneously in these realisations. In figure 8, we plot the mass of the lightest Higgs boson against m are preferred to produce a lightest Higgs state near the experimental mass. As noticed before, the result found in eq. (5.2) might be in the limit of radiative natural SUSY which establishes mt 1 3 TeV for ∆ EW < 30 [42,91]. 5 Similar results are found for the PMNS-like case.

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In conclusion, ∆a µ and the Higgs mass are accomplished simultaneously for nonuniversal models where distinct scalar soft masses are assumed for the first-two and third generation. The lightness of the former allows for a suitable supersymmetric contribution that accounts for the measured discrepancy in the muon anomalous magnetic moment whereas stops within the few-TeV regime are enough to produce a Higgs mass at the observed central value, 125 GeV.

Conclusions
We have investigated how to explain ∆a µ in supersymmetric models with right-handed neutrinos while keeping indirect flavour bounds under control.
In section 2, the main ingredients of a supersymmetric type-I seesaw model were presented together with some relations that arise between the Yukawa couplings of different species of sfermions within GUT realisations. In particular, we discussed how a Georgi-Jarlskog factor between the Yukawa couplings of up-type quarks and neutrinos may make the latter significantly smaller than the former. We provide an specific example where this happens. As exposed in this section, we focus our analysis on GUT realisations where nonuniversal conditions for the scalar soft masses are assumed at the unification scale. This kind of models are favoured from naturalness considerations since they succeed at reproducing the observed Higgs mass value while producing a low amount of fine-tuning, providing that the Higgs mass parameter is µ 1 TeV. A direct consequence of this condition is that the lightest neutralinos and charginos are higgsino-like and therefore quasi-degenerate in mass. The resulting phenomenology is poorly constrained from direct searches at LHC, so that it remains a highly attractive scenario to explore in the next generation of linear colliders.
In section 3, we recalled a well-known fact: right-handed neutrinos in SUSY models induce off-diagonal elements in the slepton soft mass matrices during the running down to the EW scale through the RGEs. The impact of the these terms in the flavour-changing transitions i → j has been investigated in section 4. Analytical expressions have been derived together with a simple relation between the amplitudes of a susy µ and BR(µ → eγ) that holds for our models.
Section 5 was dedicated to our numerical scan. We observe that non-universal models allow for a Higgs boson mass at 125 GeV and a suitable contribution to the anomalous magnetic moment of the muon which can account for the discrepancy between the experimental and the theoretical SM value. For those solutions, charged-lepton flavour-violating processes remain below the experimental limits if the neutrino Yukawas at the GUT scale are suppressed compared to the up-type quark couplings. Due to our little knowledge about the neutrino couplings, this is possible even within a GUT unification context, with a Georgi-Jarlskog factor, k GJ , as explained in section 2. The required factor between up and neutrino Yukawas depends on the mixing associated with the left-handed neutrinos, small (CKM-like) or large (PMNS-like). While in the first case a 1/3 factor, which can be easily accomodated within a complete theory of flavour, is sufficient, the PMNS-case requires a very small factor, which might be perceived as quite unnatural. Nonetheless, a future change in the experimental results could be favourable for these unified models, if A Loop functions

B Mass insertion approximation
We follow the conventions in [36] for the neutralino-sfermion-fermion and charginosfermion-fermion couplings:

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where O N is the unitary matrix that diagonalises the neutralino mass matrix, O L and O R are the unitary matrices that diagonalise the chargino mass matrix (which, in general, is not hermitian but complex) through a biunitary transformation and U , ν are the mixing matrices for charged sleptons and sneutrinos.

B.1 Flavour-expansion theorem
The flavour-expansion theorem [98], on which the mass insertion approximation is based [93][94][95][96][97], states that for any hermitian matrix that is diagonalized by an unitary matrix as then any real analytic function of D, which must be analytic around zero and should be possible to express in terms of its McLaurin series, can be expanded as For small enough off-diagonal elements in M 2 , the sum in the right-hand side can be fairly good approximated by the first terms. From eq. (B.6), one may also derive the following relation for fermionic amplitudes [98]: where M is the mass matrix that is diagonalized by two unitary matrices as and With this prescription, a suitable expression for a susy µ and BR( j → i γ) can be computed.
B.2 Expressions for BR(µ → eγ) and a susy µ By Lorentz invariance, the dipole amplitude for the transition j → i γ can be written as The decay rate of the process µ → eγ is then given by while the anomalous magnetic moment of the muon is described by First, we compute the chargino amplitude. Inspecting eqs. (B.3)-(B.4), one may observe that and anticipate that the leading contributions will come from C R e C L µ in µ → eγ and from C L µ C R µ in a µ . We expand those terms following the MIA guidance. As we are not considering complex phases,

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Applying the MIA, eq. (B.6), for the sneutrino propagator and recalling that in our models mν e mν µ : where in the last equalities we have neglected those terms proportional to cos β. The functions F c X (x 1 , x 2 ) are defined as F c X (x 1 , x 2 ) ≡ f (c) X (x 2 ). We can repeat the same procedure for the neutralino amplitude. From eqs. (B.1)-(B.2) and eqs. (B.14) and (B.17), we observe that the dominant term for µ → eγ comes from the combination N L µ N R e whereas, in the case of a µ , it is produced by N L µ N R µ . Then, following the same steps than for the chargino loop, one would obtain:    In the models considered here, however, we notice the following hierarchies between masses in the region where a susy µ is within the 3σ range: µ M 2 mμ, mν µ . Evaluating the chargino and neutralino loop functions, figure 9 (left panel), one may see that the former dominates for x = µ 2 /m 2 νµ 1. Therefore, we expect that the total amplitudes are governed by the chargino terms given in eqs. (B.30)-(B.31). In figure 3, we compared the approximate expressions against the exact results obtained with SPheno. Both cases display a good agreement for the region of interest. Deviations from the exact result are observed only for small values of a µ and the region of large BR(µ → eγ), where other loops may compete and become important.

C Majorana mass matrix
Here we detail how the Majorana mass matrix can be extracted once the neutrino Dirac matrix is fixed. In the CKM-like case, the neutrino Yukawa matrix is symmetric and can be expressed as: A M R mass matrix that correctly reproduces the light neutrino masses and mixing can be inferred from eq. (2.8) as: with Y ckm ν fixed by eq. (C.1). In the PMNS-like case, the neutrino Yukawa matrix is given by: The M R mass matrix that automatically satisfies all neutrino observables is diagonal and it is simply given by: Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.