R-parity violation in SU(5)

We show that judiciously chosen R-parity violating terms in the minimal renormalizable supersymmetric SU(5) are able to correct all the phenomenologically wrong mass relations between down quarks and charged leptons. The model can accommodate neutrino masses as well. One of the most striking consequences is a large mixing between the electron and the Higgsino. We show that this can still be in accord with data in some regions of the parameter space and possibly falsified in future experiments.

1 Introduction and outline SU(5) is the minimal and the simplest among supersymmetric grand unified theories (GUTs). It is thus of particular interest to test it in detail. In order to be predictive we will stick to its minimal renormalizable version, whose underlying problems are twofold. The discrepancies are of order one, and so cannot be easily accounted for without changing the theory, for example its physical content. The second problem is the absence of neutrino masses, similarly as in the standard model (SM). The purpose of this work is to investigate whether the fermion mass ratio problem can be resolved by R-parity violating (RPV) [1] couplings in the SU(5) model. This idea has been first proposed long ago [2] (for some other works in this direction see for example [3,4]), but never systematically worked out. We will show that R-parity violation can correct all the bad mass relations (1.2). This will immediately open up a solution also for the neutrino mass problem.
It is known that extra vector-like matter fields can correct the SU (5)  However, we do not want to enlarge the field content of the model. An obvious (and well known) candidate for a vector-like pair is provided in the MSSM by the two Higgs doublets with bilinear RPV terms. But with them only we can, according to (1.3) with θ i D = 0, just increase the mass ratio: for this reason bilinear R-parity violation can be useful in the MSSM only to correct the first generation mass ratio.
The next logical possibility is to allow also color triplets d c i to mix with the heavy SU(5) partners of the MSSM Higgses. At first glance this idea looks hopeless, since the mixing would induce the trilinear RPV couplings λ and λ from the SU(5) Yukawas after rotation and the d = 4 proton decay rate is proportional to λ λ and suppressed just by the soft supersymmetry (susy) scale. Moreover, SU(5) symmetry at the renormalizable level predicts for the RPV trilinear couplings (before rotation) λ = λ = λ , (1.4) so that it seems impossible to disentangle λ from λ . However (1.4) is valid in the original (flavour) basis, but not necessarily in the mass eigenbasis. Since we want to rotate the quarks d c i with the heavy color anti-tripletT , we can avoid (1.4). At this point, special care must be taken to cancel λ = 0, effectively preserving the baryon number below the GUT scale. This can be obtained by taking a very specific value of the trilinear RPV couplings. The requirement of λ = 0 will uniquely determine the other trilinear RPV couplings as a function of the mixings.
The mixings (i.e. the angles θ i D and θ i E ) on the other side will be fixed by (1.3) with the additional simplifying assumption that, since they go the opposite way, at a given generation i either the quark θ i D or the lepton θ i E contributes, but not both. By comparing (1.1) with (1.3) we conclude that d c quarks of the second and third generation will mix with the heavy triplet, while only the first generation lepton will require a mixing with the Higgs doublet. In the conclusions we will shortly comment on what happens if we relax these assumptions.
The resulting model is very much constrained. Not only one needs to do more than the usual single doublet-triplet fine-tuning, the original choice of the trilinear couplings must also magically combine in order to project to vanishing baryon number violating couplings after triplet rotation. Also, large lepton number violating couplings will induce tree and loop order neutrino masses, which will typically be too large unless under special conditions. We will not even attempt to understand or explain all these fortuitous relations among model parameters. But we will (shamelessly) use such possibility whenever needed by experimental data. This exercise must be thus interpreted as a purely phenomenological possibility to avoid various constraints already in the minimal SU(5) model, and not as a proposal for a theoretically attractive theory.
In spite of this, or better, because of this, the model predicts a phenomenologically very interesting situation of a large mixing between the electron (neutrino) and the charged (neutral) Higgsino. The seemingly ad-hoc assumption of only quark or lepton mixing in the same generation will at this point help in avoiding strong phenomenological constraints due to large (order 1) lepton number violating couplings present in the low-energy MSSM Lagrangian. In particular, we will see that the tiny neutrino masses predict in this scenario a fixed (negative) ratio between the wino and bino masses, provided they are not much larger than the sfermion masses. Another interesting prediction is a large suppression, with respect to its SM value, of the decay rate of the Higgs boson into electrons. Finally, the same large RPV couplings only allow a slowly decaying gravitino lighter than about 10 MeV as a dark matter (DM) candidate.
The paper is organised as follows: in Sect. 2 we discuss the general structure of the RPV SU(5) model and show how RPV interactions can correct the bad mass relations of the original SU(5) model. Most of Sect. 3 is instead devoted to checking whether the required amount of R-parity violation is still allowed by data. In particular, we discuss proton decay bounds, electroweak symmetry breaking, neutrino masses, modifications of SM couplings to leptons, lepton number and lepton flavour violating processes and gravitino DM. We conclude in Sect. 4 by recalling the main predictions of the model, while more technical details on the diagonalization of the relevant mass matrices are collected in Appendix A.

The RPV SU(5)
The field content of the minimal SU(5) model is given by 5, 5 α (α = 0, 1, 2, 3), 10 i (i = 1, 2, 3) and 24. The decomposition of the SU(5) supermultiplets under the SM gauge quantum numbers reads where 3 ( 2 ) schematically denotes the Levi-Civita tensor in the SU(3) (SU(2)) space and for the adjoint (which also spontaneously breaks SU(5) into the SM gauge group) The indices of φ stand for the SM gauge quantum numbers, while the part proportional to V denotes the GUT vacuum expectation value (vev) (hence φ = 0). The most general renormalizable superpotential can be written in one line where SU(5) contractions are understood. In particular, Λ αβk = −Λ βαk and Y 10 ij = Y 10 ji . The last three terms in Eq. (2.3) are standard (Y 10 is responsible for the up-quark masses, while M 24 and λ participate to the GUT symmetry breaking). So from now on we will focus our attention on the remaining pieces.
From the first term in Eq. (2.3) we see that one combination of5 α will get a vector-like mass with 5. Physically we know that such a mass will have to be large in the triplet sector and light in the doublet one. This means a multiple fine-tuning instead of the single one as in the R-parity conserving (RPC) case. The mass terms in the doublet-triplet sector of the superpotential become where (2.6) The doublet-triplet splitting (assuming low-energy susy) means the following: for all α = 0, 1, 2, 3, while for at least one α.

The issue of the doublet basis
Since in this setup there is no real difference between the four doublet superfields2 α = (N α , E α ) T what do we mean by the names (s)neutrino, charged or neutral Higgs(ino) and charged (s)lepton? In other words, what is the difference between neutral Higgs-sneutrino, neutral Higgsino-neutrino, charged Higgs-slepton and charged Higgsino-charged lepton? Although the results can always be written in a basis-independent way [5,6] and so these names are strictly speaking not really necessary, we will still define such names for the sake of clearness. We will choose a convenient basis, in which only one among the SM doublets2 α ⊂5 α (let it be the one with index α = 0) gets a nonzero vev v d . This can be obtained by an SU(4) rotation of the5 α which affects the relations (2.5)-(2.6) as well. One could argue that the new, rotated, M α and η α cannot be completely arbitrary, since the vevs themselves depend on them. However, it is not hard to imagine (and we will show it in more detail in Sect. 3.2) that the freedom in the choice of soft terms allows us to consider M α and η α arbitrary with 5 i = 0. Since we will not employ any particular spectrum of the soft terms, this is what we can (and will) do.
In particular, there are essentially four class of fields we have to specify: the neutral bosons, the neutral fermions, the charged bosons and the charged fermions. These are fixed in the following way: • The flavour basis of neutral bosons is defined such that the sneutrinos' vevs vanish: i.e. we define the neutral Higgs vevs as in the RPC case: More details about the electroweak symmetry breaking sector and the composition of the lightest Higgs boson in terms of the flavour basis can be found in Sect. 3.2.
• The neutral fermions mass matrix is incorporated into the neutralino quadratic part of the lagrangian: The 4 × 4 lower-right block in (2.11) is the seesaw contribution from the SM singlet and weak triplet states living in 24. In particular, we find where M 24 denotes the superpotential parameter defined in Eq. It is clear from (2.11) that in the flavour basisH 0 d is the fermionic superpartner of H 0 d that gets the vev in (2.10). The mass basis is obviously obtained by diagonalizing the matrix in Eq. (2.11) and neutrinos are the three lightest eigenstates.
• The charged fermions are part of the chargino sector H − d and e i are the weak partners of the previously definedH 0 d and ν i , respectively. In particular, the charged lepton mass eigenstates correspond to the three lightest eigenvalues of the matrix in Eq. (2.13).
• Finally the charged bosons: in the flavour basis they are just the SU(2) partners of the neutral bosons defined through (2.9) and (2.10), or, equivalently, the bosonic superpartners of the charged fermions defined in (2.13). We will denote them by H − d andẽ i .
This quadratic part of the Lagrangian, plus the analogous one for color triplets in (2.15), is RPC if M i = µ i = 0. Of course, the whole Lagrangian, or even this part of it at higher loops, is not RPC due to nonzero trilinear terms, but in the basis we use, ν i = 0, these trilinear terms do not appear in the mass matrices at the tree order.
At this point, we are still free to rotate in the 3 × 3 subspace and we use this freedom to diagonalize the Yukawa matrix: (2.14) Consequently, Eqs. (2.5)-(2.6) get rotated as well, but we will not keep track of it.

The color triplet mass eigenstates
The mass matrix for color triplets is The states3 α are still in the flavour basis. Let us rotate them into the mass eigenstates , we can easily disentangle the single heavy state from the light ones:3 In fact the triplet states are projected into the heavy direction 17) with a unitary matrix (assuming everything real for simplicity) Or, in components: Then the light 3 × 3 mass matrix (of the down quarks) is It turns out that the mass eigenvalues are always smaller than d j v d , so the optimal choice is x 1 = 0 (no mixing with the lightest family), since the down quark is required to be heavier than the electron (cf. Eq. (1.1)) and we do not want to make the problem even worse. The situation is the other way around for the strange and bottom quarks, which are a bit too large in the RPC version of SU(5).

The weak doublets in the gaugino decoupling limit
We will now show that we can realistically describe the masses of the SM fermions in the decoupling limit of large gaugino masses M 1,2 . In this case the chargino mass matrix which remains is (cf. Eq. (2.13)) i.e. analogous to (2.15). Although the Higgsino mass is presumably much lighter than the GUT scale, it is still much heavier than the light charged leptons, so a similar rotation as in the case of the triplets can be used to integrate out the heavy Higgsino. The light charged lepton mass matrix is thus in this limit with Here it would be convenient to decrease the electron mass, but not the muon or the tau (see Eq. (1.1)). Hence, the optimal choice is to take y 2 = y 3 = 0 (no mixing with the second and third generation).
In general, we are interested in the correlation between down quarks (Eq. (2.25)) and charged leptons (Eq. (2.27)). It is known, see for example [7], that with arbitrary x i , y i and d i , one can fit all down quark and charged lepton masses. In fact, defining the Yukawa Alternatively, one can define the angles The masses are then and their ratios Finally, let us consider the neutrinos: from Eq. (2.11) in the gaugino decoupling limit we get  By perform a global SU(4) transformation which also rotates µ α → µ α (the modulus µ = µ 2 0 + µ 2 k being invariant), we obtain the Higgsino and neutrino mass eigenvalues, given by So we are pretty close to the realistic case, just one non-zero neutrino mass is missing. This can be got by including the contribution of gauginos (which is actually non-negligible in the neutrino sector), and eventually adding loops. As we will see in Sect. 3.3, small enough neutrino masses can be obtained only at the expense of a heavy fine-tuning of model parameters, more precisely gaugino masses.

Numerical example
As a numerical benchmark let us consider the case of MSSM with tan β = 7 and low susy scale. From the experimental values at m Z one can use the renormalization group equations (RGEs) to get the charged lepton and down quark Yukawa couplings at the GUT scale [7] (λ exp e , λ exp µ , λ exp τ ) = (0.000013, 0.0028, 0.047) , respectively, by properly choosing the various x i , y i (see Eqs. (2.29)-(2.31)): Notice that we fit all the masses at M GU T . Although this is a correct procedure for the quarks, since we are integrating out the heavy (GUT scale) color triplet, the lepton (electron) corrections should be determined in principle at low energy, when the Higgsino is integrated out. But since the RGEs for the light Yukawas are essentially linear (dλ e /dt ∝ λ e ), the result is practically the same.
As a final remark, the r.h.s. of Eq. (2.29) for the electron mass is only approximate, since the full mass matrix in Eq. (2.13) contains mixings with gauginos as well. It is easy to check its consistency. The result is that the error by taking the approximate formula (2.29) is always below 2% for M 2 > 1 TeV.

The trilinear RPV couplings
Let us define the RPV superpotential of the low-energy MSSM effective theory as The trilinear RPV couplings are then obtained by decomposing the SU (5) (2.54) By enforcing the safe condition 3 we can calculate the other trilinear couplings. To this end, we use the choice of basis in (2.14), the explicit form of U αβ in (2.22)-(2.24) and the relation which allows to compute the inverse of U ij . Hence, after some algebra we obtain 57) or explicitly (for the numerical example discussed in Sect. 2.4) where we used for our fit x 1 = 0. The only relevant matrix element (i.e. ∝ d 3 = λ τ ) is then Similarly, for the other trilinear term we get Even in this case the piece proportional to λ τ goes never through the first generation, i.e. λ ijk ∝ λ e if any among i, j, k equals 1, since x 1 = 0. This is important, since in this way many dangerous processes, like for example neutrinoless double β decay, get automatically suppressed (cf. Sects. 3.5-3.6). Numerically we get To summarize, the L 1 lepton number is strongly broken by the O(1) parameter µ 1 /µ 0 , L 2 by the O(0.1) couplings λ 233 and λ i23 , i = 2, 3, and L 3 by the O(0.1) values of λ i33 , i = 2, 3. Neutrino masses are thus generically expected to be large (see Sect. 3.3). On the other hand, baryon number is effectively preserved below the GUT scale, thanks to the condition λ = 0.

Phenomenology
The low-energy RPV parameters considered so far are strongly correlated. In general they are parametrized by x i (= M i /M 0 ) and y i (= µ i /µ 0 ). In order to simplify our analysis and minimize the corrections to make, we assumed that the RPV parameters which make the fermion mass problem more severe are not present. Namely we took x 1 = y 2,3 = 0. The vanishing baryon number violating couplings in the MSSM and the correct fit to fermion masses then determined the numerical values of all the RPV couplings µ i /µ 0 , λ ijk and λ ijk .
To study the phenomenological consequences of such a model we need to specify also the other RPV couplings which did not enter in the analysis so far, but which can still have a strong phenomenological impact: the soft mass terms B i , m 2 0j as well as the trilinears A ijk , A ijk and A ijk . Since it is not our intent to do here a full phenomenological study of the most general case, but just to show the existence of a realistic model, we will take further simplifying assumptions: let • the RPV bilinear soft terms point in the direction 1, similarly as the µ i in the superpotential Although one would be tempted to make both r.h.s. in (3.1) and (3.2) to vanish, electroweak symmetry breaking constraints do not allow such choice, see Sect. 3.2; • the RPV trilinear terms vanish We are now ready to study the phenomenology. We will first consider proton decay, where, due to the assumption λ = 0, R-parity violation does not bring any really new issue compared to the RPC case. The analysis is however necessary in order to determine relations among scales. After that we will systematically go through strict RPV consequences.

Proton decay and unification constraints
Although we will not dwell too much on the proton decay issue, some remarks are due. Unification of gauge couplings [10][11][12][13] in the minimal renormalizable SU(5) model seems at odds with the experimental limits on proton decay if one assumes order TeV susy spectrum [14], albeit playing with the flavour structure of soft terms allows to solve the problem [15,16]. Another logical possibility is simply to increase the susy scale. Nowadays, following ugly experimental facts and neglecting beautiful theoretical ideas, this is not a taboo anymore. In the usual RPC case it is enough to increase the susy scale to the multi-TeV region for low tan β in order to get the d = 5 proton decay channel under control [17,18]. The point is [18] that by increasing the susy scale the color-triplet mass rises as well due to gauge coupling unification constraints. On the other side, this reduces the combination of the heavy gauge boson mass squared times the mass parameter of the adjoint. The gauge boson mass cannot be too low due to the d = 6 proton decay channel, but in the RPC case the mass of the adjoint can practically take any value and so can be diminished at will.
Once however RPC is abandoned, and the η i are of order one due to the doublet-triplet fine-tuning (2.5)-(2.6), the adjoint mass cannot be too small because it mediates the rank-1 type I + III seesaw mechanism for neutrino masses (see Eq. (2.40)), so it is bounded from below by around 10 13 GeV. This means that we cannot increase the susy scale at will and so we may have some problem with proton decay constraints.
Let us now estimate these scales. Denoting by mf the common sfermion mass (taken also as the matching scale between SM and MSSM), by m λ the common gaugino mass, by µ the Higgsino mass (µ 0 ≈ µ 1 ), by M T the heavy color triplet mass, by M V the heavy gauge boson mass (taken also as the matching scale between MSSM and SU(5)) and M 24 the common mass of the heavy adjoint fields (differences due to order one Clebsches are neglected), we can write the approximate relations [18,19]  (3.14) In such a case the d = 5 proton decay channel is the leading one and to be seen soon.
In all other solutions, the susy spectrum must be split with possibly light Higgsino and/or gauginos. It has to be stressed though that all we said so far is valid at most as an order of magnitude estimate, so that factors of few are possible.
Finally, let us notice that we could also have proton decay contributions due to a slightly nonzero λ . This would open up new decay channels, for example B + L conserving [20], not present in the usual Weinberg classification (although B + L conserving proton decay could be mediated by d > 6 operators even in RPC GUTs, see for example [21]). However, due to the required smallness of λ , nothing else except baryon number violating processes would change in our analysis.

Electroweak symmetry breaking
Our potential is (everything is real) 4 where α, β run from 0 to 1 (with m 2 01 = m 2 10 ) and we consider the basis The stationary equations give: This correctly reproduces the RPC case (m 00 = m H d , µ 1 = 0, B 1 = 0 and m 2 01 = 0). Notice that due to (3.21) we cannot take both B 1 and m 2 01 vanishing. This was the motivation for the assumptions (3.1) and (3.2).
By expanding H 0 u,d = v u,d + h 0 u,d , the mass matrix of the neutral (real) scalars in the (h 0 u , h 0 d ,ν 1 ) basis is found to be where we also substituted the stationary conditions in Eqs. (3.19)-(3.21) and we neglected O(v 2 ) terms. It is easy to see then, that the lightest eigenvalue (massless in the v → 0 limit) is associated with the eigenvector (tan β, 1, 0). Hence, in the decoupling limit the light Higgs has no projections on the sneutrino direction. In the finite v case the component of the light Higgs in the sneutrino direction is thus proportional to v 2 /m 2 susy .

Neutrino masses from finite gaugino masses
In this section we will see which constraints must be satisfied in order for neutrino masses to be in the ballpark. The analysis will be far from complete. We will calculate the tree level rank-1 contribution and estimate the leading one-loop corrections. In doing this, we will use the mass insertion approximation for the RPV bilinear couplings as in [5,6,23]. Although this is unjustified in the present context due to large RPV couplings, we assume that they give the right order of magnitude. The purpose of this calculation is not predicting neutrino masses but rather check their consistency with experimental data. Finite gaugino masses cannot be neglected in the neutrino sector, due to tiny neutrino masses. We can still however assume that the electroweak vev is small compared to the other mass parameters in (2.11), as has been done for example in [24] (see also Appendix A). In this limit and neglecting the typically much smaller type I + III seesaw contribution, the tree-level expression for neutrino masses reads This can be made small for our choice of parameters (µ 1 /µ 0 of order 1) only assuming a very strong cancellation i.e. having gaugino masses with opposite sign and fine-tuned ratio. Notice that the combination of gaugino masses in Eq. (3.24) is proportional to the photino mass parameter, mγ = M 1 c 2 W + M 2 s 2 W , and that the exact determinant of the generalized neutralino mass matrix in Eq. (2.11) (after restricting to the nontrivial rank-5 subspace and for η α = 0) is still proportional to mγ. Though mγ → 0 can be effectively used to suppress large tree-level neutrino masses, this limit does not seem to be associated with any new symmetry of the Lagrangian.
In fact, already at one loop this fine-tuning is not enough anymore, since the rank of the matrix will change. The most relevant diagrams are shown in Fig. 1. ii) Here the diagrams include the external neutrino mixing with both bino and wino through Higgsino; after summing all contributions and choosing a renormalization scheme such that the wino-neutrino mixing is canceled at the one-loop level [6], one gets various contributions each of the order of

26)
A more detailed calculation [6] gives an exact cancellation in the degenerate down squark case (m 2 d L =m 2 d R ). Similar diagrams with λ → λ and sleptons in the loop require degenerate sleptons (m 2 τ L =m 2 τ R ) for an exact cancellation.
iii) + iv) These contributions can be written as [5,23] where we assumed m W M 2 mf ≈ m H u,d . Notice the m 2 W /m 2 f suppression in Eq. (3.27), which is a remnant of an exact cancellation of the loop functions in the decoupling limit [23]. These contributions are in the same direction as the fine-tuned tree-level one. So all one needs is doing just a slightly different fine-tuning.
In conclusion, neutrino masses can be (admittedly barely) under control assuming a strong fine-tuning among wino and bino mass parameters (3.24) to suppress the tree-level contribution, heavy sfermions or small left-right sfermion mixings to suppress (3.25), and an approximate degeneracy in the sfermion spectrum to suppress the one-loop contribution (3.26).

Modifications of SM couplings to leptons
The mixing between leptons and higgsinos/gauginos is also constrained by the measurement of the SM couplings to the lightest lepton mass eigenstatesê 1,2,3 andν 1,2,3 . The relevant couplings to be considered here are: Zê iêj (precision measurement at the Z pole and lepton flavour violating charged lepton decays), Zν iνj (invisible Z width), Wê iνj (charged lepton universality) and Hê iêj (Higgs boson decay into charged leptons).
Assuming real parameters and denoting the deviation from a SM coupling g SM as δg SM , the modified SM couplings to leptons are found to be (see also [24][25][26]): • Zê iêj couplings: where U L,R are the bi-unitary matrices which diagonalize the generalized chargino mass matrix (cf. Appendix A), while i and j run over the three lightest eigenvalues. In particular, in the susy-decoupling limit considered in Appendix A we get: The constraints from the Z-pole observables are typically given in terms of δg V,A = 1 2 (δg L ± δg R ) and are at most at the 0.07% level for the flavour diagonal case [27][28][29].
On the other hand, the bounds on the flavour violating couplings are less strict, with the only exception of those coming from the measurement of µ → eee c , which sets δg 12 V,A 10 −6 [25,30]. The latter bound is evaded by our specific flavour orientation of the µ i vector, e.g. µ i ∝ δ 1i .
Hence, all the relevant bounds due to the modification of the Z boson couplings to charged leptons are satisfied by M 2 5 TeV and µ i ∝ δ 1i .
• Zν iνj couplings: where U 0 is the unitary matrix which diagonalizes the generalized neutralino mass matrix (cf. Appendix A), while i and j run over the three lightest eigenvalues.
At the leading order in the expansion of Appendix A, we find For µ > m Z , the typical signature is the reduction of the invisible width of the Z boson. However, even for moderate (non-decoupled) values of M 1,2 , the inferred bound on µ 1 is very mild [25].
• Wê iνj couplings: Defining the current eigenstate matrices the modified SM couplings read where i and j run over the three lightest eigenvalues. Charged lepton universality in charged current processes, such as the decay of pions and leptons, is experimentally verified at the 0.2% level [31]. This typically yields less stringent bounds than those derived from Z couplings [25].
• Hê i e c j couplings: A large mixing between e 1 andH − d leads to sizeable modifications of the Higgs boson coupling to electrons. In the decoupling limit of small Higgs vev compared to other mass scales the light Higgs is the same linear combination of H u and H d as in the RPC case (see Sect. 3.2), so that the only difference comes from the projection of the flavour electron onto the mass electron final state, namely e 1 → µ 0 µê 1 (see Appendix A). We hence expect that the decay rate of the Higgs boson into electrons is reduced with respect to the SM value by a factor Currently, only H → τ + τ − has been measured with a 30% error [32], while the sensitivity of LHC on the channels H → µ + µ − and H → e + e − is respectively a factor 7 and 4 × 10 5 above the SM predictions [33]. The electron channel is clearly inaccessible at LHC. However, there are already ongoing studies on the possibility of probing the SM electron Yukawa coupling in resonant s-channel Higgs production at future e + e − circular colliders [34]. Any estimate, however, strongly depends on the assumptions in the soft sector, so we will not discuss them further here. In any case, this effect cannot explain the recent 2.5σ hint of CMS of a non-zero BR(H → τ + µ − ) above the 10% level [40], being smaller than a permil of it due to the v 2 /m 2 susy suppression of the amplitude.
Summarising, the couplings of the Z and W bosons to the three lightest lepton mass eigenstates can be easily made compatible with the SM values by a moderate decoupling of gaugino masses (say M 1,2 5 TeV) and for µ i ∝ δ i1 . This was indeed to be expected, since in the gaugino decoupling limit we are mixing only representations with the same gauge quantum numbers (GIM-like mechanism), and hence gauge couplings have to be SM-like. On the other hand, the modifications of the Higgs boson couplings to leptons are not suppressed for heavy gaugino masses. The assumption made in this paper that µ i ∝ δ i1 (in order to only correct the electron mass eigenvalue) helps in suppressing potentially large effects in Higgs boson decay channels, which in the case of electrons in the final state are still far from being measured.

Other lepton number violating processes
On top of neutrino masses there are also other lepton number violating effects which are worth to be discussed. First of all, LHC can produce via a Drell-Yan process a pair of winos which can subsequently decay through lepton number violating couplings into same-sign dileptons [35] and 4 jets with no missing energy (ideally, a background-free process): pp → W * ± →W ±W 0 → (e ± Z)(e ± W ∓ ) → (e ± jj)(e ± jj) . (3.41) This is completely analogous the the production and decay of a light weak triplet fermion pair from type III seesaw [36][37][38][39]. Since winos are unstable the cross section σ(pp → W ±W 0 ) gets multiplied with an approximate factor This is small due to the (m W /M 2 ) 2 suppression of the Γ (see Eq. (A.15)) and eventually because E max < M 2 . Hence, in spite of the fact that the RPV coupling µ 1 /µ 0 is much larger than in the usual case, this lepton number violating process will not be easily accessible at LHC because the ratio m W /M 2 1/50 is too small, giving for Eq. (3.43) a suppression of ≈ 10 −7 . The next lepton number violating process we consider is neutrinoless double β decay. Following [1] the limits on the trilinear RPV couplings are (k = 1, 2, 3) which are easily satisfied in our case, even for relatively low super-partner masses. On the other hand, the parameter µ 1 /µ 0 contributes to the process only through the light neutrino masses, whose suppression has been already discussed in Sect. 3.3. Finally, other potentially relevant lepton number violating processes like e.g. µ + → e − conversion in nuclei, K + → µ + µ + π − orν e emission from the Sun, do not bring any really important constraint on the model parameters since the experimental limits on the branching ratios are still too weak.

Lepton flavour violation
In this section we analyse in more detail lepton flavour violating processes like µ → e conversion in nuclei, µ → eee c and µ → eγ (other processes involving the τ lepton are worse measured and their bounds can be easily evaded). At leading order ( 0 ) in = O(m W /M 2 , m 1 /m W ) 10 −2 there is no mixing between generations, i.e. the electron mass eigenstate mixes just with Higgsino, while the muon does not mix at all (µ 2 = 0), see Appendix A. In other words, at order 0 and tree level the λ and λ couplings are already in the mass eigenbasis. In particular, all the lepton flavour changing amplitudes involving electrons vanish at order 0 . Following for example the computation and notation of [41] for µ → e conversion and [42] for the other two processes, we can summarize the results as follows (λ and λ corresponding to the values determined in Sect. 2.5): • µ → e conversion: the coefficients in front of the possible operators of the typeēµqq are at tree order • µ → eee c : the coefficients in front of the possible operators of the typeēµēe are at tree order • µ → eγ: the coefficients in front of the possible operators are at one-loop order Next we want to check what happens beyond the leading order. Without doing a full calculation for the order or at higher loops, we can consider the following: 1. Either or an extra loop factor contribute with a suppression factor of at least 10 −2 ; 2. Although L 1 violation is in principle order 1, L 2 violation is of order 10 −1 (cf. discussion below Eq. (2.64)); 3. The propagator gets a suppression (m W /mf ) 2 compared to the Fermi constant G F Putting all this together, we schematically find for the generic coefficient A in Eqs. (3.46)-(3.50) relative to the different processes: • µ → e conversion: comparing theoretical expectations [43] with the experimental constraint on Titanium [44] which can be satisfied for sfermion masses of order 10 TeV or more.
• µ → eγ: following again [29] we find (notice that here we started already at one-loop) which is evaded already for mf 300 GeV.

Gravitino dark matter
In the presence of sizeable RPV interactions the only DM candidate is a slowly decaying gravitino. For m 3/2 < m Z the main decay channel of the gravitino is [46] Γ(G → γν) = 1 32π where Uγ ν = c W UB ν + s W UW ν is the photino-neutrino mixing and M P = 2.4 × 10 18 GeV is the reduced Planck mass. From Eq. (A.27) we read where tan β 1 and we already considered the fine-tuning in Eq. (3.24) in order to suppress neutrino masses. This has to be compared with the standard case where the smallness of neutrino masses is due to a tiny mixing with gauginos, yielding [46]   which is safe, as long as m 3/2 10 GeV (for M 1 ≈ 10 TeV and tan β ≈ 10). The decay of the gravitino is expected to leave an imprint on the extragalactic diffuse high-energy photon background in the form of a monochromatic line centred at m 3/2 /2. This is because m 3/2 is very light, contrary to what happens with multi-TeV gravitino masses where a continuum signal in the spectrum is expected, see for example [45]. The photon number flux, F max γ , at the peak of the maximum photon energy E γ = m 3/2 /2, is estimated to be [46]  which is compatible with the bounds coming from diffuse X-and gamma-ray fluxes [47][48][49], as long as m 3/2 10 MeV (for M 1 ≈ 10 TeV and tan β ≈ 10). The latter values correspond to a lifetime τ 3/2 > 10 27÷28 s, which is indeed the typically constraint for decaying DM into photons [50]. The last point we want to address is a possible constraint related to the reheating temperature. Assuming thermal production in the early Universe, the gravitino relic density is constrained by (see e.g. [51][52][53][54]) (3.60) where approximate equality holds when the gluino contribution can be neglected. Notice that for m 3/2 10 MeV and M 2 ≈ 30 TeV (M 1 ≈ −M 2 g 2 /g 2 ≈ 9 TeV), the reheating temperature can still be above the electroweak phase transition. On the other hand, gravitino masses lighter than already 1 MeV (or, equivalently, too large gaugino masses) would imply a reheating temperature well below the electroweak phase transition, which is difficult to reconcile with an high-energy mechanism of baryogenesis. 5 From this point of view, a gravitino mass close to the upper limit of 10 MeV (compatible with the measured photon fluxes) is theoretically favourable. This is, of course, also the most interesting region for a possible experimental discovery.

Discussion and conclusions
Among grand unified theories only renormalizable SO(10) [55][56][57] is able to derive exact R-parity conservation [58][59][60] at low energies [61][62][63], while there is no reason to assume it in SU (5). There are of course strong phenomenological constraints that make especially the baryon number violating couplings practically zero. In this work we tried to see if the remaining R-parity violating interactions in the minimal renormalizable SU(5) can be of any utility for the down quark vs. charged lepton mass problem of the original setup. The outcome of our analysis is positive: these couplings are able to reproduce the SM fermion masses and so avoid large susy breaking threshold corrections which would make our vacuum metastable [64]. At first sight a weak point of the whole setup are neutrino masses, which tend to be orders of magnitude too large and can be only controlled by a fine-tuned choice of the soft parameters. The most stringent one is given by the relation (3.24) between gaugino masses. Is this a prediction of the theory? Relation (3.24) holds at tree order and it gets corrections at higher loops, (3.27) being the dominant one. The question is thus, how exactly must M 1 /g 2 = −M 2 /g 2 hold? Let us see what we need for this relation to be for example 10% exact, i.e. suppose This is equivalent to say that the loop contribution is at most 10% of the non-fine-tuned value in Eq. (3.23), i.e.
In usual perturbation theory δm/m is loop suppressed, so small, provided the same couplings as at tree order are used. But in our case we have more like a Coleman-Weinberg situation [65], where new couplings not present at tree level, in our case B 1 , start contributing. So there is no limitation from perturbation theory and at least in principle loops could dominate over tree-level contributions. Is this what happens here? According to (3.27), and assuming a split susy spectrum µ 1 ∼ M 2 mf ∼ |B 1 | we find that very roughly the 10% correlation between bino and wino mass (4.1) is valid if M 2 10 cos 2 β mf . , where we assumed that chiral superfields contribute in SU(5) multiplets. The change of the SU(5) beta function equals ∆b chiral = 17/2 on the threshold. Getting an half-integer seems impossible to obtain: a complex representation needs always to come in pairs to be vector-like and satisfy anomaly constraints, while real representations have an integer Dynkin index. Evading this conclusion needs more sophisticated scenarios. However, if (4.1) is relaxed a bit (by M 1,2 10 mf and/or large tan β), then we can get with an integer ∆b chiral (for example 8 or 9) opposite sign bino and wino masses. Another possibility is to consider gravity mediation. From [69] 6 we see that relation (4.1) is obtained for example in SO(10) if a 210 is coupled to gauge field strength bilinears and its parity odd Pati-Salam singlet gets a non-zero F-term. Although amusing, it is unclear what this means in the context of our renormalizable SU(5) model.
On top of Eq. (4.1), there are two other predictions of this theory which make it falsifiable at future experimental facilities. A smoking-gun signature is the reduction of the H → e + e − branching ratio to the ≈ 15% of its SM value. Although this represents an extremely difficult measurement, there are already ongoing studies on the possibility of probing the SM electron Yukawa in resonant s-channel Higgs production at future e + e − circular colliders [34]. The other prediction is a gravitino dark matter candidate lighter than approximately 10 MeV, preferably closer to the upper limit in order to be reconcilable with baryogenesis. A gravitino mass in the region favoured by baryogenesis is also the most interesting one from an experimental point of view. The main signature being a monochromatic line in the diffuse extragalactic photon background picked around 5 MeV.
In this work we only used the RPV mixing effects to correct the wrong SU(5) mass relations. In practice, however, the solution to this problem could arise from different sources, partially from susy threshold corrections and partially from RPV mixings, thus modifying the numerical values of the RPV parameters here considered. Also, the ad-hoc assumption of setting to zero those couplings that make the wrong mass relations worse, is not really needed, although a generic situation might be forbidden by data. For example, a large mixing between the Higgsino and the third generation lepton doublet would be strongly constrained by the measurement of the H → τ + τ − decay.
Whereas the study of the most general parameter space is beyond the scope of this paper, let us stress that, under the working hypothesis of no susy threshold corrections (which would make our vacuum metastable), the main predictions of the model are quite solid. For instance, a non-zero mixing between the heavy color triplet and the first generation down quark would require an even larger mixing between the Higgsino and the first generation lepton doublet, in order to fit the electron mass. Consequently, the H → e + e − branching ratio would be further reduced with respect to the 15% of its SM value, which should be hence understood at most as an upper bound. On the other hand, the required fine-tuning between the gaugino masses in Eq. (4.1) and the photino-neutrino mixing in Eq. (3.55) responsible for the gravitino decay would be only slightly modified due to the larger mixing in the leptonic sector.
Although the model is a bit stretched and many tunings of parameters are needed, the phenomenology itself seems interesting: the electron mass eigenstate (or other leptons as well in a more general framework) may not be what we usually think of, but rather an order half-electron and half-Higgsino flavour state.
For simplicity we will consider real parameters and limit ourselves to the case where only µ 1 = 0 (µ 2 = µ 3 = 0). For a more general case see e.g. [25]. Then the relevant squared mass matrices in the chargino sector read which is diagonalized by U R , and which is relevant for the determination of U L . The 7 × 7 neutralino mass matrix is given in Eq. (2.11), with µ 2 = µ 3 = 0 and η α = 0. We hence neglect the contribution of the type I + III seesaw, since it can be easily made subleading (e.g. by properly decoupling the mass of the GUT-scale seesaw mediator).
Working in the phenomenological limit M 1,2 ≈ µ 0,1 v u,d = O(m W ) m 1 , at the first order in the expansion parameter = m W /M 1,2 or m 1 /m W 1 we find: