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.


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1 Introduction and outline SU(5) is the minimal and the simplest among supersymmetric grand unified theories (GUTs) [1]. It is thus of particular interest to test it in detail. In this work we will stick to its minimal renormalizable version with three matter copies of 5 ⊕ 10 and Higgs supermultiplets in 5 ⊕ 5 ⊕ 24. In fact, by allowing either non-renormalizable operators or extra superfields, many new unknown parameters enter in the superpotential thus making the model unpredictive. Yet, it is well known that the minimal renormalizable SU(5) GUT suffers from two main drawbacks. First, it predicts the equality at the GUT scale of the down quark and charged lepton masses (i = 1, 2, 3 runs over generations) (1. 2) 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 issue of charged fermion masses in minimal renormalizable SU (5) can be solved by large supersymmetry (susy) breaking threshold corrections [3][4][5][6][7][8][9][10][11][12]. The prize to pay, however, is large A-terms which make the MSSM vacuum metastable [13]. Also, this does not bring any new ingredient for the solution of the neutrino mass problem.
In this work, we want to take instead an orthogonal approach. We neglect the contribution of susy threshold corrections and investigate whether the fermion mass ratio problem can be fixed by R-parity violating (RPV) [14] couplings in the SU(5) model. This idea has been first proposed long ago [15] (for some other works in this direction see for example [16,17]), but never systematically worked out. We will show that R-parity violation can correct all the wrong mass relations (1.1). This will immediately open up a solution also for the neutrino mass problem. Let us now briefly describe the idea, while the details will be worked out in the body of the paper.
It has been long known that giving up the minimal field content and allowing for extra vector-like matter fields it is possible to correct the SU(5) fermion mass relations (for an incomplete list of references see [2,[18][19][20][21][22][23][24][25][26][27][28][29]). As we will show later, the mixing of 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 [30][31][32][33][34]. According to (1.3) with θ i D = 0 the mass ratio can only increase, so bilinear R-parity violation can be useful in the MSSM just for the first generation (see eq. (1.2)).
The next logical possibility is to allow also for 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 (cf. the superpotential in eq. (2.49)) from the SU(5) Yukawas after rotation. This would make the proton to decay too fast, since the d = 4 proton decay amplitude is proportional to λ λ and suppressed just by the susy breaking scale. Moreover, SU(5) symmetry at the renormalizable level predicts for the RPV trilinear couplings (before rotation)

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so that it seems impossible to disentangle λ from λ . However (1.4) is valid in the original (flavour) basis, and not in the mass eigenbasis. Since the rotation of quarks d c i with the heavy color anti-tripletT makes the mass and flavour eigenbasis different, 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. As we will see, the requirement of λ = 0 will uniquely determine the other trilinear RPV couplings as a function of the mixings.
From eq. (1.3) it is clear that only the relative misalignment between doublet and triplet rotations matters for the correction of the mass eigenvalues. Hence, we will take the additional simplifying assumption that at a given generation i either the quark θ i D or the lepton θ i E angle contributes, but not both. This will then uniquely determine the mixings (i.e. the angles θ i D and θ i E ). By comparing (1.2) 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. We will hence have In the conclusions we will shortly comment on what happens if we relax these assumptions.
The resulting model turns out to be 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. In order to accommodate all these constraints our soft terms will not be subject to SU (5) invariant boundary condition at the GUT scale. We will hence assume that susy breaking is mediated below the GUT scale (for more comments on that see section 4). This exercise must be thus interpreted as a purely phenomenological possibility in order 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 one) 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. 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 section 2 we discuss the general structure of the RPV SU(5) model and show how RPV interactions can correct the wrong mass relations of the original SU(5) model. Most of section 3 is instead devoted to checking whether the JHEP07(2015)123 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 section 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). The most general renormalizable superpotential is where SU(5) contractions are understood. In particular, Λ αβk = −Λ βαk and Y 10 ij = Y 10 ji . The usual R-parity conserving (RPC) case is recovered in the limit The terms in the second line of eq. (2.4) and eq. (2.3) coincide: Y 10 is responsible for the up-quark masses, while M 24 and λ 24 participate to the GUT symmetry breaking and are related by the minimum equation to the SU(5) breaking vev in eq. (2.2) via the relation V = M 24 /λ 24 . Moreover, in the RPC case Y 5 leads to the usual Yukawa unification condition (1.1) which we want to correct with the more general superpotential in (2.3).

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Let us now focus on the first line of eq. (2.3). From the first term we see that one combination of5 α gets a vector-like mass with 5. Physically we know that such a mass has to be large in the triplet sector and light in the doublet one. This can be achieved thanks to SU(5) breaking via the vev contribution in eq. (2.2). Then the mass terms in the doublet-triplet sector of the superpotential become where The doublet-triplet splitting (assuming low-energy susy) means the following: for all α = 0, 1, 2, 3, while for at least one α. Notice that while in the usual RPC case one fine-tuning is enough, in the generic RPV case four fine-tunings are needed in order to satisfy eq. (2.9) for all four possible choices of α. Finally, the terms in Λ αβk contain, on top of the above mentioned Yukawas, the trilinear RPV couplings which will be discussed in section 2.6.

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 [35,36] and so these names are strictly speaking not really necessary, we will still define such names for the sake of clarity.
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 (2.7)-(2.8) 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 section 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 classes 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:

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• 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: where v = 246 GeV. 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 section 3.2.
• The neutral fermion mass matrix is incorporated into the neutralino quadratic part of the lagrangian (see e.g. [37]): where we added the 4 × 4 lower-right block as the seesaw contribution from the SM singlet (1, 1) 0 [38][39][40][41][42] and weak triplet (1, 3) 0 [43] states living in 24, and • The charged fermions are part of the chargino sector (see e.g. [37]): 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. (2.15).
• Finally the charged bosons: in the flavour basis they are just the SU(2) partners of the neutral bosons defined through (2.11) and (2.12), or, equivalently, the bosonic superpartners of the charged fermions defined in (2.15). We will denote them by

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This quadratic part of the Lagrangian, plus the analogous one for color triplets in (2.17), 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 sub-matrix matrix (2.16) Consequently, eqs. (2.7)-(2.8) get rotated as well, but we will not keep track of it.

The color triplet mass eigenstates
The mass matrix for color triplets comes from the first term in (2.6) and the last term in the first line of (2.3) The states3 α are still in the flavour basis. Let us rotate them into the mass eigenstates , we can easily disentangle the single heavy stateT from the light ones d c k : where the matrix U projects the triplet states into the heavy direction 19) Assuming everything real for simplicity we have (see for example [26]) Then the light 3 × 3 mass matrix (of the down quarks) is

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Notice that U ij is not unitary, since it is just the 3 × 3 sub-matrix of the 4 × 4 unitary U αβ . This implies that the mass eigenvalue for any x.

The charged lepton mass eigenstates
In order to get the three lightest eigenvalues of the chargino mass matrix it turns out to be a good approximation to consider the gaugino decoupling limit. This will be numerically confirmed in section 2.5. In this case what remains in eq. (2.15) is which is analogous to (2.17). 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 for any y.

How to avoid Yukawa unification
We are interested in the correlation between down quarks (eq. (2.26)) and charged leptons (eq. (2.29)). It is known, see for example [2] and references therein, 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 λ = m/v d , one finds in the hierarchical limit d 1 From these equations it is clear that the most economical way to get (1.2) is to take x 1 = 0 (no mixing of the heavy color triplet with the first generation down quark) and y 2 = y 3 = 0 (no mixing of the Higgsino with the second and third generation lepton).

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Before ending, we want to make a connection with the notation of eq. (1.3). This can be done by defining the angles (2.37) Then the masses are from which eq. (1.3) follows.

A 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 [2] (λ exp e , λ exp µ , λ exp τ ) = (0.000013, 0.0028, 0.047) , As we saw in the previous paragraph, the Yukawas can only diminish if a mixing with an extra vector-like respectively, by properly choosing the various x i , y i (see eqs. (2.32)-(2.34)): Notice that we fit all the masses at M GUT . 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.

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As a final remark, the r.h.s. of eq. (2.32) for the electron mass is only approximate, since the full mass matrix in eq. (2.15) contains mixings with gauginos as well. It is easy to check its consistency. The result is that the error by taking the approximate formula (2.32) 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) superpotential (2.3) under the SM group and by matching it with eq. (2.49). This operation yields By enforcing the safe condition 1 which allows to compute the inverse of U . Hence, after some algebra we obtain 55) or explicitly (for the numerical example discussed in section 2.5)

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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 never goes 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. sections 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 section 3.3). On the other hand, baryon number is effectively preserved below the GUT scale, thanks to the condition λ = 0.

Phenomenology
To study the phenomenology, we have to define our low-energy effective theory which is the MSSM with specific RPV couplings. As we saw, 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 be done, we assumed that the RPV parameters which make the fermion mass problem more severe are not present (i.e. x 1 = y 2,3 = 0). Due to that we will limit our phenomenological analysis to the case µ = (µ 1 , 0, 0) . To study the phenomenological consequences of the model we also need to specify the other RPV couplings which did not enter in the analysis so far, but which can still have a JHEP07(2015)123 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 (3.2) and (3.3) to vanish, electroweak symmetry breaking constraints do not allow such choice, see section 3.2; • the RPV trilinear terms vanish We are now ready to study the phenomenology. We will first consider proton decay. Here there are two new issues compared to the RPC case. First, as we will see in the next section, an additional constraint must be taken into account in the unification analysis. Second, due to the huge sensitivity of proton decay to the exact value of λ ≈ 0, new decay channels might contribute as well. After that we will systematically go through leptonic 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 [46][47][48][49] in the minimal renormalizable SU(5) model seems at odds with the experimental limits on proton decay if one assumes order TeV susy spectrum [50], albeit playing with the flavour structure of soft terms allows to solve the problem [51,52]. 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 [53,54]. The point is [54] 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 R-parity conservation is abandoned, and the η i are of order one due to the doublet-triplet fine-tuning (2.7)-(2.8), the adjoint mass cannot be too small because it mediates the type I + III seesaw mechanism for neutrino masses (see eq. (2.13) and section 3.3.3), so it is bounded from below by around 10 13 GeV. This means that we JHEP07(2015)123 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 [54,55] M T 10 15 GeV  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.

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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 [56], 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 [57]). 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) 2 where α, β run from 0 to 1 (with m 2 01 = m 2 10 ) and we consider the basis The stationary equations give: 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. (3.20)-(3.22) 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
In this section we will see which constraints must be satisfied in order for neutrino masses to be in the right ballpark. In doing this, we will use the mass insertion approximation for the RPV bilinear couplings as e.g. in [35,36]. 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 that of predicting neutrino masses but rather to check their consistency with experimental data. In particular, we will estimate (in order of importance): the tree-level seesaw contribution from RPV interactions, the leading one-loop RPV corrections and the type I + III seesaw contribution from GUTscale mediators. Let us now discuss in turn the various cases.

Tree-level seesaw from RPV interactions
This is the most important contribution. By neglecting the typically much smaller GUTscale induced type I + III seesaw contribution (to be discuss in section 3.3.3), the only non-vanishing element of the neutrino mass matrix is where we expanded in v/M 1,2 , while keeping µ 1 /µ 0 of order one [37] (see also appendix A). Eq. (3.24) can be made small, for our choice of parameters, only assuming a very strong cancellation i.e. having gaugino masses with opposite sign and fine-tuned ratio. This is possible since we did not assume any specific boundary condition on the soft terms (e.g. gaugino masses unification). In section 4 we will shortly comment on possible mechanisms of susy breaking which might yield to relations close to eq. (3.25).
Notice that the combination of gaugino masses in eq. (3.25) 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. (2.13) (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 neutrino mass matrix will change as we will see in the next subsection.

One-loop contributions from RPV couplings
The most relevant diagrams for the RPV one-loop corrections to the neutrino mass matrix [35,36,[59][60][61][62] are shown in figure 1. Let us now estimate their size. 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 [36], one gets various contributions each of the order of

i) A standard computation gives
A more detailed calculation [36] 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 [35,61] where we assumed m W M 2 mf ≈ m H u,d . Notice the m 2 W /m 2 f suppression in eq. (3.28), which is a remnant of an exact cancellation of the loop functions in the decoupling limit [61]. 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.

Seesaw from GUT-scale mediators
For completeness, we estimate the rank-1 type I + III seesaw contribution from GUT-scale mediators in eq. (2.13) in the limit of no RPV mixing. This yields one non-vanishing neutrino mass eigenvalue where η = η 2 0 + η 2 k and η = O(1) in order to achieve the doublet-triplet splitting (cf. section 2). Notice that, since M seesaw could be as large as M GUT 10 16 GeV, this contribution to neutrino masses can be made subleading.
In conclusion, neutrino masses can be (admittedly barely) under control assuming a strong fine-tuning among wino and bino mass parameters (3.25) to suppress the tree-level contribution, heavy sfermions or small left-right sfermion mixings to suppress (3.26), an approximate degeneracy in the sfermion spectrum to suppress the one-loop contribution (3.27) and M seesaw ≈ M GUT .

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) and Wê iνj (charged lepton universality).
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 [37,63,64]): • 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: 34) and the modified couplings of the Z boson to charged leptons (electrons) are hence δg 11 L = O m 2 W /M 2 2 and δg 11 R = O m 2 1 /µ 2 . 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 [65][66][67].

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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 [63,68]. 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 [63].
• 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 [69]. This typically yields less stringent bounds than those derived from Z couplings [63].
Summarizing, 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.

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 [70] 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) . (3.42) This is completely analogous the the production and decay of a light weak triplet fermion pair from type III seesaw [71][72][73][74]. 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. (3.44) a suppression of ≈ 10 −7 . The next lepton number violating process we consider is neutrinoless double β decay. Following [14] 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 section 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 [75] for µ → e conversion and [76] for the other two processes, we can summarize the results as follows (λ and λ corresponding to the values determined in section 2.6): • 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:

JHEP07(2015)123
• µ → e conversion: comparing theoretical expectations [77] with the experimental constraint on Titanium [78] which can be satisfied for sfermion masses of order 10 TeV or more.
• µ → eγ: following again [67] 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 there are no long-lived states in the MSSM spectrum, so the only DM candidate is a slowly decaying gravitino. For m 3/2 < m Z the main decay channel of the gravitino is [80] 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. (3.25) 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 [80]  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 [79]. 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 [80]   which is compatible with the bounds coming from diffuse X-and gamma-ray fluxes (see figure 1 in [81], [82,83]), 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 [84]. Notice, also, that there are no observational constraints (from Big Bang Nucleosynthesis or CMB) on the decay of the next-to-lightest supersymmetric particle, due to its fast decay via large RPV interaction.
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. [85][86][87][88]) 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. 3 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) [89][90][91] is able to derive exact R-parity conservation [92][93][94] at low energies [95][96][97], while there is no reason to assume JHEP07(2015)123 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 [13]. The prize to pay are three classes of fine-tuning: i) a generalized doublet-triplet splitting (cf. eqs. (2.9)-(2.10)), ii) the vanishing of the baryonic RPV couplings λ in eq. (2.53) and iii) the suppression of neutrino masses in eq. (3.25).
Is relation (3.25) between gaugino masses a prediction of the theory? Since it gets corrections at higher loops, eq. (3.28) 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. (3.24), i.e. δm 11 1 10 × µ 2 1 v 2 cos 2 β 4(µ 2 0 + µ 2 1 ) 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 [98], 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 (3.28), 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 . For larger M 2 , there is still a strong correlation between M 1 and M 2 , but other parameters get involved too, so it is harder to make a definite statement of what to look for. But if eq. (4.3) is valid, the apparently weak point of the neutrino mass becomes a strong one, and the theory is falsifiable through a future experimental check of eq. (4.1). Suppose now that M 2 satisfies eq. (4.3). Is there any obvious theoretical reason why would eq. (4.1) hold? In other words, can one find a susy breaking and mediation mechanism which leads to it at least at the one-loop level? A natural candidate would be gauge mediation. The change in sign of the bino mass compared to wino mass can be obtained only by a combination of gauge messengers (which contribute negatively) with chiral messengers (which contribute positively). A naive simple computation shows that if an SU(5) adjoint breaks susy like for example in [99,100] 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. A half-integer ∆b chiral seems impossible to obtain: a complex representation needs always to come in pairs to be vectorlike 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 [102] 4 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), 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.
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.