A hybrid type I + III inverse seesaw mechanism in U(1)R−L-symmetric MSSM

We show that, in a U(1)R−L-symmetric supersymmetric model, the pseudo-Dirac bino and wino can give rise to three light neutrino masses through effective operators, generated at the messenger scale between a SUSY breaking hidden sector and the visible sector. The neutrino-bino/wino mixing follows a hybrid type I+III inverse seesaw pattern. The light neutrino masses are governed by the ratio of the U(1)R−L-breaking gravitino mass, m3/2, and the messenger scale ΛM. The charged component of the SU(2)L-triplet, here the lightest charginos, mix with the charged leptons and generate flavor-changing neutral currents at tree level. We find that resulting lepton flavor violating observables yield a lower bound on the messenger scale, ΛM ≳ (500 − 1000) TeV for a simplified hybrid mixing scenario. We identify interesting mixing structures for certain U(1)R−L-breaking singlino/tripletino Majorana masses. For example, in some parameter regimes, bino or wino has no mixing with the electron neutrino. We also describe the rich collider phenomenology expected in this neutrino-mass generation mechanism.


Introduction
One of the triumphs of contemporary physics is the observation of neutrino oscillations, which provides direct evidence that at least two of the neutrinos in the Standard Model (SM) have mass.Current efforts of the neutrino physics experiments yield the neutrino oscillation observables to be [1][2][3] ∆m 2  12 = 7.41 +0.21 −0.20 × 10 −5 eV 2 , |∆m 2 13 | = 2.507 +0.026 −0.027 × 10 −3 eV 2 , sin 2 θ 12 = 0.303 +0.012  −0.012 , sin 2 θ 23 = 0.451 +0.019 −0.016 , sin 2 θ 13 = 0.02225 +0.00056 −0.00059 . (1.1) It is well-known that explaining the origin of neutrino masses requires going beyond the SM.Thus, it is reasonable to seek a natural way to explain such a phenomenon.Over the past several decades, a plethora of mechanisms have been introduced to explain neutrino masses.By far, the most popular approach is the seesaw mechanism where the light neutrino masses are inversely proportional to a heavy scale associated with a right-handed (RH) neutrino mass (e.g.see [4][5][6] and the references therein).Specifically, in the type I seesaw mechanism, the RH neutrinos are SM singlets, while in the type III seesaw mechanism, they are SU (2) L -triplets.On the other hand, in the inverse-seesaw (ISS) mechanism, the RH neutrinos are pseudo-Dirac particles and the light neutrino masses are proportional to a small Majorana mass [7][8][9].In ISS, lepton number is approximately conserved, broken only by the small Majorana masses of the RH neutrinos, providing a natural way to explain the lightness of the SM neutrinos.
In [10] it was shown that pseudo-Dirac binos in a U (1) R -symmetric minimal supersymmetric SM (MSSM) can act like RH neutrinos and generate the light neutrino masses.
The pseudo-Dirac bino, called biνo , is an SM singlet;hence, this mechanism is equivalent to a type I ISS scenario.In this scenario, the small Majorana bino mass is proportional to the gravitino mass m 3/2 , which breaks the U (1) R global symmetry.In turn, the light neutrino masses are proportional to m 3/2 v2 /Λ 2 M , where v is the Higgs vacuum expectation value (vev) and Λ M is the messenger scale between a SUSY-breaking hidden sector and the visible sector.(The model is described in more detail in the next section.)Importantly, in [10] the lightest neutrino is predicted to be massless since biνo has only two degrees of freedom.
In U (1) R -symmetric MSSM, all gauginos are pseudo-Dirac fermions.In this paper, we investigate the scenario where the wino, as an SU (2) L -triplet, is also involved in neutrinomass generation.In this way, U (1) R -symmetric MSSM naturally gives rise to a hybrid type I+III ISS mechanism. 1In addition to strongly motivating a hybrid texture for the neutrinomass generation, this extension makes it possible to give mass to all three neutrinos and provides a much richer phenomenology at the LHC, which we discuss here.
The organization of this paper is as follows.In Section 2, we summarize the relevant parts of the U (1) R−L -symmetric MSSM.In Section 3, we explain the neutrino-mass generation mechanism in detail.In Section 4, we derive the low energy constraints on this scenario coming from lepton-flavor violating observables.In Section 5, we discuss the interesting phenomenology expected to be seen at the LHC.We conclude in Section 6.

Model
In this work, we study a U (1) R -symmetric SUSY model.In these models, a global U (1) R symmetry is imposed on the supersymmetric sector such that the superpartners have +1 R-charges while the SM fields are neutral under U (1) R .In contrast to MSSM, due to this charge assignment, gauginos in U (1) R -symmetric SUSY cannot be Majorana particles.Dirac gaugino masses, on the other hand, can be generated by introducing three adjoint superfields with −1 R-charges: a hypercharge singlet S, an SU (2) L triplet T , and an SU (3) c octet O [13].Furthermore, U (1) R symmetry also forbids the Higgsino mass terms.In order to give mass to Higgsinos, two inert doublets R u,d are introduced. 2(Relevant superfields and their charge assignments are given in Table 1.)The terms that are forbidden by the U (1) R symmetry in these models help to alleviate the SUSY flavor and CP problems [16].Furthermore, heavier stops require less Higgs fine-tuning in SUSY models with U (1) R symmetry.This is especially important given the stringent LHC limits on stops and gluinos, which negate the MSSM solution to the hierarchy problem [17,18].
In order to generate the neutrino masses in a minimal way in these SUSY models, we extend the U (1) R symmetry to U (1) R−L , where L is the lepton number.(See Table 1.)In this case, the SM leptons are charged under the global U (1) R−L , which allows the mixing Table 1: The relevant superfield content of the model and their charge assignments.The R−L charge is defined as the U (1) R -charge minus the lepton number of the field.L i , E c i are the lepton superfields and the subindex i indicates the fermion generation.The fermionic components of the superfields R u,d are the Dirac partners of the Higgsinos H u,d .Φ S,T are superfields that have the same SM charges as W α B, W and their fermionic components, S, T are the Dirac partners of the bino and the wino, respectively.ϕ is the conformal compensator and W ′ α is a D-term spurion field.
between the neutralinos and the active neutrinos.(see also [19,20] for a similar approach.)Therefore, the U (1) R−L symmetry can provide a natural mechanism for neutrino mass generation.
We incorporate SUSY breaking via F -and D-terms.We consider two independent sectors that break SUSY.The first sector, √ F 1 , D 1 ∼ 10 TeV, is coupled to the superfields whereas the second sector, √ F 2 , D 2 ≳ 10 4 TeV, is coupled to the SM only through gravity.(These scales are motivated by the resulting phenomenology [21].)SUSY breaking is communicated to the visible sector at a messenger scale Λ M .As shown in [13], supersoft SUSY breaking, i.e. emergence of Dirac gauginos, is incorporated by the spurion field W ′ α = θ α D, where D = ⟨W ′ α ⟩ is the SUSY-breaking vev of a D-term spurion field.Therefore, Dirac masses of the gauginos are generated through the following supersoft operator by integrating out the messenger field at a scale Λ M [13], where c j is a dimensionless coefficient that we take to be O(1), Φ η j is the chiral superfield whose fermionic component is the Dirac partner of the relevant gaugino χ j , and indexj ensures the correct gaugino-Dirac partner pair.In particular, χ j and η j are the Weyl components of the Dirac field ψ T j = ( χ j , η † j ) T .We assume that scalar adjoints only receive a finite soft mass from the operators in Equation (2.1) at one loop (for details, see, e.g., [13]).We note that although supersoft terms of the form W ′ α W ′α Φ η j Φ η j will generally contribute to the masses of the scalar adjoints, these soft terms can be forbidden [22].
As with all global symmetries, U (1) R−L is broken due to gravity.Consequently, Majorana masses of the gauginos are generated via anomaly mediation [23][24][25], where β(g χ j ) are the beta functions of the relevant SM gauge coupling g χ j , and m 3/2 is the gravitino mass.The gravitino picks up mass from all sources of SUSY breaking, Pl where M Pl is the Planck mass and F 2 i = F 2 i + D 2 i /2.We emphasize that U (1) R−L is approximately conserved as long as the gravitino m 3/2 is light.Hence, we assume the messenger scale Λ M is below the Planck scale, Λ M ≪ M Pl which leads to a large hierarchy between Dirac and Majorana gaugino masses, m χ j ≪ M χ j .Furthermore, U (1) Rbreaking Majorana masses for the Dirac partners could also be generated.We assume that the Majorana masses of the Dirac partners are also proportional to the gravitino mass and can be parameterized as where κ S and κ T are dimensionless coefficients.Thus, the pair ψ T j = ( χ j , η † j ) T becomes a pseudo-Dirac fermion.We will drop the index-j and refer to pseudo-Dirac bino-singlino and wino-tripletino pairs as biνo and wiνo as they will play a role in neutrino mass generation and mixing.Gluino and octino will be ignored since they are not relevant to this work.
After electroweak symmetry breaking (EWSB), adjoint fermions S and T participate in both neutralino and chargino mixing due to the presence of U (1) R symmetry.The relevant part of the superpotential that accounts for both neutralino and chargino mixing is given as [26] We work in the large tan β ≡ v u /v d limit, v d → 0, where v u,d ≡ ⟨H 0 u,d ⟩ are the up/downtype Higgs vevs with v 2 u + v 2 d = v 2 /2 ≃ (174 GeV) 2 .In this limit, Equation (2.4) generates the neutralino mixing matrix One state with mass µ d is already decoupled for v d → 0. We further assume λ u B, W = 0 such that biνo, wiνo and Higgsinos do not mix.Since possible mixing of neutralinos does not bring in any new degrees of freedom to the system, this assumption does not affect the neutrino mass generation scenario described in the next section.Neutralino mixing, however, is important for LHC phenomenology as discussed in Section 5. Similarly, for only LHC phenomenology purposes, we will assume a hierarchy µ u ≈ µ d > M B, W and take the Higgsinos to be decoupled in the mass spectrum.This assumption is less motivated theoretically, as naturalness favors light Higgsinos.
Under the assumptions given above, the chargino mixing matrix is also almost diagonal, in the basis ( We assume that the lightest charginos χ ± 1 are purely composed of charged weakino states and that they are degenerate with the wiνo.In the next section, we will describe a type III ISS mechanism where the wiνo acts like an RH triplet.This will introduce mixing between weak charginos and charged leptons, which we will discuss in Section 4.

Neutrino masses
The U (1) R−L -symmetric MSSM has a pseudo-Dirac SM-singlet fermion, namely the biνo, and a pseudo-Dirac SU (2) L -triplet fermion, the wiνo.In [10], it was shown that the pseudo-Dirac biνo could generate neutrino masses through an ISS mechanism.In this minimal scenario, the lightest neutrino is predicted to be massless.We extend this framework to include the wiνo in generating the neutrino masses.These new degrees of freedom allow for all three neutrinos to be massive, open up more parameter space of interest, and provide a richer phenomenology at the LHC.
Building on [10], we introduce the following U (1) R−L -conserving, dimension-6 operators: ) where f i B, W are dimensionless coefficients and i = e, µ, τ . 3In addition the U (1) R−Lconserving terms above, we also introduce the following U (1) R−L -breaking, dimension-5 terms: where d i S/T are dimensionless coefficients and ϕ = 1+θ 2 m 3/2 is the conformal compensator.While the operators in Equation (3.1) explicitly violate R-parity and lepton number, they conserve U (1) R−L .Moreover, the U (1) R−L -violating terms in Equation (3.2) are highly suppressed compared to the U (1) R−L -conserving terms because m 3/2 ≪ M B/ W .We note in passing that although the effective operators in Equation (3.2) are introduced for completeness, unlike the original scenario, they are not required to generate the correct neutrino spectrum.We will discuss the details shortly.
After EWSB, the interactions in Equations (3.1) and (3.2) generate mixing between neutrinos, biνo and wiνo.We emphasize that the mixing of active neutrinos and biνo has a type I ISS texture, whereas the mixing follows a type III ISS pattern for the wiνo case. 4ence, this is a natural scenario that gives rise to a hybrid type I+III ISS mechanism in explaining the neutrino masses.
Using the prescription given in [28], neutrino mass matrix in the (ν i , B, W , S, T ) basis after EWSB is where Here u B, W and v S,T are unit vectors, and the coefficients y B, W and g S,T read M B, W /Λ M and m 3/2 /Λ M , respectively.We ignore the CP -violating phases that will emerge in Equation (3.3) as they will not affect our discussion.The elements of this mass matrix is highly hierarchical, , solely due to the approximate U (1) R−L symmetry, determined by the hierarchy m 3/2 ≪ M B, W .Note that in the end, the Dirac gaugino masses will not be relevant for the neutrino masses.
In its most general form, the mass matrix in Equation (3.3) generates three massive light neutrinos with the correct mass splittings to match observations.Due to the numerous free parameters involved, it is not possible to analytically find the eigenvalues for the most general case.Therefore, we will focus on a simplified scenario as described below.
Scenario 1: We can first ignore the Majorana masses for the singlino and tripletino, setting m S = m T = 0.In this case, the relevant Wilson coefficient for the dimension-5 Weinberg operator is One can set either one of Y B , G S , Y W , G T to zero and still have two massive neutrinos with the correct mass splittings.Another simple case that results in one massless neutrino is when u B ∝ u W and/or v B ∝ v W . (The light neutrino masses in these two cases are identical to [10].)This kind of linear dependence might point towards the UV completion of the terms introduced in Equations (3.1) and (3.2).We emphasize that, in its most general form, this scenario gives three massive neutrinos.

Scenario 2:
We can ignore the U (1) R−L -violating, dimension-5 terms given in Equation (3.2), setting G S,T = 0. Then the relevant Wilson coefficient becomes In this scenario, there is always one massless neutrino.Furthermore, the limit where u B ∝ u W results in two massless neutrinos and is not physical.We analyze Scenario 2 in further detail because it is analytically solvable and is fairly distinct from what has been studied in [10].(See Appendix A.) Setting G S,T = 0, light neutrino masses in normal ordering (NO) are where m 2 < m 3 and the parameter β is set by the mass-squared splitting ratios r, and have the following form for both mass ordering scenarios: to zero for m T /m S ∼ 1.8 and ∼ 3.5 respectively.These sorts of cancellations can be interesting for future model building if, for example, one desires a sterile neutrino without significant mixing with the electron neutrino [34].This behavior will also affect the LHC signatures, which will be discussed in Section 5.The gray dashed vertical line corresponds to m T = m S , which effectively reproduces the scenario introduced in [10] with the identification Y W → G in Equation ( 9) of [10].We note that there is an ambiguity between what is called biνo or wiνo in these vectors.Namely, changing m S ↔ m T and u B ↔ u W leaves the neutrino phenomenology unchanged.See Appendix A for details.

Low-energy Constraints
The mixing of biνo and wiνo with light neutrinos can result in observable lepton-flavorviolating (LFV) effects, which can be constrained by (non-)observations.This is particularly important in this work since wiνo is an SU (The lightest neutrino is massless in this simplified scenario.)(Right) Constraints on the messenger scale Λ M for varying Majorana mass ratios m T /m S for normal ordering.The purple shaded region is ruled out by the current µ → eγ searches, Br(µ → eγ) < 4.2×10 −13 [29].The dashed purple line shows a projected limit for Br(µ → eγ) < 10 −14 [30].The blue shaded region is excluded by Br(µ → eee) < 1.0 × 10 −12 [31].The strongest constraints come from µ → e conversion in gold, R µ→e < 7 × 10 −13 [32], shown in orange.The orange dashed line shows the projected limit for R µ→e < 6.2 × 10 −16 expected from Mu2e [30,33] using aluminum.See text for details.For these plots, central values in Equation (1.1) were used.charginos and charged leptons, inducing flavor-changing neutral and charged currents at tree level.Here, we summarize the most important constraints in this model.(See Figure 3 for the processes we consider.) Following the framework detailed in [27,35], one can constrain the (dimensionless) coefficient of the dimension-6 Weinberg operator, where the vectors Y B, W are defined Equation (3.3) and their numerical values are given in Figure 2. Note that while the dimension-5 operators that generate the light neutrino masses are suppressed by the small Majorana masses, the coefficient above is not necessarily suppressed, as expected in an ISS texture.Furthermore, since M is independent of the Dirac biνo and wiνo masses.Consequently, the constraints will be only on the SUSY messenger scale Λ M .
Detailed calculations of the constraints on type III seesaw models can be found in, e.g., [27,35].By far the strongest constraints are on the e − µ element of Equation (4.1):  1.In this hybrid type I+III model, as in the pure type I case introduced in [10], µ → eγ decay proceeds at one loop.The most recent limit on the branching fraction comes from MEG [29], Br(µ → eγ) < 4.2 × 10 −13 .MEG II is expected to lower this limit to Br(µ → eγ) < 10 −14 .These translate into the constraints: These limits are shown as the purple shaded region and the purple dashed line in the right panel of Figure 1.
2. Chargino mixing with the charged leptons generates an effective Z−µ−e vertex at tree level proportional to ϵ d=6 eµ , which then allows for the tree level µ → eee decay.The branching fraction of this process is constrained to be Br(µ → eee) < 1.0×10 −12 [31], which gives This constraint is shown as the blue shaded region in the right panel of Figure 1.
It can be seen that this constraint will still be better than the forecasted µ → eγ constraints in the near future.
3. The strongest constraint on the messenger scale comes from µ → e conversion in nuclei.In contrast to the type I seesaw, here this process happens at tree level, facilitated by the same Z−µ−e vertex.The current best limit on this rate comes from the SINDRUM experiment using muonic gold: R µe < 7 × 10 −13 [32].The proposed Mu2e experiment is expected to lower this limit by three orders of magnitude to R µe < 6.2 × 10 −16 [33].The corresponding constraints are shown as the shaded orange region and the orange dashed line, respectively, in Figure 1.
Overall, the constraints in this scenario require Λ M ≳ 500 TeV, which is much stronger than the limits found in [10].This is due to the flavor-changing neutral currents (FCNC) induced by the chargino-charged lepton mixing at tree level.Note that every order of magnitude improvement in measurements raises the bound on Λ M by a factor of 10 1/4 ∼ 1.8 since the rates for these processes are proportional to 1/Λ 4 M .

LHC Phenomenology
In most of the generic type I seesaw models, the SM-singlet RH neutrinos are not produced efficiently at the LHC since the only relevant interactions are generated through neutrino mixing.In contrast, the comparable minimal scenario introduced in [10] has a rich collider phenomenology.In this model, The SM-singlet RH neutrino is a bino, which can be produced in squark decays.Furthermore, the lightest neutralino, which is assumed to be pure bino, decays to SM particles via its mixing with the neutrinos.The lifetime depends on the bino mass and the SUSY messenger scale.These features generate unique signatures at ATLAS and CMS, as well as at proposed experiments like MATHUSLA, CODEX-b, and SHiP [21,36,37].
On the other hand, type III seesaw models employ EW triplets, which could have observable collider signatures via their EW interactions.The scenario we consider here automatically gives rise to a hybrid type I+III texture, where the SM-singlet RH neutrino is the bino and the EW-triplet RH neutrino is the wino.In this section, we comment on the LHC phenomenology of this scenario.
As described in the previous section, low-energy LFV constraints require a messenger scale Λ M ∼ O(1000 TeV) or larger.Consequently, to account for neutrino masses, the singlet/triplet Majorana masses require to be O(10 MeV).Unlike the mechanism introduced in [10], neutrino masses do not directly translate into a requirement on the gravitino mass.However, we still expect the U (1) R−L -breaking singlet/triplet Majorana masses to be proportional to the gravitino mass.We assume this proportionality holds up to an O(1) constant.Therefore, the gravitino mass is expected to be around O(10 MeV) as well.We assume that the gravitino is the lightest supersymmetric particle (LSP) and the two of the lightest neutralinos are purely biνo-and wiνo-like.(In contrast to [21], here wiνo is not decoupled.)Since we assume an approximately conserved U (1) R symmetry, the Majorana masses are much smaller than the Dirac masses for the gauginos.Hence we take the biνo and wiνo masses to be M B , M W respectively.We take charginos to be degenerate with the wiνo.We consider gluinos to be decoupled, but sfermions need not be decoupled.The mass spectrum we assume is shown in Figure 4.

Biνo phenomenology
When gluino is decoupled, but squarks are accessible at the LHC, biνo will be produced in squark decays in pair production of squarks with the cross section σ B, B † = Br(q → Bq) 2 × σ(pp → q q † ) . (5.1) If biνo is heavier than ∼ 100 GeV, after being produced, it will decay to on-shell leptons, gauge bosons, and the Higgs through its mixing with the light neutrinos, as well to a gravitino and a photon, B → Gγ.This last process is suppressed by M 2 Pl and is not relevant for colliders.The biνo lifetime Γ B ∼ i MeV) for a messenger scale Λ M ∼ 100 TeV.Hence, unlike generic MSSM scenarios, biνo, which is the lightest neutralino, decays promptly at the LHC. 5iνo signals at the LHC include final states with jets, leptons, and missing energy.In fact, jets + / E T final states have the highest branching fraction with ∼ 25%.In the previous LHC study of the scenario where neutrino masses are generated via solely biνo interactions [21], the wiνo was decoupled, leading to Br(q → Bq) = 100%.Relevant LHC searches with √ s = 13 TeV and L = 36 fb −1 were recast to constrain the biνo-squark parameter space.
It was shown that squarks as light as 350 GeV are allowed for M B < 150 GeV and squarks above 950 GeV were not constrained.
When wiνo mass is lighther than squarks, squark decays to biνos can effectively be ignored, and the analysis in [21] becomes irrelevant.The biνo production in wiνo and chargino decays explained below will be important in this scenario.

Wiνo/chargino phenomenology
Since they are SU (2) L -triplets, wiνo and the charginos give rise to rich phenomenology at the LHC.Here we will highlight various scenarios to emphasize the novel signals of this type I+III inverse seesaw mechanism at the LHC.
If the squarks are accessible at the LHC, wiνo and charginos are produced in their decays together with a quark.Charginos decay to W ± B while wiνo decays to Z B, assuming kinematically allowed.In R-parity conserving MSSM, the lightest neutralino is stable and contributes to the signal as missing energy.Final states where gauge bosons decay hadronically and/or leptonically are considered depending on the search strategy, e.g.[17,[38][39][40][41].When R-parity violation is allowed, biνo can decay to a 3-body final state of leptons or quarks.We assume charginos are degenerate with the wiνo and that M B < M W + M Z such that wiνo to biνo decays are kinematically allowed.
There are a few distinct features of the model we introduce here.
• In addition to its normal decays to Z B, wiνo can decay via mixing with the neutrinos; W → W + ℓ − , Zν, hν.However, these channels are suppressed by the mixing angle θ 2 ∼ (y W v/M W ) 2 ∼ v 2 /Λ 2 M ∼ 10 −6 compared to the Z B channel.(See Figure 5.) • Charginos can decay through the Lagrangian terms in Equation (3.1), such as χ ± 1 → Zℓ ± induced by mixing with charged leptons.However, these mixing-induced decays are suppressed by M ∼ 10 −6 and are irrelevant when biνo is the lightest neutralino, as χ ± 1 → W ± B decays dominate (see Figure 6).More important are flavor-violating Z boson decays, e.g.Z → µe which is induced at tree level via this mixing.
• Since biνo decays promptly via the decay channels given in Equation (5.For biνo LSP, the lightest chargino χ ± 1 will primarily decay to biνo+W ± .However, charginos can also decay via their mixing with the charged leptons. Ultimately, the signals can be similar to those expected from R-parity violating (RPV) decays, for example, through the λ ijk L i L j E c k term.(In RPV SUSY one can have χ 0 1 → ℓℓν after sleptons are integrated out compared with B → Z(→ ℓℓ) ν in this model.)Such a search has been carried out by ATLAS [42] with charged lepton final states.Wino neutralino is ruled out for m(χ 0 2 ) > 1.6 TeV for bino mass m(χ 0 1 ) > 800 GeV with an integrated luminosity of 139 fb −1 .We point out that although the final states are similar, the results of this and similar searches do not directly apply to our model.Firstly, due to the U (1) R−L symmetry, SUSY particle production cross-sections are generally small compared to MSSM.Furthermore, when biνo is heavier than the gauge bosons, the final state momentum distribution will be different than what is expected from 3-body decays in RPV SUSY.In fact, requiring the charged lepton pairs to reconstruct to W/Z bosons could provide a smoking-gun signal for this scenario.Another important distinction is that the branching fractions to different lepton families (e, µ, τ ) are determined by the observed neutrino mixing structure, as shown in Figure 2.

Wiνo as the lightest neutralino
If wiνo is lighter than biνo, wiνo and charginos will only decay through the mixings induced by Equation (3.2) and explained in the previous section.(See also Figures 5 and 6.)This scenario exactly matches the LHC phenomenology expected from a gauged U (1) B−L MSSM described in [43].In this case, there is an ATLAS search for trilepton resonances [44] with an integrated luminosity of 139 fb −1 which directly applies to our model.(A decay channel would be χ ± 1 → ℓ ± Z(→ ℓ ± ℓ ∓ ) as shown in Figure 7.)This search excludes wino/chargino masses between 100 GeV and 1.1 TeV, depending on their branching fraction to different lepton flavors, which is taken to be a free parameter for the analysis.In this search, the electron and muon final states are the most constraining.This observation is interesting when considering the wiνo-lepton mixing parameters shown in Figure 2.For particular ratios of the singlet/triplet Majorana masses, m T /m S , it could be that chargino has no mixing with the electron while it mixes mostly with the tau, alleviating the constraints from this search.

Conclusion
U (1) R -symmetric MSSM conveniently alleviates the hierarchy problem without light stops.This is necessary given the stringent LHC constraints on stops and gluinos.When the global U (1) R symmetry is approximately broken by the gravitino mass, gauginos are expected to be pseudo-Dirac fermions with both Majorana and Dirac masses.We employed the pseudo-Dirac nature of the bino and the wino to generate light neutrino masses.To generate the neutrino masses, we extended the global symmetry to U (1) R−L , and we introduced U (1) R−L -conserving dim-5 operators and U (1) R−L -violating dim-6 effective operators suppressed the messenger scale Λ M .We found that this model can generate non-zero masses for all three neutrinos in its most general form.(This can be contrasted with the pure bino case in [10], where the lightest neutralino was predicted to be massless.)We then focused on a simplified scenario in which the only lepton-number violating terms are Majorana masses of the Dirac partners of the bino and the wino, m S and m T respectively.Then, the light neutrino masses are proportional to (m S + m T )v 2 /Λ 2 M .The Majorana masses, which break U (1) R−L , are expected to be proportional to the gravitino mass.The hierarchy between the gravitino mass and the messenger scale can explain the smallness of the neutrino masses.Interestingly, we found that for certain values of m S /m T , the bino or the wino does not mix with certain neutrino flavors.This property could be applied to other model-building scenarios where one needs a sterile neutrino to preferentially mix with, for example, tau neutrinos.Further work can be done by turning on all sources of lepton-number violation.In this most general case, a numerical scan of the large parameter space is needed.
The bino is a singlet, and the wino is a triplet under SU (2) L .Consequently, the model described here generates a hybrid type I+III (inverse) seesaw mechanism.The wiνoneutrino mixing term also causes mixing between charginos and charged leptons, leading to tree-level FCNCs.We calculated the constraints coming from µ → eγ, µ → eee and µ → e conversion in nuclei and found that Λ M > (500 − 1000) TeV for m T /m S ∼ 0.2 − 5. We also showed that future µ → e conversion experiments like Mu2e [33] can probe a messenger scale around 5000 TeV.
Finally, we discussed the rich LHC phenomenology expected from this model of neutrino mass generation.We assumed two mass orderings in the neutralino sector. 1) If the lightest neutralino is a pure bino and the next-to-lightest neutralino is a pure wino, winos and charginos will decay to a bino and a gauge boson.In this case, the bino is short-lived and decays to leptons and a gauge boson or the Higgs.Hence, this model can generate final states with four electroweak gauge bosons.2) If the wino is the lightest neutralino, it can directly decay into leptons, gauge bosons, or Higgs via its mixing with the neutrinos.The branching fractions to specific leptons are predicted by the neutrino mixing parameters.and it is set by the mass-squared splitting ratios, Equation (3.8).We assume the massive eigenstates have the form, ê± = N ± (a ± u B + b ± u W ), (A.5) where the coefficients a ± , b ± and N ± can be determined by solving the eigenvalue problem, O † Oê ± = m 2 ± ê± : The entries in the PMNS matrix fix the mass eigenstates to accommodate the correct mixing structure.Hence, the vectors u i B, W in the normal ordering are found to be The behavior of the vectors u i B, W is given in Figure 2 as a function of m S /m T = κ S /κ T .

β
NO = −2r(r + 1) + r(r + 1)(2r + 1) ≃ 0.13 .(3.8)We expect the singlet/triplet Majorana masses m S,T to be proportional to the gravitino mass m 3/2 up to O(1) factors when U (1) R−L symmetry is approximately broken.We show in Section 4 that low energy constraints require Λ M ≳ 500 TeV, which translates to m S + m T ∼ O(MeV).See Figure 1.For a given Majorana mass ratio m T /m S , the PMNS matrix fixes the vectors Y B, W = y B, W u B, W up to the overall factors y B, W = M B, W /Λ M .We show the mixing coefficients u i B, W , for i = e, µ, τ , in Figure 2 as a function of m T /m S for normal ordering.There are accidental cancellations which drive u e W and u τ B

Figure 2 :
Figure 2: The mixing matrix elements between light neutrinos and biνo/wiνo, as defined in Equation (3.3), in normal ordering.These are fixed by the PMNS matrix for a given value of m T /m S where the bands correspond to varying the mixing angles within their 1σ measured values given in Equation (1.1).The case where m T = m S matches the pure biνo scenario introduced in [10].There are accidental cancellations which drive u e W and u τ B to zero for m T /m S ∼ 1.8 and ∼ 3.5 respectively.

Figure 3 :
Figure 3: Feynman diagrams for the lepton-flavor violating processes that dominantly constrain the messenger scale in our model.(a) µ → eγ at one-loop via B/ W -neutrino mixing, (b) µ − e conversion in nuclei and (c) µ → eee at tree level via chargino-charged lepton mixing.

Figure 4 :
Figure 4: An example particle spectrum of the model described in Section 2.

Figure 5 :Figure 6 :
Figure 5: If kinematically allowed, wiνo can decay to either a biνo or leptons and gauge bosons via mixing with the neutrinos.These latter decays are highly suppressed by the mixing angle.

Figure 7 :
Figure 7: Representitive LHC final states with leptons and jets.(a) LSP wiνo pair production with decays mediated by W ± boson.(b) LSP chargino pair production with decays mediated by Z boson and final states are associated with a weak gauge/Higgs boson and a lepton.(c) wiνo and chargino pair production with decays mediated by LSP biνo and weak gauge bosons.