Bilinear R-parity violation with flavor symmetry

Bilinear R-parity violation (BRPV) provides the simplest intrinsically supersymmetric neutrino mass generation scheme. While neutrino mixing parameters can be probed in high energy accelerators, they are unfortunately not predicted by the theory. Here we propose a model based on the discrete flavor symmetry $A_4$ with a single R-parity violating parameter, leading to (i) correct Cabbibo mixing given by the Gatto-Sartori-Tonin formula, and a successful unification-like b-tau mass relation, and (ii) a correlation between the lepton mixing angles $\theta_{13}$ and $\theta_{23}$ in agreement with recent neutrino oscillation data, as well as a (nearly) massless neutrino, leading to absence of neutrinoless double beta decay.


Introduction
Significant progress has recently been made at the Large Hadron Collider (LHC), with the discovery of a new scalar state in the 125 GeV mass region [1,2]. Although a conclusive identification is still not possible, the properties of this new state resemble very much those expected for the long-awaited Higgs boson. This already constitutes one of the most important discoveries of modern physics and represents an incredible success for a 50-year old theory.
If this new scalar state were confirmed to be the Higgs boson, we would know that the Standard Model (SM) is indeed the correct effective description of elementary particles at least up to a scale which we still ignore. Measuring the exact Higgs boson mass would be crucial to know up to which scale the Higgs boson scalar potential is stable, or in other words at which scale we should expect new physics to emerge [3]. The hierarchy problem associated to the Higgs boson mass has suggested that new physics should appear around the TeV scale. Since the most promising extension of the SM to address the hierarchy problem is supersymmetry (SUSY), we expected that the stop or the gluino should be around the corner. However the first searches up to ∼ 5f b −1 at the LHC [4]  Open issues in this regard are the precise mechanism of SUSY breaking and whether Rparity is conserved. Indeed, supersymmetry may well be broken by a non-gravitational messenger. Similarly, one can have supersymmetry without R-parity [5,6]. Hence the need to consider alternative scenarios [7] where, in addition, the stringent bounds on the squark and gluino masses are relaxed [8,9].
Apart from stabilizing the Higgs boson scalar potential, supersymmetry could address other Standard Model puzzles for which new physics is invoked. Among these we have that supersymmetry might explain the origin of neutrino masses as well as cold dark matter.
Regarding the latter it has recently been shown that a relatively light gravitino in the few GeV range can provide a perfectly valid and interesting alternative in broken R-parity models [10]. Moreover, it provides a testable minimal mechanism for the origin of neutrino masses [11]. Regarding neutrinos it is well-known that bilinear R-parity violation offers a simple way to generate neutrino masses in supersymmetry [11]. In its "generic" formulation the model can not address issues associated to fermion mass hierarchies and mixings, such as those of neutrinos. Both Abelian [12] and non-Abelian [13] flavor symmetries have been used in the literature to constrain the R-parity violating terms. In this letter we propose a flavored version of bilinear R-parity violation. The model has a single supersymmetric Rparity violating parameter allowed by the flavor symmetry A 4 × Z 2 , where A 4 is the group of even permutations of four objects. This R-parity violating term is used to generate neutrino masses as required by current oscillation data, see [14]. We obtain predictions for the charged fermion masses as well as for neutrinos.
The paper is organized as follows: in section 2 we briefly review neutrino mass generation through low-scale supersymmetry with bilinear R-parity violation [15] and in section 3 we extend it by implementing a discrete flavor symmetry. In section 4 we present our results in the neutrino sector, where we find a correlation between the lepton mixing angles θ 13 and θ 23 . In section 5 we comment on the scalar potential and finally in section 6 we summarize the main predictions of the model.

Bilinear R-parity violation
Bilinear R-parity Violation [5,15] is the minimal extension of the Minimal Supersymmetric Standard Model (MSSM) that incorporates lepton number violation, providing a simple way to accommodate neutrino masses in supersymmetry. The superpotential is (2.1) The three i = ( e , µ , τ ) parameters have dimensions of mass and explicitly break lepton number. Their origin (and size) can be naturally explained in extended models where the breaking of lepton number is spontaneous [16]. In that sense, BRPV can be seen as an effective description of a more general supersymmetric framework for lepton number violation. In any case, the i parameters are constrained to be small ( i m W ) in order to account for the small neutrino masses. Furthermore, the presence of the new superpotential terms implies new soft SUSY breaking terms as well where the B i parameters have dimensions of mass squared. The i and B i couplings induce vacuum expectation values (VEVs) for the sneutrinos, ν L ≡ v L i , proportional to the i , hence small, as required (we assume B i = B i ).
In the presence of BRPV couplings, neutrinos and neutralinos mix, giving rise to neutrino masses [17][18][19]. In the basis (ψ 0 ) T = (−iB 0 , −iW 0 3 , H 0 d , H 0 u , ν e , ν µ , ν τ ) the neutral fermion mass matrix M N is given by where M χ 0 is the usual neutralino mass matrix and is the matrix that characterizes the breaking of R-parity. Note that its elements are suppressed with respect to those in M χ 0 due to the smallness of the i parameters. Therefore, the resulting M N matrix has a type-I seesaw structure and the effective light neutrino mass matrix can be obtained with the usual formula m 0 ν = −m·M −1 χ 0 ·m T , which can be expanded to give where a (0) is a combination of SUSY parameters and are the so-called alignment parameters. The projective form of m 0 ν implies only one eigenvalue is non-zero. A natural choice is to ascribe this eigenvalue to the atmospheric scale. In this case the required solar mass scale, ∆m 2 sol ∆m 2 atm , arises radiatively, at the 1-loop level, correcting the tree-level neutrino mass matrix in Eq. (2.5). Detailed computations of the 1-loop contributions to the neutrino mass matrix can be found in Refs. [17,18]. The corrections are of the type where the coefficients a (1) , b (1) , c (1) are complicated functions of the SUSY parameters.
This generates a second non-zero mass eigenstate associated with the solar scale, and the corresponding mixing angle θ 12 . Note that the neutrino mixing angles are determined as ratios of / R p parameters i and Λ i .
Let us say a few words about the phenomenology of BRPV. The breaking of R-parity has an immediate consequence at colliders: the LSP in no longer stable and decays typically inside the detectors. Since LSP decays and neutrino masses have a common origin, one can show that ratios of LSP decay branching ratios correlate with the neutrino mixing angles measured at low energies [20]. This establishes a tight link which allows one to use neutrino oscillation data to test the model at the LHC see e. g. [21].

The flavored BRpV model
Let us consider the MSSM particle content extended with one extra singlet superfieldŜ and a A 4 × Z 2 flavor symmetry with the assignments given in table 1. The superfieldŜ is required in order to generate the µ term, and is the only singlet under A 4 . On the other hand the Z 2 symmetry forbids all / R p operators with the only exception of the bilinear terms LH u , while the quark and charged lepton sectors are very similar to those in [22].
The assumption that all matter fields as well as the up and down Higgs doublets are in triplet representations of A 4 reduces the different BRPV parameters to only one. The superpotential of the model is where d-type quarks and charged fermions l couple to the same Higgs. Note that since all matter fields and Higgs scalars are in triplets of A 4 , the diagonal elements of the charged fermion mass matrices vanish. With the VEV alignment 1 we have in the charged fermion sector nine parameters, one of which can be reabsorbed.
These are used to fit nine masses and three mixing angles, hence four predictions emerge [22], given below as eqs.
From eq. (3.6) we can see the mass matrix for the charged fermions has only three free parameters which can be written as functions of the charged fermion masses. Now we can consider the squared mass matrix for the charged fermions, where we have assumed a f b f r f (see below). The invariants of this matrix give rise to three equations in terms of the fermion masses. From these one can find the parameters, a f , b f and r f as funtions of the charged fermion masses as [22,23] (3.10) From eqn. (3.10) we have the first prediction of the model, a quark-lepton mass relation: The second prediction is the Cabibbo angle, which follows from the fact that the matrix in eq. (3.7) is diagonalized by [23] This formula is simply the well-known Gatto-Sartori-Tonin relation [27]. Indeed from the matrix in eq. (3.7) one obtains for instance the V 12 mixing as which gives the famous Gatto-Sartori-Tonin relation.
Let us now turn to the neutrino sector. As already discussed in the previous section, the tree-level neutrino mass matrix in eq. (2.5) has rank one. However it is straightforward to show that with the VEV alignment in eqs.
where the Λ i defined in (2.6) now take the form with µ = λ S = λ v s / √ 2. Note that the i v d contributions that characterize "generic" where the following definitions have been made Note that the 1-loop contributions are dis-aligned with respect to the tree-level one, with the tree-level degeneracy lifted by radiative corrections. In the limit α = 0 the neutrino mass matrix has the massless eigenvector which corresponds to (2, −1, −1) T , called tri-bimaximal-1 for instance in Ref. [28]. In the limit of α = b = 0 the spectrum is tri-bimaximal.
Finally, in the limit b = c = 0 one recovers the tree-level mass matrix m 0 ν . This matrix has rank one, and thus only one eigenvalue is non-zero, m ν 3 = a| Λ| 2 = a(2 + α 2 ). The In conclusion, the neutrino mass spectrum is compatible with normal hierarchy, with a radiatively induced solar scale. The solar and atmospheric mass square differences as well as the solar mixing angle can be fitted as shown explicitly in Ref. [17].

Large θ 13 and deviations from maximal atmospheric mixing
In the CP-conserving case, the neutrino mass matrix in Eq.  Figure 3. Same as Fig. 1 for the analysis in Ref. [31].
From the presented correlations between the atmospheric and solar mixing angle, as well as the resulting allowed ranges of the reactor and atmospheric angle, it is clear that our model restricts the oscillation parameters in a non-trivial way, however consistent with the 1σ ranges for the neutrino oscillation parameters given by all global fits, in particular with the "large" reactor mixing angle and non-maximal atmospheric mixing hinted by the most recent oscillation data 5 . At 2 σ, predictions in our BRPV model become very weak.

Scalar potential and spectrum
The presence of additional Higgs doublets is a common feature to many flavor models and leads to a complicated structure of the scalar potential. Therefore, though non-trivial, the determination of a phenomenologically viable minimum of the potential is of fundamental importance. The scalar potential can be split as where V F and V D are the usual F-and D-terms and V A 4 soft , defined as contains soft SUSY breaking terms that preserve the flavor symmetry 6 . In addition, we allow for the existence of an additional piece, V / A 4 soft , with terms that break softly both SUSY and A 4 but preserve Z 2 . The presence of this piece is necessary in order to obtain a realistic spectrum. The A 4 × Z 2 flavor symmetry leads to accidental continuous symmetries in the scalar potential which, after they get spontaneously broken by the corresponding VEVs, imply the existence of additional Goldstone bosons. These massless states couple to the gauge bosons thus the explicit breaking of those continuous symmetries is required from a phenomenological point of view 7 .
We have considered the following soft breaking of A 4 V / for v d 3 and v u 2,3 , leading to analogous hierarchies for the corresponding soft squared masses 9 .
Finally, as discussed in section 2, the sneutrino VEVs are naturally small since these "induced VEVs" are proportional to the parameter. This can be seen in the corresponding tadpole equations, which imply that all v L i vanish in the = B = 0 limit.
The requirement of very large soft squared masses has important consequences on the mass spectrum. Dominated by the contributions from the soft terms, the spectrum contains several degenerate {H 0 , A 0 , H ± } sets, some of them with masses in the 10 − 100 TeV range.
This degeneracy is very strong in the case of H 0 and A 0 and only slightly broken for H ± due to its mixing with the charged sleptons.
Another interesting feature of the spectrum is the decrease of the mass of the scalar singlet for increasing tan β, where In fact, for tan β ∼ 10 one can easily find points in parameter space with a light scalar singlet, S 1 , in the range of 100 MeV -20 GeV. This result can be easily understood in a simple limit. Neglecting the mixing with the doublet states, the leading contribution to the squared mass of the scalar singlet is where v 2 = (246GeV) 2 = 4m 2 W /g 2 is the usual electroweak VEV. Eq. (5.8) shows that m 2 S 1 scales as 1/ tan β for sufficiently large tan β. This naturally leads to m 2  [32,33], it has been increasingly clear that the properties of the discovered particle are very close to those of a SM Higgs boson [34][35][36][37]. Although there is still room for new interactions [38], these are now constrained by the data. This imposes an important restriction on the size of our λ coupling, λ 1.
As a generic example to illustrate these properties, we provide the following parameter set and results for a particular but generic point in parameter space. • Neutrino physics can be accommodated with ∼ 10 −4 GeV, which in turn results in v L i of the same order. Similar soft terms are given in the charged scalar sector. In this parameter point one finds the VEV configuration: v 2 • Degeneracies between real and imaginary components of the sneutrinos.
• Very heavy (and degenerate) states with masses m H 0 m A 0 m H ± in the multi-TeV range.
With this generic prediction for the spectrum in the extended scalar sector the model turn outs to be safe from the constraints on the oblique S, T, U parameters. Besides the heaviness of some states, the degeneracies among their masses cancel their contributions to these precision observables. Note also that in our scenario the large supersymmetry breaking scale is related to the flavor symmetry used to get the required vaccuum alignment.

Discussion and conclusions
We have extended the (next to) MSSM by implementing a discrete non-Abelian flavor symmetry A 4 × Z 2 . The most general renormalizable allowed superpotential forbids all the trilinear RPV terms (including those violating baryon number) and has a single bilinear R-parity violating term. Three copies of up and down Higgs doublets are required, in addition to a SU(3) × SU(2) × U(1) singlet present inŜ, odd under Z 2 . When these develop VEVs both the electroweak and flavor symmetries are broken and, in addition, sneutrinos acquire tiny VEVs. The Higgs fields align so as to recover the correct charged fermion mass hierarchies and the two successful predictions Eqs. (3.11) and (3.12) [22].
As in the usual flavor-less BRPV model, the 1-loop radiative corrections is misaligned with the tree-level ones. These 1-loop contributions provide the solar mass square splitting.
Due to the flavor symmetry and vacuum alignment, one neutrino is nearly massles and there is a non-trivial restriction upon the neutrino oscillation parameters, displayed in Figs. 1-3, consistent however with the most recent experimental data presented at the recent Neutrino 2012 conference and the corresponding neutrino oscillation global fits [29][30][31]. As far as collider physics is concerned, our model predicts that LSP decays and neutrino mixing angles are tightly correlated, opening encourageing expectations for searches at the LHC [21]. Finally, even though the usual neutralino LSP is lost as dark matter candidate, one can show that a relatively light gravitino provides a very interesting alternative [10].