The Right Side of Tev Scale Spontaneous R-Parity Violation
aa r X i v : . [ h e p - ph ] A ug THE RIGHT SIDE OF TEV SCALE SPONTANEOUS R-PARITY VIOLATION
Lisa L. Everett, Pavel Fileviez P´erez, and Sogee Spinner
Department of Physics, University of Wisconsin, Madison, WI 53706, USA (Dated: November 6, 2018)We study a simple extension of the minimal supersymmetric Standard Model in which the Abeliansector of the theory consists of B − L and right-handed isospin. In the minimal model this Abeliangauge structure is broken to the standard model hypercharge gauge group by non-vanishing vacuumexpectation values of the right-handed sneutrinos, resulting in spontaneous R-parity violation. Thistheory can emerge as a low energy effective theory of a left-right symmetric theory realized at a highscale. We determine the mass spectrum of the theory, and discuss the generation of neutrino massesand R-parity violating interactions. The possibility of distinguishing between R-parity violatingmodels with a gauged U (1) B − L broken at the TeV scale at the Large Hadron Collider is discussed. I. INTRODUCTION
In models of physics beyond the Standard Model (SM) with low energy supersymmetry (SUSY), suchas the minimal supersymmetric standard model (MSSM) and its extensions, a conserved parity symmetry isusually added to prevent rapid proton decay from dimension four operators. The most common symmetryof this type is R-parity, which is defined as R = ( − B − L )+2 S = ( − S M , in which M is calledmatter parity and B , L and S stand for baryon number, lepton number, and spin, respectively. The inclusionof a conserved R-parity not only ensures the stability of the proton, but also allows for a stable lightestsuperpartner (LSP) that is a good candidate for the dark matter of the universe. Hence, understandingthe origin of R-parity, and whether it is an exact symmetry or is broken spontaneously at some scale,is of key importance for theories with low energy supersymmetry (see e.g. [1, 2] for general reviews ofsupersymmetry and [3, 4, 5, 6, 7, 8, 9] for discussions of R-parity violation in supersymmetric models).To understand if R-parity is conserved or broken at energies relevant for tests of supersymmetry at theLarge Hadron Collider (LHC), one can see from the definition of R-parity that it is useful to investigatescenarios in which B − L is an exact symmetry of the theory at some scale (note that if B − L is conserved,matter parity is also conserved). Such scenarios are also well-motivated from the theoretical point of viewin that B − L emerges naturally in the context of many grand unified theories. However, for minimalmodels in which B − L is a local symmetry, if the matter content of the theory includes only the MSSMstates (including right-handed neutrinos) such that no additional fields are introduced by hand, the inevitableconsequence is that R-parity is spontaneously broken.Scenarios with a spontaneously broken gauged B − L symmetry have the attractive feature that the R-parity breaking in the low energy theory is obtained in a controlled manner. In such scenarios, rapid protondecay can be avoided because baryon number can remain as a symmetry that is conserved at the perturbativelevel. Though the LSP neutralino dark matter framework is not applicable once R-parity is broken, it is wellknown that the gravitino can be a viable dark matter candidate. These models not only have the distinctivecollider signatures associated with R-parity violation, but also if the gauged B − L symmetry is broken atthe TeV scale, the gauge structure of this sector can be probed in detail at the LHC.To this end, two of the authors have studied extensions of the MSSM in which B − L is part of the localgauge symmetry, but is broken at TeV energies by non-vanishing sneutrino vacuum expectation values.These models have included both left-right symmetric models [10] and simple U (1) B − L extensions of theMSSM [11, 12]. Of these scenarios, the left-right symmetric model of [10] is particularly minimal andeconomical in that it contains the simplest possible Higgs sector needed for the breaking of the left-rightsymmetry. However, for this scenario to be viable, the soft-breaking terms must not respect the discrete left-right parity symmetry, and hence it is not possible to simultaneously understand the origin of parity violationand R-parity violation in this context. It is also difficult to satisfy all of the constraints arising from flavorviolation due to additional light Higgses in the theory. In contrast, the simple U (1) B − L extensions of theMSSM studied in [11, 12] are anomaly-free U (1) ′ theories with distinctive collider phenomenology dueto R-parity breaking and the TeV scale Z ′ gauge boson associated with the gauged B − L symmetry. Forphenomenological and cosmological aspects of the R-parity violating interactions see [15].In this work, we investigate spontaneous R-parity violation within a model which naturally includes thebest features of each of these scenarios. The model is a simple extension of the MSSM in which the Abeliansector of the theory consists of U (1) B − L and U (1) I R , the third component of right-handed weak isospin(the SU (2) R of left-right models). This theory is thus a simple U (1) B − L extension of the MSSM, but it canalso be obtained from a left-right symmetric theory that is realized at some high scale when the SU (2) R isbroken to U (1) I R by an SU (2) triplet Higgs with zero B − L charge. We determine the full spectrum ofthe theory, and discuss the generation of neutrino masses and the properties of the Higgs sector. We alsodiscuss ways to distinguish between this class of models via their Z ′ phenomenology at the LHC.This work is organized as follows. In Section II, we present our model and discuss spontaneous R-parity violation. In Section III, we compute the full mass spectrum (including neutrino masses), and outlinethe details of the R-parity violating interactions. We discuss the methods for distinguishing between thesedifferent models for spontaneous R-parity violation in Section IV. Finally, we summarize our main resultsand conclude in Section V. II. SPONTANEOUS R-PARITY VIOLATION IN A SIMPLE EXTENSION OF THE MSSM
We consider an anomaly-free extension of the MSSM in which the electroweak gauge sector consistsof SU (2) L N U (1) I R N U (1) B − L , in which U (1) I R is defined to be the third component of right-handedisospin ( SU (2) R ). More precisely, the quark and lepton superfields have the electroweak gauge charges: ˆ Q T = (cid:16) ˆ U , ˆ D (cid:17) ∼ (2 , , / , ˆ U C ∼ (1 , − / , − / , ˆ D C ∼ (1 , / , − / , ˆ L T = (cid:16) ˆ N , ˆ E (cid:17) ∼ (2 , , − , ˆ E C ∼ (1 , / , , ˆ N C ∼ (1 , − / , . (1)The MSSM Higgses transform as ˆ H u ∼ (2 , / , , ˆ H d ∼ (2 , − / , . (2)This minimal set of superfields as described above is the full matter content within our model. Therefore, inorder to break U (1) I R N U (1) B − L → U (1) Y , the sneutrinos must acquire non-zero vacuum expectationvalues, resulting in spontaneous R-parity violation (for previous discussions of sneutrino vacuum expecta-tion values, see e.g. [4, 6]). Hence, within the minimal model (in terms of particle content) with this Abeliangauge structure, R-parity should be broken spontaneously.Before exploring the details of the gauge symmetry breaking and resulting R-parity violation in thismodel, we pause to comment on the connection of this scenario to left-right symmetric models (for previousdiscussions, see [8]). The gauge structure described above can be obtained as a low energy limit of aleft-right symmetric theory based on the gauge group SU (3) C N SU (2) L N SU (2) R N U (1) B − L . The SU (2) R symmetry is broken at a high scale by a triplet Σ C ∼ (1 , , , (generically present in this classof models as part of a pair of triplets, together with Σ ∼ (1 , , , ) to the Abelian subgroup U (1) I R [8].The origin of the parity violating interactions present in the MSSM (or in the SM) can then be understoodonce the left-right symmetry is spontaneously broken at the high scale.Indeed, spontaneous R-parity violation has been previously explored within a minimal fully left-rightsymmetric model in which the Higgs sector is composed only of Higgs bi-doublets, Φ i ∼ (2 , , [10].As discussed in [10], the scenario can be consistent only if soft supersymmetry breaking terms do notrespect the discrete left-right symmetry ( i.e. , if parity is broken in the soft breaking sector). To avoid strongmixing between the leptons and the charginos/neutralinos, the left-right symmetry should be broken at theTeV scale. However, in this case the several light Higgses present in the minimal version of the theorycan result in unacceptably large flavor violation. In contrast, within the SU (2) L N U (1) I R N U (1) B − L scenario described in this paper, the scales of R-parity violation and left-right symmetry breaking ( i.e. ,parity violation) are logically separated, and issues of flavor violation can be safely avoided.Given the particle content in Eqs. (1)–(2), the superpotential is given by W R = W MSSM + Y ν ˆ L ˆ H u ˆ N C , (3)in which the MSSM superpotential takes the form W MSSM = Y u ˆ Q ˆ H u ˆ U C + Y d ˆ Q ˆ H d ˆ D C + Y e ˆ L ˆ H d ˆ E C + µ ˆ H u ˆ H d . (4)In the above, we have suppressed flavor, gauge, and Lorentz indices. We do not adhere to any specific modelof SUSY breaking, and hence the soft breaking terms are: V soft = M N C | ˜ N C | + M L | ˜ L | + M E C | ˜ E C | + m H u | H u | + m H d | H d | + (cid:18) A ν ˜ LH u ˜ N C + Bµ H u H d + 12 M BL ˜ B ′ ˜ B ′ + 12 M R ˜ W R ˜ W R + h . c . (cid:19) + V MSSMsoft . (5)In Eq. (5), M BL and M R denote the gaugino masses for the U (1) B − L and U (1) I R gauginos, and V MSSMsoft represents terms that are not relevant for our discussion of spontaneous R-parity violation in this model ( i.e. ,the remaining gaugino masses, soft mass-squared parameters, and trilinear couplings of the MSSM).We now discuss gauge symmetry breaking and its connection to R-parity violation in this model. It isstraightforward to compute the different contributions to the potential once one generation of left-handedand right-handed sneutrinos ( ˜ ν and ˜ ν C ), and the Higgses ( H u,d ), acquire vacuum expectation values (VEVs) h ˜ ν i = v L / √ , h ˜ ν C i = v R / √ , and h H u,d i = v u,d / √ , respectively. These contributions read as h V F i = 14 ( Y ν ) (cid:0) v R v u + v R v L + v L v u (cid:1) + 12 µ (cid:0) v u + v d (cid:1) − √ Y ν µ v d v L v R , (6) h V D i = 132 h g (cid:0) v u − v d − v L (cid:1) + g R (cid:0) v d + v R − v u (cid:1) + g BL (cid:0) v R − v L (cid:1) i , (7) h V soft i = 12 (cid:16) M L v L + M N c v R + m H u v u + m H d v d (cid:17) + 12 √ A ν + A † ν ) v R v L v u − Re ( Bµ ) v u v d , (8)in which g R , g and g BL are the gauge couplings for U (1) I R , SU (2) L and U (1) B − L , respectively. Mini-mizing the potential in the limit v R , v u , v d ≫ v L yields: v R = s − M N C + g R (cid:0) v u − v d (cid:1) g R + g BL , (9) v L = v R B ν M L − g BL v R − g (cid:0) v u − v d (cid:1) , (10) µ = − (cid:0) g + g R (cid:1) (cid:0) v u + v d (cid:1) + M H u tan β − M H d − tan β , and Bµ = 2 µ + m H d + m H u , (11)in which M H u = m H u − g R v R , M H d = m H d + 18 g R v R , and B ν = 1 √ Y ν µv d − A ν v u ) . (12)From Eqs. (9)–(10), we see that while v L = 0 if v R = 0 , v L is generically suppressed compared to v R . We also see from Eq. (9) that the desired symmetry breaking pattern requires negative soft breakingterms for the right-handed sneutrinos, M N C < . Such negative soft mass-squares can in principle beachieved via a radiative mechanism in the context of a high energy theory or in some gauge mediationmechanism. However, it is also worth noting that this requirement can be circumvented in the presence of aFayet-Iliopoulos (FI) terms for the Abelian symmetries, as shown in Appendix B. For our purposes for thispaper, we will restrict ourselves to the case without such FI terms and simply assume the presence of therequired soft breaking terms at low energies. The remaining minimization condition, Eq. (11), is similar tothe MSSM case except that Eq. (11) now has extra contributions from the I R D -terms. As indicated byEq. (12), these contributions are consistent with having a negative mass squared for the up-type Higgs. III. BILINEAR RPV INTERACTIONS AND MASS SPECTRUM
Once the sneutrinos acquire VEVs, R-parity is spontaneously broken and the low energy theory is sim-ilar to the bilinear R-parity violating MSSM. The traditional bilinear R-parity violating term is given by Y ν v R L ˜ H u , with coefficient suppressed by neutrino mass parameters. The kinetic term of the right-handedneutrino leads to a new R-parity violating mixing with the gaugino g R v R ν C ˜ B ′ , which is not suppressedby neutrino masses, but does not directly lead to dangerous low energy observables. Further bilinears canbe derived in a similar fashion and will be proportional to v L and therefore suppressed by neutrino massparameters. Effective trilinear terms appear once the neutralinos are integrated out and will be doubly sup-pressed and proportional to the ratio of neutrino mass parameters to neutralino masses. It is important toemphasize once more that the baryon violating λ ′′ will not be generated and therefore neither will dimensionfour proton decay operators.When considering the gauge sector, we will work in the limit that v L → for simplicity. Defining v ≡ v u + v d as usual, the neutral gauge boson mass matrix in the ( B ′ , W R , W L ) basis is given by M Z = g BL v R − g BL g R v R − g BL g R v R g R (cid:0) v R + v (cid:1) − g g R v − g g R v g v , (13)in which Tr M Z = M Z ′ + M Z = ( g BL + g R ) v R / g R + g ) v / . The neutral gauge boson massesare given by M Z,Z ′ = 18 (cid:2)(cid:0) g + g R (cid:1) v + (cid:0) g BL + g R (cid:1) v R ∓ q(cid:0)(cid:0) g + g R (cid:1) v − (cid:0) g BL + g R (cid:1) v R (cid:1) + 4 g R v v R (cid:21) (14)and, of course, the massless photon. Expanding in ǫ ≡ v /v R results in: M Z ≈ (cid:18) g R g BL g R + g BL + g (cid:19) v − g R ǫ (cid:0) g R + g BL (cid:1) ! , (15) M Z ′ ≈ (cid:0) g R + g BL (cid:1) v R g R ǫ (cid:0) g R + g BL (cid:1) ! , (16)where one can define g ≡ g R g BL / ( g R + g BL ) as the analogue of the standard model hypercharge gaugecoupling. The ρ parameter constrains ǫ < ∼ − .We proceed in the basis A = − cos θ W sin θ R B ′ + cos θ W cos θ R W R + sin θ W W L , (17) Z L = sin θ W sin θ R B ′ − sin θ W cos θ R W R + cos θ W W L , (18) Z R = cos θ R B ′ + sin θ R W R , (19)where, θ W is the weak mixing angle and θ R is the equivalent mixing angle of the right-handed sector with tan θ R = − g R /g BL . In this basis, the massless photon ( A ) is a mass eigenstate and is projected out. Theremaining × submatrix mixes Z L and Z R . This mixing angle ξ can be parameterized as tan 2 ξ = 2 M Z L Z R M Z R − M Z L , (20)where M Z L Z R = 14 g R q g BL + g R v q g + g , M Z L = 14 ( g + g ) v , (21)and M Z R = 14 (cid:0) g R + g BL (cid:1) v R . (22)Keeping terms to order ǫ yields: ξ ≈ g R p g + g (cid:0) g BL + g R (cid:1) ǫ , (23)with mass eigenstates Z = Z L cos ξ + Z R sin ξ, (24) Z ′ = Z R cos ξ − Z L sin ξ. (25)Experimentally, the Z − Z ′ mixing angle ξ must be smaller than ∼ − , which places a similar constrainton ǫ as the ρ parameter. To zeroth order in ξ , the couplings of the Z ′ gauge boson to the fermions are g Z ′ ¯ ff = q g R + g BL (cid:2) cos θ R Y ( f ) − I R ( f ) (cid:3) , (26)where Y ( f ) and I R ( f ) denote the hypercharge and the third component of the right-handed isospin.Once R-parity is spontaneously broken, mixing between the neutralinos and the neutrinos is inducedin the low energy theory. It is well known, and has been extensively been discussed in the literature,that this mixing contributes to neutrino masses (for previous discussions, see e.g [3, 4, 13, 14]). In the (cid:16) ν, ν c , ˜ B ′ , ˜ W R , ˜ W L , ˜ H d , ˜ H u (cid:17) basis, the mass matrix is given by M N = √ Y ν v u − g BL v L g v L √ Y ν v R √ Y ν v u g BL v R g R v R √ Y ν v L − g BL v L g BL v R M BL g R v R M R g R v d − g R v u g v L M g v d − g v u g R v d g v d − µ √ Y ν v R √ Y ν v L − g R v u − g v u − µ . (27)Integrating out the neutralinos generates masses for both the right-handed neutrinos and the light neutrinos.The right-handed neutrinos can then also be integrated out, resulting in further contributions to the lightneutrino masses. The light neutrino masses are therefore generated through a double-seesaw mechanism,which includes both R-parity violating and Type I seesaw contributions. For phenomenologically acceptableneutrino masses, the neutrino Yukawa coupling Y ν must be of the order of ∼ − , even for the thirdgeneration. It is important to note that within left-right models, it is a challenge to obtain such a smallneutrino Yukawa coupling and simultaneously generate appropriate charged lepton masses, although it isnot impossible. Such models require two bi-doublets, both of which contains an up-type and down-typeHiggs doublet and each with a different lepton Yukawa coupling. The Dirac masses of the charged leptonsand the neutrinos are then different linear combinations of the product of the lepton Yukawa couplings andthe appropriate VEVs of these bi-doublet fields and it is possible for one of these linear combinations to bequite small, while keeping the other large enough for the charged lepton masses. In the context of this work,in which we focus primarily on the effective TeV-scale theory involving the gauged B − L and I R ratherthan its parent left-right theory, we will not address this issue further.Therefore, in the limit that Y ν is neglected for simplicity, the right-handed neutrino masses are given by M ν C ≈ g R v R /M R + g BL v R /M BL . Hence for neutralino masses below a TeV, the right-handed neutrinomasses are on the order of 100 GeV. Recall that in the case of the Type I seesaw [19], the light neutrinomasses are given by M ν = M Dν M − ν C ( M Dν ) T . In this case, M Dν has two different contributions: the first isgoverned by Y ν , and the second results from integrating out the neutralinos. Aside from the light neutrinos,there will be five electroweak mass neutralinos. Notice from the mass matrix above that substantial mixingexists between the right-handed neutrinos and the new Abelian gauginos and is of order TeV.In the scalar sector of the theory, the sleptons and the Higgs scalars also mix as a result of R-parity viola-tion. Defining the √ (cid:0) ˜ ν, ˜ ν c , H d , H u (cid:1) basis for the CP-odd neutral scalars, the √ (cid:0) ˜ ν, ˜ ν c , H d , H u (cid:1) basis for the CP-even neutral scalars, and the (cid:0) ˜ e ∗ , ˜ e c , H −∗ d , H + u (cid:1) basis for the charged scalars, the scalarmass-squared matrices are given by Eq. (28), Eq. (29) and Eq. (30), respectively. The mass matrix of theCP-odd neutral Higgses is given by M P = v R v L B ν B ν − √ Y ν µ v R − √ A ν v R B ν v L v R B ν − √ Y ν µ v L − √ A ν v L − √ Y ν µ v R − √ Y ν µ v L v u v d Bµ + Y ν µ v L v R √ v d Bµ − √ A ν v R − √ A ν v L Bµ v d v u Bµ − A ν v L v R √ v u . (28)From Eq. (28), it is straightforward to show that the expected two Goldstone bosons are obtained. For theCP-even neutral scalars, the mass matrix is: M S = S ν S νH (cid:0) S νH (cid:1) T S H , (29)in which S ν , S νH and S H are given in the Appendix. The mass matrix for the charged Higgses is given by M C = C e C eH (cid:0) C eH (cid:1) T C H , (30)in which the definitions of C e , C eH and C H are also given in the Appendix.It is most illuminating to the study the scalar sector in the limit of zero neutrino masses ( v L , Y ν , A ν → , since the Higgs mass-squared matrices greatly simplify. In particular, the left-handed sneutrino decou-ples from the CP-even and CP-odd Higgses with a mass that is given by m ν = M L − g BL v R − g (cid:0) v u − v d (cid:1) . (31)Imposing the experimental lower bound on the sneutrino masses results in a constraint on the relationbetween M L and the B − L contribution. The remaining × CP-odd mass matrix has two zero eigenvalues;the corresponding states are the Goldstone bosons that are eaten by the Z and Z ′ . The nonzero eigenvaluecorresponds to the CP-odd MSSM Higgs, A , which has the same expression for its mass as in the MSSM.The remaining × CP-even mass matrix can be further studied in the limit of large tan β and decou-pling of the heavy CP-even MSSM Higgs. This leaves a × mass matrix with potentially large mixingbetween the right-handed sneutrino and the MSSM Higgs: M S → (cid:0) g BL + g R (cid:1) v R − g R v u v R − g R v u v R (cid:0) g + g R (cid:1) v u . (32)This matrix has the same trace and determinant as Eq. (13), indicating that at zeroth order, the masses inthis limit are given by m Re ˜ ν C = M Z ′ ∼ (cid:0) g R + g BL (cid:1) v R , (33) m h = M Z ∼ (cid:0) g + g (cid:1) v . (34)In the limit of vanishing neutrino masses, the charged Higgs mass matrix of Eq. (30) decouples into a blockdiagonal form, in which the upper × block resembles the MSSM slepton mass matrix and the lower × block is identical to the MSSM charged Higgs mass matrix. Assuming small mixing in the sleptonsector, the slepton masses are given by m e L = M L − g BL v R + 18 g (cid:0) v u − v d (cid:1) , (35) m e R = M E C + 18 g BL v R − g R (cid:0) v R + v d − v u (cid:1) . (36)The leading order corrections to the scalar masses typically scale as Y ν v R . Such contributions are quitenegligible as expected, since they are suppressed by the neutrino mass parameters.As in any theory with R-parity violation, mixing between the charged leptons and the charginos of theMSSM will occur in the charged fermion sector, (cid:16) e c , ˜ W + L , ˜ H + u (cid:17) and (cid:16) e, ˜ W − L , ˜ H − d (cid:17) . In this basis, themass matrix is given by M ˜ C = − √ Y e v d √ Y e v L √ g v L M √ g v d − √ Y ν v R √ g v u µ . (37)Since the mixing between the charginos and the charged leptons of the MSSM is proportional to v L and Y ν , small corrections to the charged lepton masses can be generated once the charginos are integrated out.However, this contribution is always small once the neutrino constraints are imposed.The squark mass matrices in this theory differ from those obtained in the MSSM due to new D termcontributions, as follows:0 M u = M Q + m u + g BL v R − g (cid:0) v u − v d (cid:1) A u v u − y u µ v d A u v u − y u µ v d M u C + m u − g BL v R + g R (cid:0) v R + v d − v u (cid:1) , (38) M d = M Q + m d + g BL v R + g (cid:0) v u − v d (cid:1) A d v d − y d µ v u A d v d − y d µ v u M d C + m d − g BL v R − g R (cid:0) v R + v d − v u (cid:1) , (39)in which m u and m d are the up-type and down-type quark masses, respectively. Let us assume for simplicitythat the LR mixings in the squark sector is small and thus can be neglected. The diagonal (LL and RR)terms must then be positive definite to avoid tachyonic squarks, as well as satisfy the experimental bounds.Therefore M d C > − m d + g BL v R + g R (cid:0) v R + v d − v u (cid:1) and M u C > − m u + g BL v R − g R (cid:0) v R + v d − v u (cid:1) , in order to have a realistic spectrum. IV. DISTINGUISHING BETWEEN DIFFERENT SRPV SCENARIOS
The collider phenomenology of this model at the LHC has distinctive features that are associated withthe gauged B − L at TeV energies and R-parity violating signatures. The possibility of testing this class ofmodels of spontaneous R-parity violation via the multilepton channels with different leptons, eeee , eµµµ , eeµµ and others, from the sneutrinos decays ˜ ν → e + i e − j has been discussed in [12]. Within this classof models, the sneutrinos can be pair produced through the Z ′ gauge boson that present in these models;see [16] for a detailed study of this production mechanism at the LHC.The models of spontaneous R-parity violation proposed here and in [11, 12] that involve a gauged B − L symmetry have many similar phenomenological features. Recall that these other scenarios are a minimalextension of the MSSM with U (1) B − L [11], and an extended version in which U (1) B − L is augmented by anadmixture of U (1) Y [12]. Given the challenges of predicting the detailed decay patterns of the neutralinos, ˜ χ → W e, Zν and the other superpartner decays, we believe that the easiest way to distinguish betweenthese models (at least as a first step) is through the properties of their Z ′ gauge bosons. In Table I, wehave displayed the couplings of the Z ′ gauge boson to fermions in each scenario. Note that these threescenarios are simple, anomaly-free models with minimal particle content in which spontaneous R-paritybreaking can be achieved and the properties of the U (1) ′ gauge boson are prescribed. Hence, if a Z ′ gaugeboson with some component of U (1) B − L is discovered at the LHC, the comparison of its properties fromthe pure B − L case can in principle reveal if any of these scenarios for spontaneous R-parity violationmight be relevant (or other possibilities that we have not yet studied). Of course, we are considering the1 Models Z ′ ¯ f f Couplings U (1) Y N U (1) B − L g BL ( B − L ) / U (1) Y N U (1) X g X ( a Y + b ( B − L )) / U (1) I R N U (1) B − L p g R + g BL (cid:2) cos θ R Y ( f ) − I R ( f ) (cid:3) TABLE I: The Z ′ couplings to the fermions in the SRpV models involving a gauged U (1) B − L broken at the TeVscale studied here and in [11, 12]. In the above, X = aY + b ( B − L ) . Slepton Masses U (1) Y N U (1) B − L m e L M L − g BL v R + ( g − g )( v u − v d ) m e R M E C + g BL v R + g ( v u − v d ) m ν L M L − g BL v R − ( g + g )( v u − v d ) m ν C M Z B − L TABLE II: The minimal B − L model. ideal case where SUSY with R-parity violating interactions is discovered at the LHC. For further studies ofhow to characterize the couplings of a Z ′ at the LHC using leptonic channels as well as additional channelsinvolving top and bottom quarks, we refer the reader to [17, 18].We now discuss the main differences between the mass spectrum of the sleptons in each SRpV model.In Tables II, III and IV, we show the mass spectrum in the limit v L → for simplicity. Within a givenSUSY breaking scenario that predicts M L and M E C , the main features of the spectrum in each case canbe determined once values for v R , g BL , g X , and tan β are assumed. Hence, if SUSY is discovered at theLHC, measurements of the detailed properties of the slepton sector can indicate whether the MSSM or anyof these extended models is suggested by the data. Slepton Masses U (1) Y N U (1) X m e L M L + g X ( − v R + v u − v d ) + ( g − g )( v u − v d ) m e R M E C + g X ( − v R + v u − v d ) + g ( v u − v d ) m ν L M L + g X ( − v R + v u − v d ) − ( g + g )( v u − v d ) m ν C M Z X TABLE III: The U (1) Y N U (1) X model, in which X = aY + b ( B − L ) with a = 1 and b = − / . Slepton Masses U (1) I R N U (1) B − L m e L M L − g BL v R + g ( v u − v d ) m e R M E C + g BL v R − g R ( v R + v d − v u ) m ν L M L − g BL v R − g ( v u − v d ) m ν C M Z LR TABLE IV: The LR-inspired model described here, in which the SM hypercharge is g = g R g BL / p g BL + g R . V. SUMMARY AND DISCUSSION
In this paper, we have discussed spontaneous R-parity violation in the context of a simple extension ofthe minimal supersymmetric Standard Model where the Abelian sector of the theory is composed of B − L and right-handed isospin. This model can be obtained as a low energy limit of left-right symmetric scenariosin which the scale of left-right symmetry breaking is much higher than the scale of R-parity breaking. Wehave analyzed the gauge symmetry breaking and low energy mass spectrum of the theory, including thegeneration of neutrino masses in this scenario. The nature of the resulting R-parity violating interactionsand their phenomenological implications are also presented. We also described how to distinguish betweenthis and other simple scenarios which include a gauged U (1) B − L as part of the Abelian gauge sector of thetheory at the LHC via their Z ′ phenomenology.Within models with a gauged U (1) B − L , if a minimal particle content is assumed then R-parity will bespontaneously broken. Indeed, in this class of models, R-parity is only an exact symmetry if the Higgs sectoris extended to include additional SM singlets which have even B − L quantum numbers. Only experimentswill tell us if R-parity is an exact symmetry at low energies, but in our view, the possibility of spontaneousR-parity violation as investigated in this paper is quite appealing and provides phenomenological signaturesthat should be considered in searches for supersymmetry at TeV energies.The simple and economical model studied in this work not only has direct connections with grandunification, but as with other scenarios with a gauged B − L symmetry that is broken at TeV energies, itprovides a simple and calculable scenario in which to study R-parity violation at the LHC. Such modelsof spontaneous R-parity violation avoid the standard concerns of supersymmetric theories associated withrapid proton decay, and can provide in the gravitino a viable dark matter candidate. Furthermore, thisclass of models also yield simple, anomaly free U (1) ′ gauge extensions of the MSSM with distinctivecouplings of the Z ′ to the fermions in the theory. The Z ′ phenomenology of such simple U (1) ′ scenariosis certainly worthy of further study. Given these attractive features, such well-motivated alternatives to R-3parity conserving models such as the MSSM (and many of its extensions) warrant further attention in thisera of unprecedented exploration of TeV scale physics at the LHC. Acknowledgments
P. F. P. is supported in part by the U.S. Department of Energy contract No. DE-FG02-08ER41531 andthe Wisconsin Alumni Research Foundation. L. E. and S. S. are supported in part by the U.S. Departmentof Energy under grant No. DE-FG02-95ER40896, and the Wisconsin Alumni Research Foundation.
APPENDIX A: GENERAL SCALAR POTENTIAL AND MASS MATRICES
Here we discuss the properties of the scalar potential, taking into account the possibility of vacuumexpectation values for all three generations of the sneutrinos. Neglecting CP violating effects and defining < H u > = v u / √ , < H d > = v d / √ , < ˜ ν i > = v iL / √ , and < (˜ ν C ) i > = v iR / √ , the scalar potential is V = 14 (cid:16) v iL Y ijν v jR (cid:17) + µ v u + v d ) − µ √ v iL Y ijν v jR v d − v u (cid:16) Y ijν v jR Y ikν v kR + v iL Y ijν v kL Y kjν (cid:17) + g
32 ( v iL v iL − v u + v d ) + g R
32 ( v iR v iR − v iL v iL ) + 12 v iR ( M N C ) ij v jR + 12 v iL ( M L ) ij v jL + 12 m H u v u + 12 m H d v d + 12 (cid:18) √ v iL A ijν v jR v u − Bµv u v d + h . c . (cid:19) . (A1)In the case in which only one generation of sneutrinos acquire vacuum expectation values and in the limitthat the Yukawa coupling and the trilinear term are flavor diagonal, the results of Section II are reproduced.In the √ (cid:0) ˜ ν, ˜ ν c , H d , H u (cid:1) basis for the CP-even scalars and the (cid:0) ˜ e ∗ , ˜ e c , H −∗ d , H + u (cid:1) basis for thecharged scalars, the mass matrices for these two sectors are given by Eq. (29) and Eq. (30), in which S ν ≡ (cid:0) g + g BL (cid:1) v L + v R v L B ν − g BL v L v R + ( Y ν ) v L v R − B ν − g BL v L v R + ( Y ν ) v L v R − B ν (cid:0) g BL + g R (cid:1) v R + v L v R B ν , (A2) S νH ≡ g v d v L − √ Y ν µ v R − g v L v u + ( Y ν ) v L v u + √ A ν v R g R v d v R − √ Y ν µ v L − g R v u v R + ( Y ν ) v u v R + √ A ν v L , (A3) S H ≡ (cid:0) g + g R (cid:1) v d + v u v d Bµ + Y ν µ v L v R √ v d − (cid:0) g + g R (cid:1) v u v d − Bµ − (cid:0) g + g R (cid:1) v u v d − Bµ (cid:0) g + g R (cid:1) v u + v d v u Bµ − A ν v L v R √ v u . (A4)4 C e ≡ C B e B e C , (A5) C eH ≡ g v d v L − Y e v d v L − Y ν µ v R g v L v u − Y ν v u v L − √ A ν v R Y e Y ν v u v R + √ A e v L Y e Y ν v d v R + √ Y e µ v L , (A6) C H ≡ g (cid:0) v u − v L (cid:1) + Bµ v u v d + Y e v L + Y ν µ v R v L √ v d Bµ + g v u v d Bµ + g v u v d g (cid:0) v d + v L (cid:1) + v d v u Bµ − ( Y ν ) v L − A ν v L v R √ v u , (A7)with C = 14 g (cid:0) v u − v d (cid:1) + 12 Y e v d − Y ν v u + v R v L B ν , (A8) C = M E c − g R (cid:0) v R + v d − v u (cid:1) + 18 g BL (cid:0) v R − v L (cid:1) + 12 Y e (cid:0) v d + v L (cid:1) , (A9)where B ν ≡ √ ( Y ν µv d − A ν v u ) , and B e ≡ √ ( Y e µ v u − A e v d ) . APPENDIX B: FAYET-ILIOPOULOS TERMS
We now describe the case in which the potential includes Fayet-Iliopoulos D terms: V F I = ξ R D R + ξ BL D BL . In this case the minimization conditions take the form v R = s − M N C − g R ξ R − g BL ξ BL ) + g R (cid:0) v u − v d (cid:1) g R + g BL , (B1) v L = v R B ν M L + g BL ξ BL − g BL v R − g (cid:0) v u − v d (cid:1) , (B2) µ = − (cid:0) g + g R (cid:1) (cid:0) v u + v d (cid:1) + M H u tan β − M H d − tan β , (B3) Bµ = 2 µ + m H d + m H u , (B4)in which M H u = m H u − g R ξ R − g R v R , M H d = m H d + 12 g R ξ R + 18 g R v R , B ν = 1 √ Y ν µv d − A ν v u ) . (B5)If the B − L FI-term, ξ BL , is large and positive, note that we can have a consistent mechanism for sponta-neous R-parity violation even with positive soft mass-squared terms for the sneutrinos. In this case, C = M E c −
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