Dynamical R-parity Breaking at the LHC
Shao-Long Chen, Dilip Kumar Ghosh, Rabindra N. Mohapatra, Yue Zhang
PPrepared for submission to JHEP
Dynamical R-parity Breaking at the LHC
Shao-Long Chen, a,b
Dilip Kumar Ghosh, c Rabindra N. Mohapatra, a Yue Zhang d a Maryland Center for Fundamental Physics and Department of Physics,University of Maryland, College Park, Maryland 20742, USA b Institute of Particle Physics, Huazhong Normal University, Wuhan 430079, China c Department of Theoretical Physics, Indian Association for the Cultivation of Science,2A&2B Raja S.C. Mullick Road, Kolkata 700 032, India d Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, I-34014 Trieste, Italy
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
In a class of extensions of the minimal supersymmetric standard model with(B-L)/left-right symmetry that explains the neutrino masses, breaking R-parity symmetry isan essential and dynamical requirement for successful gauge symmetry breaking. Two con-sequences of these models are: (i) a new kind of R-parity breaking interaction that protectsproton stability but adds new contributions to neutrinoless double beta decay and (ii) anupper bound on the extra gauge and parity symmetry breaking scale which is within thelarge hadron collider (LHC) energy range. We point out that an important prediction ofsuch theories is a potentially large mixing between the right-handed charged lepton ( e c ) andthe superpartner of the right-handed gauge boson ( (cid:102) W + R ), which leads to a brand new classof R-parity violating interactions of type (cid:101) µ c † ν cµ e c and (cid:101) d c † u c e c . We analyze the relevant con-straints on the sparticle mass spectrum and the LHC signatures for the case with smuon/stauNLSP and gravitino LSP. We note the “smoking gun” signals for such models to be leptonflavor/number violating processes: pp → µ ± µ ± e + e − jj (or τ ± τ ± e + e − jj ) and pp → µ ± e ± b ¯ bjj (or τ ± e ± b ¯ bjj ) without significant missing energy. The predicted multi-lepton final states andthe flavor structure make the model be distinguishable even in the early running of the LHC. ArXiv ePrint: a r X i v : . [ h e p - ph ] N ov ontents pp → e + e − µ ± µ ± jj pp → µ ± e ± b ¯ bjj π → e + e − decay 24 Supersymmetry (SUSY) is one of the popular and best motivated candidates for physicsbeyond the standard model (SM). It stabilizes the gauge hierarchy and provides a dark mattercandidate in a natural manner. An intuitive requirement to stabilize the dark matter inMSSM is the existence of the R-parity symmetry under which all standard model particlesare even and their superpartners are odd. The lightest supersymmetric partner field (LSP),e.g., either the neutralino or the gravitino, which is odd under R-parity is therefore suitableas the dark matter candidate. If R-parity is a global symmetry of the MSSM, it is logicalto think of it as a remnant of some high scale physics. It will of course be interesting if thehigh scale physics is motivated by further reasons. A shortcoming of R-parity conservingMSSM is the zero neutrino mass. Understanding the origin neutrino masses then requires itto be part of a larger theory. An example of extension to the MSSM is to gauge the B − L – 1 –lobal symmetry, where anomaly freedom requires introducing a right-handed neutrino toeach generation. The breaking of B − L symmetry gives Majorana masses to neutrinos. If thebreaking is accomplished by Higgs fields with B − L = ±
2, it not only helps to explain thesmall neutrino masses via the seesaw mechanism, but also leaves the R-parity as an unbrokensymmetry at the level of the MSSM [1], thereby providing a stable dark matter candidate.Extending MSSM by a B − L symmetry therefore “kills two birds with one stone”.Two possible classes of models with B − L gauge symmetry are: (i) SU (2) L × U (1) Y × U (1) B − L , and (ii) its left-right (LR) symmetric generalization based on SU (2) L × SU (2) R × U (1) B − L . Breaking B − L by two units in the second case is more appealing since it canexplain the origin parity violation, and leads to a number of interesting phenomenologicalimplications for LHC searches including the W R boson as well for low energy weak processes.We discuss them in this paper.If the gauge symmetry is to be broken by a pair of Higgs superfields ∆ c (1 , , +2) ⊕ ¯∆ c (1 , , −
2) which are required to implement the seesaw mechanism and gauge anomaly can-cellation, two interesting results follow [2]. First, even though a priori the model is expectedto have a remnant R-parity after symmetry breaking, in its minimal version, exactly theopposite happens, i.e., R-parity must be necessarily broken spontaneously in order for thefull gauge symmetry to break down to the MSSM gauge group. If R-parity is exact, gaugesymmetry cannot break [2]. If the model is extended to include singlets, there is a range ofparameters where one can still have unbroken R-parity [3]. In the minimal model, however,R-parity breaking is mandatory. The right-handed (RH) sneutrino field, ˜ ν c , has to pick up avacuum expectation value (VEV) along with the neutral member of the B − L = 2 triplet,breaking the parity symmetry and contributing to the mass of the gauge bosons and gauginosassociated with right-handed currents. Since ˜ ν c is an R-odd particle, its VEV breaks R-parity.We call this class of models “dynamical R-parity breaking” models, since R-parity breakingis forced on the theory at the global minimum of the Hamiltonian. Other examples of modelswhere R-parity breaking by ν c vev are the minimal U (1) B − L extensions of the MSSM [4, 5].In this note we will focus on the SUSYLR case.A consequence of dynamical R-parity breaking in minimal SUSYLR model is the pre-diction of an upper bound on the mass scale of the right-handed W R boson, i.e., M W R (cid:46) M SUSY /f [6], which is in the range accessible at the LHC. Here M SUSY is a generic softSUSY breaking mass scale, f is the Yukawa coupling responsible for right-handed neutrinomasses and has to be (cid:38) .
1. Similar relations are also found in SUSYLR models where (cid:101) ν c vevs break left-right symmetry[7, 8].Due to spontaneous R-parity breaking, neutrino masses arise not only from the usualtype-I seesaw mechanism, but also via mixing with the neutralinos. Another consequenceis that neutralino is no longer a stable particle and cannot therefore play the role of darkmatter. However, if gravitino is the LSP, it can have an extremely long lifetime ( ≥ sec)and play the role of dark matter [9]. Implications for such a dark matter particle have beenstudied extensively in connection with cosmic ray anomalies [10].Since the scales of both superpartners and the new gauge interactions are predicted to lie– 2 –n the few TeV range, this theory could in principle be testable at the hadron colliders [11].In this paper therefore, we study the genuine signals from dynamical R-parity breaking anddiscuss how it can be distinguished from usual R-parity breaking models at LHC.We point out that the most important consequence of dynamical R-parity breaking inSUSYLR models is a large mixing between the RH charged lepton and RH wino, i.e., thephysical RH charged lepton after symmetry breaking is generically denoted asˆ (cid:96) c = θ (cid:96)(cid:96) (cid:96) c + θ (cid:96)W (cid:102) W + R + · · · , (1.1)where θ (cid:96)(cid:96) , θ (cid:96)W ∼ O (1) for (cid:104) (cid:101) ν c (cid:105) (cid:39) M SUSY and the · · · represents the contributions of otherHiggsino fields if the corresponding Higgses VEV’s also violate parity. The physical chargedlepton field contains a large RH wino component and in turn induces new R-parity violatingterms of K¨ahler type. This is characteristic of dynamical R-parity breaking models [2, 7, 8]with the presence of gauged SU (2) R and it leads to new effects absent in usual R-parityviolating MSSM or other models of spontaneous R-parity breaking, such as [12, 13]. Inparticular, it leads to effective R-parity violating interactions of the form (cid:101) µ c † ν cµ e c , (cid:101) τ c † ν cτ e c and (cid:101) d c † u c e c for all generations of quarks/squarks. We show that these kinds of vertices add newcontributions to neutrinoless double beta decay and imply constraints on the parameters ofthe model.These new interactions bring rich phenomenology at the LHC. In the context of a realisticmodel based on left-right symmetry, we study the single production of a slepton NLSP fromthe RH neutrino decays, which is produced via an on-shell W R boson resonance at LHC. TheNLSP single production and decay yield multi-lepton final states of type pp → µ ± µ ± e + e − jj (or τ ± τ ± e + e − jj ) and pp → µ ± e ± b ¯ bjj (or pp → τ ± e ± b ¯ bjj ) which break both lepton numberand flavor and have no missing energy. The parent states could therefore be reconstructedup to the oringinal W R decay. The lepton final states are predicted to have distinct flavorstructures. We further point out that the (cid:96) c − (cid:102) W + R mixing also leads to the production ofrighthanded polarized top quarks from down-type squark decay, which is distinguishable fromthe λ (cid:48) QLd c trilinear couplings in the usual R-parity violating MSSM [14].In section II, we study the general features in a class of models where R-parity is brokentogether with extra gauge symmetries. We derive new R-parity breaking terms from theK¨ahler potential and point out how to distinguish this class of model from others, e.g. theMSSM with usual R-parity breaking terms. In section III, we review the symmetry breaking inthe context of minimal supersymmetric left-right (SUSYLR) model, emphasizing the necessityof spontaneous (dynamical) R-parity violation for SU (2) R gauge symmetry breaking. Wediscuss the flavor issues of R-parity breaking and its implications to neutrino mass and insection IV, we study the signatures of the model at the LHC. We mainly focus on the singleproduction and decay of slepton NLSP via a heavier RH neutrino. The predicted multi-leptonfinal states and the flavor structure make the model distinguishable even in the early runningof the LHC. Finally in section V, we point out a new contribution to the neutrinoless doublebeta decay in the model and conclude. – 3 – Spontaneous R-parity Violation with Extended Gauge Symmetry
Unlike explicit R-parity violation, spontaneous R-parity violation (SRPV) has the advantagethat it introduces only one new parameter into the R-parity conserving theory – the VEVof an R-parity-odd field. Furthermore, if R-parity violating scale is at the TeV range, abovethis temperature, R-parity is exact and therefore it is less constrained by cosmology.SRPV can be realized in various ways: in the first model where the idea was discussed [12],the superpartner of SM neutrino was given a non-zero VEV. Since lepton number is not agauge symmetry of the MSSM, this leads to a doublet majoron which contributes to the Z -boson width and LEP measurements therefore have ruled out this scenario. One couldof course implement SRPV by the VEV of a right-handed sneutrino [13] in extensions ofthe MSSM that explain neutrino masses. Since the right-handed sneutrino field is a stan-dard model singlet, the majoron does not couple to the Z -boson and therefore escapes theconstraints set by the LEP data.In this section, we will pursue the implications when the R-parity is spontaneously brokentogether with some extra gauge symmetry beyond G SM = SU (2) L × U (1) Y . Here we focus onthe class of models where the extended gauge group G contains a subgroup SU (2) R . Clearly,the RH neutrino and its superpartner will be charged under the SU (2) R . Giving a non-zeroVEV to (cid:101) ν c will therefore give rise to new interactions. Models with dynamical R-paritybreaking belong to this category. Furthermore, such model predicts that the scale of newgauge interactions is tied to the soft SUSY breaking scale. This leads to several interestingnew features as we show below.The key distinguishing prediction of such a model is the existence of a large mixingbetween RH charged leptons and the gaugino superpartner of the W R boson. Since we workwith the gauge group G = SU (2) L × SU (2) R × U (1) B − L , the RH neutrino and charged leptonform a doublet under SU (2) R .After the RH sneutrino developing a VEV (cid:104) (cid:101) L c (cid:105) = (cid:34) (cid:104) (cid:101) ν c (cid:105) (cid:35) , (2.1)it breaks both the SU (2) R gauge group as well as the R-parity, at the scale of (cid:104) (cid:101) ν c (cid:105) . Due tothe Higgs mechanism, the heavy gauge bosons acquire their mass by absorbing the scalars I m (cid:101) ν c and (cid:101) (cid:96) c as the longitudinal components. Due to supersymmetry, one would expectthat a corresponding large Dirac mass would develop between the ν c − (cid:101) Z (cid:48) and (cid:96) c − (cid:102) W + R .Here we are interested in the chargino–lepton mixing, which is the new source of R-paritybreaking effects. We explicitly write down the charged fermion mass matrix in the basis of( (cid:102) W + R , (cid:96) c + ) − ( (cid:102) W − R , (cid:96) − ), M C = (cid:34) M / M W R m (cid:96) (cid:35) , (2.2)– 4 –here M W R = g R (cid:104) (cid:101) ν c (cid:105) is the W R gauge boson mass and M / is the soft SUSY breakingmass for the chargino. The (1-2) element is absent because neither the RH neutrino nor theHiggs VEV couples (cid:96) − and (cid:102) W + R . The determinant of this mass matrix is proportional to the(light) charged lepton mass m (cid:96) because the RH sneutrino VEV is an electroweak singlet andtherefore does not break chirality. This means that there must be a physical state with themass m (cid:96) and identifiable as the charged lepton.Diagonalization of this mass matrix leads to a mixing between (cid:96) c and (cid:102) W + R . This mixing θ (cid:96)W is large if M / (cid:39) M W R . The physical charged lepton state is then given byˆ (cid:96) c = θ (cid:96)(cid:96) (cid:96) c + θ (cid:96)W (cid:102) W + R , (2.3)where θ (cid:96)W ∼ O (1). We note there is no such mixing induced for (cid:102) W − R . Due to this mixing, onecan derive two new classes of R-parity violating interactions from the right-handed gauginomatter coupling terms: First from the gaugino coupling √ g (cid:102) W + R ν c(cid:96) (cid:48) (cid:101) (cid:96) (cid:48) c † + h . c . , we get L (cid:96) (cid:48) new = √ gθ (cid:96)W (cid:104) ˆ (cid:96) c ν c(cid:96) (cid:48) (cid:101) (cid:96) (cid:48) c † + ¯ˆ (cid:96) c ¯ ν c(cid:96) (cid:48) (cid:101) (cid:96) (cid:48) c (cid:105) . (2.4)Analogously, using the right-handed gaugino interaction with quarks and squarks, one canalso write down the new RPV couplings for the squark-quark sector L q new = √ gθ (cid:96)W (cid:104) ˆ (cid:96) c u c (cid:101) d c † + ¯ˆ (cid:96) c ¯ u c (cid:101) d c (cid:105) . (2.5)Notice there is no supersymmetric counterpart of above terms generated, unlike those fromthe superpotential: λLLe c , λ (cid:48) QLd c , etc.. The new couplings we obtain here break not onlyR-parity but also supersymmetry, since we started from a mass matrix Eq. (2.2) includingthe SUSY breaking gaugino mass.As we will point out in Sec. 5, the most stringent constraints on the couplings in Eq. (2.5)are from neutrinoless double beta decay and HERA experiment [15], which tend to push thesquark and gluino masses to TeV. On the other hand, the LEP2 Z-pole observables give auniversal constraint [16] on the mixing parameter θ (cid:96)W in both Eqs. (2.4) and (2.5), but itturns out to be rather mild. Therefore, the sleptons masses are still allowed to be not farabove 100 GeV.From these new interactions derived from dynamical R-parity breaking, one would expectthe following distinctive signatures at the LHC. • Single production of slepton NLSP via Eq. (2.4) and subsequent decays (see Fig. 1),which is the main topic being studied in this paper. • Top quark produced from down-type squark decay via Eq. (2.5), whose polarization isopposite to the components from the SM background, as well as the conventional λ (cid:48) term.We want to point out that these predictions are common to the models with R-parity bro-ken together with extended gauge symmetries, as well as to those breaking SU (2) R symmetrywithout the Higgs triplets, as long as the RH neutrino mass lies in the proper range.– 5 – igure 1 . Single production of RH slepton from RH sneutrino decays. The black box represents RPV e c − (cid:102) W R mixing. In this section, we present a model based on the group SU (2) L × SU (2) R × U (1) B − L wherein the absence of R-parity breaking, the gauge symmetry does not break[2]. Thus the gaugedynamics dictates R-parity breaking. Hence it explicitly provides an example for dynamicalR-parity breaking .The model considered in this section does not have the discrete parity symmetry. In theappendix, we will comment if such model can be built in a completely parity symmetric form.Here, we first review the salient features of the SU (2) L × SU (2) R × U (1) B − L model (withoutparity symmetry) for completeness and show how dynamical R-parity breaking occurs. The minimal SUSYLR model has the gauge group G LR = SU (3) c × SU (2) L × SU (2) R × U (1) B − L . The particle content and their representations under the gauge group for thecompletely parity symmetric case are listed in Table 1. The SU (2) R Higgs triplets ∆ c , ¯∆ c have been introduced to give mass to the RH neutrinos and facilitate the seesaw mechanism.For the sake of simplicity, we assume that the left-handed triplets ∆, ¯∆ are decoupled at highscale and do not exist in the TeV theory. The superpotential of the model is W = Y u Q T τ Φ τ Q c + Y d Q T τ Φ τ Q c + Y ν L T τ Φ τ L c + Y l L T τ Φ τ L c + if (cid:0) L cT τ ∆ c L c (cid:1) + µ Φ ab Tr (cid:0) Φ Ta τ Φ b τ (cid:1) + µ ∆ Tr (cid:0) ∆ c ¯∆ c (cid:1) , (3.1)where Y ’s are Yukawa couplings, f is the Majorana coupling of leptons and µ ∆ is the µ -termfor triplets. Note that in the model there is no gauge singlets introduced. For recent papers where hidden dynamics breaks R-parity, see [17] and also some experimental implicationsof such models see [18]. However, in these works, the R-parity is not broken together with extended gaugesymmetries that couple to SM fermions, and therefore, does not predict the phenomenology of dynamicalR-parity breaking being discussed in this paper. – 6 – U (3) c SU (2) L SU (2) R U (1) B − L Q (cid:3) (cid:3) / Q c (cid:3) (cid:3) − / L (cid:3) − L c (cid:3) , (cid:3) (cid:3) c (cid:3)(cid:3) − c (cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3) − Table 1 . Particle content in the minimal SUSYLR model. In this section, we first concentrate on thecase with the left-handed Higgs triplets ∆, ¯∆ do not exist in the TeV theory for simplicity. We willcomment on the fully parity symmetric theory in the appendix.
The corresponding soft terms are V soft = m (cid:101) Q (cid:16) (cid:101) Q † (cid:101) Q + (cid:101) Q c † (cid:101) Q c (cid:17) + m l (cid:16)(cid:101) L † (cid:101) L + (cid:101) L c † (cid:101) L c (cid:17) + m Tr(∆ c † ∆ c ) + m Tr( ¯∆ c † ¯∆ c )+ 12 ( M L λ aL λ aL + M R λ aR λ aR + M λ BL λ BL + M λ g λ g )+ (cid:101) Q T τ A qi φ i τ (cid:101) Q c + (cid:101) L T τ A (cid:96)i φ i τ (cid:101) L c + iA f (cid:101) L cT τ ∆ c (cid:101) L c + B Φ ab Tr (cid:0) τ φ Ta τ φ b (cid:1) + B ∆ Tr (cid:0) ∆ c ¯∆ c (cid:1) + h . c . . (3.2)The D-term potential as well as the scalar potential can be found in Refs. [2, 9].The desired symmetry breaking pattern is SU (2) R × U (1) B − L → U (1) Y at the first step,giving definite meaning to the hypercharge Y = I R + ( B − L ) /
2, followed by the electroweaksymmetry breaking. The key point to note is that the potential does not break any gaugesymmetry in supersymmetric limit [19]. Even if the soft SUSY breaking terms are included,the gauge symmetry still remains unbroken as long as the RH sneutrino has zero VEV. Parityand SU (2) R × U (1) B − L → U (1) Y breaking become possible only if the RH sneutrino picks upa non-zero VEV. The RH sneutrino being superpartner field has odd R-parity and thereforeits VEV breaks R-parity – hence the claim [2] that there is no parity breaking without R-parity breaking in the minimal SUSYLR model. Furthermore, it was shown in [6] that RHsneutrino VEV is tied to the soft mass scale M SUSY . This implies an upper bound on the W R gauge boson mass of order of the SUSY breaking scale.To see this explicitly, we write down the potential including all VEV’s given below; (cid:104) (cid:101) L ce (cid:105) = (cid:34) (cid:104) (cid:101) ν ce (cid:105) (cid:35) , (cid:104) ∆ c (cid:105) = (cid:34) v R (cid:35) , (cid:104) ¯∆ c (cid:105) = (cid:34) v R (cid:35) , (3.3)where we choose to break the R-parity along the RH electron sneutrino (cid:101) ν ce direction, forphenomenological consideration to be explained in Section 3.2. The scalar potential involving– 7 – igure 2 . Correlations among the VEVs. The left-panel tells us that R-parity is broken as much asparity, (cid:104) (cid:101) ν ce (cid:105) ≈ v R + ¯ v R . The middle panel shows that the minimum always points towards the flatD-term potential direction, (cid:104) (cid:101) ν ce (cid:105) = 2( v R − ¯ v R ). The right panel tells the value of the VEV is related toparameters in the potential as (cid:104) (cid:101) ν ce (cid:105) ≈ A/ f . This agrees with the upper bound obtained in Ref. [9]. the Higgs triplets and RH sneutrinos is V = M v R + M ¯ v R − Bv R ¯ v R + | f | (cid:104) (cid:101) ν ce (cid:105) + (cid:104) | f | v R + m − | A | v R − | f | µ ∆ ¯ v R (cid:105) (cid:104) (cid:101) ν ce (cid:105) + 18 ( g + g (cid:48) )( (cid:104) (cid:101) ν ce (cid:105) − v R + 2¯ v R ) , (3.4)where M = µ + m , M = µ + m and B = B ∆ . For simplicity, we have assumed thematrices f , A f , m (cid:101) (cid:96) to be flavor diagonal and f = f ee , A = ( A f ) ee and m = ( m (cid:101) (cid:96) ) ee . TheVEVs of the Higgs bidoublets have been neglected in the first stage of symmetry breaking.The potential should satisfy B < M M to be bounded from below. This can be seen byconsidering D-flat directions (cid:104) (cid:101) ν ce (cid:105) = 0, (cid:104) ∆ (cid:105) = (cid:104) ∆ c (cid:105) = vτ and (cid:104) ¯∆ (cid:105) = (cid:104) ¯∆ c (cid:105) = ¯ vτ , where τ isthe Pauli matrix. This scalar potential has the property that on (cid:104) (cid:101) ν ce (cid:105) = 0 surface, there is nosymmetry breaking, i.e., v R = ¯ v R = 0 at the minimum. The acceptable minimum that breaksparity and the gauge symmetries therefore necessarily breaks R-parity.We have carried out numerical study of the minimization of the scaler potential, byscanning over the bulk of parameter space ( M , M , √ B, A, µ ∆ , m ) ∈ [100 , f ∈ [0 . , . V min < (cid:104) (cid:101) ν ce (cid:105) (cid:54) = 0. It turns out that there areinteresting correlations among the VEVs of RH neutrino and the Higgs fields. They are shownin Fig. 2. Typically, we find the D-term potential always vanishes, i.e., (cid:104) (cid:101) ν ce (cid:105) = 2( v R − ¯ v R ).Therefore, the physics at the RH scale does not bring additional terms to the Higgs potential.The sneutrino VEV and the Higgs triplets VEV’s are of the same order, (cid:104) (cid:101) ν ce (cid:105) ≈ v R + ¯ v R , aswell as an approximate relation (cid:104) (cid:101) ν ce (cid:105) ≈ A/ f . This agrees with the upper bound obtainedin Ref. [9]. The key point is that, in SUSYLR model, the right-handed scale is dynamicallygenerated through the SUSY breaking soft mass scale [2], v R (cid:46) M SUSY f , (3.5)– 8 –here M SUSY ∼ O (100) GeV corresponds to the generic soft SUSY breaking mass scale.
Figure 3 . Left panel: The potential V as a function of v R for given (cid:104) (cid:101) ν ce (cid:105) = 2055 GeV, ¯ v R = 1607 GeV(solid curve) and (cid:104) (cid:101) ν ce (cid:105) = ¯ v R = 0 (dashed curve). Right panel: Contour plot of the potential V in the v R − ¯ v R plane for (cid:104) (cid:101) ν ce (cid:105) = 2055 GeV. In order to illustrate the role of R-parity violation in symmetry breaking, we choose thefollowing set of parameters M = 213 GeV , M = 251 GeV , √ B = 150 GeV ,µ ∆ = 517 GeV , A = 240 GeV , m = 376 GeV , f = 0 . . (3.6)The resulting VEV’s and the minimum potential value are (cid:104) (cid:101) ν ce (cid:105) = 2055 GeV , v R = 2063 GeV , ¯ v R = 1607 GeV , V min = − . × GeV . (3.7)The configuration of the potential around the vacuum is shown in Fig. 3. Clearly, the globalminimum of the potential breaks R-parity, i.e., (cid:104) (cid:101) ν ce (cid:105) (cid:54) = 0. Because the R-parity and thelepton number are broken simultaneously with the gauge symmetries, no massless Majoron ispresent. On the other hand, the dynamical R-parity breaking associated with gauge symmetrybreaking at few TeV scale offers rich phenomenology. Neutrino masses in this model have been discussed extensively in [9]. We review the salientpoints for completeness and, in particular, constraints on flavor of the R-parity violation. InSUSYLR model, the matter fields obtain their Dirac masses from the coupling to the Higgsbidoublets Φ , . Generally, there are four SU (2) L Higgs doublets at the electroweak scale.The additional neutral Higgs bosons will lead to flavor changing neutral currents at tree level.– 9 –his can be suppressed either by proper doublet-doublet splitting or by cancellations [20].Another way to avoid the flavor changing Higgs effects is by replacing the second B − L = 0bidoublet with a B − L = 2 bidoublet as has recently been suggested [21]. We do not discussthis here.We assume that the bidoublet Higgs fields take the following form of VEV’s,Φ = (cid:34) κ (cid:35) , Φ = (cid:34) κ
00 0 (cid:35) , (3.8)where κ gives Dirac masses to the up-type quarks and neutrinos, while κ contributes todown-type quarks and charged leptons masses. With this VEV structure, the W L − W R gauge bosons do not mix with each other. Here we mainly focus on the lepton sector. Wewill attribute the hierarchies among the charged lepton masses to the Yukawa couplings. Inparticular, we focus on the low tan β = κ /κ ∼ O (1) regime. In this case, the Yukawacoupling constants are set as y τ ≈ − , y µ ≈ − , y e ≈ − . and ( Y ν ) ij ≈ − . Eventhough there are four SM Higgs doublets (or two MSSM Higgs pairs), since only two of themcontribute to fermion masses and the other two play the role of spectators, our tan β is sameas the MSSM one. The µ Φ and B Φ play a similar role as the µ and B µ parameters in theMSSM for the electroweak symmetry breaking.Because of R-parity violation, there are additional contributions to neutrino masses, ontop of the Type-I seesaw mechanism. They arise from neutrino-neutralino mixing which hasbeen calculated in Ref. [9] with R-parity breaking in the RH electron sneutrino direction,( M ν ) ij ≈ − g L g (cid:48) κ M L M (cid:101) B ( µ µ − µ ) (cid:18) M L g L + M R g R + M g (cid:19) (3.9) × (cid:20)(cid:18) µ µ (cid:19) ( Y ν ) i ( Y ν ) j + ( Y ν ) i y e δ j + ( Y ν ) j y e δ i − (cid:18) µ µ (cid:19) y e δ i δ j (cid:21) (cid:104) (cid:101) ν ce (cid:105) , assuming the LH sneutrino VEV’s are negligible. M L,R and M BL are soft supersymmetrybreaking gaugino masses. The corresponding Feynman diagram is shown in the left panelof Fig. 4. Choosing the R-parity violation along the (cid:101) ν ce direction helps to avoid too largecontributions through the y τ , y µ couplings, while the electron and neutrino Yukawa couplingsare sufficiently small to keep the neutrino mass scale O (0 .
1) eV in tact.We also notice that there are radiative corrections to the neutrino mass [22, 23], whichare also proportional to the corresponding Yukawa coupling and are loop suppressed (see theright panel of Fig. 4, as well as Fig. 2 in [24]). So they are safely small as long as (cid:104) (cid:101) ν cµ (cid:105) and (cid:104) (cid:101) ν cτ (cid:105) are vanishing and tan β is low. The above discussions justify our choices of VEV configuration in Eq. (3.3).
Before closing this section, we comment on the flavor violations. The next section willmainly concentrate on the scenario with slepton NLSP singly produced from the W ± R gaugeboson resonance, hence we need to understand the existing experimental constraints on the– 10 –elevant mass scales. In the minimal non-SUSY left-right model, the famous neutral K -meson mixing tends to push to W R mass to be above 2.4 TeV [25]. In the supersymmetricversion, loop diagrams mediated by superpartners also make additional contribution to bothquark and lepton flavor violation processes. They are safely small if the relevant mass scalesare high enough. Otherwise, in order to optimize the discovery prospects at the LHC, thesuperpartners and W R -boson masses have to lie in the (sub-)TeV regime, which requires fine-tuning the flavor structures of the model in a similar way to the MSSM. This is nothingbut the SUSY flavor problem, and the constraints on scales are quite model dependent.In principle, there could also be contribution to flavor violations from higher dimensionaloperators controlled by unknown physics in the UV.Therefore, in the following we shall only adopt the bounds M W R > m (cid:101) (cid:96) >
100 GeV from Tevatron and LEP2 direct searches, respectively.
Figure 4 . Contributions to neutrino masses from R-parity violation. Left panel: tree-level contri-bution due to neutrino-neutralino mixing. Right panel: loop-suppressed radiative correction to theneutrino masses. The ˜v represents neutral gaugino fields. The black dots are the usual Higgs VEVinsertions.
In this section, we start exploring the LHC implications of this model. First we need to knowthe R-parity violating (RPV) interactions that induce the decays of the sparticles produced.
In general, spontaneous R-parity breaking through the RH sneutrino VEV generates thebilinear terms in the superpotential and the soft potnetial W (cid:26) R = µ i L i H u , V (cid:26) R − soft = B i (cid:101) L i H u + h . c . . (4.1)The bilinear term facilitates the R-parity breaking decay of the lightest neutralino, (cid:101) χ asfollows: (cid:101) χ → Z ν , (cid:101) χ → W ± (cid:96) ∓ or (cid:101) χ → (cid:96) +1 (cid:96) − ν [5]. In the literature, more complete colliderphenomenologies of R-parity violation from the superpotential has been studied in detail andreviewed in Refs. [22, 26–28]. – 11 –n the SUSYLR model, the bilinears arise from the electron and neutrino Yukawa cou-plings and the corresponding A -term once the RH sneutrino VEV is inserted [9]. Therefore µ i µ Φ (cid:39) B i B Φ (cid:39) y e , y ν (cid:39) − . (4.2)These bilinear terms will induce trilinear R-parity breaking terms λLLe c , λ (cid:48) QLd c . The mostimportant terms for the following study are those associated with third generation fermions, λ (cid:48) i t(cid:96) i b c + λ i ν(cid:96) i τ c , (4.3)where λ (cid:48) i = y t µ i /µ Φ , λ i = y τ µ i /µ Φ and i = 1 , ,
3. On the other hand, the λ (cid:48)(cid:48) term will notbe generated, since baryon number symmetry is respected by the sneutrino VEV, therebyguarantee the proton stability.As already stated in previous sections, a distinct feature that arises when R-parity isdynamically broken together with an SU (2) R gauge symmetry is the large mixing betweenthe RH electron e c and the SU (2) R gaugino, i.e., (cid:102) W + R . This is not present in the MSSM withgeneral R-parity violation. The form of charged fermion mass matrix and the obtained mixingin SUSYLR model is given explicitly in the Appendix. From the usual gaugino Yukawa-likecoupling term for µ and τ flavors, one obtains the new couplings (similar to Eq. (2.4)) L (cid:96) new = √ gθ eW (cid:104) e c ν cµ (cid:101) µ c † + ¯ e c ¯ ν cµ (cid:101) µ c (cid:105) + √ gθ eW (cid:104) e c ν cτ (cid:101) τ c † + ¯ e c ¯ ν cτ (cid:101) τ c (cid:105) . (4.4)As we will see, these interactions open new channel for the single production of a slepton athadron colliders (Fig. 1). Similarly, from the neutralino mass matrix, one can also obtainlarge mixing between the (cid:101) Z (cid:48) and ν ce , and in turn the couplings L ν c new = √ g Z (cid:48) θ NZ (cid:48) (cid:16) ν ce µ c (cid:101) µ c † + ν ce τ c (cid:101) τ c † + ν ce ν cµ (cid:101) ν c † µ + ν ce ν cτ (cid:101) ν c † τ (cid:17) + h . c .. (4.5)In principle, sparticle single production could also happen through the mixing between ν ce and (cid:101) Z (cid:48) [24] which, however, calls for some tuning between M Z (cid:48) and M / .Contrary to the usual R-parity breaking term from superpotential, these new R-paritybreaking sources come from the gaugino Yukawa-like couplings (in the K¨ahler potential). Aswe illustrate in the below, such theories could be tested at the LHC where the new gaugeinteractions are accessible. From the previous sections one learns that, in the SUSYLR model under discussion, a newclass of RPV couplings Eq. (4.4) emerge due to the mixing between e c and (cid:102) W + R . To studyits implications for hadron colliders, we need to know the sparticle spectrum. If one takesthe assumption of universal scalar masses at high scale, the RH sleptons are likely to be thelightest among matter superpartners in the MSSM due to the smaller Yukawa couplings as wellas the smaller weak gauge couplings [29]. The situation would be similar in SUSYLR models.– 12 –he lightest slepton could be a stau or the smuon depending on detailed parameter range.In our study, we will assume that smuon is the lightest superpartner above the gravitino, thelatter in our model could be the very weakly unstable dark matter.As promised, we study the implications of scenario at the LHC for the case where smuonor stau is the NLSP among the superpartners. Due to the relatively low tagging efficiency ofthe tau lepton, we would focus on the smuon.The new LHC signals originate from the production of W R in pp collision and its subse-quent decay to muon and RH muon neutrino which subsequently decays. In the non-SUSYLR models with type I seesaw the RH neutrino decays mostly to the three body final state (cid:96) ± (cid:96) ± jj via W R exchange [30, 31]. However in the SUSY version, if the smuon, (cid:101) µ is lighterthan ν c , an interesting new two body final state channel emerges: RH neutrino decays to a (cid:101) µ and an electron. Since this is two-body decay, for the smuon sufficiently light, it will certainlydominate over the three body non-SUSY mode, as shown in Fig. 5. Therefore, the smuonsingle production could take this advantage and be large enough to be probed at the LHC. Figure 5 . Production of the RH neutrino and its decays. Left-panel: the usual same-sign leptondiagram for W R discovery. Right-panel: Single production of the smuon through RH neutrino RPVdecay. In this case, there is equal possibility to break the muon lepton number twice through theMajorana mass of ν cµ or not, so one can get either (cid:101) µ + e − R or (cid:101) µ − e + R from its decay. The black boxrepresents sneutrino VEV insertion as indicated in Fig. 1. As the NLSP, the smuon (cid:101) µ NLSP = cos α (cid:101) µ R + sin α e iβ (cid:101) µ L will decay dominantly throughR-parity breaking interactions rather than the Planck scale suppressed decay to the gravitino.In the case where there is a large mixing between LH and RH smuons, i.e., sin α ∼ O (1), (cid:101) µ + can decay to t ¯ b or τ ¯ ν through the induced trilinear RPV terms as shown in Eq. (4.3), althoughsuppressed by the small y e or y ν Yukawa couplings. Note that electroweak symmetry forbidsthe direct coupling of RH smuon to ¯ f f , and the RH sneutrino VEV does not help because itis also a singlet. Only the LH and RH smuon mixing term ( m (cid:101) µ ) LR which is proportional tothe Higgs doublet VEV, can facilitate this decay.On the other hand, if the mixing term ( m (cid:101) µ ) LR is severely suppressed, i.e., sin α (cid:28)
1, thesmuon is almost purely RH. Therefore, it decays through a four-body channel, as shown inthe right panel of Fig. 6. Such decay rate is proportional to the gauge coupling instead of the Since the RH neutrino mass matrix is proportional to the matrix f , which we have taken to be diagonalin the basis of physical charged leptons, there is no further flavor changing in ν c mass matrix (propagator). – 13 – igure 6 . Two- and four-body decay modes of the smuon NLSP. The black box represents sneutrinoVEV insertion as indicated in Fig. 1. The black dot stands for the usual Higgs VEV insertions.Hereafter, we denote the RH smuon as (cid:101) µ c ≡ (cid:101) µ + R . small y e or y ν couplings. It could be comparable or even dominate over the above two-bodydecays when the latter is further suppressed by the LR smuon mixing. In principle, (cid:101) µ c candecay to both e + µ ± jj final states. However, since the intermediate RH neutrino is off-shell,the probablity to break the muon lepton number is larger than that conserving the leptonnumber. This point can be see from the blue and brown curves in the left panel of Fig. 7.The branching ratios to different final states of the smuon decay has been plotted in Fig. 7with the following parameters chosen, M W R = 2 TeV, M ν cµ = 500 GeV and gθ eW = 0 .
2. Inthe suppressed LR slepton mixing case (left panel, here and below, we will take sin α ≈ − as a benchmark point), the four-body decay of smuon NLSP always dominates over all thetwo-body channels. In the large mixing case (right panel), the t ¯ b channel will dominate ifit is kinematically allowed, while τ ¯ ν and the four-body channels e + µ − jj could respectivelydominate in certain low smuon mass windows. In both cases, since the Majorana RH neutrinosare involved in the production and/or decay processes, lepton number can be broken, whichleads to the most promising discovery channels at the LHC. The expected signatures arelisted in Table 2. large (cid:101) µ − (cid:101) µ c mixing suppressed mixing M (cid:101) µ c > m t + m b pp → µ − e − t ¯ b, µ + e + ¯ tbpp → µ ± µ ± e + e − jjM (cid:101) µ c < m t + m b pp → µ − e + τ + + (cid:0)(cid:0) E T pp → µ ± µ ± e + e − jj Table 2 . Expected final states in the single production of the NLSP (cid:101) µ c with M (cid:101) µ c < M ν c assumed.Large and suppressed (cid:101) µ − (cid:101) µ c mixing cases are both listed. In the following two subsections we will discuss the signature of the single production anddecay of smuon NLSP via a heavier right-handed neutrino. We also discuss possible standardmodel backgrounds and elaborate on the selection criteria necessary for such signals to besignificantly observed over the standard model background. The large number of diagramsinvolved in the standard model background processes are calculated using the helicity am-– 14 –litude package MadGraph [32] and CalcHEP 2.5.4 [33]. To estimate the number of signaland background events as well as their phase space distribution(s), we use a parton-levelMonte-Carlo event generator. In our numerical analysis, we use the CTEQ6L parton distri-bution function [34] and fix the factorization scale Q = ˆ s/
4. In our parton-level simulationof both signal and background events, we smear the leptons and jet energies with a Gaussiandistribution according to δEE = a (cid:112) E/ GeV ⊕ b (4.6)with the CMS parameterization, a (cid:96) = 5% , b (cid:96) = 0 .
55% and a j = 100% , b j = 5%, ⊕ denotes asum in quadrature. pp → e + e − µ ± µ ± jj This particular final state dominates when the mixing between LH and RH smuon is sup-pressed (or in the low mass ( < ∼ M top ) region for a large mixing). In this section, we will denotesmuon NLSP as (cid:101) µ c since it is mainly the RH component. The most striking feature of thisfinal state is the three same sign leptons and one opposite sign lepton associated with twojets without missing energy. Assuming the narrow width approximation for ν c and (cid:101) µ c , wecan simply write down the signal cross-section σ s ( pp → e + e − µ + µ + jj ) as σ ( pp → e + e − µ + µ + jj ) ≈ σ ( pp → W + R → µ + ν cµ ) (4.7) × (cid:104) Br( ν cµ → ˜ µ c e − ) × Br(˜ µ c → e + µ c jj ) + Br( ν cµ → ˜ µ c † e + ) × Br(˜ µ c † → e − µ + jj ) (cid:105) , Figure 7 . Branching ratios for the smuon NLSP decay. The left panel represents the suppressed LHand RH slepton mixings ( ∼ − ) case, while in the right panel, we take an unsuppressed O (1) suchmixing. Charge conjugated final states are not listed but also possible. – 15 –here the red (dotted) arrow indicates lepton number violation by two units on the involvedRH neutrino propagator. The charge conjugated final state σ ( pp → e + e − µ − µ − jj ) which ismediated by the intermediate W − R boson can be similarly approximated. In our analysis, wecombine both these two final states. We define the signal identification with four chargedleptons and two jets. The events are further selected by the following set of cuts1. We require that both jets and leptons should appear within the detector’s rapiditycoverage, namely | η ( (cid:96) ) | < . , | η ( j ) | < . (4.8)2. The leptons are ordered according to their transverse momentum ( p T ) hardness and the p T of the leading lepton must satisfy p T ( (cid:96) ) >
100 GeV , (4.9)and for rest of the leptons p T ( (cid:96) ) >
15 GeV . (4.10)For two associated jets we demand that p jets T >
25 GeV . (4.11)3. We must also ensure that the jets and leptons are well separated so that they can beidentified as individual entities. To this end, we use the well-known cone algorithmdefined in terms of a cone angle ∆ R αβ ≡ (cid:113) (∆ φ αβ ) + (∆ η αβ ) with ∆ φ and ∆ η beingthe azimuthal angular separation and rapidity difference between two particles. Wedemand that ∆ R jj > . , ∆ R (cid:96)j > . , ∆ R (cid:96)(cid:96) > . . (4.12)4. In our analysis, We use simplified definition for the missing transverse energy: E/ T = (cid:113) ( (cid:80) p x ) + ( (cid:80) p y ) , where the sum goes over all observed charged leptons and jets.We demand that there is no significant missing energy in our signal E/ T <
30 GeV . (4.13)Our choice of p T cut on the leading lepton (Eq. (4.9)) can be well justified from the p T distribution of all four leptons as displayed in Fig. 8 assuming M W R = 1 TeV, M ν cµ = 500 GeVand M ˜ µ c = 300 GeV and at √ s = 14 TeV. Here, one should note that while generating p T distributions (Fig. 8), we impose an uniform loose cut ( p T >
15 GeV) on all four leptons,however, rest of the cuts remain unchanged. From the choice of mass parameters and simplekinematics of the production and decay chain, it is very obvious that the leading lepton ( (cid:96) )comes from the two body decay of heavy W + R → µ + + ν cµ , while rest of the leptons originat-ing from the cascade decay chain of ν c and ˜ µ c have relatively softer transverse momentum– 16 – igure 8 . p T distributions of all four leptons in the process pp → e + e − µ ± µ ± jj at √ s = 14 TeV. Theleptons are ordered according to their p T hardness ( p T ( (cid:96) ) > p T ( (cid:96) ) > p T ( (cid:96) ) > p T ( (cid:96) )). We havefixed M W R = 1 TeV, M ν cµ = 500 GeV, and M ˜ µ c = 300 GeV. Figure 9 . Signal cross sections σ ( pp → e + e − µ ± µ ± jj ) (after all cuts as mentioned in the text) as afunction of smuon mass at the LHC with √ s = 7 TeV, 10 TeV and 14 TeV. Three curves from topto bottom in each panel correspond to M W R = 1 TeV, 1.5 TeV and 2 TeV respectively. M ν cR is keptfixed at 500 GeV. compared to the p T of the leading lepton. On the other hand, as the RH neutrino mass isincreased to a value closer to the W R mass, the first lepton becomes softer. However, in thiscase, the lepton from the decay ν cµ → e − (cid:101) µ c merits the highest p T and will serve as the hardestlepton ( (cid:96) ).In Fig. 9 we show the total signal cross-section σ s (after imposing all the cuts mentionedabove) for the process shown in Eq. (4.8), as a function of the smuon (cid:101) µ c mass at the LHC for7 TeV, 10 TeV and 14 TeV energies. In each panel, three curves from top to bottom correspondto M W R = 1 TeV, 1.5 TeV and 2 TeV respectively. We fix the RH neutrino mass M ν cµ = 500GeV and the mixing parameter gθ eW = 0 . • In all three panels, irrespective of M W R , the σ s first rises with the increases of smuonmass and then becomes almost flat and finally drops sharply as M (cid:101) µ c becomes degeneratewith right-handed neutrino mass M ν cµ . • The initial rise of the cross-section with the smuon mass can be understood from thefact that for lighter smuon mass ( M (cid:101) µ c ∼ −
200 GeV), the decay products of smuons (cid:101) µ c → e + µ + jj are more collimated and fail to satisfy our isolation criteria for leptonsand jets as shown in Eq. (4.12). As the smuon mass increases, leptons and jets whichoriginate from the cascade decay of smuon tend to appear with larger ∆ R , thus satisfyingthe isolation criteria as displayed in Eq. (4.12). As a consequence, the σ s for heaviersmuon mass ( M (cid:101) µ c (cid:46) M ν cµ ) is significantly larger than for lower smoun mass region. • The signal cross secion σ s strongly depends on √ s , mass M W R and off course on M ˜ µ c .There is a possibility that the LHC may also run at √ s = 10 TeV, before attaining toits designed √ s = 14 TeV. Keeping this in mind, we decided to provide our observationfor √ s = 10 TeV also. In is very interesting to note that for all the choices of M W R and √ s the smallest cross-section always correspond to M (cid:101) µ c = 100 GeV, while the largestone correspond to M (cid:101) µ c which lies between 400 −
430 GeV as shown in Table 3. M W R Table 3 . The range of minimum and maximum σ ( pp → e + e − µ ± µ ± jj )(fb) at the LHC for √ s =7 , ,
14 TeV and M W R = 1 , . , M ν cµ = 500 GeV and gθ eW = 0 . Mass reconstruction : The most important feature of our signal events is the effectivereconstruction of all three heavy particle masses from the final state charged leptons andjets. We first select two softest leptons (satisfying our selection criteria) from the four leptonset and then recombine these two leptons with the two jets to reconstruct the smuon mass, M jj(cid:96) (cid:96) ≈ M ˜ µ c . After the obtaining the smuon resonance, we attempt to reconstruct the RHneutrino mass by combining two jets, two softest leptons with one of the two hardest leptons (cid:96) or (cid:96) . In this case, we face the complication due to combinatorics with two choices ofpairing for (cid:96) , with M jj(cid:96) (cid:96) . Finally, W R can be reconstructed by combining all four chargedleptons and two jets. In Fig. 10, we display the invariant mass distribution for (cid:101) µ c , ν cµ and W R at 7 TeV LHC. Fitting the mass distribution with a Gaussian, we get the following values M fit (cid:101) µ c = 301 . ± .
74 GeV , M fit ν cµ = 500 . ± .
75 GeV , M fit W R = 999 . ± .
10 GeV , (4.14)– 18 – igure 10 . Invariant mass distributions for M (cid:101) µ c , M ν cµ and M W R in the pp → e + e − µ ± µ ± jj process at √ s = 7 TeV with 3 fb − data for M (cid:101) µ c = 300 GeV, M µ cR = 500 GeV and M W R = 1 TeV respectively.The error-bars shown are statistical only for the indicated luminosity. Results of Gaussian fitting arealso shown. where the input masses considered for this mass reconstruction procedure are the following M true (cid:101) µ c = 300 GeV , M true ν cµ = 500 GeV , M true W R = 1000 GeV . (4.15)SM background σ (pb) σ (cid:96) ± (cid:96) ± (fb) pp → b ¯ bb ¯ b pp → t ¯ t
448 0.09 pp → Z b ¯ b × − pp → W ± W ± W ∓ Z . × − × − σ totalB Table 4 . The list of leading-order SM backgrounds that could mimic our signal. σ and σ (cid:96) ± (cid:96) ± aredefined in the text. These numbers correspond to √ s = 14 TeV. SM backgrounds : In principle, there is no intrinsic standard model background to the∆ L = 2 processes. However, there some standard model processes which could mimic oursignal if the missing transverse momentum of neutrinos are balanced. One of the dominantbackground is pp → b ¯ bb ¯ b , followed by semileptonic decay of all the b-quarks. We generate thisbackground using with the following basic cuts p T ( b ) >
25 GeV, | η ( b ) | < . R bb > . √ s = 14 TeV. After hadronization, one of the B or B has to oscillate before decay, in order to get a pair of same-sign dileptons. Theprobablity of having b ¯ b → e ± µ ± , µ ± µ ± is about P b ¯ b(cid:96) ± (cid:96) ± ≈ × − , as estimated in [35]. Aftertaking into account the semileptonic branching ratio ∼
10% for the other two b -quarks we findthis background cross-section ∼ − (fb). The other aparently looking very severe standardmodel background is pp → t ¯ t . At leading order, the top pair production cross-section ( σ t ¯ t )is 448 pb at the LHC with √ s = 14 TeV. After taking into account the leptonic branching– 19 –raction of two W bosons (from t → bW + ) and P b ¯ b(cid:96) ± (cid:96) ± , the rate goes down to ∼ − (fb).Here, we would like to mention that if we take into account the higher order QCD effects, σ t ¯ t becomes ≈
900 (pb), which means our final background cross-section from t ¯ t process mayincrease atmost by a factor of two. The other sub-leading standard model processes whichmy fake our signal processes are pp → Zb ¯ b , pp → W ± W ± W ∓ Z and pp → W ± W ± W ∓ h .In the case of pp → Zb ¯ b , process, Z → (cid:96) + (cid:96) − , (cid:96) = e, µ and same sign leptons will comefrom b ¯ b pair by oscillation of one of the B meson before decay. As a result of this, the σ ( pp → Zb ¯ b ) will be suppressed by Br( Z → (cid:96) + (cid:96) − ) , (cid:96) = e, µ and P b ¯ b(cid:96) ± (cid:96) ± . The rate for samesign leptons from remaining two processes are negligibly small. In Table 4, we summarize thestandard model background cross-sections, where, σ and σ (cid:96) ± (cid:96) ± correspond to the leadingorder cross-sections before and after folding with different suppression factors arising fromleptonic branching ratios of W ± , Z bosons, semi-leptonic branching ratio of b (¯ b ) quark andfinally P b ¯ b(cid:96) ± (cid:96) ± respectively. From this very simple minded exercise, we conclude that our signalis almost SM background free. pp → µ ± e ± b ¯ bjj The smuon heavier than top quark and with large mixing L-R mixing O (1) can lead to thisfinal state. Here we call the smuon NLSP as (cid:101) µ , without definite chirality. As shown in Table 2,the smuon will dominately decay to t ¯ b via the λ (cid:48) coupling in this case. For this signal topology,we select events with two same sign different flavoured (SSDF) charged leptons and four jets.The cross section for this channel in the narrow width approximation can be expressed as σ ( pp → µ + e + b ¯ bjj ) ≈ σ ( pp → W + R → µ + ν cµ ) · Br( ν cµ → ˜ µ − e + ) · Br(˜ µ − → ¯ tb ) · Br(¯ t → ¯ bjj ) . (4.16)The signal also includes the charge conjugated final state σ ( pp → µ − e − b ¯ bjj ) via intermediate W − R boson.Our selection cuts are same as shown in Eqs. (4.8) − (4.13), except for the transversemomentum cut on the jets. After ordering all four jets according to their p T , we imposefollowing cut on the hardest jet ( j ): p T ( j ) >
60 GeV (4.17)and for rest of the jets p T ( j , j , j ) >
25 GeV . (4.18)In Fig. 11 we display the p T distribution of two leptons and four jets respectively afterordering them according to their p T . While generating these distributions, we impose thefollowing cuts on the p T of leptons and jets, rest of the cuts remain unchanged, p T ( (cid:96) ) >
10 GeV , p jets T >
15 GeV . (4.19)– 20 – igure 11 . p T distributions of two leptons (left-panel) and all four jets (right-panel) in the process pp → µ ± e ± b ¯ bjj at √ s = 14 TeV. The leptons and jets are ordered according to their p T hardness( p T ( (cid:96) ) > p T ( (cid:96) ) and ( p T ( j ) > p T ( j ) > p T ( j ) > p T ( j ). The other model parameters are same asin Fig.8. We take the same set of mass parameters as in the previous subsection. In this case too, theleading lepton comes from the two body decay W R → (cid:96) + ν cµ . On the other hand, the leadingjet j mainly comes from the two body decay of the smuon, while the second hardest jet isproduced from the top quark decay. From the nature of the p T spectrum of leptons and jetsas shown in Fig. 11, we can justify our choice of p T cuts (Eqs. (4.17) and (4.18)) used in thisanalysis.In Fig. 12 we show the signal cross section (after all cuts on final state leptons and jetsas mentioned above) for this channel as a function of the smuon mass at the LHC for 7 TeV,10 TeV and 14 TeV energies. In each panel, three curves from top to bottom correspond to M W R = 1 TeV, 1.5 TeV and 2 TeV respectively. M ν cµ is kept fixed at 500 GeV and gθ eW = 0 . • In this case, since we look for (cid:101) µ + → t ¯ b , we focus on the smuon mass above the top quarkthreshold, as is displayed in all three panels of Fig. 12. As the smuon mass increases,the leptons and jets originating from the cascade decay of smuon tend to appear withlarger ∆ R between each other, satisfying the isolation criteria shown in Eq. (4.12). • The signal cross section begins to drop for heavier smuon mass ( ≥
350 GeV) irrespectiveof M W R and choice of the LHC energy. This is mainly due to the branching ratiosuppression of the (cid:101) µ + → t ¯ b decay mode, as can be seen in the right panel of Fig. 7.Secondly, there is also the phase space suppression when M (cid:101) µ becomes close to right-handed neutrino mass M ν cµ . • In Table 5, we show the range of signal cross sections for different values of M (cid:101) µ at theLHC with √ s = 7 , ,
14 TeV and M W R = 1 , . , M ν cµ = 500 GeV and gθ eW = 0 .
2. We quote the minimum and– 21 – igure 12 . Signal cross sections σ ( pp → µ − e − b ¯ bjj ) (after all cuts as mentioned in the text) as afunction of smuon mass at the LHC with √ s = 7 TeV, 10 TeV and 14 TeV. Three curves from top tobottom in each panel correspond to M W R = 1 TeV, 1.5 TeV and 2 TeV respectively. M ν cµ is kept fixedat 500 GeV. maximum values of the signal rate. For all three choices of M W R and √ s the smallestcross section always correspond to the value of M (cid:101) µ which is colse to M ν cµ , while thelargest cross section correspond to M (cid:101) µ lying between 260 −
300 GeV. The signal getsenhanced by more than factor of 2 as the LHC energy increases from 7 TeV to 10 TeV,and by another factor of 2–3 up to 14 TeV.
Mass reconstruction : We now discuss the mass reconstruction strategy of all three heavyparticles from the final state charged leptons and jets. From the sample of four jets, thehadronically decaying SM W -boson is reconstructed from pair jets whose invariant mass( m jj ) is closest to M W . The top quark is then reconstructed from the reconstructed W andone of the two remaining jets. We select the one which gives a invariant mass closest to M t .The smuon mass is reconstructed from this M t and with the last jet M (cid:101) µ ≡ m tj . Next, weattempt to reconstruct the right-handed neutrino mass by combining with one of the twoleptons (cid:96) or (cid:96) . In this case, we are facing the combinatorical background with two choices m tj(cid:96) , m tj(cid:96) . Finally, the W R -boson mass can be reconstructed by combining all four jets andtwo charged leptons. We will not explicitly show the reconstruction figure here, which looksvery similar to Fig. 10. SM background : In this case, the standard model process which can mimic our signal is pp → t ¯ tW ± → b ¯ bW + W − W ± → jjb ¯ b(cid:96) ± (cid:96) (cid:48)± , (4.20)where (cid:96), (cid:96) (cid:48) = e, µ . In our analysis, we do not impose the requirement of b tagging, since thestandard model background also contains b -jets, and b tagging would not improve the signalsignificance considerably. The standard model background cross sections from pp → t ¯ tW ± process is shonw in Table. 6, at different LHC energies. We expect this rate would further godown significantly (by several orders of magnitude) once we impose our selection criteria onthe final state leptons and jets. – 22 – W R Table 5 . The pp → µ ± e ± b ¯ bjj signal cross sections (in fb) at the LHC for √ s = 7 , ,
14 TeVand M W R = 1 , . , M ν cµ = 500 GeV and gθ eW = 0 .
2. Here we quote the minimum and maximum values of the signal rate and the correspondingsmuon masses are shown in the text. √ s σ t ¯ tW ± (fb) σ (cid:96) ± (cid:96) ± bkg (fb)7 TeV 99 1.3210 TeV 206 2.7714 TeV 377 5.05 Table 6 . The dominant SM background pp → t ¯ tW ± that could mimic our signal. σ t ¯ tW ± correspondto the production cross-section of t ¯ tW ± and σ (cid:96) ± (cid:96) ± bkg represents cross-section for b ¯ bjjµ ± e ± final statebefore any cuts. We also comment on the other standard model background pp → b ¯ bjj , which has a hugecross section ∼ pb after basic cuts. Taking into account of the oscillation of b ¯ b to getsame-sign e ± µ ± P b ¯ b(cid:96) ± (cid:96) ± will reduce it down to the order of ∼ b -jet will produce charged leptons which will be very close to the associated c -jet, as a result of this, lepton-jet isolation criteria will play a decisive role in reducing thisbackground further. Therefore, we conclude that this background will be also under control.The remaining backgrounds pp → W ± W ± W ∓ Z , pp → W ± W ± W ∓ h and pp → jjjjW ± W ± are much smaller [36]. In this model, there are several new contributions to neutrinoless double beta decay in additionto the usual light neutrino contribution. The contribution from the RH neutrino exchange asin the non-SUSYLR models was already discussed [37].In our model, there are two new contributions arising from the e c − ˜ W R mixing. The firstone is given in left panel of Fig. 13 below. Its contribution to the effective neutrino mass isgiven by m νββν ≈ θ eW (cid:18) M W L M W R (cid:19) p F m (cid:102) W R , (5.1)– 23 –here p F ≈ −
100 MeV is the typical momentum transfer in this process. For θ eW ∼ O (1),it is of same order as the RH neutrino contributions to this process in non-supersymmetriccase.The second contribution is given in the right panel of Fig. 13 and the correspondingeffective neutrino mass is m νββν ≈ θ eW (cid:18) α s α (cid:19) (cid:18) M W L m (cid:101) d c (cid:19) p F m (cid:101) g . (5.2)Note that for this to be consistent with the current limits on the neutrinoless double betadecay amplitude, we must have M ˜ d c , M ˜ G ≥ gθ eW ∼ . Figure 13 . New contribution to neutrinoless double beta decay due to (cid:102) W + R − e c mixing. π → e + e − decay This new R-parity violating interaction also has interesting consequences for rare leptonicdecays neutral pion and Kaon decays. We see from Eq. (2.5) that via t-channel (cid:101) u c exchangethis leads to the process π → e + e − with an amplitude given by A (cid:39) g θ eW /M u c . Thecurrent PDG bound [38] on this process is Br( π → e + e − ) ≤ × − . Using the bounds fromneutrinoless double beta decay, we predict that in our model we have Br( π → e + e − ) ≤ − .Note that if there is mixing in the right-handed charged current of the same order as theCKM mixings, then we would predict for the K → e + e − branching ratio at the level about 25times smaller than corresponding pion decay. This is about 3 times smaller than the currentPDG quoted bound. In the LHC search described above, we already restrict ourselves to thisallowed parameter range. In summary, we have studied the phenomenology of a class of minimal SUSYLR models with dynamical R-parity breaking , i.e., R-parity must necessarily break in order for parity and gauge– 24 –ymmetry breaking to occur. This induces a new class of R-parity violating interactions dueto the mixing between e c and (cid:102) W + R , which are not present in the usual MSSM with explicit orspontaneous R-parity violation. These interactions lead to a new contribution to neutrinolessdouble beta decay which restricts the squark/gluino masses to be in the TeV range. The modelhas its characteristic signature at LHC which consists of final states of type e + e − µ ± µ ± jj or µ ± e ± b ¯ bjj for smuon as the NLSP. We estimate the background for this process and find thatfor M W R not far above a TeV, the model should be testable once LHC reaches its full energyand luminosity. Incidentally, in this model there is also an upper limit on the mass of theright-handed W R boson in the low TeV range for symmetry breaking to occur. A large partof the mass range could be accessible even in the early running at the LHC. Acknowledgments
We would like to thank K.S. Babu, B. Bajc, S. Biswas, I. Gogoladze, T. Han, X. Ji, G.Senjanovi´c, S. Spinner and J. Zupan for fruitful discussions. The work of S.L.C. is partiallysupported by the US DOE grant DE-FG02-93ER-40762. D.K.G. acknowledges the hospi-tality provided by the ICTP High Energy Group, Trieste, Italy and the Regional Centrefor Accelerator-based Particle Physics (RECAPP), Harish Chandra Research Institute, Al-lahabad, India where part of this work was done. D.K.G. also acknowledges partial supportfrom the Department of Science and Technology, India under the grant SR/S2/HEP-12/2006.The work of R.N.M. is supported by the NSF grant PHY-0968854. The work of Y.Z. is par-tially supported by the EU FP6 Marie Curie Research and Training Network UniverseNet(MRTN-CT-2006-035863).
A Fully parity symmetric version
In this appendix, we consider the full parity symmetric version of the model. We now keepthe ∆ and ¯∆ multiplets in our model of Table 1. The Yukawa superpotential is given for thiscase by the same expression as Eq. (3.1) with two additional terms: L T τ ∆ L and µ ∆ Tr∆ ¯∆.The full potential that is parity symmetric is given below: V soft = m (cid:101) Q (cid:16) (cid:101) Q † (cid:101) Q + (cid:101) Q c † (cid:101) Q c (cid:17) + m l (cid:16)(cid:101) L † (cid:101) L + (cid:101) L c † (cid:101) L c (cid:17) + m (cid:104) Tr(∆ † ∆) + Tr(∆ c † ∆ c ) (cid:105) + m (cid:104) Tr( ¯∆ † ¯∆) + Tr( ¯∆ c † ¯∆ c ) (cid:105) + 12 ( M L λ aL λ aL + M R λ aR λ aR + M λ BL λ BL + M λ g λ g )+ (cid:101) Q T τ A qi φ i τ (cid:101) Q c + (cid:101) L T τ A (cid:96)i φ i τ (cid:101) L c + iA f (cid:16)(cid:101) L TL τ ∆ L (cid:101) L L + (cid:101) L cTR τ ∆ cR (cid:101) L cR (cid:17) + B Φ ab Tr (cid:0) τ φ Ta τ φ b (cid:1) + B ∆ Tr (cid:0) ∆ ¯∆ + ∆ c ¯∆ c (cid:1) + h . c . . (A.1)The D-term potential as well as the scalar potential can be found in Refs. [2, 9]. The argumentsfor the existence of the dynamical R-parity breaking is same as in the parity asymmetric– 25 –ersion discussed in sec. 2. So we do not repeat this discussion here. The only question weaddress here is the status of a possible parity symmetric vacuum with dynamical R-paritybreaking.First, we would like to understand why the symmetry breaking in the SUSYLR modelwithout Higgs triplets [8] is not compatible with the parity symmetry. The point is theLH and RH sneutrinos have opposite B − L charges, so the D-term potential contributes anegative cross term V D ∼ − g BL (cid:104) (cid:101) ν (cid:105) (cid:104) (cid:101) ν c (cid:105) , (A.2)which tends to minimize the potential in the parity conserving (cid:104) (cid:101) ν (cid:105) = (cid:104) (cid:101) ν c (cid:105) . This is why theauthors of Ref. [8] have to start with parity asymmetric soft mass squared for sneutrinos.In contrast, the corresponding term in model with Higgs triplets becomes V D ∼ − g BL (cid:18) (cid:104) (cid:101) ν c (cid:105) − v R + 2¯ v R (cid:19) (cid:18) (cid:104) (cid:101) ν (cid:105) − v L + 2¯ v L (cid:19) , (A.3)where (cid:104) ∆ (cid:105) = v L and (cid:104) ¯∆ (cid:105) = ¯ v L . According the D-flat condition found out in Fig. 2, eachbracket is very close to vanishing. Therefore, such D-term potential does not play significantrole in forcing the vacuum to preserve parity and it is still possible to start with a symmetricpotential. It has been was shown in [2] that if leptonic Yukawa couplings Y (cid:96) satisfy the bound Y (cid:96) ≥ f ( M − B ∆ ) M , (A.4)the parity violating minimum is indeed lower than the parity conserving. By choosing M ∆ and B ∆ appropriately, we can satisfy this bound so that the parity violating and R-parityviolating minimum is the global minimum. B Explicit form of charged fermion mass matrix
In this appendix, we present the explicit form of charged fermion mass matrix in the SUSYLRmodel. The spontaneous R-parity violation induces a mixing between the new chargino (cid:102) W R ,higgsino (cid:102) ¯∆ c + and the usual electron field.To see this explicitly, first note that parity violation at the TeV scale requires spontaneousR-parity breaking at a similar scale, i.e., (cid:104) (cid:101) ν ce (cid:105) (cid:39) v R (cid:39) ¯ v R . We can write down the chargedfermion mass Ψ T M (cid:101) C Ψ + h . c . , in the basis of Ψ = [( (cid:102) W + R , (cid:102) ¯∆ c + , e c + ) , ( (cid:102) W − R , (cid:101) ∆ c − , e − )] T , M (cid:101) C = (cid:34) MM T (cid:35) , M = M / −√ g R v R √ g R ¯ v R − µ ∆ g R (cid:104) (cid:101) ν ce (cid:105) f (cid:101) (cid:104) ν ce (cid:105) m e . (B.1) We thank S. Spinner for raising this point. – 26 –ollowing the similar arguments below Eq. (2.2), one finds the physical electron field massterm can be written as L m = − em e ( θ ee e c + θ eW (cid:102) W + R + θ e ∆ (cid:102) ¯∆ c + ) + h . c . ≡ − em e ˆ e c + h . c . , (B.2)where θ ee , θ eW , θ e ∆ are order 1 mixing parameters.In this model, the role played by ∆ c , ¯∆ c Higgses is to give mass to the RH neutrinos.Meanwhile, their superpartners enter in the above mixing matrix, but it does not change thegeneric prediction of large e c − (cid:102) W + R mixing. References [1] R. N. Mohapatra, Phys. Rev. D , 3457 (1986); S. P. Martin, Phys. Rev. D , 2769 (1992).[2] R. Kuchimanchi and R. N. Mohapatra, Phys. Rev. D , 4352 (1993).[3] K. S. Babu and R. N. Mohapatra, Phys. Lett. B , 404 (2008) [arXiv:0807.0481 [hep-ph]];[4] R. N. Mohapatra, Phys. Rev. Lett. , 561-563 (1986).[5] V. Barger, P. Fileviez Perez and S. Spinner, Phys. Rev. Lett. , 181802 (2009)[arXiv:0812.3661 [hep-ph]].[6] R. Kuchimanchi and R. N. Mohapatra, Phys. Rev. Lett. , 3989 (1995).[7] M. J. Hayashi, A. Murayama, Phys. Lett. B153 , 251 (1985).[8] P. Fileviez Perez and S. Spinner, Phys. Lett. B , 251 (2009) [arXiv:0811.3424 [hep-ph]].[9] X. Ji, R. N. Mohapatra, S. Nussinov and Y. Zhang, Phys. Rev. D , 075032 (2008)[arXiv:0808.1904 [hep-ph]].[10] A. Ibarra and D. Tran, JCAP , 002 (2008) [arXiv:0804.4596 [astro-ph]]; K. Ishiwata,S. Matsumoto and T. Moroi, Phys. Rev. D , 063505 (2008) [arXiv:0805.1133 [hep-ph]];S. L. Chen, R. N. Mohapatra, S. Nussinov and Y. Zhang, Phys. Lett. B , 311 (2009)[arXiv:0903.2562 [hep-ph]]; W. Buchmuller, A. Ibarra, T. Shindou, F. Takayama and D. Tran,JCAP , 021 (2009) [arXiv:0906.1187 [hep-ph]]; A. Ibarra, D. Tran and C. Weniger, JCAP , 009 (2010) [arXiv:0906.1571 [hep-ph]]; B. Bajc, T. Enkhbat, D. K. Ghosh, G. Senjanovi´cand Y. Zhang, JHEP , 048 (2010) [arXiv:1002.3631 [hep-ph]].[11] For some early studies concerning the Higgs/Higgsino sector of SUSYLR model, see K. Huitu,M. Raidal and J. Maalampi, arXiv:hep-ph/9501255; K. Huitu and J. Maalampi, Phys. Lett. B , 217 (1995); K. Huitu, J. Maalampi and M. Raidal, Nucl. Phys. B , 449 (1994) andreferences therein.[12] C. S. Aulakh and R. N. Mohapatra, Phys. Lett. B , 136 (1982); G. G. Ross andJ. W. F. Valle, Phys. Lett. B , 375 (1985).[13] A. Masiero and J. W. F. Valle, Phys. Lett. B , 273 (1990).[14] M. Arai, K. Huitu, S. K. Rai and K. Rao, JHEP , 082 (2010).[15] M. Hirsch, H. V. Klapdor-Kleingrothaus and S. G. Kovalenko, Phys. Rev. Lett. , 17 (1995). – 27 –
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