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Testing the Mechanism for the LSP Stability at the LHC

The lightest supersymmetric particle (LSP) is a natural candidate for the cold dark matter of the universe. In this Letter we discuss how to test the mechanism responsible for the LSP stability at the LHC. We note that if R-parity is conserved dynamically one should expect a Higgs boson which decays mainly into two right-handed neutrinos (a"leptonic"Higgs) or into two sfermions. The first case could exhibit spectacular lepton number violating signals with four secondary vertices due to the long-lived nature of right-handed neutrinos. These signals, together with the standard channels for the discovery of SUSY, could help to establish the underlying theory at the TeV scale.

underlying theory at the TeV scale.
Theoretical Framework: We begin by reviewing some details of the MSSM. As it is well-known, the superpotential is given by where W RpC is the R-parity conserving part, and contains the R-parity violating terms. The simplest way to imbed R-parity conservation into a gauge symmetry is through B − L where the new gauge group is Now the terms in W RpV are not allowed at the B − L scale. Gauging B − L requires the addition of three copies of righthanded neutrinos to satisfy anomaly cancellation, enhancing the superpotential to where the last term is dependent on the Higgs sector. There are two simple Higgs sector that allow for R-parity conservation: Model I: A pair of Higgses: X,X ∼ (1, 1, 0, ±2) is added to the theory so that the extra term in the above superpotential becomes In this model B − L is broken by the vevs of X andX, and R-parity is an exact symmetry at the TeV scale. Due to the second term in the above equation, the neutrinos are Majorana fermions and the new physical Higgses, X 1 , X 2 and A BL , can decay at tree level into two right-handed neutrinos. These decays are the key to revealing the properties of the B −L breaking Higgses, i.e. indicating if R-parity is really conserved at the TeV scale. In this case radiative symmetry breaking could explain the origin of symmetry breaking at the TeV scale [4]. Model II: S,S ∼ (1, 1, 0, ±n S ) with even n S = 2 , are introduced and the extra term in the superpotential is Here the neutrinos are Dirac fermions and the new physical Higgses, S 1 , S 2 and A S , do not couple to SM fields at tree level. These Higgses can decay at tree level into two sfermions and give rise to peculiar signals in specific SUSY scenarios. It is important to mention that an odd |n S | = 1/2, 1, will have higher-dimensional operators which will affect the stability of the LSP, although it may still live long enough to be a dark matter candidate. Symmetry Breaking: In order to simplify the symmetry breaking discussion we introduce the generic fields, φ,φ ∼ (1, 1, 0, ±n φ ). Then, φ(φ) can be X(X) in Model I or S(S) in Model II. The relevant soft terms for our discussion are: whereB ′ is the B − L gaugino. Spontaneous B − L breaking and R-parity conservation requires the vacuum expectation values (VEV) of φ andφ to be non-zero. Using φ = v/ √ 2 and φ =v/ √ 2 for the VEVs, one finds Assuming that the potential is bounded from below we get: is necessary for a nontrivial vacuum. Minimizing with respect to v andv one gets with tan β ′ = v/v and m 2 ZBL = g 2 BL n 2 φ (v 2 +v 2 )/4. Notice that the minimization conditions are quite similar to the MSSM conditions.
Higgs Sector: The Higgs sector contains two extra CP-even neutral Higgs fields: H 1 and H 2 , and one CP-odd Higgs, A φ , where this general notation will be replaced by the specific notation outlined in the model subsections. The eigenvalues in the CP-even neutral Higgs sector read as with and the mixing angle obeys the following relation: Notice that in the limit, Then, assuming g BL ∼ O(1) and the experimental limit, m ZBL /g BL > 3 TeV [5], one expects two light Higgses, H 1 and A φ , and a heavy one, H 2 , when m A φ is small: a technically natural limit as a massless pseudoscalar corresponds to an enhanced global symmetry of the Lagrangian. Production Mechanisms and Higgs Decays: Testing these scenarios for R-parity conservation requires an understanding of how to produce the B − L Higgses at the LHC and their decays. The following production mechanisms are possible: a) Single Production via Gluon fusion: The single production is quite model dependent while the associated production and the vector boson fusion are very small due to the experimental limits on the Z BL mass. Therefore, we focus mainly on the pair production, which also has the interesting property that while it is not a SUSY process, it is absent in minimal non-SUSY B − L models. We postpone the study of the other production channels for a future publication [6].
The pair production cross section for Model I, pp → X 1 A BL , versus the CP-odd scalar Higgs mass is shown in we show two different Z BL masses: 1 TeV (solid) and 2 TeV (dashed) (assuming g BL at its maximum allowed value in each case), and three different values for the Higgs mass, m X1 = 100 GeV, 200 GeV and 400 GeV. This process also has a mild dependence on the SUSY spectrum through the Z BL width. All results are shown for center of mass energy of 14 TeV. Larger cross sections for larger values of m X1 (for m X1 +m ABL below the m ZBL threshold) correspond to larger value of tan β ′ , a parameter that is uniquely determinable at each point on the curves. For details see [6]. This plot shows that when the pseudoscalar mass, m ABL , is smaller than 500 GeV, the hadronic cross section can be above 1 fb for these input parameters and in the most optimistic region (m X1 < 200 GeV and m ABL < 250 GeV) can go as high as 100 fb indicating a large number of events. Cross sections for Model II do not simply scale as n 2 S as it also effects the width of Z BL , see [6].
With these values for the cross section we are now ready to study the possible decays in these models, focusing on treelevel two-body decays for H 1 and A φ as the most accessible fields. In Model I the CP-even physical Higgs, X 1 , can decay at tree level into two right-handed neutrinos, into two sfermions or into two B − L neutralinos, while the CP-odd Higgs can decay into two right-handed neutrinos, two B − L neutralinos, and X 1 and Z BL . However, since the collider bounds on the Z BL mass are strong and we're focusing on a light Higgs, the decays into two Z BL s and sfermions are suppressed. Therefore, we are mainly interested in the decays into two right-handed neutrinos which would reveal the nature of the B − L Higgses. Notice that in this case the righthanded neutrinos are Majorana fermions and their decays produce lepton number violating signals.
In Model II the physical Higgses, S 1 , S 2 and A S do not have couplings to the fermions at tree level. Then, the CPeven Higgses can decay mainly into two sfermions or two B − L neutralinos. The CP-odd Higgs can decay into Z BL and S 1 or two B − L neutralinos. One possible interesting scenario corresponds to the case where S 1 can decay mainly into selectrons and A S through the three body decays, The possible signals in this model depend of the particular SUSY spectrum. The simplest scenarios will be discussed in the next section.
Since in Model I the Higgses can decay mainly into two right-handed neutrinos, their properties are important when discussing signals at the LHC. The possible decays for N are: N → W ± e ∓ , Zν, νh i , where h i are the MSSM Higgses. The partial width for these decays are proportional to the mixing between the light SM neutrinos and N. The relevant mixing matrix is given by Here V P MN S is the PMNS mixing matrix, m = diag(m ν1 , m ν2 , m ν3 ) are the physical neutrino masses, and Ω is a complex orthogonal matrix, which conveniently parameterizes some of the unknown degrees of freedom of the neutrino sector. In Fig. 3 we show lines of constant decay length for the right-handed neutrinos in the right-handed neutrino mass−lightest left-handed neutrino mass plane for a normal hierarchy (NH) spectrum for the light neutrinos. Here we have used the central values for the atmosphere and solar mass squared difference, the tri-bimaximal ansatz and assumed Ω = 1. As can be appreciated from Fig. 3, the righthanded neutrinos are long-lived in the full parameter space considered implying the existence of secondary vertices. For a detailed study of the right-handed decays see Ref. [7].
Signals at the LHC: In Model I we will focus on the light Higgs bosons scenario, with m X1 > 2m N , which decay only into two right-handed neutrinos. This corresponds to the most optimistic case where one has lepton number violation through the right-handed neutrino decays. For single production of the B −L Higgs boson X 1 , the following lepton number violating signals at the LHC are possible: Since the single production depends heavily on the SUSY spectrum we shift our focus to the pair production as the arena for testing the stability of the LSP. In this case one has leading to four same-sign leptons and eight jets in the final state and allowing for observation of lepton number violation. These signals are even more spectacular once we consider the fact that the right-handed neutrinos are long-lived, which lead to four secondary vertices. The topology of these event are shown in Fig. 1.
We can perform a naive estimate to gain an appreciation for the number of events for this four same-sign leptons and 8 jets signal using a cross section of 100 fb and an integrated luminosity of 100 fb −1 : Where the BR (X 1 → N 1 N 1 ) = 1/3 due to the three possible generations of right-handed neutrinos and BR (N 1 → e + W − ) ≈ 3/10 due to model specific parameters and kinematics, see [6] and the benchmark defined below. In order to make a realistic calculation of the number of events we pick a benchmark scenario: The sfermion masses effect the Z BL width. In this case σ pp→X1ABL = 65.7 fb, and we display the predicted number of events in Table I for the five possible final states with an e and/or a µ. We also show the combinatorics factor which takes into account the branching ratios of the Higgses into righthanded neutrinos, right-handed neutrinos into leptons and W s into jets. This number can be multiplied by any cross section and integrated luminosity to yield the number of events. In this case, if we ignore the displaced vertices, the main SM background is ttW ± ttW ± and it has a negligible cross section so that while there are only a few number of events, they are background free. This does not change the fact though that the reconstruction would be quite challenging due to the   µ) for a luminosity of 100 fb −1 and a pair production cross section of 65.7 fb corresponding to benchmark I. We also display the combinatorics factor which combines the branching ratios for the Higgses into right-handed neutrinos, right-handed neutrinos to specific leptonic final states and W s into jets.
presence of eight jets in the final state. Imposing the condition that the invariant mass of two jets, |M (jj) − M W | < 15 GeV [7], can improve the reconstruction process as well as the order millimeter displaced vertices due the long lifetimes of the right-handed neutrinos.
In Model II the Higgs, S 1 , can decay mainly into two sfermions. Here, for simplicity we focus on a scenario where the gravitino is the LSP with a simplified spectrum: mG < mχ 1 < mẽ i , mq i , mν i < m S1 /2. This type of spectrum could be obtained in gauge mediation. Assuming that the neutralino is Bino-like, the pair production could lead to signals with mutileptons, multiphotons and missing energy: pp → S 1 A S →ẽ * ẽ S 1 e + i e − i → e ± e ∓ e ± e ∓ e + i e − i γγγγ + E miss T In principle, these type of signals are quite spectacular because they could also include displaced vertices due to the lifetime of the neutralinos. Unfortunately, the branching ratio of the Higgs into sfermions is quite spectrum dependent. For a recent study of long-lived neutralinos see Ref. [8]. We postpone the study of these channels for a future publication.

Summary and Outlook:
In this letter we have discussed how to test the mechanism responsible for the LSP stability at the LHC. We note that if R-parity is conserved dynamically, a Higgs boson which decays mainly into two right-handed neutrinos ("leptonic" Higgs) or into two sfermions is likely. In the first case one could have spectacular lepton number violating signals with four secondary vertices due to the existence of long-lived right-handed neutrinos. The second case could have peculiar signals with multileptons and multiphotons. A more detailed study will appear in a future publication [6]. These signals, together with the standard channels for the discovery of SUSY, could help us to establish the underlying theory at the TeV scale.