The $R$-parity Violating Decays of Charginos and Neutralinos in the B-L MSSM

The $B-L$ MSSM is the MSSM with three right-handed neutrino chiral multiplets and gauged $B-L$ symmetry. The $B-L$ symmetry is broken by the third family right-handed sneutrino acquiring a VEV, thus spontaneously breaking $R$-parity. Within a natural range of soft supersymmetry breaking parameters, it is shown that a large and uncorrelated number of initial values satisfy all present phenomenological constraints; including the correct masses for the $W^{\pm}$, $Z^0$ bosons, having all sparticles exceeding their present lower bounds and giving the experimentally measured value for the Higgs boson. For this"valid"set of initial values, there are a number of different LSPs, each occurring a calculable number of times. We plot this statistically and determine that among the most prevalent LSPs are chargino and neutralino mass eigenstates. In this paper, the $R$-parity violating decay channels of charginos and neutralinos to standard model particles are determined, and the interaction vertices and decay rates computed analytically. These results are valid for any chargino and neutralino, regardless of whether or not they are the LSP. For chargino and neutralino LSPs, we will-- in a subsequent series of papers --present a numerical study of their RPV decays evaluated statistically over the range of associated valid initial points.


The B-L MSSM
In this section, we briefly review the contents of the B − L MSSM theory relevant to a phenomenological discussion of its R-parity violating decay processes and their potential signatures at the LHC. The low energy manifestation of the "heterotic standard model", that is, the B − L MSSM, arises from the breaking of an SO(10) GUT theory via two independent Wilson lines, denoted by χ 3R and χ B−L , associated with the diagonal T 3R generator of SU (2) R and the generator T B−L of U (1) B−L respectively. These specific generators are chosen since it can be shown that there is no kinetic mixing of their respective Abelian gauge kinetic terms at any energy scale-thus simplifying the RG calculations [42]. However, identical physical results will be obtained for any linear combination of these generators. Associated with these Wilson lines are two mass scales, M χ 3R and M χ B−L , with three possible relations between them; 1) M χ B−L > M χ 3R , 2) M χ 3R > M χ B−L and 3) M χ 3R = M χ B−L . As discussed in [42], the masses in the first two relations can be adjusted so as to enforce exact unification at one loop of all gauge couplings at the SO(10) unification scale M U , whereas gauge unification cannot occur for the third mass relationship without accounting for threshold effects at the unification scale or the SUSY scale [23,42,58]. For this reason, we will -6 -not consider the third option in this paper. The gauge coupling RG equations associated with each of the first two mass relations were discussed in detail in [42] and, as far as low energy LHC phenomenology is concerned, give almost identical results. For specificity, therefore, in this paper we will focus on the first relationship and, without loss of accuracy, choose M χ B−L = M U . The lower scale M χ 3R , which we henceforth denote by M I , is adjusted so as to obtain exact gauge coupling unification. We emphasize, however, that the low energy results predicted for the LHC are almost unchanged even if M I is chosen to yield only "approximate" gauge unification -with moderate sized gauge "thresholds". Conventionally, the scale of supersymmetry breaking is defined to be where mt 1 and mt 2 are the lightest and heaviest stop masses respectively; see ,for example, [44].
Suffice it here to say that for supersymmetry breaking to occur between the electroweak scale and 10 TeV, which will be the case in this paper, the unification scale M U is found to be O(3 × 10 16 GeV). Over the same range of supersymmetry breaking, however, the intermediate scale M I changes from O(2 × 10 16 GeV) to O(3 × 10 15 GeV) respectively [23]. The details of the symmetry breaking and the respective mass spectra for this choice of Wilson line hierarchy were given in [42]. Here, we simply note that in the mass regime between M U and M I , the gauge group is broken from SO(10) to SU (3) C × SU (2) L × SU (2) R × U (1) B−L with the spectrum shown in Figure 1. This theory is referred to as the "left-right" model [59,60]. As discussed above, for the supersymmetry breaking scales of interest in this paper, this mass regime will on average be considerably smaller than one order of magnitude in GeV. At the "intermediate" scale M I , the second Wilson line breaks this "left-right" model down to the exact B − L MSSM. This theory has the SU (3) C × SU (2) L × U Y (1) gauge group of the standard model augmented by an additional U (1) B−L Abelian symmetry. As mentioned above, it is convenient-and equivalent -to use the Abelian group U (1) 3R with the generator in the RGE's since the associated gauge kinetic term cannot mix with the gauge kinetic energy of U (1) B−L . That is, the B − L MSSM gauge group is chosen, for computational convenience, to be 3) The associated gauge couplings will be denoted by g 3 , g 2 , g R and g BL . The spectrum, as shown in Figure 1, is exactly that of the MSSM with three right-handed neutrino chiral multiplets, one per family; that is, three generations of matter superfields The superpotential of the B − L MSSM is given by where flavor and gauge indices have been suppressed and the Yukawa couplings are three-bythree matrices in flavor space. In principle, the Yukawa matrices are arbitrary complex matrices. However, the observed smallness of the three CKM mixing angles and the CP-violating phase dictate that the quark Yukawa matrices be taken to be nearly diagonal and real. The charged lepton Yukawa coupling matrix can also be chosen to be diagonal and real. This is accomplished by moving the rotation angles and phases into the neutrino Yukawa couplings which, henceforth, must be complex matrices. Furthermore, the smallness of the first and second family fermion masses implies that all components of the up, down quark and charged lepton Yukawa couplings-with the -8 -exception of the (3,3) components -can be neglected for the purposes of the RG running. Similarly, the very light neutrino masses imply that the neutrino Yukawa couplings are sufficiently small so as to be neglected for the purposes of RG running. However, the Y νi3 , i = 1, 2, 3 neutrino Yukawa couplings cannot be neglected for the calculations of the neutralino, neutrino and chargino mass matrices, as well as in decay rates/branching ratios. The µ-parameter can be chosen to be real, but not necessarily positive, without loss of generality. We implement these constraints in the remainder of our analysis. Spontaneous supersymmetry breaking is assumed to occur in a hidden sector-a natural feature of both strongly and weakly coupled E 8 × E 8 heterotic string theory -and be transmitted through gravitational mediation to the observable sector and, hence, to the B − L MSSM. Since the B − L MSSM first manifests itself at the scale M I , we will begin our analysis by presenting the most general soft supersymmetry breaking interactions at that scale. That is, at scale M I , the soft supersymmetry breaking Lagrangian is given by (2.7) The b parameter can be chosen to be real and positive without loss of generality. The gaugino soft masses can, in principle, be complex. This, however, could lead to CP-violating effects that are not observed. Therefore, we proceed by assuming they all are real. The a-parameters and scalar soft mass can, in general, be Hermitian matrices in family space. Again, however, this could lead to unobserved flavor and CP violation. Therefore, we will assume they all are diagonal and real. Furthermore, we assume that only the (3,3) components of the up, down quark and charged lepton a-parameters are significant and that the neutrino a parameters are negligible for the RG running and all other purposes. For more explanation of these assumptions, see [44]. As discussed in [44], without loss of generality one can assume that the third generation righthanded sneutrino, since it carries the appropriate T 3R and B − L charges, spontaneous breaks the B − L symmetry by developing a non-vanishing VEV This VEV spontaneously breaks U (1) 3R ⊗U (1) B−L down to the hypercharge gauge group U (1) Y . We denote the associated gauge parameter by g . However, since sneutrinos are singlets under the SU (3) C ⊗ SU (2) L ⊗ U (1) Y gauge group, it does not break any of the SM symmetries. At a lower mass scale, electroweak symmetry is spontaneously broken by the neutral components of both the up and down Higgs multiplets acquiring non-zero VEV's. In combination with the right-handed sneutrino VEV, this also induces a VEV in each of the three generations of left-handed sneutrinos. The notation for the relevant VEVs is where i = 1, 2, 3 is the generation index. The neutral gauge boson that becomes massive due to B − L symmetry breaking, Z R , has a mass at leading order, in the relevant limit that v R v, of The second term in the parenthesis is a small effect due to mixing in the neutral gauge boson sector. A discussion of the neutrino masses is presented in the next section, where they are shown to be roughly proportional to the Y ν ij and v Li parameters. It follows that Y ν ij 1 and v Li v u,d , v R . In this phenomenologically relevant limit, the minimization conditions of the potential are simple, leading to the VEV's Here, the first two equations correspond to the sneutrino VEVs. The third and fourth equations are of the same form as in the MSSM, but new B − L scale contributions to m Hu and m H d shift their values significantly compared to the MSSM. Eq. (2.12) can be used to re-express the Z R mass as This makes it clear that, to leading order, the Z R mass is determined by the soft SUSY breaking mass of the third family right-handed sneutrino. The term proportional to v 2 /v 2 R is insignificant in comparison and, henceforth, neglected in our calculations.
Recall that R-parity is defined as It follows that a direct consequence of generating a VEV for the third family sneutrino is the spontaneous breaking of B − L symmetry and, hence, R-parity. The R-parity violating operators induced in the superpotential are given by and Y ei is the ith component of the diagonal lepton Yukawa coupling. This general pattern of Rparity violation is referred to as bilinear R-parity breaking and has been discussed in many different contexts [61][62][63][64]. In addition, the Lagrangian contains bilinear terms generated by v Li and v R in the super-covariant derivatives. These are The consequences of spontaneous R-parity violation are quite interesting, and have been discussed in a number of papers [45][46][47][48][49][50][65][66][67]. In this paper, we will present the decay channels for arbitrary mass charginos and neutralinos, and analytically determine their decay rates. However, in a series of following works, we will explore the phenomenological consequences of the R-parity violating (RPV) decays of the lightest, and next-to-lightest, supersymmetric particles; referred to as the LSP and NLSP respectively. These decays are potentially observable at the ATLAS detector of the LHC. Hence, if detected, these explicit decays could verify the existence of low energy N = 1 supersymmetry, shed light on the structure of the precise supersymmetric model-such as the B − L MSSM -and, as will become apparent, even constrain whether the neutrino mass hierarchy is "normal" or "inverted". However, as is clear from expressions (2.19) and (2.21), these results will depend explicitly on the values of the parameters i , i = 1, 2, 3 and v L i , i = 1, 2, 3 defined in (2.20) and (2.13) respectively. In turn, these parameters are dependent on the present experimental values of the neutrino masses. These reduce the number of independent RPV parameters from six to one and potentially restrict the value of the remaining independent coefficient. For that reason, we will discuss the neutrino masses and their direct relationship to the i and v L i parameters in the next section.

Neutrino Masses and the RPV Parameters
As discussed in [38,43,44,52,66], it follows from the above Lagrangian that the third family right-handed neutrino and the three left-chiral neutrinos ν i , i = 1, 2, 3 mix with the fermionic superpartners of the neutral gauge bosons and with the up-and down-neutral Higgsinos. In other words, the neutralinos now mix with the neutral fermions of the standard model. The mixing with the third-family right-handed sneutrino, through terms proportional to i = Y ν i3 v R / √ 2 and v L i , allows the third-family right-handed sneutrino to act as a seesaw field giving rise to Majorana neutrino masses. This is reviewed in this section.
First, we note that this paper is focused on the consequences of RPV decays at the LHC. There is the possibility that the RPV parameters are so small that the LSP decay length is too long for it to decay within the detector. Then the LSP would be effectively stable within the detector. For certain cases, such effectively stable sparticles have been searched for in [68]. If the LSP decay length is small enough to decay within the detector, but greater than about 1 mm, this would lead to "displaced" vertices, such as those searched for in, for example, [69]. In the present paper, we will -11 -choose parameters so that the decay length of the LSP, whatever sparticle that may be, is less than about 1 mm. We refer to such decays as "prompt" decays. Therefore, even though the analysis in this work is valid for any mass chargino and neutralino, should we choose the initial conditions so that they are the LSP, then their RPV decays will be prompt.
As was shown in the case of stops and sbottoms in [38,52], prompt decays require the RPV parameters to be large enough to allow for significant Majorana neutrino masses. We expect the same to hold true for a variety of LSPs. Therefore, in this paper we focus on the case of significant Majorana neutrino masses. Note that, in addition to these Majorana neutrino masses, there can be pure Dirac mass contributions coming from the neutrino Yukawa coupling. The components Y ν i3 , which couple the left-handed neutrinos to the third-family right-handed neutrino, allow the third-family right-handed neutrino to act as a seesaw field and give rise to Majorana neutrino masses. The other components, Y ν i1 and Y ν i2 , couple the left-handed neutrinos to the first-and second-family right-handed neutrinos. Note that in this model, the heavy third-family right-handed neutrino acts as a seesaw field, while the first-and second-family right-handed neutrinos remain as light sterile neutrinos. This means that the Dirac mass terms related to Y ν i1 and Y ν i2 can give rise to active-sterile oscillations in the neutrino sector. There have been some experimental hints of such oscillations, see [70] for review. However, it is not yet clear that these results are due to true active-sterile oscillations. Hence, we proceed under the assumption that no such oscillations exist and that the Y ν i 1 and Y ν i 2 components of the neutrino Yukawa coupling must, therefore, be negligible, so they do not appear in the neutralino mass matrix below. It may be interesting to revisit the question of active-sterile neutrino oscillations in the B − L MSSM in the future, perhaps after there is more experimental data.
In the basis where Mχ0 is a six-by-six matrix of order a TeV given by and m D is a six-by-three matrix of order an MeV. This allows the mass matrix to be diagonalized perturbatively. Note that we have suppressed all terms of the form v L i Y νij in Mχ0 since both v L i and the neutrino Yukawa parameters are small. In addition, we emphasize that since only the third family right-handed sneutrino gets a non-vanishing VEV, only ν c 3 couples to the gauginos/Higgsinos. It follows that only the Dirac mass of the third-family neutrino enters the above mass matrix, whereas the the first and second family Dirac neutrino masses are excluded.
The entire mass matrix Mχ0 in (3.1) can be diagonalized to where N is the matrix that diagonalizes Mχ0 given in eq. (3.2). Requiring that M D χ 0 be diagonal yields The second matrix on the right-hand side of N rotates away the neutralino/left-handed neutrino mixing, whereas the first matrix diagonalizes the six neutralino/third family right-handed neutrino states as well as the three left-chiral neutrino states. In this section, we will consider the diagonal 3 × 3 left-handed neutrino Majorana mass matrix only, returning to the diagonal neutralino mass matrix later in the paper.
The diagonal left-chiral neutrino Majorana mass matrix is found to be The 3 × 3 matrix m ν is given by [38] and (3.12) As will be discussed in detail below, the soft mass parameters are all initialized statistically at the scale M I , whereas the measured values of the gauge couplings are introduced at the electroweak scale. All of these parameters are then run to the appropriate energy scale using the RGEs discussed in detail in [44]. Additionally, the value of tanβ will be chosen statistically within a physically relevant interval and, for a given value of tanβ, the parameters v u and v d are the measured Higgs VEVs. Finally, for any given set of statistical initial data, we fine-tune the value of the parameter µ using equation (2.14), so as to obtain the experimental value of the electroweak gauge boson Z 0 and, hence, the measured values for W ± as well. The 3 × 3 Pontecorvo-Maki-Nakagawa-Sakata matrix is with c ab (s ab ) = cos θ ab (sin θ ab ). The mixing angles and phases are determined by neutrino experiments. For the mixing angles, we use the values and uncertainties from [71]. They are sin 2 θ 12 = 0.307 ± 0.013 , sin 2 θ 13 = (2.12 ± 0.08) × 10 −2 (3.14) for both the normal and inverted neutrino mass hierarchies. For θ 23 , however, the best-fit values depend on the hierarchy, and the data admits multiple best-fit values. In the normal hierarchy, one finds sin 2 θ 23 = 0.417 +0.025 In this paper, we will do a complete study of all four of the cases in equations (3.15) and (3.16).
Regarding the CP-violating phase, δ, we use the recent results in [71] that in the normal hierarchy In addition, note that there is only one "Majorana" phase, that is, parameter A, since in both the normal and the inverted hierarchy one of the neutrinos is massless and, therefore, does not have a Majorana mass. The value of A is unknown and, hence, in this paper we will simply throw it statistically in the interval [0 • , 360 • ]. The mathematical expressions for the mass eigenvalues of the Majorana neutrino mass matrix m D νij can be constructed from the A,B,C components of m νij given in (3.9), (3.10) and (3.11) respectively, as well as from the PMNS matrix given in (3.13). This has been done in detail in [38], to which we refer the reader for details. Given values for all the relevant parameters discussed above, and the measured values for the neutrino mass eigenvalues for the normal and inverted hierarchies, this allows one to solve for the RPV parameters i , v L i i = 1, 2, 3. Respectively, the experimental values of the mass eigenvalues of the normal and inverted hierarchies are [70] • Normal Hierarchy: In each case, all three v L parameters as well as two of the parameters can be determined in terms of a third parameter. The explicit expressions, of course, differ in the normal and inverted hierarchy cases, and are presented in detail in [38]. These are encoded into the computer program by which we determine all decay rates and branching ratios and won't be presented here. Suffice it to say that, in each case, which parameter i is inputted is undetermined. Thus, we will statistically decide which of the three dimension one i parameters is selected. Furthermore, we choose its value by randomly throwing it to be in the interval [10 −4 , 1.0] GeV with a log-uniform distribution. We limit the upper bound to 1.0 GeV to avoid excessive fine-tuning in the neutrino masses. Furthermore, we cut off the lower bound at 10 −4 GeV-although this could be taken to 0 -to enhance the readability of our branching ratio plots. It is important to note that, having statistically chosen one of the parameters in the above range, the other two parameters are determined by the computer code and are not necessarily bounded by this interval. For example, at least one of the remaining two parameters could, in principle, be considerably larger than 1.0 GeV. If so, this could have important consequences for for the suppression of lepton number violating interactions. The reason is the following.
It was shown in [54], and discussed in [38], that the experimental bound on the decay of µ −→ eγ leads to the constraint on 1 and 2 that Scanning the initial parameters in the B − L MSSM, using (2.10), (2.17) and the values for the gauge parameters discussed in [42], we find that this becomes Therefore, to adequately suppress lepton number violating decays, it is essential to show that this bound is satisfied for any physically interesting set of initial data in this analysis. At the end of Section 4, we will demonstrate that for the LSPs of interest in this and in the follow-up papers, that is, for charginos and neutralinos, constraint (3.22) is easily satisfied.  The sign of µ and the various soft parameters of the form M and a are chosen randomly to have either a + or -sign.
• Randomly Scattered Choice of tanβ: The upper and lower bounds for tanβ are taken from [51] and are consistent with present bounds that ensure perturbative Yukawa couplings.
In addition to being subject to the above constraints, physically acceptable initial conditions are those which lead to the following phenomenological results. First, B −L gauge symmetry must be spontaneously broken at a sufficiently high scale. Presently, the measured lower bound on the Z R mass is given by [72] M Z R = 4.1 TeV . (4.4) Secondly, electroweak (EW) symmetry must be spontaneously broken so that the Z 0 and W ± masses have the measured values of [70] M Z 0 = 91.1876 ± 0.0021 GeV, M W ± = 80.379 ± 0.012 GeV . Third, the remaining sparticles must be above their measured lower bounds [44] given in Table 1 400 GeV Sbottom LSP 500 GeV Gluino 1300 GeV Table 1: Current lower bounds on the SUSY particle masses.
Finally, the Higgs mass must be within the 3σ allowed range from ATLAS combined run 1 and run 2 results [73]. This is found to be We now want to search for physically acceptable initial data, subject to all of the constraints and phenomenological conditions introduced above. Before applying any of these constraints, the number of parameters appearing in the B − L MSSM greatly exceeds 100. However, subject to the constraints discussed above, this number is significantly reduced-down to only 24 soft SUSY breaking parameters, as well as tanβ and µ. The RG code [44] that we use in this analysis involves all 24 SUSY breaking parameters. It is, however, helpful to point out that many of the RGE's are dominated by two specific sums of these parameters given by where the traces are over generational indices. This is helpful in that one can now reasonably plot initial data points in two-dimensional S BL − S 3R space, rather than in the full 24-dimensional space of all parameters. At the electroweak scale, we randomly set the value of tanβ. Furthermore, and importantly, we do not run the parameter µ. Rather, after running all other parameters down to the electroweak scale, we fine-tune µ to give the measured values for the electroweak gauge bosons, as discussed above. Searching for physically acceptable initial data, subject to all of the constraints and phenomenological conditions above, we find the following. For 100 million sets of randomly scattered initial conditions, it is found that 4,351,809 break B − L symmetry with the Z R mass above the lower bound in equation (4.4). These are plotted as the green points in Figure 2. Running the RG down to the EW scale, one finds that of these 4,351,809 appropriate B − L initial points, only 3,142,657 break electroweak symmetry with the experimentally measured values for M Z 0 and M W ± given in equation (4.5). These are shown as the purple points in the Figure. Now applying the constraints that all sparticle masses be at or above their currently measured lower bounds presented in Table 1, we find that of these 3,142,657 initial points, only 342,236 are acceptable.
These are indicated by cyan colored points in the Figure. Finally, it turns out that of these 342,236 points, only 67,576 also lead to the currently measured Higgs mass given in equation (4.6). That is, of the 100 million sets of randomly scattered initial conditions, 67,576 satisfy all present phenomenological requirements. In Figure 2, we represent these "valid" points in black. That is, of the 100 million randomly scattered initial points, approximately .067% satisfy all present experimental conditions. Although this might-at first sight -appear to be a small percentage, it is worth noting that these initial points not only break B − L symmetry appropriately and have all sparticle masses above their present experimental lower bounds, but also give the measured experimental values for the mass of the EW gauge bosons and, remarkably, the Higgs boson mass as well! From this point of view, this percentage of valid black points seems remarkably high. The electroweak gauge boson masses were obtained, as discussed above, by fine-tuning the parameter µ. For example, a typical value of the fine-tuning of µ is of the order of 1 in 1000 [43]. However, one might also be concerned that getting the Higgs mass correct might require some other fine-tuning of the 24 initial parameters that may not be apparent. However, in previous work [44] it was shown that the 24 parameters associated with any given black point are generically widely disparate with no apparent other fine-tuning. Figure 2: Plot of the 100 million initial data points for the RG analysis evaluated at M I . The 4,351,809 green points lead to appropriate breaking of the B−L symmetry. Of these, the 3,142,657 purple points also break the EW symmetry with the correct vector boson masses. The cyan points correspond to 342,236 initial points that, in addition to appropriate B − L and EW breaking, also satisfy all lower bounds on the sparticle masses. Finally, as a subset of these 342,236 initial points, there are 67,576 valid black points which lead to the experimentally measured value of the Higgs boson mass.
We conclude that the B − L MSSM, in addition to arising as a vacuum of heterotic M-theory and having exactly the mass spectrum of the MSSM, satisfies all present experimental low-energy -18 -physical bounds for a remarkably large number of disparate initial data points. Given this, it becomes of real interest to determine whether the RPV decays of the B − L MSSM can be directly observed at the LHC at CERN. These decays are most easily observed in the lightest sparticles in the mass spectrum; that is, the LSP has the best prospects for RPV detection in general. There are, however, cases in which the next lightest supersymmetric particle (NLSP) is highly degenerate in mass with the LSP-see examples presented in [44] -and, hence, their RPV decay channels become relevant as well. Hence, in the sections to follow, we compute the decays of charginos and neutralinos without making any assumptions regarding their masses. As discussed in detail in [44], the particle spectrum of each of the 67,576 valid black points is exactly determined by the computer code. It follows that we can compute the LSP associated with each valid black point. It turns out that there are many possible different LSPs. Before enumerating these, however, we must be more specific about the definition and structure of any LSP. Although the original fields entering the B − L MSSM Lagrangian are "gauge" eigenstates, the LSP associated with a given valid black point is, by definition, a "mass" eigenstate-generically a linear combination of the original fields. For example, as will be discussed in detail in subsection 5.1, the lightest mass eigenstate chargino of either charge, which we denote byχ ± 1 , is found to be an R-parity conserving linear combination of the charged Wino,W ± , and the charged Higgsino,H ± , added to RPV terms proportional to the left and right chiral charged leptons. As discussed in subsection 5.1, the RPV coefficients are very small and, hence, can be ignored in the discussion of the masses of the charginos. Therefore, in this section, since we are analyzing the possible LSPs, we will consider the R-parity conserving part of the chargino states only. It then follows from the discussion in subsection 5.1 that when M 2 < |µ| the lightest chargino is given byχ whereas for |µ|< |M 2 |χ (4.10) The angles φ ± are exactly determined for any given black point. It follows that for some black points the mass eigenstateχ ± 1 is predominantly a charged Wino, whereas for other black points it is mainly a charged Higgsino. We will, henceforth, denote the first and second type of mass eigenstates byχ ± W andχ ± H , and refer to them as "Wino charginos" and "Higgsino charginos" respectively. That is, instead of labelling a chargino LSP simply as χ ± 1 , and counting the number of valid black points associated with it, we can be more specific-breaking the chargino LSP into two different types of states,χ ± W andχ ± H respectively, and counting the number of black points associated with each type individually. This gives additional information about the structure of the LSPs.
With this in mind, we have calculated the LSP associated with each of the 67,576 valid black points and plotted the results as a histogram in Figure 3. The notation for the various possible LSPs is specified in Table 2. For example, out of the 67,576 valid black points, there are 4,858 that have aχ ± W Wino chargino as their LSP. Similarly, out of all the valid black point initial conditions, 4,869 have aχ 0 W Wino neutralino as their LSP. And so on. Notice that the cases in which the chargino LSP is dominantly a charged Higgsino-that is,χ ± H -are rare. In fact, in Figure 3 there is precisely one such black point. As discussed above and shown in Section 5, the lighter chargino state is dominantly Wino if |M 2 |< |µ|, and dominantly Higgsino if |µ|< |M 2 |. The little hierarchy problem tells us that µ is generally large, of the order of a few TeV. However, the M 2 parameter generally takes smaller values in our simulation. For this reason, the instances in which |µ|< |M 2 |required for the Higgsino chargino to be the LSP -are scarce. showing the percentage of valid black points with a given LSP. Sparticles which did not appear as LSPs are omitted. The y-axis has a log scale. The notation and discussion of the sparticle symbols on the x-axis is presented in Table 2.
For any given choice of LSP, we can plot the number of such points as a function of their masses in GeV. As an example, Figures 4 (a) and (b) present such a mass distribution for Wino chargino and Wino neutralino LSPs respectively. We obtain viable supersymmetric spectra with Wino chargino and Wino neutralino LSP masses ranging from about 200 GeV to 1700 GeV. A striking feature of the Wino chargino and Wino neutralino LSP mass distributions in Figure 4 is the peak towards the low mass values. Higher LSP masses are exponentially less probable. The reason is that we sample all soft mass terms log-uniformly in the interval [200 GeV, 10 TeV]. This includes the M 2 gaugino mass term, which gives the dominant contribution for both the Wino chargino and Wino neutralino masses, see (5.6) and (5.47) respectively. If we would plot all the Wino chargino or Wino neutralino masses for all the viable points in our simulation, we would obtain an almost uniform mass distribution. However, for the Wino charginos or Wino neutralinos to be the LSPs, their masses must be lower than all the other random soft masses in our simulation. Conversely, it demands that all the other random soft mass terms be larger than a Wino chargino or Wino neutralino mass value for each viable point. This is exponentially less likely as this mass value increases, following a Boltzmann distribution. We point out that this discussion is a simplification of what actually happens, since it omits the running of the soft mass terms, as well as their mixing in the final mass eigenstates. These details, however, do not effect the essence of the above argument, since the mass runnings and the mass mixing couplings are generically very small. Associated with a given choice of LSP, there are a fixed number of valid initial points. For example, as mentioned above, a Wino chargino LSP arises from 4,858 black points. As discussed at the end of Section 3, for each such black point, we 1) statistically throw one of the parameters i , i = 1, 2, 3 in the interval [10 −4 , 1.0] GeV, 2) choose the neutrino mass hierarchy to be either normal or inverted and, having done so, choose the associated value of θ 23 , 3) then, using (3.8), determine the remaining two epsilon parameters and the three v L parameters using the computer code. Let us denote the maximum one of the three parameters by max . By running over all 4,858 black points subject to a fixed choice of the neutrino hierarchy and θ 23 , one can create a histogram of the number of valid points associated with a given value for max . For example, the results of, first, choosing a normal neutrino hierarchy and θ 23 such that sin θ 23 = 0.597 and, second, choosing an inverted neutrino hierarchy and θ 23 with sin θ 23 = 0.529 are graphically depicted in Figure 5. We find only a statistically insignificant number of points, in the case of the normal hierarchy 1 point and in the case of the inverted hierarchy 4 points, that exceed √ 68 GeV. If max = 3 , then constraint (3.22) is immediately satisfied. Even if this parameter is, say, 2 , it remains statistically extremely likely that constraint (3.22) remains satisfied. It follows that, for the choice of a normal neutrino hierarchy and sin θ 23 = 0.597 and an inverted hierarchy with sin θ 23 = 0.529, lepton number violation via µ → eγ is statistically highly suppressed in our theory. We find similar results for each of the other two choices of θ 23 . The identical conclusion can be drawn for the Wino neutralino. We conclude that lepton number violation is highly suppressed in the B − L MSSM-at least when the LSP is a Wino chargino or a Wino neutralino.
In previous papers [38,52], we have analyzed in detail the RPV decays of the "admixture" stop. Stops have a very high production cross section from proton-proton collisions. Furthermore, their decay products are relatively easy to observe at the LHC detectors. For these reasons, the ATLAS group at the LHC did a detailed study of the RPV decays of the admixture stop LSPs [73][74][75][76]. However, it is clear from Figure 3 that neutralinos and charginos are much more prevalent as -21 - LSPs of the B − L MSSM. Therefore, in the present paper, we begin a study of the RPV decays of neutralinos and charginos. Their RPV decay channels will be analyzed in detail.

Chargino mass eigenstates
After EW breaking, the Higgs fields acquire a VEV which induces off-diagonal couplings between the charged gauginos of the theory. The terms that enter the chargino mass matrix, in the absence of RPV effects, are The first terms come from the supercovariant derivative of the Higgs chiral fields, the Wino mass term originates in the soft SUSY breaking Lagrangian, while the last term is introduced in the superpotential W . Combining the charged Higgsinos and the charged Winos into ψ + = (W + ,H + u ) Mostly third generation right-handed neutrino. 1st and 2nd generation right-handed sneutrinos. τ Third generation left-handed stau. e c ,μ c 1st and 2nd generation right-handed sleptons. LSPs evenly split among two generations. τ c Third generation right-handed stau. Table 2: The notation used for the LSP states on the x-axis of Figure 3.
, we can write the previous terms in the form where Mχ± is the 2 × 2 matrix given by The mass eigenstatesχ with Mχ± 1 and Mχ± 2 positive. One can solve analytically for the eigenvalues and obtain correspond to the − and + sign in front of the square root respectively.
We will always choose the square root to be positive, so that the "minus" sign-and, hence,χ ± 1 -corresponds to the lighter mass eigenstate. That is, with this convention Mχ± The expressions for the mass eigenvalues can be simplified by noting that the lower bounds on sparticle masses are well above M W ± . It follows that M 2 W ± M 2 2 , µ 2 . Therefore, the mass eigenvalues depend primarily on the parameters M 2 and µ. When |M 2 | |µ|, we find that whereas for |µ| |M 2 |, the expressions for the mass eigenvalues are simply exchanged; that is, The mixing matrices U and V , defined by also are dependent on the relative sizes of M 2 and µ. For |M 2 | |µ| they are found to be The Pauli matrix σ 3 is inserted so that the diagonal entries of M D are always positive. The angles φ ± are given by respectively. On the other hand, when |µ| |M 2 |, we find that (5.11) remains the same, as do the expressions (5.13) and (5.14) for the angles φ ± . However, the matrix O ± now becomes It is important to note that the |µ| |M 2 | results for the U , V matrices can be obtained from the |M 2 | |µ| expressions (5.11), (5.12), (5.13) and (5.14) simply by replacing in all expressions. We will use this replacement, when required, in the main numerical analysis to follow.
It is useful to note that when |M 2 | |µ|, it follows from (5.12) that the lightest chargino eigenstate isχ , µ 2 , we see from (5.13) and (5.14) that tan 2φ ± 1 and, hence, |cos φ ± |> |sin φ ± |. It follows thatχ We say thatχ ± 1 is "predominantly" a charged Wino and, regardless of the exact value of φ ± , denote it byχ ± W . On the other hand, when |µ| |M 2 | it follows from (5.15) that Expressions (5.13) and (5.14) again tell us that |sin φ ± |< |cos φ ± | and, hencẽ That is,χ ± 1 is "predominantly" a charged Higgsino and, regardless of the exact value of φ ± , we denote it byχ ± H . Having made this analysis, we note that the results could have been read off directly from the leading term in the expressions for the mass eigenvalues given in (5.6) and (5.8). Specifically, for |M 2 | |µ|, the leading term in (5.6) is |M 2 |, the soft mass associated with the charged Wino-indicating thatχ ± 1 W ± . Similarly, for |µ| |M 2 |, the leading term in (5.8) is |µ|, the parameter associated with the charged Higgsino-indicating thatχ ± 1 H ± , as above. As stated previously, we will refer to the mass eigenstatesχ ± W andχ ± H simply as a Wino chargino and Higgsino chargino respectively, even though they are only "predominantly" the pure states of W ± and H ± respectively.
Let us now include the R-parity violation terms in the Lagrangian. These will not significantly affect the chargino masses, but they do introduce mixing between the charginos and the standard model charged leptons. These mixings are central to our study because they allow RPV chargino decays. In the extended bases ψ + = (W + ,H + u , e c i ) and ψ − = (W − ,H − d , e i ), the mixing matrix can again be written in the form of eq. (5.2), and m e , m µ , and m τ denote the Dirac masses of the standard model charged leptons. This matrix can be expressed in a schematic form that will be useful in diagonalizing it. Let us write (5.24) and m e i is the 3 × 3 matrix with diagonal entries (m e , m µ , m τ ). The G, Γ, and m e i matrices have entries that are much smaller than the entries of the Mχ± matrix and, therefore, can be used to perturbatively diagonalize the Mχ± matrix. The mass eigenstates are related to the gauge eigenstates by unitary matrices V and U defined by (5.25) They are chosen so that Next, requiring that U * Mχ±V −1 is diagonal allows one to compute the ξ + and ξ − matrices. To first order, they are given by The values of all the U and V matrix elements are presented in the Appendix B.1. These matrices will be crucial in calculating the decay rates of the charginos via RPV processes. In general, the exact expressions for the five charged mass eigenstates are complicated, and will be dealt with numerically in our calculations. However, it of interest to present the complete analytic expressions, including the RPV terms, for the mass eigenstatesχ ± 1 . We find that the positive eigenvectorχ + 1 is where one sums over i = 1, 2, 3. When |M 2 |< |µ|, the V coefficients are given by and The result for |µ|< |M 2 | is found by replacing φ + → φ + + π 2 in these expressions, as discussed above. Similarly, the negative eigenvectorχ − 1 is found to bẽ where one sums over i = 1, 2, 3. When |M 2 |< |µ|, the U coefficients are given by and The result for |µ|< |M 2 | is found by replacing φ − → φ − + π 2 in these expressions. Note that the first two terms ofχ ± 1 are independent of RPV and are identical to the expressions given in (5.17) and (5.19) for |M 2 |< |µ| and |µ|< |M 2 | respectively. Also, note that these terms dominate over the RPV terms and, hence, are the main contributors to the mass eigenvalues. However, although they are numerically smaller, the RPV terms inχ ± 1 play a crucial role in the R-parity violating decays of chargino LSPs and, hence, cannot be ignored.

Neutralino mass eigenstates
In the absence of the RPV violating terms proportional to i and v L i , the neutral Higgsinos and gauginos of the theory mix with the third generation right handed neutrino. In the gauge eigenstate basis The mass eigenstates are related to the gauge states by the unitary matrix N whereχ 0 = N ψ 0 . N is chosen so that To see this, consider the limit M 2 BL -that is, when the EW scale is much lower than the soft breaking scale so that the Higgs VEV's are negligible. In this limit, the mass matrix in eq. (5.36) becomes The first, fifth, and sixth columns, corresponding to the Blino, the Rino and the third generation right-handed neutrino, are now decoupled from the others and mix only with each other. In the reduced basis ν c 3 ,W R ,B , the mixing matrix is (5.39) The limit in which the gaugino masses are much smaller than M Z R is phenomenologically relevant due to the lower bound on M Z R being much higher than typical gaugino mass lower bounds. This limit is also motivated theoretically because RG running makes the gauginos masses lighter. In this limit, the mass eigenstates can be found as an expansion in the gaugino masses. At zeroth order, they areB =W R sin θ R +B cos θ R , (5.40) with masses, calculated to first order, given by We then get a new neutralino mass matrix, which is in agreement with the MSSM model after B-L breaking. It is given by Since the EW scale is generally much lower than the gaugino mass scale, the off-diagonal terms are small, where θ W is the Weinberg angle. Unlike for charginos discussed previously, our labels do not imply any mass ordering. The exact eigenstates are more difficult to compute than in the chargino case. They are, generically, linear combinations of the six gauge neutralino states. However, as -29 -was discussed in detail for charginos, the dominant gauge neutralino in an eigenstate can be read off directly from the leading term in the associated mass eigenvalue. Using this, as well as explicit numerical computation, we find that As with the charginos, we henceforth denote these mass eigenstates byχ 0 respectively; and refer to them as a Bino neutralino, a Wino neutralino and so on, even though they are only "predominantly" the pure neutral state.
Let us now add the RPV couplings i and v L i . This introduces mixing between the neutralinos and the neutral fermions of the standard model -the neutrinos. As discussed at the beginning of Section 3, mixing with the first-and second-family right-handed neutrino would lead to activesterile neutrino oscillations. Unless and until there is more experimental evidence of such oscillations, we will continue to assume that they do not exist and, therefore, that the mixing with the firstand second-family right-handed neutrinos is negligible. Therefore, the neutrino mass matrix given below includes only mixing with the three families of left-handed neutrinos-the seventh column -and the third-family right-handed neutrino-the sixth column. As in the case of the charginos, the effect of adding these RPV couplings is important because it will allow RPV decays of the neutralinos. It's effect on the physical masses of the neutralinos, however, is negligible. The new basis is then extended to W R ,W 0 ,H 0 d ,H 0 u ,B , ν c 3 , ν 1 , ν 2 , ν 3 . The extended mass matrix is given by This is the matrix that was introduced in eq. (3.1). Just as for charginos, we can write the neutralino matrix in a schematic form that will help us diagonalize it perturbatively. As discussed in detail in Section 3, Mχ0 can be expressed as N can be written in the perturbative form where N is the unitary matrix introduced below eq. (5.36). It is a 6 × 6 matrix, analogous to the 2 × 2 matrices U and V in Section 5.1. However, while eq. (5.11), (5.12) and (5.15) provide simple analytic expressions for U and V in terms of the rotation angles φ ± , it is much harder to solve for N without approximations. In this paper, we will compute N numerically, using the the relevant soft mass terms and couplings as input.
The equation ξ 0 = M −1 χ 0 m D is obtained by requiring that M D χ 0 be diagonal. We denote the mass eigenstates asχ 0 = N ψ 0 . The entries of N are central in calculating the neutralino decay rates and are all presented in Appendix B. However, exactly as in the case of the charginos discussed above, the physical masses of the "proper" neutralinos are not significantly changed by introducing the RPV couplings, so eqs. (5.46) -(5.51) remain valid. The states χ 0 6+i for i = 1, 2, 3 are the three left-handed neutrinos which now receive Majorana masses. This process has been discussed in detail in Section 3. As in the case of charginos, it is important to note that, although small compared to the R-parity preserving coefficients, the RPV terms make important contributions to the RPV decays of the neutralino LSPs and, hence, cannot be ignored.

.1 General B-L MSSM Lagrangian
The general B − L MSSM Lagrangian, written in terms of chiral multiplet component fields, (φ i , ψ i ), and vector multiplet components, (A a µ , λ a ), has the generic form (superpotential scalar-fermion-fermion Yukawa coupling) The interaction vertices that allow the charginos and neutralinos to decay into standard model particles are encoded in its complicated interactions. In order to read these RPV vertices from this Lagrangian, one needs to follow a series of steps: • In the first step, the component fields are pure gauge states. After B − L and electroweak symmetry breaking, these states mix to form massive states. In Section 5, we discussed how massive chargino and neutralino states are constructed. The second step then, is to write all gauge eigenstates in the Lagrangian in terms of their mass eigenstate expansion.
• After the second step is completed, one can identify the RPV vertices that couple a single chargino or a single neutralino to two standard model particles, typically a boson and a lepton. However, so far the theory has been written in terms of 2-component Weyl spinors, while the physical fermions are described by 4-component spinors. The final step, then, is to write the identified RPV vertices in 4-component spinor notation.
In the following sections, we will identify the RPV decay amplitudes for the charginos and neutralinos displayed in Table 3, using the three steps described above. We find that such sparticle decays are due entirely to the RPV couplings proportional to i and v L i , i = 1, 2, 3 that mix the three generations of leptons and the gauginos of the MSSM inside the chargino and neutralino mass matrices. That is, the mass eigenstate charginos and neutralinos can decay into SM particles precisely because they have lepton components. Only after we express the B − L MSSM Lagrangian in terms of the mass eigenstates, will the decay processes in Table 3 become apparent. Henceforth, we use χ ±,0 when referring to chargino and neutralino 2-component Weyl fermions and X ±,0 when referring to chargino and neutralino 4-component Dirac fermions. Furthermore, we use e i , i = 1, 2, 3 for the three families of charged leptons Weyl fermions, and i , i = 1, 2, 3 for the three families expressed as Dirac fermions. The Dirac fermion states are defined in eq. (6.33).

Mass eigenstate expansion
The chargino mass eigenstatesχ ± = (χ ± 1 ,χ ± 2 ,χ ± 3 ,χ ± 4 ,χ ± 5 ) are related to the gauge eigenstates ψ + = (W + ,H + u , e c 1 , e c 2 , e c 3 ) and ψ − = (W −H − d , e 1 , e 2 , e 3 ) via the unitary matrices U and V defined in Section 5 and given in Appendix B.1 . That is, There are two things worth pointing out here. First, only the mass eigenstatesχ ± 1 andχ ± 2 are considered to be the actual charginos. They have dominant contributions from the MSSM gauginos and only small SM lepton components. Moreover, the mass eigenstatesχ ± 3 e 1 , e c 1 ,χ ± 4 e 2 , e c 2 , χ ± 5 e 3 , e c 3 are considered to be the three generations of charged leptons (to be more precise, the left-handed Weyl spinors of the negatively and positively charged leptons). Second, the U and V matrices are defined so that the chargino statesχ ± 1 are lighter than the chargino statesχ ± 2 ; that is, The stateχ ± 1 can be dominantly charged Wino or charged Higgsino, but it will be always be less massive thanχ ± 2 . In terms of the chargino mass eigenstates, the gauge eigenstate can be expressed as ψ − = U †χ− and ψ + = V †χ+ . We then have the following mass eigenstate decomposition: Similarly, the Wino and Higgsino gauge eigenstate can be expressed as: The neutralino mass eigenstatesχ 0 = (χ 0 1 ,χ 0 2 ,χ 0 3 ,χ 0 4 ,χ 0 5 ,χ 0 6 ,χ 0 7 ,χ 0 8 ,χ 0 9 ) are related to the gauge eigenstates ψ 0 = (W R ,W 0 ,H 0 d ,H 0 u ,B , ν c 3 , ν e , ν µ , ν τ ) via the unitary matrix N , defined in Section 5 and given in Appendix B.2. That is, Just as for the chargino states, it is important to remember that only the first six statesχ 1,2,3,4,5,6 are considered actual MSSM neutralinos, since their dominant contributions are from sparticles. The statesχ 7,8,9 are the three generations of left handed neutrinos, which obtain Majorana masses after the neutralino matrix diagonalization. However, the notation of the six neutralino mass eigenstates differs from that of the chargino mass eigenstates. In the case of charginos, the states are defined such that Mχ± , where bothχ ± 1 andχ ± 2 could be dominantly charged Wino or charged Higgsino. As discussed above, usually |M 2 |< |µ|, in which caseχ ± 1 would be dominantly charged Wino, whileχ ± 2 would be dominantly charged Higgsino. In the rare case when |µ|< |M 2 |, χ ± 1 would be dominantly charged Higgsino, whileχ ± 2 would be dominantly charged Wino. For the neutralino mass eigenstates, however, we always haveχ 0 1 mostly Bino,χ 0 2 mostly Wino,χ 0 3,4 mostly Higgsino,χ 0 5,6 mostly right-handed third generation neutrino. We don't know, a priori, which state is the lightest, nor how to order them in terms of mass. Their masses are computed after we diagonalize the 6 × 6 neutralino mixing matrix Mχ0neglecting RPV couplings -an operation significantly more complicated than the diagonalization of the 2 × 2 chargino mass matrix, Mχ±, in the absence of RPV couplings.
In terms of the neutralino mass eigesntates, the gauge eigenstates are given by The Higgs scalar fields in the MSSM consist of two complex SU (2) L doublets; that is, eight degrees of freedom. When electroweak symmetry is broken, three of them become the Goldstone bosons G 0 , G ± , where G − = G + * . The rest will be Higgs scalar mass eigenstates; that is, CPeven neutral scalars h 0 and H 0 , a CP-odd neutral scalar Γ 0 and a charged H + and a conjugate H − = H + * . They are defined by [51].
where R α , R β 0 , R β ± are rotation matrices. Specifically, the matrix in front of the Standard Model Higgs Boson h 0 is R α = cos α sin α − sin α cos α , (6.15) while, to lowest order, the other matrices are where tan β = v u /v d . The mixing angle α is, at tree level: where the masses of the Higgs eigenstates are and -34 -

Interaction vertices
We now express Lagrangian (6.1) in terms of all the matter and gauge fields in our B − L MSSM theory, and then replace all gauge eigenstates with their mass eigenstate expansion. However, the full B−L MSSM Lagrangian is complicated when expressed in its most general form. We proceed, therefore, by looking only for the terms that can lead to chargino or neutralino decays into standard model particles. We identify the following tri-couplings: • gψ i †σµ T a A a µ ψ i , from the covariant derivative of the fermionic matter fields from the superpotential Yukawa couplings We now want to write these interactions in terms of the B − L MSSM component fields. The procedure is non-trivial. Hence, we split these interactions terms into two categories : 1) those responsible for the neutralino or chargino decays into a gauge boson (Z 0 -boson, W ± -boson or photon γ 0 ) and a lepton; that isχ and 2) those responsible for the decays into a Higgs boson and a lepton; that is The terms with Yukawa couplings and those from the supercovariant derivatives are relevant only for the Higgs boson-lepton decay channel, as we will show.
6.3.1χ ±,0 →Z 0 , γ 0 , W ± -lepton The part of the Lagrangian responsible for the gauge boson-lepton decay channels is (6.23) where this expression represents the sum over SU (2) L and U (1) Y . The i = 1, 2, 3 represents the three lepton families. Expressed in terms of the MSSM component fields this becomes where we sum over the i index, W a µ , a = 1, 2, 3 are the three vector bosons of the SU (2) L group and B µ the vector boson of the hypercharge U (1) Y group. Here, g 2 and g are the SU (2) L and -35 -U Y (1) couplings. In addition, L i represents the i-th SU (2) L left chiral lepton doublet defined in (2.4). We now make the replacements (6.25) where θ W is the Weinberg angle, and rearrange the previous expression to obtain J µ , J µ n and j µ EM are the usual weak charged, neutral and electromagnetic currents from the standard model theory of EW breaking, while J µ H , J µ nH and j µ EM H are the equivalent currents of the Higgsino fermionic fields. Also note that e is the electromagnetic coupling e = g 2 g In 2-component Weyl notation these currents are: • Weak charged currents, coupling to W ± bosons • Electromagnetic currents, coupling to the photon γ 0 • Neutral currents, coupling to the Z 0 boson where in J µ , j µ EM and J µ n we sum over i = 1, 2, 3. Plugging these currents into eq. (6.26), and arranging the couplings in terms of W ± , Z 0 and γ 0 respectively, we get where we have used the notation s W = sin θ W , c W = cos θ W and summed over i = 1, 2, 3. Finally, we are in a position to expand all gauge eigenstates in terms of the mass eigenstates, as in equations (6.3)-(6.8) and (6.11)-(6.12). After this procedure, the Lagrangian (6.32) is expressed in terms of the mass eigenstatesχ ± 1 ,χ 0 n , e i , ν i for i = 1, 2, 3, and their hermitian conjugates. Notice that we keep onlyχ ± 1 to simplify our results, sinceχ ± 1 are always lighter thanχ ± 2 and, hence, have better prospects to be detected. Their charged Wino and charged Higgsino content are determined from the rotation matrices U and V in eq. (5.11). At the same time, we study the vertices of general neutralinoχ 0 n states for n = 1, 2, 3, 4, 5, 6, since we have no a priori mass ordering for these states. We remind the reader that n = 1 means a mostly Bino neutralinoχ 0 1 =χ 0 B , n = 2 means a mostly Wino neutralinoχ 0 2 =χ 0 W , n = 3, 4 means a mostly Higgsino neutralinoχ 0 3,4 =χ 0 H and n = 5, 6 means a mostly third generation right-handed neutrino neutralinoχ 0 5,6 =χ 0 ν c

3
. Until now, we have used 2-component Weyl spinor notation for all of our matter fields. Since we are interested in the decays of physical particles, we will henceforth introduce and use 4component spinor notation for the initial and final states of the interacting particles. The 4component spinors are defined in terms of the 2-component Weyl spinors as (6.33) In our model, ± i , X ± 1 are Dirac fermions, while ν i , X 0 n are Majorana fermions. Note that, for simplicity, we use the same symbol, ν i , for both a Weyl and Majorana neutrino. The Lagrangian (6.32) then becomes where we sum over all neutralino states n = 1, 2, 3, 4, 5, 6, all lepton families i = 1, 2, 3 and j = 1, 2, 3. P L and P R are the projection operators 1−γ 5 2 and 1+γ 5 2 respectively.
6.4χ ±,0 →h 0 -lepton The part of the Lagrangian responsible for the Higgs boson-lepton decay channels is Notice that we have kept only the terms with neutral Higgs scalar components, since only those have a Higgs boson mass eigenstate component h 0 . Other terms responsible for this decay channel arise from the supercovariant derivatives of the Higgs fields of the type in Lagrangian (6.35). For the B − L MSSM, these produce the terms (6.38) -38 -It follows that the part of the Lagrangian responsible for theχ ±, 0 → h 0 − lepton decays is the sum of the equations (6.36) and (6.38). Note that these expressions are written in terms of the gauge eigenstates. As in the previous section, we expand the gauge eigenstates in terms of the mass eigenstatesχ ± 1 ,χ 0 n , e i , ν i for n = 1, 2, 3, 4, 5, 6, i = 1, 2, 3 and their hermitian conjugates. Once this step is completed, we group the terms into 4-component spinors to get where the angle α is defined in equation (6.17) and we sum over all lepton families i, j = 1, 2, 3. One now has the information required to compute the amplitude for each of the processes listed in Table 3. To do this, we need the exact expression for the vertex coefficient associated with each such process. These can be read off from the Lagrangians in Eqs. (6.34) and (6.39). For example, consider the the decay channelX − 1 → Z 0 − i . Then it follows from (6.34) that the vertex coupling is gX− where (6.42) However, their form and derivations are somewhat cumbersome. With this in mind,we provide a series of diagrams to pictorially express the origin of the interaction terms in the Lagrangian.
-39 -6.5 Chargino decay diagrams 6.5.1X + 1 → W + ν i The vertices associated with positively charged chargino decays are shown in Figures 6a and 6b. The vertices associated with the negatively charged chargino decays are the hermitian conjugates of those. The diagrams in Figure 6a are expressed in terms of gauge eigenstates written as 2component Weyl spinors. The diagrams in Figure 6b are the Feynman diagrams of the same vertices, in terms of 4-component mass eigenstates. -40 - The vertices associated with positively charged chargino decays are shown in Figures 7a and 7b. The vertices associated with the negatively charged chargino decays are the hermitian conjugates of those. The diagrams in Figure 7a are expressed in terms of gauge eigenstates written as 2component Weyl spinors. The diagrams in Figure 7b are the Feynman diagrams of the same vertices, in terms of 4-component mass eigenstates.  Figure 8b are the Feynman diagrams of the same vertices, written in terms of 4-component mass eigenstates. h 0 Figure 9: a) We show the vertices as they appear in the MSSM Lagrangian in terms of the gauge eigenstates, expressed as 2-component Weyl fermions. Vertices (a1), (a2) and (a3) arise from the covariant derivatives of the lepton and Higgsino matter fields, respectively. Vertices (a4) and (a5) come from the covariant derivative of the non-Abelian gaugino fields. b) We express the interactions in terms of the mass eigenstates relevant for the chargino decay into SM particles, expressed as 4-component Dirac fermions. We assumeX 1 is the lightest chargino and is either dominantly charged Higgsino or charged Wino. The "red" matrix elements have small values and are proportional to i /M sof t , while "blue" matrix elements are of order unity with small RPV corrections of the form 1 − i /M sof t . At first order, the decay amplitudes are proportional to -43 -6.6 Neutralino decay diagrams 6.6.1X 0 n → Z 0 ν i The vertices associated with neutralino decays are shown in Figures 10a and 10b. The diagrams in Figure 10a are expressed in terms gauge eigenstates written as 2-component Weyl spinors. The diagrams in Figure 10b are the same vertices in terms of mass eigenstates, expressed as 4-component spinors.
The vertices associated with neutralino decays are shown in Figures 11a and 11b. The diagrams in Figure 11a are expressed in terms gauge eigenstates written as 2-component Weyl spinors. The diagrams in Figure 11b are the same vertices in terms of mass eigenstates, expressed as 4-component spinors.
6.6.3X 0 n → h 0 ν L i The vertices associated with neutralino decays are shown in Figures 12a and 12b. The diagrams in Figure 12a are expressed in terms gauge eigenstates written as 2-component Weyl spinors. The diagrams in Figure 12b are the same vertices in terms of mass eigenstates, expressed as 4-component spinors. - 46 -In the previous discussion, we presented all relevant RPV decay channels of charginosχ ± 1 and neutralinosχ 0 n , n = 1, 2, 3, 4, 5, 6 into standard model particles and gave the associated Lagrangian interactions. In this section, we will use these results to calculate the decay rates associated with each such process. The calculations are carried out using the dominant linear terms in the RPV couplings i and v L i only, since higher order terms are highly suppressed. The analysis is completely general, regardless of whether or not the charginos or the neutralinos are the LSPs. However, only the lightest sparticles decay exclusively via RPV processes into SM particles. Furthermore, they have best prospects for detection at the LHC. Therefore, in subsequent publications, we will use these results to compute the RPV branching ratios of chargino and neutralinos LSPs and NLSPs.

Wino/Higgsino Chargino
We are ultimately interested in LSP and NSLP decays via RPV channels. Thus, we calculate the decay rates of theχ ± 1 charginos only, which are defined to be lighter than theχ ± 2 charginos. As discussed in detail at the end of subsection 5.1, a chargino mass eigenstate is a superposition of a charged Wino, a charged Higgsino and an RPV sum over left-chiral and right-chiral charged leptons gauge eigenstates. In the present analysis, we will consider the decays of a generic chargino with arbitrary mixed charged Wino, charged Higgsino and RPV charged lepton content. More specialized decays involving predominantly Wino chargino or Higgsino chargino mass eigenstates, will be considered in future publications.
•X ± 1 → W ± ν i The terms in the Lagrangian associated with these decay channels have been calculated in eq. (6.34) and illustrated in Figure 6. Summing all the vertices, one finds and Next, using the expressions for the matrix elements of U, V and N given in Appendices B.1 and B.2, we get and The decay width Γ is proportional to the square of the amplitude of this process. Note that we account for the longitudinal degrees of freedom of the resultant W ± gauge bosons (Goldostone equivalence theorem) in calculating this decay width. This results in an amplification of this channel, which becomes more significant as the mass of the decaying chargino increases. The result is are proportional to the RPV couplings i and v L i at first order. Therefore, the decay of the chargino into the SM particles W ± boson and neutrino would vanish in the absence of RPV.
The terms in the Lagrangian associated with these decay channels have been calculated in eq. (6.34) and illustrated in Figure 7. Summing all the vertices, one finds Using the expressions for the matrix elements of U and V from the Appendix B.1, we get and where there is no sum over the i in v L i m e i .
The decay width Γ is proportional to the square of the amplitude of this process. We note that we have accounted for the longitudinal degrees of freedom of the resultant Z 0 gauge bosons (Goldstone equivalence theorem) in calculating this decay width. This results in an amplification of this channel which becomes more significant as the mass of the decaying chargino increases. We find that (7.10) Note that both GX ± →Z 0 ± i coefficients are proportional to the RPV couplings i and v L i at first order. Therefore, there would be no decay of the chargino into the SM Z 0 boson and charged leptons if the RPV effects were non-existent.
The terms in the Lagrangian associated with these decay channels have been calculated in eq. (6.34) and illustrated in Figure 8. Summing all the vertices, one finds and gX− Using U and V in Appendix B.1, we find that and 14) The decay width Γ is proportional to the square of the amplitude of this process. The Goldstone equivalence theorem does not apply in this case, as the photon is massless. The result is Note that GX ± →γ 0 ± i is proportional to the RPV couplings i and v L i at first order. Therefore, there would be no decay of the chargino into the SM photon and charged leptons if there was no RPV.
The terms in the Lagrangian associated with these decay channels have been calculated in eq. (6.39) and illustrated in Figure 9. Summing all the vertices, one finds Next, using the expressions for the matrix elements of U and V from the Appendix B.1, we get and where we do not sum over the i in either of these expressions. The decay width Γ is proportional to the square of the amplitude of this process, and is found to be (7.20) Again, note that GX ± →h 0 ± i are proportional to the RPV couplings i and v L i at first order. Hence, there would be no decay of the chargino into the SM Higgs boson and charged leptons if there would be no RPV effects.

Neutralinos
Recall that the index n indicates the neutralino species as follows: For a general neutralino state n, we found expressions for the parameters of the decay to two standard model particles.
When replacing a charged Wino gaugino with the lightest chargino mass eigenstate, we need the elements U 1 1 and V 1 1 given by and their conjugates. Similarly for replacing a charged Higgsino, one needs U 1 2 and V 1 2 , which we find to be U 1 2 = sin φ − , V 1 2 = sin φ + (B.5) and their conjugates. We also need the elements U 2+i 1 and V 2+i 1 and their complex conjugates when replacing a charged Wino state with a charged lepton, where The elements U 2+i 2 and V 2+i 2 and their complex conjugates when required when replacing a charged Higgsino state with a charged lepton, The angles φ ± are defined in Section 5.1. They express the charged Wino and charged Higgsino content of the chargino mass eigenstates, in the absence of the RPV couplings i and v L ĩ χ ± 1 = cos φ ±W ± + sin φ ±H ± (B.8) andχ ± 2 = − sin φ ±W ± + cos φ ±H ± . (B.9) Hence, for φ ± = 0, we have purely Wino chargino statesχ ± 1 and purely Higgsino chargino statesχ ± 2 . Conversely, for φ ± = π/2, we have purely Higgsino chargino statesχ ± 1 and purely Wino chargino statesχ ± 2 .

B.2 Neutralino mass matrix
The N matrix can be written schematically as The rows of ξ 0 are the gaugino gauge eigenstates, whereas the columns correspond to the neutrino gauge eigenstates. These are explicitly labeled and presented below. They are N is the matrix that diagonalizes the neutralino mass matrix in the absence of RPV couplings to the three families of left-handed neutrinos. If the soft masses in the neutralino mass matrix, eq. (B.20), are much larger than the Higgs VEV's v u and v d , then, at zeroth order, we have However, in the regimes with small chargino and neutralino masses that we analyze, this approximation is no longer valid. The elements of N will, in general, have complicated expressions and we choose to evaluate them numerically. We use as input the numerical values of the neutralino mass matrix. We expect, however, based on the zeroth order form of N , that for n = 1, 2, 3, 4, 5, 6 and i = 1, 2, 3. Elements of the bottom left block V † P M N S ξ † 0 are computed in a similar fashion. One can then determine N 6+i n as