The Minimal SUSY $B-L$ Model: From the Unification Scale to the LHC

This paper introduces a random statistical scan over the high-energy initial parameter space of the minimal SUSY $B-L$ model--denoted as the $B-L$ MSSM. Each initial set of points is renormalization group evolved to the electroweak scale--being subjected, sequentially, to the requirement of radiative $B-L$ and electroweak symmetry breaking, the present experimental lower bounds on the $B-L$ vector boson and sparticle masses, as well as the lightest neutral Higgs mass of $\sim$125 GeV. The subspace of initial parameters that satisfies all such constraints is presented, shown to be robust and to contain a wide range of different configurations of soft supersymmetry breaking masses. The low-energy predictions of each such"valid"point - such as the sparticle mass spectrum and, in particular, the LSP - are computed and then statistically analyzed over the full subspace of valid points. Finally, the amount of fine-tuning required is quantified and compared to the MSSM computed using an identical random scan. The $B-L$ MSSM is shown to generically require less fine-tuning.

The minimal supersymmetric standard model (MSSM) is the simplest possible N = 1 supersymmetric extension of the standard model of particle physics. It has the gauge group SU (3) C × SU (2) L × U (1) Y , replaces each gauge field by a vector supermultiplet, each matter fermion by a chiral supermultiplet, has a conjugate pair of Higgs chiral superfields and has no right-handed neutrino multiplets. The MSSM was introduced in various contexts in [1,2] and has been extensively reviewed in [3]. However, without further modification, the most general superpotential for the MSSM contains cubic interactions that explicitly violate both lepton and baryon number and lead, among other things, to rapid proton decay-which is unobserved. The traditional solution to this problem is to demand that, in addition to the gauge group, the MSSM be invariant under an ad hoc Z 2 finite symmetry-R-parity-which acts on individual component fields as (−1) 3(B−L)+2s . Although there have been many attempts to explain how R-parity can arise in the MSSM [4][5][6][7][8] or be spontaneously broken [9][10][11]. These are all "non-minimal" in the sense, for example, that they require additional matter multiplets, have other ad hoc assumptions and so on. Is there a more natural and minimal approach to R-parity in the MSSM?
One begins by noting that, within the context of supersymmetry, R-parity is a finite subgroup of U (1) B−L -see for example [12]. It follows that R-parity might arise as a consequence of demanding that the MSSM be invariant under a global U (1) B−L symmetry. This would be consistent with current bounds on baryon and lepton number violation, and can also be imposed on the MSSM extended to include three families of right-handed neutrino chiral supermultiplets . However, one of the implications of the standard model is that a global continuous symmetry group is likely to appear in its local form-that is, as a gauge symmetry. With this in mind, one might ask if R-parity could arise as the consequence of extending the MSSM gauge group to include a new gauged U (1) B−L symmetry. It has long been known that the standard MSSM is anomalous with respect to gauged B − L symmetry, whereas the MSSM extended by three families of right-handed neutrino supermultiplets is anomaly free. Furthermore, this is the minimal such extension of the MSSM. We will call this anomaly free, minimal content theory the B − L MSSM, and propose that this is a more "natural" way in which R-parity can arise in low energy supersymmetric particle physics.
The B − L MSSM was identified from a low-energy "bottom-up" point of view in [13][14][15] 1 . In several papers [17,18], these authors explored its structure and some phenomenological consequences. In addition, these ideas, and other directions, have recently been reviewed in [19]. Interestingly, the B − L MSSM was also discovered from a high-energy "top-down" viewpoint in [20][21][22], where it was shown that this model arises within the context of heterotic superstring theory [23,24]. More specifically, the B − L MSSM is the low-energy effective theory associated with compactifying the E 8 × E 8 heterotic string on a Schoen Calabi-Yau threefold [25] with a specific class of SU (4) vector bundles [26]. An important aspect of this highenergy point of view is that the parameters of the theory are specified near the gauge coupling unification scale, and then run down to the electroweak scale using the renormalization group (RG). This allows one to explore fundamental aspects of the theory-such as B − L and electroweak symmetry breaking. First steps in this direction were taken in [27][28][29], where it was shown-for a restrictive set of initial parametersthat radiative breaking of both of these symmetries can indeed occur. A further study of the Wilson lines, spectra, mass scales and the unification of gauge couplings from the high-energy superstring point of view was presented in [30]. Combining the bottom-up and top-down approaches to the B − L MSSM, various aspects of both LHC and neutrino phenomenology were studied in the special case where a stop or a sbottom sparticle is the lightest supersymmetric particle (LSP) [31,32].
However, all of the previous analyses involved specific assumptions-either about high-energy initial conditions or the low-energy structure of the theory. For the B − L MSSM to be a realistic contender for the low-energy theory of particle physics, it is essential that its initial parameter space be explored in a generic way, and that its low-energy predictions be compared with all present experimental data. This will be carried out in detail in this paper. We perform a statistical scan over a well-defined and wide range of initial parameters and, for each fixed set of such parameters, scale the theory down to low energy using the RGEs with specified threshold conditions. The results will be examined to determine the subset of the parameter space that, sequentially, 1) breaks B − L symmetry at a scale consistent with experiment, 2) breaks electroweak symmetry, 3) has all sparticle masses above their present experimental lower bounds and 4) predicts the mass of the lightest neutral Higgs scalar to be within 2σ standard deviation from the ATLAS measured value of 125.36 GeV. A small subset of important results from this statistical scan were given in [33]. Here, we give present the details of the method use, as well as a wide array of new results and experimental predictions.
This paper is organized as follows. In Section II, the B − L MSSM model at the TeV scale is described in detail. This includes a discussion of both B − L and electroweak symmetry breaking and much of our notation. The connection between the UV picture at the scale of gauge coupling unification and the TeV scale is outlined in Section III. Specifically, the important mass scales and energy regimes are fully described and the relationships between the different mass scales are presented. The content of these two sections is expanded upon in Section IV. Here, the details of the renormalization group equation (RGE) evolution of the parameters of the theory between the UV and TeV scales are discussed. This includes the appropriate input values of the parameters and the relevant equations-with reference to Appendix A when necessary. Special attention is given to the right-handed sneutrino RGE which drives radiative B − L symmetry breaking. The relationship between the different running parameters and scales-introduced in Section III-is further discussed. The latter part of this section outlines the experimental bounds used in our analysis. First, collider bounds are discussed. These correspond to lower bounds on the physical sparticle masses-which are closely related to the running mass parameters. The section finishes by describing-and implementing-the well-known bounds from flavor changing neutral currents and CP violation.
This paper approaches the connection between UV and TeV physics in a novel way. Specifically, instead of assuming some universal conditions or relationships between UV soft SUSY breaking parameters, we simply allow all such parameters to be within about an order of magnitude of some chosen SUSY breaking scale. Our approach will then be to scan all relevant SUSY breaking parameters, at the high scale, over this possible range. These parameters will then be RG evolved to the TeV scale, following the discussion in Section IV. This very general approach, as well as other details of our scan, are described in Section V.
Such an approach is especially applicable to the string realization of the B − L MSSM model, since, in that case, each chiral supermultiplet arises from a different 16 representation of SO (10). Therefore, their soft SUSY breaking masses do not obey boundary conditions at the high scale. Our analysis is also valid for a wide range of pure GUTs, which can impose disparate boundary conditions at the unification scaleincluding none at all. Finally a "meta" scan is conducted to choose the optimal value for the range of SUSY breaking mass scales.
The optimal range, arrived at in Section V, is used to generate all subsequent results in this paper-the bulk of which are presented in Section VI. Relating these results to the twenty-four phenomenologically relevant scanned parameters is daunting at best. Fortunately, a cohesive picture can be presented in terms of two so-called S-parameters. These S-parameters, which are each the sum of the squares of SUSY breaking mass parameters, play an important role in radiative B − L symmetry breaking. Their role in this capacity is presented in Fig. 5. The results for all other experimental constraints can also be expressed in terms of these two parameters-see Fig. 6 and Fig. 7. A central result of this work is given is Fig. 8, which displays the frequency at which particles appear as LSPs in our scan. The section closes with histograms of the spectra, as well as some spectrum plots to help characterize specific features of our results.
Fine-tuning in our approach is addressed in Section VII. While fine-tuning in the B − L MSSM is not drastically different than in the MSSM, there are several key issues to highlight. First, while one might expect that the scale associated with B − L symmetry breaking could introduce new contributions to finetuning, it is shown that this is not the case. Second, the MSSM, analyzed using the same methods as in this paper, typically yields equal or more fine-tuning than in the our model for similar initial points. Finally, an LSP analysis similar to Fig. 8, but with fine-tuning constrained to be better than one part in a thousand, is presented in Fig. 17. We conclude in Section VIII.
In addition to the main sections of this paper, three Appendices are included to help elucidate various topics. Appendix A contains all one-loop RGEs for this model in the different regimes. The first part of Appendix B specifies how to relate the running soft mass parameters to the physical masses of the SUSY particles. The second part of Appendix B describes the procedure used to calculate the SM-like Higgs mass.
Those readers interested in the details of how the random scan in this paper was conducted, are directed to Appendix C.

II. THE TEV SCALE MODEL
Motivated by both phenomenological considerations and string theory, we analyze the minimal anomaly free extension of the MSSM with gauge group As discussed in [30], we prefer to work with the Abelian factors U (1) 3R ⊗ U (1) B−L rather than U (1) Y ⊗ U (1) B−L -although they are physically equivalent. This is motivated by the fact that the former is the unique choice that does not introduce kinetic mixing between the associated field strengths at any scale in their renormalization group equation (RGE) evolution. The gauge covariant derivative can be written as where I 3R is the U (1) 3R charge and the factor of 1 2 is introduced in the last term by a redefinition of the gauge coupling g BL -thus simplifying many equations. As discussed in [30] and throughout this paper, a radiatively induced vacuum expectation value (VEV) for a right-handed sneutrino will spontaneously break the Abelian factors U (1) 3R × U (1) B−L to U (1) Y , in analogy with the way that the Higgs fields break SU (2) L ⊗ U (1) Y to U (1) EM in the SM. For simplicity, we will refer to this as "B − L" symmetry breaking-even though it is technically the breaking of a linear combination of the U (1) 3R and U (1) B−L generators, leaving the hypercharge group generated by invariant. The particle content of the minimal model is simply that of the MSSM plus three right-handed neutrino chiral multiplets. That is, three generations of matter superfields We refer to this model throughout the remainder of this paper as the B − L MSSM.
The superpotential of the B − L MSSM is given by where flavor and gauge indices have been suppressed and the Yukawa couplings are three-by-three 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 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, and lepton Yukawa couplings-with the exception of the (3,3) components-can be neglected for the purposes of this paper. Similarly, the very light neutrino masses imply that the neutrino Yukawa couplings are sufficiently small so as to be neglected for the purposes of this paper. The µ-parameter can be chosen to be real, but not necessarily positive, without loss of generality. The soft supersymmetry breaking Lagrangian is then given by 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, and lepton a-parameters are significant and that the neutrino a parameters are negligible. For more explanation of these assumptions, see Section IV B.
Spontaneous breaking of B − L symmetry results from a right-handed sneutrino developing a nonvanishing VEV, since it carries the appropriate I 3R and B − L charges. 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.
To acquire a VEV, a right-handed sneutrino must develop a tachyonic mass. As discussed in [16,34,35], a Here and throughout this paper we use the term "tachyon" to describe a scalar particle whose m 2 parameter is negative. Although all m 2 parameters at high scale will be chosen positive, one or more can be driven negative at lower energy by radiative corrections. This signals dynamical instability at the origin-although a stable VEV may, or may not, develop.
VEV can only be generated in one linear combination of the right-handed sneutrinos. Furthermore, beyond the fact that its VEV breaks B − L symmetry, in which combination it occurs has no further observable effect. This is because there is no right-handed charged current to link the right-handed neutrinos to a corresponding right-handed charged lepton. Therefore, without loss of generality, one can assume that it is the third generation right-handed sneutrino that acquires a VEV. 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 where The second term in the parenthesis is a small effect due to mixing in the neutral gauge boson sector. The hypercharge gauge coupling is given by where Since the neutrino masses are roughly proportional to the Y ν ij and v Li parameters, it follows that In this phenomenologically relevant limit, the minimization conditions of the potential are simple and worthwhile to note. They are 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 contribution to m Hu and m H d shift their values significantly compared to the MSSM. Eq. (13) 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.
A direct consequence of generating a VEV for the third family sneutrino is the spontaneous breaking of R-parity. The induced operators in the superpotential are This general pattern of R-parity violation is referred to as bilinear R-parity breaking and has been discussed in many different contexts, especially in reference to neutrino masses-see references [36][37][38][39] for early works. In addition, the Lagrangian contains additional bilinear terms generated by v Li and v R from the super-covariant derivative. These are The consequences of spontaneous R-parity violation are quite interesting, and have been discussed in a variety of papers. For LHC studies, see [17,18] as well as recent work on stop and sbottom LSP's in this context and the connection between their decays and the neutrino sector [31,32]. Predictions for the neutrino sector were discussed in [16,34,35]. It was shown that the lightest left-handed, or active, neutrino is massless and that the model contains two right-handed neutrinos, referred to as sterile neutrinos, that are lighter than the remaining two active neutrinos. Sterile neutrinos can influence the cosmological evolution of the universe due to their role as dark radiation. This effect was studied in [18].
In this section, we have focussed on the TeV scale manifestation of the B − L MSSM. However, the main content of this paper will be to study the connection between this low energy theory and its possible origins in E 8 ⊗ E 8 heterotic string theory, thereby linking some of the B − L MSSM phenomenology to high scale physics. In this context, our model is the remnant of an SO(10) unified symmetry broken by two Wilson lines. Although the details of the various physical regimes of this theory, and the renormalization group scaling between them, were given in [30], we review them in the next section for completeness.

III. JOURNEY FROM THE UNIFICATION SCALE
The goal of this section is to review the physics associated with the string construction of the B − L MSSM-from unification to the electroweak scale. After compactification to four-dimensions, the unified gauge group is SO (10). This is then further broken to the B − L MSSM gauge group by the turning on of two Abelian Wilson lines, denoted by χ 3R and χ B−L respectively. The energy scales associated with these Wilson lines need not be the same. In fact, exact gauge coupling unification at one-loop, which we will assume throughout this paper, requires that the scales be different-implying there is a two-step symmetry M χ 3R and M χ B−L respectively, we find the following two initial symmetry breaking patterns.
These scales will be further discussed below.
In the intermediate regime, the particle content of the left-right model consists of nine copies of the matter family two copies of a Higgs bi-doublet, which contains the MSSM Higgs fields, and a pair of color triplets Once the second Wilson line turns on, the extra particle content integrates out and one is left with exactly the spectrum of the B − L MSSM.
At this point, it is important to make a quick note on notation for the B − L gauge coupling. Thus far, we have discussed the gauge parameter g BL , which couples to 1 2 (B − L) charge. As is well known, this gauge coupling has to be properly normalized so as to unify with the other gauge parameters. We use g BL defined by to denote the properly unifying coupling. The parameter g BL couples to 3 8 (B −L) charge and will appear in the RGEs. For quantities of physical interest, such as physical masses, g BL will be used.
To fully understand the evolution of this model from unification to the electroweak scale, it should be noted that there are five relevant mass scales of interest, two of which were mentioned briefly above. All five are described in the following: • M U : The unification mass and the scale of the first Wilson line. We assume that all gauge couplings unify at this scale. That is, g 3 = g 2 = g R = g BL = g U .
• M I : The intermediate scale associated with the second Wilson line and the symmetry breaking SU (2) R → U (1) 3R -that is, the right-handed isospin breaks into its third component. Since the gauge coupling of SU (2) R slightly above M I is equal to the U (1) 3R gauge coupling slightly below M I , we use g R for both SU (2) R and U (1) 3R . All gauge couplings have trivial thresholds at this scale.
• M B−L : The B − L scale is the mass at which the right-handed sneutrino VEV triggers U (1) 3R ⊗ U (1) B−L → U (1) Y . Physically, this corresponds to the mass of the neutral gauge boson Z R of the broken symmetry and, therefore, the scale of Z R decoupling. Specifically where M Z R depends on parameters evaluated at M B−L -see Eq. (9). Substituting Eq. (9) into this relation yields a transcendental equation that must be solved using iterative numerical methods to obtain the correct value for M B−L .
At this scale, we also evaluate the hypercharge gauge coupling using its relationship to the gauge parameters of B − L and the third component of right-handed isospin. This is given by where Note that Eq. (29) is just a restatement of Eq. (11) with gauge couplings properly normalized for unification, including a rescaled hypercharge gauge coupling g 1 defined by This scale is useful because when the stops decouple, the parameter that controls electroweak symmetry breaking, that is, the soft H u mass parameter, effectively stops running-see [40] for more details. Like the B − L scale, the SUSY scale must be determined using iterative numerical methods because the physical stop masses in Eq. (32) depend implicitly on the SUSY scale.
• M EW : The electroweak scale. This is the well-known scale associated with the Z and W gauge bosons of the SM. We will make the identification For correct electroweak breaking, one must satisfy the conditions The first constraint guarantees that the Higgs potential is bounded from below while the second indicates that the trivial vacuum is not stable.
With the relevant mass scales appropriately defined, we can now discuss the physical regimes that exist in between them. To begin with, we will be interested in the evolution of the gauge couplings-since our assumption that they unify will help relate these disparate scales to each other. We present below, for each regime, the slope factors b a appearing in the gauge RGE's We will refer to this scaling interval as the "left-right regime" and, when required, denote the associ- We will refer to this scaling interval as the "B-L MSSM regime" and, when required, denote its b-coefficients by b BL a .
The remaining two regimes depend on which of the following two cases occurs: M B−L > M SUSY -the "right-side-up" hierarchy-and M SUSY > M B−L -the "upside-down" hierarchy.
right-side-up hierarchy: • M B−L − M SUSY : In this case B − L has been broken but SUSY is still a good symmetry, thereby giving an MSSM-like theory-that is, the MSSM plus two light right-handed neutrino chiral multiplets. Another possible deviation might occur in the composition of the bino-more about this late.
In general, however, this is the MSSM. Specifically, the gauge couplings in this regime evolve like the well-known MSSM gauge couplings with b a coefficients We refer to this interval as the "MSSM" regime and denote the associated b-coefficients by b MSSM a .
• M SUSY − M EW : In this regime, one simply has the SM with two sterile neutrinos. It has the wellknown slope factors We refer to this as the "SM" regime and denote the b-coefficients by b SM a .
upside-down hierarchy: • Given the above information, and the demand that all gauge couplings unify, we can solve for a given mass scale in terms of the others. First consider the unification mass-corresponding to the scale at which the four gauge couplings become equal to each other. Practically, it is derived as the energy-momenta at which g 3 = g 2 . As is well-known, this will not be influenced by any scale that acts as a threshold for complete multiplets of a minimal group that unifies SU (3) and SU (2)-for example, SU where the superscripts on the slope factors indicate their regime of relevance and the α i take their experimental values at M Z [41]: Inserting all the coefficients, the unification scale becomes Substituting for M U using Eq. (44) gives These relationships are displayed in Fig. 1.

IV. THE FRAMEWORK
The purpose of this paper is twofold: (i) to analyze how the high scale parameter space is associated with the TeV scale phenomenology of the B − L MSSM and (ii) to introduce a new way of analyzing this question. More specifically, (i) is an attempt to understand the kind of general UV conditions that yield valid, experimentally consistent points at the TeV scale-especially conditions that lead to B − L breaking and a viable B − L/electroweak hierarchy, a topic which has not been widely studied before. Such a study requires a framework for evolving parameters through the relevant regimes. The skeleton of this framework was discussed in the previous section. This section will continue this discussion and review the well-known experimental constraints on the SUSY parameters. Utilizing these constraints, the next section presents the motivation and the strategy for our scan. This addresses (ii).
The approach to the RG evolution of the parameters is similar to other such work, with several deviations that will be highlighted below. The RGEs of interest are calculated using reference [42] and are presented in . This is important and will be discussed later. Since these new Yukawa couplings are unknown, RG running them through this regime would not contribute to the predictability of this study. In practice, we implement these calculations piecewise starting with the analytically tractable equations first. These are the gauge couplings, gaugino mass parameters and the first and second generation sfermion mass parameters, as well as all sneutrino mass parameters. We then numerically calculate the evolution of the remaining parameters.
As is traditional, we begin by inputting the experimentally determined parameters-that is, the gauge couplings and Yukawa couplings derived from fermion masses-at the electroweak scale. The initial values of the gauge couplings were given above in Eq. (43). For the purposes of this paper, the SM Yukawa couplings, which are three-by-three matrices in flavor space, can all be approximated to be zero except for the three-three elements which give mass to the third generation SM fermions. We use the initial conditions For details on relating fermion masses to Yukawa couplings, see [45]. Here the lower case y represents Yukawa couplings in the non-SUSY regime. These can be evolved to the SUSY scale, both in the rightside-up hierarchy, Eqs. (A5) -(A7), and up-side-down hierarchy, Eqs. (A8) -(A10). At the SUSY scale, one has the non-trivial boundary conditions The The gauge couplings in the various regimes were discussed in the previous section. With those solutions in hand, the RGE evolution of the gauginos can be easily derived. Gaugino masses are inputted at the unification scale and evolved down. Naively, one might expect gaugino mass unification. However, this is not always the case-as has been discussed in a number of contexts, see for example [43,44]. Therefore, and to be as general as possible, we impose no relationship between the different gaugino masses at the unification scale. The general RGE for a gaugino mass parameter is where a indexes the gauge groups. These equations can be solved analytically. For the gauginos associated The bino, however, is a treated somewhat differently for each of the two possible hierarchies between the B − L and SUSY scales. For the right-side-up hierarchy, at the M B−L scale, we have three neutral fermions that mix: the third generation right-handed neutrino ν c 3 , the B − L gaugino (blino) and the I 3R gaugino (rino). This is a direct consequence of R-parity violation in the B − L MSSM. As we will see, it is possible for a neutralino LSP mass eigenstate to have a significant ν c 3 component. The mixing between the third-family right-handed neutrino and the U (1) gauginos is described in the (ν c 3 ,W R ,B ) basis by the mass matrix  This mass matrix neglects mixing with the Higgsinos through the electroweak breaking Higgs VEV. This is a safe approximation since the lower bound on the ZR mass implies that the electroweak Higgs VEV will be negligible compared to the third-family right-handed sneutrino VEV.
Due to the RGEs monotonically pushing the values of M R and M BL down, they will typically be significantly lighter than M Z R . It is, therefore, instructive to perturbatively diagonalize this mass matrix in the limit M R , M BL M Z R . At zeroth order, the mass eigenstates arẽ with masses At first order, the effect of adding M R and M BL back into the mass matrix is to give the bino a mass of This shows that, in the right-side-up hierarchy, between the scales M B−L and M SUSY we have the gauge group and particle content of the MSSM plus two right-handed neutrino supermultiplets-that is, the two sneutrino generations that do not acquire a VEV. Below the B − L scale, the bino mass is In the upside-down case, all neutralinos are diagonalized at the SUSY mass scale. B − L breaking as was discussed in detail in [27,28]. The right-handed sneutrino mass RGE is At some points in parameter space, it is possible that the required limit will not be satisfied and there will not be a mass eigenstate that can clearly be identified as the bino. However, since the scaling regime between MB−L and M SUSY is always small, the errors introduced by assuming the existence of a bino are insignificant. where Despite the lack of a large Yukawa coupling, the right-handed sneutrino mass can still be driven tachyonic by appropriate signs and magnitudes of the S-terms defined in Eqns (59,60). To emphasize this, the analytic solution to the sneutrino mass RGE is presented here. It is Recall that the value of any Abelian gauge couplings grows larger at higher scale. Therefore, we see that a tachyonic sneutrino is only possible when S R (M I ) is negative and/or S BL (M I ) is positive. This demonstrates the central role played by the S-terms in the breaking of B − L symmetry. Note that in typical unification scenarios all soft masses are "universal" and, hence, both S-terms vanish. However, it was mentioned earlier that, in this string construction, different elements of a given generation arise from different 16 representations of SO (10). Therefore, the soft masses of a given generation are generically non-degenerate. Hence, the S-terms can be non-zero.
is used to iteratively solve for the B − L scale. The SUSY mass scale must also be solved for iteratively using the equation where mt 1 < mt 2 are the physical stop masses. The relationships between the soft mass parameters and the physical masses are given in Appendix B 1. The soft mass squared parameter for the up-type Higgs is driven tachyonic, as usual, by the large top Yukawa coupling. Furthermore, the decoupled values of the soft Higgs mass squared parameters are used to calculate the µand b-terms using Eqs. (15) and (16).
The soft mass parameters have non-trivial boundary conditions at the B − L scale due to the effects of the B − L and I 3R D-terms: where M − B−L and M + B−L indicate a scale slightly below and slightly above the B − L scale respectively, and I 3R and Y are the third component of right-handed isospin and hypercharge of a generic scalar φ.
Once the mass parameters have been properly evolved to their appropriate scales, the physical masses can be evaluated. For much of the spectrum, this has been discussed in the literature, see for example [3], and has been included in Appendix B 1. The new element here is the mass of the scalar associated with the third generation right-handed sneutrino-the B − L Higgs-degenerate with the Z R mass. In addition, the calculation for the SM-like Higgs mass is crucial since the experimentally measured value of ∼125 GeV requires substantial radiative corrections from the stop sector in the MSSM. In this paper we follow the approach of references [46][47][48]-taking into account the decoupling scales of the two stops, matching the quartic Higgs coupling at those scales and RGE evolving the quartic coupling to the electroweak scale to calculate the Higgs mass. Full details are given in Appendix B 2.
Once a given physical mass is calculated, it is compared to current lower bounds or, in the case of the SM Higgs, the experimentally measured value. If a given initial set of parameters predicts a physical mass that is inconsistent with current bounds, it is rejected as being an invalid point. These lower bounds are discussed in the next subsection.

A. Collider Constraints
The bounds placed by collider data on SUSY masses are, in general, model dependent. That is, they depend on the spectrum and decay modes. Despite the much larger energy of the LHC, LEP 2 still has competitive bounds on colorless particles that couple to the Z and/or the photon-including sleptons in scenarios with both R-parity conservation [49,50] and violation [51], bounds on charginos [52,53] and bounds on sneutrinos in the case of R-parity violation [51]. As one may expect, due to the relatively clean environment at LEP, these bounds are close to one half the center of mass energy of LEP 2. Therefore, for simplicity, we proceed with the bound that all colorless fields that couple to the photon must be heavier than 100 GeV. That is, where˜ is any charged slepton. Colorless states that couple to the Z, the left-handed sneutrino, must be heavier than half the Z mass: Colorless states that do not couple to the Z, such as right-handed sneutrinos/neutrinos and the bino, have such small collider production cross-sections that they do not have collider-based lower bounds. Wino and Higgsino neutralinos are degenerate with their chargino partner, thereby effectively putting a lower bound of 100 GeV on those states as well.
The bounds from the LHC are much more dependent on the parameters. For example, if one investigated the bound on, for example, degenerate squarks in this model with a neutralino LSP, those bounds could be significantly different than in the case of a sneutrino, or some other, LSP. Allowing the squark masses to split would further alter the lower bounds. In fact, a full treatment would involve calculating the signatures of a given point in parameter space, comparing the number of events to the most recent LHC bounds on such events, and determining if the parameter point is valid. We do not expect the details of these lower bounds to heavily affect our results. We will, therefore, simply use the naive bounds which are based on recent CMS [54] and ATLAS [55] studies of the R-parity conserving MSSM. In these studies, the colored states decay into jets and missing energy-possible final states in our model whenever the LSP decays into neutrinos. In this paper, we impose these bounds except in the case of a stop or sbottom LSP. These two cases were explicitly studied in [31,32] and yielded the following lower bounds: wheret 1 (b 1 ) denotes the lightest stop (sbottom). Here, right-handed refers to a stop that is almost completely right-handed-that is, a stop mixing angle, θt > 85 • or, equivalently, a state composed of 99% righthanded stop-while admixture stop refers to all other stops. This distinction is based on the phenomenology of the stops; right-handed stops have significant decays into a top quark and neutrinos while admixture stops decay almost exclusively to a bottom quark and a charged lepton.
The lower bound on the Z R mass from LHC searches is 2.5 TeV [56,57]. Finally, we require that the Higgs mass be within the 2σ allowed range from the value measured at the ATLAS experiment at the LHC.
We naively obtain the two sigma range by adding in quadrature the systematic and statistical uncertainties from [58], and multiplying the result by two: See [59] for comparable data from CMS. A summary of the collider bounds mentioned above is given in • Soft sfermion mass matrices are diagonal.
• The first two generations of squarks are degenerate in mass.
• The trilinear a-terms are diagonal.
• The gaugino masses and trilinear a-terms are real.
In addition, it is typically assumed that the soft trilinear a-terms are proportional to the Yukawa couplings, that is, generically a = Y A for each fermions species. Each A is a dimensionful real number on the order of a TeV, while each Y factor is a dimensionless matrix in flavor space. This condition effectively makes all non-third generation trilinear terms insignificant. Note that this assumption does not immediately follow from the above experimental constraints. However, significant radiative corrections to fermion masses, proportional to the a-term, can arise in SUSY, as first discussed for fermions in references [68][69][70]. For example, a down quark mass is modified by gluino exchange, through the diagram in Fig. 2, as follows: where I(x, y, z) = xy ln x y + yz ln y z + xz ln z and Mg is the gluino mass. If a d is on the order of a TeV, this radiative correction can be quite large, possibly larger than the down quark mass. If this were the case, the radiative correction would have to be fine-tuned against the tree-level contribution to reproduce the correct down quark mass. This motivates allowing only the third generation a-terms to be significant. Hence, our assumption that This makes all a-parameters, except for those associated with t, b and τ , insignificant. For simplicity, we will choose all other a-parameters to be zero. Summarizing the above, we employ the following constraints motivated by low-energy physics: ,q =Q,Ũ ,D , Note that all of these constraints can be implemented at the high scale, since RG evolution to the SUSY scale will not spoil these relations. Furthermore, we do not assume here that the first and second generation slepton masses are degenerate-unlike the squark masses-since this is not required by low energy experiments. The degeneracy or non-degeneracy of these states will not, however, greatly effect the results of this paper.

V. THE PARAMETER SPACE AND SCAN
The previous section reviewed the framework used in this paper for connecting the high scale to LHC accessible physics. It remains at this point to discuss the input values for the SUSY breaking parameters.
In this section, we introducing a novel way to analyze the initial parameter space of a SUSY model. While there have been many studies of specific, fixed boundary conditions at the high scale, and some recent interesting discussions of random parameter scans at the TeV scale [71,72], a study that combines both has not-until now-been undertaken. Specifically, our approach in this paper is to make a statistical scan of input parameters at the high scale-followed by a RG evolution of each set of those parameters to the TeV scale and an analysis of which of these high scale initial conditions lead to realistic physics. While the soft SUSY breaking sector contains over 100 dimensionful parameters, the constraints of low energy experiments discussed in the previous section only allow collider significant values for about a fifth of these-24 to be specific. These, along with tan β and a discussion of the sign of certain parameters, are presented in Table II.
The high scale initial values of the 24 relevant SUSY breaking parameters are determined as follows.
To conduct our scan, we make the assumption that there is only one overall scale associated with SUSY breaking. This assumption does not require that the soft mass parameters be equal to each other, or even have similar values. It does, however, require that these parameters be at least within an order of magnitude, or so, of each other. To quantify this, we demand that any dimension one soft SUSY breaking parameter be chosen at random within the range where M is the mass setting the scale of SUSY breaking and f is a dimensionless number satisfying 1 ≤ f 10. We will further insist that any such parameter be evenly scattered around M ; that is, that M be the average of the randomly generated values. Clearly, this will not be the case if parameters are chosen from a uniform probability distribution in the range ( M f , M f )-referred to as a "flat prior". Instead, a "log prior" is adopted. This means that the natural logarithm of a given soft SUSY breaking parameter is chosen from a uniform distribution in the range With a log prior distribution, M is the geometric mean of the randomly generated parameters. In addition to the dimensionful soft masses, we also scan tan β as a flat prior in the range (1.2, 65), thus selected so that all Yukawa couplings remain perturbative through the entire range. Furthermore, we randomly generate the signs of µ, the three tri-scalar couplings a t,b,τ , and the four gaugino masses M 3,2,R,BL .
In this paper, we are interested in the low energy spectra being accessible at the LHC or a next generation collider. Therefore, in addition to the experimental constraints mentioned in the previous section, we further demand that all sparticle masses be lighter than 10 TeV. We call any point that satisfies this, as well as all previous criteria, a "valid accessible" point. The parameters M and f are chosen in such a way so as to maximize the number of such points. To determine the values of M and f which yield the greatest number of valid accessible points, we begin by making a ten by ten grid in the M − f plane. At each of these hundred points, we randomly generate one hundred thousand initial points in the 24-dimensional parameter space discussed above, RG scale them to low energy, and count the subset that satisfies the experimental checks discussed above. We then plot curves corresponding to a constant number of valid accessible points in Fig. 3. The plot shows a broad peak or plateau, the center of which maximizes the number of such points.
This maximum occurs approximately for These values will be used to generate the results in the remainder of this paper. Note that for these values, the smallest soft parameter is maximally about an order of magnitude away from the largest soft parameter.
Specifically, for M and f in Eq. (76), the ranges for the random scan of each parameter are given in Table II.

VI. RESULTS
All of the following results arise from a scan consisting of ten million randomly generated points with M = 2700 GeV and f = 3.3. We will refer to this as the "main scan". Recall that in the previous section, in addition to the experimental constraints, we imposed an extra condition that all masses be lighter than remove this additional condition. That is, the only constraints in the main scan are the experimental ones given in Section IV. All valid points must satisfy these constraints as well as some other checks, which are reviewed here briefly.
In order to be valid, a point must break B − L symmetry, the Z R mass must be above the lower bound, electroweak symmetry must be broken and since the stops play a crucial role in electroweak breaking, we designate a check that the stops are not tachyonic. Since our numerical analysis uses an iterative process to solve for the SUSY and B − L scales, it is possible that a point may pass a check on the first iteration but fail it on the final iteration, although this is very uncommon. Therefore, we include a "spill" check of the Z R bound, electroweak breaking check and non-tachyonic stop checks. Points that pass these spill checks did so on the final iteration of solving for the B − L and SUSY scales. Furthermore, a valid point must have B − L and SUSY scales that converge to a value in the iterative solution process. We also check that-in addition to the stops-all other SUSY sparticles are not tachyonic and satisfy the imposed mass bounds.
Finally, we check that the Higgs mass matches its experimental value.
All of these conditions are listed in the first column of A striking feature of Table III is  A third important conclusion drawn from Table III is that, for the main scan, a large percentage of the initial points that have B − L breaking consistent with the Z R mass lower bound, also lead to the radiative breakdown of electroweak symmetry. Note that the Z mass can always be adjusted-albeit by low energy tuning-to its experimental value of 91.2 GeV. Again, further analysis-for other values of M and f -shows that electroweak breaking, like B − L breaking, is also roughly independent of M . However, unlike B − L, small f favors electroweak symmetry breaking. As is well known from the literature [73][74][75], electroweak breaking occurs for universal boundary conditions-that is, for f = 1. On the other hand, as f increases, the randomly generated parameter m 2 Hu (M I ) can be considerably larger than the square of the initial stop masses. In this case, the RGE evolution to the SUSY scale may be insufficient to render m 2 Hu (M SUSY ) tachyonic. However, since the initial soft masses are randomly generated, the electroweak breaking survival rate will decrease with increasing f , but will not go to zero.
Whether or not stop masses remain non-tachyonic at low scale depends on the randomly chosen values of several of the initial parameters. As can be seen from Table III,  to that of the Z R mass bound. Further analysis shows that this rate is also controlled by the choice of M -a higher value for M resulting in a higher survival rate for this check. The Higgs mass survival rate for the main scan is, perhaps, surprisingly high-given that we were checking that the Higgs mass for a randomly generated point matches an experimentally measured value within an error of less than one percent. The reason this rate is so high is that the measured value of the Higgs mass is within the range expected for TeV scale supersymmetry breaking.
Since the initial soft SUSY breaking parameter space is 24-dimensional, graphically displaying the subspaces associated with each survival check in Table III is very difficult. However, as can be seen from the RGEs and has been discussed in the text, much of the scaling behavior of the parameters is controlled by the two S-terms, S R and S BL , defined in Eqs. (A24) and (A23). It follows that the results in Table III can be reasonably displayed in the two-dimensional S BL (M I ) -S R (M I ) plane. We begin by presenting in Fig. 5 the initial points in the S BL (M I ) -S R (M I ) plane that satisfy, sequentially, the first two fundamental checks in Table III;  phenomena. Furthermore, it shows the strong dependence of B − L breaking and the Z R mass on the values of the S-terms. There is a line in the S BL -S R plane-between the yellow and red regions-below which B − L breaking is not possible. Note that this includes the origin, which corresponds to vanishing S-terms and, hence, universal soft masses. This shows that at least a small splitting from sfermion universality is required for B − L breaking. Another line exists-between the green and yellow regions-below which Z R is always lighter than its lower bound.
Proceeding sequentially, we present in Fig. 6 the initial points in the S BL (M I ) -S R (M I ) plane that, in addition to breaking B − L with a Z R mass above the experimental bound, also break EW symmetry.
The entire colored region encompasses the green points shown in Fig. 5. Those points that also break EW symmetry are displayed in purple. This plot reaffirms the conclusion drawn from Table III that most of the points that break B − L with a Z R mass above the experimental bound, also break EW symmetry. Note that a small density of green points that do not break EW symmetry are obscured by the purple points.
In Fig. 7, we reproduce Fig. 6 but now, in addition, sequentially indicate the points that are consistent FIG. 6. A plot encompassing the green region in Fig 5. The green points in this plot correspond to those which appropriately break B − L symmetry, but which do not break electroweak symmetry. However, the purple points, in addition to breaking B − L symmetry with an appropriate Z R mass, also break EW symmetry. Note that a small density of green points that do not break EW symmetry are obscured by the purple points.
with the remaining checks-that is, non-tachyonic stops/spill checks/convergence/all lower bounds on sparticles masses satisfied and, finally, that they reproduce the Higgs mass within the experimental uncertainty.
Points that appropriately break B − L symmetry but do not satisfy electroweak symmetry breaking are still shown in green. Points that, additionally, do break electroweak symmetry are again shown in purple. Such points that also have non-tachyonic stops, pass all spill checks and convergence, and which satisfy all lower bounds on sparticles masses, but do not match the known Higgs mass, are now indicated in cyan. Finally, points that satisfy all checks, including the correct Higgs mass, are shown in black. These are the valid points. The density of black points indicates, as observed above, that there is a surprisingly high number of initial parameters that satisfy all present low energy experimental constraints. The distribution of black points can be explained from the fact that, while B − L breaking favors non-zero S-terms, very large Sterms can effect the RGE evolution of sfermion masses adversely. Since the effect of the S-terms depends on the charge of the sfermion in question, some sfermions will become quite heavy while others light or tachyonic. Therefore, in general, the valid points in our scan are a compromise between large S-terms, needed for a Z R mass above its lower bound, and small S-terms needed to keep the sfermion RGEs under control. The most important property of the inital SUSY parameter space in determining low-energy phenomenology is the identity of the LSP. Recall that when R-parity is violated, no restrictions exist on the identity of the LSP; for example, it can carry color or electric charge. Our main scan provides an excellent opportunity to examine the possible LSP's and the probability of their occurence . To this end, a histogram of possible LSP's is presented in Fig. 8-with the possible LSP's indicated along the horizontal axis, and log 10 of the number of valid points with a given LSP on the vertical axis. The notation here is a bit condensed, but is specified in more detail in Table IV. The notation is devised to highlight the phenomenology of the different LSP's, specifically their decays, which are also presented in Table IV.
The most common LSP in our main scan is the lightest neutralino,χ 0 1 . However, not allχ 0 1 states are created equal. LHC production modes for the lightest neutralino depend significantly on the composition of the neutralino-a bino LSP cannot be directly produced at the LHC, but the other neutralino LSP's can.
This is the basis we use for the division of these states. The stateχ 0B designates a mostly rino or mostly blino neutralino,χ 0W a mostly wino neutralino andχ 0H a mostly Higgsino neutralino. Here, the word mostly indicates the greatest contribution to that state. As an unrealistic example, ifχ 0 1 is 34% wino, 33% bino and 33% Higgsino, it is still labeledχ 0W . The chargino LSP's are similarly separated into wino-like and higgsino like charginos, and the stops and sbottom divisions are as in our earlier papers, references [31,32].
Note that this notation for the stops,t ad andt r , are only used to describe stop LSP's. For non-LSP stops, we use the conventional notationt 1 andt 2 .
To make Fig. 8 more readable, we have made an effort to combine bins that have similar characteristics.
The first and second generation left-handed sneutrinos are combined into one bin, where about 50% of the LSP's are first generation sneutrinos. The same holds true for the first and second generation righthanded sleptons, while the first generation right-handed sneutrino is always chosen to be lighter than the second generation right-handed sneutrino. This similarity between the first and second generation sleptons is expected, since their corresponding Yukawa couplings are not large enough to distinguish them through the RG evolution. For both sleptons and squarks, more LSP's exist for the third generation-as expected from the effects of the third generation Yukawa couplings, which tend to decrease sfermion masses in RGE   there have been several works on the phenomenology of explicit bilinear R-parity violation, which has some similarities to this model. References to such papers are mentioned below although see [76][77][78][79] for general discussions. Table IV provides some basic information on the most probable decay modes of each of the possible LSP's. Note that signifies a charged lepton of any generation and j a jet-implying a light quark.
Interesting aspects of Table IV are  This is true whether they are directly produced or occur at the end of a cascade decay. The generation of depends on the neutrino mass hierarchy, as discussed in [31,32]. In the normal hierarchy, muons and taus are most likely, while in the inverted hierarchy all charged leptons are possible.
• Gluino: Most of the LSP decay products mentioned in Table IV mimic well-known hypothetical states-for example, neutralinos decay like TeV scale right-handed neutrinos and squarks decay like leptoquarks. The same can not be said of the gluino, making it an interesting candidate for further study. However, its decays depend strongly on the identity of the next to lightest supersymmetric particle (NLSP). Also, bounds on the gluino are the strongest because of its large production cross section. Therefore, when the gluino is the LSP, it is likely that it is the only LHC-accessible SUSY particle. As with the neutralinos, the gluino's Majorana nature allows same-sign dilepton final statesindicating lepton number violation.
• Squarks: All squark LSP's act like leptoquarks in this model, meaning they are pair-produced and decay into a lepton and a quark. Stop and sbottom LSP's in this model were studied in [31,32]. For both a down-and an up-type non-third generation squark LSP, there will be two highly degenerate LSP states-either a degenerate down and strange squark pair or a degenerate up and charm squark pair-as required by phenomenology. In the inverted hierarchy, these can decay into an electron and jet or a neutrino and a jet, making them tempting explanations for the recent CMS excess, see [80], in the eejj and eνjj channels [81]. However, the branching ratios seem to be inconsistent with the cross section [82,83]. See reference [84] for a study of stop in trilinear R-parity violation.
• Left-handed sneutrinos: Left-handed sneutrinos decay like heavier neutral Higgs bosons, that is, H 0 and A 0 , due to their R-parity violating mixing with the Higgs sector. In general, decays into heavier Higgses are also possible but, of course, this depends on kinematics. The final state + − represent a lepton flavor violating final state, such as µ + e − . Sneutrinos LSP decays were studied in the case of explicit bilinear R-parity violation, which has some similarities to the B − L MSSM, in reference [85,86].
• Right-handed sneutrinos: These states decay into missing energy and, therefore, cannot be easily distinguished from the R-parity conserving MSSM. However, since the sneutrino is spin 0, as opposed to spin half neutralinos, a detailed collider study might reveal some differences. It is also interesting to note that it may be possible to pair produce right-handed sneutrinos through a Z R resonance, although the cross section would probably be small. Yukawa and tri-scalar couplings (see Appendix B 1) have the potential to make its mass lighter than the third-family left-handed sneutrino. Therefore, the left-handed stau is the only left-handed charged slepton capable of being the LSP. Slepton LSPs with explicit R-parity violation were discussed in reference [87].
To get a sense of the non-LSP spectrum, we produce histograms of the masses of the sparticles from the main scan. In the following histograms, there will be quite a few pairs of fields that will be highly degenerate; these will be represented by only one curve. This includes SU (2) L sfermion partners, which are only split by small electroweak terms. First generation squarks are also degenerate with second generation squarks with the same isospin, due to phenomenological constraints. A consequence of this is that all first and second generation left-handed squarks are highly degenerate. In viewing these histograms, it is helpful to remember that aside from the usual RGE effects of the MSSM, there are two additional effects involved. The first of these is the boundary conditions at the B − L scale, corresponding to the B − L and I 3R D-terms which are given in Eq. (64). The second is the new RGE effects of the S R and S BL terms.
Although the signs of these terms are not fixed, Fig. 5 shows that S BL is typically positive while S R is typically negative. This indicates that S R will tend to increase (decrease) sfermion masses for sfermions with a positive (negative) I 3R charge, while S BL tends to increase (decrease) sfermion masses for sfermions with negative (positive) B − L. Figure 9 shows histograms of the squark masses. Because they come in SU (2) doublets and the firstand second-family squarks must be degenerate, all four of the first-and second-family left-handed squarks have nearly identical mass and the histograms coincide. The degeneracy of first-and second-family squarks is also evident in the right-handed squark masses. The first and second family right-handed down squarks are generally lighter than their up counterparts because of the effect of the U (1) 3R charge in the RGEs.

VII. FINE-TUNING
Fine-tuning in supersymmetric models arises from Eq. (15), . This cancellation can come either from other soft masses or the µ 2 term. The delicate cancellation between the parameters on the right side to yield the smaller term on the left side is "fine-tuning". The necessity of such fine-tuning in supersymmetric models has been referred to as the "little hierarchy problem".
Here we explain the little hierarchy problem within the context of the B − L MSSM, using a rough analytic argument along the lines of that presented in [88]. Although we discuss it using the language and where the ellipsis represents neglected higher order terms and terms due to other contributions in Eq. (A42).
Additionally, there are corrections due to the boundary condition Eq. (64). The m 2 Q 3 and m 2 t c themselves receive large contributions through their RGEs, Eqs. (A44) and (A46). Focusing on the contributions from the gluino mass yields As discussed in Section IV A, the stops and gluino have relatively high mass bounds from LHC searches.
Additionally, as discussed in Appendix B 2, satisfying the observed value of the Higgs mass tends to require heavy stops. This means that the stop and gluino soft mass contributions in Eq. (79) must be relatively large and give large contributions to the right-hand side of Eq. (77). For example, if Y t = 0.9, g 2 3 = 1, The Substituting Eq. (61) into Eq. (17) allows us to write the S-term contribution to M Z R . It is given by Substituting this into equation Eq. (80) yields S-term contributions to the H u soft mass. In addition, the S-terms also influence the running of the H u soft mass through the RGEs. Including both of these contributions, the value of m 2 Hu at the SUSY scale is  Fine-tuning in supersymmetric models has historically [89][90][91][92] been quantified using the Barbieri-Giudice (BG) sensitivity, introduced in [93] and [94]. This quantifies the sensitivity of some observable quantity to changes in any of the fundamental parameters of a theory. The delicate cancellation between TeV-scale supersymmetry parameters in Eq. (77) results in the electroweak scale, M Z , having a large BG sensitivity. The BG sensitivity of the electroweak scale is defined as where a i is any of the fundamental parameters of the theory. This says that a fractional change in a i would produce a fractional change in M 2 Z that is F a i times larger. The overall degree of fine-tuning is usually taken to be the largest of all the F a i 's ; that is, The BG sensitivity F will be used to quantify the fine-tuning required in the B − L MSSM.
It is worth mentioning that some authors have pointed out drawbacks to the BG sensitivy and suggested other quantifications of fine-tuning. For example, as discussed in [91], the BG sensitivity and the overall degree of fine-tuning depend on how the fundamental parameters of the theory a i are chosen. Furthermore, one could reasonably use the BG sensitivity of M Z , rather than M 2 Z , as the indicator of fine-tuning. This would result in fine-tuning that is smaller by a factor of two. Such ambiguities in the way fine-tuning is calculated from the BG sensitivity suggest that it is not a precise way to quantify fine-tuning. A separate paper, [95], points out that the relationship between the proton mass and the strong coupling constant at a high scale exhibits high BG sensitivity, but is not actually finely tuned. They propose a more precise quantification of fine-tuning and show that the BG sensitivity actually overestimates the fine-tuning in some sample points in the MSSM.
Despite the possible shortcomings, the BG sensitivity remains the most widely used tool for making rough quantitative analyses of fine-tuning in supersymmetric models. We, therefore, proceed using the BG sensitivity to quantify fine-tuning in the B − L MSSM. For each of the valid points, we compute F . We allow a i to span all of the soft mass parameters of the theory, as well as µ. In the case of scalar soft masses, we take a i to be the mass squared, while in the case of gaugino soft masses and µ we take a i to be the mass to the first power. This choice corresponds to how these parameters appear in the Lagrangian. We then create a histogram of F for all of the valid points in our main scan. This data is shown as the blue line in Fig. 16. Note that the fine-tuning required by the highest percentage of valid points is F ∼ 5000. Be that as it may, a reasonable number of valid points need significantly less fine-tuning-with about 2% requiring With the fine-tuning of each randomly generated point in the B − L MSSM now quantified, we are equipped to produce results for just the most natural points-that is, those requiring minimal fine-tuning. Figure 17 shows a histogram of the LSP's for those points with F < 1000, corresponding to the least fine-tuned ∼ 0.1% of points, from a larger scan of four hundred million points. We refer to these points as "natural" valid points. There are three notable differences between Fig. 17 and Fig. 8. First, stop LSP's are more common. This includes both admixture and mostly right-handed stop LSP's. Stop LSP's are more common because heavy stops tend to cause fine-tuning, so low fine-tuning favors lighter stops and stop LSP's. Second, sbottom LSP's are more common. This is due to the fact that first, both the stop and sbottom masses depend on the soft mass m 2 Q 3 and second, because the right-handed stop and sbottom soft masses have similar terms in their RGE's. These two facts imply that favoring light stops tends to favor light sbottoms as well. Third, Fig. 17 does not have the gluino LSP's shown in Fig. 8, and it does have somẽ d R LSP's not found in Fig. 8. However, the disappearance and appearance of these states in the F < 1000 histogram is not statistically significant and, hence, these states can be ignored. The prevalence of stop and sbottom LSP's is the only significant difference between the natural valid points and the valid points. Note that the physical implications of having a stop or sbottom LSP in the B − L MSSM have been studied in [31,32].

VIII. CONCLUSION
In this paper, we presented a novel approach for relating UV physics to TeV scale physics and applied A central result of this work is the general region of initial parameter space that leads to radiative B − L symmetry breaking. While this depends on multiple parameters of the theory, it can be expressed in terms of the two S-parameters and is presented in this context in Fig. 5. A subsequent figure, Fig 7, shows how additional constraints, such as electroweak symmetry breaking and lower bounds on new sparticle masses, depend on the S-parameters. These two plots indicate that a significant amount of the initial parameter space leads to experimentally viable results. They are followed by various spectrum graphs which show that acceptable spectra are relatively general and do not depend on a specific hierarchy of initial masses. FIG. 17. A histogram of the LSP's for the "natural" valid points with F < 1000. Sparticles which did not appear as LSP's are omitted. The y-axis has a log scale. The notation for the various states, as well as their most likely decay products, are given in Table IV. Note that the natural valid points favor stop and sbottom LSP's more than the valid points presented in Fig. 8. Note that we have combined left-handed first and second generation sneutrinos into one bin and each generation makes up about 50% of the LSP's. The same is true for the first and second generation right-handed sleptons and sneutrinos.
The phenomenology of a given point at the LHC strongly depends on the identity of the LSP. Therefore, another central result of this paper is the calculation of the probability that a given SUSY particle can be the LSP. This was addressed in Fig. 8. As might be expected, a mostly bino neutralino is the most likely candidate. However, since binos cannot be directly produced at the LHC, signals associated with bino LSPs also depend on the rest of the SUSY spectrum. Therefore, an interesting future direction might be to investigate the phenomenology of mostly wino or Higgsino neutralinos. Mostly wino or Higgsino neutralinos can be directly produced at the LHC, independently of the rest of the SUSY spectrum, and have relatively large cross sections for colorless particles. The signals associated with different LSPs are summarized in Table IV. Finally, the fine-tuning associated with this statistical scan was investigated. While it is not drastically different than the fine-tuning in the MSSM with a similar UV completion, one might think that the new mass scale associated with B − L breaking could introduce new contributions to fine-tuning. We showed that it does not. In fact, a given point in this model is typically less fine-tuned than a similar point in the MSSM.
In addition, we explored possible LSPs for points with fine-tuning better than one part per thousand-in a way analogous to Fig. 8. We found that stops and sbottoms become much more likely LSP candidates, as one might expect-see Fig. 17. The signals of stop and sbottom LSPs were discussed previously in [31,32]. It is perhaps useful to separate the scalar mass RGEs into those that are analytically tractable and those that are not. In the B −L MSSM scaling regime, the first-and second-family and sneutrino soft mass RGEs, analytically solvable, are In the MSSM scaling regime, which is only relevant to the case of the right-side-up hierarchy, the RGEs are The right-handed sneutrinos masses do not run in this regime because they are not charged under the MSSM gauge group. In the upside-down case the the right-handed sneutrinos are present in the brief scaling regime between M SUSY and M B−L . Their soft mass RGEs are For the third family sfermions (excluding the sneutrinos) and for the MSSM Higgs, all of which are not along with the notation for the physical masses.
mũ L = mũ + ∆Q 1 , mũ R = mũc + ∆ũc, mc L = mc + ∆Q 2 , mc R = mcc + ∆sc, md L = md + ∆Q 1 , md R = md c + ∆d c , ms L = ms + ∆Q 2 , ms R = msc + ∆sc, The third-family right-handed sneutrino physical state (referred to asν c R ) mass is different because it acquires mass through the B − L symmetry breaking mechanism and is degenerate with the Z R mass.
The Yukawa and tri-scalar couplings associated with third-family squarks and charged sleptons contribute non-negligible mixing terms among these scalars. These effects are captured in the stop, sbottom, and stau mixing matrices. Here we use the conventional notation a t,b,τ = Y t,b,τ A t,b,τ . The stop mixing matrix in the basis (t,t c * ) is The eigenstates of this matrix will be referred to ast 1 andt 2 with mass eigenvalues defined such that The sbottom mixing matrix in the basis (b,b c * ) is The eigenstates of this mass matrix will be referred to similarly to the stops. The stau mixing matrix in the basis (τ ,ẽ c * ) is The eigenstates of this matrix will be referred to similarly to the stops and sbottoms. All of the running parameters in these matrices are evaluated at the SUSY scale.
We present these matrices in terms of the fermion masses M t,b,τ for simplicity. However, for numerical evaluation these fermion masses are replaced with the appropriate Higgs VEV times Yukawa coupling evaluated at the SUSY scale.
For any of these matrices,  the relevant mixing angle is given by where the angle θf may always be chosen to be between 0 • and 90 • . Defined this way, a mixing angle close to zero means the lighter mass eigenstate consists of mostly the left-handed gauge eigenstate and a mixing angle close to 90 • means the lighter state is mostly right-handed.
The chargino content is identical to that of the MSSM in the approximation of vanishing R-parity violation. This is a good approximation for calculating masses but the mixing with the charged leptons need to be take into account when calculating decays, see [31] for example. Continuing with the approximation of vanishing R-parity violation, the results of [3] may be used. Those results, in our own notation, are m 2 In the basis (ν,W R ,B ,W 0 ,H 0 u ,H 0 d ), the neutralino mass matrix is  where c θ ≡ cos θ and s θ ≡ sin θ etc. As with the charginos, we have assumed that mixing with the lefthanded neutrinos, due to R-parity violation is 0. This is good approximation for calculating masses and will be used here, but cannot be used when calculating decay rates. As discussed in Section IV some of the eigenstates of this matrix have masses associated with the B − L scale while others have masses associated with the SUSY scale. A conventional approach to this situation would be to perturbatively diagonalize the matrix in the limit M SUSY M B−L for the right-side-up case or M B−L M SUSY for the upside-down case. However, these two scales may be comparable so the entire mass matrix must be diagonalized without the use of perturbative methods. This has the potential to introduce errors since it doesn't account for the fact that some states should be integrated out at different scales. However, the errors will always be small because the B −L and SUSY scales are always of comparable size. We choose to evaluate all of the running parameters in this matrix at the SUSY scale. The error introduced by doing this should be smaller than the error introduced by associating the entire SUSY spectrum with a single scale, M SUSY . The mass eigenstates are referred to asχ 0 1 · · ·χ 0 6 in a mass ordered basis with eigenvalues mχ0 1 · · · mχ0 6 . The physical gluino mass, Mg is simply equal to the running gluino mass evaluated at the SUSY scale. (B11)

Higgs Masses
Supersymmetric models such as the MSSM and this B − L MSSM contain five Higgs particles. The most important for the present discussion is the lightest neutral SM-like Higgs, h 0 , which we refer to as "the Higgs" throughout this paper. This one is important because its mass is known and can be used to constrain some of the SUSY parameter space. The other four Higgses are the heavy Higgs, H 0 , the Higgs pseudoscalar, A 0 , and the charged Higgses, H ± .
The Higgs mass is calculated using methods discussed in [46][47][48]. The physical Higgs mass is with the Higgs quartic coupling, λ, evaluated at the scale of the physical Higgs mass. Above the SUSY scale, λ comes from the D-terms and is thereby fixed. Below the SUSY scale, RGE effects will cause λ to deviate from its supersymmetric value. These effects come mainly from one-loop graphs involving the top quark. They are contained in the RGE for λ in the SM scaling regime. We employ results from [48]. Here we re-state the relevant equations in our own notation. The supersymmetric boundary condition on λ is λ(M SUSY ) = 1 4 g 2 L + 3 5 g 2 1 cos 2 2β + δλ.
The parameter δλ contains threshold corrections applied at the SUSY scale. Including only the dominant stop contributions from [48], where we define X t = A t − µ cot β (note that this definition is different from that used in [48]) and F (x) = 2x ln x x 2 − 1 (B15) combination of analytic solutions (discussed throughout this paper) and numerical methods (not discussed in this paper). For the purposes of this Appendix, it is sufficient to know that the solution can indeed be calculated. It is also useful to note that, with the exception of the spill and convergence checks, the checks in Table III are applied sequentially. For example, a point is subjected to the EW breaking check if and only if it passes the preceding B − L breaking and Z R bound checks. This means that a point that fails a particular check a) has implicitly passed all previous checks and b) is immediately discarded and never subjected to subsequent checks. The sequential nature of these checks is what enables us to define the survival rates given in Table III. The spill checks and the convergence check, however, are different because they are not necessarily applied in a particular order and may even be applied multiple times to a single point. Nevertheless, if any point fails a spill or convergence check, at any step in the iterative process, we count that point as having passed all spill checks that appear above the failed check in Table III. This removes any ambiguity about how to define survival rates for the spill and convergence checks. If the guess for the SUSY scale satisfies its definition to within 1%, the rest of the steps in this paragraph are skipped. If not, the guess for the SUSY scale is changed to mt 1 (M SUSY )mt 2 (M SUSY ) and we rerun the RGEs. If the point now does not satisfy EW breaking, it is counted as failing the EW breaking spill check. If the point now does not satisfy the non-tachyonic stops check, it is counted as failing the nontachyonic stops spill check. If it does pass these checks, but M SUSY does not satisfy its definition to within 1%, then the steps in this paragraph are repeated. If they have been repeated more than 300 times without a conveniently chosen number which provides adequate opportunity for the iteration to converge. success, the point is counted as failing the convergence check.
Having successfully passed all of the previous criterion, we now must check the remaining checks. If the point does not satisfy the Z R bound, it is counted as failing the B − L bound spill check. If the point does not satisfy the sparticle bounds, it is so counted. If the point does not satisfy the Higgs mass check, it is so counted. If it does, however, satisfy all of these experimental checks, it is a valid point.
The procedure described in the previous five paragraphs is represented pictorially by the "flow chart" in   and SUSY scales works. Every block that begins with the word "Check" has an outgoing red and green arrow. The green arrow is followed if the check is satisfied and the red arrow is followed if the check is not satisfied.