The Minimal SUSY $B-L$ Model: Simultaneous Wilson Lines and String Thresholds

In previous work, we presented a statistical scan over the soft supersymmetry breaking parameters of the minimal SUSY $B-L$ model. For specificity of calculation, unification of the gauge parameters was enforced by allowing the two ${\mathbb Z}_{3}\times {\mathbb Z}_{3}$ Wilson lines to have mass scales separated by approximately an order of magnitude. This introduced an additional"left-right"sector below the unification scale. In this paper, for three important reasons, we modify our previous analysis by demanding that the mass scales of the two Wilson lines be simultaneous and equal to an"average unification"mass $\left$. The present analysis is 1) more"natural"than the previous calculations, which were only valid in a very specific region of the Calabi-Yau moduli space, 2) the theory is conceptually simpler in that the left-right sector has been removed and 3) in the present analysis the lack of gauge unification is due to threshold effects--particularly heavy string thresholds, which we calculate statistically in detail. As in our previous work, the theory is renormalization group evolved from $\left$ 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 soft supersymmetry breaking masses that satisfies all such constraints is presented and shown to be substantial.

Within the context of the heterotic superstring and heterotic M-theory [1][2][3][4], there have been a number of vacuum states whose four-dimensional low energy effective field theory [5] has the exact spectrum of the MSSM-with or without right-handed neutrino chiral multiplets-and, to prohibit rapid proton decay, contains R-parity [6][7][8]-either as a discrete symmetry or as a subgroup of an anomaly free U (1) extension of the standard model gauge group [9][10][11][12][13][14][15][16][17][18]. One such vacuum was presented in [19][20][21][22] and will be referred to as the B − L MSSM. Be that as it may, this is only the first step in finding a realistic heterotic string vacuum. Any such theory must also also be compatible with all presently observed low-energy phenomenology; that is, it must spontaneously break electroweak (EW) symmetry at the observed scale, must be compatible with the newly discovered Higgs particle with mass ∼ 125 GeV [23,24], have all sparticle masses above the present observational lower bounds and-assuming R-parity is contained in an additional U (1) symmetry-spontaneously break that Abelian group with an associated gauge boson mass in excess of the present experimental lower bound.
In a series of papers [25][26][27][28][29], the B −L MSSM was examined in detail-using a random statistical sampling of the initial set of soft supersymmetry (SUSY) breaking parameters-and the results confronted with these phenomenological requirements. It was shown, within a restricted region of the compactification moduli space, that the B −L MSSM easily passed each of these requirements for a large and basically uncorrelated set of initial conditions. Furthermore, this analysis led to a series of low energy predictions-for example, directly relating the lightest stop decay channels and branching ratios to the neutrino mass hierarchy and mixing angles-thus linking LHC experimental results to neutrino measurements. That is, the B − L MSSM is a possible candidate for a phenomenologically acceptable theory of the real world-a statement that will be directly testable as its structure and predictions are confronted with upcoming data from the LHC, neutrino experiments and cosmological observations. For these reasons, in this paper we extend the results of [25][26][27][28][29] to a more general, and more natural, region of the B − L MSSM moduli space.
To quantify this, we should briefly discuss the structure of the B − L MSSM. It arises in the observable sector of heterotic M-theory compactified to four-dimensions on a Shoen Calabi-Yau (CY) threefold [30] with first homotopy group π 1 = Z 3 × Z 3 . This manifold admits a specific slope-stable holomorphic vector bundle [31] with structure group SU (4) ⊂ E 8 , as well as two Wilson lines-each wrapped over a two-cycle associated with a different Z 3 homotopy factor. This theory has three, in principle distinct, mass scales-M U at which the gauge bundle spontaneously breaks E 8 to SO (10) and two Wilson line mass scales, which we denote by M χ 3R and M χ B−L , associated with the inverse radii of their respective two-cycles. In our previous analysis, we worked in a restricted region of CY moduli space where the radius of one two-cycle is distinctly smaller than that of the other; that is, we chose M U M χ B−L > M χ 3R . By taking the scale of separation of the two Wilson lines to be approximately an order of magnitude, one can exactly unify all gauge coupling parameters-thus specifying boundary conditions in the renormalization group equations (RGEs). However, this specificity comes at the cost of introducing an additional scaling regime.
Between M χ B−L and M χ 3R the effective theory is that of the "left-right" model [12,16,25,32] with gauge group SU (3) C × SU (2) L × SU (2) R × U (1) B−L and a specific particle spectrum that can be computed from string theory. It is only for energy-momentum below the lightest Wilson line mass M χ 3R that one obtains the spectrum and SU In this paper, we generalize and simplify the phenomenological analysis of the B − L MSSM by working in a generic region of CY moduli space where the radii of the two Wilson lines and the average radius of the CY manifold are all approximately equal: that is, with M U M χ B−L M χ 3R . This generalization is significant in that 1) the region of moduli space is much larger and more "natural" than that used previously and 2) the "left-right" scaling region is eliminated, with the B − L MSSM emerging immediately below the compactification scale-thus simplifying the scaling regimes. Of course, the four B − L MSSM gauge couplings will no longer unify near the scale of the CY radius. This does somewhat complicate the RG analysis. However, it opens the door for a discussion of unification of all gauge couplings with the gravitational coupling at the "string scale"-as has been discussed by many authors in [33][34][35][36][37][38][39][40][41][42][43][44][45]. More specifically, such unification should take place at tree level. However, at the one-loop (and higher) level one expects such unification to be split by "threshold" corrections. These are due to several effects, such as the inclusion of field theory thresholds at each of the SUSY, B − L and "unification" scales, and genus-one string theory corrections. Since in this analysis the latter is expected to be the largest, we will focus exclusively on them. By running the four B − L MSSM gauge parameters up to the string scale, we will 3) statistically compute the heavy string threshold corrections for each gauge coupling. Furthermore, we will statistically compute the hypercharge gauge threshold and, by subtracting various thresholds, analyze the moduli dependent sub-component of each. Finally, there is yet another important benefit of analyzing the B − L MSSM at the generic region of its CY moduli space-although we will not pursue this in the present paper. In our previous work and the present analysis, the scale of the soft SUSY breaking parameters is chosen to be in the TeV region. This is done in order for our low energy phenomenological predictions to be LHC accessible. However, unlike in the case of unified gauge couplings enforced by splitting the Wilson line masses discussed in [25][26][27][28][29]-which, for reasons elucidated in the Conclusion, is essentially restricted to the TeV region-for the simultaneous Wilson line masses discussed in this paper, the SUSY breaking mass scale can be taken to be arbitrarily large. This has a number of important applications, both in particle phenomenology and in early universe cosmology [46]. We will pursue this in future work [47].
The present paper is structured as follows. In Section II we review the salient parts of the B − L MSSM theory, presenting the spectrum, the supersymmetric and the soft SUSY breaking Lagrangians, discussing the generic structure of spontaneous B − L and EW symmetry breaking and setting our notation. Section III is devoted to defining the exact meaning of the "simultane-ous" Wilson line analysis presented in this paper-as opposed to the "split" Wilson line approach in our previous work. It then discusses, in detail, the four relevant mass scales from "unification" to electroweak symmetry breaking. The statistical definitions of the "unification" mass and gauge coupling are given in our present context. In Section IV, the three scaling regimes-for both the "right-side-up" and "upside-down" scenarios-along with the associated gauge coupling beta function parameters are presented. The RG running of the Yukawa couplings, including their transition at the SUSY scale, is discussed. Section V gives a brief review of the "statistical" approach to setting the initial soft supersymmetry breaking parameters at the "unification" scale presented in detail in our previous work. In Section VI, the experimental constraints on the sparticle masses, the heavy vector boson mass and the lightest neutral Higgs mass are presented. We then solve the RGEs-for randomly chosen initial soft SUSY breaking parameters-sequentially from the "unification" scale down through the EW breaking scale subject to these constraints. Plots-and the exact number-of the initial points that sequentially satisfy the experimental constraints are given; ending with the robust number of phenomenologically "valid" black points that satisfy all experimental constraints. A brief analysis of both the LSP and non-LSP spectra is then given. In Section VII, we briefly discuss fine-tuning in the B − L MSSM. In Section VIII, we statistically calculate the heavy string threshold corrections for each of the four B − L MSSM gauge couplings, as well as the hypercharge gauge threshold, and analyze the moduli dependent differences of these quantities. Finally, in Section IX, we present our conclusions.

II. THE MINIMAL SUSY B − L MODEL
In this section, we briefly review the minimal anomaly free extension of the MSSM with gauge group whose structure was motivated by heterotic string theory in [20] and by phenomenological considerations in [48,49]. Although this model has been discussed in our previous papers [25][26][27][28][29], we outline its main features in this section for specificity and to set our notation. The Abelian gauge factors U (1) 3R × U (1) B−L can be rotated into physically equivalent charge bases, such as However, as shown in [25], this comes at the cost of introducing kinetic mixing between the gauge fields. We therefore prefer to work in the basis where I 3R , I BL and g 3R , g BL are the generators and couplings for the U (1) 3R and U (1) B−L groups respectively. The gauge boson associated with U (1) B−L is denoted B to distinguish it from the gauge boson associated with U (1) Y , which is normally denoted B. The factor of 1 2 in the last term is introduced by redefining the gauge coupling g BL , thus simplifying many equations. A radiatively induced vacuum expectation value (VEV) of a right-handed sneutrino will break the Abelian factors U (1) 3R × U (1) B−L to U (1) Y , in analogy with the way the MSSM Higgs fields break SU (2) L × U (1) Y to U (1) EM . This process is referred to as "B − L" symmetry breaking, although technically it breaks a specific combination of the groups generated from I 3R and I BL , leaving invariant the usual hypercharge group generated by The particle content of the model is simply that of the MSSM plus three right-handed neutrino chiral multiplets. This amounts to three generations of matter superfields along with the usual two Higgs supermultiplets where we have displayed their SU (3) C × SU (2) L × U (1) 3R × U (1) B−L quantum numbers. The superpotential of the B − L MSSM is given by where both generational and gauge indices have been suppressed. In principle, the Yukawa couplings are three-by-three complex matrices. However, the observed smallness of the CKM mixing angles and CP-violating phase imply that the quark Yukawa matrices can be approximated as diagonal and real for the purposes of RG evolution in this paper. The charged lepton Yukawa coupling can be made diagonal and real by moving the PMNS angles and phases into the neutrino Yukawa couplings. The small size of neutrino masses implies that the neutrino Yukawa couplings can be neglected for the purposes of RG evolution in this paper. The smallness of first-and secondgeneration fermion masses implies that first and second-generation Yukawa quark and charged lepton Yukawa couplings can also be neglected. The µ-parameter can be chosen to be real without loss of generality.
The soft supersymmetry breaking Lagrangian is where generation and gauge indices have been suppressed. The a-parameters and sfermion softmass terms can, in principle, be hermitian matrices in family space. However, this tends to lead to unobserved CP violation. Therefore, we proceed assuming that they are diagonal and real.
Furthermore, as discussed in Section V, we assume that the a-parameters are proportional to the Yukawa couplings. This implies that all the a-parameters can be neglected except for the (3,3) component of the quark and charged lepton a-parameters. The b-parameter can be chosen to be both real and positive without loss of generality. Although the gaugino soft masses can be complex in principle, this tends to lead to unobserved flavor and CP violation. Therefore, we proceed assuming that they are real.
The B − L symmetry is spontaneously broken by the VEV in a right-handed sneutrino, which carries the appropriate I 3R and I B−L charges to break those symmetries while preserving hypercharge symmetry. This VEV is brought about by a sneutrino soft-mass term becoming tachyonic 1 at the TeV scale due to the RGE evolution. As discussed in [17,50,51], this VEV will be purely in one of the three right-handed sneutrino generations -not in a linear combination of them.
Furthermore, the three generations of right-handed sneutrinoes can be relabeled without loss of generality. Therefore, we henceforth assume that it is the third-generation right-handed sneutrino that acquires a VEV. Electroweak symmetry is broken by VEVs in the neutral components of the up and down Higgs multiplets. The electroweak breaking VEVs and the B − L breaking VEV together lead to small VEVs in all three generations of left-handed sneutrinos. The above VEVs will be denoted by where i = 1, 2, 3 is the generation index.
The neutral gauge boson that becomes massive due to B − L symmetry breaking is referred to and assuming that v 2 v 2 R , Z R acquires to leading order a mass of The hypercharge gauge coupling is given by where The smallness of the neutrino masses implies, first, that the neutrino Yukawa couplings are small and, second, that the left-handed sneutrino VEVs are much smaller than the electroweak scale. In this limit, the minimization conditions of the potential simplify to Noting from above that |v 2 (9) and (12) can be combined to give The VEV in the third-generation right-handed sneutrino induces spontaneous bilinear R-parity violation through the operators Bilinear R-parity violation has been discussed extensively, including its relevance to neutrino masses. See, for example, some early works [52][53][54][55]. The Lagrangian of this model contains additional bilinear terms due to the sneutrino VEVs: The R-parity violating terms in this model have a variety of interesting consequences that have been studied in a number of different contexts. These include LHC studies [48,49,56,57], predictions for neutrinos [17,50,51], and connections between the two [26,27]. It has been shown that the R-parity violation can give rise to Majorana neutrino masses, with the lightest lefthanded neutrino being massless. There is also a pair of sterile right-handed neutrinos that can have cosmological implications [57].
The minimal supersymmetric B − L model, reviewed in this section, will be referred to simply as the B − L MSSM throughout the rest of this paper. We now turn to connecting the phenomenology of the B − L MSSM to its high-scale origins. Specifically, we are considering the possibility that the B − L MSSM is the observable sector of the low-energy effective theory of an E 8 × E 8 heterotic string theory. In this context, the B − L MSSM gauge group unifies into an SO (10) gauge group, which is itself the commutant of the SU (4) structure group of the observable sector E 8 vector bundle on the CY threefold. We have previously studied the B − L MSSM in this context [29]. In this paper, however, we will study the effects of string threshold corrections on gauge unification. This requires a discussion of gauge unification-to which we now turn. unification scale. That is, one "naturally" expects requires that these scales be different. For specificity of the RGE calculation, it was convenient to impose precise gauge coupling unification. Hence, in [25] we studied the two cases with separated Wilson line masses-even though this can occur only in special regions of moduli space. Under the assumption that the soft SUSY breaking masses be of TeV order-to assure that sparticle masses potentially be LHC accessible-we found that gauge coupling unification dictates that the Wilson line scales must be separated by less than/approximately an order of magnitude in either case.
Additionally, we found that both cases lead to similar low energy phenomenology. Hence, for specificity, we carried out our analysis using the first of these symmetry breaking patterns; that is, the intermediate regime containing the "left-right" model. We refer the reader to [25] for that analysis. Here, for concreteness, we simply show in Figure 1  In this paper, we turn to the analysis of the generic region of moduli space where equation (20),  [29], we continue to use the same notation for all quantities. In particular, it is important to use identical notation for the B − L gauge coupling. Thus far in this paper, we have discussed the gauge parameter g BL , which couples to the I BL 2 generator. However, as was discussed in [25], this gauge coupling has to be properly normalized so as to unify with the other gauge parameters in the split Wilson line scenarios. The appropriate coupling was denoted g BL and defined by Even though the four gauge couplings, including g BL , will not unify in the simultaneous Wilson line scenario in this paper, we will continue to use this parameter when appropriate. Note that g BL couples to the 3 8 I BL generator 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 four relevant mass scales of interest. All four are described in the following: M U : the unification and the first and second Wilson lines mass scale.
Since, as discussed above, exact unification of the four gauge couplings no longer occurs for simultaneous Wilson lines, it is essential to give a justification-and an explicit definition-of what we mean by the "unification mass" in the present context. In [29], every phenomenologically valid point in the space of randomly chosen initial soft supersymmetry breaking parameters corresponds to an explicit unification mass M U and unified coupling α u . Both the unification scale and unification parameter vary for different valid points. The associated statistical histograms for these quantities are shown in Figures 2 and 3 respectively, along with their average values. These are found to be In this paper, we will refer to the average values M U and α u as the "unification" mass and "uni- is M B−L : the mass at which the right-handed sneutrino VEV triggers 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 Note that M Z R itself depends on parameters evaluated at M B−L . This results in a transcendental equation that can be solved using for M B−L using numerical methods. The boundary condition relating the hypercharge coupling to the gauge couplings of U (1) 3R and U (1) B−L at this scale is nontrivial. It is given by where As with the B − L gauge coupling, the hypercharge coupling has been rescaling to allow for unification in the split Wilson line scenarios. The rescaled hypercharge gauge coupling, g 1 , is defined by M SUSY : the soft SUSY breaking scale.
This is the scale at which all sparticles are integrated out, with the exception of the right-handed sneutrinos, which are associated with B −L breaking and, therefore, are integrated out at the B −L scale [29]. While the sparticles do not all have the same mass, we use the scale of stop decoupling as a representative scale associated with all sparticles. That is, The scale of stop decoupling is the best choice for the SUSY scale since the stops give the dominant radiative corrections to phenomenologically important quantities such as the electroweak scale and the Higgs mass. See, for example, [58] for more details. Note that the physical stop masses depend on quantities evaluated at M SUSY . Therefore, this equation must be solved using iterative numerical methods for the correct value of M SUSY .
M EW : the electroweak scale.
This is the well-known scale associated with the Z and W gauge bosons of the standard model (SM). We identify this scale with the mass of Z boson, as is conventional. That is,

PARAMETERS
Having defined the relevant mass scales, we turn to a brief discussion of RG evolution that occurs between them. The gauge coupling RGEs are  right-side-up hierarchy: • M B−L − M SUSY : In this regime, the gauge group and particle content is that of the MSSM plus two right-handed neutrino supermultiplets. The gauge couplings in this regime evolve with the slope factors We refer to this regime as the "MSSM" regime.
• M SUSY − M EW : In this regime, the sparticles are integrated out, leaving the SM with an additional two sterile neutrinos. It has the well-known slope factors We refer to this regime as the "SM" regime.
upside-down hierarchy:  (25), (26). In previous work [25], exact unification conveniently specified sin 2 θ R ≈ 0.6. However, in the present scenario we are not requiring exact unification. Hence, this specificity is lost and sin 2 θ R is a free parameter. We proceed by simply setting in order to make the results of this paper more directly comparable to those of [25]. Similarly to the gauge parameters, the Yukawa couplings run differently under the RG through each of the above scaling regimes. Before discussing them, we must first decide which Yukawa couplings are relevant to our analysis. As discussed in [29], we begin by inputting the experimen- For details on relating fermion masses to Yukawa couplings, see [60]. We use lower case y to denote Yukawa couplings in the non-SUSY regime. The one-loop RGEs for these Yukawa couplings were presented in Appendix A of [29], to which we refer the reader. The Yukawa couplings in

V. THE SOFT SUPERSYMMETRY BREAKING PARAMETERS
The remaining parameters of the B − L MSSM are the massive coefficients appearing in Eqn. (7) which are responsible for softly breaking supersymmetry. Their RGEs in each physical regime were presented in detail in [29] and won't be discussed in this paper. Here, we simply note that flavor and CP-violation experimental results place well-known limits on these quantities.
Generically, the implication of these constraints are, approximately, as follows: • 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.
It is typically assumed that the soft trilinear a-terms are proportional to the Yukawa couplings.
That is, a = Y A for each fermions species. Each A is real and associated with the SUSY scale.
Each Y factor is a dimensionless matrix in family space. This condition effectively makes all non-third-generation trilinear terms insignificant. The above constraints are summarized as These constraints can be implemented at the scale M U , since RG evolution to the SUSY scale will not spoil these relations. Note that we do not assume that the first and second generation slepton masses are degenerate, unlike the squark masses, since this is not required by experiments.
The degeneracy or non-degeneracy of the first and second generation sleptons will not, however, greatly effect the results of this paper.
We now turn to the input values for the SUSY breaking parameters. Unlike the cases of the gauge and Yukawa couplings, these soft SUSY breaking parameters are not experimentally determined. In [29], we introduced a novel way to analyze the initial parameter space of a SUSY model. We will follow the same approach in the present analysis of simultaneous Wilson lines.
Specifically, we run a statistical scan of input parameters at the scale M U . The randomly generated input parameters are then RG evolved to the SUSY scale. We conduct an analysis of which of these high-scale initial conditions lead to realistic physics. Although the soft SUSY breaking Lagrangian contains over 100 dimensionful parameters, the phenomenologically motivated assumptions discussed briefly above only allow significant values for 24 of them. These, along with tan β and the sign of certain parameters, are presented in the first column of Table I.
The high-scale initial values of the 24 relevant dimensionful SUSY breaking parameters are determined as follows. We make the assumption that there is only one overall scale associated with SUSY breaking, requiring that these parameters be separated by no more than an order of magnitude, or so, from 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 overall 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. In [29], we found that in the case of split Wilson line masses, the maximal number of phenomenologically acceptable "valid" initial points were obtained by statistically scattering within the interval defined by To allow direct comparison of the results of this paper to those of [29], we will continue to use these values in the present context. This is shown in the second column of Table I, along with the scattering interval associated with tanβ and the allowed signs of various parameters.

VI. THE PARAMETER SCAN AND RESULTS
The technical details of our statistical scan over the interval of soft supersymmetry breaking parameters, the complete set of all RG equations, the evolution of all parameters under the RGEs and a discussion of the sparticle and the Higgs masses were presented in detail in both the text and Appendices of [29]. We will not repeat them here and refer the reader to that paper. In this section, we will simply apply these methods to the more "natural" case of simultaneous Wilson lines satisfying Eqn. (23). As we did in [29], we will perform a scan over 10 million random  mass be within the 2σ allowed range from the value measured at the ATLAS experiment at the LHC [23,24]: Since the initial soft SUSY breaking parameter space is 24-dimensional, graphically displaying the results is, in principle, very difficult. However, as was discussed in both the text and Appendices of [29], much of the scaling behavior of the parameters is controlled by the two S-terms, S BL and S 3R , defined by where "Tr" implies a sum over the three families. It follows that our results can be reasonably We begin by presenting in Fig. 5  Proceeding sequentially, we present in Fig. 6 the initial points in the 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 indicates 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. Specifically, we find that out of the 697,886 green points that break B − L with • 485,952 -the purple points-also break EW symmetry.
In Fig. 7, we reproduce Fig. 6 but now, in addition, sequentially indicate the points that are consistent with the remaining checks-that is, 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 satisfy all lower bounds on sparticles masses, but do not match the known 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 LSP Spectrum:
An important property of the initial 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 oc- Sparticles which did not appear as LSP's are omitted. The y-axis has a log scale. The dominant contribution comes from the lightest neutralino, as one might expect. The notation for the various states, as well as their most likely decay products, are given in Table III. Note that we have combined left-handed first and second generation sneutrinos into one bin, and that 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.
currence . 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 III.
The notation is devised to highlight the phenomenology of the different LSP's, specifically their decays 2 , which are also presented in Table III.
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 2 Recall that when R-parity is violated, as it is in this paper, the LSP can decay to lighter non-supersymmetric states. First and second generation right-handed sleptons. eν, µν LSP's are split evenly between these two generations. other neutralino LSP's can. This is the basis we use for the division of these states. The statẽ χ 0B designates a mostly rino or mostly blino neutralino,χ 0W a mostly wino neutralino andχ 0H a mostly Higgsino neutralino. Here, the subscript 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 labeled χ 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 [26,27]. 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 right-handed 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 thirdgeneration Yukawa couplings, which tend to decrease sfermion masses in the RGE evolution.
The myriad of possible LSP's leads to a rich collider phenomenology. This phenomenology is not the main focus of this paper, but it is worthwhile to briefly review it here. In models where R-parity violation is parameterized by bilinear R-parity breaking, such as the B − L MSSM, SUSY particles are still pair produced and cascade decay to the LSP. At this point, the bilinear R-parity violating terms allow the LSP to decay. While only a few studies have been done on the phenomenology of the minimal B − L MSSM [26,27,56,57], there have been several works on the phenomenology of explicit bilinear R-parity violation, which has some similarities to this model. See [61][62][63][64] for general discussions. Table III 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. Some interesting aspects of Table III were discussed in [29].
The Non-LSP Spectrum: To get a sense of the non-LSP spectrum, we produce histograms of the masses of the sparticles associated with the valid points in 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.  Figure 10 shows histograms of the masses of the sneutrinos and sleptons. The third-family sleptons and left-handed sneutrinos tend to be the lighter because of the influence of the τ Yukawa coupling. The right-handed sneutrinos are labeled such thatν R 1 is always lighter thanν R 2 . Figure 11 presents histograms of the CP-even component of the third-generation right-handed sneutrino, the heavy Higgses, the neutralinos, the charginos, and the gluino. The CP-even component of the third-generation right-handed sneutrino is degenerate with Z R . It is always heavier than 2.5 TeV because we have imposed the collider bound on Z R . The neutralinos and charginos are labeled from lightest to heaviest as is canonical in SUSY models. Theχ 0 5 andχ 0 6 are typically Higgsinos. We emphasize that all of the above histograms are calculated using our main scan; that is, for the choice of M = 2700 GeV and f = 3.3. We

VII. FINE-TUNING
A detailed discussion of the the little hierarchy problem, fine-tuning and the Barbieri-Giudice (BG) method of quantifying the degree of fine-tuning was presented in [29]. Here, we simply give the results in the simultaneous Wilson line scenario discussed in this paper. Unlike the quantitites presented above, which can differ substantially from the split Wilson line results in [29], the BG

VIII. STRING THRESHOLD CORRECTIONS
As discussed in the Introduction and Section III, and graphically illustrated for a valid initial point in Figure 4, the four gauge couplings of the B − L MSSM do not unify at M U for simultaneous Wilson line masses; that is, when However, as described in the Introduction, the B−L MSSM arises on the observable orbifold plane of heterotic M -theory compactified on a Schoen Calabi-Yau threefold with π 1 = Z 3 × Z 3 and a holomorphic vector bundle with SU (4) ⊂ E 8 structure group. That is, the B − L MSSM is a low energy effective theory of heterotic string theory. Hence, as discussed in numerous papers [33][34][35][36][37][38][39][40][41][42][43][44][45], it is expected that at string tree level all four gauge couplings, along with the dimensionless where G N is Newton's constant and α is the string Regge slope, unify to a single parameter g string at a "string unification" scale M string = g string × 5.27 × 10 17 GeV .
The string coupling parameter g string is set by the value of the dilaton, and is typically of O(1). A common value in the literature, see for example [37,41,45], is g string = 0.7 which, for specificity, we will use henceforth. Therefore, we take α string and the string unification scale to be as in Eqn. (31). However, the RGEs are now altered to become 3 Note that the one-loop running couplings no longer unify exactly at M string . Rather, they are "split" by dimensionless threshold effects. These arise predominantly from massive genusone string modes that contribute to the correlation function F a µν F aµν and, hence, to the α a gauge couplings. This is depicted graphically in Figure 15.
Recall that in this paper we have found 44, 884 valid initial points in the space of soft supersymmetry breaking dimensionful couplings-each of which satisfies all low energy phenomenological criteria. For each of these points, we can calculate-by scaling from the electroweak scale to M U -the four gauge couplings α 3 ( M U ), α 2 ( M U ), α 3R ( M U ) and α BL ( M U ). Note that in this analysis, we have defined an "average" SUSY scale M SU SY , the B − L breaking scale M B−L , as well as an "average" unification scale M U , in (28), (24) and (22) respectively. The RGEs have been scaled through the requisite intermediate regimes with the appropriate beta-function coefficients. That is, we have already taken into account the predominant threshold effects associated with each of these scales. For statistically "average" valid initial points-the vast majority of the phenomenologically acceptable initial soft SUSY breaking parameters-possible additional threshold effects arising from the "splitting" of particle masses around these scales are expected to be relatively small-and will be systematically ignored relative to the heavy string thresholds. That is, the∆ a , a = 3, 2, 3R, BL parameters in (50) will closely approximate the four heavy string gauge thresholds. With this input, using eqn. (49), p = M U = 3.15 × 10 16 GeV and α string , M string given in (48), one can calculate the associated heavy string thresholds from (50); that is, for each a = 3, 2, 3R, BL . Of course, these thresholds are expected to differ for each different valid initial point. It follows that one should analyze the thresholds statistically-graphing the dispersion of each as one runs over the 44, 884 valid initial points. The histograms associated with each of these four thresholds are presented in Figure 16. To better understand the relationship of these different thresholds, we find it useful to plot all four of them in a single histogram. This is presented in Figure 17.
It is also useful to calculate the string threshold associated with the Abelian hypercharge coupling α 1 defined, using (25) and (26), by 4 The associated statistical histogram is given in Figure 18. It is well-known [33,35,37,38] that each string threshold breaks into two parts, where Y is a "universal" piece independent of the gauge group and ∆ a records the contributions of all massive string states as they propagate around the genus-one string worldsheet torus diagram well-known-see the previous section-that string threshold corrections can be responsible for such non-unification, providing a consistent theoretical framework for this analysis.
Our results indicate that a substantial region of the initial soft SUSY breaking parameter space is consistent with all low-energy experimental data-see Figs. 5-7. We also presented our results for important physical quantities; histograms of the LSP species-see Fig. 8 paper share many common features. In addition, they show that all low-energy phenomenological constraints can be satisfied for a wide range of initial soft SUSY breaking parameters over a more natural and generic region of CY moduli space. The fundamental difference between the calculations in this paper from those of our previous work is that the gauge couplings of the B − L MSSM can no longer unify at an average mass associated with the inverse CY radius. Rather, the gauge couplings remain split at that scale-a splitting that allows one to compute the heavy string threshold corrections due to heavy states on the genus-one string worldsheet. These string threshold corrections and their differences are analyzed statistically, and predict what one should obtain using a a more direct worldsheet formalism. It would be interesting to see the results of such formal string calculations done in the heterotic B − L MSSM context.
Finally, having achieved these more generic results, we are now in a position to arbitrarily raise the mass scale of the soft SUSY breaking parameters-without encountering the above mentioned