A Note on Modulus-dominated SUSY-breaking

In models where supersymmetry-breaking is dominated by the Kahler moduli and/or the universal dilaton, the B-parameter at the unification scale should be consistent with the value of tan(beta) at the electroweak scale determined by minimization of the Higgs potential triggering REWSB. We study such models employing a self-consistent determination of the B-parameter. In particular, we study the viability of a generic model, as well as M-theory and Type IIB flux compactifications with modulus-dominated supersymmetric soft-terms from the GUT scale, M_{GUT}=2x10^{16}GeV.


I. INTRODUCTION
If TeV-scale supersymmetry is discovered at LHC, it will open a window in which to explore physics at higher-energy scales. In particular, the measurement of superpartner masses can provide a test of different proposed mechanisms for breaking supersymmetry. Moreover, it may allow us to probe the underlying theory which provides the UV completion of known low-energy physics.
In particular, in various string theory compactifications where the effective low-energy N = 1 supergravity approximation holds true, it is possible to generate superpartner spectra which may be compared to whatever may be observed at LHC. The most studied model of supersymmetry breaking is minimal supergravity (mSUGRA), which arises from adopting the simplest ansatz for the Kähler metric, treating all chiral superfields symmetrically. In this framework, N = 1 supergravity is broken in a hidden sector which is communicated to the observable sector through gravitational interactions. Such models are characterized by the following parameters: a universal scalar mass m 0 , a universal gaugino mass m 1/2 , the Higgsino mixing µ-parameter, the Higgs bilinear B-parameter, a universal trilinear coupling A 0 , and tan β. One then determines the B and |µ| parameters by the minimization of the Higgs potential triggering REWSB, with the sign of µ remaining undetermined. Thus, we are left with only four parameters.
Although, mSUGRA is one of the most generic frameworks that can be adopted, many string compactifications typically yield expressions for the soft terms which are even more constrained, in particular, when supersymmetry breaking is dominated by the Kähler moduli and/or dilaton. As is well-known, the Kähler moduli of Type I, IIB orientifold, and heterotic string compactifications have a classical no-scale structure [1,2,3,4], which guarantees that the vacuum energy vanishes at tree-level. The no-scale structure corresponds to having non-vanishing expectation values for the auxiliary fields of the Kähler moduli. The generic appearance of the no-scale structure across many string compactifications combined with the highly-constrained and thus predictive framework strongly motivates the consideration of modulus-dominated supersymmetry breaking, although there are some string models for which the soft-terms are not as constrained (see [5,6,7,8] for a model of this kind).
For modulus-dominated supersymmetry breaking, we generically have m 0 = m 0 (m 1/2 ) and A = A(m 1/2 ). This reduces the number of free parameters compared to mSUGRA down to two, m 1/2 and tanβ. In fact, adopting a strict no-scale framework, one can also fix the B-parameter as B = B(m 1/2 ), and thus we are led to a one-parameter model where all of the soft terms may be fixed in terms of m 1/2 . However, for this framework to be consistent, the value of tanβ at the electroweak scale should be consistent with B at the string scale.
In a previous paper, we studied a generic one-parameter model and found its viable parameter space [9]. However, in this work we did not require that tanβ obtained at the electroweak scale be consistent with the value of B = B(m 1/2 ) defined at the GUT scale. For the present work, we impose this constraint for a generic one-parameter model and find that there is no viable supersymmetry parameter space, assuming the standard RGE running between the electroweak scale and the GUT scale. Furthermore, we find the same result for M-theory and Type IIB flux compactifications. In addition, we consider different modular weights for some of the chiral fields, again with negative results. We conclude that modulus-dominated supersymmetry breaking is not viable, in the case of a standard RGE running of the soft terms starting from the GUT scale.

II. MODULUS-DOMINATED SUSY-BREAKING
For certain classes of string compactifications, the soft-terms are of the form m 0 = m 0 (m 1/2 ) and A = A(m 1/2 ) if supersymmetry is dominated by the Kähler moduli and/or the universal dilaton.
In particular, much work has been done in the past to study two generic cases inspired by no-scale supergravity in the framework of the free-fermionic class of heterotic string compactifications. The first of these two cases is referred to as the special dilaton scenario, while the second is referred to as the strict moduli scenario, In previous work, it was found that there is no viable parameter space for the strict moduli scenario which satisfies experimental constraints. However, in the case of the special dilaton scenario there is a small allowed parameter space.
Moreover, the soft-terms for many string compactifications will also be of similar form. In particular, the soft terms for heterotic M-theory compactifications take the form [10] while for dilaton dominated supersymmetry breaking they take the form These expressions reduce to the above moduli and dilaton scenarios respectively in the limit In addition, the so-called large-volume models have been studied extensively [11] [12] in recent years and the generic soft terms for this framework have been calculated in [13]. These models GeV. In such models, the soft terms can take the form where M is a universal gaugino mass.
As can be seen for these different string compactifications, the soft terms can generically be of the form where c 1 , c 2 , and c 3 are constants. In addition, we will take the string scale to be M GU T = 2 × 10 16 GeV. However, we should note that the string scale at which the soft-terms are defined could be different from the conventional GUT scale. In particular, we can see for the case of the M-theory compactifications, the unification scale can be higher than the GUT scale, while for the large-volume Type IIB flux compactifications, the string scale could be substantially lower.

III. IMPOSING THE B CONSTRAINT
As stated in the introduction, the value of the µ parameter and tanβ are determined at the electroweak scale by imposing the conditions and which follow from the minimization of the Higgs potential triggering REWSB. From these equations, one can calculate the value of the B-parameter at the electroweak scale. In order for this to be a true one-parameter model, B at the electroweak scale should be consistent with the ansatz The usual procedure to find the viable parameter space is to calculate the sparticle masses using the parameters m 0 , m 1/2 , A 0 , sgn(µ), and tanβ, and plot m 0 vs. m 1/2 for a specific tanβ, and further scan the entire tanβ space for solutions that satisfy the current experimental constraints and corresponding relic neutralino density. In particular, such an analysis was performed for a generic one-parameter model in [9]. However, the consistency constraint between the B-parameter at the electroweak scale and the GUT scale has not been imposed in this analysis. For the present work, we perform a scan of the parameter space, including tanβ, and filter the results through the latest experimental constraints and dark matter density, and in addition, compare the allowed parameter space with the value of the B-parameter at M High . For the present work, we will identify M High with M GU T . This determines whether the allowed parameter space calculated from tanβ can also satisfy the constraint on the B-parameter at the unification scale (see [14] for a similar study in the case of F-theory compactifications). We take the top quark mass to be m t = 173.1 GeV [17] and leave tan β as a free parameter, while µ is determined by the requirement of REWSB. However, we do take µ > 0 as suggested by the results of g µ − 2 for the muon. The resulting superpartner spectra are filtered according to the following criteria: 1. The 5-year WMAP data combined with measurements of Type Ia supernovae and baryon acoustic oscillations in the galaxy distribution for the cold dark matter density [18], 0.1109 ≤ Ω χ o h 2 ≤ 0.1177, where a neutralino LSP is the dominant component of the relic density. In addition, we look at the SSC model [19], in which a dilution factor of O(10) is allowed [20], where Ω χ o h 2 1.1. For a discussion of the SSC model within the context of mSUGRA, see [21]. We also investigate another case where a neutralino LSP makes up a subdominant component, allowing for the possibility that dark matter could be composed of matter such as axions, cryptons, or other particles. We employ this possibility by removing the lower bound.
2. The experimental limits on the Flavor Changing Neutral Current (FCNC) process, b → sγ.
The results from the Heavy Flavor Averaging Group (HFAG) [22], in addition to the BABAR, Belle, and CLEO results, are: There is also a more recent estimate [23]  4. The process B 0 s → µ + µ − where the decay has a tan 6 β dependence. We take the upper bound to be Br(B 0 s → µ + µ − ) < 5.8 × 10 −8 [25].
To determine the B-parameter at M High = M GU T , B is determined at m Z from Eqns. (8) and (9) prediction. Additionally, it is also necessary for these points of intersection between the B curves and predictions to lie within the range of points within the experimentally allowed parameter space.
These points just described will satisfy not just the aforementioned five experimental constraints, but also the constraint on the B-parameter at the unification scale. However, as we will show here, it is very difficult to satisfy all these constraints simultaneously for a model with universal soft-supersymmetry breaking parameters.
We compute the B-parameter at the GUT scale here for two models: a generic one-parameter model [9,27,28,29] and an M-Theory model [10]. We find that for the models with a predicted B- For the one-parameter model, we begin with the minimal model in the special dilaton scenario with the soft terms of the form and construct a method of varying the modular weights. To accomplish this, we modify the expressions above and introduce three new parameters ξ, δ, and η that will allow us to investigate more general cases: Using these expressions, the minimal one-parameter model is the case (ξ, δ) = (1, 1). We now let ξ = 1 2 , 1, 3 2 , 2 and δ = 1 2 , 1, 3 2 , 2, which will give us 16 different cases to examine. The 16 cases shall be divided up into four data sets such that each data set will be plotted independently. Each data set will have constant ξ, and thus constant m 0 , while δ is varied, and hence A is varied.
Therefore, each of the four plots will contain four sets of curves, where each set of curves pertains to one (ξ, δ). The gaugino mass is incremented from 100 GeV to 1000 GeV in steps of 100 GeV, The minimal one-parameter model and the M-Theory model with x = 0 is the case (ξ, δ) = (1, 1). In these plots, all the allowed points highlighted in black satisfy the relic neutralino density in the SSC scenario.
Those points satisfying only the WMAP relic density are not highlighted. As the plots show, the points experimentally allowed do not intersect the predictions for B, hence, the B-parameter constraint cannot be satisfied by the models displayed in this Figure. whereas tanβ is varied in increments of one from 2 to 60. From these specifications, a list of soft supersymmetry breaking terms is generated and the B-parameter at the GUT scale is calculated for each set of soft terms. As shown in Fig. 1, there are solutions to the one-parameter model when only the experimental constraints are considered, though when the B-parameter constraint is applied, the experimentally allowed parameter space is nullified. There are no intersections between the B-parameter curves and the horizontal lines representing the predictions for the B-parameter.
Note that η for the three predictions are To further ensure that we have examined all possibilities for the minimal one-parameter model, we computed an additional case with an independent modular weight for the Higgs scalars at the unification scale. Our motivation for attempting this is that since the Higgs typically come We scan for real solutions that give m 0 > 0, A < 0, and B < 0, and find these solutions only exist for 0 ≤ x ≤ 0.6742. The case x = 0 is shown in Fig. 1, so we run the three additional In fact, as the unknown variable x increases toward the upper end of its range, the discrepancy becomes larger. Here again, as in the one-parameter model, the M-Theory model cannot produce any viably allowed parameter space that satisfies both the experimental constraints and the Bparameter constraint.

IV. CONCLUSION
A well-motivated framework for studying supersymmetry breaking is to assume that it is dominated by the Kähler moduli and/or the universal dilaton. Such scenarios give rise to very constrained supersymmetry breaking soft-terms which depend only on a universal gaugino mass. In addition, modulus-dominated supersymmetry breaking appears as a generic feature of many string compactifications. We find that the simplest models are not viable, at least under a standard