Abstract
We provide a detailed study of the phenomenology of orbifold compactifications of the heterotic string within the context of supergravity effective theories. Our investigation focuses on those models where the soft Lagrangian is dominated by loop contributions to the various soft supersymmetry breaking parameters. Such models typically predict non-universal soft masses and are thus significantly different from minimal supergravity and other universal models. We consider the pattern of masses that are governed by these soft terms and investigate the implications of certain indirect constraints on supersymmetric models, such as flavor-changing neutral currents, the anomalous magnetic moment of the muon and the density of thermal relic neutralinos. These string-motivated models show novel behavior that interpolates between the phenomenology of unified supergravity models and models dominated by the superconformal anomaly.
Introduction
The recent interest in models where supersymmetry breaking is transmitted from a hidden sector to our observable world via the superconformal anomaly [1, 2, 3] has served as a reminder that there are occasions where working at one loop order is more than a theoretical luxury, but an absolute necessity if one is to understand even the gross features of some models. Recently the complete one loop supergravity correction to soft supersymmetry breaking terms has been obtained [4, 5] , a subset of which are the aforementioned "anomaly-mediated" terms. The broad features of these corrections, in particular for gaugino masses, was studied in a heterotic string context in [6] . In the current work we wish to study in greater detail the low energy phenomenology of models based on orbifold compactifications of the weakly-coupled heterotic string in which these loop corrections are significant. To that extent we will choose two "laboratories" in which tree level soft supersymmetry breaking terms are either absent or greatly suppressed.
Our first example will involve an observable sector in which the matter fields are assumed to be from the untwisted sector of the orbifold compactification. In the event that supersymmetry breaking is transmitted by the compactification moduli T α , whose vacuum expectation values determine the size of the compact manifold, then this model will exhibit a no-scale pattern of soft terms with vanishing soft supersymmetry breaking terms at tree level. Clearly the phenomenology of such models will depend critically on the form of the loop induced soft terms. Indeed, the "sequestered sector" models considered in [2] were of just such a form.
In our second example we will switch our focus to cases in which it is the dilaton field S, whose vacuum expectation value determines the magnitude of the (unified) coupling constant g str at the string scale, that participates exclusively in supersymmetry breaking. We will work in the context of models in which string nonperturbative corrections to the Kähler potential act to stabilize the dilaton in the presence of gaugino condensation [7, 8] . Such theories predict a suppression of dilaton contributions to soft terms relative to those of the supergravity auxiliary field -thereby suppressing tree level contributions to supersymmetry breaking relative to certain loop contributions.
After describing our framework and introducing both the tree level and one loop soft supersymmetry breaking terms in Section 1, we introduce the parameter space for our two broad classes of models in Section 2. The subsequent section describes the analysis tools we employ and the types of low-energy phenomena we will study before describing the implications of low-energy experimental constraints on each of our models in Section 4.
In general we find that these examples have a phenomenology that combines the features of minimal supergravity (mSUGRA) models with those of anomaly-mediated models. Parameters related to the orbifold compactification and the stabilization of string moduli interpolate between these regimes. Current observations and particle mass limits from collider experiments already shed light on the nature of moduli stabilization. Future discoveries of superpartners will likely provide strong evidence in favor of one stabilization mechanism or another in the context of weakly-coupled heterotic string models.
1 The structure of heterotic orbifold models at one loop
We work in a framework in which the chiral superfields Z M can be divided into two classes: observable sector superfields, noted Z i , charged under the observable sector gauge symmetries; and hidden sector superfields, noted Z n . Since our interest here is primarily on broad issues of phenomenology, the hidden sector fields we will consider are the dilaton S in the chiral multiplet formulation and the three diagonal Kähler moduli T α .
In the orbifold compactifications we study here, the tree-level Kähler potential for the moduli and the matter superfields is known. For the moduli sector we have
where we prefer to leave the form of the dilaton Kähler potential k unspecified at this point. Its precise form depends on how one stabilizes the dilaton. We will return to this issue when we discuss the dilaton-dominated scenario of Section 2.2. As for the observable sector matter fields Z i with modular weights n α i associated with each of the three T α , we assume a diagonal Kähler metric given by
2)
In the interests of simplicity, we assume that the three Kähler moduli T α can be treated as equivalent so that K(S, S; T, T ) = k(S + S) − 3 ln(T + T ); 4) where n i = α n α i . The tree-level gauge kinetic functions f a (Z n ), one for each gauge group G a , are given in the weak coupling regime by f To (1.5) will be added certain string threshold corrections when we exhibit the one loop soft supersymmetry breaking terms below.
The scalar potential, written in terms of auxiliary fields, is given by the expression 2 6) with K IJ = ∂ 2 K/∂Z I ∂ZJ being the Kähler metric. The fields F I in (1.6) are the auxiliary fields associated with the chiral superfields Z I while the field M is the auxiliary field of the supergravity multiplet. Solving the equations of motion for these auxiliary fields yields 8) with K M N being the inverse of the Kähler metric. Note that these expressions are given in terms of reduced Planck mass units where we have set M pl / √ 8π = 1. The supergravity auxiliary field is related to the gravitino mass by
We adopt the ansatz of Brignole et al. [9] in which one assumes that the communication of supersymmetry breaking from the hidden sector to the observable sector occurs through the agency of one of the moduli -in this case either the dilaton S or the (universal) Kähler modulus T -by the presence of a non-vanishing vacuum expectation value of their auxiliary fields F S or F T . In principle both types of moduli could participate in supersymmetry breaking, and so one typically introduces a "Goldstino angle" θ to parameterize the degree to which one sector or the other feels the supersymmetry breaking. While this is a common practice, all known explicit models of moduli stabilization in the heterotic string context predict that this "angle" will be some integer multiple of π/2; that is, only one of the two classes of moduli participates in supersymmetry breaking.
If these are the only sectors with non-vanishing auxiliary fields in the vacuum, then the further requirement that the overall vacuum energy at the minimum of the potential (1.6) be zero allows us to immediately identify (up to phases, which we set to zero in what follows)
sin θ, (1.10) 11) where the last equality holds for the vacuum expectation values, using (1.9) above. We should note that the condition of vanishing vacuum energy is a necessary one to employ the above parameterization. In this work we will avoid discussing specific models of dynamical supersymmetry breaking and moduli stabilization, but any such model must include some mechanism for engineering a vanishing vacuum energy in order to make contact with the results presented here.
Modular invariance and tree level supersymmetry breaking terms
The soft supersymmetry breaking terms in string-derived supergravity depend on the moduli through the observable sector superpotential and this is, in turn, determined by insisting on mod-ular invariance of the low-energy effective Lagrangian. The diagonal modular transformations 12) leave the classical effective supergravity theory invariant, though at the quantum level these transformations are anomalous [10, 11, 12, 13] . This anomaly is cancelled in the effective theory by the presence of a universal Green-Schwarz counterterm and model-dependent string threshold corrections [14, 15] , which we describe below. A matter field Z i of modular weight n i transforms under (1.12) as
while the Kähler potential of (1.4) undergoes a Kähler transformation K → K + 3(F + F ), with F = ln(icT + d), under (1.12). Therefore the classical symmetry will be preserved provided the superpotential transforms as [16] 
(1.14)
To ensure this transformation property the superpotential of string-derived models has a moduli dependence of the form
where 16) and it has a well-defined transformation under (1.12) given by
We will also have to introduce the modified Eisenstein function
which vanishes at the self-dual points t = 1 and t = e iπ/6 . We are now in a position to give the tree level soft supersymmetry breaking terms. The tree level gaugino mass for canonically normalized gaugino fields is simply
We define our trilinear A-terms and scalar masses for canonically normalized fields by
20)
i z i is a normalized scalar field, and by
With these conventions our tree level expressions are
If we specialize now to the case of (1.10) and (1.11) with moduli dependence given by (1.4), (1.5) and (1.15), then the tree level gaugino masses (1.19), A-terms (1.22) and scalar masses (1.23) become
(1.24)
The expressions in (1.24) are to be understood as vacuum values, though we will drop the cumbersome brackets . . . from here onwards. These tree level soft terms have been studied extensively in the past and we wish to understand what impact the less-studied loop corrections can have on the phenomenology of orbifold models. We can now appreciate the importance of the two cases we intend to study in Sections 2.1 and 2.2. In the case where the modular weights of the observable sector are universally n i = −1, as would be the case for the untwisted sector of orbifold compactifications, then from (1.15) it is clear that the moduli do not couple to the observable sector through the superpotential and we have a true noscale model. Indeed, substituting (1.11) into (1.24) we find that if the dilaton does not participate in supersymmetry breaking all (θ = 0) tree level soft terms are precisely zero in such a case. This is the situation we will investigate in Section 2.1.
Alternatively, if the dilaton is the primary source of supersymmetry breaking in the observable sector, loop-level corrections to at least the trilinear A-terms and gaugino masses will continue to be important provided the magnitude of F S is suppressed relative to that of M , as it is in models where nonperturbative corrections to the Kähler potential are used to stabilize the dilaton [17] . In such a scenario the modular weights of the matter fields are irrelevant at the tree level, so we will continue to assume n i = −1 for the sake of convenience when we investigate these models in Section 2.2. We thus need to turn our attention to the study of soft supersymmetry breaking at one loop.
General one loop corrections to soft supersymmetry breaking
In this section we aim to provide sufficient background to justify the form of the one loop soft term expressions which are our goal, as well as explain some notation we will need for our phenomenological analysis. For more complete explanations of what is contained in this section, including expressions with arbitrary modular weights and three independent Kähler moduli, the reader should consult the precursor to this work [6] .
We begin with gaugino masses which can be understood as a sum of loop-induced contributions from the field theory point of view, and terms that can be thought of as one loop stringy corrections. The field theory loop contribution can be derived completely from the superconformal anomaly and is given by [4, 18] 25) where C a , C i a are the quadratic Casimir operators for the gauge group G a in the adjoint representation and in the representation of Z i , respectively, and b a is the beta-function coefficient for the group G a :
(1.26)
As mentioned in the previous section one expects modular anomaly cancellation to occur through a universal Green-Schwarz counterterm with group-independent coefficient δ gs . Such a term can be thought of as a loop correction that contributes to gaugino masses in the form
(1.27)
In addition string threshold corrections generally appear in the effective theory, which may be interpreted as one loop corrections to the gauge kinetic functions of the form 28) which generate loop contributions to gaugino masses given by
Putting together the tree level gaugino mass with the loop contributions (1.25), (1.27) and (1.29) gives
where we have defined the quantity
(1.31)
To understand the form of the one-loop A-terms and scalar masses it is necessary to describe how field theory loops are regulated in supergravity, seen as an effective theory of strings. The regulation of matter and Yang-Mills loop contributions to the matter wave function renormalization requires the introduction of Pauli-Villars chiral superfields Φ A = Φ i , Φ i and Φ a which transform according to the chiral matter, anti-chiral matter and adjoint representations of the gauge group and have signatures η A = −1, +1, +1, respectively. These fields are coupled to the light fields Z i through the superpotential
where T a is a generator of the gauge group, and their Kähler potential takes the schematic form 33) where the functions κ A are a priori functions of the hidden sector (moduli) fields. These regulator fields must be introduced in such a way as to cancel the quadratic divergences of the light field loops -and thus their Kähler potential is determined relative to that of the fields which they regulate. The PV mass for each superfield Φ A is generated by coupling it to another field Π A = (Π i , Π i , Π a ) in the representation of the gauge group conjugate to that of Φ A through a superpotential term 34) where µ A (Z N ) can in general be a holomorphic function of the light superfields. There is no constraint on the Kähler potential for the fields Π A as there was for those of the Φ A . However, if we demand that our regularization preserve modular invariance then we can determine the moduli dependence of the regulator fields Φ A
and we can determine the moduli dependence of the supersymmetric mass µ A (Z N ) in (1.34) 36) with n A and q A being the modular weights of the fields Φ A and Π A , respectively. Furthermore, we can postulate the form of the moduli dependence of κ A for the mass-generating fields
So at this point the dilaton dependence in the superpotential term (1.34) and the functions h A , as well as the modular weights q A of the fields Π A , are new free parameters of the theory introduced at one loop as a consequence of how the theory is regulated. Given (1.34) we can extract the Pauli-Villars masses that appear as regulator masses in the logarithms at one loop 38) with m A = (m i ,m i , m a ) being the masses of the regulator fields Φ i , Φ i , Φ a , respectively. In terms of these regulator masses, the complete one loop correction to the trilinear A-terms and scalar masses in a general supergravity theory was given in [5] . In this survey we will simplify the expressions and reduce the parameter space by making some reasonable assumptions. Let us first assume that the functions µ A (Z N ) that appear in (1.34) and (1.38) are proportional to one overall Pauli-Villars scale µ pv so thatμ i = µ a = µ i ≡ µ pv . This scale is presumed to represent some fundamental scale in the underlying string theory. Let us further assume that there is no dilaton dependence of the PV masses so that h A (S + S) is trivial and µ pv is constant. With these simplifications the complete trilinear A-term at one loop is given by
where we have defined the following combinations of modular weights from the Pauli-Villars sector
which we will refer to as "regularization weights" in reference to their origin from the PV sector of the theory. 3
In (1.39) M 0 a and A 0 ilm are the tree level gaugino masses and A-terms given in (1.24) and the parameters γ determine the chiral multiplet wave function renormalization
We have implicitly made the approximation that generational mixing is unimportant and can be neglected in (1.39) , and that motivates the definitions
The scalar masses are obtained similarly and take the form
with M 0 i being the tree level scalar masses of (1.24). To put these expressions into a less cumbersome form we will consider the case where the various regularization weights p ia and p jk can be treated as one overall parameter p. Then inserting the tree level soft terms (1.24) into (1.39) and (1.43) yields these soft terms in the final form we will use in the following analysis:
where γ i is a shorthand notation for
The adoption of one overall regularization weight p makes it possible to identify the quantity ln(µ 2 pv /µ 2 R ) − p ln (t +t)|η(t)| 4 as a stringy threshold correction to the overall PV mass scale, or effective cut-off, µ pv . The one loop expressions of (1.30), (1.44) and (1.45) will be our starting point in the following two sections where we investigate their phenomenological significance in scenarios where tree level soft supersymmetry breaking is suppressed or vanishing.
Classification of heterotic orbifold models
As a result of the assumptions made in Section 1 our general parameter space is defined by seven quantities, if we assume a tree-level Kähler potential for the dilaton of the form k(S, S) = − ln(S+S) with s +s = 2/g 2 str ≃ 4. Three of these parameters represent scales: the gravitino mass M 3/2 , the boundary condition scale µ uv at which the soft supersymmetry breaking terms appear and the Pauli-Villars cutoff scale µ pv . The other four are parameters describing the moduli sector: the vacuum expectation value of the real part of the (uniform) Kähler modulus t +t , the modular weights of the PV regulators parameterized by p, the value of the Green-Schwarz coefficient δ gs and the Goldstino angle θ which determines the degree to which the dilaton and moduli F-terms participate in the transmission of supersymmetry breaking.
Moduli dominated scenarios
This section assumes that i) all fields have modular weight n i = −1, which gives a no-scale structure, and ii) we have moduli-dominated supersymmetry breaking (θ vanishes, cos θ = 1) so that the tree level soft terms are precisely zero. In addition, those loop-induced soft terms in (1.45) proportional to tree level quantities are also zero, leaving only those arising from the superconformal anomaly and those related to the Pauli-Villars sector:
Note that in the special case where the moduli are stabilized at one of their two self-dual points t = 1 and t = e iπ/6 we recover the universal "anomaly-mediated" results
The scenario that has come to be referred to as the "Anomaly-Mediated Supersymmetry Breaking" (AMSB) scenario [1, 2, 3] , in which scalar masses first appear only at two loops, is obtained in the limit as the phenomenological parameter p approaches unity. This would be the case, for example, if the mass-generating Pauli-Villars fields Π A had the same Kähler metric as the regulator fields Φ A and thus q i = −1,q i = 1 and q a = −3. In any event, it is clear that the parameter p defines a family of "anomaly mediated" models for the case of moduli stabilized at self-dual points. Importantly, for p < 1, as would be typical in orbifold models, the anomaly-mediated masses for the matter scalars are positive with only the Higgs fields having negative squared masses at the scale µ uv . The contrast between these results and those of the standard AMSB case was discussed in [5, 6] , but it is important to point out that the vanishing of the scalar masses at one loop in the p → 1 limit clearly demands a two-loop analysis of these soft terms. A treatment of supergravity radiative corrections, regularized by the Pauli-Villars mechanism, at two loops would be a massive undertaking. Some subset of these corrections would undoubtedly be the soft terms found in [2, 3] using a spurion technique. As is now well known, these two-loop terms arising from the superconformal anomaly would imply negative squared masses for the scalar leptons. To remedy this situation, the AMSB model, as it has been institutionalized in studies such as the Snowmass Points and Slopes [19] , adopts an otherwise ad hoc scalar mass contribution that is universal and sufficiently large. Not surprisingly, these models are often characterized as having light sleptons as a key feature of the phenomenology.
While several extensions of the original (or "minimal") AMSB model now exist which address the slepton mass problem in a variety of ways, 4 the generalized anomaly mediated model we present here suffers from no such problem when p = 1. While the phenomenology associated with the gaugino mass spectrum is familiar from past studies of the AMSB paradigm, the scalar mass sector looks quite different. We will begin our analysis in Section 4.2 with this generalized anomaly mediated model (which we will refer to as the "PV-AMSB model") before enlarging the parameter space to include all moduli-dominated models in Sections 4.3 and 4.4.
Dilaton dominated scenarios with Kähler suppression
We next turn to the opposite extreme, where the dilaton is the primary source of supersymmetry breaking in the observable sector (sin θ = 1). In such a scenario we would ordinarily expect the loop corrections such as those in (1.25), (1.39) and (1.43) to be small perturbations on the dominant (and universal) tree level contributions and thus safely neglected. The phenomenology of the dilaton-domination scenario at tree level has been extensively studied in the literature [21] .
The bulk of these studies, however, have considered only the case of the standard Kähler potential k(S, S) = − ln(S + S) derived from the tree level string theory. It is well known that when this form is used, stabilizing the dilaton at acceptable weak-coupling values is extremely difficult. In the case where a dilaton potential is imagined to arise through field theory nonperturbative effects (such as gaugino condensation) it has been shown that one condensate alone can never succeed in stabilizing the dilaton in such a way as to reproduce the weak coupling observed in nature [22, 23] . Furthermore, while multiple condensates can be used to provide the requisite stabilization [24, 25] , these models tend to predict a moduli-dominated scenario such as those in Section 2.1.
In contrast, if one postulates a nonperturbative correction of stringy origin to the dilaton Kähler potential, as was first motivated by Shenker [26] , then one condensate can indeed stabilize the dilaton at weak coupling while simultaneously ensuring vanishing vacuum energy at the minimum of the potential. The power of this approach was first demonstrated in an explicit model of modular invariant gaugino condensation in [7, 8] and confirmed in [27] , providing a concrete realization of the so-called "generalized dilaton domination scenario" [23] .
The key feature of such models is the deviation of the dilaton Kähler metric from its tree level value. If we imagine the superpotential for the dilaton having the form W (S) ∝ e −3S/2b + , with b + being the largest beta-function coefficient (1.26) among the condensing gauge groups of the hidden sector, then it is clear from (1.7) and (1.8) that requiring the potential (1.6) to vanish implies
The condition in (2.49) is independent of the means by which the dilaton is stabilized and is a result merely of requiring a vanishing vacuum energy in the dilaton-dominated limit. The explicit model 4 For some early solutions, see [20] .
of [8] , however, was able to achieve precisely this relation with k s = −g 2 str /2, using a correction to the dilaton action that involved tuning O(1) numbers only. 5 We can parameterize the departure that (2.49) represents from the tree level Kähler metric (k tree ss ) 1/2 = 1/(s +s) = g 2 str /2 ≃ 1/4 by introducing the phenomenological parameter
so that the auxiliary field of the dilaton chiral supermultiplet can be expressed as
The importance of (2.49) for our purposes is to recognize that the factor of b + , containing as it does a loop factor, will suppress the magnitude of the auxiliary field F S relative to that of the supergravity auxiliary field M through the relation (2.51). Therefore tree level soft terms for the gaugino masses and trilinear A-terms will be of comparable magnitude to the loop-induced soft terms arising from the superconformal anomaly (though scalar masses will be only negligibly altered from their tree level values). We are thus led to consider the phenomenology of models given by the following pattern of soft supersymmetry breaking terms:
where we have dropped terms of O 1/(16π 2 ) 3 in the scalar masses. These models will be studied in Section 4.5.
Experimental and cosmological constraints
In order to translate the spectra defined by the soft supersymmetry breaking terms in Sections 2.1 and 2.2, a low energy soft Lagrangian is obtained by solving the renormalization group equations (RGEs) from some initial boundary condition scale to the electroweak scale. Having thus extracted the soft supersymmetry breaking parameters at the low-energy scale, the physical mass spectra for the superpartner and Higgs sector can be obtained and various direct collider constraints and indirect non-collider constraints can be applied to examine the region of phenomenological viability of these models. In this section we describe the tools we use and the various observational constraints we impose.
Electroweak symmetry breaking and physical masses
We have used the Fortran code SuSpect [28] to solve the RGEs for the soft supersymmetry breaking parameters between the initial high energy scale µ uv and the scale given by the Z-boson mass. While the initial scale µ uv should itself be treated as a model-dependent parameter, we have chosen µ uv = µ gut throughout. We use tan β and the sign of the supersymmetric µ parameter in the superpotential as free parameters, defined at the low-energy (electroweak) scale. The magnitude of the µ parameter is determined by imposing electroweak symmetry breaking (EWSB) at the scale defined by the geometric mean of the two stop masses (mt
, which minimizes the one-loop scalar effective potential [29, 30] . The one-loop corrected µ term is obtained from the conditionμ [31, 32, 33] . SuSpect includes the corrections from all the third generation fermion and sfermion loops as well as loops from gauge bosons, Higgs bosons, charginos and neutralinos. As the superpartner spectrum depends on the value of the µ parameter computed in (3.53), the proper inclusion of superpartner thresholds in the RG evolution requires that the procedure be iterated until a coherent and stable value for µ is obtained. Usually, SuSpect requires only two or three iterations to have a relative precision of the order of 10 −4 . We have used three. The soft supersymmetry breaking parameters at the weak scale are then passed to the C code micrOMEGAs [34] to perform the calculation of physical masses for the superpartners and various indirect constraints, to be described in the following section.
The specific algorithm we employ can be described as follows. The one-loop order SUSY breaking parameters of the heterotic orbifold models obtained in the previous section are entered into SuSpect at the high energy scale. Other parameters, including Standard Model fermion masses and gauge weak couplings as well as certain EWSB parameters such as tan β are input at the low energy scale. SuSpect runs the renormalization group evolution of the parameters back and forth between the low energy scales such as M Z and the electroweak symmetry breaking scale, and the high-energy scale such as the GUT scale. This is the case for the soft SUSY-breaking terms (scalar and gaugino masses, bilinear and trilinear couplings and tan β) and µ. This procedure has to be iterated in order to include SUSY threshold effects or radiative corrections due to Higgs and SUSY particles. In the first step, these thresholds are only guessed since the spectrum has not been calculated yet, and the radiative corrections are not implemented. The thresholds are properly taken into account in subsequent iterations. At the electroweak scale, the consistency in the calculation of the µ term is checked using the expression in (3.53). We then call micrOMEGAs, which calculates the SUSY spectrum at one loop order using FeynHiggs for the Higgs sector. micrOMEGAs also calculates the b → sγ branching ratio as well as the relic density of the lightest neutralino and the anomalous magnetic moment of the muon. We have checked the consistency between the two spectra generated by SuSpect and by micrOMEGAs: the maximum deviation was of the order of 4% and was confined to specific regimes such as high tan β.
The first condition that we require on a given set of soft supersymmetry breaking masses is that an appropriate vacuum state appears -in particular that EWSB occurs. Clearly it is possible for the value of µ 2 in (3.53) to become negative for a particular choice of model parameters. In such a situation EWSB does not occur. The precise domain of parameter space where this occurs is sensitive to the values of the Yukawa couplings -and hence to the choice of quark pole masses assumed. This makes the SUSY corrections to the bottom quark mass quite important. We have performed the analysis using the heavy fermion masses in SuSpect of 54) and gauge couplings at the Z-mass of
We also reject all points in the parameter space which give a tachyonic Higgs boson mass (in particular, where m 2 A < 0) or negative sfermion squared masses. We next require that the lightest supersymmetric particle (LSP) be neutral, and thus also reject all model parameters where the lightest neutralino χ 0 1 is not the LSP. In most of the rejected cases it is the lightest stau τ 1 or the gluinog which becomes lighter than the χ 0 1 . The remaining parameter space is further reduced by limits on superpartner and Higgs masses from various collider experiments.
Concerning the mass bounds we use, we take the most recent results combined by the LEP Working Group [35, 36] . For the light CP-even neutral Higgs mass (m h ), we assume that a 95% confidence level (CL) lower limit on m h is harrishoward@example.org GeV. The search for an invisibly decaying Higgs boson in hZ production has allowed a 95% CL lower limit on m h to be harrishoward@example.org GeV, assuming a production cross section equal to that in the Standard Model and a 100% branching fraction to invisible decays [37] . We believe the value of 113.5 GeV serves as a good reference point, and this is the value we use in our analysis. Concerning the chargino limit, we take 103.5 GeV, bearing in mind that in some degenerate cases and for light sleptons the limit can go down to 88
GeV [38] . For the squark sector the limit of 97 GeV [39] is implemented. For all mass bounds we should keep in mind that experimental limits are always given in the context of a particular SUSY model which is not generally a string motivated one. The bounds we use could possibly be weakened in some cases.
Indirect constraints on supersymmetric spectra
Various non-collider observations can be used to further reduce the allowed parameter space of these loop-dominated orbifold models. We will focus our attention on the three such processes that yield the most stringent constraints: the density of relic neutralino LSPs, the branching ratio for decays involving the process b → sγ and the measurement of the anomalous magnetic moment of the muon.
The relic neutralino density
The existence of dark matter is one of the first glimpses of possible physics beyond the Standard Model. It is probable that dark matter consists of some stable or extremely long-lived particle left over from the hot early universe. For any given extension of the Standard Model containing stable or long-lived particles, the present-day relic density of such particles can prove quite constrainting. The basic calculation of a relic density from thermal considerations is standard [40] . Here we review the steps in such a calculation under the explicit assumption that the lightest neutralino constitutes the cold dark matter particle.
The initial number density of the neutralino is determined because the particle is assumed to have been in thermal equilibrium. When the particle begins in thermal equilibrium with its surroundings, interactions that create neutralinos usually happen as frequently as reverse interactions which destroy neutralinos. Once the temperature drops below T ≃ m χ , most particles no longer have sufficient energy to create neutralinos. Now neutralinos can only annihilate, and these annihilations occur until about the time when the Hubble expansion parameter becomes larger than the annihilation rate, H ≥ Γ ann . When expansion dwarfs annihilation, neutralinos are being separated apart from each other too quickly to maintain equilibrium. This happens at the freeze-out temperature, usually at T F ≃ m χ /20 for cold dark matter.
In most neutralino relic density calculations, the only interaction cross sections that need to be calculated are annihilations of the type χχ → X where χ is the lightest neutralino and X is any final state involving only Standard Model particles. However, there are scenarios in which other particles in the thermal bath have important effects on the evolution of the neutralino relic density. Such a particle annihilates with the neutralino into Standard Model particles and is called a coannihilator [41] . To serve as an effective coannihilator, the particle must have direct interactions with the neutralino and must be nearly degenerate in mass. Such degeneracy happens in the MSSM, for instance, with possible coannihilators being the lightest stau [42] , the lightest stop [43] , the second-to-lightest neutralino or the lightest chargino [44, 45] . When this degeneracy occurs, the neutralino and all relevant coannihilators form a coupled system. In this section, we will denote particles belonging to that coupled system by χ i . Now all interactions involving particles in this coupled system come into play, including χ i χ j → X, χ i X → χ j Y , and χ i → χ j X. Here both X and Y denote states including Standard Model particles. Decays once again enter the calculation because the coannihilators are generally not stable and eventually decay into the lightest neutralino.
For the case without coannihilations, evolution of the relic particle number density, n, happens in accordance with the single species Boltzmann equation
where n eq is the equilibrium number density, H is the Hubble parameter at time t, and σv is the thermally averaged annihilation cross section. The number density is modified by Hubble expansion and by direct and inverse annihilations of the relic particle. The relic particle is assumed to be stable, so relic decay is neglected. In the above expression, we have also assumed T invariance to relate annihilation and inverse annihilation processes.
In the presence of coannihilators, the Boltzmann equation becomes more complicated but can be simplified using the stability properties of the relic particle and the coannihilators (using n = N i=1 n i ). Application of these simplifications leads to
To a very good approximation, one can use the usual single species Boltzmann equation for the case of coannihilations if the following replacement is made for the thermally averaged cross section:
For scenarios involving coannihilations, the expression for the thermally averaged cross section begins as a six dimensional integral, seven dimensional including the integration over the final state angle. This integral has been conveniently put into the form of a one-dimensional integral over the total squared center of mass energy [46] . Computer codes exist which numerically perform the center of mass momentum integration and the final state angular integration. Some codes include only a subset of all possible coannihilation channels [47, 48] , while the program micrOMEGAs includes all relevant coannihilation channels. Analytical expressions for the cross sections after integration over final state angle also exist in the literature [49] . In practice, the Boltzmann equation is generally not given a full numerical treatment since accurate approximate solutions exist [40] .
Once the effective thermally averaged cross section has been derived as a function of temperature, it is used to iteratively determine the freeze-out temperature. In practice, the dimensionless inverse freeze-out temperature, x F = m χ /T F , is calculated using
Here m pl is the Planck mass, g is the total number of degrees of freedom of the χ particle (spin, color, etc.), g * is the total number of effective relativistic degrees of freedom at freeze-out, and the thermally averaged cross section is evaluated at the freeze-out temperature. For most cold dark matter candidates, x F ≃ 20.
When the freeze-out temperature has been determined, the total (co)annihilation depletion of the neutralino number density can be calculated by integrating the thermally averaged cross section from freeze-out to the present temperature (essentially T = 0). The thermally averaged cross section appears in the formula for the relic density in the form of the integral
Here g * S is the number of effective relativistic degrees of freedom contributing to the entropy of the universe and h is the reduced Hubble parameter, defined by H 0 = 100 h km sec −1 Mpc −1 . The above expression is commonly given in the less explicit form:
This is the expression that one compares with an experimental determination of the dark matter abundance. Recent evidence suggests [50] that Ω χ ∼ 0.3 with h 2 ∼ 0.5. We will take as a conservative favored region, 0.1
The lower bound comes from the requirement that χ 0 1 should at least form galactic dark matter, and the upper bound is a very conservative interpretation of the lower bound on the age of the Universe. Recently, WMAP [51] has elegantly confirmed the composition of the Universe to be 73% dark energy and 27% matter. WMAP determines a total matter density Ω M h 2 = 0.135 +0.008 −0.009 and a total baryon density Ω B h 2 = 0.0224 ± 0.0009, from which one can extract a 2σ range for the density of cold dark matter: Ω CDM h 2 = 0.1126 +0.0161 −0.0181 . We do not use this new result in our analysis. The effect of the WMAP measurement has been studied in the case of neutralino dark matter from mSUGRA [52] and rSUGRA [53] and also in the case of scalar dark matter from 'little Higgs' theories [54] . Let us stress that the requirement of (3.62) should not be treated as a constraint, but rather as an indication of the region preferred by cosmological considerations. Theoretical assumptions made to extract the a present relic density of neutralino LSP need not hold. In fact, the missing non-baryonic matter in the universe may not consist of relic harrishoward@example.org.
The b → sγ constraint
Another observable where the supersymmetric contribution can be important and measurable is the flavor changing decay b → sγ [55] . In the Standard Model, this process is mediated by virtual isospin +1/2 quarks and W -bosons. In supersymmetric theories, the spectrum allows new contributions involving loops of charginos and squarks or top quarks and charged Higgs bosons. As these two contributions appear at the same order of perturbation theory, the measurement of the inclusive decay B → X s γ is a powerful tool in the search for physics beyond the Standard Model. For our analysis, we will use the results given by the CLEO and BELLE collaborations [56] .
The Particle Data Group [57] summarizes these results and gives the current limit as:
BR(b → sγ) = (3.37 ± 0.37 ± 0.34 ± 0.24 63) where the three first errors represent respectively the statistical errors, the systematic error, and the estimated error on the model describing the behavior of the quarks in the B-meson decay. The fourth uncertainty is the error made by an extrapolation to the entire energy range for the photon (harrishoward@example.org GeV for the experiment). The last error value is an estimate of the theoretical uncertainties. To be as conservative as possible, one could add linearly all the uncertainties of (3.63). But we will adopt the procedure taken in the recent benchmark study of Battaglia et al. [58] and choose to impose the constraint
Let us note that we perform these calculations under the assumptions of minimal flavor violation.
The muon anomalous magnetic moment
The relation between the spin s of the muon and its magnetic moment µ is given classically by µ = g µ eh 2mµc s where g µ = 2(1 + a µ ). Following [59] we introduce the parameter δ µ to quantify the difference between theoretical and experimental determinations of a µ :
Recently, the Brookhaven collaboration has given a new measurement of the anomalous magnetic moment of the muon [60] (g exp µ − 2) 2 = a exp µ = (11 659 202 ± 14 ± 6) × 10 −10 , 66) where the first error is statistical uncertainty and the second is systematic uncertainty. From this the current experimental determination of the parameter δ µ is δ exp µ = 203 ± 8. To understand the implications of this result for supersymmetry one needs to know the Standard Model contribution. Unfortunately, the computation of δ sm µ in the Standard Model is complicated by our poor understanding of the hadronic contributions to certain vacuum polarization diagrams. The theoretical result depends on whether one uses the e + e − annihilation cross section or τ decay data to estimate these contributions [61] : 68) corresponding to a discrepancy between the Standard Model prediction and experiment of respectively 3.0 and 0.9 standard deviations. In our discussion we will be less conservative than the authors of [59] and consider a 2 standard deviation region about the anomalous moment of the muon based on the τ decay analysis:
The contribution of SUSY particles to the anomalous moment of the muon mainly comes from neutralino-smuon and chargino-sneutrino loop-induced processes. As the chargino-slepton-lepton couplings have the same form as the chargino-squark-quark couplings, the parameter space restriction imposed by the constraint (3.69) will be similar in many respects to the one given by the b → sγ process. Thus, the SUSY contribution to (g µ − 2) is large for large values of tan β and small values of the soft-breaking masses. It is interesting to note that the sign of the SUSY contribution is equal to the sign of the µ parameter. With our conventions this implies that positive µ is favored, as in the b → sγ constraint [55] .
Let us briefly summarize the above-mentioned constraints that are to be applied below. First, we demand that electroweak symmetry be correctly broken, as defined by deriving a positive value for µ 2 from (3.53). It should be noted that we use parameters of the Standard Model as defined in (3.54) and (3.55). Next, we demand that the mass of the lightest scalar Higgs boson be greater than or equal to 113.5 GeV. Additionally, we require that the mass of the lightest chargino be greater than or equal to 103.5 GeV. We also require that the lightest slepton mass be larger than 88 GeV and the lightest squark mass be greater than 97 GeV, but these two scalar fermion mass limits exclude no regions of parameter space in the plots resulting from our analysis. We also indicate regions of parameter space that result in the right thermal abundance of relic neutralinos: 0.1 ≤ Ω χ h 2 ≤ 0.3. While we technically do not exclude any regions solely for failing to produce the cosmologically preferred relic density, we do require that the lightest neutralino be the LSP. In terms of the flavor changing decay b → sγ, we take a conservative approach and require the branching ratio to obey 2.33 × 10 −4 < BR(b → sγ) < 4.15 × 10 −4 , given in equation 3.64. Finally, we follow the recent τ decay analysis of the muon anomalous magnetic moment and require −11.6 < δ new physics µ < 30.4, given above in equation 3.69.
Results and analysis
We are now in a position to investigate the way in which low energy observations can distinguish different regions of the weakly-coupled heterotic string parameter space. We first recall the standard analysis in the context of the minimal supergravity (mSUGRA) parameter space. While the unified mSUGRA paradigm is unlikely to find a manifestation in realistic string-based models, it nonetheless provides a useful benchmark for calibrating results from one study to the next.
Revisiting mSUGRA
The minimal supergravity paradigm is based on the assumption that the soft supersymmetry breaking Lagrangian is determined by only five parameters. These include a universal gaugino mass M 1/2 , a universal scalar mass M 0 and a universal trilinear soft parameter A 0 , as well as the sign of the µ parameter and tan β. A quick survey of the soft terms presented in Section 2 indicates that there is no point in the parameter space of the string models that we consider that generates such a universal outcome. A unified scenario is possible only in the event that the tree-level dilaton domination case arises [21] , though no explicit realization of this outcome exists in a complete model. Even if such a scenario were realized, the soft terms would be constrained to a particular point of the mSUGRA parameter space such that
Despite this lack of theoretical motivation, the extreme simplicity of the mSUGRA approach makes it an attractive scenario for phenomenological study [62] . We will here just present the main outcomes of our analysis. There are still large areas of parameter space in the (M 1/2 , M 0 ) plane which are allowed, particularly for large tan β where the Higgs mass bound is less constraining. The constraint from the process b → sγ can be the dominant constraint for low gaugino masses. Nevertheless, it is never very severe for positive µ, even for high tan β (note the change in scale in Figure 2 relative to Figure 1 ). The situation would change in the case of large trilinear A-terms because of the possibility of light third generation squarks [43] . However, in this case charge and/or color breaking (CCB) problems often arise.
In both plots of Figure 1 the exclusion regions for the chargino and Higgs masses are given. These exclusion regions implement the experimental limits of 103.5 and 113.5 GeV, respectively, that were discussed in Section 3.1. We also provide contours of constant Higgs mass of m h = 115.5 GeV and 117 GeV (the dashed lines from left to right) as well as a contour of chargino mass m χ ± 1 = 500 GeV (dotted line). For small and moderate values of tan β the requirement that the density of the lightest neutralino accounts for the dark matter density is very constraining and sensitive to the precise value of tan β. For moderate and low values of this parameter, one usually finds too much relic density in the mSUGRA model. For example the "bulk" region at low scalar and gaugino masses is excluded by the Higgs mass limit. τ -LSP exclusion zone, a large Higgs pole region at extremely high tan β, and the "focus point" [63] region at high values of M 0 (the thin strip along the "no EWSB" region in Figure 2 ). The region next to the τ -LSP exclusion zone provides for efficient τ 1 χ 0 1 coannihilation, considerably reducing the neutralino density. For high values of tan β, interesting areas exist where Ω χ h 2 becomes smaller due to near-resonant s-channel annihilation through the heavy Higgs states (A or H). Here m A and m H become smaller and their couplings to the b quark and the τ lepton increase. In this region χ 0 1 χ 0 1 → bb, τ + τ − dominates and causes significant depletion of the relic density. This can clearly be seen in the plot for tan β = 50. It should be noted that these results are strongly dependent on the treatment of the radiative correction for the bottom mass and the Higgs masses. As stated earlier, we have taken M b = 4.62 GeV. Small changes in these values have dramatic consequences on the density. At large M 0 and small M 1/2 , there is another region (the "focus point" region) where the lightest neutralino as well as the lightest chargino acquire relatively small mass and large Higgsino components. The coupling to the bosons W and Z is large enough to make the annihilation process into W W or ZZ efficient. Our results are in agreement with the previous studies of mSUGRA [62] . The only discrepancies appear in the high tan β regimes, mostly due to the different treatment of the b−quark mass and associated Yukawa coupling, and the Higgs sector masses. These differences can be understood in the light of the work done by Allanach et al. [64] . Indeed, the treatment of the Yukawa coupling of the b−quark (λ b ) in SuSpect results in a higher value for λ b than from the other publicly available codes (a 4 to 8 percent effect). This discrepancy affects the Higgs masses (m A in particular) via the RGEs which are dominated by λ b for high tan β. For instance, for tan β = 50, we can easily find a 20 to 50 GeV difference for m A , explaining differences in the parameter space allowed by the relic density in the A−pole sector. In the remainder of our paper we will restrict ourselves to tan β ≤ 35. In this range, the discrepancies between codes are smaller than 3 percent. In terms of relic densities, the differences between results using the codes DarkSusy [48] and micrOMEGAs [34] come from both the b-mass treatment and the selection of included coannihilation channels: previous versions of DarkSusy did not include any coannihilation channels with sleptons and scalars while micrOMEGAs does. 6 To summarize, in the case of the mSUGRA model, even if large areas of the (M 1/2 , M 0 ) parameter space are still allowed by present accelerator data, combining the b → sγ limits, the neutralino relic density constraints and the Higgs mass limit selects the τ coannihilation region, the "focus point" region, or extremely high values of tan β. These three regions represent the extreme boundaries of mSUGRA. Thus, it is natural go beyond mSUGRA to see how these constraints affect orbifiold models.
Moduli domination: the anomaly-mediated limit
The phenomenology of the original (minimal) AMSB model has been extensively studied (for some of the earlier work, see [65] ). However the phenomenology of the string-based PV-AMSB model, defined by the soft terms given in (2.48) , has yet to be thoroughly explored. We view this anomalydominated regime as a particular limit of string models where supersymmetry is broken by the F-term of some compactification modulus. Note that this anomaly-mediated limit is one in which all tree level soft supersymmetry breaking terms vanish. We thus choose to begin our study of string-based heterotic orbifold phenomenology at one loop with this particularly simple regime.
Like the minimal AMSB scenario, the PV-AMSB model has two free parameters, apart from tan β and the sign of the µ term. One parameter is the overall scale, given by the gravitino mass M 3/2 . The other parameter is the regularization weight p which determines the size of the scalar mass terms relative to the gaugino masses and A-terms. In this sense the weight p plays a role comparable to the bulk M 0 postulated in the minimal AMSB scenario but it is not an ad hoc parameter and it appears elsewhere in the spectrum. Note however that the gaugino masses are independent of this weight. Thus achieving a sufficiently heavy chargino implies a lower bound on the absolute scale as displayed in Figure 3 .
If we look at the branching ratio b → sγ we observe than it becomes important in two situations: small values of the gravitino masses and/or large values of p. As previously mentioned, small gravitino masses give small values of m χ , which leads to a big contribution of the diagram involving χ ±q in the loop for b → sγ. Having a large value for p also leads to small squark masses and thus a large contribution from the squark exchange diagrams. Moreover, the same limit p → 1 also leads to lightχ 0 3,4 ,χ ± 2 , which can now also contribute substantially to b → sγ (because of the small µ parameter).
Concerning the Higgs mass, our findings agree with intuition gained from mSUGRA. One needs significant loop contributions from relatively heavy squarks to satisfy the LEP constraint, especially at low tan β. Thus regions where squarks are light are ruled out. This includes the region with p → 1 and smaller gravitino mass scales. In Figure 3 we show the region excluded by our Higgs mass constraint in the left plot, as well as a contour of m h = 114 GeV for comparison. In the right plot for tan β = 35 there is no constraint in this gravitino mass range arising from the Higgs search limit and we provide contours of m h = 117 GeV and 118 GeV. In previous AMSB studies it was found that the neutralino thermal relic density is generically too small to explain the amount of dark matter [66] . Due to the low ratio of M 2 /M 1 in the AMSB scenarios, the wino content of the LSP is quite high. Additionally, coannihilation between the LSP and the lightest chargino is also very efficient. Both of these effects combine to make the thermal relic density of LSP negligible. Throughout the parameter space of Figure 3 the thermal relic density is around Ω χ h 2 ∼ 10 −4 . Thus the anomaly-mediated character of the gaugino sector in this model necessitates a non-thermal production mechanism for neutralino LSPs, or another candidate for the cold dark matter must be postulated. We sum up these constraints in Figure 4 where we have shown the allowed region in the (p, tan β) plane for a gravitino mass of M 3/2 = 40 TeV. We clearly see that high tan β area is excluded by b → sγ, and the low tan β area is disfavored because of the Higgs mass limit. In every case, p cannot be too large (must be less than ∼ 0.85). This excludes the "minimal" AMSB scenario (p = 1) up to the two-loop corrections to soft terms which have not been fully calculated in these string-based models. Most of the interior of the parameter space in Figure 4 has sufficiently heavy Higgs masses, as shown by the dashed contours of m h = 115.5 GeV and 117 GeV. The supersymmetric contributions to the b → sγ rate in the interior are small as well. The solid contour is BR(b → sγ) = 3.53 × 10 −4 . The chargino masses are very light throughout and we have provided contours of m χ ± 1 = 112.5 GeV (right-most dotted line) and 113.5 GeV (left-most dotted line). We note that much of this parameter space predicts a relatively small supersymmetric contribution to the muon anomalous magnetic moment due to relatively heavy scalar masses (i.e. δ susy µ ∼ 0). The upper shaded region is ruled out by improper electroweak symmetry breaking (µ 2 < 0).
In summary, the Pauli-Villars regularization weight p generates a whole class of anomalydominated models at the one-loop level. While this new degree of freedom solves the tachyonic slepton problem, the minuscule relic density still does not allow the neutralino to be a thermal dark matter candidate. Additionally, the constraints coming from accelerator physics exclude high values of p and low values of M 3/2 and predict small contributions to the muon anomalous magnetic moment.
The general moduli dominated case
We will now relax the assumption that the various Kähler moduli are stabilized at self-dual points, while maintaining the assumption that the modular weights of all matter fields are equal and given by n i = −1. When the vacuum value Re t is not fixed at Re t = 1 we must use the soft terms in (2.47) . Note that the value of the Green-Schwarz coefficient δ gs becomes relevant for the determination of gaugino masses in this case. For the time being we will take δ gs = 0 in order to keep the parameter space to manageable size.
For the study of the moduli dominated case we have looked at the behavior of the model as a function of the vacuum expectation value (vev) of the real part of the (universal) modulus field t = T | θ=0 , for different values of p and tan β. To be more precise, we have taken two values of p, 0 and 0.95, and two values of tan β, 5 and 35. We have chosen to use the value p = 0.95 because at this value the gaugino masses and scalar masses have roughly the same magnitude. It is also very near the pure AMSB limit. We first stress the common features of the moduli-dominated case, and extract the corresponding phenomenological consequences, before looking at specific points in the parameter space.
We choose to investigate the region 0.1 ≤ Re t ≤ 5. Since the vacuum value of this modulus is related to the radius of the compact dimensions, negative values have no physical meaning. Let us note that the combination (t +t)G 2 (t,t), which appears in the gaugino mass when the equation of motion (1.11) is substituted into (2.47), is invariant (up to a sign) under the duality transformation t → 1/t as shown in Figure 5 . However, since this term competes with the anomaly-mediated contribution to gaugino masses at the one loop level, gaugino masses will not display this duality symmetry. Any value Re t = 1 represents a spontaneous breaking of modular invariance and the phenomenology of the theory for Re t < 1 will be different from the case Re t > 1.
Looking again at the gaugino masses in (2.47) it is clear that when δ gs = 0 the gaugino masses are proportional to beta-function coefficients. Thus we expect the phenomenology of this scenario to be very similar to that of the PV-AMSB scenario of the previous section. For example, we continue to have
independent of the value of Re t or p. At the low scale this implies M 2 ≪ M 1 and the lightest neutralino χ 0 1 and the lightest chargino χ ≃ M 2 , and the relic density of the LSP neutralino is almost entirely depleted by (co)annihilation channels involving χ 0 1 and χ ± 1 . Unlike the case of Section 4.2, however, there is now a value of Re t such that the contribution from the Eisenstein function exactly counterbalances the contribution from the superconformal anomaly giving us vanishing gaugino mass soft breaking terms M a to this order. For the choice of phase conventions in (1.7) and (1.8) this occurs when 7 (t +t)G 2 (t,t) = 1 → ζ(t) = 0 → Re t = 0.523.
(4.72)
Not surprisingly, then, we find that much of the parameter space in the vacinity of Re t = 0.5 is ruled out by the chargino mass bounds for any reasonable choice of scale M 3/2 in Figures 6 and 7 .
In the extreme cases where Re t → 0.1 and Re t → 5 the absolute value of the combination (t +t)G 2 (t,t) becomes large and dominates over the anomaly contribution in the gaugino masses. Here gauginos will typically be similar in size to the scalar masses at the high scale, particularly as the value of p is increased towards its limiting value p = 1 ( Figure 7 easily satisfied here and the increased contribution from M 3 in the RG evolution of the scalar masses makes the Higgs mass constraint easier to satisfy, particularly for large tan β. In Figure 6 we show the Higgs and chargino exclusion regions in the tan β = 5 panel and the contours m h = 115 GeV (dashed line) and m χ While the mass contours and exclusion regions in Figure 6 show the underlying modular symmetry t → 1/t in a very broad sense, it is clear that these contours do not obey this symmetry in the strict sense. We have indicated the self-dual point Re t = 1 by a heavy vertical line. The exclusion regions do not show the expected duality symmetry about this point. As already mentioned above, this is due in large part to the competition between contributions to the gaugino masses from the moduli sector and contributions from the superconformal anomaly. The relative sign between these contributions depends on which side of the self-dual point the (overall) modulus is stabilized. Having chosen the conventions of (1.7), (1.19) and (1.20) this implies a zero in the gaugino mass formula (1.30) on a definite side of the self-dual point (with the conventions here, the smaller value side). The conventions can be changed to make this zero occur on the other side, but only at the expense of changing the convention for the sign of the µ parameter. As we keep this sign always fixed at positive values, the relative sign between the gaugino masses -in particular, the gluino mass M 3 -and the µ term is the key variable in distinguishing the two sides of the self-dual point. This is reflected in the fact that at tan β = 35 in Figure 6 the b → sγ constraint applies to only one side of the figure. Other observables, such as the muon anomalous magnetic moment, will be sensitive to this relative sign as well, making any spontaneous breaking of modular invariance a true observable, at least in principle. 8 As we approach the minimal AMSB limit of p → 1 the relevant constraints change as scalar masses diminish. This is displayed in Figure 7 . The Higgs mass limit is important over much of the parameter space, requiring gravitino mass scales well in excess of 50 TeV for Re t ≃ 1 in order to compensate for the rapidly diminishing stop masses. Note that Figures 6 and 7 correctly reproduce our previous results in the limit of Re t → 1. The light squarks in the p ≃ 1 regime tend to contribute too much to the b → sγ process -a process that provided no such constraint in this region of Re t in Figure 6 due to the relatively large scalar masses in those cases.
Nearly the entire region of the parameter space for tan β = 35 shown in Figure 7 is ruled out by this constraint. The remaining slender allowed region near Re t ≃ 0.25 allowed by the experimental constraints is the result of a relatively light charged Higgs that helps to cancel the contributions from the light chargino. Here the scalars and the gauginos are light -in particular, the gluino has a vanishing soft mass at the high scale near Re t ≃ 0.3 when δ gs = 0 (see Figure 8 below) . This leads to a very small µ term value in this narrow range, and consequently a small charged Higgs mass. For example, as we move from Re t ≃ 0.1 to Re t = 0.3 along the line M 3/2 = 15 TeV in the right panel of Figure 6 the value of µ at the electroweak scale drops from 1700 to 1100 GeV due to the diminishing gluino mass. But in the right panel of Figure 7 µ goes from 1700 to 380 GeV over the same range, leading to a charged Higgs mass of approximately the same value. Scalar masses are around 200 GeV at this point.
In the right panel of Figure 7 with p → 1 (light squarks) and large tan β (large couplings in the Higgs sector) there also appears a zone excluded by too large of a supersymmetric contribution to the anomalous magnetic moment of the muon (a black region labeled "µ") due to the presence of very light squarks and gauginos. This region is already excluded, however, by multiple observables such as the b → sγ rate, the chargino mass and the Higgs mass bound. The shaded region in the upper right of the tan β = 35 plot in Figure 7 is disfavored by the presence of CCB minima deeper than the preferred electroweak vacuum. The presence of such a region is due to the very large trilinear couplings for these values of Re t which are larger than the scalar masses when p ≃ 1.
In conclusion we see that in this simple moduli-dominated scenario the phenomenology is similar to that of anomaly mediation. The closer one approaches the minimal AMSB limit of p = 1 the more problematic the scenario becomes, particularly at large tan β. Other points in the PV-AMSB class have some areas of phenomenological viability, though the chargino and Higgs mass bounds can be quite constraining, particularly near the self-dual point Re t = 1. All of these cases will be unable to explain the cold dark matter content of the universe, at least in terms of thermal relic neutralinos.
Our study of the PV-AMSB scenario of Section 4.2 had one effective parameter, the regularization weight p, apart from the overall scale given by the gravitino mass M 3/2 . This regularization weight controls the size of the scalar mass relative to the fundamental scale M 3/2 while the gaugino masses were fixed by their beta-function coefficients. In the present section the parameter space has been expanded to allow for Re t = 1, allowing the gaugino masses to be varied relative to the scalars and gravitino mass M 3/2 in an independent way. Small, or even vanishing, gaugino masses are now possible, as is the possibility of relatively large gaugino masses as one moves far from the self-dual point for the compactification moduli.
Despite this new degree of freedom the parameter space is still quite constrained. In particular, obtaining the correct relic density for the neutralino LSP is impossible in this framework, because of its extreme wino-like nature. But, as we will see in the next section, the possibility of non-vanishing Green-Schwarz counterterm can open a new region of parameter space completely in accord with present experimental data.
The influence of the Green-Schwarz counterterm
In this section we wish to introduce the possibility that δ gs = 0, as would typically be the case in realistic heterotic orbifold constructions. As we will see, this parameter allows one to interpolate between regions with a phenomenology similar to anomaly mediated models (specifically the PV-AMSB scenario of equation 2.48) and regions where the phenomenology resembles that of the minimal supergravity paradigm.
In the conventions defined by the string threshold correction (1.28) the value of δ gs is a negative integer between 0 and -90. For any given orbifold construction the precise value of this coefficient can be worked out, and indeed such an exercise has been performed for realistic Z 3 models already [67] . In the figures that follow we will treat this variable as a continuous parameter. At values of the Kähler modulus far from its self-dual point, where the Eisenstein function is not negligible, the term in (2.47) proportional to δ gs can give rise to substantial (universal) contributions to the gaugino masses. Thus points in parameter space with large values of δ gs and values of Re t far from unity will likely have a pattern of gaugino masses similar to mSUGRA -and hence are more likely to provide a viable dark matter candidate than cases with δ gs ≃ 0. Furthermore, once |δ gs | ∼ √ 16π 2 ∼ O(10), the gaugino masses and scalar masses will be of approximately the same size. This is very different from the hierarchical situation of the previous sections.
Because of the difference of sign between the beta functions b a of the Standard Model gauge groups, M 3 has a particular behavior that differs from the other gaugino masses. This in turn drives the phenomenology of the model, particularly through the gluino's RG effects on the rest of the superpartner spectrum. With a non-vanishing Green-Schwarz coefficient the various gaugino masses will no longer have zeros at the same value of Re t. In particular, for certain combinations of Re t and δ gs it is possible for the gluino mass to vanish at the boundary condition scale while the other gauginos have non-vanishing masses. These combinations are shown in Figure 8 . For parameter choices near these contours we might expect the phenomenologically unacceptable result of a gluino LSP. Even when the gluino is not the LSP, its small value relative to other gauginos may provide too little radiative correction to squark masses and the Higgs mass bound may be difficult to satisfy. On the other hand, a relatively light gluino is likely to reduce the amount of fine-tuning required to obtain M Z = 91.2 GeV [68] . In Figures 9, 10 and 11 we look at this model in the (δ gs , M 3/2 ) plane for p = 0 and Re t = 0.5, Re t = 1.23 and Re t = 2, respectively. Beginning with the case Re t = 0.5 in Figure 9 we see that the limit as δ gs → 0 reproduces the cases studied in the previous section. In particular, such models require very large M 3/2 in these regions to satisfy the chargino and Higgs mass constraints. This behavior is seen in all three of the Re t presented here. A maximum value of |δ gs | can in general be obtained by requiring that lightest neutralino be heavier than the lightest stau. For any values of tan β the maximum value of |δ gs | is determined by the requirement of keeping the lightest neutralino as the LSP. The quantity |δ max GS | is smaller for higher tan β because the stau is in this case lighter. In between these disallowed regions the increasing values of the gaugino masses for fixed scalar mass, as the Green-Schwarz coefficient increases in absolute value, results in smaller contributions to the b → sγ branching ratio and a smaller (in absolute value) anomalous magnetic moment of the muon. In Figure 9 we have chosen a value of Re t near the value where gaugino masses vanish when δ gs → 0. As is evident from the figure, gaugino masses are indeed falling as this limit is reached. Note that in this case there is no point in the plane for which the gluino is the LSP.
Concerning the relic density, it is negligible for small values of the coefficient for the GreenSchwarz counterterm, as we have seen previously: the neutralino is extremely Wino-like in nature and is nearly degenerate with the lightest chargino. Almost all of the relic neutralinos are depleted through coannihilation channels with the charginos. When we increase |δ GS |, however, we increase the gaugino masses through the increased importance of the G(t, t) terms in (2.47) . This universal contribution can come to compete with the (nonuniversal) anomaly-mediated term, driving the gaugino mass terms M i=1,2 far away from the previous "degenerate" situation. The gaugino mass sector then begins to look similar to that of mSUGRA and a more bino-like LSP can develop, leading to an increased relic density. Just as in the mSUGRA case there continues to be a region of stau coannihilation near the excluded stau-LSP area. And -again, just as in mSUGRA -much of the "bulk" area is ruled out by the chargino and Higgs mass bounds, or the b → sγ constraint.
In Figure 10 we investigate the same parameter space for the case where Re t = 1.23. When multiple gaugino condensates are utilized to stabilize the dilaton, with the tree level Kähler potential given by K(S, S) = − ln(S + S), then a modular invariant treatment of the resulting nonperturbative potential for the moduli fields will lead to the conclusion that the auxiliary field for the dilaton F S must vanish in the vacuum. The remaining potential for the Kähler moduli leads to their stabilization at this value [69] . Specific points in the space shown in Figure 10 were singled out for a more detailed study of their collider signatures in a recent set of string-inspired "benchmark" models [70] . Here a more complete survey of the parameter space is possible. In accordance with Figure 8 we note a region centered about δ gs = −15 where the gluino is the LSP. Interestingly, this is precisely the region of values favored by semi-realistic Z 3 orbifold constructions [67] . To the right, as the absolute value of the Green-Schwarz counterterm diminishes, the chargino mass bound is eventually reached. The value Re t = 1.23 is sufficiently close to the self-dual value that the theory near δ gs = 0 is anomaly mediated-like in character: very large values of the gravitino mass are generally required to achieve a sufficiently heavy lightest chargino and lightest Higgs (note the difference in scale between Figures 9 and 10 ). For tan β = 35 the Higgs mass constraint is less important, being only relevant for low gravitino masses near the gluino LSP region. In addition, the LSP is predominantly wino-like on the right side of the gluino LSP region, and the relic density of these neutralinos is negligible.
As we move to larger absolute values of δ gs we interpolate between this anomaly mediated region and an mSUGRA-like region. For certain critical values of this parameter the wino content of the LSP is sufficiently reduced to result in the correct annihilation efficiency to produce the cosmologically preferred relic abundance. This change in the character of the LSP is the result of the increasing importance of moduli contributions to the gaugino masses (the first term in the gaugino mass parameter in equation 2.47) versus the anomaly mediated contributions (the last term in that expression). The ability to achieve the right relic density, independent of the scalar mass values, is a general property of models with this sort of non-degeneracy among gaugino mass parameters [44, 45] . We investigate the case Re t = 2.0 in Figure 11 . The behavior at small tan β is not dissimilar from the nominally "dual" case of Re t = 0.5 in Figure 9 . The stau LSP constraint now arrives at much smaller values of the Green-Schwarz coefficient. Again, the region with the promising relic density of neutralinos is largely constrained by the Higgs mass limit.
The parameter space for the higher tan β regime is even more tightly constrained (at least at these relatively low values of the gravitino mass). We again see a region of gluio LSP -a narrower region at lower absolute values of δ gs consistent with Figure 8 . At the bottom of the plot the values are ruled out by the presence of CCB vacua while the region to the left has a stau LSP. The light charginos for this value of the gravitino mass lead to large contributions to the muon anomalous magnetic moment (the black region labeled "µ"). Finally, all but the uppermost corner of the space is ruled out by the Higgs mass limit and the upper bound on BR(b → sγ).
We combine all of these observational and theoretical constraints to explore the relation between δ gs and Re t in Figure 12 . We plot the allowed parameter space in the (δ gs , Re t ) plane in a similar fashion to Figure 8 for the specific gravitino mass of M 3/2 = 10 TeV. On either side of the self-dual point at large negative values of δ gs the stau is the LSP. Near these regions there is a strip of preferred neutralino relic density. The area at small values of |δ gs | and near Re t = 1 is ruled out by improper EWSB. Here again there is a small strip of preferred relic density, though this is within the region ruled out by the Higgs mass, chargino mass or b → sγ constraint. In the interior there continues to be viable parameter space, including points with acceptable relic neutralino densities.
The generalized dilaton domination case
If we now consider the other extreme case (θ = π/2) the general features are somewhat different. Referring to the soft terms of equation 2.52, we can clearly see that the soft scalar mass terms will be roughly dominated by the gravitino mass scale (up to radiative corrections). We anticipate, therefore, that the relevant gravitino mass scale will be approximately one order of magnitude lower than the one necessary in the moduli-dominated case because of the absence of the loop suppression factors on the scalar masses. The gaugino mass soft supersymmetry breaking terms will be determined by the dilaton auxiliary field vev F S .
In the case where nonperturbative corrections to the dilaton Kähler potential are imagined, the minimum of the combined (modular invariant) dilaton/Kähler moduli potential now occurs at F S = 0 and compactification moduli stabilized at self-dual points where F T = 0. As described in Section 2.2, the requirement of vanishing vacuum energy naturally leads to a suppression of this auxiliary field vev relative to the auxiliary field of the supergravity multiplet. Thus the two contributions to the gaugino mass in equation 2.52 each involve a loop suppression factor (unlike the scalar masses) and the dilaton and conformal anomaly contributions are comparable. Indeed, the beta-function coefficients b a of the Standard Model gauge groups are of the order of 10 −2 . Moreover, the F S term will be driven by b + , the largest beta-function coefficient among the condensing gauge groups of the hidden sector, from (2.51) and (2.49) . In fact, if we look in more detail at the expression for F S in equation 2.51, it is apparent that for not too large values of b + we can consider that F S has a linear evolution as a function of that parameter. direct consequence on the values of the gaugino breaking terms, while increasing M 3/2 has direct consequences on the general size of all soft breaking terms -in particular soft scalar masses. We can consider the effect of the parameter b + as a "fine structure" on top of the gross feature of a hierarchy between scalars and gauginos in this regime.
The general features of this situation are shown in Figure 13 . When the suppression of the gaugino masses is large from small values of b + (small a np ), the gaugino sector looks increasingly like that of anomaly-mediation. This is true of any value of tan β. As this value is increased the wino content of the LSP diminishes and we approach the overwhelmingly bino-like LSP of mSUGRA. However, the value of b + cannot increase without limit. As we are considering the weakly coupled E 8 × E 8 heterotic string, we cannot imagine a condensing group larger than the complete E 8 of the hidden sector, for which b E 8 = 90/16π 2 = 0.57. It is more likely that the same Wilson-line mechanism that breaks the observable sector gauge group to the product of sub-groups at the compactification scale also breaks the hidden sector group as well. Thus a more typical range for this parameter might be b + ≤ 15/16π 2 = 0.095, where this value corresponds to a pure Yang-Mills sector of SU (5) .
For this range of condensing group beta-function, only small values of tan β are consistent with the stringent demands of electroweak symmetry breaking. This fact was observed in this model some time ago [71] , but it is here displayed explicitly in Figure 13 . The large regions marked "EWSB" in this figure haveμ 2 < 0. This is a manifestation of the "focus point," or hyperbolic nature of this model [63] : the scalar masses are generally much larger than gaugino masses throughout the parameter space. The light gluinos imply that third generation squarks must have large masses at the boundary condition scale to ensure a sufficiently heavy Higgs mass. Thus M 3/2 > ∼ 4 TeV is needed in the tan β = 5 case and M 3/2 > ∼ 1.3 TeV is needed in the tan β = 35 case.
Note also that the very large (and nearly universal) scalar masses will imply heavy sleptons at the electroweak scale. Therefore both direct neutralino annihilation through slepton exchange, as well as τ 1 /χ 0 1 coannihilation, will be ineffective in reducing the relic neutralino density in the early universe. Nevertheless, acceptable relic densities are possible in this model, without the need for resonant annihilation through a heavy Higgs eigenstate. The cosmologically preferred region is the shaded strip labeled "Ω" in Figure 13 . Between this strip and the region ruled out by improper EWSB the relic density is below the preferred amount. In this region the LSP has a sufficiently large wino content to annihilate efficiently. Below the marked strip the LSP is bino-like and the relic density is too large. We note that the region with the correct relic density can continue indefinitely to large scalar masses by maintaining the correct wino-content of the LSP.
Conclusion and Perspectives
Even within the restricted context of the weakly-coupled heterotic string compactified on an orbifold, the phenomenology of such supergravity effective theories is far richer and varied than that of the standard "minimal" supergravity approach. While our current indirect knowledge on the nature of supersymmetry already constrains these models significantly, there still exist large regions of parameter space for all the cases studied here -even for low values of tan β. We anticipate that future measurements of, or limits on, superpartner masses and kinematic distributions will constrain the parameters of these models further. So too will future cosmological or astrophysical measurements. In fact, these sorts of measurements form a useful complementarity that will be crucial in unraveling the nature of supersymmetry breaking and transmission in such models in the manner begun here. Future studies of string-based models should focus on extending this initial survey to true collider signatures in both hadron and lepton machines, as well as computing event rates for astrophysical processes.
Our analysis in this paper was somewhat cursory -we consider the constraints that arise principally from the chargino and Higgs searches at LEP as well as the measurement of the branching ratio for b → sγ in the minimal flavor violation scenario. We have indicated, where applicable, the region that seems to be favored by cosmological observations of dark matter relic densities. Interpreting this observation as a constraint on a model implies the assumption that the dark matter is the result of thermal processes in the early universe and is composed entirely of relic neutralino LSPs. In string models many other dark matter candidates are known to exist: axions, superheavy exotic string states, hidden sector gauge and matter condensates, etc., so care should be exercised in applying the cosmological bounds. We have therefore kept the range we show within the wider region 0.1 ≤ Ω χ h 2 ≤ 0.3 rather than the far more restrictive result from the recent WMAP experiment. We have also used only the most conservative possible interpretation of the measurement of the muon anomalous magnetic moment as to not prematurely prejudice one model versus another.
Despite the uncertainties that must be borne in mind in such an analysis arising from the computational tools employed (e.g. the choice of quark pole masses utilized, uncertainties in the determination ofμ at large tan β etc.) some general statements can certainly be made. Among these is the observation that explicit models of moduli stabilization utilizing field-theoretic methods tend to predict that at least some sector of the theory obtains soft supersymmetry breaking terms only at the one-loop level. This is especially true of the gaugino sector. Furthermore, this suppression leads to the general requirement that the gravitino be relatively heavy in these cases -perhaps alleviating the cosmological problems so often associated with models that contain gravitinos.
Parameters that are related to the orbifold can have a large impact on these broad features. This is a welcome result -implying that experimental data can indeed probe the nature of the underlying theory within a class of models. For example, in cases where supersymmetry breaking is transmitted predominantly by the moduli associated with compactification the relative sign of terms proportional to their auxiliary field and that of the conformal anomaly can in principle be measured. This parameter, in turn, is related to the spontaneous breakdown of modular invariance that may occur in such models.
The phenomenology, even in the simple initial approach taken here, is neither that of minimal supergravity nor minimal anomaly mediation but is often a hybrid of the two. This makes a detailed study of collider signatures all the more urgent as search strategies are often based on one or the other of these paradigms. Given that the era of experimental supersymmetry may well be at hand -and that weakly coupled heterotic strings still represent the best motivated string-based approach to understanding supersymmetry and its breaking in the low-energy world -the time is right for this exciting undertaking.

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