Kahler Independence of the G2-MSSM

The G2-MSSM is a model of particle physics coupled to moduli fields with interesting phenomenology both for colliders and astrophysical experiments. In this paper we consider a more general model - whose moduli Kahler potential is a completely arbitrary G2-holonomy Kahler potential and whose matter Kahler potential is also more general. We prove that the vacuum structure and spectrum of BSM particles is largely unchanged in this much more general class of theories. In particular, gaugino masses are still supressed relative to the gravitino mass and moduli masses. We also consider the effects of higher order corrections to the matter Kahler potential and find a connection between the nature of the LSP and flavor effects.

From several theoretical points of view, the existence of moduli fields seems inevitable. For instance, supersymmetry may be the mechanism responsible for stabilizing the scale of the Standard Model. Supersymmetry requires supergravity, whose only (known) reasonable UV completion seems to be String theory; and along with string theory come extra dimensions and their moduli. In fact, since string/M theory contains no dimensionless parameters, moduli appear necessary to explain the observed values of various couplings in nature. From the bottom up, moduli appear in various theories with "dynamical couplings" as well as in Inflation -the inflaton field is usually a neutral scalar field aka a modulus. For all of these reasons and more, moduli physics and phenomena must be considered seriously.
In a series of papers, [1], [2], [3], a very detailed model of moduli physics coupled to matter has been described. The G 2 -MSSM model, largely inspired by M theory compactifications on manifolds of G 2 -holonomy, is a model in which strong gauge dynamics in the hidden sector generates a potential which both stabilizes all the moduli fields and simultaneously generates a hierarchically small scale -thus solving (most of) the hierarchy problem. The model has an interesting spectrum: moduli have masses in the 50-100 TeV region, scalar superpartners and higgsinos have masses in the 10's of TeV region, whilst gauginos, which are the lightest BSM particles have masses of order 100's of GeV. Direct production of gluinos and electroweak gauginos are the dominant new physics channels at the LHC. The nature of the LSP is also very interesting as it is a neutral Wino. Moreover, its production in the early universe is dominated by the decays of the moduli fields (ie non-thermal production) and can naturally account for the observed fraction of dark matter today. The moduli and gravitino problems are avoided due to the gravitino mass scale being one to two orders of magnitude larger than the TeV scale. One drawback of the model is the fine tuning between the 10's of TeV scale and M Z and is the reason the model solves most of the hierarchy problem and not all of it.
However, the G 2 -MSSM model, as defined in [2], is based on some specific assumptions about the moduli and matter Kahler potentials, albeit with the claim that these are general enough to incorporate all of the essential ingredients of more general Kahler potentials (and hence G 2 -manifolds). Thus far, there has been no serious study of these assumptions and it is the main aim of this paper to undertake this. The main result that we prove here is that the mass spectrum of the theory depends very weakly on the specific form of the moduli Kahler potential; in fact the spectrum depends on the Kahler potential for moduli only through the fact that it is the Log of a homogeneous function (the volume of the extra dimensions); the precise nature of this homogeneous function is fairly irrelevant as we will see.
We also discuss the Kahler potential for charged matter fields. We give three consistent arguments for calculating the moduli dependence of the matter kinetic terms in 4d Einstein frame. Whilst nontrivial, these modifications do not change the results of [1], [2] much. More importantly, we also consider higher order terms in the matter Kahler potential, in particular the terms which are usually considered troublesome for flavor physics in theories of gravity mediated susy breaking. Whilst we expect that such operators will be suppressed, if they enter with large coefficients they can affect the mass spectrum: 1) they can directly alter the scalar and higgsino masses, which are typically large; 2) they can indirectly (via threshold effects from the higgsinos) alter the nature of the LSP. In particular, we find that the LSP can also be a Bino in some cases. This provides a connection between flavor physics and the nature of the LSP in models of this sort.
The paper is organized as follows. The next section describes some simple properties of the moduli space metric for general G 2 -manifolds which will be important for our later considerations. Section III is devoted to the Kahler potential for charged matter fields. Following this, we re-do the analysis of Moduli stabilization from [1] in this much more general context. In section V we compute the mass spectrum and susy breaking couplings in the minimum of the potential and demonstrate that it is almost identical to that of the original G 2 -MSSM. In section VI we present a further generalization of the construction. In section VII we renormalize the Lagrangian down to the Electroweak scale and give the spectrum there.

II. GENERAL PROPERTIES OF MODULI SPACE METRICS ON G 2 HOLONOMY MANIFOLDS
In this section we describe some very general and simple properties of the moduli space metric of G 2 -manifolds. It is these simple properties, which will allow us to draw very general conclusions.
The metric g(X) on a G 2 holonomy manifold X can be expressed in terms of the associative three-form Φ as with Expanding Φ in terms of basis harmonic three-forms φ i ∈ H 3 (X, Z) (modulo torsion) we obtain where s i are geometric moduli corresponding to the perturbations of the internal metric. The complexified moduli space M(X) of a G 2 holonomy compactification manifold X has holomorphic coordinates z i given by where t i are the axions parameterizing the zero modes of the 11-dimensional supergravity three form C 3 . The classical moduli space metric (not including possible quantum corrections) can be derived from the following Kahler potential [4] where the dimensionless volume V X ≡ V ol(X)/l 7 M is a homogeneous function of s i of degree 7/3 and l M is the 11d Planck length. The homogeneity of V X is the key property that we will utilize in what follows. In terms of the associative three-form Φ, the volume is given by [4] Define the following derivatives with respect to the modulî The matrixK ij , the Hessian ofK is related to the actual Kahler metricĜ ij which controls the kinetic terms as 4Ĝ ij =K ij , where in the Hessian we simply replace index j withj. Since V X is a homogeneous function of degree 7/3, the first derivative ofK defined above has the following property Differentiating (8) with respect to s j we obtain an important property of the metricK ij N i=1 s iKij = −K j , and sinceK ij is symmetric Let us now introduce a set of dual coordinates {τ i } defined by Note that the variables {τ i } are homogeneous functions of {s i } of degree 4/3 . Using the homogeneity of the volume together with the definition (10) we can express the volume V X as Combining (3), (6) and (11) we can reexpress the dual variables as where for each harmonic basis three-form φ i ∈ H 3 (X, Z) we introduced a Poincare dual four-cycle [τ i ] ∈ H 4 (X). We now use the above duality to make a particularly convenient choice of the basis harmonic three-forms. In particular, we choose a basis {φ i } ∈ H 3 (X) such that the periods of the fundamental co-associative four-form * Φ over the Poincare dual four cycles are positive definite: This choice of a basis becomes obvious when we recall that for a generic basis four-cycle where the above relation becomes an equality if and only if the corresponding four-cycle is co-associative. All geometric moduli describing the fluctuations of the internal metric must be massive in order to satisfy constraints from fifth force experiments and cosmology. At the same time the vacuum expectation values of the moduli (coordinates s i on the moduli space) must be fixed in the region of the moduli space where the geometric description makes sense. In Section IV we describe a way to stabilize the moduli, which ensures that we find isolated minima that satisfy these conditions automatically.
It turns out that for our purposes it is convenient to introduce a set of "angular" variables a i defined by We see that a i are scale-independent and satisfy Thus, we can also parametrize the moduli space M(X) by a subset of N − 1 variables a i plus one volume, e.g. the volume of the manifold V X . Differentiating the a i allows us to introduce the matrix which has components P ij has the following contraction properties, which follow from (17) and the fact that a i are homogeneous of degree zero We can then writeK where the matrix ∆ ij is defined as and satisfies the following contraction properties where we used (20) to derive (23). Note that parameters a i defined in (16) are the components of an eigenvector a of the non-Hermitian matrix ∆ with unit eigenvalue. We can compute the formal inverse of the Hessian metric,K ij . By definition of the inverse it must satisfy and using (21) it can be expressed asK where the inverse matrix ( Symbolically we can express ∆ −1 as which in terms of components translates into Using (20) and (28) we derive the following properties of the inverse matrix ∆ −1 which could have also been obtained directly from (23). Note that although we do not have a closed form expression for the components (∆ −1 ) ij , the contraction properties in (29) are what will ultimately allow us to derive explicit expressions for the terms in the soft breaking lagrangian -since such couplings depend only on the contractions and not the precise details of the functional form of V X . Before going on to the details of these calculations, we first must consider the Kahler potential for matter fields in M theory.

III. KAHLER POTENTIAL FOR CHARGED CHIRAL MATTER
In this section we re-visit the Kahler potential for charged matter fields in M theory. In practice, the absence of a useful microscopic formulation makes it difficult to compute the moduli dependence of the Kahler potential for these fields in general. Below we outline three arguments for the structure of the Kahler potential -first from dimensional reduction, second based on the scaling properties of physical Yukawa couplings and the third based on the form of the threshold corrections to the physical gauge coupling. Happily, all three methods agree.

A. Kahler potential from dimensional reduction
In M theory, charged chiral matter is localized near conical singularities [5], [6], [7], [8]. These are literally points in the seven extra dimensions. Because of this, we expect that the kinetic terms for the chiral matter fields should be "largely independent of bulk moduli fields" that the G 2 manifold X has. The precise meaning of this statement will be clarified below in terms of the scaling property of the kinetic term. They could, of course depend on local moduli inherent to the conical singularity, but, since, in a supersymmetric theory, a single chiral multiplet in a complex representation of the gauge group usually has no D or F -flat directions [9], there are typically no such local moduli.
There is a subtlety in the above general arguments. Since, in four dimensions, a scalar field kinetic term is not invariant under Weyl rescalings of the metric, one has to pick a Weyl gauge. We will argue that the correct Weyl gauge for the statement above is NOT the 4d Einstein frame. Therefore, the kinetic term for chiral matter will be non-trivial in the 4d Einstein frame, which is the standard one in which to define the Kahler potential.
Since the physics of a conical singularity in M theory does not introduce any new scale, asides from the 11d Planck scale, the only reasonable Weyl frame is the 11d Einstein frame. Therefore the lagrangian density in the 11d frame is where δ 7 is a delta function peaked at the position of the matter multiplet containing the scalar field φ and has mass dimension seven. κ(s i ) is a homogeneous function of the moduli of degree zero which will generally be of order one and vary adiabatically.
The above property implies that κ(s i ) remains invariant when the moduli are rescaled as s i → λs i , thus explicitly implementing the idea that in the 11d frame the kinetic term of a matter field localized at a point p ∈ X is "largely independent of bulk moduli". A particularly simple example satisfying (31) is when κ(s i ) = const. Integrating this over X leads to a 4d density where V X is the volume of the extra dimensions in 11d units. This is the Lagrangian in 11d Einstein frame. If we now Weyl rescale into the 4d Einstein frame we find where the subscript E indicates that we are using the 4d Einstein frame metric. We have only considered the Einstein-Hilbert and kinetic terms of the matter fields. Including all the other terms would give the 4d supergravity Lagrangian in Einstein frame. In particular, from this we would read off that the Kahler metric for the multiplet containing φ is If we introduce dimensionless fieldsφ as in φ = m plφ , the Kahler potential is As we will see, this is consistent with the arguments given in the next subsection.

B. Kahler metric from the properties of the physical Yukawa couplings
Here we will describe an alternative way of deducing the volume dependence of the Kahler metric for charged chiral matter. This method is due to Conlon, Cremades and Quevedo [11] and utilizes the relation between the physical (normalized) Yukawa couplings Y αβγ and the unnormalized Yukawa couplings Y ′ αβγ that appear in the supergravity superpotential. Recall that in G 2 compactifications of M theory, a superpotential Yukawa coupling Y ′ αβγ between the multiplets α, β, γ that are localized at three co-dimension seven singularities is induced by an M2-brane instanton wrapping a supersymmetric three-cycle connecting the three singular points. The absolute value of the Yukawa coupling is given by where is the volume of the supersymmeric three-cycle. After we diagonalize the Kahler metric for the matter fields and go to the canonical basis, the relation between the absolute values of the physical and unnormalized Yukawa couplings is simply a rescaling by e K/2 K αKβKγ On the other hand, one can construct perfectly well-defined seven-dimensional local models where the G 2 manifold is non-compact, e.g. an ALE-fibration over a three-sphere or a quotient thereof, in which case V X → ∞ ⇒ m pl /M 11 → ∞ and gravity is effectively decoupled. Such models can also contain charged chiral matter fields and since their interactions are determined locally, the corresponding physical Yukawa couplings should not vanish when gravity is decoupled. Local models of this type can be obtained via lifting effective theories on intersecting D6-branes in Type IIA to M theory.
Therefore, locality implies that the physical Yukawa couplings should be independent of the overall volume V X in the limit V X → ∞. For that to happen, the Kahler metricsK α ,K β ,K γ in (38) must scale with the volume V X asK which is in perfect agreement with the form of the Kahler metric derived in the previous subsection.

C. Consistency check for the Kahler metric
In this section we confirm the form of the Kahler metric for charged chiral matter by comparing the threshold corrections to the physical gauge couplings in G 2 compactifications of M theory with the general results in N = 1 D = 4 supergravity.
Let us first consider a hidden sector containing a pure glue SU (N ) supersymmetric Yang-Mills theory. Using the notation in [12] we have the following relation for the gauge coupling at one loop where S are the one-loop threshold corrections, g(µ) is the physical gauge coupling and g M is the tree-level Wilsonian gauge coupling. In our convention g M is related to the gauge kinetic function f as Recall that the Wilsonian gauge coupling gets renormalized at one loop only. On the other hand the physical coupling g(µ) is renormalized to all orders. The threshold corrections come from massive states and are independent of the scale µ. Based on the topological arguments [12], the threshold corrections due to Kaluza Klein modes in G 2 compactifications of M theory are rather simple and can be calculated even without knowing the G 2 metric! Friedmann and Witten [12] explicitly computed one-loop threshold corrections due to the heavy Kaluza-Klein modes living on a supersymmetric cycle Q with b 1 (Q) = b 2 (Q) = 0 and a non-trivial fundamental group. Such corrections come in a form of linear combinations of Ray-Singer analytic torsions [13], which are topological invariants of Q. For the case at hand, the threshold corrections are given by where V Q is the volume of the supersymetric cycle Q, T i are the Ray-Singer torsions corresponding to different irreducible representations of the fundamental group and Q are the generators of SU (N ).
Here, the cutoff dependence appears as a correction due to the zero mode contributions transforming in the trivial representation of π 1 (Q). Once the threshold corrections are included explicitly, a somewhat unexpected cancellation of the Λ-dependence occurs [12] and the one-loop relation can be written as In fact, the cancellation of the Λ-dependence occurs for any supersymmetric cycle Q with b 1 (Q) = b 2 (Q) = 0 [12]. Now we would like to consider a more general case when the gauge theory is a supersymmetric QCD with N f flavors of chiral matter fields Q α transforming in N of SU (N ) plus N f flavors ofQ α transforming in N . Each chiral matter field transforming in a complex representation arises from a separate co-dimension seven conical singularity on X with each singular point P i ∈ Q. It was argued in [12] that the singularities producing charged chiral matter fields have no effect on the KK harmonics of the seven-dimensional vector multiplet. Moreover, since the conical singularities introduce no new scale below the eleven-dimensional Planck scale M 11 , the effective cutoff scale for these multiplets is naturally M 11 . Including such multiplets into the running is straightforward and results in In addition to the KK thresholds, there may be some unknown corrections due to possible charged massive matter fields with masses of order M 11 . At this point we cannot say with certainty whether such massive charged M theory modes are present in the spectrum but we cannot exclude this possibility either. Just like the KK thresholds, these corrections cannot be holomorphic functions of the chiral multiplets z i describing the moduli of X since the axion partners of the geometric moduli decouple from the computations of the threshold corrections. However, there may be some non-holomorphic as well as constant contributions from such massive charged states. For now we will simply assume that they are constant and result in a slight shift of the tree-level gauge coupling. On the other hand, moduli dependent contributions may arise from non-perturbative corrections due to membrane instantons but they will be exponentially suppressed and can be safely neglected. Our next task is to independently verify that the Kahler metric for the charged chiral matter fields matches the previously obtained result (34). Here we will use a strategy similar to the one in [14], [10] and compare (45) with the corresponding one-loop expression in N = 1 D = 4 supergravity given by [15] In the above expression,K = −3 ln 4π 1/3 V X is the Kahler potential for the moduli andK αᾱ is the Kahler metric for the charged chiral matter fields. We can use the definition of the four-dimensional Newton's constant κ 4 = √ 8πG N = 1/m pl in terms of the eleven-dimensional gravitational coupling κ 11 in combination with the common convention 2κ 2 11 = (2π) 8 M −9 11 and M 11 = 2π/l M to obtain Using the above relations together with 4π/g 2 The appearance of the last term is most likely due to the convention used to define M 11 in terms of κ 11 as well as the ambiguity in defining the relation between l M and M 11 . Thus, we shall regard this term as an artifact and ignore it in further discussion. Comparing (49) with the expression on the right hand side in (45) we conclude that up to a constant multiplicative factor, Kahler metric for the charged chiral matter fields Q α isK which precisely matches the result obtained in the previous subsections. On the other hand, the constant term S ′ in (45) has no corresponding analog in (49) and represents a genuine threshold correction to the Wilsonian gauge coupling g M . In the framework of N = 1 D = 4 supergravity, the RG-invariant scale where super QCD with N > N f becomes strongly coupled is where the second exponential factor is due to the local SUSY. The Affleck-Dine-Seiberg effective superpotential [16] W should be identified with where the gauge coupling insideΛ is complexified. Using (51), up to an overall numerical constant we obtain In (53), the dimensionful chiral matter fields 2 Q can be expressed in terms of dimensionless fieldsQ as Then, the superpotential becomes whereC is an overall numerical constant. In our further notation, we also define the following constants Let us now consider the case of N f = 1 flavors. Introducing an effective meson degree of freedom we can rewrite the superpotential in terms of φ as where we have absorbed the factor of 2 1/(N −1) into the normalization constant C. Along the D-flat direction we haveQ =Q and the Kahler potential for the matter fields can be rewritten in terms of the effective meson fields φ asK

D. Higher order terms
Based on three independent arguments we have been able to deduce the volume dependence of the Kahler metric for charged chiral matter fields localized at co-dimension seven singularities. Denoting the visible sector charged chiral matter fields by Q α their Kahler potential is then given bỹ In the regime where the size of the supersymmetric cycle supporting the visible sector is large (this assumption is justified in the context of the MSSM where the corresponding volume is α −1 GU T ≈ 25 ) we can perform a systematic expansion in the inverse volume of the cycle (weak coupling) so that in the leading order κ αβ (s i ) is a homogeneous function of s i of degree λ, satisfying Based on the property that a given charged chiral matter multiplet is localized at a point p ∈ X we expect that λ = 0. Nevertheless, for the sake of generality we will keep λ as a free parameter for the time being. In our derivation of the Kahler potential we so far neglected possible higher order contributions to the visible sector matter Kahler potential of the form 3 which gravitationally couple the hidden sector meson to the visible sector fields Q α . In the above expression, the subscript c denotes canonically normalized matter fields in the 4-dimensional Einstein frame. Such couplings can create problems if the meson F -term is quite large (which is true in the G 2 -MSSM) because they can induce flavor changing neutral currents. This is the flavor problem of gravity mediated susy breaking models 4 . These terms were neglected in our previous work [2]. Technically, computing the unknown coefficients c αβ (s i ) from the underlying theory is difficult, goes well beyond the scope of this work and our aim here is not to explain the flavor structure of the supersymmetry breaking Lagrangian. Rather, we would like to understand the effect that the presence of such terms might have on other sectors of the theory, e.g. their effect on superpartner masses and couplings. For these purposes it is sufficient to assume that the flavor structure of the Kahler metric is completely determined by the matrix κ αβ (s i ), so that A sequestered Kahler potential has the form and the Kahler metric for the visible sector is given by Absorbing the factor of 4π 1/3 into the definition of the fields and expanding the above expression in powers of φ 2 /V X we obtain Comparing the above expression with (62) we can read off the coefficients which corresponds to (63) when c(s i ) = 1. Hence, function c(s i ) in (63) is the measure of deviation of the matter Kahler potential from the exactly sequestered form. As was pointed out in [18], sequestering is not at all generic in string/M theory and presumably G 2 compactifications of M theory are no exception. We thus will regard the value of c(s i ) in a given vacuum as a parameter and consider the theory for various values of c(s i ).
Combining all of the previous considerations, the visible sector matter Kahler metric and its inverse take the formK Combining (61) with the above we conclude that κ αβ (s i ) is a homogeneous function of the moduli of degree −λ. Function c(s i ) will be typically assumed to take values in the range However, as long as the Kahler metric is positive-definite, one may also consider the regime when c(s i ) < 0. Diagonalizing the Kahler metric of the visible sector we obtaiñ where the eigenvaluesK α are given bỹ and κ α (s i ) are homogeneous functions of degree λ that satisfy (61). In computing the anomaly mediated contribution to the gaugino masses, it will be necessary to compute various derivatives of lnK α . For this purpose, it turns out that it is very convenient to express lnK α as where K is the Kahler potential in (78).

IV. MODULI STABILIZATION
In this section we reconsider the problem of moduli stabilization with the much more general moduli and matter Kahler potentials introduced in the previous section. We will be working in the framework of N = 1 D = 4 effective supergravity and will demonstrate that all the moduli can be stabilized self-consistently in the regime where the supergravity approximation is valid. Recall that in the compactifications we study here, non-Abelian gauge fields arise from co-dimension four singularities [19], [20], [21], [22], [5]. In other words, there exist three-dimensional submanifolds Q inside the G 2 -manifold X, along which there is an orbifold singularity of A-D-E type. The basic idea is that strong dynamics in the hidden sector breaks supersymmetry, stabilizes the moduli and generates a small scale. The simplest possibility consistent with the supergravity approximation is a hidden sector with two gauge groups SU (P + N f ) and SU (Q) where the first is super QCD with N f = 1 flavor of quarks Q andQ transforming in a complex (conjugate) representation of SU (P + 1) (the corresponding associative cycle Q contains two isolated singularities of co-dimension seven) and the second hidden sector with the gauge group SU (Q) is a "pure glue" super Yang-Mills theory. One can easily consider more general gauge groups without much qualitative difference. One can also consider a setup with charged matter in both hidden sectors. However, as was demonstrated in [1], in such cases, one of the two F -terms coming from the matter fields in the hidden sectors is always suppressed relative to the other and thus does not contribute to the quantities relevant for phenomenology. A single hidden sector gauge theory is also enough to stabilise the moduli, though the vacuum is not in a place where supergravity is trustable! Therefore, the non-perturbative effective superpotential generated by the strong gauge dynamics in the hidden sectors is given by The matter field φ represents an effective meson degree of freedom defined in (57) in terms of the chiral matter fieldsQ andQ. The coefficients b 1 , b 2 and a are In [1] it was explained that if one uses a superpotential of the form (74), de Sitter vacua arise only when Q > P (if we include matter in both hidden sectors dS vacua exist without such condition). Hence, we will keep this in mind from now on.
In (74) we explicitly assumed that the associative cycles supporting both hidden sectors are in proportional homology classes which results in the gauge kinetic function being given by essentially the same integer combination of the moduli z i for both hidden sectors were is the volume of the corresponding associative cycle with the integers N i specifying the homology class. While one can certainly consider possibilities where the gauge kinetic functions f 1 and f 2 are not proportional, the results in [1] taught us that unless f 1 ∝ f 2 it is more difficult to stabilize all the moduli in the regime where the supergravity approximation is valid. Thus, obtaining solutions which we can trust is the main reason for choosing to consider the case where f 1 = f 2 = f . Obviously, progress in the more general cases would be welcome. Typical examples for 3-cycles supporting non-Abelian gauge fields in G 2 -manifolds are spheres and their quotients such as Lens spaces S 3 /Z q considered in [12]. The expression in (74) can in principle contain many additional non-perturbative contributions if X contains other rigid associative cycles. In that respect, the two terms included in (74) should be regarded as the leading order exponentials. As long as Q and P are large enough compared to the Casimirs from the other gauge groups, the remaining terms will be exponentially suppressed in general. This is particularly true for the membrane instanton corrections to (74) which come with exponentials containing b i = 2π. On the other hand, some such instantons induce Yukawa interactions among the visible sector matter fields and are therefore implicitly assumed to be part of the full superpotential.
The total Kahler potential -moduli plus hidden sector matter, is given by In what follows we first consider a simplified case where the function κ(s i ) is a pure constant, i.e.
However, in section VI we will generalize our results to the case where κ(s i ) is a homogeneous function satisfying (31). The important point is that even then the functional form of the soft breaking terms remains virtually unchanged compared to the simplified case, thus validating our approach. In general, (78) must include the contributions to the Kahler potential from all matter sectors including the visible sector as described in the previous section. However, since the visible sector fields will obtain zero vacuum expectation values (vevs), they can be dropped for the purposes of stabilizing moduli. The standard N = 1 D = 4 supergravity scalar potential is given by where the F -terms are In the above we used together with the definition of a i in (16) in combination with We also parametrized the meson field φ as and fixed one combination of the axions and the meson phase θ Before we proceed to constructing de Sitter vacua it is instructive to take a step back and consider a simpler case where the first non-perturbative term in the superpotential is also a pure gaugino condensate arising from a "pure glue" supersymmetric Yang-Mills theory. In this case one possible solution corresponds to a supersymmetric AdS extremum described by the following set of equations which is equivalent to Using the contraction property (17) we can find from (88) that the volume N · s of the hidden sector three-cycle can be determined by solving the following transcendental equation In the limit when N · s ≫ 1, the approximate solution is given by The moduli vevs can then be found from where the seven-dimensional volume is stabilized at Note from (91) that at the extremum, the periods of the co-associative four-form τ i ∼ N i up to a positive constant. Recall that here, N i are the integers representing the homology class of Q: where the harmonic three-form φ i is Poincare dual to a four-cycle [τ i ]. Therefore, from (91) we see that extremization of the supergravity scalar potential dynamically stabilizes the co-associative four-form * Φ to be proportional to the integral homology class of the associative three-cycle Q: Therefore, in the basis specified by (14) the integers N i must be positive definite In order to determine the values of a i at the minimum we substitute our expressions for s i (91) into the definition of a i in (16) to get a system of N transcendental equations, which then completely determine a i in principleK Note that the dependence on V Q in (96) is gone due to the scaling property of the volume V X . Hence, we have recast the problem of determining the moduli vevs at the minimum into a problem of determining the values of a i . Obviously, obtaining general analytic solutions for a i from (96) is impossible in practice, since V X has not been specified. However, precisely because the moduli vevs at the minimum are given by (91), it turns out that in order to compute the quantities relevant for particle physics, one does not need to know the values of a i explicitly. All one actually needs to know are the contraction properties (17) and (29). Therefore, the results we derive will be valid for any singular manifold of G 2 holonomy containing an associative three-cycle Q that contributes to the non-perturbative superpotential in a form of at least two gaugino condensates, whose integral homology class in the basis (14) is specified by positive integers. By explicitly checking in explicit toy examples, both numerically and analytically it seems that, for a given form of V X , an isolated solution indeed exists.
In principle, there exists an alternative way of determinimg a i more directly, although in the long run it may be more practical to solve the system (96). Namely, suppose one can reexpress the volume V X in terms of the dual variables τ i defined in (10). 6 With respect to τ i the volume V X is a homogeneous function of degree 7/4. Then, we find where we used the property that s i are homogeneous functions of τ i of degree 3/4 and the symmetry of the Jacobi matrix Then, using (16), (97) and (91) we obtain where in the final step we used the property that the "angular" variables a i do not scale. Then we can re-express the moduli vevs (91) as In order to illustrate how the system (96) is realized in practice we give a couple of explicit examples, though we stress that we have checked many more general examples than just those given here. Let us first consider a particularly simple N -parameter family of Kahler potentials consistent with G 2 holonomy where the volume V X is given by In this case the solutions to (96) are simply constants given by In fact, this example represents the class of Kahler potentials considered in the previous work [1] and the solutions are discussed in detail there. One may consider more complicated examples such as In this case system (96) translates into In these cases one can check numerically that, for very generic sets of parameters {n k i , c k , N i }, the system of equations (104) where for both examples which explicitly demonstrates that having positive solutions for a i is fairly generic and more importantly is guaranteed when V X is not just a randomly picked homogeneous function of degree 7/3 but represents an actual volume of a G 2 manifold X.
We now go on to consider de Sitter vacua by including the charged chiral matter fields Q andQ into the hidden sector. The superpotential and the Kahler potential are given by (74) and (78). In order to compute the scalar potential we need to compute the inverse Kahler metric. Using the Kahler potential (78) together with (16), (21), (83) and (84) we first obtain the following components for the Kahler metric Note that on the right hand side of the above expressions aj and ∆ ij are the same real quantities defined previously with index j replaced byj. The inverse Kahler metric must satisfy the following set of equations After a little bit of work we obtain the following components for the inverse Kahler metric Note that despite the fact that the matter part of the Kahler potential in (78) is only given up to the quadratic order in φ 2 0 V X , we decided to keep all the higher order terms inside the inverse Kahler metric. This is self-consistent as long as the combination φ 2 0 3V X appearing in the inverse Kahler metric is stabilized at a value sufficiently smaller than one such that the quartic and higher order terms are suppressed. Now, putting all the pieces together we obtain the scalar potential To understand the minima of the potential we will use the techniques developed earlier in [1]. Namely, we will work in the regime when the volume of the hidden sector associative cycle V Q = N · s is large and expand our solutions in the inverse powers of this volume. This is equivalent to an expansion in the UV weak hidden sector gauge coupling. In this long a tedious procedure we utilize the methods developed in [1], yet with some important modifications.
Since we are considering the simplified case by setting κ(s i ) = 1 in the Kahler potential for the effective meson field, the supersymmetry breaking F -term contributions are functions of V X and N · s only and therefore the scale invariant "angular" coordinates a i will remain the same as in the supersymmetric case. On the other hand, the "radial" coordinate parametrized by V Q (or V X ) will be shifted. Reintroducing the notation of [1] α we therefore make the following ansatz for the moduli vevs at the minimum In this notation, the volume of the associative cycles supporting the hidden sector gauge groups is given by in which case the moduli ansatz (112) can be rewritten as Let us first assume that L is non-zero and finite when y → 0. This assumption will be verified in this section by determining L explicitly. Then, we get from (113) and the definitions above This fixes the value of the volume V Q of the hidden sector three-cycle. We now go on to demonstrate that the ansatz for the moduli vevs (112) indeed represents the correct solution at the minimum of the scalar potential. In particular, we must verify our assumption that L is non-zero and finite in the limit y → 0 by determining L self-consistently in this limit. Hence, we will now derive the equation for L and demonstrate explicitly that one of the possible solutions is indeed non-zero and finite in this limit. After minimizing the potential with respect to the moduli s i and using the definitions (111) we obtain the following system of equations where in one of the intermediate steps we simplified Multiplying (116) by s k a k x 2 and using the explicit expression for the potential (110) in terms of the quantities (111) we obtain At first sight it appears that finding an analytic expression for L from (118) is hopeless since a closed form for (∆ −1 ) ij is unknown and a i have not been determined explicitly. However, upon further examination we notice that in order to find L from (118) we only need to know the contraction rules (17), (20) and (29). Indeed, using the the ansatz (112) together with the definition (18) and applying (17), (20) and (29) we first evaluate the terms and then use the same ansatz (112) and contraction identities for the rest of the terms in (118) to obtain the following equation for L Multiplying the above equation by 3 28 (1 + φ 2 3V X ) y 2 zx and taking the limit y → 0 we obtain where we dropped terms of O(y 2 ) and higher. A non-trivial solution can be obtained by solving the corresponding quadratic equation which is analogous to the equation in the second line in (126) of [1]. Solving (122) to the first subleading order in y results in Hence, we see that this solution is non-zero and finite when y → 0 and therefore is self-consistent. This is the solution describing the minimum of the potential. We must note that there is another possible solution of (122) for which L ∼ y → 0. In fact this other solution corresponds to the extremum at the top of the potential barrier and we will not discuss it further. Using (123) we can now compute the first subleading order correction to α to obtain Using (124) we can express the solution for L from (123) as In the leading order, the moduli vevs are given by .
We note that since a i , N i are positive, we need P ef f > 0 if Q > P , so that there exists a local minimum with s i > 0. The next step is to determine the vev of the effective meson field by minimizing the potential with respect to φ 0 . Let us first compute the potential at the minimum as a function of the meson. The result is given in equation (140) and the reader not interested in its derivation may proceed directly there. It turns out that since the moduli vevs at the minimum are proportional to a i /N i as in (114), explicit computation of the F -terms at the minimum and various contractions thereof while using the rules (17) and (29) becomes possible. Let us demonstrate some of these computations in detail. First we need to identify the gravitino mass in terms of our notation in (111). Using the usual definition in combination with (111) we have Because the existence of de Sitter vacua requires Q − P > 0 (see [1] for details) we obtain using (124) that On the other hand, since m 3/2 > 0 we can express the following combination in terms of the gravitino mass We now multiply F i in (81) by e K/2 and using (111) and (130) express where γ W denotes the overall phase of the superpotential. Using the ansatz (112) for s i we obtain from (131) where in the second line we used obtained from (113). Similarly, we find from (81) using (111) together with (130) Before computing e K/2 F i we would like to express the K ij components of the inverse Kahler metric at the minimum using the ansatz (112) for sj as follows Contracting (132) and (134) with the inverse Kahler metric and using the solution for L from (123) we then obtain where in the last line we used (124) to plug into x, y, and z defined by (111) except for the combination 1 + aαV X φ 2 0 x and kept the leading term in 1/P ef f . Note that in order to get from the second to third line in (167) we used the second contraction property in (29). Similarly, contracting (132) and (134) with the corresponding components of the inverse Kahler metric (109) we obtain Using the results (132), (134), (136) and (137) together with (123) and (124) we can compute the following contributions where we also used N · s = V Q while performing the computations in the first line of (138). Then, the potential at the minimum is given by Using (124) and dropping the terms of order O(1/P 2 ef f ) we obtain the following expression for the leading contribution to the vacuum energy as a function of the meson field The polynomial in the square brackets in (140) is quadratic with respect to the canonically normalized meson vev squared φ 2 c ≡ φ 2 0 /V X with the coefficient of the (φ 2 0 /V X ) 2 monomial being positive (+1) and therefore, the minimum V 0 is positive when the corresponding discriminant is negative. Tuning the cosmological constant to zero is then equivalent to setting the discriminant of the above polynomial to zero, which boils down to a simple condition Note that P ef f defined in (125) is actually dependent on φ but because of the smallness of a and the Log dependence, it was safe to use the approximation P ef f ≈ const. This approximation turned out to be self-consistent since P ef f is fairly large. From (141) we see immediately that Minimizing (140) with respect to φ 2 c and imposing the condition that the expression in the square brackets in (140) is tuned to zero, we obtain the meson vev at the minimum in the leading order .
If we tune the leading contribution to the vacuum energy and set Q − P = 3 we obtain Recalling the factor of two in the definition of the meson field (57) we note that along the D-flat direction, the bilinears of the canonically normalized charged matter fields that appear in the original Kahler potential have a somewhat smaller vev which makes it a bit easier to justify the truncation of the higher order terms in the Kahler potential for hidden sector matter. We find numerically that for the minimum value Q − P = 3, the tuning of the cosmological constant by varying the constants A 1 and A 2 inside the superpotential results in fixing the value of P ef f at while the canonically normalized meson vev squared is stabilized at thus confirming the analytical results above. For example, we obtain the values in (146) and (147)  It is instructive to compare the moduli vevs obtained above numerically with the values obtained by using the analytic expression (127). However, before we can apply (127) we need to determine the values of a i at the minimum. This can be done by plugging the values of N i , n k i and c k into the system (104) and solving it numerically. To compute s i from (127) we use P ef f from (146) in order to get a better accuracy. As a result, we obtain: which confirms explicitly that the analytic expression (127) for the moduli vevs at the minimum is indeed very accurate and reliable. Here we also verified that the Hessian of the volume has Lorentzian signature.
Although the value in (147) is not much smaller than one, the combination which is small enough to make the quartic and higher order terms which we kept inside the inverse Kahler metric much smaller. As we will see in the computations that follow, the value of P ef f will enter into many quantities relevant for particle physics, such as tree-level gaugino masses, etc. Here we note that small changes in P ef f do not lead affect the supersymmetry breaking masses much, but do change the cosmological constant significantly. For instance, while changing the value of P ef f in the range 61 ≤ P ef f ≤ 62 hardly affects the values of the soft breaking terms, as will be evident from the corresponding explicit formulas, such small changes in P ef f result in vastly different values of the vacuum energy: Therefore, once we coarsely tune P ef f to its approximate value, the cosmological constant problem becomes completely decoupled from the rest of particle physics. Even though this should be the case, it is satisfying to see it explicitly in a complete example of moduli coupled to matter. Note also that in the original paper [1] we obtained P ef f ≈ 83. This is due to the different matter Kahler potential considered there. As we will see, this numerical difference will result in slightly different values for the soft breaking terms if compared to those obtained in [1], [2].
Recall that for a stable minimum to exist it is necessary that Q − P ≥ 3. We have seen that when Q−P = 3 and the minimum of the potential is approximately tuned to zero, the value of P ef f ≈ 60 which ensures that the moduli (127) can be reliably fixed at values large enough to satisfy the supergravity approximation. On the other hand, when Q − P = 4, from (141) we get P ef f ≈ 20, in which case if Q is fixed, the moduli vevs become smaller by about a factor of four. Thus, unless the ranks of hidden sector gauge groups are incredibly large, situations when Q − P > 3 may put our solutions well outside of the supergravity approximation. Therefore, from now on we will only consider the case when Q − P = 3 to ensure the validity of the regime where our construction is reliable.

A. Gravitino mass
In supergravity the bare gravitino mass is defined as and can now be computed since we stabilized V Q explicitly. It is given by When the cosmological constant is tuned such that (146) is satisfied, for Q − P = 3 we obtain where C 2 ≡ A 2 /Q was defined in (56). Calculating C 2 goes beyond the scope of this paper. Here we will treat C 2 as a phenomenological parameter with values C 2 ∼ O(0.1 − 1) since it may experience a mild exponential suppression as in (56).
On the other hand, the actual value at which the volume V X must be stabilized can be almost uniquely determined from the scale of Grand Unification. In particular, we can use equation (4.12) in [12] to express where the factor L(Q) is due to the threshold corrections from the Kaluza-Klein modes and is given by such that 5w is not divisible by q. For typical values we obtain In Table I we list a few typical benchmark values for the volume and the resulting gravitino mass up to the overall factor C 2 .
Interestingly, the gravitino mass scale naturally turns out to be constrained to m 3/2 ∼ O(10)TeV. While this is presumably large enough to alleviate the gravitino problem, it is also small enough to give some of the superpartners masses which can be easily accessible at the LHC energies. As we will see below, this is possible because of the significant suppression of the tree-level gaugino masses relative to m 3/2 .
In addition to the gravitino mass it is instructive to compute the scale of gaugino condensation. Using (146) the volume of the hidden sector associative cycle for Q − P = 3 is given by From (51) the scale of gaugino condensation in the second hidden sector is

B. Moduli masses
In order to compute the masses of the moduli we first need to evaluate the matrix V mn with m, n = 1, N + 1, with components given by at the minimum of the potential. However, because the Kahler metric in (107) is not diagonal, we also need to find a unitary transformation U which diagonalizes the Kahler metric. We denote all the components of the Kahler metric as K mn . Then, by diagonalizing K mn we obtain After that, we need to rescale the fluctuations of the moduli around the minimum by the corresponding 1/ √ 2K k factors so that the new real scalar fields have canonical kinetic terms. At the end, finding the moduli mass squared eigenvalues boils down to diagonalizing the following matrix Unlike most of the other masses, the detailed form of the moduli mass matrix does depend upon the detailed form of V X . Therefore we have resorted to numerical analyses in this case and found that there is one heavy modulus whose mass mainly depends on Q and for Q = 30 and N lighter moduli with masses The heavy modulus arises from the fluctuation which deforms the volume of the three-cycle V Q , while N -1 light moduli originate from the fluctuations approximately preserving the volume and tangential to the hyperplane defined by The remaining light modulus represents the fluctuations of the hidden sector meson φ mixed with the geometric moduli.

C. Gaugino masses
The universal tree-level contribution to the gaugino masses can be computed from the standard supergravity formula [24] m tree where the visible sector gauge kinetic function is another integer combination of the moduli Note that, since the dominant F -term is that of the meson field, the gaugino masses at tree level will be suppressed wrt the gravitino mass. Since the scalar masses typically get contributions of order m 3/2 the expectation is to have light gauginos and heavier scalars, as we will indeed verify shortly.
Plugging the solution for α (124) into (136) while using the definitions (111) we obtain where we dropped the overall phase factor e −iγ W . It is now straightforward to compute the tree-level gaugino mass It is interesting to note that this formula is identical to the leading order expression previously obtained in [1] when one replaces the combination φ 2 0 /V X by the canonically normalized meson field. Here, again the suppression coefficient is completely independent of the number of moduli N as well as the integers N i (N vis i ) appearing inside either the hidden sector (76) or the visible sector (166) gauge kinetic functions. Moreover, all the detailed dependence on the individual moduli is completely buried inside the volume V X and the gravitino mass m 3/2 (which also depends on V X ) and therefore expression (168) is universally valid for any G 2 manifold that yields positive solutions of the system of equations in (96). Hence, despite the presence of a huge number of unknown microscopic parameters, the tree-level gaugino masses in (168) depend on very few of them. Moreover, when the cosmological constant is tuned to a small value and Q − P = 3, the gaugino mass suppression coefficient becomes completely fixed! Indeed, using (146) and (147) for Q − P = 3 we obtain This result gets slightly corrected by the threshold corrections to the gauge kinetic function from the Kaluza-Klein modes computed in [12] α −1 In the above formula, T ω is a topological invariant (Ray-Singer torsion) where w and q are integers such that 5w is not divisible by q. In this case, the tree-level gaugino mass is given by where

D. Anomaly mediated contribution to the gaugino masses
Because of the substantial suppression of the universal tree-level gaugino mass, it makes sense to take into account the anomaly mediated contributions which appear at one-loop. The anomaly mediated contributions are given by the following general expression [25]  Plugging the solution for α (124) into (137) while using the definitions (111) we obtain where we dropped the overall phase factor e −iγ W . Combining (167), (176) and using (78), (72), (73) and (61) we now compute the contributions In the above we also used (83) and (84) together with the definition of a i in (16) as well as its contraction property (17). We also dropped unknown subleading contributions proportional to Using the definition (174) we then obtain the following expression for the anomaly mediated contributions to the gaugino masses where we have explicitly separated the conformal anomaly contribution from the Konishi anomaly term using (73). Notice the appearance of the function c(s i ) which controls the size of the higher order corrections to the matter Kahler potential. As expected, when λ = 0 the Konishi anomaly vanishes in the exactly sequestered case [26], i.e. when c(s i ) = 1. Again, in the leading order in 1/P ef f , when c(s i ) = 0 the result obtained above is almost the same as the one in [1]. Just like in the case with tree-level gaugino masses, the above result is completely independent of the detailed moduli dependence of the volume V X and therefore is completely general. In what follows, we will regard the value of the function c(s i ) at the minimum of the scalar potential as a phenomenological parameter When we set λ = 0 and Q − P = 3, tune the leading contribution to the vacuum energy by imposing the constraint (146), use (147) and combine the above formula with the tree-level contribution (172), we obtain the following expression for the total gaugino masses Note that as was previously pointed out in [26], in the limit when c → 1 we obtain a particular type of a mirage pattern for gaugino masses [27]. However, as we will see below, in this limit the scalars become tachyonic and therefore, the exact mirage pattern is disfavored. An exact numerical computation confirms the above result giving Substituting the MSSM Casimirs (175) into (181) we then obtain The form of (182) allows us to see explicitly that for c = 0 the Konishi anomaly contribution is larger than the contribution from the conformal anomaly by a factor of a few, which is what made the gaugino mass spectrum in [2] very different from other known patterns. However, as we will see below, suppressing the scalar masses relative to the gravitino mass by tuning the coefficient c will automatically result in a large suppression of the Konishi anomaly.

E. Scalars
The masses of the unnormalized scalars can be computed from the following general expression [24] m ′2 Since the vacuum energy is tuned to zero we set V 0 = 0 in the above. Using (68), (78), (83), (84), the contraction properties (8), (9), (61) and the F-terms (167), (176) we obtain from (183) in the leading order where for consistency reasons we only kept contributions linear in c(s i ) and dropped unknown subleading terms proportional to the derivatives of c(s i ), such as e.g.
In the above derivation we also used the following properties Notice that despite the presence of the derivatives of the Kahler metricK αβ in the definition (183), the final expression (184) containsK αβ only as an overall multiplicative factor. This happened because the moduli F-terms (167) up to a phase are essentially given by e K/2 F i = 2s i × m tree 1/2 and the matrix κ αβ (s i ) is a homogeneous function satisfying (61). Therefore, diagonalization and canonical normalization of the corresponding kinetic terms automatically results in universal masses for the canonically normalized scalars After setting λ = 0, we tune the leading contribution to the vacuum energy by imposing the constraint (146) and use (169) to obtain from (186) where we again treat the value of the function c(s i ) for a given vacuum as a phenomenological parameter (179). A numerical computation in this case gives excellent agreement Again, for c = 0 we recover the old result in [1] where all the scalars have a flavor-universal mass equal to the gravitino mass. Furthermore, the anomaly contributions to the scalar mass squareds are suppressed relative to the gravitino mass and since we wish to consider generic O(1) values of (1 − c) we will neglect such contributions. Concretely we are going to consider only those values of 0 < c < 1 which give such that the anomaly mediated contributions to the scalar masses can be safely neglected. However, one can certainly extend our model and include such contributions. Once again, the result above is completely independent of the details of V X and therefore holds for any G 2 manifold that solves the system (96) with a i > 0 such that the Kahler metric at the minimum is positive definite.

F. Trilinear couplings
The unnormalized trilinear couplings for the visible sector fields can be computed from the following general expression [24] where {α , β , γ} label visible sector matter fields and Y ′ αβγ are the unnormalized Yukawas that appear in the superpotential. Recall that the Yukawa couplings Y ′ αβγ arise from the membrane instantons wrapping associative cycles Q αβγ , which connect isolated singularities supporting the corresponding matter multiplets. They are given by The integer combination of the moduli V Q αβγ = i m αβγ i s i gives the volume of the associative cycle Q αβγ connecting co-dimension seven singularities α, β and γ where the chiral multiplets are localized. The coefficients C αβγ are constants. The relation between the physical and unnormalized Yukawa couplings is given by Using (167) and (191) we can compute the contribution Similarly, using (68), (78), (83), (84), the contraction properties (8), (9), (61) and the F-terms (167), (176) we find where for consistency reasons we only retained contributions linear in c(s i ) and dropped unknown subleading terms proportional to e K/2 F i ∂ i c(s i ) ∼ m tree 1/2 s i ∂ i c(s i ). Again, in the above expressions we did not display the overall phase factor e −iγ W . Using the definition (190) along with (177), (193) and (194) we obtain the following expression for the physical (normalized) trilinear couplings at tree-level which gets reduced to the result in [1] when c = 0 and λ = 0. Once again, the detailed structure of the volume V X played absolutely no role in our ability to obtain the above expression for the tree-level trilinear couplings. The actual volumes of three cycles V Q αβγ do depend on the microscopic properties of G 2 manifolds and in our general framework these parameters remain undetermined. However, below we will present a good argument for dropping such volume contributions completely when the third generation trilinear couplings are computed. Setting λ = 0 and Q − P = 3, when the leading contribution to the vacuum energy is tuned we obtain for the reduced trilinears (the physical trilinears divided by the physical Yukawa couplings) From the corresponding numerical calculation we obtain the following result Since the physical Yukawa couplings for the third generation fermions are much larger than the first two generation Yukawas, one can typically neglect the trilinears for the first and second generations. Moreover, the large size of the third generation Yukawas implies that the volumes of the three-cycles of the corresponding membrane instantons are very small. In fact, because the top Yukawa is of order one, one can assume that the point p 1 supporting the up-type Higgs 5 of SU (5) coincides with the point p 2 supporting the third generation 10, so that the coupling H u 10 3 10 3 has no exponential suppression [5], [12]. At the same time the point p 3 supporting the down-type5 Higgs and the point p 4 supporting the third generation matter5 are distinct but still close to p 2 , p 1 so that the coupling of H d53 10 3 which accounts for the bottom(tau) Yukawa is slightly smaller than the top Yukawa at the GUT scale. These considerations completely justify dropping the corresponding V Q αβγ terms for the third generation trilinears which then become simplified For generic values of c the trilinears are of the same order as the gravitino mass. In the limit c → 1, the reduced trilinear couplings at tree-level become suppressed relative to the gravitino mass. Note that as c approaches one, the suppression of the trilinear couplings above is much stronger than that of the scalars. In this case, the anomaly-mediated contributions may become comparable to the tree-level ones and therefore must be taken into account. General expressions given in [28] can be simplified in the nearly sequestered limit as where the last term denotes the unknown contributions vanishing in the sequestered limit. Note that such terms are suppressed compared to the tree-level piece (198) due to the loop factor. As long as (1 − c) is small enough, they become subleading and we will drop them in further analysis. Using (177) and substituting the corresponding MSSM expressions for γ a s, where we set g 1 = g 2 = g 3 = g GU T , we obtain the following expressions for the anomaly mediated contributions to the reduced trilinear couplings When we set Q − P = 3, tune the tree-level vacuum energy by imposing the constraint (146), use (147) and combine the above formula with the tree-level contribution (197), we obtaiñ Numerical computations give the following expressions for the total reduced trilinears which demonstrate a fairly high accuracy of the analytically derived result in (201).

G. µ and Bµ -terms
The full hidden sector plus visible sector Kahler potential and superpotential can be written in the following general form Here, φ denote the hidden sector matter fields while Q α are visible sector chiral matter fields wherẽ K αβ (s i , φ,φ) is the visible sector Kahler metric and Y ′ αβγ are the corresponding unnormalized Yukawa couplings. It can be shown that the supersymmetric mass parameter µ ′ can be forbidden by requiring certain discrete symmetries which are also used in order to solve the problem of doublet-triplet splitting [29]. Hence, in our analysis we will rely on the Giudice-Masiero mechanism [30] in generating effective µ and Bµ terms where the bilinear coefficient Z αβ (s i , φ,φ) in (203) plays a key role. The general expressions for the normalized µ and Bµ are given by [24] where we can set µ ′ = 0. Unfortunately, at this point we do not have a reliable way to compute the Higgs bilinear Z αβ (s i , φ,φ) for G 2 compactifications. Therefore, in our analysis we will parameterize the µ and Bµ terms as follows and treat Z 1 ef f and Z 2 ef f as phenomenological parameters. Naturally, we expect that Z 1, 2 ef f ∼ O(1) and, as we will see in the next section, tuning µ parameter in order to get the correct value of the Z -boson mass boils down to tuning the values of Z 1, 2 ef f .

VI. GENERALIZATION TO THE CASE WHEN κ(s i ) IS A NON-TRIVIAL HOMOGENEOUS FUNCTION
Recall that the above results have been obtained assuming that the factor κ(s i ) appearing in the Kahler potential (78) of the hidden sector matter is a pure constant. In this section we briefly outline the main results for the case when κ(s i ) is a general homogeneous function of the moduli of degree zero, satisfying the property (31). Here we will not give any explicit analytic derivations (these are quite tedious and would mostly resemble the computations in the preceding sections) and instead present numerical evidence that most of results obtained for the simplified case κ(s i ) = 1 can be directly extended to the more general scenario.
The main difference from the previous case is in the form of the moduli vevs at the minimum of the scalar potential. These are now given by where r ≈ 3/2 when Q − P = 3. Note that the vev φ 2 c of the canonically normalized effective meson field at the minimum is given by the same expression as in the case when κ(s i ) = 1: where P ef f is exactly the same as in (141). Keep in mind that the analytic expression (207) is only valid when the leading contribution to the vacuum energy is tuned to zero. Parameters c i are defined as and satisfy because κ(s i ) is a homogeneous function of degree zero. At the minimum of the potential parameters a i and c i can be determined by solving a system of 2N coupled transcendental equations Once again, the structure of the soft supersymmetry breaking terms remains virtually unchanged in the leading order in 1/P ef f expansion and does not depend on the precise details of the function κ(s i ). In fact, the only modification compared to the previously derived expressions is the replacement of the following combination whose vev at the minumum is given by (207) and is exactly the same as before! However, it turns out that the soft supersymmetry breaking terms now become slightly sensitive to the compactification details via the subleading corrections. The most sensitive parameter is the tree-level gaugino mass. Up to an overall phase it is given by where we introduced a phenomenological quantity δ ∼ O(1) in order to parametrize the additional correction. In the numerical toy examples we studied, we obtain a O(1 − 10)% variation in the value of the tree-level gaugino mass, while the other soft terms vary by less than 1%. For the sake of completeness we shall list the expressions for the remaining soft breaking terms where we defined To illustrate the high accuracy of the analytical results presented here we present a simple toy example with two moduli.
For the following choice of the parameters in the superpotential where A 2 was tuned to cancel the leading contribution to the vacuum energy, we obtain numerically Notice that the above values of φ 2 c and P ef f are extremely close to those in (147) and (146), obtained numerically for the case when κ(s i ) = 1. These values are also in very good agreement with the corresponding analytical results. By solving the system (210) for the above example we obtain which demonstrate a mild dependence of the tree-level gaugino mass on the compactification-specific details. Using φ 2 c ≈ 0.746 together with P ef f ≈ 61.68 in the analytic formula (212) for Q − P = 3 we obtain which is in fairly good agreement with the numerical results. The numerical results for the remaining soft terms are given by the following expressions and are virtually unchanged compared to the numerical results in (181), (202) and (188), computed for the case when κ(s i ) = 1. The values obtained from the corresponding analytic expressions are in good agreement with the numerical values above and give essentially the same results as in the case when κ(s i ) = 1 because the values of P ef f and φ 2 c barely changed. Thus, the effect of including a non-trivial function κ(s i ) in the Kahler potential for the hidden sector matter fields can be reliably described by a single parameter δ that appears in the subleading contributions to the tree-level gaugino mass, while the remaining soft terms stay essentially unaffected.

VII. ELECTROWEAK SCALE SPECTRUM
In order to obtain the corresponding MSSM spectrum at the electroweak scale we need to RG-evolve all the masses and couplings from the GUT scale down to the electroweak scale. This procedure was described in great detail in [2]. Here we will only highlight a few important points and give the final results.
As we have seen in the previous section, at the GUT scale, the gaugino masses are non-universal and highly suppressed relative to the gravitino mass. On the other hand, unless c is very close to one, the scalars, trilinear couplings and the µ-term are all of order m 3/2 . Hence, we can define a scale m s at which all the heavy states decouple and the effective theory below that scale is the Standard Model plus gauginos. More specifically, we can choose the decoupling scale m s to be the geometric mean of the stop masses This is okay as long as the mass differences between the lightest stop and the other heavy states is not too large. Then, the running can be done at one loop in two stages with tree-level matching at the scale m s . This method, however, does not capture the two-loop effects, which may give significant contributions to the running. Thus, in what follows we will utilize the SOFTSUSY package [31] and perform the running at two-loops with the full the MSSM spectrum and accound for the effects from the heavy scalars via threshold corrections.

A. Gauginos
As one notices from (222), due to the anomaly mediated contribution, the gaugino masses are sensitive to the value of α GU T . However, the value of α GU T is only determined once we know the exact spectrum and run the gauge coupling up to the GUT scale. Therefore, there is a feedback mechanism, which allows us to completely fix the gaugino masses by imposing the gauge coupling unification. In practice, we first pick an initial value of α GU T ∼ 1/25, compute the gaugino masses, scalar masses, trilinears, etc. at the GUT scale and run them down to the electroweak scale where we compute the spectrum. We then run the gauge couplings up using two-loop RGEs to check if they unify at the same value of α GU T as we chose to compute the gaugino masses. If there is disagreement, we change the value of α GU T by a small increment and repeat the steps until there is a match. In addition, parameter η which appears inside the gaugino masses and was defined in (173) can be safely set to one. This is because as one varies the integers w and q inside (171) over a reasoble range, the torsion, unless specifically tuned, is so small that that the KK threshold corrections can be neglected.
Since M Higgsino ∼ µ ∼ O(m 3/2 ), there is a substantial threshold contribution from the Higgs-Higgsino loops which has to be taken into account when computing bino (M 1 ) and wino (M 2 ) masses [2], [32][33][34]: In the above expression we expanded the logarithm using µ 2 m 2 A ∼ 1 and used tan β ∼ O(1). The latter is especially true when 2 Z 1 ef f ≈ Z 2 ef f . We also relied on the fact that the supersymmetric µ-term almost does not change with the RG evolution so one can use (205). Since m 3/2 ∼ O(10)TeV, the above correction to M 2 can be as large as a few hunrded GeV. It turns out that this contribution is intimately related to the value of parameter c that directly affects the scalar masses and indirectly forces the value of µ to get smaller, as the scalars get lighter. For 0 ≤ c < ∼ 0.05 and 0.8 < ∼ µ/m 3/2 the contribution (224) is actually large enough to completely alter the nature of the LSP depending on the sign of µ. In this respect, the sign of δ that parametrizes the subleading corrections to the tree-level gaugino mass (221) also plays an important role. In particular, from the left plot in Fig 1 where we picked c = 0 there is a region where δ < ∼ −0.7 such that the LSP is Wino-like, while for δ > ∼ −0.7 the LSP becomes mostly Bino. Furthermore, as one can see from the plot, there exists a small range of values where M 1 and M 2 become nearly degenerate. This is certainly an intriguing possibility, which may provide for a well-tempered neutralino candidate [33]. Note that in the Wino-like LSP case, the lightest chargino and neutralino are degenerate at tree-level, i.e. χ 0 1 = χ ± 1 = M 2 . However as we take into account the 1-loop contribution from the gauge bosons [34], this degeneracy is removed, as is seen from the corresponding entries in Table  the relic density to acceptable levels by increasing the annihilation crossection of the LSPs. It turns out that for generic values of c, 0 ≤ c < 1, the higgsinos are always much heavier than the gauginos. This is because at the decoupling scale m s , the µ 2 -term must be of the same order of magnitude as m 2 Hu to give a correct value of the Z-boson mass, and since for typical values of c we get |m 2 Hu | >> M 2 1, 2 , the higgsinos do not mix with gauginos.
The only way to make the higgsinos light is to consider values of c very close to one. In this limit both scalars and higgsinos also become highly suppressed relative to the gravitino mass. As we will see in the analysis of this limit, the higgsinos can become as light as the gauginos and for certain values of tan β can give a well-tempered LSP candidate.

B. Squarks and sleptons
Recall that at the GUT scale all the squarks and sleptons have a universal mass (188), which for generic values of c (0 ≤ c < 1) is smaller but nevertheless typically of the same order of magnitude as the gravitino mass. However, as we evolve these down to the electroweak scale, the third generation scalars become significantly lighter whereas the first and second generation scalars experience a very mild change in their masses. Indeed, because the third generation Yukawa couplings are large, the stops, sbottoms, and staus are affected through the corresponding trilinear couplings (202), which are of O(m 3/2 ). As one can see from Table II, this effect is especially dramatic for the lightest stopt 1 . Yet, it is still much heavier than the gauginos and is effectively decoupled from the spectrum at the electroweak scale.
However, since gluinos can be pair produced at the LHC via gluon fusion, the gluinos (which have to decay via a quark-squark pair) have a sizeable branching fraction into top-stop -precisely because the stop is the lightest squark. This leads to events containing up to four top quarks at the LHC [39] Table III contains six benchmark points generated by SOFTSUSY. To compute the thermal relic density we used SOFTSUSY generated output as input for the micrOMEGAs program [38]. Note that the first benchmark point is experimentally excluded by the lower bound on the Higgs mass. As one can see from the entries for M 1 and M 2 , the LSP is mostly Bino. Thus, for a generic value of tan β the thermal component of the relic density turns out to be much larger than the observed value Ω c h 2 ≈ 0.1099±0.0062 [40]. However, for large values of tan β there is a window where the Higgsino component of the LSP can be large enough to reduce the relic density. Hence, for each set of m α and m 3/2 we can tune tan β so that the value of Ω c h 2 is compatible with experiment. For example, Points 2 and 3 with m α = 800 GeV and m 3/2 = 16 TeV both satisfy the experimental bound on the relic density but have different values of tan β, i.e tan β = 43 and tan β = 38. In this case, if tan β is within a window 38 < ∼ tan β < ∼ 43, the thermal component of the relic density is too low and one may have to appeal to non-thermal mechanisms [3]. Note that the SU (5) GUT scale relation Y b ≈ Y τ automatically holds in our construction. However, for large tan β the values of all third generation Yukawa couplings become comparable. Namely, for the benchmarks in Table III, we obtain