A Realistic Unified Gauge Coupling from the Micro-Landscape of Orbifold GUTs

We consider 5-dimensional supersymmetric field theories where supersymmetry is broken by the Scherk-Schwarz mechanism (or, equivalently, by the F-term VEV of the radion). In such models, the radion effective potential is calculable in terms of the 5d gauge coupling, the UV cutoff of the 5d field theory, and the field content. We provide simple, explicit formulae for the leading part of the two-loop effective potential. Our analysis applies in particular to 5d orbifold GUTs motivated by heterotic orbifold models. We focus on potentially realistic models of this type and make the additional assumption that the UV cutoff scale is identical with the strong-coupling scale of the 5d gauge theory. Given our stabilization mechanism, the 5d radius is now fixed in terms of the 5d gauge coupling and the field content of the model. This implies a prediction for the effective 4d gauge coupling only in terms of the field content of the model. Given the `micro-landscape' provided by the different possible distributions of Standard Model fields between bulk and branes, we find a subset of models with a realistic unified gauge coupling. We also discuss two possibilities for the `uplifting' of our SUSY-breaking AdS vacua: One is based on the possible presence of a weak warping, the other appeals to F-terms in an extra brane-localized SUSY-breaking sector.


Introduction
Supersymmetric grand unification at a scale of ∼ 10 16 GeV is one of the best motivated proposals for physics beyond the standard model [1,2]. It fits rather naturally into the framework of heterotic string theory, where a large class of potentially realistic constructions with gauge coupling unification can be obtained in orbifold model building [3] (see [4] for a recent review). One of the possibilities for overcoming the string-scale GUT-scale problem [5], which generically affects these scenarios, is the compactification on anisotropic orbifolds [6][7][8][9], where at least one of the compactification radii is much larger than the string length scale. Such models have a useful effective description in terms of higher-dimensional field theories valid at energies between the string scale and the compactification scale. They are also known as orbifold GUTs which provide, independently of a possible string-theoretic UV completion, some of the simplest realistic models of grand unification [10][11][12].
It is therefore essential to understand possible stabilization mechanisms for the largest compact dimensions at a quantitative level. Here, we focus on the simple case of 5d supersymmetric gauge theories on S 1 /Z 2 or S 1 /(Z 2 × Z ′ 2 ) with hypermultiplets in the bulk and chiral matter localized at the boundaries. In orbifold GUTs of this type, the 4d gauge coupling is given by where g is the 5d gauge coupling and R is the compactification radius. 1 It will be instructive to rewrite this relation in terms of the parameters g 2 4 N 16π 2 and which govern the perturbative series of an SU(N) gauge theory in 4 and in 5 dimensions [13]. 2 Note that in the 5d case, loop corrections are proportional to positive powers of (Λ/M), where Λ is the cutoff scale. Hence M can also be viewed as the 'fundamental scale' or 'strong-coupling' scale of the 5d theory: It is the highest scale to which the cutoff can be raised in perturbative effective field theory.
In terms of the proper expansion parameters given in Eq. (2), the expression for the 4d gauge coupling, Eq. (1), takes the form This formulation shows a rather precise connection between 4d and 5d perturbativity: A strongly coupled 4d effective theory emerges when the compactification scale is raised to 1 We view the 4d theory as resulting from a projection applied to the 4d spectrum of an S 1 -compactified 5d theory. Hence the S 1 volume 2πR rather than the orbifold volume πR or πR/2 appears. 2 It has recently been shown [14] (footnote 4) that considering specifically the vacuum polarization and using Pauli-Villars regularization, the 5d loop suppression factor is actually 24π 2 rather than 24π 3 . This may not be completely unexpected since in odd dimensions, after the usual rewriting 2 |k| d−1 d|k| = (k 2 ) d/2−1 dk 2 , there appears a root in the integrand of the loop integral, leading to a factor of π. We continue to use the standard loop suppression factor, keeping in mind that a more detailed analysis may be necessary if one wants to fix the strong-coupling scale M more precisely. the 5d strong-coupling scale M. Hence, when it comes to numbers, it is more convenient to think of 1/M rather than of g 2 as of the parameter defining the 5d gauge theory.
For the phenomenological value α GUT ≃ 1/25 and an SU(5) gauge group, the l.h. side of Eq. (3) takes a value ≃ 1/60. Thus, we need a corresponding hierarchy between 1/R and M. We will see that such a mild hierarchy 3 is relatively easy to achieve: Given the discrete set of models provided by different distributions of matter fields between bulk and branes, one finds many situations where the Casimir energy stabilizes the radius at the desired scale.
Before describing our specific results in more detail, we recall the generic situation: It is well-known that the compactification radius R (i.e. the radion field in 4d language) is a modulus at tree-level. Loop corrections lift the flatness of its effective potential [15]. If all bulk fields are massless, this 'Casimir energy' is ∝ 1/R 4 at one-loop order on dimensional grounds. Radius stabilization requires a more complicated functional form of the effective potential and hence the presence of another mass scale. This scale can be provided, for example, by warping [16], by massive bulk matter or by brane-localized kinetic terms for bulk fields [17]. These and other mechanisms for radion stabilization have also been discussed by many authors in the context of models with spontaneously broken supersymmetry (see e.g. [18][19][20][21][22]). In the present, orbifold-GUT motivated context, Casimir stabilization has recently been analyzed in 6d, using brane-localized soft terms and FI-terms to provide the required mass scale [23].
We base our analysis on the observation that Casimir stabilization can occur even in the minimal realistic setting of a 5d gauge theory [24]. If it does, one has more predictive power than in many of the more elaborate constructions mentioned above. The idea is simply to use the two-loop effective potential, which is of the form 1/R 4 + g 2 /R 5 for an S 1 compactification. For appropriate numerical coefficients, a perturbatively controlled minimum at relatively large R can arise. 4 For an S 1 /Z 2 or S 1 /(Z 2 ×Z ′ 2 ) orbifold, the twoloop contribution is enhanced by a factor ln(ΛR), where Λ is the UV cutoff scale of the 5d field theory. This enhancement originates in the UV divergence of brane localized gaugekinetic terms. For Λ ≫ 1/R, the logarithm is large and predictivity is maintained even in the presence of unknown tree-level brane operators (as long as they are not unnaturally large). In [24], these ideas were worked out in the case of S 1 for supersymmetric and non-supersymmetric models and in the case of S 1 /Z 2 , but without supersymmetry or gauge symmetry breaking by orbifolding.
If we make the assumption that the cutoff or UV-completion scale Λ takes its highest possible value, Λ ≃ M, the potential takes the form The numerical coefficients of the two competing terms, which have been suppressed for brevity, can have different signs and values. Their ratio, which depends only on the field content of the model, determines the position of the minimum. For appropriate field content, the minimum is at R ≫ M −1 , rendering our analysis self-consistent.
It is clear that the value of R at the minimum is proportional to g 2 or, equivalently, to 1/M. The proportionality factor is calculable in terms of the field content of the model. Hence Eq. (1) provides a prediction of the 4d gauge coupling, even though we cannot determine the values of M and R independently. Of course, there are good reasons to believe that R −1 is of the order of M GUT ∼ 10 16 GeV, which would require the 5d model to be characterized by M ≃ 45M GUT . However, we emphasize again that the overall uncertainties of these scales do not affect our prediction of g 4 . This prediction is based only on the quantity MR, which is calculable in terms of the gauge group, symmetry breaking pattern and matter content of the 5d orbifold model.
In the present paper, we analyze two-loop Casimir stabilization in the potentially realistic case of supersymmetric S 1 /Z 2 or S 1 /(Z 2 × Z ′ 2 ) orbifolds with gauge symmetry breaking by boundary conditions. Although both supersymmetry (with Scherk-Schwarz breaking [27]) and gauge symmetry breaking have a significant effect on the Casimir energy, the potential can be derived essentially without new loop calculations. This is achieved using simple arguments based on the N = 2 SUSY case and elementary group theory.
It is essential for our analysis that SUSY breaking is dominated by the F -term VEV of the radion superfield T , which contains R as the real part of its scalar component. This situation corresponds to Scherk-Schwarz breaking in the rigid SUSY approximation [28,29]. The SUSY breaking scale is proportional to the (dimensionless) Scherk-Schwarz twist parameter or, equivalently, F T . This is a small number, the square of which enters the radion potential as an overall prefactor. Hence, the precise scale of SUSY breaking is irrelevant for the position of the minimum.
We apply our general results to some SU(5) orbifold GUT models. We find that the possibility of Casimir stabilization depends crucially on the distribution of matter fields between bulk and branes. (This freedom corresponds to what we previously called the 'field content' of the model.) As described in more detail above, the stabilization radius in units of g 2 directly determines the value of the 4d gauge coupling at the compactification scale. Since we aim at potentially realistic models, we need to ensure that this value is consistent with the phenomenological value of the unified gauge coupling. Within the 'micro-landscape' arising from the possible localization of matter fields, several examples with a realistic gauge coupling can be found.
Our paper is organized as follows. In Sect. 2, we review the most important results of Ref. [24] and explain our strategy for determining the unified gauge coupling in more detail. In Sect. 3, we derive general formulae for the radion potential for supersymmetric gauge theories on S 1 /Z 2 with charged hypermultiplets in the bulk. This analysis is extended to situations with broken gauge symmetry in Sect. 4 and to S 1 /(Z 2 × Z ′ 2 ) compactifications with gauge symmetry broken at one of the branes in Sect. 5. We apply these results to simple realistic GUT models in Sect. 6 and identify models in which a realistic 4d gauge coupling is dynamically realized. Since the Casimir energy at the stabilization point is negative, some form of 'uplifting' is required. This issue is addressed in Sect. 7. The conclusions, given in Sect. 8, are followed by an Appendix, where we describe part of the underlying component field calculation in more detail. Although our results, as emphasized above, can be obtained without any explicit new loop calculations, we find this useful in view of a disagreement with some of the component-field results of [30].

Perturbatively controlled radius stabilization by Casimir energy
Let us first consider 5d gravity (with vanishing cosmological constant) and pure gauge theory, compactified on S 1 : The parameters M P,5 and g are defined in the uncompactified 5d effective theory at zero momentum. We view this as an effective quantum field theory in which the cutoff can be taken as high as the strong-interaction scale. Compactifying this theory on S 1 then corresponds to an IR modification which should not introduce any new infinities and hence no cutoff dependence. Thus, the 4d effective potential for R (i.e. the Casimir energy) can only depend on g and M P,5 . Gauge loop effects are suppressed by powers of g 2 /R, while gravitational loop effects are suppressed by powers of 1/(M P,5 R) 3 . Hence, the latter are subdominant in situations where 1/g 2 M P,5 (M P,5 R) 2 . Neglecting gravitational interactions, we have where c (i) is the coefficient of the ith loop order contribution. 5 As explained above, the c (i) s are cutoff-independent calculable numbers.
Given that c (1) < 0 and c (2) > 0, one finds a minimum at at two-loop order. 6 Unfortunately, if all c (i) s are O(1), the loop expansion parameter g 2 /R is also O(1) in the vicinity of R ∼ R min . Hence, this minimum is not perturbatively controlled.
However, in special cases where c (1) is negative and O(1) while c (2) is large and positive, the loop expansion factor g 2 /R is small for R close to R min . Higher-loop contributions to the effective potential are then suppressed by powers of g 2 /R. Perturbative control is guaranteed if the possible growth of c (i) (for i > 2) with i is overwhelmed by the increasing powers of g 2 /R. This is indeed easily realized in simple models [24].
In essence, this strategy of finding a perturbatively controlled minimum by tuning the coefficients c (1) and c (2) via the field content also applies to orbifold compactifications. 5 For simplicity, we work with the effective potential in the Brans-Dicke frame, i.e., we do not absorb the prefactor R of the Einstein-Hilbert term into the metric. Since we will eventually only be interested in models with vanishing cosmological constant, this will not affect the position of the minimum. 6 The Casimir energy V (R min ) = c (1) 5R 4 min is negative. In non-supersymmetric theories, one can simply introduce an appropriate brane tension (for models with branes or boundaries) in order to get a vanishing vacuum energy. In supersymmetric theories, this is less obvious, especially if one is not willing to compromise radion mediation as the dominant SUSY breaking mechanism. We will discuss this in detail in Sect. 7.
However, the power-law behavior of the effective potential, Eq. (6), is in general modified. The reason is the presence of brane-localized operators in the effective action. The prime example is a brane-localized contribution to the gauge-kinetic term. Such terms were first studied in the context of orbifold GUTs, where they can be employed to achieve gauge coupling unification by a modified logarithmic running above the compactification scale [11,[31][32][33]. These terms are logarithmically UV-sensitive. Unless the UV-completion of our model is known, the coefficients at the cutoff scale Λ are free parameters. For reasons of naturalness, we assume these to take O(1) values. However, their values at the scale M c = 1/R are enhanced by an additive contribution ∼ ln(Λ/M c ), the coefficient of which is calculable in the low-energy effective field theory. Thus, the calculability of the Casimir energy is unspoiled as long as Λ and M c are at least a few orders of magnitude apart.
The log-enhanced brane-operators discussed above affect the Kaluza-Klein mass spectrum, resulting in an extra contribution to the one-loop Casimir energy. This contribution is enhanced by ln(ΛR) because of the running of the brane-operators but suppressed by g 2 /R because it is a brane-effect. Alternatively, this can be viewed as a two-loop effect since it arises in the interplay of the one-loop running and the one-loop Casimir energy calculation. This point of view is also consistent with the fact that this contribution is proportional to g 2 . Due to the log-enhancement it dominates over the two-loop Casimir energy from the bulk (which is ∝ g 2 /R 5 ). In summary, the 4d effective potential at leading two-loop order has the form where c (br) (or, equivalently,c (br) ) is the coefficient of the brane-induced contribution. This coefficient will be determined for supersymmetric gauge theories in sections 3, 4 and 5.
As already discussed in the Introduction, we will assume that the cutoff scale Λ takes its highest possible value -the strong-coupling scale M of the 5d gauge theory (cf. Eq. (2)). Equation (8) then determines the stabilization radius in terms of M and the calculable ratio c (1) /c (br) . This is the basis of our explicit determination of the unified gauge coupling.
Obviously, Λ is in principle an independent parameter of our 5d effective theory. For example, in an orbifold compactification of the heterotic string, Λ would depend on the values at which the dilaton and the 5 remaining compact dimensions are stabilized. As a further constraint, we would have to require that the 4d Planck mass is correctly reproduced. As discussed in some detail in [8], the present setting with a relatively large 5th dimension and a maximally extended validity range of the 5d gauge theory is one of the more appealing options for solving this complicated problem. This may be viewed as an extra motivation for our assumption Λ ≃ M.
Even if Λ < M, the main message of the present analysis remains unchanged: Equation (8) will determine the compactification radius in terms of M, the field content, and Λ. The 'micro landscape' of orbifold GUTs will then allow us to tune the field content in such a way that a realistic 4d gauge coupling is obtained. In fact, Λ enters the Casimir energy only logarithmically and hence g 2 4 will also only have an (approximately) logarithmic dependence on Λ. Of course, our analysis breaks down if Λ is so small (i.e. the validity range of the 5d theory is so limited) that unknown O(1) terms are of the same size as ln(Λ/M c ).
3 Casimir energy for S 1 /Z 2 Before determining the Casimir energy let us briefly review 5d N =1 SUSY and its breaking by orbifold boundary conditions. The 5d vector multiplet (VMP) consists of a real vector A M , a real scalar Σ and a Dirac spinor λ, corresponding to two 4d Weyl spinors λ L , λ R . Under 4d N =1 SUSY, it decomposes into a 4d vector multiplet V = (A µ , λ L ) and a 4d chiral multiplet χ = (Σ + iA 5 , λ R ). The 'gauginos' λ L , λ R can also be written as an SU(2) R doublet of symplectic Majorana spinors which makes the SU(2) R symmetry of the theory manifest [34]. The 5d hypermultiplet (HMP) consists of an SU(2) R doublet of scalars H 1 , H 2 and a Dirac spinor ψ (which is equivalent to two Weyl spinors ψ L and ψ R ). Under 4d N =1 SUSY, it decomposes into a 4d chiral multiplet H = (H 1 , ψ L ) and another chiral multiplet H c = ((H 2 ) * , ψ R ) in the conjugate representation of the gauge group.
The Z 2 -parities of the fields can only be assigned consistently in a way that breaks 4d N = 2 SUSY to N = 1: Invariance of the action under Z 2 transformations demands V to be Z 2 -even and χ to be Z 2 -odd, while H and H c must have opposite parities. Hence, only V and either H or H c have Kaluza-Klein (KK) zero modes. The massive KK modes of the 5d VMP at each KK level form a 4d N = 1 massive vector multiplet (which has twice as many degrees of freedom (d.o.f.) as a massless 4d vector multiplet in Wess-Zumino gauge). On the other hand, the massive KK modes of H and H c form pairs of massive 4d chiral multiplets.
The residual SUSY can be broken by a Scherk-Schwarz twist. It has been shown that this leads to the same spectrum as in radion mediated SUSY breaking [28,29,35]. Hence the latter scenario can be viewed as a dynamical realization of Scherk-Schwarz breaking. The bosons A M , Σ in the VMP and the fermions ψ L , ψ R in the HMP are SU(2) R singlets and hence have 'untwisted' boundary conditions. On the other hand, the gauginos and hyperscalars are SU(2) R doublets which have a nontrivial twist-matrix T = exp(2πiωσ 2 ). Here, ω is the Scherk-Schwarz parameter (which can be identified with the radion F -term VEV) and σ 2 is the second Pauli matrix. As a consequence, the KK masses of the gauginos and hyperscalars receive a shift n/R → (n + ω)/R, which lifts the mass degeneracy of the 4d N = 1 SUSY multiplets (see Fig. 1). Even though the masses of bosons and fermions do not agree at any KK level, the UV-divergent part of the quantum corrections respects 4d N = 1 SUSY, which is locally unbroken. In other words, Scherk-Schwarz breaking is a global (=IR) effect which does not modify UV properties. In particular, the logarithmic divergences at the boundaries are supersymmetric, so that the resulting log-enhanced corrections to the KK masses are the same for all component fields within a 4d N = 1 supermultiplet.
To display the generic formula for the one-loop Casimir energy, we consider an SU(2) R doublet of complex scalars with opposite Z 2 parities on S 1 /Z 2 . Such a doublet forms the bosonic part of a hypermultiplet. The KK spectrum in the presence of a Scherk-Schwarz parameter ω (allowing also for ω = 0) is m Figure 1: KK masses of the components of the VMP and HMP. The arrows denote the states whose masses are shifted for ω = 0.
a Casimir energy where the factor 1/2 comes from the Z 2 projection, the factor 4 counts the d.o.f. of two complex scalars, and k E is a Euclidean d-momentum. The finite, R-dependent part of the above expression is where ζ ω (s) ≡ ∞ k=1 k −s cos(2πkω) is a generalization of the Riemann zeta-function ζ ω=0 . In a theory with Scherk-Schwarz SUSY breaking, each d.o.f. with ω = 0 has a superpartner with non-zero ω, so that the following quantity is useful: Note that c ω ≥ 0, since ζ(s) > ζ ω (s) for ω = 0.

Casimir energy for a pure gauge theory
The one-loop bulk coefficient for a pure gauge theory with gauge group G and supergravity is obtained by adding the contributions from all physical d.o.f., taking into account a minus sign for fermions and the respective Scherk-Schwarz parameter ω of each field. One finds 8 [36,38] Let us now determine c (br) . This contribution comes from brane-localized operators. Such operators induce a shift δm n of the KK masses. The relative mass shift ∆ ≡ δm n /m n 7 The one-loop vacuum diagrams can be split into an R-dependent finite part and a divergent part which is linear in R and represents a contribution to the 5d cosmological constant [15] (see also [37]). The latter cancels between bosons and fermions in supersymmetric models. 8 Let us fix here our group theory conventions. The generators T a R for an irreducible representation R are normalized such that tr(T a The quadratic Casimir operator is denoted by C 2 (R). For the dimension of the representation R we use the notation d R . The adjoint representation is denoted by R = G. We often use the identity d R C 2 (R) = d G C R . is independent of the KK level n. For one bosonic d.o.f. with tree level KK spectrum m (ω) n , the extra contribution to V (R) due to brane-localized operators is given by [24] To obtain the entire contribution to the Casimir energy, one has to sum over all d.o.f.
The brane-localized gauge-kinetic terms which are induced in case of a gauge theory are equivalent to a shift of the 4d effective gauge coupling g 4 by [11,[31][32][33]39] This in turn corresponds to a relative shift of the KK masses of the VMP by Note that, in this equation, a factor 1/2 arising from the fact that m 2 is corrected by 2∆ vmp cancels a factor 2 arising from the enhanced sensitivity of cosine-modes to brane terms as compared to the zero mode. Since ∆ vmp is the same for all d.o.f. of the VMP, using Eq. (13) and adding the contributions from all d.o.f. we find Using this result and Eqs. (12) and (15) one can easily read off Note that this is always negative so that perturbatively controlled radius stabilization cannot be achieved in a pure super-Yang-Mills theory.

Including hypermultiplets
Including the physical d.o.f. of HMPs, the one-loop bulk coefficient is [36,38] We now determine c (br) in the presence of HMPs. As mentioned before, c (br) is due to brane-localized operators induced by quantum fluctuations above the compactification scale. More precisely, c hmp denote the contribution from the one-loop selfenergy of the VMP respectively HMP. First of all, notice that, for both cases, the contribution from the HMP in the loop vanishes. To see why, recall that H and H c have opposite Z 2 parities. Fields with opposite parities however lead to brane-localized operators of opposite signs. 9 Hence, the contributions from H and H c cancel each other and only the VMP in the loop contributes to the selfenergy. This means that c (br) vmp , as given in Eq. (17), is unchanged. In the following we derive c (br) hmp . To start with, let us consider a HMP in the adjoint representation. A 5d supersymmetric gauge theory with a VMP and an adjoint HMP has 5d N = 2 SUSY, corresponding to N = 4 in 4d. 10 Now, note that one of the four 4d SUSY parameters is invariant under modding out by reflections and translations: Two of the SUSY parameters are even under Z 2 -reflections and one of these furthermore has no Scherk-Schwarz twist (cf. Fig. 1 where ψ L is an invariant Weyl spinor). This means that after compactification on S 1 /Z 2 the theory still has some unbroken SUSY so that the vacuum energy must vanish. Thus, the contribution of the adjoint HMP has to cancel that of the VMP: This corresponds to a KK mass shift ∆ adj.hmp = ∆ vmp .
Next we generalize this to a HMP in an arbitrary irreducible representation R. The only quantity that can change is the 'group theory factor' d G C G in Eq. (19). Clearly, there is more than one expression for a general R which in the special case R = G reduces to Eq. (19) (e.g. both C 2 (R) and C R reduce to C G ). The correct generalization of Eq. (19) is found by recalling that the brane-localized operators arise from the one-loop selfenergy of the HMP with a VMP in the loop. Since the corresponding 'coupling matrices' are given by (T a R ) ij , the mass shift for the HMP component with gauge group index i (the result has to be independent of i, of course, due to the unbroken gauge symmetry) is Hence, the correct generalization of Eq. (19) is The total contribution of both a VMP and a HMP takes the simple form As an example, consider the case G = SU(N) and h HMPs in the fundamental representation (so that C R → hC F = h/2). One can easily check that for any number h it is impossible to have c (1) = 2c ω (hN − N 2 − 1) < 0 and at the same time This situation may be improved if some of the gauge symmetry is broken. We discuss this in the following. of generators Tā of H and a set of 'broken generators' Tâ. The HMP representation R of G splits into k R k where each R k is a representation of H = H 1 ⊗ . . . ⊗ H n .
The one-loop bulk coefficient, Eq. (18), remains unchanged for S 1 /Z 2 compactifications with broken gauge symmetry. The reason is simply that the KK mass spectrum is unchanged in comparison to the unbroken case (even though the wavefunctions of the higher KK modes of some components of the VMP and HMP have flipped Z 2 -parity). As we will see in Sect. 5, this is different for S 1 /(Z 2 × Z ′ 2 ) compactifications, where also the one-loop bulk Casimir energy 'feels' the gauge symmetry breaking.
Since at the boundaries the gauge symmetry is broken, the boundary coefficient, Eq. (22), is modified. Let us first consider c (br) vmp . As argued in Sect. 3, only the VMP in the loop gives a contribution. Z 2 -invariance of the action implies that the structure constants f abc have an even number of 'broken indices' [31]. The brane-localized operators leading to c (br) vmp are thus determined from diagrams with the following gauge group indices: A crucial point is that the prefactor of the brane-localized terms induced by diagrams of type 1.) is minus that of diagrams of type 2.). The reason is the same which allowed us to argue that the contribution from the HMP in the loop vanishes: Fields with opposite parities give contributions of opposite signs. Thus, the mass shift of the unbroken VMP components with indexc is (up to a prefactor which doesn't depend on group theory indices) where we defined ηā = 1, ηâ = −1. Since the tree-level KK masses of all d.o.f. are the same, one has where we used ā,b fābĉfābĉ − â,b fâbĉfâbĉ = 0. We thus have 11 In a similar way we can infer c (br) hmp , which is due to the selfenergy of the HMP with a VMP in the loop. Let us split the indices {i} of the HMP representation into two sets 11 In order to calculate c (br) vmp one could also proceed as in Sect. 3 and use the shifts δ(g −2 4,i ) of the 4d gauge couplings of the unbroken subgroups H i , which can be extracted from Ref. [39]: One arrives at the result The above formula however depends on all factors H i of H and hence conceals the fact that really the only information from the gauge symmetry breaking entering the result is the number d H .
{ī} and {î} by defining that, say, Hī is Z 2 -even and Hî is Z 2 -odd. Then, only elements (T a R ) ij with an even number of 'hatted indices'â,î or are nonvanishing in order for the interaction terms to be Z 2 -invariant. Thus, the group theory factor is determined by: Note that, in contrast to VMP loops, the effect of HMP loops is non-zero even if the external index is Z 2 -odd (cf. diagrams 3.) and 4.)).
For the same reason as above for c (br) vmp , 1.) and 2.), respectively 3.) and 4.), have prefactors of opposite signs. Thus we have Note that for η a = 1 ∀a, both expressions reduce to C 2 (R).
In order to calculate the Casimir energy, we furthermore need to know the (relative) prefactors which are missing in Eq. (26). To this end recall that H and H c have opposite parities and H c transforms underR if H transforms under R. Thus, 1.) corresponds to even modes in representation R (from H) and odd modes in rep.R (from H c ), while 3.) corresponds to odd modes in rep. R (from H) and even modes in rep.R (from H c ). This shows that the interchange ↔ corresponds to R ↔R. Since (T ā R ) ij = −(T a R ) ji , the proportionality factors between ∆ī hmp , ∆î hmp and the r.h. sides of Eq. (26) are the same. Thus we have The final result is then c Adding c We stress that this does not depend on the details of the gauge symmetry breaking, but only on the dimension of H.
To illustrate this result we consider again G = SU(N) and h fundamental HMPs. One has C R − C G → h/2 − N in this case. This is negative, unless h ≥ 2N which would however imply c (1) > 0. Hence, (2d H −d G ) needs to be negative in order to obtain c (br) > 0 and c (1) < 0. The only possible breaking pattern of SU(N) by Z 2 inner automorphisms is SU(p + q) → SU(p) × SU(q) × U(1) [40] (see also [41]), for which 2d H − d G = (p − q) 2 − 1. This is negative only for p = q. A potentially phenomenologically interesting example for this is SU(6) → SU(3) × SU(3) × U(1) [42]. On the other hand, for the important case SU(5) → SU(3) × SU(2) × U(1), the term (2d H − d G ) is zero, i.e. the contribution of the bulk fields to the leading order two-loop term vanishes. 5 S 1 /(Z 2 × Z ′ 2 ) with broken gauge symmetry at one brane is obtained by modding out by a second Z ′ 2 parity. In this case, fields can have different boundary conditions at the two inequivalent fixed points. If some of the gauge symmetry is broken at both of the boundaries, there are additional massless fields at tree level besides the radion, namely the zero modes of some of the higher-dimensional components of the gauge bosons. 12 They acquire masses from radiative corrections [43]. In this section we restrict our attention to situations where the gauge symmetry remains unbroken at one of the branes. In that case the radion is the only modulus.

Pure gauge theory
As in the previous sections we start with a pure gauge theory. The (Z 2 , Z ′ 2 )-parities and the KK levels of the 4d superfields (V, χ) which form the VMP are shown in the following table (cf. [11,31,32] Note that the KK masses of broken and unbroken components of a multiplet are displaced. This is an important difference to the S 1 /Z 2 case, which has the consequence that also the one-loop bulk Casimir energy feels the breaking of the gauge symmetry as we will see. The Casimir energy for one bosonic d.o.f. on S 1 /(Z 2 × Z ′ 2 ) with KK spectrum 2(n + ω)/R is given by f (ω, R/2) while for one d.o.f. with KK spectrum [2(n + ω) + 1]/R it is f (ω + 1/2, R/2). Using the duplication formula Li s (z) + Li s (−z) = 2 1−s Li s (z 2 ) for polylogarithms Li s (z) ≡ ∞ k=1 k −s z k , one finds ζ ω+ 1 2 (5) = −ζ ω (5) + 1/16 ζ 2ω (5). This, together with the inverse quartic scaling of f (ω, R) with R, results in Fields with odd KK spectrum give an almost opposite contribution to the Casimir energy as fields with even KK spectrum.
As a simple check, for the unbroken case H = G this becomes c vmp = −32d G c ω , which is 16 times the result for S 1 /Z 2 . The relative factor of 16(= 2 4 ) arises, because the length of the physical space for S 1 /(Z 2 × Z ′ 2 ) is one half of that for S 1 /Z 2 .
We now determine the brane-coefficient. The UV-divergent contribution to the branelocalized operators is induced by fluctuations of the bulk fields 'close to' the brane. It can therefore not depend on the boundary conditions at the other brane. This implies that for the unbroken brane of S 1 /(Z 2 × Z ′ 2 ) we can use the mass shift for S 1 /Z 2 with unbroken gauge symmetry (cf. Eq. (15)), and for the broken brane we can use the mass shift for S 1 /Z 2 with broken gauge symmetry (cf. Eq. (23)). More precisely, we have to take one half of Eq. (15) respectively Eq. (23), since we need the contribution of only one brane. The argument of this paragraph is also valid for the HMP contribution discussed below.
The gauge coupling correction, or equivalently the mass shift of the KK modes (Eq. (15)), which determines the coefficient c (br,G) vmp for the unbroken brane, is G-universal. Thus, Eq. (16) applies, and together with Eq. (31) we immediately get In order to determine the coefficient c vmp for the brane where the gauge symmetry is reduced to H, we use Eq. (23). Moreover, since no brane-localized terms are induced for the broken components of the VMP (i.e. those with 'shifted' KK spectrum (2n + 1)/R) at that brane, the summation Eq. (24) still applies and we get The factor 8 in comparison to Eq. (25) arises, since we have to multiply by 2 4 (for the reduced length) and divide by two (for one brane).

Including hypermultiplets
The (Z 2 , Z ′ 2 )-parities of the 4d superfields H and H c which form the HMP follow from the (Z 2 , Z ′ 2 )-parities of the VMP. H and H c necessarily have opposite Z 2 -parities and opposite Z ′ 2 -parities, leading to a chiral spectrum for the zero modes. The Z ′ 2 -parities depend on the gauge group index. We still have the freedom to choose an overall sign of the Z ′ 2 action on the HMP. Similar to the S 1 /Z 2 -case, we define, by the following The brane-coefficient from the unbroken brane, denoted by c (br,G) hmp , follows from the results of Sect. 3: From Eqs. (18) and (21), together with Eq. (16), one reads off that (for one brane) This and Eq. (16) lead to c (br,G) Regarding the contribution c (br,H) hmp , Eq. (26) together with the subsequent paragraph implies (for i =ī and i =î) Using this and Eq. (13), one arrives at As a simple check, one can verify that for an adjoint HMP, the expression a,î η a (T a R T a R )îî vanishes, so that in this case one indeed has c 6 Application to 5d SUSY-GUT models For the application of our results to realistic models, we need to consider also the effect of charged chiral multiplets located at a boundary (see e.g. [34] for an explicit Lagrangian). This can for instance be an MSSM matter-or Higgs-sector. The inclusion of their effects in the Casimir energy is straightforward: The contribution of brane fields to the running of the gauge coupling is the usual one of 4d gauge theories. For fields at a brane (of an Here, b H i is the β-function coefficient for the unbroken subgroup H i . For the case of a brane with unbroken gauge symmetry this becomes c S 1 /(Z 2 × Z ′ 2 ), the only difference is an extra factor of 16 due to the reduced length of the interval. Observe that c (br) loc is always positive since the β-function coefficient of chiral multiplets is positive. Hence, in situations where the contribution of the bulk field content alone to c (br) is not positive and large as it needs to be, brane-localized chiral multiplets can help to assure a perturbatively controlled radion effective potential.
Let us now apply our results to supersymmetric SU(5)-GUTs on S 1 /(Z 2 × Z ′ 2 ) such as those which were proposed in Ref. [11] (see also [10]). These models have an unbroken SU(5) brane as well as a brane where the gauge symmetry is broken to the Standard Model (SM) gauge group SU(3) × SU(2) × U(1). The gauge sector resides in the bulk and the Higgs sector is located on the SM brane (this avoids the doublet-triplet splitting problem). In Ref. [11], the matter sector is assumed to be located either completely on the SM brane or in the bulk. As explained in detail in Ref. [11], in the latter case each bulk matter family consists of two copies of a (10 +5) with opposite Z ′ 2 -parities, such that there is a full MSSM matter family at the zero mode level.
We generalize the models of Ref. [11] by allowing for an arbitrary distribution of the MSSM matter to the bulk and the SM brane, which is our 'micro-landscape'. For any of the three families, we allow the freedom to have, instead of a pair with opposite Z ′ 2 -parities, just one copy of a5 = (3, 1) 1/3 ⊕ (1, 2) −1/2 in the bulk. Depending on the Z ′ 2 -parity of this5, there is either a D R = (3, 1) 1/3 or an L L = (1, 2) −1/2 at the zero mode level. Analogously, for one HMP transforming as a 10 = (3, 2) 1/6 ⊕ (3, 1) −2/3 ⊕ (1, 1) 1 , one has either a Q L = (3, 2) 1/6 or a U R ⊕ E R = (3, 1) −2/3 ⊕ (1, 1) 1 at the zero mode level. Let us in the following consider the most general situation where the zero modes of HMPs lead to r generations of U R ⊕ E R , s generations of L L , t generations of Q L and u generations of D R in the bulk (where r, s, t, u ∈ {0, 1, 2, 3}). Consequently, the remaining (3 − r) U R ⊕ E R generations, (3 − s) L L generations, (3 − t) Q L generations and (3 − u) D R generations must be located at the SM brane.
We now determine the Casimir energy for this situation. Applying Eqs. (31) and (34) and adding c (1) grav = −64c ω from the supergravity multiplet, one finds where we used c 2ω = 4c ω + O(ω 4 ). Here and in the following we neglect O(ω 4 ) effects.
The brane contribution to the Casimir energy which is induced by the VMPs and HMPs in the bulk is given by Eqs. (32), (33), (36) and (38). 14 Adding all terms, one finds after some algebra that On the other hand, using Eq. (40), the brane effect induced by the matter and Higgs fields on the brane is found to be so that the total brane coefficient is One can now determine the position of the minimum of V (R) in units of g 2 or equivalently in units of 1/M ≃ g 2 N/(24π 3 ), for all choices of r, s, t, u. Note that the condition c (1) < 0 for the existence of a minimum is not satisfied for many choices of r, s, t, u, while c (br) > 0 is always satisfied. If a minimum exists, one easily finds that it is given by Here, W −1 (x) is the Lambert W -function, which is defined as the inverse function of x e x . More precisely, since the Lambert W -function is double-valued on (−1/e, 0), W −1 (x) denotes the branch which satisfies W −1 (x) ≤ −1 for −1/e ≤ x < 0 (see e.g. [44]). By scanning all 4 4 = 256 possible choices of r, s, t, u, we find that V (R) has a minimum at large radius, say MR > 10, for about a third of them.
Since the minimum R is given in units of 1/M (or equivalently g 2 , as explained in the Introduction), by stabilizing the radius we determine the value of the 4d gauge coupling For some orbifold GUT models (albeit not the most simple ones) we predict a realistic size for the unified gauge coupling! As a comparison, for the most simple model where all matter is located at the SM brane (i.e. r = s = t = u = 0) we find α ≃ 1/40. This is too small. One should keep in mind however that the correction due to unknown nonvanishing but not unnaturally large coefficients of the brane-localized operators at the scale M is expected to be roughly of the order 1/ ln(MR) ∼ 25%.

Uplifting to a small cosmological constant
The Casimir energy we find gives a negative contribution of the order ω 2 /R 4 ∼ m 2 1/2 M 2 c to the vacuum energy. In order for the theory to be potentially realistic, there need to be other effects canceling this negative contribution, such that a tiny positive Λ 4 is obtained. This will certainly involve fine-tuning, otherwise we would have solved the cosmological constant problem. 15 In this section, we briefly discuss how such an 'uplifting' could be realized. We also check that our proposals are consistent with the assumptions of our Casimir energy calculation, namely a (sufficiently) flat 5d background and Scherk-Schwarz SUSY breaking.
Note that there are no loop-contributions to Λ 4 coming from fields other than the radion. This is easily seen in the language of 4d supergravity: Our Casimir energy calculation is equivalent to the calculation of a loop correction to the no-scale Kähler potential of the radion. In the presence of a constant brane-localized superpotential W 0 , which is related to the Scherk-Schwarz parameter ω by this correction turns into a potential energy. Loop effects will generically also correct the Kähler potential of matter and Higgs fields, which is canonical at tree level. Since these fields do not develop a VEV, their Kähler corrections do not induce a contribution to the vacuum energy. Furthermore, there are no perturbative corrections to the superpotential. Thus, the negative vacuum energy which we find at the minimum of the radion effective potential has to be taken seriously and some compensating effect is required.

Uplifting by small warping and a brane-superpotential
Let us allow for a 5d cosmological constant, which of course has to be small enough not to affect our flat-space Casimir energy calculation. In other words, we assume that we are dealing with a supersymmetric Randall-Sundrum model [45], but with very weak warping. The drawback of this proposal is that we do not know how such a warping could arise from the heterotic orbifold perspective. One may hope that it can be realized, e.g., by fluxes in the five compact dimensions, which would make it discretely tunable. In any case, what follows should be consistent from the point of view of 5d supergravity coupled to gauge fields and charged matter.
In the presence of a constant superpotential W 0 at the IR brane, the corresponding 4d theory is defined by [46] The AdS curvature scale k is related to the 5d cosmological constant by Λ 5 = −6k 2 M 3 P,5 . Working on S 1 /Z 2 , the T = πR + · · · is the radion superfield, while Ω and W are the 'superspace kinetic function' and the superpotential of 4d supergravity. A Kähler-Weyl rescaling brings them to the equivalent form For weak warping, k(T +T ) ≪ 1, we have which can then be treated as a small correction to the basic no-scale structure of the model. This puts us into the setting of 'Almost no-scale supergravity' of Luty and Okada [18], where the corresponding correction to the Brans-Dicke-frame scalar potential, has been given. 16 One immediately sees that, in order for δV to cancel the negative Casimir energy, one needs a warping of the order Thus, the warping required for the uplifting is indeed small whenever the stabilization radius is large in units of the 5d Planck scale (this is anyway necessary for gravity to be perturbative at the compactification scale). We conclude that our flat space calculation remains justified in spite of the fact that we are really dealing with a Randall-Sundrum type model.
We finally note that our complete stabilization and uplifting proposal can be formulated within the framework of [18]: Our two-loop Casimir energy can be reinterpreted as a Kähler correction with the structure ∆Ω Casimir ∼ 1 (T +T ) 2 + g 2 (T +T ) 3 ln(M(T +T )) .
Adding this to the warping-induced correction of Eq. (49), we find that Eq. (50) generates a scalar potential the minimum of which can be tuned to zero by adjusting the ratios of k, M P,5 and g.

Uplifting in a detuned Randall-Sundrum model
The uplifting proposal of the last subsection can be reformulated in terms of the 'supersymmetric detuned Randall-Sundrum model' of Bagger and Belyaev [47]. According to [47], the UV and IR brane tensions Λ 0 and Λ π , which are normally given by can take arbitrary values in a consistent 5d supergravity model, as long as they obey the constraint |Λ 0,π | ≤ Λ. Thus, it is natural to attempt to uplift our previous stabilized flat 5d model by allowing for a small warping together with a small detuning of the IR-brane tension, 0 < Λ π + Λ ≪ Λ .
The naive expectation is that, as long as warping is small, this will give a constant and positive contribution to the radion effective potential in the Brans-Dicke frame. Note that this is not inconsistent with general theorems concerning the possible vacuum states of supergravity theories: The detuned Randall-Sundrum model has an AdS 4 ground state at a certain radius. Since we stabilize the radius by the Casimir energy at a different value, we are actually forcing the theory into a metastable state with a tiny positive Λ 4 .
To confirm the above expectation, we utilize the 4d supergravity description of the detuned Randall-Sundrum model derived in [48]. The Kähler potential K and the superpotential W are explicitly given in Eqs. (6.1) of [48]. We first rewrite K in terms of Ω = −3 exp(−K/3) and perform a (constant) Kähler-Weyl transformation bringing Ω to the form given in Eq. (47). We then work out W in the limit Λ 0 → Λ, making also use of the relation Λ π + Λ ≪ Λ. The functional form agrees with our Eq. (47) and we determine Calculating δV according to Eq. (50), we find as expected (see also [49]). We conclude that the uplifting proposals of Sect. 7.1 and of the present section are equivalent. The underlying technical result is the superfield formulation of the detuned Randall-Sundrum model of [48]: It implies that including a constant IR-brane-localized superpotential in a supersymmetric Randall-Sundrum model is equivalent to a weak detuning of the IR brane tension. In both cases, a non-zero scalar potential is induced. This potential is positive and approximately constant at small values of the radion (i.e. for a warp factor close to one).
We note that the 4d superfield description of the detuned Randall-Sundrum model has also been considered, e.g., in [49] (independently of [48]) as well as in [50] and [22]. In particular, one-loop corrections to the radion potential have been analyzed in [49,50]. Our 'uplifting' proposal of Sects. 7.1 and 7.2 differs in that we use a two-loop effective potential to stabilize the radion at a very small (from the Randall-Sundrum model perspective) value. The warping is then irrelevant for the loop calculation and its only effect is to provide, in its interplay with a small detuning, an approximately constant uplifting contribution. For related applications of the detuning in supersymmetric Randall-Sundrum models see, e.g., [20,51].

F -term uplifting
Alternatively, one may insist that Λ 5 is exactly zero. In that case, Scherk-Schwarz breaking predicts a negative vacuum energy and there needs to be another source of SUSY breaking, in the spirit of 'F -term uplifting' (see e.g. [52]). Following again [18], we assume that via some unspecified dynamics there arises an F -term VEV of a branelocalized singlet S of the right order of magnitude to obtain a small cosmological con-stant: F S ∼ ω/R 2 . This can potentially affect the SUSY-breaking mass splitting of bulk fields and hence our Casimir energy calculation.
In the spirit of gaugino mediation, S may couple to the gauge-kinetic term via a brane-localized higher-dimension operator, suppressed by the fundamental scale M. The induced gaugino masses are of the order [53] Similarly, S may couple to the kinetic term of bulk hypermultiplets via a higherdimension brane-localized operator. This induces scalar masses of the same order of magnitude as the gaugino masses given in Eq. (57), m 0 ∼ m 1/2 .
By contrast, the 'radion-mediated contribution' to the gaugino masses and to the scalar masses of bulk multiplets is of the order so that in comparison the effect of F S is suppressed by 1/(MR) 2 . We conclude that our Casimir energy calculation and the corresponding stabilization mechanism remain under quantitative control in the presence of brane-localized F -term uplifting. One may nevertheless feel that the uplifting mechanism of Sects. 7.1 and 7.2 is more elegant since it does not require an extra SUSY breaking sector.

Conclusions
We have analyzed Casimir stabilization of supersymmetric gauge theories on 5d orbifolds with Scherk-Schwarz SUSY breaking and gauge symmetry breaking by boundary conditions. Depending on field content and symmetries of the 5d theory, a minimum in the Casimir energy (as a function of the radius R) can arise from the interplay of the one-loop and two-loop contribution. The dominant two-loop effect comes from logarithmically divergent operators localized at the boundaries. We rely only on the log-enhanced part of the coefficients of these operators. Our results are therefore independent of the UV completion as long as the compactification scale 1/R is much smaller than the cutoff scale Λ. Then, the Casimir energy -including the one-loop term -is given by (see Eq. (8)) We provide general formulae for the one-loop coefficient c (1) and for the coefficient c (br) of the brane-induced two-loop effect. The coefficient c (1) is well-known for the case of unbroken gauge symmetry (Eq. (18)). It is unchanged for S 1 /Z 2 with broken gauge symmetry. For S 1 /(Z 2 × Z ′ 2 ) with gauge symmetry broken at one of the branes, it is given by Eq. (31) for vector-and by Eq. (34) for hypermultiplets.
For S 1 /Z 2 , the coefficient c (br) is given by Eq. (22) for unbroken gauge symmetry and by Eq. (29) for gauge symmetry breaking by boundary conditions. In the most relevant case of S 1 /(Z 2 × Z ′ 2 ) with gauge symmetry breaking at one of the boundaries, c (br) is given by Eqs. (32), (33) for vector multiplets and by Eqs. (36), (38) for hypermultiplets.
We were able to obtain these coefficients without explicit new loop calculations, relying only on the Z 2 parity transformation properties of the fields, supersymmetry and the group theoretic structure of the model.
We have applied the above formulae to SU(5) orbifold GUT models on S 1 /(Z 2 × Z ′ 2 ), with gauge-symmetry breaking to the Standard Model at one of the boundaries. For simplicity, we have assumed that the cutoff Λ takes its highest possible value -the strong coupling scale M ≃ 24π 3 /(5g 2 ) of the 5d gauge theory. Furthermore, we have focused on scenarios where the Higgs sector is located at the Standard Model brane, allowing the matter sector to be distributed between bulk and Standard Model brane in various ways. The resulting 256 possibilities form our 'micro-landscape'. Of course, using our formulae, one could also consider models which, for instance, have Higgs fields in the bulk instead of on the brane and/or have additional vector-like matter. Similarly, SO (10) or other unified gauge groups could be studied. Our analysis is clearly incomplete from this perspective. Nevertheless, restricting ourselves to the simple class of GUT models defined above, we find a minimum in the effective potential at large MR for many of the possible matter distributions. The position of the minimum determines the 4d gauge coupling at the compactification scale. For several of the models, we find results which are in agreement with the phenomenological value of α GU T ≃ 1/25. Thus, two-loop Casimir stabilization offers a simple 'explanation' (within our modest realization of the landscape paradigm) for the appearance of a relatively large 5th dimension and a correspondingly small gauge coupling at the GUT scale.
We have finally discussed two mechanisms for uplifting our AdS 4 vacua without affecting the underlying stabilization mechanism: One possibility is to allow for a small warping, i.e. a small negative 5d cosmological constant. In combination with a brane localized constant superpotential, this induces the required positive contribution to the scalar potential. Alternatively, one can include an extra SUSY-breaking sector on one of the branes (independently of the radion, which is our dominant source of SUSY breaking). Such a brane-localized F -term can provide the required uplift. It would be important to find a string-theoretic realization of these uplifting mechanisms and, more generally, to work out in more detail to which extent our stabilization proposal is consistent with a full-fledged underlying heterotic string construction. due to the Yukawa coupling instead of a negative one (cf. Eq. (45) in Ref. [30]).