AdS-phobia, the WGC, the Standard Model and Supersymmetry

Journal of High Energy Physics, Jun 2018

Abstract It has been recently argued that an embedding of the SM into a consistent theory of quantum gravity may imply important constraints on the mass of the lightest neutrino and the cosmological constant Λ4. The constraints come from imposing the absence of any non-SUSY AdS stable vacua obtained from any consistent compactification of the SM to 3 or 2 dimensions. This condition comes as a corollary of a recent extension of the Weak Gravity Conjecture (WGC) by Ooguri and Vafa. In this paper we study T 2/Z N compactifications of the SM to two dimensions in which SM Wilson lines are projected out, leading to a considerable simplification. We analyze in detail a T 2/Z4 compactification of the SM in which both complex structure and Wilson line scalars are fixed and the potential is only a function of the area of the torus a2. We find that the SM is not robust against the appearance of AdS vacua in 2D and hence would be by itself inconsistent with quantum gravity. On the contrary, if the SM is embedded at some scale M SS into a SUSY version like the MSSM, the AdS vacua present in the non-SUSY case disappear or become unstable. This means that WGC arguments favor a SUSY version of the SM, independently of the usual hierarchy problem arguments. In a T 2/Z4 compactification in which the orbifold action is embedded into the B − L symmetry the bounds on neutrino masses and the cosmological constant are recovered. This suggests that the MSSM should be extended with a U(1)B−L gauge group. In other families of vacua the spectrum of SUSY particles is further constrained in order to avoid the appearance of new AdS vacua or instabilities. We discuss a possible understanding of the little hierarchy problem in this context.

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AdS-phobia, the WGC, the Standard Model and Supersymmetry

Received: May AdS-phobia, the WGC, the Standard Model and Supersymmetry Eduardo Gonzalo 0 1 2 Alvaro Herraez 0 1 2 Luis E. Iban~ez 0 1 2 0 Cantoblanco , 28049 Madrid , Spain 1 Universidad Autonoma de Madrid 2 Departamento de F sica Teorica and Instituto de F sica Teorica UAM/CSIC It has been recently argued that an embedding of the SM into a consistent theory of quantum gravity may imply important constraints on the mass of the lightest neutrino and the cosmological constant 4. The constraints come from imposing the absence of any non-SUSY AdS stable vacua obtained from any consistent compacti cation of the SM to 3 or 2 dimensions. This condition comes as a corollary of a recent extension of the Weak Gravity Conjecture (WGC) by Ooguri and Vafa. In this paper we study T 2=ZN compacti cations of the SM to two dimensions in which SM Wilson lines are projected out, leading to a considerable simpli cation. We analyze in detail a T 2=Z4 compacti cation of the SM in which both complex structure and Wilson line scalars are xed and the potential is only a function of the area of the torus a2. We nd that the SM is not robust against the appearance of AdS vacua in 2D and hence would be by itself inconsistent with quantum gravity. On the contrary, if the SM is embedded at some scale MSS into a SUSY version like the MSSM, the AdS vacua present in the non-SUSY case disappear or become unstable. This means that WGC arguments favor a SUSY version of the SM, independently of the usual hierarchy problem arguments. In a T 2=Z4 compacti cation in which the orbifold Compacti cation and String Models; Superstring Vacua; Supersymmetric - action is embedded into the B L symmetry the bounds on neutrino masses and the cosmological constant are recovered. This suggests that the MSSM should be extended with a U(1)B L gauge group. In other families of vacua the spectrum of SUSY particles is further constrained in order to avoid the appearance of new AdS vacua or instabilities. We discuss a possible understanding of the little hierarchy problem in this context. 3 The SM on the T 2=Z4 orbifold The SM on the circle I (R & 1=me) Fixing the Wilson lines. The segment S1=Z2 1 Introduction 2 to delimit the swampland are those given by the Weak Gravity Conjecture (WGC) [2, 3], which loosely speaking implies that the gravitational interaction must be weaker than gauge interactions in any consistent theory of quantum gravity. In its simplest incarnation of a U(1) theory coupled to gravity it implies that there must be at least one charged particle with mass m and charge Q such that m Q in Planck units, and is motivated by blackhole physics. There are di erent versions of the conjecture and also extensions to multiple U(1)'s and to the antisymmetric gauge tensors of string theory and supergravity, see [2{32]. In the latter case it is the tension of the branes of string theory which are bounded in terms of the charge of the antisymmetric RR or NS tensors, so that T Q. Recently Ooguri and Vafa (OV) put forward a sharpened version of the WGC [33] (see also [34{37]) stating that in non-SUSY theories the inequality is strict, T < Q. When applied to string ux so far no counterexample to this conjecture in string theory.1 The authors of [33] go also beyond the string theory setting and conjecture that non-SUSY AdS/CFT duality is in the swampland in general, and not only for ux vacua. We will call AdS-phobia for short the condition of the absence of any AdS stable nonSUSY vacuum in a given theory, as suggested by the more general OV condition. AdSphobia, if correct, would be a very powerful constraint on physics models. For example it would drastically reduce the possibilities in the string landscape: only Minkowski and de Sitter vacua would be consistent with quantum gravity and only those (in addition to SUSY ones) would in principle count in addressing the enormous multiplicity of vacua in string theory. It has been recently pointed out that AdS-phobia, if applied to the SM, would have important implications on SM physics [33, 39, 40]. The point is that, as described in [41], compactifying the SM to 3 or 2 dimensions one may obtain AdS vacua, depending on the values of the cosmological constant 4 and the neutrino masses. On the other hand the assumption of background independence implies that if a theory is consistent in 4D, all of its compacti cations should be consistent also, since we have only changed the background geometry. Thus the existence of these lower dimensional AdS vacua (if stable) would imply the inconsistency of the SM itself [33, 39]. A detailed analysis of these constraints on neutrino masses and the cosmological constant was presented in [39]. Four interesting conclusions were obtained by imposing the absence of AdS minima, Majorana neutrinos (two degrees of freedom per neutrino) are not possible [33]. 4 10 3 eV (normal hierarchy) or The lightest neutrino mass is bounded as m 1 m 3 1 10 3 eV (inverted hierarchy) [39]. The cosmological constant (Dirac) neutrino, 4 & m4i [39]. 4 is bounded from below by the mass of the lightest 1There are examples of non-SUSY theories with AdS/CFT duals but, unlike the SM coupled to gravity, nite number of elds, see e.g. [38] and references therein). The upper bound on the lightest neutrino mass implies an upper bound on the Electro-Weak scale (for xed neutrino Yukawa couplings) [39, 40]. The last point is obvious from the fact that the neutrino mass depends on the Higgs vev. Any increase in the EW scale above a scale ' 1 TeV would make the lightest neutrino mass to be too large and violate the bound to avoid AdS. This is remarkable because it would imply that values of the EW scale much above the observed scale would belong to the swampland. Thus, in this sense, the EW ne-tuning problem would be a mirage, since the EW scale is xed in order to avoid AdS vacua, and its scale is secretly dictated by 4 and the neutrino Yukawa couplings. The smallness of the EW scale compared to the versus the Planck scale [39, 40]. In [39] the neutrino AdS vacua in 3D and 2D were analyzed in detail. These minima appear from an interplay between the (positive) dimensional reduction of the cosmological constant term, the (negative) Casimir contribution of the photon and graviton and the (positive) Casimir contribution of the lightest fermions of the SM, the neutrinos. If the lightest neutrino is su ciently light, its contribution can destroy the AdS vacuum and make the SM AdS-safe. This is the origin of the bound on the lightest neutrino. The Casimir contribution of the other SM particles is exponentially suppressed with their mass. However as one explores the radion potential (e.g. in 3D in a circle) at shorter radius R, the thresholds of the other SM particles are reached and their contribution needs to be taken into account. So the question arises whether there are further local minima or runaway directions at shorter radii. In particular, as one goes up in energies (smaller R) additional scalar moduli appear associated rst with the Wilson line of the photon and then, above the hadron threshold, two Wilson lines in the Cartan subalgebra of QCD. At even shorter distances the Wilson line of the Z0 appears. So the full moduli space in 3D involves one radion and four Wilson lines, with a scalar potential depending on all of them. An analysis of the existence of these additional Wilson line dependent minima was recently carried out in [42], concluding that indeed runaway eld directions at smaller R exist for non-trivial photon Wilson line. The R ' 1=m AdS vacua are bona- de local minima but if they could tunnel into these eld directions there would be no constraints on neutrino masses nor the EW hierarchy. In this paper we reexamine and extend the study of the AdS minima of the SM both to smaller compact radii and geometries other than the circle and the torus. It is useful to classify the vacua we nd in three categories: Type D (Dangerous). Vacua which contain a stable AdS minimum which cannot be avoided by constraining free parameters of the theory. Type S (Safe). Vacua which contain no stable AdS minima. Type P (Predictive). Vacua which contain no stable AdS minima for a certain range of the free parameters of the theory. Our attitude concerning instability will be conservative in the sense that we will consider unstable any potential which includes any runaway direction in the elds, even though we { 3 { are not certain how the tunneling from local minima to those runaway directions could take place in detail. We nd minima in the three categories. After examining the case of the compacti cation on a circle, we discuss SM compacti cations in which Wilson line moduli are xed. After a brief discussion of the compacti cation on the segment S1=Z2, we discuss the compacti cation on toroidal orbifolds T 2=ZN . We specialize to the case of the T 2=Z4 orbifold, but most implications apply to ZN , N = 2 6 in general. The reason to study this particular background is twofold. On the one hand, unlike the case of the parity re ection in the segment, a Z4 rotation is a symmetry of the uncompacti ed 4D SM. Secondly, in such a background all the Wilson lines as well as the complex structure are xed, so that the scalar potential depends only on the torus area. This simpli es enormously the study moduli disappear. It turns out that, in order to obtain predictive T 2=Z4 vacua one needs to embed the discrete rotation into internal symmetries of the SM. We nd that di erent embeddings into the SM gauge group lead necessarily to stable AdS vacua, rendering the SM inconsistent with quantum gravity. This is true irrespective of the value of neutrino masses or any other free particle physics parameter. Thus some of the SM vacua are of Type D and make the SM as such not viable. The underlying reason why the SM has these AdS vacua is that it is too fermionic, it has many more fermions than bosons. The dangerous AdS vacua could become unstable Type S if there were enough bosons to compensate in the Casimir potential. A situation in which this naturally happens is a SUSY embedding of the SM like the MSSM. Thus, unlike the SM, the MSSM seems robust against the generation of stable AdS vacua to the extent that we have not found vacua of Type D. Furthermore we nd Z4 embeddings into a discrete subgroup of the U(1)B L gauge symmetry leading to AdS vacua of Type P which may be evaded for appropriate neutrino masses, like the original 3D vacua in the circle, and hence are predictive. This suggests that the MSSM should be extended at some scale with an extra U(1)B L gauge symmetry. Another class of embeddings into discrete R-symmetries, in addition to neutrinos, also constraint the masses of the SUSY particles in a way essentially dictated by the sign of the supertrace SS = P ( 1)(ni)m2. More i i generally, in SUSY models avoiding AdS vacua from charge-colour breaking AdS minima inherited from the 4D potential impose additional constraints. Thus an AdS-safe MSSM should pass both these constraints, leading potentially to interesting conditions for SUSY model building. As discussed in [39, 40], avoiding the AdS neutrino vacua imply an upper bound on the EW scale close to its experimental value. On the other hand absence of lower 4D chargecolour breaking minima requires typically a relatively massive SUSY spectrum. This could provide for an explanation for the little hierarchy problem of the MSSM: absence of AdS neutrino vacua requires a EW scale close to its experimental value, while absence of chargecolour breaking AdS minima pulls up the value of SUSY masses. It is important to remark that the absence of consistent AdS non-SUSY vacua is at the moment only a conjecture, although in agreement with the string examples examined up to now. Furthermore, the results rely on the assumption that the obtained AdS SM vacua are stable. Although indeed the obtained minima appear stable, if the theory e.g. is { 4 { embedded into a string landscape, one cannot rule out transitions into other lower energy vacua which might exist in the landscape. This issue is discussed in [39]. It is di cult to quantify, given our ignorance of the structure of the landscape. In the present paper we assume that those transitions are absent and the AdS minima are stable, and explore what the physical consequences would be. The structure of this paper is as follows. In the next section we review the case of the compacti cation of the SM on a circle and the origin of the corresponding bounds on neutrino masses and the cosmological constant, exploring also values of the radii up to the EW scale. We then consider the structure of the potentials at small R with xed Wilson lines, as would appear in a compacti cation on the segment S1=Z2. In section 3 we discuss the compacti cation of the SM on T 2=Z4 and show how the SM has vacua of Type D and hence would be incompatible with quantum gravity. In section 4 we show how such vacua are not present in the SUSY case and how one recovers the neutrino mass bounds if U(1)B L (or a discrete subgroup) is gauged at some level. New vacua may also lead to constraints on the SUSY spectrum. We leave section 5 for our conclusions and outlook. In appendix A we discuss the computation of the Casimir potential in the SM on the circle, including also the EW degrees of freedom. In appendix B we work out some results for the compacti cation on the T 2=Z4 orbifold. Appendix C contains a discussion on the consistency of assuming a at non-compact background in the computation of Casimir energies. Finally, appendix D contains a table with the MSSM spectrum used to draw some of the plots. 2 The SM in 3D In this section we rst study the compacti cation of the SM coupled to Einstein gravity with a cosmological constant down to 3D on the circle. We begin by exploring the regions with the compact radius larger than the electron wave length and then regions with smaller radius, going up to the EW scale. The resulting theory has one radius, one Higgs and four Wilson line scalar variables. We reproduce the local minima associated to the neutrino region and study the potential at shorter radii. The potential features runaway directions in the presence of Wilson line moduli that may lead to decay of the neutrino vacua through tunneling. We also discuss the structure of the radius e ective potential in the case with frozen Wilson lines. This would happen in particular in theories compacti ed in the segment S1=Z2 and serves as an appetizer to the case of compacti cations on T 2=ZN which are discussed in the next section. In this case it is easier the search for minima for the radius potential, since then all Wilson lines are xed. 2.1 The SM on the circle I (R & 1=me) We rst brie y summarize and extend some of the results in [39, 41] so that the reader better understands the further results presented in this paper. In [41] the compacti cation of the SM Lagrangian to lower dimensions was considered. One of the purposes of that work was to show how the notion of landscape of vacua should not be associated exclusively to string theory, but even the SM itself has a wealth of vacuum solutions when compacti ed to 3D or { 5 { 2D. They concentrated in the deep infrared sector of the SM, below the electron threshold, in which the only relevant particles are the photon, the graviton and the neutrinos. They found that, depending on the value of the neutrino masses, the SM possesses local minima both on 3D and 2D, with very large compacti cation radii of order R ' 1=m . The appearance of these minima goes as follows. The action of the pure gravity action reduced to 3D in a circle is given by SGR = Z d x Here Mp is the 4D reduced Planck mass, Mp = (8 GN ) 1=2 and 4 is the 4D cosmological HJEP06(218)5 constant. The action of the graviphoton of eld strength W is also shown. There is a runaway potential inherited from the 4D cosmological term. Nevertheless, the cosmological constant is so small that the quantum contribution of the lightest SM modes to the e ective potential becomes relevant. Their contribution at one-loop level can be identi ed with the Casimir energy, which for a non-interacting massless eld with periodic boundary conditions is found to be: Vp(R) = np 720 1 R6 ; where np counts the number of degrees of freedom of the particle. For bosons the negative sign applies, whereas for fermions the sign is positive. The only massless states in the SM are the photon and the graviton, so in the absence of any other light particle it is clear that the potential becomes unstable and negative for small R. However, if there are fermions with a mass of order m & 14=4, minima can develop. The Casimir potential for a non-interacting massive particle of spin sp and mass mp has the form [41] Vp(R) = ( 1)2sp+1np m2p 8 4R4 X1 K2(2 nRmp) : n=1 n2 Here K2(x) is a modi ed Bessel function of the second kind. This gives us the leading quantum contribution to the radion potential. As long as couplings remain perturbative, higher loop corrections will be small and unimportant. This excludes the region close to the QCD scale in which non-perturbative techniques would be appropriate. We will thus concentrate in the potential above or below the QCD scale. In the latter case we will count pions and kaons as elementary. The Bessel function decays exponentially for large argument. For this reason, to work out the structure of the potential around the scale nos will be relevant, since heavier particles like the electron contribute in a negligible way to the local potential. Neutrino oscillation experiments tell us what are the mass di erences between neutrino masses, but not their Dirac or Majorana character. We do not know whether the hierarchy of the masses is normal (NH) or inverted (IH). Experimental constraints tell us for the neutrino mass di erences that [43] 4 1=4 within the SM, only the neutri{ 6 { (b) massless and a minimum develops. For higher energies, the neutrinos behave as massless particles. We have 4 bosonic and 6 fermionic degrees of freedom so, in the massless limit, R6V (in units of degrees of freedom) tends to 2. (b) Radion e ective potential for Dirac neutrinos. The di erent lines correspond to several values for the lightest neutrino mass m 1 . We have 4 bosonic and 12 fermionic degrees of freedom so, in the massless limit, R6V tends to 8. It is important to remember that positive valued minima in this plot do not necessarily correspond to dS minima of V . However, negative minima do correspond to AdS minima of V . We do not know what the mass of the lightest neutrino is, and it is not experimentally excluded that it could be massless. We use the above neutrino constraints in computing the radion potential and plotting the gures. We always plot the potential divided by the contribution of a single massless degree of freedom, so that we can interpret it as the number of e ective degrees of freedom. Thus, the potential in our plots appears multiplied by R6 times a certain constant. Care must be taken when extracting physical conclusions from these plots, since minima of R6V need not be minima of V . However, an AdS minima of R6V which eventually turns positive (in all directions), always corresponds to an AdS minima of V . If neutrinos are Majorana and are given periodic boundary conditions, we have six fermionic degrees of freedom which at small R dominate over the photon/graviton four degrees of freedom, so that the potential grows. However when going to values R ' 1=m 1 an AdS minimum is always created, irrespective of the neutrino masses, since only two neutrino degrees of freedom can become su ciently light and those cannot overwhelm the bosonic contribution at large R, see gure 1(a). On the other hand if neutrinos are Dirac, up to 4 neutrino degrees of freedom may be su ciently light to compensate for the four bosonic ones. In this case minima forms or not depending on the value of the lightest neutrino mass, see gure 1(b). If we apply the principle of no AdS non-SUSY vacua we reach the conclusion that neutrino Majorana masses are not possible and we obtain bounds for the lightest neutrino { 7 { Dirac mass m 1 > 7:7 for details.2 10 3 eV for (NI) and m 3 > 2:56 10 3 eV for (IH), see [39] As we mentioned before, another interesting implication is that within this scheme there is a lower bound on the value of the 4D cosmological constant [39], qualitatively 4 & m41 . This is interesting because it is the rst argument implying a non-vanishing 4 only on the basis of particle physics, with no cosmological input. A further implication concerns the ne-tuning issue of the Electro-Weak (EW) scale in the SM. Indeed, the mentioned upper bounds on neutrino masses imply an upper bound on the Higgs vev, for xed Yukawa coupling. Thus SM versions with a Higgs sector leading to Higgs vevs above 1 TeV would be in the swampland, since larger vevs implies larger neutrino masses and hence AdS vacua would form in 3D or 2D [39]. In particular for Dirac neutrino masses one has a bound jhHij . 1:6(0:4) 4 1=4 Y 1 ; (2.7) 1=4 4 ' 2:25 for NH(IH) neutrinos. Here Y 1 is the lightest neutrino Yukawa coupling and 4 the 4D cosmological constant. With values of 4 as observed in cosmology, larger values for the EW scale would necessarily lead, for xed Yukawa, to undesired AdS minima. From the low-energy Wilsonian point of view the smallness of the EW scale looks like an enormous ne-tuning, but this would be a mirage since eld theory parameters leading to larger EW scale would not count as possible consistent theories. The smallness of the EW scale compared to the Planck scale is here tied up to the smallness of the cosmological constant. Note that here we are taking the value of 4 xed to its observed value while varying H. This may be motivated by the fact that anthropic arguments from galaxy formation require 10 3eV within a factor of order 4 [44, 45]. So we vary H while keeping the cosmological constant around its measured present value. Having Dirac neutrinos rather than Majorana departs from the most common scenario to understand the smallness of neutrino masses, the seesaw mechanism. In the present approach the lightest neutrino is very light to avoid that an AdS vacuum forms. Its mass is bounded from above by m 1 . 41=4, giving an explanation for the apparent numerical coincidence of neutrino and cosmological constant scales. Strictly speaking, it is only necessary for one of the three neutrinos to be that light. However, if neutrino Yukawa couplings of the di erent generations are correlated, as they are for the charged leptons and quarks, this condition will drag all of the neutrinos to be very light, as observed.3 Still, one would like to have in addition an explanation to why the neutrino Yukawa couplings are 2Note that what counts is that the number of degrees of freedom of the lightest neutrino should be 4. Thus it may also be pseudo-Dirac, i.e. it may have both Dirac and Majorana mass contributions of the same order of magnitude. In this case -less double -decay could be posible. 3Note in this respect that obtaining Majorana neutrino masses in string compacti cations has proven to be a notoriously di cult task both in heterotic and Type II orientifold constructions. In the latter, Majorana neutrino masses may in principle appear from string instanton e ects [46, 47]. However, a dedicated search [48] for such instanton e ects in a large class of MSSM-like Gepner orientifolds was unsuccesful. On the other hand R Majorana masses may appear from non-renormalizable couplings of right-handed neutrinos to scalars breaking the B L symmetry in extended gauge symmetry models. { 8 { so small. As emphasized in [39], neutrino Majorana masses are still allowed if we go beyond the SM and assume there is an additional singlet Weyl fermion with mass m Then the AdS vacua is again avoided if the lightest (Majorana) neutrino is light enough. . 10 3 eV. In this case the bound on the EW symmetry scale has the form [39, 40] (Y 1 hHi) 2 M the EW scale is bounded from above by the geometric mean of the cosmological constant scale and the lepton number violation scale M . Thus, e.g. for Y 1 ' 10 3 and M ' 1010 1014 GeV, one gets hHi . 102 104 GeV. This possibility is interesting because, since it uses the see-saw mechanism, no hierarchically small Yukawas are needed. In the rest of this paper it will not be relevant whether neutrinos are Dirac or Majorana + , but we think it is interesting to keep in mind the di erent possibilities. All these constraints obtained imposing the absence of AdS local minima assume that these vacua are absolutely stable. A possible source of instabilities may in principle arise if there are lower minima or runaway directions at smaller radion values (higher thresholds) contributing to the Casimir potential [42]. This is an important motivation to go to the orbifold case, as we will discuss in section 3. 2.2 The SM on the circle II (1=me & R & 1=MEW) In this section we study the one-loop e ective potential of the full Standard Model with massive Dirac neutrinos and minimal coupling to gravity compacti ed in a circle. Our calculations were carried out using the Background Field Method, in the spirit of [49]. Some details on the computations can be found in appendix A. We parameterize the metric as in the usual Kaluza-Klein ansatz: g = 1 " R2 gij + R4BiBj R4Bi # : R4Bj R4 We will set the graviphoton Bi = 0 from the start, since its vev is zero and it does not contribute to the e ective potential. The scalar elds in the 3D E ective Action are the radion R, the Higgs eld H (whose vev is v ' 246 GeV) and the Wilson lines of all the gauge bosons in the Cartan Subalgebra of SU(3)C lines (G1,G2), the photon A and the Z boson. To perform the compacti cation we need to expand our elds in a basis that respects the boundary condition: (xi; y + 2 ) = ei2 ( 1 2 z ) (xi; y); since y 2 (0; 2 ). Fermions can have either periodic z = 1 or antiperiodic z = 0 boundary conditions, while bosons are only allowed to have periodic ones. We use Fourier harmonics: { 9 { (2.8) HJEP06(218)5 (2.9) (2.10) (2.11) The relevant classical contribution to the e ective potential in three dimensions consists of a term coming from the 4D cosmological constant, the Higgs potential and a mass term for the Z0 boson Wilson line (Z). We will neglect the e ects of a non-vanishing 3D curvature in the Casimir computation. We argue in appendix C that those e ects may be safely neglected as long as the radii are not exceedingly small. The one-loop potential can be consistently identi ed with the Casimir energy Vp of the di erent particles. One then has V [R; H; Z; A; G1; G2] = 1 As explained in appendix A, the contribution of each particle to the Casimir potential can always be written as: where sp, np, mp are the spin, number of degrees of freedom and mass of the particle and is a function of the Wilson lines and zp. Note that the masses of the particles appear multiplied by the Higgs divided by its vev, v. The only exception is the Higgs eld, whose mass is changed by the non-linear couplings to the Z0 Wilson line and to itself. Using the same techniques as e.g. [42, 50] we regularize the integration and the sum using the analytic continuation of generalized functions. After regularization, the contribution to the Casimir energy for each particle reads: Vp = ( 1)2sp+1np m2p 8 4R4 n2 X1 K2(2 nRmp) cos (2 n ) n=1 ( 1)2sp+1npVC [R; mp; ] : (2.14) K2(x) decays exponentially for large x, which means that at low energies, the contribution of very massive particles to the Casimir energy is highly suppressed. For massless particles + 1 2 MZ2 Hv22 Z2 + X Vp: p (2.12) = 0 we have while for = 12 we have VC [R; 0; 0] = 1 We cannot use this one-loop formula to study the minima of the potential around since perturbation theory breaks down for the strong interaction. Below the QCD scale, we consider an e ective eld theory of pions and kaons. For completeness, we give here the formula above the QCD scale: V C R; mH 2 1 + 3 2 c2wZi 3VC hR; MZ Hv ; 0i H 2 v + 2mm2Z2H Z2 1=2 # ; 0 : The rst two lines include the contribution of the graviton, photon and the 8 gluons. The third line include the one of leptons and the following three lines the 3 colours of quarks. The seventh line gives the contribution of massive W 's and the Z0. The last line is the contribution of the Higgs. Here gL;R give the left and right Z0 couplings to fermions, cw = cos W and Z; A are the Z0 and photon Wilson lines. This formula is the generalization of the one in appendix B1 in [41], since it includes the Z0 Wilson line and the Higgs eld. Notice that our gluon Wilson lines are expressed in terms of the basis chosen in [41] as G1 = G1 G2 = G1 + G2 : G2 ; (2.17) (2.18) eq. (2.17) in principle allows for a detailed study of the SM action on the circle. In practice, it is a complicated function of six scalar elds and the fermions boundary conditions, and a full analysis is a formidable task. However a number of conclusions may already be drawn without fully analyzing the complete potential. In gure 2 we plot the value of R6V in the Higgs-Radion plane with Wilson lines xed to zero and periodic boundary conditions for fermions. As expected the minima stays always at H = v. Moreover, this conclusion also holds for di erent values of the Wilson lines. Thus, we will set the Higgs equal to its tree level vev from now on. Even tough the potential in terms of the four Wilson lines is still complicated, we can see with a simple example that the neutrino vacua may become unstable to tunneling due to the existence of runaway directions at smaller radius R. In gure 3 we show the e ective potential as a function of the photon Wilson line and the Radion, with the Higgs eld at the minima and all other Wilson lines set to zero. The point illustrated by this plot is that the e ect of the Wilson lines on the Casimir potential of a particle is to gradually change the Higgs. The tree level potential dominates and the Higgs is not displaced from its tree level minimum by the one-loop corrections. This behavior is independent of the particular value of the Wilson lines Although not very visible in the plot, the Higgs minimum remains at the same location as R 1 increases. Higgs eld at the minima and all other Wilson lines set to zero. Notice that a runaway develops along the Radion direction for a non-zero value of the Wilson line. its sign. More precisely, it transforms periodic particles into antiperiodic ones (there is also a factor 78 involved). The results of [39] correspond to the A = 0, R 1 < me part of the plot, where we still see the neutrino minima. Moving on to higher energies along A = 0, we see a small bump corresponding to the four degrees of freedom of the electron. After going through the QCD transition, where we do not know the shape of the potential, we observe a fast increase caused by the copiously unleashed quark degrees of freedom. At around 100 GeV, the W; Z0 and the Higgs slow down a bit the potential and, nally, the top quark increases it a bit more. Unless new physics is introduced, R6V remains constant for higher energies, so V keeps increasing, at least while our one-loop approximation remains valid. On the other hand, if we displace the Wilson line A from 0, these conclusions are changed. in In particular, for A = 12 or 34 , the sign of each contribution is reversed. As we can see gure 3 this leads inevitably to a negative potential. After the top quark scale, R6V remains constant for higher energies, so V keeps decreasing. This runaway behaviour of the potential could mean that the neutrino minima is actually metastable, as pointed out in [42]. In fact, to be sure that this vacuum is unstable we should be able to nd the bounce interpolating from the vacuum to the runaway direction, which is a complicated question in such a complicated potential. We will however conservatively assume that indeed the vacua are metastable. This would be consistent with the OV conjecture and would lead to no constraints on neutrino masses nor on the hierarchy. If we want to search for constraints on observable physics using the OV conjecture we would need Type P SM vacua in which this potential tunneling in the Wilson line directions is absent. That would be the case of vacua in which the Wilson lines are frozen or projected out in such way that these decay directions disappear. In the rest of this paper we will focus on this class of models. For reasons soon to be explained, a good option is the compacti cation on the orbifold T 2=Z4. But before going to that case we explain in the next section the case of 3D vacua with Wilson line moduli xed, which will give us intuition for the more elaborate T 2=Z4 case. 2.3 Fixing the Wilson lines. The segment S1=Z2 In this section we describe the structure of the radius scalar potential at shorter distances, up to the EW scale, setting the Wilson line moduli to zero by hand, looking for new features beyond the neutrino local minima. In gure 4(a) we show the product R6V for the Standard Model with all fermions having periodic boundary conditions, using the formulae introduced in the previous section for the circle. We see that the only minimum we may obtain is the one associated with the bounds in the neutrino masses (in the Majorana case). Above the neutrino scale the potential grows smoothly until it reaches the QCD region, around one GeV, where non-perturbative physics become important and the oneloop Casimir computation is not a good description (this is denoted by the red, vertical band in the plots). Above the QCD scale perturbation theory again makes sense and the potential keeps growing inde nitely. The small well in the upper part of the gure corresponds to the W ; Z0; H0 threshold, while the nal increase corresponds to the top quark threshold. Consequently, the neutrino minima seems stable and the bounds on neutrino masses would apply. HJEP06(218)5 Let us consider now the case in which some of the fermions are instead assigned antiperiodic boundary conditions. In principle one could naively say that we can assign arbitrary boundary conditions to each fermion multiplet of the SM and each generation. However, due to the presence of Yukawa couplings and generation mixing all quarks must have the same boundary conditions, and likewise for the leptons. So there are four options according to the (quark; lepton) boundary conditions: (P; P ), (AP; P ), (P; AP ) and (AP; AP ). The last possibility, with all fermions antiperiodic does not produce any locally stable vacuum since the potential always decreases and even if they appeared they seem unstable to decay into a bubble of nothing [51]. It is easy to see that the (AP; P ) case will lead to a runaway potential. In the SM there are more degrees of freedom associated with quarks than leptons and, since for small radius the antiperiodic fermion contribution picks up a factor 78 , the potential will grow large and negative. The remaining case with antiperiodic leptons and periodic quarks is illustrated in gure 4(b). As R decreases the potential decreases steadily until the QCD transition. Once passed the hadron region, the potential grows up monotonously due the appearance of all quark degrees of freedom, which dominate the potential as R ! 0. Although the proximity of the QCD decon ning region does not allow for a computation of the precise location of the minimum, one certainly expects a minimum to develop. No lower minima develops for other regions of R and hence that minimum seems stable. This gives a Type D minimum since it cannot be avoided by xing any free parameter of the Standard Model. For example, the masses of all the SM particles around the QCD scale are known, so there is no way to eliminate it by varying masses, as it happens in the neutrino case. As it is, the SM would be inconsistent with the OV conjecture and would not be embeddable into a consistent theory of quantum gravity. Still, up to here we have not provided a reason why the Wilson lines should be xed. Those are xed in the class of T 2=ZN vacua considered in the next section, and again this class of AdS vacua will appear. Moreover, we will see in section 3 that there are even more AdS vacua which will con rm the di culties in embedding the SM into quantum gravity. A way to avoid the AdS vacua above would be to go beyond the minimal SM and add e.g. additional massive bosons above the EW scale so that the potential develops a runaway behavior as R ! 0, making the dangerous minima unstable. Indeed, consider the limit mR ! 0 of the Casimir potential with no Wilson lines. To rst order in the masses of the particles one obtains V = (2 + 2 + 2 8) where we denote by P the particles with periodic boundary conditions and by AP the fermions with antiperiodic boundary conditions. The rst term corresponds to the massless graviton, photon and gluons. It is clear that if we add a su ciently large number of periodic bosons the potential will become negative and unbounded from below as R ! 0, making the potential unstable, and hence leading to no violation of the OV conjecture. This scenario is not particularly well motivated. boundary conditions for all particles, the Higgs xed at its minimum and all Wilson lines turned o . There is no runaway solution. The only AdS minimum we have is the one that would be associated with neutrino bounds. (b) E ective potential of the Standard Model with periodic boundary conditions for all particles except for the leptons. The Higgs is xed at its minimum and all Wilson lines are turned o . There is no runaway solution. We nd an AdS minimum which cannot be avoided by constraining any free parameters of the Standard Model. We will argue in section 4 that a more elegant option is to embed the SM into a SUSY version like the MSSM. It turns out that in the SUSY case this vacuum is not possible since quarks and leptons are forced to have the same boundary conditions. But we will also see that SUSY also avoids new classes of AdS vacua which exist in the orbifold case. Considering SUSY is an attractive possibility because of another important reason. It is well known that the Higgs potential of the SM may have a second high energy minimum at scales above 1010 GeV [52{57]. This 4D minimum is would be in AdS and, if stable, would be again inconsistent with the AdS-phobia condition. Although it would be easy to save the SM by the addition of some BSM physics like e.g. additional singlet scalars eliminating the high energy vacuum, SUSY seems to be a more attractive option. If the SUSY breaking scale is below 1010 GeV the high energy Higgs minimum does not develop (see e.g. [58, 59]), since the SUSY potential is positive de nite. We will come back to the SUSY case in section 4. Up to now we have just set the Wilson lines to zero by hand and have analyzed the resulting potential as a function of R and the Higgs eld. We discuss in what follows how the Wilson lines could be xed. The simplest option seems to be to compactify on the segment S1=Z2, where Z2 is a re ection with respect to one spatial dimension y. We have boundary conditions under y ! y for scalar, vector and spinor elds: (xi; y) = (xi; y) A (xi; y) = fAi(xi; y); A3(xi; y)g (2.20) (2.21) (2.22) We see that the Wilson lines are projected out, they are forced to be zero. Furthermore, half of the Fourier modes are projected out, and one can check that the Casimir potential is reduced by a factor 2. We can use directly the formulas for the circle if we take into account these two points since nothing else changes. In this orbifold the contribution to the Casimir potential of each particle VC [R; mp; ] is thus very simple. The results for the scalar potential would be qualitatively identical to the case of the circle with Wilson lines xed to zero. We could envisage to compactify the SM on the segment and obtain the results above with xed Wilson lines. However there is a technical obstruction in this simple situation. Indeed, re ection with respect to a single coordinate (or in general, parity) is not a symmetry of the SM (since it reverses chirality) and we cannot twist the theory with respect to it, unless we do some modi cation or extension. Such an extension could be the embedding of the SM in a theory which is left-right symmetric like SU(3) SU( 2 )R or SO(10). But then we have to include the Higgsing down to the SM in the computations. We will prefer instead to go to 2D orbifold compacti cations T 2=ZN , with ZN a discrete rotation in two dimensions, which is always a symmetry of the 4D SM. We will rst discuss the case of the torus, from which it is easy to obtain the results for the orbifolds. 3 The SM on the T 2=Z4 orbifold In this section we consider the case of the compacti cation of the SM on the torus T 2 and then on the orbifold T 2=Z4 which is closely related. 3.1 The SM on the T 2 torus In this section we rst review some of the most relevant features and results of the compacti cation of the SM on a torus. Compacti cations of the SM on a torus have been considered in [41, 60, 61] and [39, 42]. As in the circle, to obtain the potential of the moduli we will consider the scalar potential of the SM plus Einstein gravity compacti ed on the torus and the one-loop contribution coming from the Casimir energy of the corresponding particles. We take for the distance element in 4 dimensions, ds2 = g dx dx + B idx dyi + tij dyidyj ; with g the 2-dimensional metric, B 2 and B 3 the two graviphotons, which will be set to zero from now on for the same reason as in the circle, and tij the metric on the torus, which reads tij = a2 " 1 2 # 1 1 j j2 : (3.1) (3.2) Here = 1 + i 2 and a2 are the complex structure and area moduli, respectively. The relevant piece of the e ective action after dimensional reduction on the torus takes the form SGR+SM = Z d x 2 p 1 g2 2 Mp2(2 a)2 R( 2 ) 1 2 V ; (3.3) where V includes the 4D cosmological constant, the tree-level and the Casimir contributions to the potential of the scalars in the theory. Notice that, as expected from the fact that the 4-dimensional graviton has two degrees of freedom, only the two complex structure moduli propagate in two dimensions, whereas the area moduli does not propagate any additional degree of freedom (i.e. it has no kinetic term in the 2D action). Moreover, unlike in the case of the circle, now we cannot perform a Weyl transformation to express the action in the Einstein frame, due to the conformal invariance of 2D gravity. This means that the function V should not be interpreted as a canonical potential, and that we have to resort to the equations of motion in order to study the vacua of the lower dimensional theory. Even though V is not minimized with respect to all the variables at the vacua of the theory, we will still call it potential. The conditions for AdS2 vacua are (see [41, 60] for details): where the j represent all the other 2-dimensional scalars in the theory. Notice that these conditions are the usual minimization conditions for the potential with respect to all the scalars in the theory except a. These two atypical constraints (the ones in the rst column) come from requiring constant eld solutions to the equations of motion. Variation of the action with respect to a yields the constraint (3.4) (3.5) so that the curvature in 2D is xed by the derivative of the potential with respect to a. Thus an AdS vacuum is obtained for a negative derivative. A possible new degree of freedom in the torus case is the addition of magnetic uxes along some U(1) in the Cartan subalgebra of the SM. However, uxes contribute enormously to the vacuum energy and do not lead to interesting vacua, see [42, 60]. Regarding the rest of 2D scalars that enter the potential, we have the Higgs, two Wilson lines (corresponding to the two cycles of the torus) for each of the neutral gauge elds in the SM, that is, two for the photon A, two for the Z0 boson and two more for each of the two gluon elds belonging to the Cartan subalgebra of SU(3)C . The potential that we consider has the following structure V [a; ; H; Zi; Ai; G1i; G2i] = (2 a)2 4 + (2 a)2VSM(H; Z1; Z2) + X Vp(a; ; H; Zi; Ai; G1i; G2i): (3.6) As in the circle compacti cation, it consists of a piece that depends on the 4D cosmological constant 4, a tree level piece for the Higgs and the Z Wilson lines (which comes from the 4D Higgs potential and the 4D mass term of the Z boson, respectively) and a contribution from the Casimir energy of the particles in the spectrum. Again, this last piece is the sum of the contributions coming from each particle and introduces the dependence on the geometric moduli, the Higgs and the Wilson lines to which it couples. In order to compute p the Casimir energy we need to expand our 4D elds in a basis that respects the periodic (zi = 1) or antiperiodic (zi = 0) boundary conditions along the two cycles of the torus, that is (x ; yi) = 1 X After inserting this expansion into the one-loop quantum e ective action and integrating over the compact dimensions, the Casimir contribution of a particle with spin sp, np degrees of freedom and mass mp turns out to be: Vp = ( 1)2sp+1np 2 i 1 X Z n1;n2= 1 The i are functions of the appropriate Wilson lines and boundary conditions zi. Using dimensional or -function regularization techniques one arrives, after renormalizing the cosmological constant, at the following expression for the Casimir energy of each particle [42, 60]: (n+ 1)2 22 +m2a2 2 fLi2 (e + )+Li2(e )g+fLi3(e + )+Li3(e ) g (3.7) (3.8) # (3.9) (3.10) 2 2 q Vp = ( 1)2sp+1np ( 1)2sp+1npVC[a; ; m; 1; 2]; where Lis is a polylogarithm and = 2 i f (n + 1) 1 + 2g q (n + 1) 2 22 + M a2 2 : There are ve scalar elds that must be stabilized at low energies (the three geometric moduli and the two photon Wilson lines), and the number increases as the energy grows: four extra gluon Wilson lines once the QCD scale is surpassed and the Higgs plus the two Z0 Wilson lines when the EW scale is reached. In the case of the circle we saw that the Wilson lines tend to create runaway directions that non-perturbatively destabilize any possible AdS vacua of the lower dimensional theory [42], hence ruining the possibility to obtain any constraint from AdS-phobia, we only obtain Type S vacua. As we saw in subsection 2.3, setting the Wilson lines to zero simpli es things in such a way that constraints are expected to come up. This is our main motivation to consider an orbifold compacti cation of the torus, as the action of the ZN will project out the Wilson lines from the spectrum. Let us explain why we choose to study, in particular, the Z4 orbifold. First of all the complex structure is only xed if the quotient is by ZN with N 3. The quotient by Z2 would project out the Wilson lines but would not x the complex structure. In this case it is necessary to nd the minima in the direction of the complex structure. This is easy to do when all particles have periodic boundary conditions. In this case the potential is invariant under SL(2; Z) modular transformations and therefore the extrema of the potential must be at the stationary points of the complex structure, that is, = 1 or = 1=2 + ip3=2. In fact, as it is concluded in [60], only the latter allows for the existence of minima. However, if there are both P and AP boundary conditions the combined minimization of complex structure and radius becomes less trivial. As shown in subsection 2.3, the appearance of antiperiodic boundary conditions is of some interest, because they may generate additional \exotic"AdS minima, that come out as a generic feature of the SM as we know it. For this reason, we prefer an orbifold in which the complex structure is xed from the start. Finally, among the remaining orbifolds, the formulas for the Z4 orbifold are the easiest ones to work with. Still the general results obtained in the next section are expected to apply for any ZN orbifold acting crystallographically on the torus. SM compacti cation on the T 2=Z4 orbifold The most relevant di erence between T 2 and T 2=Z4 is that we need to introduce additional boundary conditions to prevent the Lagrangian from becoming multivalued as a result of the new identi cations. Instead of using a; it is sometimes useful to use the torus radii R1; R2 and the angle between them as parameters: = ei R2 R1 2 = a 2 R2 1 1 = 1 q R2 1 R12R22 a4 a2 = R1R2 sin (3.11) Remember that, for consistency of the Z4 identi cations [62], the geometrical action of the orbifold must act crystallographically on the torus lattice, xing the complex structure to = i and R1 = R2 R or a2 = R2. We have two symmetry transformations associated with the torus T and three non-trivial ones with the orbifold P. Their action on the compact coordinates is given by: + n1+n2 (xi) sin 2 n1y1 cos 2 n2y2 + n1n2 (xi) sin 2 n1y1 sin 2 n2y2 1 X n1;n2=0 Under any of these symmetries the scalar elds transform as (x; y) is enough to impose boundary conditions for the two translations and for the smallest ! (x; y0). It rotation, of angle 2 expand the modes in the usual Fourier series: . The boundary conditions for the translations imply that we can (xi; y1; y2) = n+1+n2 (xi) cos 2 n1y1 cos 2 n2y2 + n+1n2 (xi) cos 2 n1y1 sin 2 n2y2 TA(y1; y2) = (y1 + R; y2) TB(y1; y2) = (y1; y2 + R) P4(y1; y2) = ( y2; y1) P42(y1; y2) = ( y1; y2) P43(y1; y2) = (y2; y1) (3.12) (3.13) (3.14) (3.15) (3.16) (3.17) which cancels if all MSSM particles have KK towers. Here G is Catalan's constant. One has to be slightly more careful, though. Since the leading contribution as a ! 0 cancels, we have to examine the sub-leading terms to decide whether a runaway behavior exists or not. In the case of the segment one would have for a SUSY spectrum from eq. (2.19) a behaviour for small R V (R)R!0 ! 48 R4 1 X( 1)2sp npm2p 1 48 R4 SS + O(1=R2) where SS is the supertrace over all masses. In the torus and T 2=Z4 cases something analogous but slightly more complicated is obtained. The sub-leading mass-dependent contribution from a bosonic KK tower turns out to be (see appendix B) easy to check that the complete spectrum of the MSSM get KK towers. In addition to the SM particles there will be contributions from KK towers of squarks, sleptons, gauginos, Higgsinos and the gravitino. We are thus including 48 (complex) scalars from the sfermions, 4 new Higgs (real)scalars, 4 Higgsinos, 12 gauginos and one gravitino. Altogether an addition of 60 net bosonic degrees of freedom which just balances the mostly fermion dominated SM. We show in gure 6(b) the function a2V (a) for this case, in which we have chosen a speci c SUSY mass spectrum, provided in appendix D. The qualitative structure does not depend on the details of the spectrum. For values 1=a & MW , the SUSY thresholds open up, and there is a cancellation between fermions and bosons so that the potential goes to zero as a ! 0. Indeed we nd that the leading contribution to the Casimir potential in this limit is given by and the opposite for a fermion. In the (ma) ! 0 limit we are considering the bosonic contribution is negative and the fermionic is positive. Numerically the situation is rather similar to case of the segment. In particular, the log (2 ma) gives even more importance to the most massive particles in the spectrum with respect to the case when only the supertrace appeared. When it comes to understanding the nal sign of the potential it must be noted that, if the sign of the supertrace is dictated by the most massive particle in the spectrum (as it is usually the case), it is guaranteed that the sign of the potential will be actually given by the one of the supertrace. There are then essentially two model-dependent possibilities, depending on the particular structure of the SUSY spectrum: SS 0. In this case there is again an instability at small a, and we have a Type S vacuum. The WGC constraints are evaded and there is no inconsistency. 1 p ( (4.1) (4.2) (4.3) SS > 0. In this case there is no runaway direction and the possibility of stable neutrino minima is recovered, as well as the corresponding predictions, we have a Type P vacuum. This time not only the neutrino masses are constrained but also the SUSY masses. Although both signs may lead to consistent theories, the SS > 0 case is particularly attractive since the interesting constraints from the non-existence of neutrino AdS vacua are preserved. The sign of the superstrace is model dependent. In particular it depends on the values of the SUSY-breaking gaugino, squark, slepton and Higgssino/Higgs masses. One could think that in practically any MSSM constructed to date one has a dominant positive contribution to SS , since there are many more massive SUSY bosons than SUSY fermions, due to family replication. This is true, however, only if one restricts oneself to the e ect of the SUSY partners of the observed SM particles. But here it is also relevant the mass of the gravitino m3=2. The minimum SUSY breaking sector should involve at least a goldstone chiral multiplet, with a goldstino and a s-goldstino. The gravitino becomes massive combining with the goldstino and the s-goldstino typically also gets a mass of the same order. Thus one can also obtain SS > 0 by having e.g. msg heavier than all the rest of the spectrum. The sign also obviously depends on the possible presence of further particles beyond the MSSM and gravitino/goldstino sectors. Still, the condition SS > 0 may be an important constraint on speci c SUSY extensions of the SM. In particular, consider a MSSM model in which the gluino is the heaviest SUSY particle, larger than all other MSSM sparticles but also larger than m3=2 and msg. If the gluino is heavy enough one will violate the SS > 0 condition and, although possibly consistent with the AdS-phobia condition, the radion potential would be unstable and the predictions from the neutrino AdS vacua would be lost. Note however that this can be avoided and the neutrino conditions would survive if e.g. the s-goldstino mass turns out to be heavier than the gluino. Since the s-goldstino and gravitino sector depends strongly on each model we cannot get a rm prediction that the heaviest observable SUSY particle should be a boson, since the SUSY-breaking is in general precluded from the observable sector. On the other hand SS > 0 may be an interesting constraint to test in speci c SUSY extensions of the SM. 4.2.1 New AdS vacua for particular choices of SUSY spectra Although the SS > 0 condition guarantees stability of the potential, it turns out that there can be new AdS minima at nite 1=a for particular choices of SUSY masses. Let us show here a couple of examples. The general structure of these examples requires three ingredients. First, to have a boson as the heaviest particle in order to ful ll the SS > 0 condition and ensure local stability of the minimum. Second, to have an energy scale at which bosons dominate in order to be in the V < 0 region and have a chance to form a minimum. Third, between this scale and the mass of the heaviest boson, we must have some fermionic degrees of freedom that can lift the potential to create the AdS minimum. To make things simple, we will consider here all the supersymmetric particles to have masses U(1)s. The masses of all superpartners of the SM particles are set to 1 TeV except for the gravitino (4 d.o.f.), m3=2 = 19 TeV and the s-goldstino (2 d.o.f.), msgolds = 33 TeV. (b) displays a zoom in of (a) and it can be seen that a Type P AdS minimum forms since it depends on the masses of the SUSY spectrum, hence giving some constraints from AdS-phobia. around 1 TeV except the boson that ensures SS > 0 and the fermion that gives a positive contribution to the potential. The rst example of these kind of vacua is shown in gure 8 and we can constraint the relation between the mass of the gravitino and the mass of the s-goldstino (for xed masses of all the other superpartners). In particular, an AdS minimum forms from m3=2 = 19 TeV and msgolds = 33 TeV onwards, excluding these kind of spectra in SUSY models with U(1)S symmetries. Another example of these kind of vacua can be seen in gure 9, in which we consider the most massive particle to be a squark of mass msquark = 23 TeV and the fermions that lift the potential to be the gluinos, with mass mg = 18 TeV. From these masses on an AdS minimum forms and we can exclude these kind of spectra in the SUSY extensions of the SM. More generally, given a SUSY version of the SM, computing the supertrace gives us information on the stability of the lower dimensional compacti cation and possible constraints on the SUSY spectrum and other parameters, i.e. neutrino masses and the cosmological constant. Furthermore one has to check whether additional AdS vacua can form depending on particular choices of SUSY masses. 4.2.2 Charge-colour breaking AdS minima It is well known that the 4D MSSM has, apart from the standard EW minima in which the Higgs elds get a vev, a plethora of other possible minima in which squarks or sleptons are the ones who get a vev. [84, 85]. The space of these other minima is in general complicated and strongly dependent on the SUSY-breaking parameters. Many of these other minima are driven by the trilinear scalar couplings of the form e.g. Atht(t~RHut~L) + h:c:, where At U(1)s. The masses of all superpartners of the SM particles are set to 1 TeV except for the gluinos (16 d.o.f.), mg = 18 TeV and 1 squark (12 d.o.f.), msquark = 23 TeV. (b) displays a zoom in of (a) and it can be seen that a Type P AdS minimum forms since it depends on the masses of the SUSY spectrum, hence giving some constraints from AdS-phobia. is a soft parameter with dimension of mass, see [85] for a detailed analysis and references. One can derive necessary conditions on soft masses in order to avoid these minima to be lower than the standard Higgs minimum. A well known bound for minima derived from this trilinear stop coupling is jAtj2 < 3ht2 m2Hu + mt2~L + m2~ tL : (4.4) Weaker bounds may be derived by allowing minima lower than the Higgs one but imposing that the minimum is su ciently stable at the cosmological level. In fact most of the examples of SUSY spectra discussed in the literature belong to this class in which the standard Higgs MSSM vacuum is metastable. These charge/colour breaking minima to which the Higgs vacuum is unstable to decay are AdS 4D minima, and the deepest of them will be stable. If we want to forbid AdS non-SUSY vacua altogether these 4D AdS minima should be absent from the start. The logic of AdS-phobia then requires the Higgs vacua of the MSSM to be strictly stable and not just metastable. In fact full stability has been already advocated on di erent grounds, see e.g. [86] and references therein for a recent discussion. A general consequence of imposing strict stability and a Higgs mass around 125 GeV is a quite heavy SUSY spectrum [86]. On the other hand this agrees well with the non-observation so far of any SUSY particle at LHC. In going to 2D on T 2=Z4, a necessary condition in order not to get any of these AdS vacua inherited is to impose as a strict condition the stability of the Higgs MSSM vacuum in 4D. In addition, one expects that some of the stable vacua in 4D may become unstable in 2D. So the actual conditions on SUSY breaking mass parameters will be stronger than the parent 4D conditions. Thus one should impose that 1) the MSSM Higgs vacuum is stable in 4D against the decay into charge/colour breaking minima and 2) it remains stable against decay in 4D. Unfortunately these conditions can only be checked in a case by case basis. Summarizing, with the SM embedded into a SUSY completion the AdS vacua which appeared in the non-SUSY SM become unstable and the theory is safe. The constraints for the neutrino masses, cosmological constant and EW hierarchy are recovered from the existence of a Z4 vacua with action embedded into a discrete subgroup of B L in the R-parity preserving MSSM. Further constraints on the SUSY spectrum appear from a Z4 embedding into a discrete subgroup of the R-symmetry present in the R-parity preserving MSSM, if the supertrace SS > 0. Note that the latter constraint on the MSSM spectrum would only arise if the discrete subgroup of U(1)s is a gauge symmetry of the underlying theory. We see that the EW scale is bounded to be close to its experimental value in order to avoid the generation of stable 2D AdS neutrino generated vacua. On the other hand, avoiding 4D and/or 2D AdS charge-colour breaking minima typically requires relatively large SUSY masses in the multi-TeV region. Note that this could explain the so called little hierarchy problem: the SUSY spectrum needs to be relatively heavy in order to avoid these dangerous minima. On the other hand the EW scale is kept smaller by the condition that the lightest neutrino is su ciently light to avoid AdS neutrino vacua. We must emphasize, though, that from this discussion the scale of SUSY breaking needs not be in the multi-TeV region. It could be much larger, up to a scale e.g. of order 1010 GeV and the stability properties would still persist. Twisted sectors In all the previous sections we have not discussed the possible existence of twisted sectors. The presence of those is expected if there are orbifold xed points in the compacti cation. Fixed points may be absent if the twist in the torus is accompanied by some translation in the compact six dimensions of the original 4D compacti cation. If they are present, in general there will be 2D particles from twisted sectors. Thus e.g. they explicitly arise in heterotic orbifold compacti cations to 2D (see e.g. [87{93]). In Type II orientifolds there may be branes (D-strings in the 2D case) localized on the xed points, leading to additional massless particles in the 2D theory. A full discussion of these twisted sectors would require a full knowledge of the underlying string compacti cation which leads to the SM or the MSSM in the rst place. Fortunately for our purposes, to leading order we can ignore the e ect of possible twisted sectors in our Casimir energy computations. This is because any twisted particle is localized in the singularities and hence does not have any KK tower. As we have emphasized, only states with KK towers contribute to the Casimir potential and hence twisted sectors do not modify to leading order the structure of the radius potential. In this respect their e ect is similar to untwisted zero modes which also do not contribute to the Casimir potential. Notice however that in principle a radius independent constant contribution could arise from the twisted sector. This could be interpreted as a contribution of the tension of the objects localized in the singularities. Like e.g. in orientifold compacti cations in which the tension of branes and orientifolds cancel, one may expect such local contribution to cancel at leading order. We are assuming that vacua exist in which such constant contribution is either absent or small. A small constant term would not alter signi cantly our discussion. At large a the bulk 4D 4 clearly dominates (it is multiplied by a2) whereas at small a the Casimir energy contributions grow rapidly like 1=a2. Therefore, a constant piece may postpone the growth of a negative potential due to the photon and the graviton but it cannot avoid the appearance of an AdS minima. The compacti ed models here discussed are in general chiral in 2D. Thus e.g. the model constructed by twisting by a Z4 subgroup of B L have zero modes transforming like quarks under the SM group, with di erent 2D chiralities and has SU( 2 )L 2D anomalies. The model obtained by twisting by a U(1)s subgroup also has chiral zero modes transforming like all fermions of the MSSM. In 2D there are in general gauge and gravitational anomalies [94] and indeed these spectra by themselves have 2D anomalies. In particular, gauge anomalies are given by the quadratic Casimir eigenvalue Ta of each chiral fermion, with sign depending on whether the zero mode is left- or right-moving. In principle one could use 2D anomaly cancellation conditions to try to gure out what the quantum numbers of the 2D twisted sectors could be. Indeed, one may easily obtain anomaly-free 2D theories by adding appropriate representations, which would be 2D lepton-like objects in our case. See e.g. [87] for speci c 2D examples of anomaly cancellation. Although, as discussed above, the twisted sectors play no role in the computation of the Casimir potential, it would be interesting to study further the 2D anomaly constraints in speci c 2D compacti cations of the SM, MSSM or generalizations, i.e. compactifying a 10D theory directly to 2D. We leave this for future work. 5 Conclusions and outlook In this paper we have studied compacti cations of the SM to 3D and 2D, looking for stable AdS vacua, completing and generalizing previous work in [39, 41]. In those works it was shown how the Casimir energy of the lightest sector of the SM gives rise to a radius dependent potential which may have AdS minima. Our motivation was the conjecture in [33] that posits that no theory with stable, non-SUSY AdS vacua can be embedded into a consistent theory of quantum gravity. In [39] constraints on neutrino masses were obtained from this condition applied to the SM compacti ed on the circle and on the torus. Here the assumption of background independence is crucial so that the constraints can be applied to the theory obtained upon compacti cation. This is also an important assumption in the present paper. For the minimal SM one nds that neutrinos cannot be Majorana (as in fact already suggested in [33]) and upper bounds on the lightest neutrino mass m 1 4:1 10 3 eV(NH) or m 3 1 10 3 eV(IH) are obtained. Furthermore, it was found that the 4D cosmological constant is bounded from below by the scale of neutrino masses, m41 . 4. This is an attractive prediction, since it is the rst argument implying a non-vanishing 4 only on the basis of particle physics, with no cosmological input. Finally, the upper bound on the neutrino mass implies an upper bound on the EW scale (for xed Yukawa coupling), eqs. (2.7), (2.8). This could give an explanation for the stability of the Higgs mass against quantum corrections, i.e. the hierarchy problem. Larger values of the EW scale allow for the generation of AdS vacua. Although the AdS minima which may arise are perturbatively stable, non-perturbative instabilities could arise towards decay into lower minima or runaway directions, if present at smaller compact radius R < 1=me, with me the electron mass [42]. Indeed, as we decrease R, new thresholds of leptons and quarks become relevant to the Casimir energy and the potential becomes more and more complex. In particular, it was shown that the presence of photon Wilson line moduli may give rise to non-perturbative instabilities [42]. If such instabilities exist, there would be no contradiction with AdS-phobia, although then we would lose the attractive constraints summarized above. Searching for predictive vacua, in this paper we study SM compacti cations in which Wilson line moduli of the SM gauge group are projected out. This is known to happen in orbifold T 2=ZN compacti cations and we consider in particular the Z4 case because of its simplicity and because for N > 2 the complex structure of the torus is also projected out and we are left with a potential depending only on the area of the torus and the Higgs scalar. We classify the di erent vacua we nd as Type D, which contain a stable AdS minimum which cannot be avoided by constraining free parameters of the theory, Type S, which have no stable AdS minimum and Type P, which have AdS vacua or not depending on some free parameters of the theory. Twisting only by ZN gives rise only to Type S vacua and no constraints. More interesting vacua are obtained if we embed the Z4 symmetry into U(1) gauge (or gaugable) degrees of freedom of the theory. Embedding Z4 into the SM gauge group we obtain necessarily some stable AdS vacua for the SM. Thus, remarkably, the SM as it stands would not be embeddable into a consistent theory of quantum gravity. This is true for any value of neutrino masses. Note that a further problem for the minimal SM is the possible existence of a second 4D high energy Higgs vaccum at scales above 1010 GeV. This minimum would be AdS and, if stable, would again be in contradiction with the OV conjecture. Interestingly, if the SM is embedded into a SUSY version like the MSSM, those AdS vacua become automatically unstable and the theory is consistent with WGC constraints. Furthermore, the second high energy Higgs vacuum also disappears if the SUSY breaking scale is not above 1012 GeV. We study further possible Type P vacua of the MSSM compacti ed in T 2=Z4 which could lead to constraints on particle physics. In particular, an interesting vacuum is obtained if we embed Z4 into a discrete subgroup of the U(1)B L symmetry, which is a global symmetry of the R-parity conserving MSSM and may be gauged at higher energies. The resulting vacuum has a potential with just a possible AdS minimum around the neutrino region. This is the same AdS minimum found in [39, 41], and may be avoided if one of the (Dirac) neutrinos is su ciently light. Thus we again have the constraints on the cosmological constant and the gauge hierarchy described in [39, 40]. This suggests that the MSSM should be extended to include a U(1)B L gauged symmetry. It is interesting to see whether there are other Z4 compacti cations leading to further phenomenological restrictions e.g., on the values of the SUSY masses. The other familyindependent global symmetry of the R-parity preserving MSSM is the R-symmetry U(1)s of HJEP06(218)5 theories obtained in string compacti cations. If we embed Z4 into a discrete subgroup of this U(1)s we obtain a 2D model in which all MSSM particles have a KK tower. In this case, depending (essentialy) on the mass2 supertrace, new conditions on the SUSY spectrum appear. However in this latter case the constraints would apply only if the underlying theory contains such discrete R-symmetry. Additional constraints come from avoiding the presence of charge and/or colour breaking AdS minima both in 4D and 2D. This typically requires a relatively heavy SUSY spectrum in the multi-TeV region [86]. These arguments could provide a possible explanation for the little hierarchy problem, i.e. the fact that, if low energy SUSY is correct, the SUSY particles seem to be relatively heavy compared to the EW scale. From the present point of view the EW scale is small due to the constraint in eqs.(2.7), (2.8) whereas the SUSY spectrum is forced to be relatively large to avoid charge/colour breaking AdS minima. An important question is whether the scale of SUSY-breaking MSS and the masses of SUSY particles are close to the EW scale and LHC energies or not. It would be extremely interesting if the AdS-phobia condition as applied to the MSSM could give us a hint on what the scale of SUSY breaking is. In particular it is conceivable e.g. that avoiding the appearance of AdS vacua when compactifying the MSSM down to 2D could require a lowenergy SUSY spectrum around a few TeV, low-energy SUSY. Or else that avoiding such vacua could require a very massive SUSY spectrum. Note in this respect that, as we said, SUSY cannot be arbitrarily high, since if MSS > 1010 GeV the second lower SM Higgs minimum may develop, which would be in AdS. To test whether there is a preference for low-energy SUSY or not coming from AdS-phobia we have to improve the present analysis. In particular we should consider the renormalization group improved couplings and masses. When going to very high energies (or rather very small a) large logs will appear which cannot be ignored at the quantitative level. The soft masses run and e.g the sign of SS may change as masses and couplings run. We leave this important analysis for future work. We have used the 2D vacua in this paper as auxiliary tools in order to derive constraints on the parent SM or MSSM 4D model. However some of these 2D vacua could be cosmologically interesting in the following sense. Consider the 2D vacua obtained from embedding into U(1)s. It contains zero modes for all the particles in the MSSM except for the Higgs bosons. Once the AdS neutrino (and/or SUSY) bounds are respected, the potential is monotonously decreasing into large 2D volume, the KK towers become massless and we recover the 4D MSSM (including the Higgs bosons) as a ! 1. So one could speculate that the universe started two-dimensional and became 4D well before experimentally constrained cosmological events took place. In such a model the 4D cosmological constant would have been 2D volume dependent. The situation would be consistent with the conjecture in [32] that no stable dS vacuum should exist and the universe should have a runaway behavior. It would be interesting to explore whether a sensible cosmology could be constructed in this scheme. our ndings: Since the casuistic above could confuse the reader, let us conclude with a brief list The SM as it stands necessarily has stable AdS vacua in 2D and hence would be in the swampland. In the MSSM those AdS vacua become unstable and lead to no incompatibility with quantum gravity. If the MSSM is extended by a U(1)B L gauge group (or a discrete subgroup), the theory has 2D AdS vacua which can be avoided if neutrinos are Dirac and the lightest is su ciently light. The four predictions listed in the introduction are recovered. If in addition the MSSM 4D vacua has a gauged discrete R-symmetry, subgroup of the global U(1)s R-symmetry of the R-parity preserving MSSM, further constraints on the SUSY spectra, depending on the supertrace, are obtained The hierarchy problem is solved by imposing absence of AdS neutrino vacua. But SUSY is needed to avoid additional AdS minima which would otherwise be present. Thus the SUSY spectrum could be substantially above the EW scale, but also possibly in the multi-TeV region. Finally, it would be important to improve our understanding of the stability of this kind of SM compacti cations to lower dimensions. In particular in the 2D vacua the radius a does not propagate and the Higgs is the only propagating degree of freedom (along with the sfermions if present). It would be interesting to study how the tunneling towards lower minima or runaway directions happens in this class of theories. More generally the results depend on the validity of the assumption of the Ooguri-Vafa conjecture of AdS-phobia and it would be important to bring additional evidence in favour or against it. In the meantime we think it is well motivated to study what the consequences of its validity would be, and they look, indeed, quite intriguing. Acknowledgments We thank A. Font, F. Marchesano, V. Martin-Lozano, A. Uranga, I. Valenzuela, C. Vafa and especially M. Montero for useful discussions and suggestions. This work has been supported by the ERC Advanced Grant SPLE under contract ERC-2012-ADG-20120216320421, by the grant FPA2012-32828 from the MINECO, and the grant SEV-2012-0249 of the \Centro de Excelencia Severo Ochoa" Programme. A.H. is supported by the Spanish FPU Grant No. FPU15/05012 and E.G. by the Spanish FPU Grant No. FPU16/03985. HJEP06(218)5 Model 1 p 2 We denote the quantum e ective action by and the classical, background elds by an overline. We are interested only in the ground state of the theory. For this reason we can set the classical, background elds of the fermions to zero from the start. We perform the computation in the background- eld gauge, because we will use the background- eld method to compute the e ective action and this choice enables us to maintain gauge invariance in the background, gauge-boson elds. If we parameterize the Higgs doublet as , then the gauge xing term is given by: i + g"ijkW j W k + i g 2 1 h 2 i g0 4 yY v vyY 2 2 y i v 2 1 2 vy 2i i2 @ Ga + gfabcG bG c : (A.1) ! 1. Using the available We work in the unitary gauge, obtained by taking the limit ; background gauge invariance one can gauge away all components except the one along the compact direction of those bosons in the Cartan Subalgebra of SU(3)C We denote these four Wilson lines as Z; A; G1; G2, corresponding to the vevs of the Z boson, the photon and the gluons associated with the two diagonal Gell-Mann matrices. We de ne the Wilson lines so that they absorb the gauge coupling constants in the covariant derivative. Using the background eld method, we must compute the following path integral, neglecting terms associated with diagrams which are not connected and 1PI. At one-loop order we need only study the terms quadratic in the quantum elds, since linear terms would never give 1PI diagrams and zero order terms correspond to the classical (tree level) action. Denoting by to all fermions and by to all the necessary ghosts, and denoting in general as to all elds in the theory, the measure would then be D = Dg DA DZ DW +DW DH D D D D and the path integral to compute ei [R;H;Z;A;G1;G2] = D eiS[ + ]: Z (A.2) 1PI, Connected For completeness, we will write in some detail the SM Lagrangian. We will omit from the Lagrangian the part of the gauge- xing term (A.1) that depends on the background elds (of course, it must included in the calculations of the e ective action). We do not write in any detail the ghost Lagrangian LSFMP, since it was shown in [49] that they do not contribute to the e ective action at one-loop order. The computation of the e ective potential of the Einstein-Hilbert Lagrangian compacti ed in S1, to one-loop order, was performed in [49], so we also omit the details here. We will also omit the details of the counterterm Lagrangian and the renormalization procedure. To regularize we will use either Dimensional Regularization with extra at dimensions or Zeta Function Regularization techniques. Note that ; ; n; m space-time indices run from 0; 1; 2; 3; i; j space-time indices from 0; 1; 2; i; j colour indices run from 1; 2; 3 and a; b colour indices run from 1; 2 : : : 8. We will replace all space-time covariant derivatives by parf 1 2 1 2 tial derivatives from the start, since we will be interested only in the one-loop corrections to the tree level potential. Finally, f denotes sum over all fermions and A sum over the fermions families: eA = (e; ; ), pA nA ! u c t d s b ! . L = Lasymptotic +Lbasic interaction +LSFMP +LcSoMunterterms +LEinstein-Hilbert + 4 SM SM Lasymptotic = LFermions +L(Y2M) +L(S2B)S SM LFermions = X mf ) f ~ = en n = nmenem L(Y2M) = W [ M W2 ]W + + Z [ 1 2 + A [ 2 L(S2B)S = Lbasic interaction = LNC +LCC +L3YM +L4YM +L(S3B+S4) +LYW SM LNC = X f ~ (eQf A + g cW (gLfPL +gRfPR)Z +gsGa Tfa) f ; MZ2 ]Z HJEP06(218)5 Tfa = LCC = p X A ( 0 colour singlet ) colour triple h:c] 4 LYM = W + Z Z W + Z A W W W +Z Z ] e2[W W + A A W W +A A ] W +(Z A +A Z )] +g2(W W + W W + W +W + W 2 LSBS W + + 2c12w Z Z ] W vH3 1 2 H4 4 )+ 4 gs fabcfadeGb Gc G dG e LYW = gL;R = T3 L;R s2wQf H X mf v f f f : Taking into account the boundary conditions, each eld is expanded in a Fourier series, as explained in the text. Since we are staying at cuadratic order, the integral over each eld will be gaussian. Each gaussian integral reduces to computing the determinant of a certain operator. Using det A=elog TrA; we transform the determinant into a trace. After computing the traces for the di erent particles one nds that the contribution of each particle to the Casimir potential can always be written as: 1 Z d3p p2'2 + H mp v 2 ! + (n + )2 '2 : (A.3) , , e, , u,c,t d,s,b 2 3 1 1 3 1 2 1 2 12 + s2w s 2 2 3 w 12 + 13 s2w 3 w s 1 2 3 w U(1)Y quantum numbers of the particles in the Standard Model. B One-loop e ective potential in T 2=Z4 The rst steps of the computation are essentially the same as in the previous appendix. Besides, some details were given in the text so we do not repeat them here. In the text we also mentioned that, to gain a better understanding of the Casimir potential, it is interesting to study the ultraviolet behaviour. In this appendix we will use the expressions from the text to extract this UV behaviour. To take this am ! 0 limit, it will be useful to rewrite the general formula for the Casimir potential in the torus using the following formulas from [95] Lis(rei ) = Lis(r; ) + i Tis r sin 1 r cos Tis r sin 1 r cos ; tan where Li2(r; ) = Li3(r; ) = 1 Z r log 1 2 0 Z r Lis(x; ) 0 x dx; 2x cos x + x2 dx; and Tis are generalized inverse tangent integrals which verify: Tis ( x) = Tis (x), Tis ( x; y) = Tis (x; y). Thus, Lis(rei ) + Lis(re i ) = 2Lis(r; ). Introducing this last equation in (3.9) we nd: VC [a; t; mp; 1; 2] = e + = e 2 p(n+ 1)2t22+m2a2t2 ei2 [ (n+ 1)t1+ 2] = rei : 2 p=1 1 X + 1 2 t2 n= 1 (B.2) (B.3) # (B.4) (B.5) Secondly, we set t1 = 0 and t2 = 1 in eq. (B.4) and eq. (B.5), 1 = 2 , and take the massless limit: p=1 1 2 1 X n= 1 p4 2 jn + j Li2(e 2 jn+ j; 2 ) + Li3(e 2 jn+ j; 2 P1 p=1 zpps we can do the geometric sums in n using Next, we can use that for periodic boundary conditions Lis(r; 0) = Lis(r), and for antiperiodic Lis(r; ) = Lis( r). Finally, using the de nition of the polylogarithm Lis (z) = (B.6) ) o # : (B.7) (B.8) (B.9) (B.10) (B.11) 3 3 1 X n= 1 1 X n= 1 jnj (e 2 p)jnj = (e 2 p)jnj = 2e 2 p e 2 p)2 = 1 2 csch2 p 1 1 + e 2 p e 2 p = coth p for the periodic case and X n=1 j2n + 1j (e 2 p)j2n+1j = X(e 2 p)j2n+1j = n=1 2 e 6 p + e 2 p e 4 p)2 1 4 1 = csch2 p + sech2 p 2e 2 p e 4 p = csch2 p 1 4 = csch2 p coth 2 p for the antiperiodic case. For both cases we have been able to reduce the result to one single summation. The nal step is to rewrite these sums in terms of known ones. For this purpose we rely on identities such as: 1 2 X1 coth p p=1 p3 7 2 360 which can be found in [96]. We have checked numerically the validity of the identities we use with arbitrary precision, but we could not nd a reference in the literature where they were proven. The massless Casimir function for periodic boundary conditions gives: VC [a; 0] = 1 1 1 2 2 1 1 2 2 1 X p=1 1 p4 + 1 2 1 X p=1 (2 a)2 4 2 Li4(1) + n= 1 X X n= 1 p=1 2 jnj Li2(e 2 jnj) + Li3(e 2 jnj)o5 2 p2 jnj (e 2 p)jnj + p3 (e 2 p)jnj 5 1 2 1 2p2 sinh2 p coth p p3 3 5 1) + (2 a)2 3 1 1 1 p=1 1 X1 ( 1)p + 1 p4 2 1 X n= 1 1 2 Li4( 1)+ 1 X1 n ( 1)p 2 1 2 p=1 n=1 2 = (2 a)2 6 1 G = 1 2 VC [a; 0] : where G ' 0:915966 is Catalan's constant. For the case of antiperiodic boundary conditions the Casimir energy reads: VC ha; 12 i j2n+1j Li2( e j2n+1j)+Li3( e j2n+1j)o5 2p2 + p3 3 5 In the ultraviolet region of supersymmetric models, all particles are essentially massless, so the sign of the total casimir potential is determined by the next order in the expansion. Retaining the next order in the polylogarithms we nd the following cuadratic terms in am. For the periodic case we have: C = (am)2 log (2 ma) 2 6 1 4 1 Li2(1) + X 2 log 1 e 2 n 5 2 + log n=1 1 ) 4 2 3=4 Finally, for the antiperiodic case we nd: C V ( 2 ) a; m; 1 2 = (am)2 = (am)2 Li2( 1) + 2 12 3 4 1 X n= 1 log 2 log 1 + e j2n+1j ) The sign of the second order terms is opposite to the leading order sign. This means that it is positive for bosons and negative for fermions. The sign of the potential is controlled by the most massive particles in the spectra. If the fermionic degrees of freedom win, the potential will be negative and it will develop a runaway solution. (B.12) 3 # (B.13) ) (B.14) (B.15) Consistency of the expansion about at background in the noncompact dimensions In this appendix, we show that taking the Minkowski metric as the background metric for the calculation of the one-loop potential of the 3D and 2D theories is a valid approximation up to the energy scales that we have studied in this paper. When computing the one-loop quantum e ective action, one obtains, in the general case, extra contributions from the curvature of spacetime that take the form of quadratic terms in the Riemann tensor, the Ricci tensor and the Ricci scalar. In particular, since we want to study the vacua of the theory, we are interested in the cases in which the background metric is dS, Minkowski or AdS. The typical energy scale associated to these terms is then given by the inverse of the radius of curvature, that is, the dS or AdS length. Hence, neglecting them (i.e. taking the background metric to be Minkowski) is justi ed as long as this energy scale stays well below the energy scales of the other contributions to the potential. In our case, the typical energy scale associated with the e ective potential is the KK scale of the compacti cation and in this way we can say that our calculations are consistent and meaningful as long as we ful ll the condition lAdS/dS lKK : Note that it is by no means the intention of this appendix to give a detailed calculation of the contributions to the one-loop e ective action coming from the non-zero curvature of the background metric but just to show that they are negligible for the cases we have studied. Before going to the two cases that are relevant for us, let us recall the general relation between the Ricci scalar and the radius of curvature (i.e. the dS or AdS length), l, which takes the form R(d) = 1) R(3) = 6 (3) = 6 V Mp(3) ; l = p 2 RMp(4) : V 1=2 (C.1) (C.2) (C.3) (C.4) In particular, it is interesting to rewrite this equation in terms of quantities that we can easily obtain from our plots. For that purpose we recall that in our gures we plot R6V where the positive sign corresponds to dS whereas the negative one corresponds to AdS. 3D potential. In the circle compacti cation, we can use Einstein's equations to relate the Ricci scalar with the 3D cosmoical constant, (3), which at the same time is related with the potential, V , obtaining where Mp(3) = p 2 RMp is the 3D reduced Planck mass. In this way, since both the Planck mass and the potential depend on the radion, we can use eq. (C.2) to obtain an expression for l as a function of the radion eld and from there work out which values of R ful ll the condition (C.1). This expression takes the form in units of degrees of freedom, so n = 720 R6V , the e ective degrees of freedom, is the quantity that we read from the plots. In terms of these variables, eq. (C.4) can be reexpressed as l = r 720 2 where, r is just an arbitrary energy scale that ensures the right units for the potential, so we can take it to be 1 GeV without loss of generality. From the gures we can safely say that, in these units, the potential takes values from n = 10 to n = 70 in all the cases we have studied (see gures 1{5), so we can take it to be n 100 to be safe. After plugging in the numbers, one obtains that taking Minkowski as the 3D background metric in the one-loop quantum e ective action calculation is justi ed as long as as in all our models. 2D potential. Due to conformal invariance of 2D gravity, we have seen that we cannot go to the Einstein frame in order to de ne a canonical potential or cosmological constant in 2D. However, the equations of motion for the T 2=Z4 orbifold allow us to express the 2D Ricci scalar in terms of the potential as follows 107 GeV; R( 2 ) = (2 Mp)2 a 1 l = 23=2 Mp a As before, we can use eq. (C.2) to obtain a relation between the curvature radius and the potential and it takes the form Following the same reasoning as in the 3D case, we want to express this in terms of variables that we can easily read from our plots, that is, in terms of n = 12 2a2V =G and its derivatives with respect to log(a 1[GeV]). Taking all this into account, we can rewrite the previous expression as l2 = 8 2Mp2 G a 4 : log e 12 2 2a2V + @log(a 1)(a2V ) From gures 6{9 it can be seen that n 100 is again a safe value and that the slopes can also be overestimated by taking them to be 100. Plugging in these values, one obtains that 2D Minkowski background is a safe approximation if we ful ll a 1 1018 GeV: (C.10) This again implies that, with a conservative estimation, the potential we have considered would be changed so slightly by the corrections coming from a non-vanishing 2D curvature that we can safely ignore them. 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Eduardo Gonzalo, Alvaro Herráez, Luis E. Ibáñez. AdS-phobia, the WGC, the Standard Model and Supersymmetry, Journal of High Energy Physics, 2018, 51, DOI: 10.1007/JHEP06(2018)051