Uber-naturalness: unexpectedly light scalars from supersymmetric extra dimensions
aa r X i v : . [ h e p - t h ] M a y Preprint typeset in JHEP style - HYPER VERSION
IC/2010/013DAMTP-2010-31 ¨Uber-naturalness: unexpectedly light scalarsfrom supersymmetric extra dimensions
C.P. Burgess, , Anshuman Maharana and F. Quevedo , Department of Physics & Astronomy, McMaster University, Hamilton ON, Canada. Perimeter Institute for Theoretical Physics, Waterloo ON, Canada. DAMTP/CMS, University of Cambridge, Cambridge CB3 0WA, UK. Abdus Salam ICTP, Strada Costiera 11, Trieste 34014, Italy.
Abstract:
Standard lore asserts that quantum effects generically forbid the occurrence oflight (non-pseudo-Goldstone) scalars having masses smaller than the Kaluza Klein scale, M KK ,in extra-dimensional models, or the gravitino mass, M / , in supersymmetric situations. Weargue that a hidden assumption underlies this lore: that the scale of gravitational physics, M g ,( e.g the string scale, M s , in string theory) is of order the Planck mass, M p = √ πG ≃ GeV. We explore sensitivity to this assumption using the spectrum of masses arising withinthe specific framework of large-volume string compactifications, for which the ultravioletcompletion at the gravity scale is explicitly known to be a Type IIB string theory. In suchmodels the separation between M g and M p is parameterized by the (large) size of the extradimensional volume, V (in string units), according to M p : M g : M KK : M / ∝ V − / : V − / : V − . We find that the generic size of quantum corrections to masses is of the orderof M KK M / /M p ≃ M p / V / . The mass of the lighest modulus (corresponding to the extra-dimensional volume) which at the classical level is M V ≃ M p / V / ≪ M / ≪ M KK is thusstable against quantum corrections. This is possible because the couplings of this modulusto other forms of matter in the low-energy theory are generically weaker than gravitationalstrength (something that is also usually thought not to occur according to standard lore).We discuss some phenomenological and cosmological implications of this observation. ontents
1. Introduction 12. Generic loop estimates 3
3. Large-volume (LV) string models 13
4. Radiative corrections in LV models 15
5. Scenarios 206. Conclusions 23A. Extra-dimensional kinematics vs KK sums 25
1. Introduction
Light scalar fields play a disproportionate role in our search for what lies beyond the StandardModel. On one hand, there are a variety of reasons why scalar fields are very useful: theirexpectation values provide the Lorentz-invariant order parameters for spontaneously breakingsymmetries, at least within the weakly coupled limit that is under the best theoretical control.They are ubiquitous in supersymmetric and extra-dimensional theories, which are among thebest motivated we have, where they arise as symmetry partners of particles having other (4D)spins. And if they are sufficiently light, scalars can do interesting things: they play importantroles in many of the various cosmological scenarios that have been proposed to explain themysteries of Cosmic Inflation (and its alternatives), Dark Matter and/or Dark Energy.However, scalars are notoriously difficult to keep light enough to be relevant to present-day phenomenology. Because their masses are difficult to protect from receiving large quan-tum corrections, they are usually very sensitive to the ultra-violet (UV) sector. For instance,– 1 – light scalar φ coupled to a heavy field ψ through a coupling g φ ψ , generically generates(see § δm φ ≃ (cid:18) gM π (cid:19) , (1.1)from the graph shown in Fig. 1. Here M is the mass of the heavy ψ particle, and the factorsof 4 π are those appropriate to one loop (in four dimensions). Because of such contributions, itis often only possible to obtain m φ ≪ gM/ π if there is a conspiracy to cancel very precisely– often to a great many decimal places – these kinds of large loop contributions against otherparameters in the underlying microscopic theory describing the ultra-violet physics. ✫✪✬✩s φ φψ Figure 1 : A large mass correction to a light scalar from a quartic coupling.Because of this, light scalars are rarely found in a theory’s low-energy limit, with the rareexceptions corresponding to when a (possibly approximate) symmetry protects the scalar fromreceiving large quantum corrections. On one hand, the comparative rarety of such naturallylight scalars can be regarded as a feature and not a bug: it could explain why no fundamentalscalars have yet been found experimentally. But on the other hand, this makes it difficult tokeep scalars light enough to be useful for understanding the origin of electroweak symmetrybreaking, or to be relevant for cosmology. It is this observation that lies at the core of theelectroweak hierarchy problem, among others.A great deal of attention has therefore gone towards exploring those cases where sym-metries are able to protect light scalar masses without conspiracy. The known symmetriesof this kind are: ( i ) approximate shift symmetries, such as φ → φ + c and its nonlinearextensions, as appropriate for Goldstone and pseudo-Goldstone bosons; ( ii ) supersymmetry,for which cancellations between superpartners of opposite statistics suppress contributionsfrom heavy particles whose mass is higher than the relevant supersymmetry-breaking scale(such as the gravitino mass, M / ); and ( iii ) extra dimensions, for which higher-dimensionalsymmetries (like gauge invariance or general covariance) can protect masses from receivingquantum corrections larger than the Kaluza-Klein (KK) scale, M KK .Since both the supersymmetry breaking scale and the KK scale cannot be too low withoutrunning into phenomenological difficulties, it is usually expected that the only scalars likelyto be light enough to be relevant at very low energies are Goldstone (or pseudo-Goldstone)bosons, despite the fact that many candidates for fundamental theories count an abundanceof scalars among their degrees of freedom. In the absence of a protective symmetry noneof these scalars would survive to be light enough to observe experimentally. This is true inparticular for string theory, which has both supersymmetry and extra dimensions as sourcesfor its many scalars. – 2 –ntil recently this expectation has proven hard to test, because of the technical difficultiesassociated with exploring the spectrum of excitations near ‘realistic’ vacua, far from thesupersymmetric configurations for which calculations are best under control. What is newin recent times is the ability to explore non-supersymmetric vacua to see how massive thevarious would-be light scalars become once supersymmetry breaks.Particularly interesting from this point of view are large volume (LV) flux compactifica-tions [1] of Type IIB string vacua. These have the property that the same flux that fixes thescalar moduli also breaks supersymmetry, producing a very predictive spectrum of scalarswhose masses depend in a predictable way on the (large) extra-dimensional volume. SinceLV models in particular predict (at the classical level) the existence of non-Goldstone scalarsthat are parametrically much lighter than both the KK and the gravitino mass, commonlore would lead one to expect that there are large loop corrections to these classical masspredictions, which lift their masses up to either M KK or M / .Our purpose in this paper is estimate the generic size of radiative corrections to scalarmasses in LV models, in order to estimate how these compare with the predicted classicalvalues. We find that since loop effects are smaller than the classical predictions, the classicalmasses really do provide a good approximation to the full result. LV models therefore provideone of the few examples of light (non-Goldstone) scalars whose masses are naturally smallerthan both the SUSY breaking and KK scales. It turns our that their masses are nonetheless( ¨uber ) natural because of an interesting interplay between the scale of supersymmetry breakingand the size of the extra dimensions.We organize our observations as follows. First, § § §
4, by a discussion of how large the radiative corrections areto these masses in LV models, with the estimates compared with (and shown to agree with)the results of explicit string loop calculations, when these can be done. Finally we summarizeour conclusions in §
2. Generic loop estimates
We start with a discussion of the generic size of loop effects in four- and extra-dimensionalmodels, contrasting how the supersymmetric case differs from the non-supersymmetric one.This section is meant to provide tools for later application, and because it does not containnew material (although perhaps presented with a slightly different point of view) it can beskipped by the reader in a hurry.
To set the benchmark for what should be regarded to be a generic quantum mass correction,consider a model involving two real scalar fields, one of which ( φ ) is light while the other ( ψ )– 3 –s heavy: − L ( φ, ψ ) := 12 h ( ∂φ ) + ( ∂ψ ) i + 12 h m φ + M ψ i + 124 h λ φ φ + 6 g φ ψ + λ ψ ψ i , (2.1)where M ≫ m > ψ is heavy, we may integrate it out to obtain an effective theory describing theself-interactions of φ at energies below M . One of the contributing graphs is given in Fig. 1,which when evaluated at zero momentum in dimensional regularization leads to δm ≃ g µ ǫ Z d n k (2 π ) n (cid:18) k + M (cid:19) + · · ·≃ c µ ǫ ǫ (cid:18) gM π (cid:19) + · · · , (2.2)where 2 ǫ = n − → /ǫ divergence is explicitly dis-played by factoring it out of the dimensionless coefficient c . This divergent contribution isrenormalized into the light-scalar mass parameter, m ( µ ), along with its other UV-divergentcontributions.The renormalized parameter m acquires a renormalization-group (RG) running that atone loop is of order µ ∂m ∂µ ≃ h c g M + ( c λ φ + c g ) m i , (2.3)where c i are dimensionless constants (into which factors of 1 / (4 π ) are absorbed), and the m term arises from the graph of Fig. 2 using the quartic φ interaction, and from the (wave-function) renormalization of the kinetic terms. This may be integrated to give m ( t ) = e R t dx ( c λ φ + c g ) (cid:20) m + c Z t d x g M e − R x du ( c λ φ + c g ) (cid:21) ≃ m (cid:18) µµ (cid:19) c λ φ + c g − c g M c λ φ + c g " − (cid:18) µµ (cid:19) c λ φ + c g , (2.4)where m = m ( µ = µ ) and t = ln( µ/µ ). Here the second, approximate, equality neglectsthe µ dependence of λ φ , g and M .This expression shows that m ( µ ) can only be much smaller than M at scales µ ≪ M if m ( M ) is large – of order gM / (4 π ) – in order to very precisely cancel with the evolutionfrom µ = M to µ ≪ M . A parameter like this, whose small size cannot be understood at anyscale µ where one chooses to ask the question, is called ‘technically unnatural.’ Although We deliberately do not phrase this discussion in terms of a momentum cutoff, Λ, as is often done, for reasonsdescribed in more detail elsewhere [2]. Rather than meaning that naturalness arguments are irrelevant, it meansas this section shows, that they can be usefully recast in terms of ratios of renormalized mass parameters. Just how repulsive this is depends on how detailed the cancellation must be. Although reasonable peoplecan differ on how repelled they are by cancellations of 1 part in 100 or 1000, most would agree that cancellationsof 5 decimal places or more would be unprecedented. – 4 –e know of many hierarchies of mass in Nature, we know of none for which the small scaleemerges in this kind of technically unnatural way. Instead, either m really is measured to beof order gM / (4 π ) (and so is not unnaturally small), or there exists a (possibly approximate)symmetry [3] that ensures that corrections to the parameter m are never larger than of order m itself.The goal of this and later sections is to estimate the size of these loop corrections to scalarmasses from several sources in extra-dimensional models, viewing them as lower bounds onthe masses of the physical scalars that can naturally emerge from such models. Relevant and irrelevant interactions
In the previous example the coupling between light and heavy sectors was through a marginalinteraction, described by the dimensionless coupling g . But there are also other kinds ofdangerous interactions that can generate large corrections to small scalar masses. One classof these consists of super-renormalizable – or relevant, in the RG sense – interactions, forwhich the corresponding couplings have dimension of a positive power of mass (in unitswhere ~ = c = 1). ✫✪✬✩s s φ φψ Figure 2 : A graph contributing to a scalar mass through cubic couplings.For instance, if one were to supplement the above theory with super-renormalizable cubicinteractions, that break the symmetry φ → − φ , − ∆ L = 16 h ξ φ φ + 3 h φ ψ i , (2.5)then evaluating the graph of Fig. 2 gives the new contribution δm ≃ h Z d n k (2 π ) n (cid:18) k + M (cid:19) + · · · , (2.6)and so µ ∂m ∂µ ≃ c ′ (cid:18) h π (cid:19) + · · · . (2.7)Barring cancellations one expects m to be larger than either ( h/ π ) or ( gM/ π ) , whicheveris largest. A possible exception is the 4D cosmological constant, for which no completely convincing technicallynatural proposal has been made. We regard the jury to be out on whether a technically natural solution tothis particular problem is possible. – 5 –n general the underlying theory could also include non-renormalizable – or irrelevant, inthe RG sense – effective interactions, such as − ∆ L nr = 148Λ φ ψ , (2.8)where Λ ≫ M ≫ m is some still-higher scale that is already integrated out to obtain ourstarting lagrangian, eq. (2.1). This kind of interaction generates a contribution to the lightscalar mass (see Fig. 3) that is of order δm ≃ (cid:20)Z d n k (2 π ) n k + M (cid:21) ∝ (cid:18) π (cid:19) M Λ , (2.9)and so contributes contributions to m that are small relative to those already considered. ✫✪✬✩s✫✪✬✩ φ φψψ Figure 3 : A large mass correction to a light scalar from an irrelevant coupling.
Two new issues that arise when considering naturalness in extra-dimensional models (seerefs. [4] for other discussions of loops in extra dimensions) are higher-dimensional kinematicsand symmetries. An extra-dimensional generalization of the two-scalar model described aboveprovides the simplest context for discussing the first of these, while higher dimensional gravityprovides the most commonly encountered framework for the second. We therefore considereach of these examples in turn.
Scalar fields
Start first with a light and heavy scalar field in D = 4 + d dimensions, with lagrangian −L = 12 h ( ∂ Φ) + ( ∂ Ψ) i + 12 h m Φ + M Ψ i + g + g Ψ + · · · , (2.10)and use (for now) a flat background metricd s = η µν d x µ d x ν + L δ mn d y m d y n , (2.11)whose extra-dimensional volume is V = L d . Since a canonically normalized scalar fieldin D dimensions has engineering dimension (mass) ( D − / , the cubic and quartic couplingsgenerically are RG-irrelevant, having negative mass dimension g ∼ (cid:18) (cid:19) ( D − / and g ∼ (cid:18) (cid:19) ( D − / . (2.12)– 6 –e next integrate out the massive field Ψ, assuming the masses satisfy the hierarchy M ≫ m ≫ M KK := 1 /L . In this limit the sum over discrete KK modes is well describedby a continuous integral, and so the contribution of the cubic interaction (from the graph ofFig. 2) is of order δm ≃ g Z d n k (2 π ) n (cid:18) k + M (cid:19) ∝ g M D − ≃ (cid:18) M Λ (cid:19) D − M , (2.13)where this time n = D − ǫ → D rather than 4. The result obtained from using the quarticinteraction in Fig. 1 is similarly estimated to be δm ≃ g Z d n k (2 π ) n (cid:18) k + M (cid:19) ∝ g M D − ≃ (cid:18) M Λ (cid:19) D − M . (2.14)These both differ by the factor ( M/ Λ) D − = ( M/ Λ) d relative to the corresponding 4D ex-ample (for which d = 0), and these extra powers of M arise ultimately due to the additionalphase space available for the extra-dimensional loop momenta. If M ≃ Λ then the correctionsare generically of order M in any dimension, but a proper exploration of contributions at Λshould really be done using whatever UV completion kicks in at this scale. The main lessonhere is that extra-dimensional kinematics make most interactions irrelevant in the technicalsense, leading to higher powers of the heavy scale M in the generic contribution to δm . Gravity
In practice, the fields of most interest in higher dimensions are various types of gauge andgravitational fields. The crucial difference between these and the scalar fields described hereto-fore is the existence of local gauge symmetries (or general covariance) that keep these fieldsmassless in the extra-dimensional theory. Their only masses in 4D are therefore those due tothe KK reduction. We now explore the sensitivity of these masses to integrating out particlesthat are both heavier and lighter than the KK scale.To see the effects of integrating out a heavy field in this case consider a massive scalarcoupled to gravity, − L p − g ( D ) = λ + 12 κ R + 12 ( ∂ Ψ) + M Ψ , (2.15)where R = g MN R MN denotes the metric’s Ricci scalar, λ is a (bare) cosmological constant,and κ = 8 πG D := M − ( D − / g is the (bare) reduced gravitational coupling constant, whichalso defines the higher-dimensional Planck scale, M g . A semiclassical treatment [5] assumeswe require λ ≪ M Dg and that the heavy particle mass satisfies M ≪ M g .As before, integrating out the massive field Ψ leads to many new effective interactions forthe remaining light field, which are local so long as the heavy particle’s Compton wavelength, We do not attempt to track factors of 2 π , so ignore these in the definition of M KK . Alternatively these additional powers of M relative to the 4D results found earlier can be regarded asbeing due to the necessity to sum over the tower of KK modes (see Appendix A). – 7 –s much shorter than the size of the extra dimensions, M ≫ M KK . These can be organized inorder of increasing dimension, with the coefficient of each involving the appropriate power of M , as required on dimensional grounds. Because in the present case the low-energy field isthe metric, the form of these interactions is strongly restricted by general covariance to takethe form of a curvature expansion, − L eff p − g ( D ) = λ + 12 κ R + a M D − R MNPR R MNPR + · · · , (2.16)where all possible terms involving curvatures and their derivatives are contained in the ellipses,while λ and κ are appropriately renormalized λ = λ + a M D and 12 κ = 12 κ + a M D − , (2.17)with a i dimensionless constants, and so on.In general, the condition M ≪ M g implies κ ≃ κ , making the Einstein term largelyinsensitive to the effects of integrating out heavy particles. The same is usually not true for λ and λ because the Einstein equations imply R ≃ O ( κ λ ), and calculability demands R ≃ /L = M KK be much smaller than M g . For non-supersymmetric theories taking M ≫ /L and κ λ ≃ O ( M KK ) usually means fine-tuning λ to ensure that λ ∼ O ( M KK M D − g ) ≪ M D .The upshot is this: integrating out a field with mass M ≫ M KK just leads to the additionof local interactions to the higher-dimensional action. These do not qualitatively change thelow-energy consequences so long as it is the lowest-dimension interactions that are of physicalinterest (like the Einstein action) and provided that the most general interactions are includedin the action from the start. This is because the condition for the validity of the semiclassicaltreatment in terms of an extra-dimensional field theory requires M ≪ M g , ensuring that thenew contributions are swamped by those involving M g . An understanding of the contributionsat scale M g then should be done using the UV completion at the gravity scale, which withinstring theory would potentially involve a full string calculation. Integrating out KK modes
The effective theory becomes four-dimensional at energies of order M KK and below, so oncethese scales are integrated out we can no longer use higher-dimensional symmetries (likeextra-dimensional general covariance) to restrict the form of the result.We wish to track how the integrating out of modes with masses at the KK scale andbelow depends on the underlying scales M KK or M g . Within the semiclassical approximationany such an integration arises as an expansion about a background field, g MN , so that g MN = g MN + κh MN . (2.18)It is also useful to explicitly scale out the local linear size of the extra dimensions, e u ( x ) ,(measured using the background geometry) from the total metric g MN d x M d x N = ω e − du ˆ g µν d x µ d x ν + e u ˆ g mn d y m d y n + off-diagonal terms , (2.19)– 8 –here the factor pre-multiplying the 4D metric is chosen to ensure no u -dependence in the4D Einstein action ( i.e. the 4D Einstein frame).The factor ω := ( M p /M g ) numerically converts to 4D Planck units, and the vacuumvalue h e u i ∝ M g L provides a dimensionless measure of the extra-dimensional linear size (and so h e du i ≃ ( M g L ) d := V ). Recall the 4D Planck scale is related to the dimensionlessvolume, V = V M dg , by M p = V M D − g = V M g , and so ω = M p /M g = V , and M g ≃ M p / V / .Using p − g ( D ) = p − ˆ g (4) p ˆ g ( d ) ω e − du and R d d y ∝ M − dg , we find the 4D Einstein termbecomes M D − g Z d d y p − g ( D ) g µν R µν = ω M g q − ˆ g (4) ˆ g µν ˆ R µν + · · · = M p q − ˆ g (4) ˆ g µν ˆ R µν + · · · , (2.20)as required.Expanding the action in powers of fluctuations and focussing on the 4D scalar KK modescontained within h mp := g mn h np — generically denoted ϕ i — similarly gives the followingdimensionally reduced kinetic terms −L kin ≃ M D − g Z d d y p − g ( D ) g µν R µν = ω M g Z d d y q − ˆ g (4) ˆ g µν H mnpq ∂ µ h pm ∂ ν h qn + · · · := M p q − ˆ g (4) ˆ g µν G ij ( ϕ ) ∂ µ ϕ i ∂ ν ϕ j + · · · . (2.21)where H mnpq are a set of coefficients depending on the h mp (but not their derivatives), while G ij ( ϕ ) denotes the target-space metric for the dimensionless 4D fields ϕ i . The detailed formof G ij is not important beyond the fact that it contains no additional dependence on V , andso is generically O (1) in the large- V limit.By contrast, the contributions to the scalar potential for the ϕ i instead scale with M g and L according to −L pot ≃ M D − g Z d d y p − g ( D ) g mn R mn = ω M g Z d d y e − du q − ˆ g (4) ( e − u ˆ g mn ) K pqrs ∂ m h rp ∂ n h sq + · · · := M p V /d q − ˆ g (4) U ( ϕ )= M KK M p q − ˆ g (4) U ( ϕ ) , (2.22)which uses e ( d +2) u = V /d and M KK /M p = ( M KK /M g )( M g /M p ) ≃ V − /d V − . Again, thedetailed form of K pqrs and U ( ϕ ) are not important, beyond the fact that they generically donot contribute to the V dependence of the result. We take V ≃ L d and R ≃ /L for the same scale L , and by so doing ignore features that might arisewithin strongly warped spacetimes (for which the low-energy sector can be strongly localized), or within somenegatively curved spaces (for which the scales associated with volume and curvature can radically differ). – 9 –nce canonically normalized, ϕ ∝ φ/M p , we find the following schematic mass terms,cubic and quartic interactions − L p − g (4) ≃ ( ∂φ ) + M KK φ − L p − g (4) ≃ M p φ ( ∂φ ) + M KK M p φ (2.23)and − L p − g (4) ≃ M p φ ( ∂φ ) + M KK M p φ , and so on. These show that the generic mass is M KK ≃ M p / V (1+2 /d ) / (as expected), andalthough the low-energy derivative interactions are Planck suppressed, those in the scalarpotential have a universal additional suppression by a factor of M KK /M p = 1 / V /d =( M g /M p ) /d relative to generic Planck size.A similar analysis for the curvature-squared terms shows that these introduce threekinds of 4D interactions: ( i ) O (1) 4-derivative interactions, ∼ k ( ϕ )( ∂ϕ ) ; ( ii ) O ( M KK ) two-derivative interactions, ∼ M KK k ( ϕ )( ∂ϕ ) ; and O ( M KK ) potential terms, ∼ M KK k ( ϕ ). Eachis therefore suppressed relative to its counterpart (if this exists) coming from the Einsteinterm by an additional factor of M KK /M p = 1 / V /d .Provided λ is chosen to allow classical solutions for which R ≃ κ λ ≃ /L ∼ M KK , asdiscussed above, the contributions of the cosmological constant term to the 4D dimensionallyreduced interactions scale in the same way as do those coming from the Einstein term. Naturality of the KK couplings
It is noteworthy that, from a 4D perspective, the couplings of the KK modes amongst them-selves are weaker than Planckian, yet they are stable against radiative corrections.As we’ve seen, the dominant contribution from the scales much larger than M KK comefrom scales M > ∼ M g , which must be performed within the theory’s UV completion. If this isa string theory, the contribution to the low-energy action from integrating out string statesis what gives the initial higher-dimensional gravity action, plus a variety of higher-derivativecorrections involving powers of the curvature and other low-energy fields. Since our initialestimate for the size of the interactions comes from the higher-dimensional Einstein term,the leading corrections in the UV come from dimensionally reducing terms involving morederivatives than this. If it is a curvature-squared term that dominates, then we expect thelargest corrections from these scales to be suppressed relative to the leading terms by of order M KK /M g .The sole exception to this happy picture of insensitivity to higher-scale physics is the con-tribution to the extra-dimensional cosmological constant, although this is also not dangerousonce λ ≃ M KK /κ ≃ M KK M D − g is tuned to be small enough to allow the extra dimensionsto be large in the first place. As we shall see, even this problem need not arise when couchedin a supersymmetric context, since in this case higher-dimensional supersymmetry usuallyrequires λ = 0. – 10 –inally, what of the naturality of the 4D scalar masses from loops with M < ∼ M KK ? Sincethese are not protected by higher-dimensional symmetries, they should be analyzed withinthe effective 4D theory. As shown above, it is the contributions of the relevant and marginal – i.e. cubic and quartic – interactions that are then the most dangerous. But because all of theinteractions in the scalar potential are suppressed by M KK /M p relative to Planck strength,their use in loop graphs generates interactions that are generically suppressed relative to thosewe start with by similar factors. This implies they are of similar order to the contributionsof higher-derivative corrections in the extra-dimensional theory just considered.Similarly, graphs using the derivative interactions give much the same result, since al-though these involve couplings unsuppressed (relative to M p ) by powers of 1 / V , they are moreUV divergent and so depend more strongly on the mass of the heaviest 4D state that cancirculate within the loop. But this mass is again M KK , since for masses much higher thanthis the restrictions of higher-dimensional general covariance limit the result to one of thelocal interactions considered above. For instance, using these estimates in Fig. 2 then gives δm ≃ M p Z d k (2 π ) k ( k + m ) ≃ cµ ǫ ǫ (cid:18) M KK M p (cid:19) , (2.24)again implying corrections that are suppressed by M KK /M p .These arguments indicate that the dominant corrections to the 4D scalar potential areof order δV ≃ O ( M KK ) δU ( ϕ ). Notice that this implies the corresponding contributions tothe 4D vacuum energy are δV ≃ M KK ≃ /L , in agreement with explicit Casimir energycalculations [7]. Moduli
It often happens that accidental symmetries in the leading order action imply that some ofthe KK scalars are massless once dimensionally reduced. This would happen for the volumemodulus, e u , in particular, if the equations of motion coming from the leading action werescale invariant (as would be true for the Einstein equations in the absence of a cosmologicalterm, or for the lowest-derivative terms in most higher-dimensional supergravities).In this case the dimensionally reduced mass for the corresponding moduli comes fromnext-to-leading effects that break the relevant symmetry. These could be from loop effects inthe low-energy 4D theory, in which case the above estimates indicate their masses would beexpected to be of order M mod ≃ M KK /M p ≪ M KK rather than being precisely massless. Al-ternatively, the dominant corrections could come from loop effects in the higher-dimensionaltheory, which we’ve seen are equivalent to use of sub-dominant terms in the derivative ex-pansion (such as from curvature-squared or higher) when compactifying. If it is curvature-squared terms that dominate, then the discussion above shows we again expect masses of Notice that if M KK ≃ − eV then M mod < ∼ − eV, showing that if the observed vacuum energydensity could be arranged to be dominated by a KK energy, then the presence of moduli would produce aquintessence cosmology, with no additional tunings required to ensure small enough scalar masses [8]. For an example where curvature-squared terms do not dominate, see the supersymmetric estimate below. – 11 –rder M mod ≃ M KK /M p . Notice that masses these size are also no larger than the genericsize of radiative corrections in the low-energy theory, based on the estimates given above. For supersymmetric systems, the estimates just given can differ in several important ways. • Extra-dimensional non-renormalization theorems:
For higher dimensional supergravityit is the expectation of one of the fields (the dilaton) that plays the role of the loop-counting parameter. Often its appearance in the leading derivative expansion of theaction is restricted by supersymmetry, in which case extra-dimensional loops must nec-essarily contribute suppressed both by powers of the coupling constant and low-energyfactors. For instance, in string theory it is often true that higher-order contributionsin the string coupling, g s , necessarily only arise for terms that are also sub-leading inpowers of the string scale, α ′ = 1 /M s [9].It is also true that supersymmetry can raise the order in the derivative expansion ofthe first subdominant contributions to the action that arise even without loops. Forinstance, in the Type IIB models of later interest, the first higher-curvature correctionsthat arise at string tree level involve four powers of the curvature, as opposed to thegeneric expectation of curvature squared [10]. • Four-dimensional non-renormalization theorems:
If the supersymmetry breaking scaleshould be much smaller than M KK , then the effective 4D description can be written asa 4D supergravity, possibly supplemented by soft supersymmetry-breaking terms. Inthis case the most dangerous relevant and marginal scalar interactions appear in theholomorphic superpotential, W ( φ ), and so are protected by 4D non-renormalizationtheorems [11, 12, 13, 14]. If supersymmetry is not broken these exclude corrections toall orders in perturbation theory. Corrections become possible once supersymmetry isbroken, but are further suppressed by the relevant supersymmetry breaking scale. Whencomputing explicit loops, these suppressions come about through the usual cancellationsof bosons against fermions, together with mass sum rules like [15] X s ( − ) s (2 s + 1)Tr M s ≃ M / , (2.25)where M / is the 4D gravitino mass. • Additional extra-dimensional fields:
A third way in which supersymmetric theories candiffer is through their extra-dimensional field content, which always involves more fieldsthan just gravity. Interactions involving these other fields sometimes provide the dom-inant contributions to quantities like the masses of moduli. The most familiar exampleof this sort occurs when the background supergravity solution involves nonzero 3-formflux fields, G mnp , whose presence gives some of the moduli masses (as in 10D Type IIBflux compactifications). In this case it is the term L = √− g G mnp G mnp ∝ / V thatdominates these masses, leading to M mod ≃ M p / V ≫ M KK /M p ≃ M p / V / .– 12 – . Large-volume (LV) string models The large-volume (LV) framework [1] is a scenario for moduli stabilization for Calabi-Yaucompactifications within Type IIB string theory. In generic Type IIB flux compactifications,the presence of background 3-form fluxes alone can stabilize the dilaton and the complexstructure moduli of the underlying Calabi-Yau geometry [17]. The K¨ahler moduli can thenbe stabilized by the non-perturbative effects localized on four cycles associated with variousbranes that source the geometries [18]. The LV scenario identifies an interesting subclass forwhich the volume modulus is naturally stabilized at exponentially large volumes,
V ∝ exp ( cτ s ) , (3.1)where τ s is the size of a (comparatively much smaller) blow-up cycle of a point-like singularityin the underlying geometry, while c is an order-unity constant.The framework applies to a large class of Calabi-Yau compactifications since there areonly two requirements for its implementation [19]: ( i ) there must be at least one of theK¨ahler moduli must be the blow-up mode, τ s , of a point-like singularity; and ( ii ) the numberof complex structure moduli have to be greater than the the number of K¨ahler moduli (inorder for the Euler number of the Calabi-Yau space to have the sign required for the potentialto have a minimum for large V ).A key ingredient is the inclusion of the leading order α ′ correction to the 4D K¨ahlerpotential, since this is what generates the potential with a minimum with compactificationvolumes that are exponentially large in τ s . Furthermore, because τ s scales as the inverse ofthe the value of the dilaton, τ s ∝ g s , (3.2)and so is given as a ratio of integer flux quanta. Thus the framework naturally generatesan exponentially large volume of compactification from integer flux quanta, with V passingthrough an enormous range of values as the fluxes range through a range of order tens.In what follows we review some relevant aspects of these compactifications with an em-phasis on the mass scales and strength of couplings. LV models enjoy a rich pattern of particle masses and couplings, for which it is useful formany purposes to express in 4D Planck units. Our goal here is to express how these quantitiesscale as functions of the dimensionless extra-dimensional volume, expressed in string units:
V ≃ α ′ R √ g ≃ ( L/ℓ s ) ≫ Masses
In terms of V and the 4D Planck scale, M p , the largest scale in the excitation spectrum is thestring scale itself, M − s = ℓ s ≃ α ′ , which (ignoring factors of g s ) is of order M s ≃ M p V / . (3.3)– 13 –his characterizes the order of magnitude of the masses of generic string excitation levels.If the linear sizes of the various extra dimensions are all of the same order, L , then since V ∝ ( L/ℓ s ) , the scaling with V of the masses of Kaluza-Klein (KK) excitations is M KK ≃ L ≃ M s V / ≃ M p V / . (3.4)The presence of fluxes stabilizes the complex structure moduli, or what would otherwisehave been massless KK zero modes. These are systematically light compared with the KKscale because of the V suppression of the various fluxes, due to the quantization conditions ofthe form R F ≃ N α ′ , for an appropriate 3-form, F mnp . Masses obtained from the flux energy, R d x √− g F ∝ V − , are of order M cs ≃ M s V / ≃ M p V . (3.5)The K¨ahler moduli survive flux compactification unstabilized to leading order, and sonaively might be expected to receive a mass that is parametrically smaller than those ofthe complex-structure moduli. For instance, small K¨ahler moduli like τ s are stabilized bythe nonperturbative terms, W ≃ e − cτ s , in the low-energy 4D superpotential, and at thepotential’s minimum these are comparable to the α ′ corrections, implying the relevant partof the potential depends on the volume like 1 / V . However, because the K¨ahler potential forthese moduli is K = − V , their kinetic term is also proportional to 1 / V , implying that thevolume-dependence of the mass of these moduli is again of order M k ≃ M s V / ≃ M p V . (3.6)The K¨ahler modulus corresponding to the volume of the compactification also receivesits mass from the leading α ′ correction. However for this modulus ∂ V depends nontrivially on V (unlike for the small moduli like τ s ), leading to a kinetic term of order ∂∂K ≃ ∂∂ V / V anda quadratic term in the potential that is of order ∂∂ V / V . The result is a volume-modulusmass whose V -dependence is of order M V ≃ M s V ≃ M p V / . (3.7)For many compactifications K¨ahler moduli come with a range of sizes, and there are oftenmany having volumes τ i ≫ τ s . These tend to be exponentially suppressed in the superpoten-tial, W ≃ e − c i τ i ≪ e − cτ s ≃ / V , and so at face value also appear to have exponentially smallmasses when computed using only the leading α ′ corrections. However for any such moduliit is the sub-leading corrections to the scalar potential (like string loops or higher orders in α ′ ) that instead dominate their appearance in the potential. Consequently these states aresystematically light relative to the generic moduli discussed above. The leading string-loopcontribution turns out to be dominant, and so these moduli generically have masses of order M ≃ M p V / . (3.8)– 14 –inally, the gravitino mass associated with supersymmetry breaking is itself of order M / ≃ M s M p ≃ M s V / ≃ M p V . (3.9)Notice for the purposes of the naturalness arguments that the masses of the volume modulusand the other large K¨ahler moduli are parametrically lighter than both the gravitino massand the KK scale. Couplings
The potential term associated with the complex structure and K¨ahler moduli scales as V − asit arises by dimensionally reducing a higher-dimensional flux energy, ∼ R d y √ g G mnp G mnp ,rather than from a curvature-squared term.Because the KK scale is larger than typical SUSY-breaking scales, like M / , the effective4D theory can be described within the formalism of N = 1 supergravity. In this context thesuppression of the couplings relative to Planck strength reflects itself in the no-scale form ofthe Kahler potential, K = − V , (3.10)since this suppresses all terms in the potential by a factor of e K W ∝ W/ V .The volume dependence of the interactions of low-energy fields with Kaluza-Klein modescan be inferred using arguments [6] very similar to those used in section 2. Specializing thepotential term obtained there to D = 10 and d = 6 implies that it scales as V − / .The couplings of these fields to ordinary matter can be estimated if the ordinary matteris assumed to be localized on a space-filling brane somewhere in the extra dimensions. Thestrength of these couplings depends on the particular brane field of interest, but a represen-tative coupling is to gauge bosons, which has the generic form L int = φf F µν F µν , (3.11)where, for example, f ≃ M p for many of the K¨aher moduli (like the volume modulus itself),although some of the ‘small’ moduli couple with strength f ≃ M s ≃ M p / V / .
4. Radiative corrections in LV models
The previous sections summarize the rich pattern of scalar masses, related to one another bypowers of V , predicted by LV models at leading order: M s ≃ M p V / , M KK ≃ M s V / ≃ M p V / , M mod ≃ M / ≃ M p V , M V ≃ M p V / . (4.1)The couplings among these states are also V -dependent, and this is important when com-puting the size of the loop-generated masses. The contributions of wavelengths longer than– 15 –he KK scale to these loop corrections can be evaluated within the context of an effective 4Dtheory, while those having shorter wavelength should be computed within the higher dimen-sions. Technical naturalness asks that both the 4D and the higher-dimensional contributionsbe smaller than the lowest-order mass of the light particle itself. Suppose a light particle has a mass, m φ ≃ M p V − a , that is suppressed by a particular negativepower of V . Suppose further that this light particle couples to a more massive particlehaving a mass m ψ ≃ M p V − b , with b < a . Imagine these particles to experience a relevantcubic coupling of the schematic form L ≃ h φ ψ∂ p ψ whose coupling strength is of order h ≃ M p (1 /M p ) p V − c , where the power of energy, E , would arise if the coupling were toinvolve the derivatives of the fields. Assuming 4D kinematics are relevant, repeating thesteps of section 2 shows that the one-loop correction to the light scalar mass generated usingthis interaction in a graph like Fig. 2 is of order δm φ ≃ h Z d k (2 π ) k p ( k + m ψ ) ≃ h m p ψ π ! ≃ π ) (cid:18) V (cid:19) p b + c ) M p . (4.2)The mass of the light particle is larger than these radiative corrections provided δm φ < ∼ m φ , and so b p + c ≥ a . In light of the discussion in §
2, since the most massive states forwhich 4D kinematics applies have m ψ ≃ M KK ≃ M p / V / , we make take as the worst case b = . Furthermore, the discussion of section 2 implies that the couplings of such states isgenerically either a Planck-strength derivative coupling ( p = 2 and c = 0), or a potentialinteraction whose strength is proportional to M KK /M p ≃ V − / ( p = 0 and c = ). Inboth cases b p + c = , implying corrections that are sufficiently small for all 4D moduli(for which a = 1) except for the volume modulus (for which a = ).Alternatively, suppose the light particle couples to the massive one through a marginalquartic coupling L ≃ g φ ψ∂ p ψ , whose coupling strength is of order g ≃ (1 /M p ) p V − c ,where the power of energy, E , again arises as derivatives of the fields. In this case the one-loopcorrection (using 4D kinematics) to the light scalar mass is of order δm φ ≃ g Z d k (2 π ) k p ( k + m ψ ) ≃ g m p +1 ψ π ! ≃ π ) (cid:18) V (cid:19) p +1) b +2 c M p , (4.3)so this contribution to the mass is technically natural provided b (1 + p ) + c ≥ a .Again, the maximum mass appropriate to 4D kinematics is m ψ ∼ M KK , correspondingto b = and there are two kinds of quartic interactions amongst the KK modes: Planck-suppressed derivative couplings ( p = 1 and c = 0) and non-derivative interactions sup-pressed by M KK /M p ( p = 0 and c = ). Both of these choices satisfy b (1 + p ) + c = ,– 16 –nd so are the same size as those obtained from cubic interactions, and not larger than anyof the moduli masses ( a = 1), except for the volume modulus ( a = ). Supersymmetric cancellations
Apart from the volume modulus, we see that moduli masses tend to be generically stableagainst 4D radiative corrections. Naively this would seem to indicate the classical approxi-mation is not a good one for the volume modulus itself, which should then be expected to bemore massive than O ( M p / V / ). However to this point we have not yet availed ourselves ofthe cancellations implied by 4D supersymmetry, which survives into the 4D theory because M / ≪ M KK .Let us reconsider the 4D contributions in this light. As usual, since particle masses aredriven by the form of the superpotential, they receive no corrections until supersymmetrybreaks. At one loop an estimate of the leading supersymmetry-breaking effects obtained bysumming the contributions of bosons and fermions, keeping track of their mass difference.For instance, keeping in mind the sum rule, eq. (2.25), gives δm ≃ g (cid:0) m B − m F (cid:1) < ∼ (cid:18) M KK M p (cid:19) M / ≃ (cid:18) V / (cid:19) M p V , (4.4)implying the generic supersymmetry-breaking mass shift is δm ≃ M p / V / . This is smallenough not to destabilize the mass of the volume modulus, M V ≃ M p / V / . Higher-dimensional contributions
For scalar mass corrections, normally it is loops involving the heaviest possible particles thatare the most dangerous, so it might be natural to expect the contributions from states having M ≫ M KK to dominate the 4D estimates just obtained. In this section we argue this not tobe the case, with the dominant contribution to modulus masses arising from loops involvingstates at the KK scale.The argument proceeds as in section 2, with the recognition that the effects of particlesabove the Kaluza-Klein scale require the use of the higher-dimensional theory, where newsymmetries like extra-dimensional general covariance come to our aid. In particular, sincethe largest mass scale in the higher dimensional theory is M g , or for Type IIB models thestring scale, M s ≃ g s M g , the contributions of states this massive are summarized by localstring-loop and α ′ corrections to the 10D action.However, any such contributions are strongly constrained by 10D supersymmetry. Theleading α ′ corrections arising at string tree level are known, and first arise at O ( α ′ ), withfour powers of the curvature. The volume dependence of this, and of the other α ′ correctionsthat arise at next-to-leading order were studied by refs. [1], with the conclusion that thesefirst contribute to the 4D scalar potential at order δ α ′ U ≃ W V U α ′ ( ϕ ) , (4.5)– 17 –here U α ′ ( ϕ ) denotes some function of the dimensionless moduli (whose detailed form is notimportant for the present purposes). Given their Planck-scale kinetic terms, this contributesof order δm ≃ O ( M p / V / ) to the low-energy moduli masses. Indeed, it is these α ′ correctionsthat lift the mass of the volume modulus in the first place, showing why it is generically oforder M p / V / .The V -dependence of the leading string-loop effects has also been analyzed [20, 21, 19],with the result that it contributes at order δ loop U ≃ W V / U loop ( ϕ ) , (4.6)with U loop another function of the dimensionless moduli. The resulting contribution is δm ≃ M p / V / , in agreement with the above estimates for the size of supersymmetric cancellationsin one-loop effects. This agreement indicates that it is states having masses of order M KK that provide the dominant contribution to these string loops. Large-volume string models have the advantage of having been studied well enough that thereare several nontrivial checks on the above estimates.
Comparison with the 4D supergravity
One check comes from comparing the above estimates of the size of cancellations in one-loopsupergravity amplitudes with the kinds of corrections that are allowed to arise within theeffective 4D supergravity describing the moduli. In this supergravity the above loop effectsmust appear as volume-dependent corrections to the K¨ahler function, K , since this is thequantity that encodes the effects of high-energy loops. The leading contributions to δK arising up to one loop has been estimated [16] , and has the form K ≃ − V + k V / + k V + k V / · · · , (4.7)where the k term first arises at the level of one string loop while k contains the α ′ contri-butions at string tree level, and the k term gives the next corrections at one string loop.Recall that the leading contribution to the 4D potential varies as U ≃ W / V . It happensthat because the k term is proportional to V − / , it completely drops out of the scalarpotential [20, 21]. The k term then gives the dominant correction to U , contributing atorder 1 / V . Finally, the k term contributes a correction to U that is of order 1 / V / , all inagreement with the above estimates. Comparison to explicit string calculations
The explicit form of one-loop effects associated with Kaluza-Klein (and string) modes areavailable for orbifolds and orientifolds of toroidal compactifications. We here briefly summa-rize the strength of these corrections and their interpretation in the low energy effective field– 18 –heory. This illustrates that the truncation to the four dimensional effective field theory to beconsistent and the quantum corrections to the mass of the volume modulus associated withKaluza-Klein modes to be subleading in an expansion in the inverse volume.We focus on N = 1 orientifold T / Z × Z analyzed by Berg, Haack and Kors in ref. [16].The untwisted moduli in this model are the three K¨ahler and complex structure moduli { ρ I , U I } axio-dilaton S, open string scalars A I associated with position of the D D K = − ln( S − ¯ S ) − X I =3 ln (cid:20) ( ρ I − ¯ ρ I )( U I − ¯ U I ) − π ( A I − ¯ A I ) (cid:21) , (4.8)while the one-loop correction was computed in [16], and found to be K (1) = 1256 π X I =1 (cid:20) E D ( A I , U I )( ρ I − ¯ ρ I )( S − ¯ S ) + E D (0 , U I )( ρ J − ¯ ρ J )( ρ K − ¯ ρ K ) (cid:12)(cid:12)(cid:12)(cid:12) K = I = J (cid:21) (4.9)where the superscripts D D E D and E D are complicated, a symmetric choice of the D D D E D (0 , U ) = 1920 X ( n,m ) =(0 , Im( U ) | n + U m | (4.10)The indices ( m, n ) can be interpreted as labelling the Kaluza-Klien momenta of theexchanged particles in the open-string loop diagrams. Note that the contribution from thehigher KK modes is suppressed by inverse powers of the KK momentum, as a result thehigher KK modes make only a small contribution. We note that while the contribution ofa single mode running in loop correction to the mass would scale as the square of the mass( n ), cancelations due to supersymmetry lead to an effective scaling of n − . We would like toemphasize that the resulting sum is ultraviolet finite. One can interpret this as the restorationof supersymmetry at the high scale.The contribution of the one-loop K¨ahler potential to the scalar potential scales as V − / which indeed is subleading in the large volume expansion. As mentioned earlier, this scalingof the volume precisely matches the estimations of the size of loop effects from KK modes inthe low energy effective field theory [21].The bottom line is that the combination of supersymmetry with the V -suppressed cou-plings coming from extra dimensions ensures the stability of the masses of the moduli againstradiative corrections in large-volume string models.– 19 – . Scenarios Given the robustness of the light scalar masses, for phenomenological purposes it is usefulto see how large the above masses are for various choices for the string scale, in order to seeprecisely how light the relevant scalars can be. Since V varies exponentially sensitively withthe parameters of the modulus-stabilizing flux potential, it varies over many orders of mag-nitude as potential parameters are changed only through factors of order 10. It can thereforebe regarded as a dial that can be adjusted freely when exploring the model’s implications. Weak-scale gravitino mass
One attractive choice is to take
V ≃ , in which case M / ≃ M p / V ≃ GeV is of orderthe TeV scale and M s ≃ M g ≃ M p / V / ≃ GeV, corresponding to the intermediate-scalestring [22].In this case the generic moduli also have masses at the TeV scale, while the volume mod-ulus is interestingly light, being of order M V ≃ M p / V / ≃ − GeV ≃ ∼ Intermediate scale gravitino mass and soft terms
In [25] a novel framework for supersymmetry breaking is put forward in the context of thelarge volume scenario. Since the main source of supersymmetry breaking is the F -term ofthe volume modulus and, since the 4 D supergravity is approximately of the no-scale type, itscontributions to soft terms can be highly suppressed in powers of the volume, the F -termsof the other K¨ahler moduli dominate the structure of the soft terms. If the brane that hoststhe standard model does not wrap the dominant cycle for supersymmetry breaking, then thesoft terms are hierarchically suppressed with respect to the gravitino mass ∆ m ∼ M p / V q with q = , m ∼ GeV. Again we have a situation in which the soft terms, in particularthe masses of the scalar partners of the standard model fields, are much smaller than thegravitino mass and loop corrections could in principle destabilize these masses. However,the same arguments as before imply that these loop corrections are at most proportional to M KK M / /M p ≃ M p / V / (see also [25, 26]).– 20 –s a result squarks and sleptons much lighter than the gravitino mass are naturally stable.A similar estimate can be made for the other soft terms, such as A -terms and gaugino masses.This implies that as long as the contributions to soft terms from the volume modulus aresuppressed, including anomaly mediated contributions [27] (see however [28]), the gravitinomass can be hierarchically heavier than the TeV scale. Notice that this ameliorates thecosmological moduli problem mentioned above since the volume modulus mass, which is oforder M p / V / , in this scenario can be as heavy as 1 TeV or heavier, instead of being orderMeV as in the previous scenario. Weak-scale strings and SUSY breaking on the brane
The largest possible volume within this scenario that is consistent with experience is
V ≃ ,for which M s ≃ M p / V / ≃ GeV is at the weak scale, while the moduli and gravitino areextremely light: M mod ≃ M / ≃ M p / V ≃ − GeV ≃ − eV. In this case the volumemodulus would appear to be astrophysically relevant, since M V ≃ M p / V / ≃ eV.More care is needed in this particular case, however, because the supersymmetry breakingscale, M / , is so very low. Because it is so low, another source of breaking must be introducedfor the model to be viable, to accommodate the absence of evidence for supersymmetryin accelerator experiments. The simplest such source of supersymmetry breaking is hardbreaking by anti-branes, with all of the observed particles living on or near these branes sothat they are not approximately supersymmetric.There are dangers to breaking supersymmetry in such a hard fashion, however. In par-ticular, for a viable scenario one must check that the potential energy of the anti-brane —which we shall find is of order M s ≃ M p / V — does not destabilize the LV vacuum. Noticethat this is a more serious problem than occurs if anti-branes are introduced to uplift theprevious cases so that their potentials are minimized to allow flat spacetime on the branes. Itis worse in this case because the anti-brane must be the dominant source of SUSY breaking,and so the value of its tension cannot be warped down to small values without also makingthe mass splittings amongst supermultiplets — which are of order M s on the brane — toosmall. In what follows we assume that this issue has been dealt with, either through inspiredmodelling or through fine-tuning.Since the splitting between the masses of the bosons and fermions localized on the braneis of the order M s , it is loops involving these brane states that are the most dangerouscorrections to M V in this picture. We now argue that these loops of brane states generatemass corrections of order M p / V , and so in this particular case lift the volume modulus to becomparable in mass to the masses of the other moduli.The estimate begins with the Born-Infield action on the brane and the ten dimensionalEinstein action in the bulk S = M g Z d x √ g R − T Z d x q det[ G αβ ] (5.1) We thank Joe Conlon for emphasizing this point. – 21 –here T ≃ O ( M s ) is the brane tension and G αβ is the pullback of the ten dimensional metricto the brane world volume. Neglecting powers of the string coupling, we take the 10D Planckscale of order the string scale: M g ≃ M s .To carry out the dimensional reduction we take, as before, the ten dimensional metric tobe of the form d s = ωe − u ( x ) g µν d x µ d x ν + e u ( x ) g mn d y m d y n , (5.2)where, as before, ω = ( M p /M s ) , u ( x ) is the volume modulus and g mn is the metric of aCalabi-Yau of unit volume in string units. Working with static embedding coordinates forthe brane the effective action for the volume modulus u ( x ) and the transverse scalars y m is − M p Z d x √− g ∂ µ u∂ µ u − ˆ T Z d x √− g e − u ( x ) q det[( δ αβ + e u ( x ) ∂ α y m ∂ β y n g mn )](5.3)where M p is the four dimensional Planck mass and ˆ T = ω T ≃ O ( M p ).Next we make field redefinitions which bring their kinetic terms to the canonical form, u ( x ) = u + ℓ ( x ) / √ M p and y m = e u φ m / p ˆ T , where u is the v.e.v. of the field u ( x ) andrelated to the volume of the compactification in string units by e u = V . This brings (5.3)to the form − Z d x √− g ∂ µ ℓ∂ µ ℓ − ˆ T V Z d x √− g e −√ ℓ/M p s det (cid:20)(cid:18) δ αβ + V ˆ T e ℓ/ √ M p ∂ α φ m ∂ β φ n g mn (cid:19)(cid:21) (5.4)We note that the derivative expansion for interactions involving the fields φ m is controlledby the scale Λ = ( ˆ T / V ) / . At the two-derivative level interactions between ℓ and φ m arePlanck suppressed, the leading order term being1 M p Z d x √− g g mn ∂ α φ m ∂ α φ n ℓ ( x ) , (5.5)The graph for the one loop correction to the mass of the volume modulus involving thisinteraction has two inverse powers of M p from the interaction vertices. The relevant integralover the virtual momenta has four powers of momenta from the integration measure, fourfrom the two interaction vertices and four inverse powers from the propagators. This leadsto a loop correction to the mass squared of the volume modulus of the order of Λ /M p . Theabove estimate in fact holds for all contributions arising from virtual loops of the fields φ m .This is most easily seen from dimensional analysis. The associated graph involves two legsof the field ℓ ( x ), and since the field always appears in the combination of ℓ/M p this leadsto two factors of 1 /M p . Next note that for processes involving loops of φ m , all the verticesand loop momenta are powers of ˆ T / V . This implies that the mass correction is of the form δm ∝ ( ˆ T / V ) p /M p , where on dimensional grounds the value of p must be p = 1. Since– 22 – T ∼ O ( M p ), the scale Λ ∼ M p / V / ∼ M s is of order the string scale, so the size of the masscorrection is δm ∝ Λ M p ≃ M s M p ≃ M p V . (5.6)This implies the mass of the volume modulus is naturally of order M mod ∼ M / ∼ M p / V when supersymmetry is badly broken on a brane, removing the hierarchy between the volumemodulus and other moduli.If the string scale is ∼ M p / V ≃ − eV, making them just on the edge of relevance to laboratory tests of the gravitationalforce law. Because it is so close, corrections can be important, and – as shown in ref. [24] –the couplings between brane matter and bulk fields give an additional logarithmic suppressionto the volume modulus mass, which then scales as M p / ( V ln V ). If this correction pushes thevolume modulus into the range probed by terrestrial fifth-force experiments, it could makethe effects of this field detectable. The existence of such a scalar as a robust consequence ofweak-scale string models within the large-volume picture provides additional motivation formore detailed calculations of its mass and properties in the presence of anti-branes.TeV string scenarios could have spectacular experimental implications at LHC (see forinstance [29], and references therein), so it is of great interest that they might be viablewithin the large volume scenario for which control over issues like modulus stabilizationand supersymmetry breaking allows a detailed prediction of the low-energy spectrum. Theexistence within this spectrum of a very light volume modulus was the main obstacle to seriousmodel building, so the existence of mass generation by explicit breaking of supersymmetryon the brane is of particular interest.
6. Conclusions
In this paper we show that extra dimensions and supersymmetry can combine to to protectscalar masses from quantum effects more efficiently than either can do by itself. Supersym-metry by itself would not necessarily protect masses lighter than the gravitino mass and extradimensions in principle need not protect scalar masses lighter than the KK scale. But bothtogether allow scalar masses to be hierarchically smaller than both the KK and gravitinomasses. We call this kind of unusual scalar hierarchy ¨uber-natural .New mechanisms for keeping scalar masses naturally light are interesting because they areboth rare and potentially very useful, both for applications to particle physics and cosmology.Because light scalars tend to have many observable consequences, their existence can helpidentify those models to which the ever-improving tests of general relativity on laboratoryand astrophysical scales are sensitive. From a model builder’s perspective, light scalars arealso useful because they can cause problems, such as the cosmological moduli problem, andthereby focus attention on those models that can deal with these problems. We thank I. Antoniadis, G. Dvali and D. L¨ust for useful conversations on this issue. – 23 –ur power-counting estimates show that ¨uber-naturally light scalars ultimately remainlight because their masses and interactions are systematically suppressed by powers of 1 / V = M g /M p , showing that they are light because the gravity scale is lower than the 4D Planckscale. But because scalars are potentially so UV sensitive, to properly establish that theirmasses are naturally small requires knowing the UV completion of gravity, within a frameworkthat includes an understanding of modulus stabilization and supersymmetry breaking. Thepossibility of studying this quantitatively has only recently become possible, within the large-volume vacua of Type IIB flux compactifications in string theory.The large volume scenario predicts clear hierarchies in the low-energy spectrum of scalarstates, within a framework of calculational control that allows us to be explicit about thesize of quantum corrections. It also allows several sub-scenarios, depending on the size of thevolume and the location of the standard model within the extra dimensions. We discuss afew of the preliminary implications of our results for several of these. • We find that soft supersymmetry breaking terms (∆ m ∝ / V / , / V ) much smallerthan the gravitino mass M / ∝ / V can be stable against quantum corrections, sincethese are smaller or equal to 1 / V / . This is important for the stability of those recentmodels where the soft supersymmetry breaking relevant to weak scale particle physicsis parametrically small compared with the gravitino mass, such as happens when themain contribution to supersymmetry breaking comes from cycles different from the cyclewrapped by the standard model brane. • We find that TeV scale string models not only can be obtained from the large volumescenario, but that their main potential observational obstacle – the existence of anextremely light volume modulus – might not be such a problem, making them muchmore appealing. The volume modulus need not be a problem because the relativelylarge mass corrections it receives from the strong breaking of supersymmetry on thebrane that is required in such models.More generally, ¨uber-naturalness provides the mechanism that underlies many of theattractive features of LV models that have proven valuable for phenomenology. For instance,LV models ultimately bring the news of supersymmetry to ordinary particles through a formof gravity mediation, yet avoid the normal pitfalls (such as with flavour-changing neutralcurrents) of gravity mediation for low-energy phenomenology [30]. ¨Uber-naturalness providesthe framework for the stability of this process against loop corrections.Similarly, inflationary models have been constructed within the LV scenario with theinflaton being a volume modulus [31] or another K¨ahler modulus [32], with the latter achievinga slow roll by virtue of the inflaton taking large field values, rather than requiring a tuningof parameters in the potential. These scenarios profit from the ¨uber-natural protection of thepotential within the LV picture, indicating that extra-dimensional symmetries like generalcovariance can provide an alternative to global shift symmetries (see for instance [33]) foraddressing the η -problem. – 24 –urther implications can well be envisaged. In particular, the suppressed corrections tothe masses of light moduli may be useful for cosmology by providing new and better models ofinflation. Perhaps new models of dark energy could become possible with this new naturalnessconcept in mind. We hope that our results will at least serve to stimulate the search, in asexplicit a manner as possible, for further suppression mechanisms for scalar fields in theorieswith supersymmetric extra dimensions. Acknowledgements
We wish to thank Ignatios Antoniadis, Joe Conlon, Matt Dolan, Gia Dvali, Sven Krippendorf,Louis Leblond, Dieter L¨ust and Sarah Shandera for useful discussions. Various combinationsof us are grateful to the Center for Theoretical Cosmology in Cambridge, Perimeter Institute,McMaster University, the Cook’s Branch Workshop in Houston and the Abdus Salam Inter-national Center for Theoretical Physics, for their support and provision of pleasant environswhere some of this work was done. We also thank Eyjafjallajokull for helping to provideus with undivided time to complete this paper. AM was supported by the EU through theSeventh Framework Programme and the STFC. CB’s research was supported in part by fundsfrom the Natural Sciences and Engineering Research Council (NSERC) of Canada. Researchat the Perimeter Institute is supported in part by the Government of Canada through IndustryCanada, and by the Province of Ontario through the Ministry of Research and Information(MRI).
A. Extra-dimensional kinematics vs KK sums
In this appendix we discuss the loops correction in the higher dimensional theory from theperspective of the lower dimensional theory. Consider a massless scalar field in D dimensionswith a quartic coupling of strength of the order of the higher dimensional cutoff −L D = −
12 ( ∂ Φ) + g
24 Φ , (A.1)with g = (cid:18) (cid:19) ( D − . (A.2)Upon dimensional reduction on a d -torusd s = η µν d x µ d x ν + L δ mn d y m d y n (A.3)and canonical normalization this gives the lower dimensional action for the KK modes of theform −L = − X n i (cid:20) ( ∂φ n i ) + 12 m n i φ n i (cid:21) + g L d X n i ,n j ,n k ,n l c n i n j n k n l φ n i φ n j φ n k φ n l , (A.4)– 25 –here i, j, k, l = 1 ..d , m n i = L P i n i and the interaction coefficients c n i n j n k n l are of orderone if the KK charge associated with the vertex is vanishing.Now let us consider the loops in the above theory, the loop contribution due to a KK ofmass m n i is δm n i ≈ g L d m n i . (A.5)In order to estimate the effect of the entire KK tower we sum the loop contributions of allKK modes up to the scale Λ i.e we restrict the KK momenta to | n i | < N max = Λ L . This gives X n i δm n i ≈ g L d (Λ L ) d = Λ , (A.6)in agreement with the discussion in section 2.We note that the estimate assumed no cancellations in the sum over the loop contribu-tions in (A.6), as is appropiate for a scalar. But, as emphasized in section 2 if one considersgravity in higher dimensions, the restrictions on the form of higher dimensional action im-posed by general covariance necessarily implies cancellations between the loop contributionsof various particles in the lower dimensional theory. Such cancellations can lower the sizeof loop corrections to scales parametrically below the cut off scale of the higher dimensionaltheory. References [1] V. Balasubramanian, P. Berglund, J. P. Conlon and F. Quevedo, “Systematics of ModuliStabilisation in Calabi-Yau Flux Compactifications,” JHEP (2005) 007[arXiv:hep-th/0502058];J. P. Conlon, F. Quevedo and K. Suruliz, “Large-volume flux compactifications: Modulispectrum and D3/D7 soft supersymmetry breaking,” JHEP (2005) 007[arXiv:hep-th/0505076].[2] For a discussion of the pitfalls of using cutoffs witin a non-gravitational context, seeC.P. Burgess and David London, “Uses and abuses of effective Lagrangians,” Phys. Rev.
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