A Note on the Magnitude of the Flux Superpotential

The magnitude of the flux superpotential $W_{flux}$ plays a crucial role in determining the scales of IIB string compactifications after moduli stabilisation. It has been argued that values of $W_{flux}$ much less than one are preferred, and even required for physical and consistency reasons. This note revisits these arguments. We establish that the coupling (g) of heavy Kaluza-Klein modes to light states scales as ${M_{KK} / M_{Pl}}$ (hence is suppressed by two third powers of the inverse volume of compactification) and argue that consistency of the superspace derivative expansion requires $gF/M^2 \sim m_{3/2}/ M_{KK}<<1$, where $F$ is the auxiliary field of the light fields and $M$ the ultraviolet cutoff. This gives only a mild constraint on the flux superpotential, $W_{flux}<<V^{1/3}$ (where V is the volume of the compactification), which can be easily satisfied for order one values of $W_{flux}$. This regime is also statistically favoured and makes the Bousso-Polchinski mechanism for the vacuum energy hierarchically more efficient.


Introduction
Flux quantisation is a general topological condition on string compactifications. It is the source of a discretuum of vacua after moduli stabilisation which is the basis of the string landscape. The flux superpotential of type IIB string theory compactified on a Calabi-Yau X is given by [1,2]: Even though W 0 is determined from quantised fluxes, over the years several arguments have been given that tend to prefer very small values of W 0 . 1 These arguments can be grouped as based on either consistency, phenomenology and statistics: 1 Notice that in supergravity the superpotential is defined up to a Kähler transformation, and so talking about small or large magnitudes of superpotentials is not a Kähler invariant statement. The Kähler invariant quantity with physical meaning is e K |W | 2 = m 2 3/2 . Hence the superpotentials discussed here are in a particular basis.
1. Argument from Consistency I: a small W 0 has been required by the following consistency argument. The use of a derivative expansion in a supersymmetric effective field theory indicates that there should also be an expansion in powers of ǫ ≡ F/M 2 where F is the auxiliary field of the relevant light fields and M an ultraviolet cutoff.
Imposing ǫ ≪ 1 implies that the superpotential which is proportional to F should be very small [3,4]. 2. Argument from Consistency II: a natural value W 0 ≃ O(1−10) has been argued to be incompatible with a four-dimensional effective field theory since it implies background fluxes with an energy density of order the string scale: . This is not true since the important quantity to look at is not the scaling of the flux potential energy but its vacuum expectation value (VEV). If the dilaton and the complex structure moduli are fixed supersymmetrically, then this VEV is vanishing at leading order, even if it would formally scale as M 4 s . In order to trust the four-dimensional effective field theory, one has to check that the effects used to fix the Kähler moduli, develop a potential whose VEV satisfies V ≪ M 4 KK .
3. Argument from Phenomenology I: a small W 0 has been argued to be necessary also for a viable phenomenology. In the original efforts to stabilise the Kähler moduli T , a non-perturbative term W np was added to W flux [5]. In order to stabilise the T -moduli at values large enough to trust the effective field theory, W np has to be of the same order as W flux , requiring the latter to be 'fine tuned' to values as small as 10 − [8,9].
In this note we revisit these arguments and argue that actually the natural values for the flux superpotential are the largest possible allowed by tadpole constraints, that is W 0 ≃ O (10). As is well known, the requirement in the original KKLT scenario that the flux superpotential be the same order as the non-perturbative superpotential no longer holds in the LARGE Volume Scenario (LVS) [10,11] where there is also no need to tune W 0 to obtain physically interesting scales. More importantly, we will argue that the consistency argument on the derivative expansion, while apparently severe, in fact only imposes the mild constraint W 0 ≪ V 1/3 which is easily satisfied for generic values of W 0 for large enough volume. Finally we also explore the statistics of W 0 for uniform distribution and its impact on the Bousso-Polchinski [12] argument for the cosmological constant and find that large W 0 is statistically preferred and in the LVS it substantially improves the tuning needed in the Bousso-Polchinski mechanism.

Arguments from Consistency
It has been argued that since the supersymmetry multiplet for a chiral superfield includes the auxiliary fields F , the standard derivative expansion in an effective field theory will also incorporate an expansion in powers of F . Concretely, if heavy fields of mass M have been integrated out, then the effective theory will naively contain an expansion in powers of F/M 2 [3,4] (here F is the normalised magnitude of the F-term: F ≡ K TT F T FT ).
In particular consider string flux compactifications where |F T | ≃ M 2 P W 0 /V. If the heavy mass is set to be the ten-dimensional Kaluza-Klein scale M KK ≃ M P /V 2/3 , then imposing |F |/M 2 KK ≪ 1 would imply: As the volume of the compact space has to be large for the effective field theory to be valid, if correct, this condition would set a strong constraint on the value of W 0 , implying We will review this argument here and readdress the original argument for the identification of the right expansion parameter. As the analysis is made after integrating out the heavy fields, the key-point is that the physical implications must depend on the strength of the coupling between heavy and light fields.
In order to get some intuition let us discuss this issue in global supersymmetry. We require a case in which supersymmetry is broken and the F-terms are not trivially small.
We therefore revisit the simplest O'Raifertaigh model.

O'Raifertaigh model revisited
Recall that this model contains three chiral superfields L, H 1 , H 2 with canonical kinetic terms. The superpotential looks like: with g a dimensionless coupling and mass parameters m, M with m ≪ M .
In the component action the couplings of light (L) and heavy (H 1 , H 2 ) states can be read explicitly from the scalar potential: from which we can easily extract the light-heavy couplings: the cubic coupling gM LH 1 H 2 of strength gM and the quartic coupling g 2 L 2 H 2 1 of strength g 2 . Supersymmetry is broken since the equations F L = F H 1 = F H 2 = 0 cannot be simultaneously satisfied. In a simple vacuum with H 1 = H 2 = 0 and F H 1 = F H 2 = 0 supersymmetry is broken by F L = −gm 2 . The splitting of the multiplets is such that for the H 1 multiplet the two scalar components acquire masses of order: with the fermionic component of mass M . Therefore the small expansion parameter is: If the coupling g were of order one, this would correspond to the naive expansion parameter quoted in the literature F/M 2 [3,4]. However in cases of parametrically weak coupling g between light and heavy sectors, this factor can play an important rôle. This will in fact be realised in string compactifications, where the heavy states are Kaluza-Klein modes and their couplings to the light sector will be suppressed by volume factors, as in gravitational interactions.

The Kaluza-Klein case
Let us now consider the case of interest in which the heavy states are Kaluza-Klein (KK) modes of mass M KK ∼ M s /V 1/6 ∼ M P /V 2/3 . We will show from three different approaches that the coupling of heavy KK modes to light states is of order g ∼ M KK /M P ∼ 1/V 2/3 ≪ 1.

Naive KK approach
As M KK ∼ M P /V 2/3 , the Lagrangian for the heavy Kaluza-Klein modes H is: We can then work out the couplings of the heavy KK states with the light volume modulus by expanding the volume around its VEV as V = V + δV. Writing the canonically normalised light field as L ∼ δV/ V , the heavy-light couplings then become: From this we can read the (approximate) coupling between the light and heavy fields to be and we have, similar to the O'Raifertaigh case, gM KK LH 2 and g 2 L 2 H 2 couplings.

Holomorphy approach
There is also an argument based on holomorphy of four-dimensional supergravity theories, that can be applied to the higher order couplings as well. As the light volume modulus Re(T ) ∼ V 2/3 has an axion for its imaginary part, it cannot appear perturbatively in the superpotential. The superpotential must then read: where f i (U ) are functions of the complex structure moduli. The fact that the mass of H scales as M KK ∼ M P /V 2/3 then enforces the unnormalised kinetic terms to look like: (2.11) The interactions between the light modulus L ∼ ln V and the heavy KK modes H are all then induced by canonical normalisation. Prior to normalisation we have: Normalisation then gives: As any power of V can be written as e −λL/M P for some λ, we extract the heavy-light couplings by expanding the exponential, to obtain: We therefore see that any heavy-light interaction is suppressed by the dimensionless cou-

Explicit dimensional reduction approach
Next, we show how this coupling arises from dimensional reduction of the higher-dimensional theory. Our discussion will be similar to that in [13]. Using −ĝ (10) = −g (4) √ g (6) ω 2 e −6u and d 6 y ∝ M −6 s , expanding the action in powers of fluctuations and focussing on the four-dimensional scalar KK modes ϕ i contained within h mn , gives the following four-dimensional Einstein term and scalar kinetic terms: (2.16) On the other hand the contributions to the scalar potential for the ϕ i fields scale as follows: where we have used e 8u = V 4/3 and M 2 In terms of the canonically normalised fields φ i determined by ϕ i ∝ φ i /M P , we find the following schematic quadratic, cubic and quartic interactions: with: Therefore each of the couplings can be clearly written in terms of the dimensionless coupling g and the four-dimensional cut-off scale M KK . Note that φ here stands for both light (L) and heavy (H) states and therefore this potential captures the light-heavy KK couplings needed in the text to show that in the supersymmetric extension the expansion Note also that the expansion of the kinetic terms is of the form (∂φ) 2 + φ M P (∂φ) 2 + φ M 2 P 2 (∂φ) 2 + · · · illustrating that the low-energy derivative interactions are Planck suppressed in contrast to those in the scalar potential that have a universal additional suppression by a factor of M 2 KK /M 2 P = 1/V 4/3 relative to the generic Planck size [13]. We note that this is in agreement with the results obtained from the requirement of holomorphy of the superpotential.
Having identified g from three different approaches we may then continue to determine the consistency expansion parameter in terms of g and F : where we have used that in supergravity theories with (almost) vanishing cosmological constant the value of F is of order F ∼ W/(M P V). 2 Imposing ǫ ≪ 1 implies: which is easily satisfied for W 0 ∼ O(1 − 10) and large volume. 3 The last equality in (2.20) can be seen as a consistency check that the identification of g and ǫ are correct since it has the same functional form as in the O'Raifertaigh case.
The fact that the expansion parameter is gF/M 2 instead of F/M 2 makes a big difference since g ∼ V −2/3 .

Integrating out dilaton and complex structure moduli
A crucial ingredient to stabilise the dilaton S and the complex structure moduli U is the turning on of three-form background fluxes G 3 which carry an energy density of order: A natural value W 0 ≃ O(1 − 10), then leads to an energy density of order the string scale which might look incompatible with a four-dimensional effective field theory. However this argument is too naive since the important quantity to look at is the VEV of the previous expression. In the context of the four-dimensional supergravity theory, ρ flux can be rewritten as: Therefore the effective field theory is under control if the S and U moduli are fixed supersymmetrically by imposing D S,U W = 0, which implies V flux = 0. Once the dilaton and the complex structure moduli have been integrated out, one has then to make sure that subleading effects needed to fix the Kähler moduli develop a potential whose VEV satisfies

Arguments from Phenomenology
In general terms, the Kähler moduli are not stabilised at tree-level and therefore quantum

Arguments from Statistics
Let us now consider the statistical distribution of flux compactifications and concentrate on how it depends on the magnitude of W 0 . Away from the tail of the distribution, W 0 is assumed to be uniformly distributed as a complex function [6]. In [6] general arguments were given to justify the uniform distribution of the Kähler invariant combination e K |W 0 | 2 before Kähler moduli stabilisation. This has been confirmed in recent studies of particular Calabi-Yau flux compactifications where complex structure moduli stabilisation was explicitly computed [14]. The O(100) moduli were reduced to only two by the use of discrete symmetries to make the problem tractable.
On the other hand different conclusions have recently been found for a simplified toy-model of the superpotential for the many moduli case without the use of discrete symmetries [7]. Furthermore, this exploration indicates the possibility that the distribution for the cosmological constant peaks at zero. However this toy model does not correspond to Calabi-Yau compactifications. In particular, it assumes a Kähler potential for the dilaton and complex structure model of the form and a linear superpotential These choices lead to a superpotential VEV that is a product of h 2,1 factors, and the peaking at zero reflects the possibility that any one of these factors may be close to zero.
However, the structure of equations eq. In what follows we assume that W 0 is uniformly distributed (in the region of values suitable for moduli stabilisation) and study implications for the distribution of the cosmological constant. Let us take the density of states to be ρ. Then the number of states in an area dA of the (Re(W 0 ), Im(W 0 ))-plane is dN = ρ dA. Thus the number of states between |W 0 | and |W 0 | + δ|W 0 | is: Therefore for uniformly distributed W 0 the number of states is proportional to its magnitude |W 0 | and a generic O(10) superpotential is also preferred statistically over the tuned small W 0 . Given this, let us compare the distribution of the cosmological constant in the KKLT and LVS constructions. Both of these scenarios give rise to AdS minima and an uplifting is needed. The uplifting mechanism can be of several sources from anti D3-branes to D-terms, non-perturbative effects, etc. (see [15] for a recent review of all these effects in LVS models). The rôle of the uplifting term is to bring the minimum to a value close to Minkowski. The tuning of the cosmological constant can then be done by variation of the fluxes as in the Bousso-Polchinski scenario. We will assume the uplifting has been done and concentrate on the tuning that is achieved by knowing there is an enormous number of flux vacua for a small range of values of |W 0 | say from 0 to 100.
For KKLT the vacuum energy Λ is of order |W 0 | 2 /V 2 . At the minimum of the scalar potential V ∼ ln |W 0 | and so the vacuum energy behaves as |W 0 | 2 . Now suppose we want the number of states between Λ and Λ+δΛ. δΛ would be 10 −120 M 4 P to have a good spacing to realise the Bousso-Polchinski scenario for the cosmological constant: The number of states between Λ and Λ + δΛ is then almost uniform: However for LVS the situation is different. In this case: implying that the cosmological constant scales as: Let us study the distribution at fixed g s . A small shift δ|W 0 | causes a shift in Λ of the order: δΛ ∼ |W 0 | −2 e −3a/gs δ|W 0 | , (4.8) and so, by using (4.3), the number of states between Λ and Λ + δΛ is: dN ∼ ρ|W 0 | 3 e 3a/gs δΛ , (4.9) which is not independent of |W 0 | anymore. Thus from the point of view of the cosmological constant, LVS models have an increasing density as |W 0 | increases. (Furthermore, recently in [16] it was found that the decay from a typical vacuum with value |W 0 | to another one with a different value of |W 0 | is such that the change in |W 0 |, |∆W 0 | (not to be confused with δ|W 0 | above) is also of order O(1)).
These results can also be rephrased in terms of the number h 1,2 of complex structure moduli required for efficient tuning of the cosmological constant. Recall that the number of flux vacua for a Calabi-Yau is given by: 10) where N tad is the number of different values that each flux quanta is allowed to take. N tad is set by tadpole cancellation and is typically of O(10), giving: Using this in (4.9) and demanding δN 1 for δΛ ∼ 10 −120 one arrives at: Note that e a/gs ∼ V/|W 0 | ∼ M P /m 3/2 , and so (4.12) can be rewritten as: Parameterising the flux superpotential as |W 0 | ∼ 10 x and the gravitino mass as M P /m 3/2 ∼ 10 y , the previous relation becomes: implying that in LVS models for a given y, larger values of x are more efficient on the tuning of the cosmological constant since they allow the existence of a reasonable density of states with the observed value of Λ for smaller h 1,2 .

Conclusions
We have revisited the question regarding the magnitude of the flux superpotential in Calabi-Yau compactifications of IIB string theory.
We found that the derivative expansion of supersymmetric field theories with broken supersymmetry can have a control parameter which is significantly smaller that F/M 2 . We have estimated the expansion parameter by three independent methods: a naive Kaluza-Klein reduction, an argument based on holomorphy and a study of the volume dependence of different terms in detailed dimensional reduction. They all give g ≃ V −2/3 . This implies that the consistency condition for the effective field theory to be valid is gF/M 2 ≃ W 0 ≪ V 1/3 which is satisfied for large volumes without imposing strong constraints on W 0 . The fact that the coupling among heavy and light KK states is of order V −2/3 may have other physical implications which may be worth exploring.
We also concluded that for moduli stabilisation, KKLT requires small values of W 0 but this is not needed in the LVS context. A generic value of W 0 ∼ O(1 − 10) is also preferred statistically and in LVS it makes the Bousso-Polchinski mechanism more efficient. What is needed to have a small cosmological constant is a small spacing δ|W 0 | and not small W 0 .
Also, since the scale of the soft terms depends on the values that both W 0 and the volume take, the smallness of W 0 is not a necessary criterion for TeV-scale soft terms. This in keeping with the fact that it is the gravitino mass, determined by e K/2 |W 0 |, that is Kähler invariant while W 0 by itself is not.