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Physics Letters B 624 (2005) 270-274
www.elsevier.com/locate/physletb
Kähler corrections for the volume modulus of flux
compactifications
G. von Gersdorffa, A. Hebeckerb
a Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA b Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16und 19, D-69120 Heidelberg, Germany
Received 3 August 2005; accepted 8 August 2005
Available online 18 August 2005
Editor: L. Alvarez-Gaume
Abstract
No-scale models arise in many compactifications of string theory and supergravity, the most prominent recent example being type IIB flux compactifications. Focussing on the case where the no-scale field is a single unstabilized volume modulus (radion), we analyse the general form of supergravity loop corrections that affect the no-scale structure of the Kahler potential. These corrections contribute to the 4d scalar potential of the radion in a way that is similar to the Casimir effect. We discuss the interplay of this loop effect with string-theoretic a' corrections and its possible role in the stabilization of the radion. © 2005 Elsevier B.V. All rights reserved.
In flux compactifications of type IIB supergravity, all complex structure moduli and the dilaton are gener-ically fixed by the non-trivial superpotential induced by the 3-form field strength [1,2]. However, this superpotential is independent of the Kahler moduli. Even if supersymmetry is broken by the non-zero vacuum expectation value of the superpotential W, one of the flat directions associated with the Kahler moduli survives. The resulting 4d model is of no-scale type and the no-scale field T is the Kahler modulus related to an overall rescaling of the compact volume. Pertur-
E-mail addresses: gero@pha.jhu.edu (G. von Gersdorff), a.hebecker@thphys.uni-heidelberg.de (A. Hebecker).
bative corrections generically renormalize the Kahler potential, destroy the no-scale structure and lift the flat directions. We will be interested in loop corrections to the no-scale Kahler potential of the volume modulus T (the radion). In the large-volume limit, such corrections should be calculable within the low-energy effective field theory. They are potentially relevant for the stabilization of the radion and the uplifting to a metastable de Sitter vacuum [3,4].
To understand the supergravity 1-loop corrections, we first focus on a situation where W = 0 and supersymmetry is unbroken. We consider the corrections to the radion kinetic term and to the Einstein-Hilbert term of the 4d effective theory. Before Weyl rescaling, these corrections are independent of the 10d Planck
0370-2693/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.08.024
mass and their form can therefore be inferred from dimensional arguments. It is then straightforward to derive the corresponding Kâhler corrections, which are of the form 1/(T + T)2, with the compact volume scaling as V ~ (Re T)3/2. After a small non-zero W has been introduced as a perturbation, they induce a potentially important contribution to the radion scalar potential.
For the purpose of our technical discussion, we first adopt a slightly more general perspective. Consider a d -dimensional supergravity theory, compactified to 4d on a k -dimensional manifold (d = 4 + k) for which its total volume V corresponds to a flat direction. We write the metric as
ds2 = gßv dx» dxv + R{x)2gmn dym dyn,
where Greek and Latin indices run over 0,..., 3 and 5,...,d, respectively, and the decomposition gmn = R2gmn is defined in such a way that the volume of the compact space measured with the metric gmn is 1. The physical volume is V = Rk .In spite of its various interesting physical effects [2], we neglect for simplicity the possible warp factor, i.e., we assume that g^v does not depend on y. This is justified in the large volume limit.
Assuming that the fundamental d-dimensional Einstein-Hilbert term has coefficient Md-2/2, the 4d action reads
S = j d4x ^g(MR)kM2T
x R + k(k - 1)(d In R)2 + •••],
where R is the 4d curvature scalar. A possible dila-ton dependence of the coefficient Md-2 has not been made manifest since we assume that the dilaton (as well as other moduli) are stabilized at a high scale.
We now set M = 1 and perform a Weyl rescaling
g»v ^ R kgßv,
which takes us to the Einstein frame action
j d4x ^g
1 k(k + 2) 2 2R - (d InR)2 + •
Note that the reason for our very explicit derivation of this familiar action is the importance of the intermediate form, Eq. (2), for the subsequent discussion of quantum corrections.
We further assume that the effective 4d theory is an N = 1 no-scale model [5] where the flat direction R is described by a no-scale field T with Re T = Ra. By comparing Eq. (4) with the kinetic term derived from the standard no-scale Kahler potential K = —3ln(T + T), we find
a = V-(- + 2)/3.
Although our analysis is general, we will primarily focus on two cases:
• d = 5 (k = 1) compactifications of minimal 5d supergravity on Sl /Z2 with supersymmetry broken by the Scherk-Schwarz mechanism [6]. In this case, the no-scale field is given by T = R + iA5, A5 being the fifth component of the graviphoton. This is in agreement with the value a = 1 implied by Eq. (5). The constant superpotential characteristic of the no-scale model is proportional to the Scherk-Schwarz parameter. This simple and familiar example will provide us with a useful consistency check for our results.1
• d = 10 (k = 6) flux compactifications of type IIB string theory, where the internal compact space is a Calabi-Yau orientifold. These indeed result in a no-scale model with no-scale field T = R4 + ib [2], where b = Im T stems from the dimensional reduction of the RR four form. Note that, again, the relation between Re T and R is correctly given by the exponent of Eq. (5).
Perturbative corrections AK to the Kahler potential generically destroy the no-scale structure. After su-persymmetry is broken by the addition of a constant superpotential W (which we consider to be a paramet-rically small effect), this Kahler correction generates a non-trivial potential for the volume modulus. A common approach in field-theoretic model building is to calculate this potential (i.e., the Casimir energy) and, if required, to infer the corresponding Kahler correction (see, e.g., [8] and, in particular, [9]).
Here, we instead consider the Kahler correction directly in the model with W = 0, i.e., before SUSY breaking. We identify the structure of AK from the
1 For the relation of 5d Scherk-Schwarz breaking to 4d no-scale models see Ref. [7].
corrections to the kinetic term of the field R and to the 4d Einstein term. These corrections are most easily understood in the 4d action before Weyl rescaling, Eq. (2).
The tree-level action is invariant under shifts in Im T and we can expect the 1-loop correction to respect this symmetry. The finite corrections, to be added to the action of Eq. (2), then take the form
^^ =J d4x ^g[F(R)U + G(R)(dR)2],
i.e., there is no explicit dependence on Im(T).
The form of the functions F and G follows from dimensional arguments. At 1-loop, the corrections arise simply from the propagation of d-dimensional free fields in the compact space. Alternatively, one may say that they arise from a summation of a Kaluza-Klein tower of 4d fields with mass splitting ~ 1/R. In any case, the d -dimensional Planck mass M does not enter these corrections and the only scale known to these corrections is the compactification radius R. Thus, from the requirement of a dimensionless 4d action, we have F(R) ~ 1/R2 and G(R) ~ 1/R4, i.e., the corrections are
R-2R and R-4(dR)2.
In the above, we have pretended that the field-theoretic one-loop corrections are finite. If they are not, a UV cutoff scale (say the string scale a') enters the result. However, such cutoff-dependent contributions can always be absorbed in a local d-dimensional action (including higher-dimension operators). The leading operators relevant for us are those of Eq. (2) (a d-dimensional cosmological constant is not generated in supersymmetric theories). Subdominant terms may be important. For example, the a' corrections of [10] (see also [11]) considered recently in this context [12] are of this type. Our present result of Eq. (7) is limited to those corrections which cannot be viewed as the dimensional reductions of d-dimensional local operators.
The terms in Eq. (7) correspond to the operators before Weyl rescaling. Going to the Einstein frame,2 we find that both operators give corrections to the kinetic
2 Note that the Weyl rescaling has to be modified in the presence of the first operator in Eq. (7).
(R-(k+4) +----)(dR)2, (8)
where '• • •' stand for terms which are suppressed by inverse powers of R in the limit R Rewriting
Eq. (8) in terms of T, we see that we need a AK which induces a kinetic term
(T + T )-
dT -dT.
We conclude that
(T + T )c
with c =
k + 2 ¡3(k + 2)
We now have all the necessary information to calculate the form of the one-loop potential for R that arises if a non-zero W is included. Using the standard su-pergravity formula for the scalar potential we find the Einstein-frame result
Vcasimir(R) |2(T + T)
"(c+3)
The numerical prefactor, which we have suppressed in the above expression, includes a term c(c - 1). This vanishes for c = 0 and c = 1, i.e., in the two cases where AK preserves the no-scale structure (at least in the large-R limit).
Returning to the frame used in Eqs. (2) and (6) (which we will refer to as the Brans-Dicke frame) by undoing the Weyl rescaling Eq. (3), we find
VCaSimir(R) - l W|2R
2 p—3a+k-2
In the example of the 5d compactification on S1 or S1/Z2 with Scherk-Schwarz SUSY breaking, this gives the well-known 1-loop potential V(R) ~ \W|2R-4. Since the Scherk-Schwarz parameter is dimensionless, this correction has to behave as one would expect in a massless non-SUSY field theory on dimensional grounds. Indeed, the R-4 behaviour is the familiar scaling of the Casimir energy, which ensures that the 4d potential has mass dimension 4.
In the case of 10d flux compactifications, which is our primary interest in this Letter, we obtain a correction
V(CaSimir(R) |2R —8.
This has to be compared with the perturbative string-theoretic (a') corrections [10,12] recently considered
in this context, which scale as
vBd(r)
|W |2R—6.
Even though our Casimir correction is subdominant, it is clearly less so than non-perturbative corrections to the superpotential, which are expected to be exponentially suppressed at large volume.
Thus, if a (meta-)stable minimum at large volume can be found in the combined potential
VBD(R) - |W|2(ca'R—6 + CCasimirR-8), (15)
one may hope that this result will survive a more detailed analysis. Naively, such a minimum appears at
p2 4|cCasimir| Rmin =
3|Ca'|
whenever car < 0 and cCasimir > 0. However, as long as this value cannot be made parametrically large, there is clearly no reason to neglect higher-order and non-perturbative corrections.
A similar situation has recently been discussed in the 5d field-theoretic context, where the interplay of 1- and 2-loop Casimir energy effects was used to stabilize a 5d model in a controlled way [13] (for earlier related ideas see, e.g., [14]). The key there was the possibility of finding a class of models with a hierarchy between the coefficients of the two leading terms in the 1/R expansion.
The obvious parameter that could create such a hierarchy in the present context is the value of a'. To see this in more detail, we write the type IIB supergrav-ity action not in terms of a' and the dilaton, but rather in terms of a' and the 10d Planck mass M. Then the tree-level part depends only on M while the a' correction (and therefore the coefficient ca) involve an explicit factor a'3. Since cCasimjr depends only on the tree-level supergravity action, we conclude that
(RminM)2 - 1/(mV3), (17)
which will be large at small a'. Unfortunately, this corresponds to the strong coupling regime of string theory. By the S self duality of the type IIB theory, the small-a' regime has a dual description with Regge slope a' — a/—1. We expect a' corrections to arise in this theory, implying that the coefficient of the R—6 term can never be made small.
If an explicit calculation of cCasimir, along the lines of [15], were available in a sufficiently large class of
models (with known ca), one could attempt to isolate geometries where the above minimum occurs accidentally at large R. Work in this direction is under way [16]. In the absence of a detailed study based on such explicit results, we can make the following proposal for how a large value of Rmin may arise: Recall that ca is proportional to the Euler number x of the underlying manifold. We can now think of a topolog-ically complicated space, where the two Hodge numbers h1,1 and h2,1 are large while x = 2(h1,1 — h2,1) is small. In this limit one might expect that, because of the large number of light fields (and the presumably large number of corresponding Kaluza-Klein towers), the coefficient cCasimjj. will be large. Thus, stabilization at large Rmin should naturally occur.
To summarize, we have derived the parametrical form of 1-loop supergravity Kahler corrections to the volume modulus of type IIB flux compactifications. We have found the leading finite correction to be of the form AK — 1/(T + T)2 with Re T — R4 — V2/3. In the presence of a non-zero vacuum value of the superpotential W, this gives rise to a scalar potential of the form |W|2/R8, which is subdominant relative to the potential contribution |W|2/R6 induced by a' corrections. We note that our correction, which resembles the Casimir energy effect discussed extensively in field-theoretic models, is dominant for manifold with vanishing Euler number. Furthermore, for specific compact spaces, this Casimir correction may combine with the a' correction to ensure volume stabilization at large R. We expect the Casimir correction discussed in this Letter to be relevant for a wide class of models and stabilization mechanisms.
Note added in proof
After submission of this Letter, a closely related string calculation appeared [17].
Acknowledgements
We would like to thank Jonathan Bagger, Boris Kors, Jan Louis, Claudio Scrucca, Minho Son, Michele Trapletti and Alexander Westphal for helpful discussions. G.G. is supported by the LeonMadansky Fellowship and by NSF Grant P420-D36-2051.
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