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On Volume Stabilization by Quantum Corrections

Marcus Berg, Michael Haack, Boris Kors

TL;DR

This work investigates stabilizing the overall volume modulus of ${\cal N}=1$ type IIB orientifold compactifications using only perturbative corrections to the Kahler potential, aiming to avoid non-perturbative effects. By combining known tree-level $\alpha'$ corrections with string-loop corrections that break the no-scale structure, the authors show it is, in principle, possible to obtain a large-volume minimum for the volume modulus, though explicit realizations are challenging and require fine-tuning of complex-structure moduli. The analysis hinges on the large-volume expansion where the leading no-scale breaking terms scale as $\frac{c_1}{(-i(\rho-\bar{\rho}))^{3/2}}$ and $\frac{c_2}{(-i(\rho-\bar{\rho}))^{2}}$, with stabilization requiring $c_1<0$, $c_2>0$, and $|c_2/c_1|\gg1$, a condition explored in concrete models such as the $\mathbb{T}^6/(\mathbb{Z}_2\times\mathbb{Z}_2)$ orientifold and heuristically in a $\mathbb{Z}_6'$-type model. The results indicate that while a perturbative, large-volume minimum is plausible in principle, achieving it in practice depends on the Euler number sign and significant tuning of the complex structure, highlighting both the potential and the delicacy of purely perturbative volume stabilization.

Abstract

We discuss prospects for stabilizing the volume modulus of N=1 supersymmetric type IIB orientifold compactifications using only perturbative corrections to the Kahler potential. Concretely, we consider the known string loop corrections and tree-level alpha' corrections. They break the no-scale structure of the potential, which otherwise prohibits stabilizing the volume modulus. We argue that when combined, these corrections provide enough flexibility to stabilize the volume of the internal space without non-perturbative effects, although we are not able to present a completely explicit example within the limited set of currently available models. Furthermore, a certain amount of fine-tuning is needed to obtain a minimum at large volume.

On Volume Stabilization by Quantum Corrections

TL;DR

This work investigates stabilizing the overall volume modulus of type IIB orientifold compactifications using only perturbative corrections to the Kahler potential, aiming to avoid non-perturbative effects. By combining known tree-level corrections with string-loop corrections that break the no-scale structure, the authors show it is, in principle, possible to obtain a large-volume minimum for the volume modulus, though explicit realizations are challenging and require fine-tuning of complex-structure moduli. The analysis hinges on the large-volume expansion where the leading no-scale breaking terms scale as and , with stabilization requiring , , and , a condition explored in concrete models such as the orientifold and heuristically in a -type model. The results indicate that while a perturbative, large-volume minimum is plausible in principle, achieving it in practice depends on the Euler number sign and significant tuning of the complex structure, highlighting both the potential and the delicacy of purely perturbative volume stabilization.

Abstract

We discuss prospects for stabilizing the volume modulus of N=1 supersymmetric type IIB orientifold compactifications using only perturbative corrections to the Kahler potential. Concretely, we consider the known string loop corrections and tree-level alpha' corrections. They break the no-scale structure of the potential, which otherwise prohibits stabilizing the volume modulus. We argue that when combined, these corrections provide enough flexibility to stabilize the volume of the internal space without non-perturbative effects, although we are not able to present a completely explicit example within the limited set of currently available models. Furthermore, a certain amount of fine-tuning is needed to obtain a minimum at large volume.

Paper Structure

This paper contains 4 sections, 22 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General remarks on the large volume expansion
  3. The $\mathbb T^6/(\mathbb Z_2\times \mathbb Z_2)$ orientifold
  4. Perturbative volume stabilization

Figures (3)

  • Figure 1: The expression $\frac{1}{2} c\, {\mathcal{E}}_2^{\rm D7}(0,U)$, where $c$ is given by $(128 \pi^6)^{-1}$ as in gg1. The oscillatory behavior for small imaginary part of $U$ is shown in more detail in figure \ref{['fig:E2_U1decrease']}. For small real part and large imaginary part, the function behaves as $\frac{1}{3} 128\pi^4 c \times {\rm Im}(U)^2$ (see equation (B.3) in gg1).
  • Figure 2: The expression $\frac{1}{2} c\, {\mathcal{E}}_2^{\rm D7}(0,U)$ along Im$(U) \in\{ 0.6,0.7,0.8\}$ in figure \ref{['fig:E2_3d']}.
  • Figure 3: Volume stabilization with the potential (\ref{['potential']}), for a few values of Im$(S)$, and Im$(U)=650$. The plot shows $10^{18}\times V/|W|^2$.