Moduli Self-Fixing

TL;DR

Casas and Ibáñez propose that in 4D no-scale vacua, the moduli-dependent species scale $\Lambda(z_i,\bar z_i)$ generates a one-loop potential that fixes no-scale moduli at desert points with $z_i\sim \mathcal{O}(1)$, creating a Minkowski minimum and a subsequent de Sitter plateau before a runaway at large moduli. The analysis combines local one-loop calculations with global expressions, showing $\delta V_{1-\rm{loop}} \simeq e^{2\phi_4} m_{3/2}^2 M_{ m P}^2 g^{i\bar i} (\partial_i \Lambda/\Lambda)(\partial_{\bar i} \Lambda/\Lambda)$ and identifying minima where $\partial_i \Lambda=0$, i.e., desert points. They illustrate the mechanism in a Type II ${\mathbb Z}_2\times{\mathbb Z}_2$ toroidal orientifold, where modular invariance can constrain non-perturbative corrections and, with D6-branes, yield a 3-generation MSSM-like spectrum with some closed-string moduli fixed. The work also discusses infrared–ultraviolet correlations and potential inflationary realizations via modular-invariant plateau potentials, suggesting new avenues for modulus stabilization beyond conventional non-perturbative superpotentials.

Abstract

In Quantum Gravity (QG), large moduli values lead to towers of exponentially light states, making the QG cut-off field-dependent. In 4D supersymmetric (SUSY) theories, this cut-off is set by the species scale $Λ(z_i, \bar{z}_i)$, where $z_i$ are complex moduli. We argue that in GKP-like 4D no-scale vacua, accounting for this field dependence generates one-loop, positive-definite potentials for the otherwise unfixed moduli, with local Minkowski minima at the desert points in moduli space with $z_i \sim \mathcal{O}(1)$. As these no-scale moduli grow, a dS plateau emerges and, for larger moduli, the potential runs away to zero, consistent with Swampland expectations. This may have important consequences for the moduli fixing problem. In particular non-perturbative superpotentials may not be necessary for fixing the Kähler moduli in a Type IIB setting. Although one loses control over non-perturbative corrections, we argue that modular invariance (more generally, dualities) of the species scale may give us information about the behaviour at small moduli in some simple cases. We illustrate this mechanism in a Type II 4D $\mathbb{Z}_2 \times \mathbb{Z}_2$ toroidal orientifold, where in some cases, modular invariance can give us control over the non-perturbative corrections. The vanishing cosmological constant reflects a delicate cancellation between IR and UV contributions at the minimum. The models may be completed by the addition of intersecting D6-branes, yielding a 3-generation MSSM (RR tadpole free) model with some (but not all) closed string moduli fixed in Minkowski, with the complex structure moduli fixed at the desert points.

Figures (5)

• Figure 1: The different towers of states and the light states with unbroken SUSY are depicted on the left of the picture. On the right, SUSY is spontaneously broken, the gap between states is given by the gravitino mass, and there are generally fewer light states in the theory.
• Figure 2: Schematic representation of the one-loop computation described in the text. The first loop corresponds to summing over light states, while the second loop corresponds to summing over the associated tower of states with masses below the cutoff.
• Figure 3: Qualitative behaviour of equation \ref{['V1loop']} in terms of a single scalar field. The figure shows how, at the field theory level, the one-loop potential contribution drives the scalar field to small vevs.
• Figure 4: The blue scalar potential corresponds to the one-loop contribution with a fixed gravitino mass. The orange potential corresponds to a modulus-dependent gravitino mass, as observed in simple flux compactification settings. In this latter case the potential vanishes at the desert points and at infinity, performing generically dS maxima in between. In this simple one modulus toy model we have fixed the $e^{\phi_4}$ to a constant.
• Figure 5: Schematic representation of the one-loop computation carried out from the F-term auxiliary field insertion in supergravity. The loop corresponds to summing over the tower of states, and the sum in $i$ corresponds to summing over the no-scale moduli contributing to SUSY breaking. The 4d dilaton $e^{\phi_4}$ factor represents the loop-counting coupling parameter $g$.