Some Properties and Uses of the Species Scale

TL;DR

This paper explores the moduli dependence of the Species Scale in Quantum Gravity, deriving differential equations for Wilson coefficients and analyzing one-loop potentials in string compactifications, with implications for Swampland conjectures.

Contribution

It demonstrates how the Species Scale influences effective actions and moduli stabilization, connecting differential equations to Swampland bounds and analyzing one-loop potentials in string models.

Findings

Wilson coefficients obey Laplace-like equations related to the Species Scale.

One-loop potential exhibits minima at desert points and decreases exponentially at large moduli.

Potential stabilization of Kahler moduli at specific points in moduli space.

Abstract

The 'Species Scale' has proved to be an important concept when studying consistent effective actions in Quantum Gravity. This is a short summary of my contribution to the Corfu Summer Institute in September 2025, in which I covered two topics, both related in different ways to the fact that the Species Scale is moduli dependent. In the first, based on work done in collaboration with C. Aoufia and A. Castellano, we show how the one-loop Wilson coefficients $\mathcal{F}_n^{(d)}$ multiplyiing BPS protected ${\cal R}^{2n}$ operators obey Laplace-like eigenvalue differential equations of the form $\mathcal{D}^2_{\bf {\cal M}} \mathcal{F}^{(d)}_n = \eta_d\, \mathcal{F}^{(d)}_n$. This is true both for $n=2$ with 32 and 16 SUSY generators in 10,9,8 dimensions and theories with 8 SUSY generators in 6,5,4 dimensions $(n=1)$. We argue that this fact is at the root of some Swampland conjectures put forward in the past, like bounds on the dumping rates for the tower scales and the exponential behaviour in the Swampland Distance Conjecture. For the second topic, based on work done in collaboration with G.F. Casas, we discuss the one loop potential of the no-scale moduli in GKP-like Type IIB 4d orientifolds. To compute this potential we sum both over light and heavy (tower) modes using the Species Scale as a UV cut-off. We find a generic form $V_{1-loop}\sim g^2m_{3/2}^2M_p^2(g^{i{\bar i}}(\partial_i\Lambda)(\partial_{\bar i}\Lambda))/\Lambda^2$, with $\Lambda$ the Species Scale. This has minima at the $desert$ $points$ in moduli space and exponentially decreases at large moduli, with a dS hill in between. We argue that this potential may lead to the stabilisation of some or all Kahler moduli at the desert points in 4d Type IIB orientifolds of phenomenological interest.