Some Properties and Uses of the Species Scale

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.