Discrete Gauge Symmetries and the Weak Gravity Conjecture

TL;DR

The paper investigates how a Weak Gravity Conjecture (WGC) analogue can apply to discrete gauge symmetries realized via spontaneous breaking of a U(1), despite the absence of massless gauge fields. By employing a dual description with two U(1) factors coupled through a topological term, it derives bounds on the unit-charge mass $m$ and the symmetry-breaking scale $v$ that saturate but do not violate black hole (BH) remnant constraints, identifying the EFT cutoff with the Higgs scale $\\Lambda \\sim v$. For $\\ ext{Z}_N$ and $\\text{Z}_2^N$ cases, the bounds take the form $m v \\\lesssim \\frac{M_{Pl}^2}{N}$, with $m, v \\\lesssim \\frac{M_{Pl}}{\\sqrt{N}}$ in the appropriate limits; these results illuminate how discrete hair must be lost through UV completion without modifying gravity. The work also discusses implications for naturalness and the weak scale within the Swampland program, including potential connections to neutrino masses in gauged B-L scenarios and the idea of lowering the gravitational cutoff via large $N$ copies of a discrete symmetry. Overall, it frames discrete gauge theories as remnants of the WGC in a Higgsed phase and suggests concrete avenues where quantum gravity constraints could impact low-energy phenomenology.

Abstract

In theories with discrete Abelian gauge groups, requiring that black holes be able to lose their charge as they evaporate leads to an upper bound on the product of a charged particle's mass and the cutoff scale above which the effective description of the theory breaks down. This suggests that a non-trivial version of the Weak Gravity Conjecture (WGC) may also apply to gauge symmetries that are discrete, despite there being no associated massless field, therefore pushing the conjecture beyond the slogan that `gravity is the weakest force'. Here, we take a step towards making this expectation more precise by studying $\mathbb{Z}_N$ and $\mathbb{Z}_2^N$ gauge symmetries realised via theories of spontaneous symmetry breaking. We show that applying the WGC to a dual description of an Abelian Higgs model leads to constraints that allow us to saturate but not violate existing bounds on discrete symmetries based on black hole arguments. In this setting, considering the effect of discrete hair on black holes naturally identifies the cutoff of the effective theory with the scale of spontaneous symmetry breaking, and provides a mechanism through which discrete hair can be lost without modifying the gravitational sector. We explore the possible implications of these arguments for understanding the smallness of the weak scale compared to $M_{Pl}$.