A Note on the WGC, Effective Field Theory and Clockwork within String Theory

TL;DR

The paper analyzes how Higgsing in $U(1)^N$ theories interacts with the Weak Gravity Conjecture within heterotic string theory. It shows that while EFT hints at WGC violations after Higgsing, the actual cutoff is governed by the string scale, which remains invariant under Higgsing, and is further constrained by the requirement that large-charge Higgs fields be part of the massless spectrum. A key result is the bound $Z^2\le k_0$, limiting parametric hierarchies and restricting iterative clockwork, with the total number of $U(1)$ factors bounded by central-charge considerations. Consequently, clockwork can exist in string-theoretic embeddings but with finite, model-dependent limits on depth and suppression.

Abstract

It has been recently argued that Higgsing of theories with $U(1)^n$ gauge interactions consistent with the Weak Gravity Conjecture (WGC) may lead to effective field theories parametrically violating WGC constraints. The minimal examples typically involve Higgs scalars with a large charge with respect to a $U(1)$ (e.g. charges $(Z,1)$ in $U(1)^2$ with $Z\gg 1$). This type of Higgs multiplets play also a key role in clockwork $U(1)$ theories. We study these issues in the context of heterotic string theory and find that, while indeed there is no new physics at the standard magnetic WGC scale $Λ\sim g_{IR} M_P$, the string scale is just slightly above, at a scale $\sim \sqrt{k_{IR}}Λ$. Here $k_{IR}$ is the level of the IR $U(1)$ worldsheet current. We show that, unlike the standard magnetic cutoff, this bound is insensitive to subsequent Higgsing. One may argue that this constraint gives rise to no bound at the effective field theory level since $k_0$ is model dependent and in general unknown. However there is an additional constraint to be taken into account, which is that the Higgsing scalars with large charge $Z$ should be part of the string massless spectrum, which becomes an upper bound $k_{IR}\leq k_0^2$, where $k_0$ is the level of the UV currents. Thus, for fixed $k_0$, $Z$ cannot be made parametrically large. The upper bound on the charges $Z$ leads to limitations on the size and structure of hierarchies in an iterated $U(1)$ clockwork mechanism.