Axion Mass from Magnetic Monopole Loops

TL;DR

This work identifies a new mechanism by which axions coupled to abelian gauge fields acquire a potential from magnetic monopole loops via the Witten effect, quantified through a $\theta F \wedge F$ coupling. It develops a worldline formalism to compute the vacuum energy, producing a $\theta$-dependent potential $V_{\rm eff}(\theta)$ with a Fourier-like sum over monopole windings, and interprets each winding as a Euclidean instanton on the monopole worldline, with a saddle action that matches known instanton values in a limiting case. The analysis remains robust against massless fermions and higher-derivative corrections, and the authors discuss phenomenological implications in a hidden sector where the loop-induced axion mass can influence dark matter relic abundances. The results bridge abelian instanton physics with UV completions and suggest intriguing deformations to nonabelian instantons or Euclidean brane instantons in broader frameworks, offering new avenues for axion cosmology and model-building.

Abstract

We show that axions interacting with abelian gauge fields obtain a potential from loops of magnetic monopoles. This is a consequence of the Witten effect: the axion field causes the monopoles to acquire an electric charge and alters their energy spectrum. The axion potential can also be understood as a type of instanton effect due to a Euclidean monopole worldline winding around its dyon collective coordinate. We calculate this effect, which has features in common with both nonabelian instantons and Euclidean brane instantons. To provide consistency checks, we argue that this axion potential vanishes in the presence of a massless charged fermion and that it is robust against the presence of higher-derivative corrections in the effective Lagrangian. Finally, as a first step toward connecting with particle phenomenology and cosmology, we discuss the regime in which this potential is important in determining the dark matter relic abundance in a hidden sector containing an abelian gauge group, monopoles, and axions.

Figures (1)

• Figure 1: Pink regions: axion oscillates with the monopole loop-induced mass dominating over the monopole background-induced mass before CMB formation at temperature $T_{\rm rec}$. Green solid line: the sum of axion abundance ($\Omega_a h^2$) and monopole abundance ($\Omega_{\textsc{m}} h^2$) today is $\Omega_{\rm tot}h^2$ = 0.12, above which the abundance overcloses the Universe. Black dashed lines indicate the fraction of axion dark matter in the total abundance of axion and monopole today. Below the blue line, the axion's mass is so small that it never oscillates. Left panel: monopole yield saturates the Kibble bound. Right panel: monopole yield is from a second order phase transition with a critical exponent $\nu = 0.5$. We fix $m_{\textsc{m}} r_c = {\pi}/{e^2}$, the critical temperature to be $T_c = 1/r_c$, and $f = 10^{15}$ GeV.