Black Holes and Abelian Instantons

TL;DR

This paper demonstrates that electromagnetic theta-terms are physical in the Standard Model coupled to gravity, introducing gravitational Abelian instantons that influence vacuum energy and break axion shift symmetry, highlighting quantum gravity's role in violating global symmetries.

Contribution

It introduces gravitational Abelian instantons in Einstein-Maxwell theory and shows their impact on vacuum energy and axion symmetry breaking.

Findings

Existence of finite-action, asymptotically flat Euclidean configurations with non-zero Chern number.

Instantons generate a theta-dependent contribution to vacuum energy.

Instantons break the axion shift symmetry by generating a potential for the axion.

Abstract

We argue that the electromagnetic $\theta$-term is a physical parameter of the Standard Model coupled to gravity. Specifically, in the context of 4-dimensional Einstein-Maxwell theory we show that there exist Euclidean field configurations that have finite action, are asymptotically flat, and feature non-zero electromagnetic second Chern number. These ``gravitational Abelian instantons" correspond to a dyonic extension of a Euclidean wormhole. We argue that these configurations should be included in the gravitational path integral, and that doing so generates a non-perturbative contribution to the vacuum energy density that is $\theta$-dependent. We provide a Lorentzian interpretation of these instantons as capturing the effect of quantum fluctuations corresponding to pair production and annihilation of charged black holes. When $\theta$ is the expectation value of a dynamical axion field, the instantons presented here generate a potential for the axion, thereby breaking the axion shift symmetry. This provides yet another example of how quantum gravity violates global symmetries through the existence of black holes.