Wormholes and masses for Goldstone bosons

TL;DR

This paper shows that Euclidean wormholes in Einstein gravity nonperturbatively break global axion shift symmetries down to a discrete subgroup, generating an exponentially suppressed effective potential with S_inst ∝ M_Pl/f_a and a measurable impact on axion phenomenology. The authors establish the stability of these wormholes by computing the quadratic action’s spectrum and demonstrate a positive definite fluctuation determinant, enabling a well-defined gravity-induced axion potential. They derive concrete phenomenological bounds for the QCD axion (m_a ≳ 4.8×10^{-10} eV, f_a ≲ 10^{16} GeV) and show that gravity can also supply mass terms for ultralight scalars relevant for dark matter, while discussing implications for black hole superradiance and electroweak-scale relaxation via clockwork. Finally, they extend the analysis to generic Goldstone cosets, finding that gravity generically endows all Goldstone bosons with a mass and that UV completions can modify the wormhole action, with broader implications for UV physics and model-building.

Abstract

There exist non-trivial stationary points of the Euclidean action for an axion particle minimally coupled to Einstein gravity, dubbed wormholes. They explicitly break the continuos global shift symmetry of the axion in a non-perturbative way, and generate an effective potential that may compete with QCD depending on the value of the axion decay constant. In this paper, we explore both theoretical and phenomenological aspects of this issue. On the theory side, we address the problem of stability of the wormhole solutions, and we show that the spectrum of the quadratic action features only positive eigenvalues. On the phenomenological side, we discuss, beside the obvious application to the QCD axion, relevant consequences for models with ultralight dark matter, black hole superradiance, and the relaxation of the electroweak scale. We conclude discussing wormhole solutions for a generic coset and the potential they generate.

Figures (15)

• Figure 1: The Euclidean wormhole solution connects two asymptotically flat regions (left panel). The minimum size of the wormhole throat defines the characteristic length $L$ in eq. (\ref{['eq:Wormhole']}). The wormhole throat is characterized by the axion charge $n$ (see eq. (\ref{['eq:Charge']})). This characteristic allows for the possibility to interpret the wormhole as the combination of an instanton with axion charge $+n$ and an anti-instanton with axion charge $-n$ (right panel).
• Figure 2: Pictorial representation of an euclidean wormhole with endpoints connected to the same asymptotically flat space. The Euclidean time runs in the direction of the arrow. At each instant of time, two space dimensions have been suppressed. The cross-section of the wormhole throat is a three-sphere $S_3$.
• Figure 3: Axion field profile corresponding to the wormhole solution in eq. (\ref{['eq:Instanton']}). The blue and red colors describe the instanton and anti-instanton part of the solution (see fig. \ref{['fig:Wormhole']}). A string of widely separated instantons and anti-instantons (right panel) provides an approximate stationary point of the Euclidean action (which becomes an exact solution of the equation of motion only asymptotically, in the limit of infinite separation).
• Figure 4: Tunneling through a potential barrier in Quantum Mechanics mediated by non-perturbative instanton solutions.
• Figure 5: Spectrum of the differential operator in eq. (\ref{['eq:Schrod']}) that coincides with the Pöschl-Teller potential well in Quantum Mechanics. The green band corresponds to the continuum spectrum. In the left (right) panel we show the parity odd (parity even) eigenfunction $q^{-}_{\lambda = 0}(\tau)$ ($q^{+}_{\lambda = -8}(\tau)$). We do not show the parity even eigenfunction $q^{+}_{\lambda = 0}(\tau)$ since it is not square-integrable.