[Paper Review] Wormholes and masses for Goldstone bosons

Rodrigo Alonso, Alfredo Urbano|arXiv (Cornell University)|Jun 22, 2017

Cosmology and Gravitation Theories80 references9 citations

TL;DR

This paper proposes that gravitational effects in Euclidean wormhole solutions can dynamically break global shift symmetries of Goldstone bosons like axions, generating an effective potential without explicit symmetry breaking. Using a semi-classical analysis of the quadratic action around wormhole backgrounds, the authors show the spectrum of perturbations has only positive eigenvalues, confirming stability and suggesting a non-perturbative mechanism for mass generation in models with ultralight dark matter, axion-like particles, and electroweak scale relaxation.

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.

Motivation & Objective

• To investigate whether gravitational effects in Euclidean wormhole solutions can break global shift symmetries of Goldstone bosons such as axions.
• To assess the stability of these wormhole solutions by analyzing the spectrum of quadratic fluctuations around the background.
• To explore phenomenological implications for QCD axions, ultralight dark matter, black hole superradiance, and electroweak scale relaxation.
• To generalize the mechanism to arbitrary Goldstone coset spaces and derive the effective potential generated by wormhole solutions.

Proposed method

• Construct a Euclidean action for a minimally coupled axion field in Einstein gravity, including the Gibbons-Hawking-York boundary term.
• Derive the quadratic action for tensor and inhomogeneous scalar perturbations around the wormhole background using a 4D de Sitter-like slicing with a conformal factor.
• Diagonalize the resulting Schrödinger-type operators for perturbations in the $ au$-direction and spatial modes on $S_3$, using the Pöschl-Teller potential form.
• Analyze the spectrum of the differential operators $ abla^2 + 3$ and $ abla^2 + 6$ for scalar and tensor modes, respectively, to confirm positivity of eigenvalues.
• Use rescaling techniques to eliminate conformal factor problems and ensure the action is bounded from below.
• Generalize the analysis to arbitrary coset spaces by considering the structure of the effective potential generated by wormhole solutions.

Experimental results

Research questions

• RQ1 Can non-trivial Euclidean wormhole solutions in gravity dynamically break global shift symmetries of Goldstone bosons?
• RQ2 Are these wormhole solutions stable under small perturbations, as indicated by the spectrum of the quadratic action?
• RQ3 What is the effective potential generated by such wormhole solutions, and how does it affect axion masses and cosmological dynamics?
• RQ4 How do these solutions impact phenomenological scenarios such as QCD axion models, ultralight dark matter, and black hole superradiance?
• RQ5 Can the mechanism be generalized to arbitrary Goldstone coset spaces beyond the $U(1)$ case?

Key findings

• The spectrum of tensor perturbations around the wormhole background is fully positive, with the differential operator $ abla^2 + 6$ having eigenvalues starting from 6.
• The inhomogeneous scalar perturbations also exhibit a fully positive spectrum, with the lowest eigenvalue of the Schrödinger-type operator $ abla^2 + 3$ starting from 3.
• The Pöschl-Teller potential in the scalar mode analysis features a double-degenerate negative eigenvalue at $E = -1/4$, but this corresponds to the lowest physical mode $ u = 3$, which is positive and stable.
• The absence of negative modes in both tensor and scalar sectors confirms the stability of the wormhole solution under small fluctuations.
• The effective potential generated by the wormhole solution is non-trivial and competes with QCD contributions, potentially affecting axion mass and vacuum structure.
• The mechanism generically applies to any Goldstone coset, suggesting a universal non-perturbative mechanism for generating masses in global symmetry breaking scenarios via gravitational instantons.