Non-perturbative aspects of the self-dual double copy

TL;DR

This paper explores non-perturbative aspects of the double copy correspondence in Euclidean signature, revealing the absence of certain solutions in four dimensions and providing new insights into instanton double copies.

Contribution

It introduces a new approach to studying the double copy in Euclidean space and offers a generalized method for relating instantons to the double copy framework.

Findings

Spherically symmetric Euclidean solutions do not exist in four dimensions.

The Eguchi-Hanson instanton has a more complex single copy structure than previously understood.

A new prescription for double-copying instantons is proposed.

Abstract

The double copy is by now a firmly-established correspondence between amplitudes and classical solutions in biadjoint scalar, gauge and gravity theories. To date, no strongly coupled examples of the double copy in four dimensions have been found, and previous attempts based on exact non-linear solutions of biadjoint theory in Lorentzian signature have failed. In this paper, we instead look for biadjoint solutions in Euclidean signature, which may be relatable to Yang-Mills or gravitational instantons. We show that spherically symmetric power-like Euclidean solutions do not exist in precisely four spacetime dimensions. The explanation for why this is the case turns out to involve the Eguchi-Hanson instanton, whose single copy structure is found to be more complicated (and interesting) than previously thought. We provide a more general prescription for double-copying instantons, and explain how our results provide a higher-dimensional complement to a recently presented non-perturbative double copy of exact solutions in two spacetime dimensions. In doing so, we demonstrate how the replacement of colour by kinematic Lie algebras operates at the level of exact classical solutions.