Exact solutions for the biadjoint scalar field

TL;DR

This paper investigates exact nonperturbative static solutions of the biadjoint scalar field $Φ^{aa'}$, a theory central to the double copy and CHY formalisms, by solving the equation $\partial^2Φ^{aa'} - y f^{abc}\tilde{f}^{a'b'c'}Φ^{bb'}Φ^{cc'} = 0$. It first obtains a fully nonperturbative spherical solution for identical Lie algebras, $Φ^{aa'} = -\frac{2 δ^{aa'}}{y T_A r^2}$, and analyzes its divergent energy requiring a short-distance cutoff. Extending to $SU(2)\otimes SU(2)$, a richer one-parameter family of nonperturbative solutions is found, with $Φ^{aa'} = \frac{1}{y r^2}[-k(δ^{aa'} - x^a x^{a'}/r^2) ± \sqrt{2k-k^2}\,ε^{aa'd} x^d / r]$ and energy $E = \frac{16π k}{y^2 r_0^3}$; these include spherical-cylindrical mixtures and are singular at the origin. The authors discuss connections to Wu-Yang monopoles in gauge theory and potential zeroth-copy structures, suggesting that nonperturbative biadjoint configurations could illuminate the nonperturbative aspects of the double copy and CHY relations.

Abstract

Biadjoint scalar theories are novel field theories that arise in the study of non-abelian gauge and gravity amplitudes. In this short paper, we present exact nonperturbative solutions of the field equations, and compare their properties with monopole-like solutions in non-abelian gauge theory. Our results may pave the way for nonperturbative studies of the double copy.