Revealing the Landscape of Globally Color-Dual Multi-loop Integrands

TL;DR

This paper advances understanding of constructing color-dual multi-loop integrands, introduces a unifying cubic theory, and demonstrates both successes and limitations of current methods in achieving global color-kinematics duality.

Contribution

It identifies a semi-abelian Yang-Mills theory for constructing one-loop numerators, constructs two-loop NLSM numerators via bootstrap, and analyzes the failure of duality in pure Yang-Mills at two loops.

Findings

Successfully constructed two-loop NLSM numerators with color-kinematics duality.

Reproduced known integrands of special theories via double-copy.

Identified the failure point in two-loop pure Yang-Mills duality, guiding future research.

Abstract

We report on progress in understanding how to construct color-dual multi-loop amplitudes. First we identify a cubic theory, semi-abelian Yang-Mills, that unifies many of the color-dual theories studied in the literature, and provides a prescriptive approach for constructing $D$-dimensional color-dual numerators through one-loop directly from Feynman rules. By a simple weight counting argument, this approach does not further generalize to two-loops. As a first step in understanding the two-loop challenge, we use a $D$-dimensional color-dual bootstrap to successfully construct globally color-dual local two-loop four-point nonlinear sigma model (NLSM) numerators. The double-copy of these NLSM numerators with themselves, pure Yang-Mills, and $\mathcal{N}=4$ super-Yang-Mills correctly reproduce the known unitarity constructed integrands of special Galileons, Born-Infeld theory, and Dirac-Born-Infeld-Volkov-Akulov theory, respectively. Applying our bootstrap to two-loop four-point pure Yang-Mills, we exhaustively search the space of local numerators and find that it fails to satisfy global color-kinematics duality, completing a search previously initiated in the literature. We pinpoint the failure to the bowtie unitarity cut, and discuss a path forward towards non-local construction of color-dual integrands at generic loop order.