Virtual Color-Kinematics Duality: 6-pt 1-Loop MHV Amplitudes

TL;DR

This paper explores 1-loop MHV amplitudes in N=4 super Yang-Mills and N=8 supergravity, revealing a new 'virtual' color-kinematics duality where physical information is independent of certain expansion coefficients.

Contribution

It introduces a novel 'virtual' color-kinematics duality framework for 6-point 1-loop MHV amplitudes, simplifying the understanding of amplitude relations without solving complex functional equations.

Findings

All physical data are independent of expansion coefficients modulo a functional equation.

The functional equation related to Jacobi identities has no solutions, leading to the concept of virtual duality.

Derived a compact formula for 6-point supergravity amplitudes via the double-copy construction.

Abstract

We study 1-loop MHV amplitudes in N=4 super Yang-Mills theory and in N=8 supergravity. For Yang-Mills we find that the simple form for the full amplitude presented by Del Duca, Dixon and Maltoni naturally leads to one that has physical residues on all compact contours. After expanding the simple form in terms of standard scalar integrals, we introduce redundancies under certain symmetry considerations to impose the color-kinematics duality of Bern, Carrasco and Johansson (BCJ). For five particles we directly find the results of Carrasco and Johansson as well as a new compact form for the supergravity amplitude. For six particles we find that all kinematic dual Jacobi identities are encapsulated in a single functional equation relating the expansion coefficients. By the BCJ double-copy construction we obtain a formula for the corresponding N=8 supergravity amplitude. Quite surprisingly, all physical information becomes independent of the expansion coefficients modulo the functional equation. In other words, there is no need to solve the functional equation at all. This is quite welcome as the functional equation we find, using our restricted set of redundancies, actually has no solutions. For this reason we call these results virtual color-kinematics duality. We end with speculations about the meaning of an interesting global vs. local feature of the functional equation and the situation at higher points.