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

TL;DR

This work introduces MHV polygons as a planar, symmetry-respecting basis for 1-loop $n$-point amplitudes in $ rightarrow$N=4 SYM and, up to 6 points, $ rightarrow$N=8 supergravity. It shows that a Del Duca–Dixon–Maltoni-type YM expression, expanded in terms of these polygons, yields correct residues and reproduces known 5-pt gravity results via BCJ double-copy, while at 6 points a single functional constraint encodes all CK-duality relations, giving a notion of virtual CK duality with no ordinary solution. Remarkably, physical information becomes independent of the auxiliary expansion coefficients modulo this functional constraint, yet the gravity amplitudes can still be obtained through double-copy. The study reveals a local-versus-global structure in CK duality and proposes a framework to extend these ideas to higher points using a compact, polygon-based expansion.

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.

Figures (2)

• Figure 1: Consistent Moves
• Figure 2: Inconsistent Moves