The Differential of All Two-Loop MHV Amplitudes in N=4 Yang-Mills Theory

TL;DR

The paper delivers an explicit analytic calculation of the differential $dR_n^{(2)}$ for planar two-loop MHV amplitudes in $N=4$ SYM, demonstrating that it can be written entirely in terms of polylogarithms $Li_k(-x)$ with $k=1,2,3$ and $x$ restricted to cluster $\mathcal{X}$-coordinates on the kinematic space. Building on Caron-Huot's differential framework and the ideas of positivity and cluster coordinates, the authors convert the problem into a tractable form and produce compact, symmetry-aware expressions for the $C^{(2)}_{2,i}$ contributions, including explicit $Li_3$ and $Li_2 Li_1$ terms and their cross-ratio arguments. They introduce a practical setup using cross-ratios and momentum-twistor-inspired variables, validate the analytic results against numerical integrations for various $n$ and points in the positive domain, and discuss parity, telescopic cancellations, and the non-uniqueness of representations. The work provides strong evidence that cluster structure governs SYM amplitudes and showcases positivity as a powerful organizing principle, offering a path toward fully analytic higher-point results with accompanying computational tools. Overall, the results advance the understanding of the analytic structure of multi-loop amplitudes and highlight cluster coordinates as a natural framework for organizing polylogarithmic expressions in gauge theory.

Abstract

We present an explicit analytic calculation of the differential of the planar n-particle, two-loop MHV scattering amplitude in N=4 super Yang-Mills theory. The result is expressed only in terms of the polylogarithm functions Li_k(-x), for k=1,2,3, with arguments x belonging to the special class of dual conformal cross-ratios known as cluster X-coordinates. The surprising fact that these amplitudes may be expressed in this way provides a striking example of the manner in which the cluster structure on the kinematic configuration space underlies the structure of amplitudes in SYM theory.