An analytic result for the two-loop seven-point MHV amplitude in N=4 SYM

TL;DR

The paper addresses constructing explicit analytic formulas for the two-loop MHV amplitudes in $\mathcal{N}=4$ SYM from symbol and differential data. It develops an algorithm that uses $A_3$ cluster polylogarithm functions to capture the non-classical piece, and classical polylogarithms with $\mathcal{X}$-coordinate arguments to complete the symbol, while fixing beyond-the-symbol and constant terms via the differential and collinear limits. The authors apply the method to obtain an explicit analytic representation for the seven-point amplitude $R_7^{(2)}$, including detailed structures from $A_3$ building blocks and weight-four coproduct data; they also check consistency with known results and positivity in the positive domain. This work provides a scalable, data-driven toolkit for constructing multi-loop, multi-point amplitudes in $\mathcal{N}=4$ SYM and offers a blueprint for extending to larger $n$.

Abstract

We describe a general algorithm which builds on several pieces of data available in the literature to construct explicit analytic formulas for two-loop MHV amplitudes in N=4 super-Yang-Mills theory. The non-classical part of an amplitude is built from $A_3$ cluster polylogarithm functions; classical polylogarithms with (negative) cluster X-coordinate arguments are added to complete the symbol of the amplitude; beyond-the-symbol terms proportional to $π^2$ are determined by comparison with the differential of the amplitude; and the overall additive constant is fixed by the collinear limit. We present an explicit formula for the seven-point amplitude $R_7^{(2)}$ as a sample application.

Figures (1)

• Figure 1: The two-loop seven-point MHV remainder function $R_7^{(2)}$ evaluated along the line segment parameterized by eq. (\ref{['eq:line']}) between $u=0$ and the boundary of the positive domain at $u\approx 0.3569$.