An analytic result for the two-loop seven-point MHV amplitude in N $$ \mathcal{N} $$ = 4 SYM

John Golden,Marcus Spradlin

doi:10.1007/jhep08(2014)154

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 $\pi^2$ 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.

Highlights

This section is a brief review of some of the more advanced mathematics that will appear throughout the rest of the paper, namely the coproduct δ and cluster algebras
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 A3 cluster polylogarithm functions; classical polylogarithms with cluster X coordinate arguments are added to complete the symbol of the amplitude; beyond-thesymbol terms proportional to π2 are determined by comparison with the differential of the amplitude; and the overall additive constant is fixed by the collinear limit

Summary

Background

This section is a brief review of some of the more advanced mathematics that will appear throughout the rest of the paper, namely the coproduct δ and cluster algebras. The cluster algebra relevant for two-loop MHV scattering amplitudes in SYM theory is the Gr(4, n) Grassmannian cluster algebra, which is related to the kinematic configuration space for n particles, Confn(P3). These coordinates come in two flavors, Aand X - coordinates. Cluster X -coordinates are a special class of cross-ratios built from A-coordinates These two topics, polylogarithms and cluster algebras, merge beautifuly in the arena of SYM theory. Before we describe our algorithm we would first like to clarify the difficulties that our cluster algebraic approach allows us to overcome

The problem of integrating symbols

Conclusion