Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory

TL;DR

This paper derives a compact analytic formula for the two-loop six-point NMHV amplitude in planar N=4 super Yang-Mills theory, expanding the understanding of multi-loop amplitudes beyond MHV cases.

Contribution

It introduces a novel analytic expression for the two-loop six-point NMHV amplitude, including new functions and a detailed analysis of parity sectors, advancing multi-loop amplitude knowledge.

Findings

Derived a simple analytic formula for the two-loop six-point NMHV amplitude.

Identified two new functions, a^{(2)} and cf4^{(2)}, characterizing non-polylogarithmic sectors.

Revealed a parity odd sector absent at one loop, uniquely determined by the even sector.

Abstract

We provide a simple analytic formula for the two-loop six-point ratio function of planar N = 4 super Yang-Mills theory. This result extends the analytic knowledge of multi-loop six-point amplitudes beyond those with maximal helicity violation. We make a natural ansatz for the symbols of the relevant functions appearing in the two-loop amplitude, and impose various consistency conditions, including symmetry, the absence of spurious poles, the correct collinear behaviour, and agreement with the operator product expansion for light-like (super) Wilson loops. This information reduces the ansatz to a small number of relatively simple functions. In order to fix these parameters uniquely, we utilize an explicit representation of the amplitude in terms of loop integrals that can be evaluated analytically in various kinematic limits. The final compact analytic result is expressed in terms of classical polylogarithms, whose arguments are rational functions of the dual conformal cross-ratios, plus precisely two functions that are not of this type. One of the functions, the loop integral \Omega^{(2)}, also plays a key role in a new representation of the remainder function R_6^{(2)} in the maximally helicity violating sector. Another interesting feature at two loops is the appearance of a new (parity odd) \times (parity odd) sector of the amplitude, which is absent at one loop, and which is uniquely determined in a natural way in terms of the more familiar (parity even) \times (parity even) part. The second non-polylogarithmic function, the loop integral \tilde{\Omega}^{(2)}, characterizes this sector. Both \Omega^{(2)} and tilde{\Omega}^{(2)} can be expressed as one-dimensional integrals over classical polylogarithms with rational arguments.