The six-point remainder function to all loop orders in the multi-Regge limit

TL;DR

This paper derives an all-orders formula for the six-point amplitude in planar N=4 super Yang-Mills theory within the multi-Regge limit, confirming previous results up to 14 loops and exploring the near-collinear behavior.

Contribution

It introduces a universal all-orders formula for the six-point amplitude in the multi-Regge limit and connects different helicity configurations using single-valued harmonic polylogarithms.

Findings

Agreement with Lipatov and Prygarin's integral formula up to 14 loops

Explicit all-orders expansion in the near-collinear limit

Leading term matches the double-leading-logarithmic approximation

Abstract

We present an all-orders formula for the six-point amplitude of planar maximally supersymmetric N=4 Yang-Mills theory in the leading-logarithmic approximation of multi-Regge kinematics. In the MHV helicity configuration, our results agree with an integral formula of Lipatov and Prygarin through at least 14 loops. A differential equation linking the MHV and NMHV helicity configurations has a natural action in the space of functions relevant to this problem---the single-valued harmonic polylogarithms introduced by Brown. These functions depend on a single complex variable and its conjugate, w and w*, which are quadratically related to the original kinematic variables. We investigate the all-orders formula in the near-collinear limit, which is approached as |w|->0. Up to power-suppressed terms, the resulting expansion may be organized by powers of log|w|. The leading term of this expansion agrees with the all-orders double-leading-logarithmic approximation of Bartels, Lipatov, and Prygarin. The explicit form for the sub-leading powers of log|w| is given in terms of modified Bessel functions.