Single-valued harmonic polylogarithms and the multi-Regge limit

TL;DR

This paper introduces single-valued harmonic polylogarithms as natural functions for describing the multi-Regge limit in six-gluon scattering within planar N=4 super Yang-Mills theory, enabling high-loop order calculations and fixing constants in the amplitude.

Contribution

It applies single-valued harmonic polylogarithms to compute multi-loop six-gluon scattering amplitudes and determine BFKL eigenvalues and impact factors with high precision.

Findings

Computed six-gluon MHV remainder function up to ten loops in LLA.

Determined BFKL kernel eigenvalues and impact factors to high logarithmic accuracy.

Fixed constants in the four-loop remainder function using multi-Regge limit matching.

Abstract

We argue that the natural functions for describing the multi-Regge limit of six-gluon scattering in planar N=4 super Yang-Mills theory are the single-valued harmonic polylogarithmic functions introduced by Brown. These functions depend on a single complex variable and its conjugate, (w,w*). Using these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine the six-gluon MHV remainder function in the leading-logarithmic approximation (LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through nine loops. In separate work, we have determined the symbol of the four-loop remainder function for general kinematics, up to 113 constants. Taking its multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix all but one of the constants that survive in this limit. The multi-Regge limit factorizes in the variables (\nu,n) which are related to (w,w*) by a Fourier-Mellin transform. We can transform the single-valued harmonic polylogarithms to functions of (\nu,n) that incorporate harmonic sums, systematically through transcendental weight six. Combining this information with the four-loop results, we determine the eigenvalues of the BFKL kernel in the adjoint representation to NNLLA accuracy, and the MHV product of impact factors to NNNLLA accuracy, up to constants representing beyond-the-symbol terms and the one symbol-level constant. Remarkably, only derivatives of the polygamma function enter these results. Finally, the LLA approximation to the six-gluon NMHV amplitude is evaluated through ten loops.