Heptagon Functions and Seven-Gluon Amplitudes in Multi-Regge Kinematics

TL;DR

This paper computes all two-to-five gluon scattering amplitudes in planar N=4 super-Yang-Mills theory within multi-Regge kinematics, providing four-loop results expressed in single-valued multiple polylogarithms, and validating the bootstrap approach against the all-orders multi-Regge conjecture.

Contribution

It presents the first complete four-loop computation of two-to-five gluon amplitudes in multi-Regge kinematics, connecting bootstrap results with the all-orders multi-Regge framework.

Findings

Results agree with the conjectured all-order central emission vertex.

Provides a resummation of Fourier-Mellin residues into single-valued polylogarithms.

Cross-validates bootstrap approach with multi-Regge predictions.

Abstract

We compute all $2\to5$ gluon scattering amplitudes in planar $\mathcal{N}=4$ super-Yang-Mills theory in the multi-Regge limit that is sensitive to the non-trivial ("long") Regge cut. We provide the amplitudes through four loops and to all logarithmic accuracy at leading power, in terms of single-valued multiple polylogarithms of two variables. To obtain these results, we leverage the function-level results for the amplitudes in the Steinmann cluster bootstrap. To high powers in the series expansion in the two variables, our results agree with the recently conjectured all-order central emission vertex used in the Fourier-Mellin representation of amplitudes in multi-Regge kinematics. Our results therefore provide a resummation of the Fourier-Mellin residues into single-valued polylogarithms, and constitute an important cross-check between the bootstrap approach and the all-orders multi-Regge proposal.