Systematics of the Multi-Regge Three-Loop Symbol

TL;DR

This paper systematically analyzes the structure of three-loop scattering amplitude symbols in the multi-Regge limit, introducing new building blocks derived from known symbols and symmetry constraints to facilitate amplitude construction.

Contribution

It introduces a systematic method for constructing multi-Regge limit amplitudes at three loops using a finite set of building blocks derived from known symbols and symmetry considerations.

Findings

Two main building blocks are sufficient at leading logarithmic order.

The seven-point building block is constructed using single-valued multiple polylogarithms.

Additional subleading terms require extra building blocks or further perturbative data.

Abstract

We review the systematics of Mandelstam cut contributions to planar scattering amplitudes in the multi-Regge limit. Isolating the relevant cut terms, we explain how the BFKL expansion can be used to construct the perturbative n-point multi-Regge limit amplitude in certain kinematic regions from a finite number of basic building blocks. At three loops and at leading logarithmic order, two building blocks are required. Their symbols are extracted from the known three-loop six-point and seven-point symbols for general kinematics. The new seven-point building block is constructed in terms of single-valued multiple polylogarithms to the extent it can be determined using the symbol as well as further symmetry and consistency constraints. Beyond the leading logarithmic order, the subleading and sub-subleading terms require two and one further building block, respectively. The latter could either be reconstructed from further perturbative data, or from BFKL integrals involving yet-unknown corrections to the central emission block.