Multi-Regge kinematics and the moduli space of Riemann spheres with marked points

TL;DR

This paper develops a new mathematical framework connecting scattering amplitudes in planar N=4 Super Yang-Mills theory with the geometry of moduli spaces of Riemann spheres, enabling explicit calculations at high loop orders.

Contribution

It introduces a novel approach using single-valued iterated integrals on moduli spaces to compute scattering amplitudes in multi-Regge kinematics, including all-loop and multi-leg results.

Findings

Explicit analytic results for MHV amplitudes up to five loops.

All L-loop MHV amplitudes determined by amplitudes with up to L+4 legs.

Non-MHV amplitudes obtained via convolution with a helicity flip kernel.

Abstract

We show that scattering amplitudes in planar N = 4 Super Yang-Mills in multi-Regge kinematics can naturally be expressed in terms of single-valued iterated integrals on the moduli space of Riemann spheres with marked points. As a consequence, scattering amplitudes in this limit can be expressed as convolutions that can easily be computed using Stokes' theorem. We apply this framework to MHV amplitudes to leading-logarithmic accuracy (LLA), and we prove that at L loops all MHV amplitudes are determined by amplitudes with up to L + 4 external legs. We also investigate non-MHV amplitudes, and we show that they can be obtained by convoluting the MHV results with a certain helicity flip kernel. We classify all leading singularities that appear at LLA in the Regge limit for arbitrary helicity configurations and any number of external legs. Finally, we use our new framework to obtain explicit analytic results at LLA for all MHV amplitudes up to five loops and all non-MHV amplitudes with up to eight external legs and four loops.