Differential equations for multi-loop integrals and two-dimensional kinematics

TL;DR

This paper develops differential equations for multi-loop integrals in planar N=4 SYM, enabling iterative solutions for complex scattering amplitudes using symbol technology and boundary conditions.

Contribution

It introduces explicit differential equations relating two-loop integrals to simpler diagrams in two-dimensional kinematics, advancing methods for calculating multi-loop amplitudes.

Findings

Explicit equations for two-loop eight-point finite diagrams.

Solution of integrals using symbol technology and boundary conditions.

Applicability to all-loop order for double pentaladders.

Abstract

In this paper we consider multi-loop integrals appearing in MHV scattering amplitudes of planar N=4 SYM. Through particular differential operators which reduce the loop order by one, we present explicit equations for the two-loop eight-point finite diagrams which relate them to massive hexagons. After the reduction to two-dimensional kinematics, we solve them using symbol technology. The terms invisible to the symbols are found through boundary conditions coming from double soft limits. These equations are valid at all-loop order for double pentaladders and allow to solve iteratively loop integrals given lower-loop information. Comments are made about multi-leg and multi-loop integrals which can appear in this special kinematics. The main motivation of this investigation is to get a deeper understanding of these tools in this configuration, as well as for their application in general four-dimensional kinematics and to less supersymmetric theories.