On the maximal cut of Feynman integrals and the solution of their differential equations

TL;DR

The paper addresses the challenge of solving coupled differential equations for multi-loop Feynman integrals by focusing on the homogeneous part of the system. It proposes the maximal cut as a direct source of homogeneous solutions, enabling the construction of inhomogeneous solutions through Euler's variation of constants, and tests the method on one-loop and two-loop examples, including elliptic cases. Key contributions include demonstrating that maximal cuts yield explicit homogeneous solutions (often in terms of complete elliptic integrals) and showing how sub-loop localization can simplify calculations, producing one-fold integral representations. The work provides a practical framework for obtaining homogeneous solutions in complex cases and suggests avenues for extending canonical-basis concepts to non-polylogarithmic integrals, with implications for efficient multi-loop computations.

Abstract

The standard procedure for computing scalar multi-loop Feynman integrals consists in reducing them to a basis of so-called master integrals, derive differential equations in the external invariants satisfied by the latter and, finally, try to solve them as a Laurent series in $ε= (4-d)/2$, where $d$ are the space-time dimensions. The differential equations are, in general, coupled and can be solved using Euler's variation of constants, provided that a set of homogeneous solutions is known. Given an arbitrary differential equation of order higher than one, there exist no general method for finding its homogeneous solutions. In this paper we show that the maximal cut of the integrals under consideration provides one set of homogeneous solutions, simplifying substantially the solution of the differential equations.