# Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph

## TL;DR

This paper develops a method using maximal cuts and elliptic integrals to solve the differential equations for three-loop massive banana Feynman integrals with equal masses, providing explicit solutions.

## Contribution

It introduces a novel approach combining maximal cuts and elliptic integrals to explicitly solve the differential equations for the three-loop massive banana graph.

## Key findings

- Homogeneous solutions expressed in terms of elliptic integrals.
- All independent solutions obtained via contour integration of the maximal cut.
- Inhomogeneous solutions derived using Euler's variation of constants.

## Abstract

We consider the calculation of the master integrals of the three-loop massive banana graph. In the case of equal internal masses, the graph is reduced to three master integrals which satisfy an irreducible system of three coupled linear differential equations. The solution of the system requires finding a $3 \times 3$ matrix of homogeneous solutions. We show how the maximal cut can be used to determine all entries of this matrix in terms of products of elliptic integrals of first and second kind of suitable arguments. All independent solutions are found by performing the integration which defines the maximal cut on different contours. Once the homogeneous solution is known, the inhomogeneous solution can be obtained by use of Euler's variation of constants.