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

Understand

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.