Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral

TL;DR

This paper demonstrates how analyzing the imaginary parts and dispersion relations of Feynman amplitudes via differential equations can effectively solve complex multi-loop integrals, with applications to the sunrise and kite diagrams.

Contribution

It introduces a novel approach leveraging dispersion relations and imaginary parts within differential equations to evaluate two-loop Feynman integrals involving masses.

Findings

Successfully evaluated the two-loop sunrise integral.

Applied the method to the QED kite graph.

Provided first-order (d-4) expansion results.

Abstract

It is shown that the study of the imaginary part and of the corresponding dispersion relations of Feynman graph amplitudes within the differential equations method can provide a powerful tool for the solution of the equations, especially in the massive case. The main features of the approach are illustrated by discussing the simple cases of the 1-loop self-mass and of a particular vertex amplitude, and then used for the evaluation of the two-loop massive sunrise and the QED kite graph (the problem studied by Sabry in 1962), up to first order in the (d-4) expansion.