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

TL;DR

The paper develops a framework that integrates dispersion relations for imaginary parts with the differential equations approach to tackle massive multiloop Feynman amplitudes. It demonstrates the method on the two-loop massive sunrise and the QED kite, deriving dispersive representations and solving master integrals up to first nontrivial order in (d-4); the sunrise reduces to one-fold integrals over square-root quartics coupled with multiple polylogarithms, while the kite benefits from sunrise dispersions to yield compact expressions. A careful basis choice for the sunrise amplitudes reveals elliptic structures, enabling a controlled (d-4) expansion and straightforward analytic continuation across physical regions. The results provide practical, reusable expressions combining polylogarithms and elliptic integrals, with potential applications to more complex three- and four-point functions in collider phenomenology.

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.