Feynman Diagrams and Differential Equations

TL;DR

This paper pedagogically reviews the differential equations method for evaluating Feynman integrals, demonstrating its application in quantum electrodynamics and complex multi-loop cases, highlighting technical aspects like Laurent expansion and boundary conditions.

Contribution

It provides a comprehensive tutorial on using differential equations for Feynman integrals, including advanced examples and technical insights not extensively covered before.

Findings

Exact computation of Vacuum Polarization tensor in D dimensions

Analysis of differential equations for two-loop three-point and four-loop two-point integrals

Discussion on Laurent expansion, boundary conditions, and relation to difference equations

Abstract

We review in a pedagogical way the method of differential equations for the evaluation of D-dimensionally regulated Feynman integrals. After dealing with the general features of the technique, we discuss its application in the context of one- and two-loop corrections to the photon propagator in QED, by computing the Vacuum Polarization tensor exactly in D. Finally, we treat two cases of less trivial differential equations, respectively associated to a two-loop three-point, and a four-loop two-point integral. These two examples are the playgrounds for showing more technical aspects about: Laurent expansion of the differential equations in D (around D=4); the choice of the boundary conditions; and the link among differential and difference equations for Feynman integrals.