Feynman Diagrams and Differential 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.

invited review article from Int. J. Mod. Phys. A

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• The iterated structure of the all-order result for the two-loop sunrise integral
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• Determining Feynman integrals with only input from linear algebra
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