Differential Equations for Feynman Graph Amplitudes

TL;DR

Remiddi presents a differential-equation framework for Feynman graph amplitudes, showing how IBP identities reduce integrals to master integrals and yield a linear system of first-order ODEs in external variables. The paper derives the central $p^2$-derivative equation for the 1-loop self-mass master integral $S(n,m_1^2,m_2^2,p^2)$, and demonstrates the utility of expansions around $p^2=0$, the pseudo-threshold, and large $p^2$, as well as a quadrature form and the $n\to4$ limit. Although the detailed demonstration is for the 1-loop case, the author argues that the method extends to multi-point and multi-loop amplitudes, producing a powerful tool for numerical evaluation and analytic study with potential intractable closed forms. The work thus provides a concrete, scalable approach to compute and analyze Feynman amplitudes by solving differential equations rather than performing loop integrations.

Abstract

It is by now well established that, by means of the integration by part identities, all the integrals occurring in the evaluation of a Feynman graph of given topology can be expressed in terms of a few independent master integrals. It is shown in this paper that the integration by part identities can be further used for obtaining a linear system of first order differential equations for the master integrals themselves. The equations can then be used for the numerical evaluation of the amplitudes as well as for investigating their analytic properties, such as the asymptotic and threshold behaviours and the corresponding expansions (and for analytic integration purposes, when possible). The new method is illustrated through its somewhat detailed application to the case of the one loop self-mass amplitude, by explicitly working out expansions and quadrature formulas, both in arbitrary continuous dimension n and in the n \to 4 limit. It is then shortly discussed which features of the new method are expected to work in the more general case of multi-point, multi-loop amplitudes.