Solving Recurrence Relations for Multi-Loop Feynman Integrals

TL;DR

The paper presents an algorithmic framework to solve IBP recurrence relations for a broad class of Feynman integrals using Baikov's parametric representation. It splits the problem into identifying master integrals and constructing coefficient functions that express any integral as a linear combination of masters, employing contour and gamma-function techniques and, when necessary, auxiliary parametric masters to maintain rational dependence on the dimension $d$. The method is illustrated with concrete one- and two-loop examples, including NRQCD-heavy-quark-potential diagrams, demonstrating explicit master sets and coefficient functions with results consistent with known analyses. The work provides a practical procedure for reducing complex multi-loop integrals and highlights its computational efficiency and potential extensions to other diagram classes or large-d expansions.

Abstract

We study the problem of solving integration-by-parts recurrence relations for a given class of Feynman integrals which is characterized by an arbitrary polynomial in the numerator and arbitrary integer powers of propagators, {\it i.e.}, the problem of expressing any Feynman integral from this class as a linear combination of master integrals. We show how the parametric representation invented by Baikov can be used to characterize the master integrals and to construct an algorithm for evaluating the corresponding coefficient functions. To illustrate this procedure we use simple one-loop examples as well as the class of diagrams appearing in the calculation of the two-loop heavy quark potential.

Figures (5)

• Figure 1: Feynman diagrams for one- and two-loop massless propagator ((a) and (b)). At two-loop order there are two master integrals which are shown in (c) and (d).
• Figure 2: (a) Generic Feynman diagram needed for the one-loop heavy quark potential. There two master integrals: one corresponds to the very diagram with all indices equal to one and the second one is shown (b).
• Figure 3: Feynman diagrams corresponding to case A (a) and case B (b).
• Figure 4: Feynman diagrams corresponding to the master integrals of case A. In addition to $I_{61}$ there is also a master integral ($I_{61a}$) containing an irreducible numerator (see text).
• Figure 5: Feynman diagrams corresponding to the master integrals of case B. In addition to $I_{6i}$ ($i=3,\ldots,7$) there are also master integrals ($I_{6ia}$) containing an irreducible numerators (see text).