Symbolic Reduction of Multi-loop Feynman Integrals via Generating Functions

TL;DR

This paper addresses the bottleneck of reducing multi-loop Feynman integrals by exploiting a generating-function framework that converts IBP relations into differential equations, yielding complete symbolic recurrence relations among integral coefficients. The authors formulate a structured algorithm (Modules I–III) that systematically derives reduction rules for differential operators, avoiding the exponential growth of traditional IBP approaches and Gröbner-basis methods. They demonstrate the method on top sectors of sunset and nonplanar double-box diagrams, achieving complete reductions to a small set of master integrals across massless and massive cases. The approach promises automation and scalability for high-rank, multi-loop computations, with potential extensions to other correlators and related perturbative problems.

Abstract

We introduce a novel, systematic method for the complete symbolic reduction of multi-loop Feynman integrals, leveraging the power of generating functions. The differential equations governing these generating functions naturally yield symbolic recurrence relations. We develop an efficient algorithm that utilizes these recurrences to reduce integrals to a minimal set of master integrals. This approach circumvents the exponential growth of traditional integration-by-parts relations, enabling the reduction of high-rank, multi-loop integrals critical for state-of-the-art calculations in perturbative quantum field theory.