An Algorithm for the Symbolic Reduction of Multi-loop Feynman Integrals via Generating Functions

TL;DR

This paper introduces a generating-function approach to symbolically reduce multi-loop Feynman integrals, transforming the problem into solving differential equations within a non-commutative algebra framework.

Contribution

It presents a novel algebraic method that reformulates integration-by-parts identities as differential equations for generating functions, enabling systematic reduction of complex integrals.

Findings

Demonstrated the method on sunset and double-box topologies.

Showed the approach can handle sectors with no master integrals.

Organized reduction rules and completeness criteria within a unified algebraic framework.

Abstract

We develop a generating-function formulation for the symbolic reduction of multi-loop Feynman integrals. In this framework, integration-by-parts identities are rewritten as differential equations for sector-wise generating functions, so the reduction problem can be studied in a non-commutative algebra of differential operators rather than only through relations among individual integrals. This viewpoint leads to an iterative algorithm that generates candidate equations, extracts symbolic reduction rules, updates the active rule set, and tests completeness on the lattice of integral indices. We illustrate the method with the sunset topology, planar and non-planar massless double-box topologies, representative subsectors, and a degenerate example in which the top sector contains no master integral. Together, these examples show how symbolic reduction rules, descendant equations, and completeness criteria can be organized within a single algebraic framework.