Direct Solution of Integration-by-Parts Systems

TL;DR

The work introduces a direct IBP-solution framework using generating vectors to produce targeted, master-compatible reduction equations for two-loop planar integrals. By focusing on irreducible invariants and employing higher-degree polynomials, it yields explicit reductions for arbitrary powers and facilitates master-integral identification without solving large linear systems. The method extends to doubled propagators and general powers, with generating-function differential equations producing closed-form sequences for key topologies (sunrise, slashed-box, and double-box) in terms of Appell and hypergeometric functions. Overall, the approach streamlines high-loop integral reductions and opens avenues for algebraic-geometry-inspired analysis of D-modules to identify masters, enabling more efficient QFT amplitude computations.

Abstract

Systems of integration-by-parts identities play an important role in simplifying the higher-loop Feynman integrals that arise in quantum field theory. Solving these systems is equivalent to reducing integrals containing numerator products of irreducible invariants to a small set of master integrals. I present a new approach to solving these systems that finds direct reduction equations for numerator terms of a given Feynman integral. As a particular example of its power, I show how to obtain reduction equations for arbitrary powers of irreducible invariants, along with their solutions.

Figures (3)

• Figure 1: The slashed-box integral $P_{1,1}$.
• Figure 2: The planar double-box integral $P^{**}_{2,2}$.
• Figure 3: The sunrise integral $P_{0,0}$.