Feynman integral relations from parametric annihilators

TL;DR

This paper explores shift relations between Feynman integrals using Mellin transforms and parametric annihilators, linking them to the Euler characteristic of Lee-Pomeransky polynomials to determine the number of master integrals.

Contribution

It introduces a novel approach connecting Feynman integral relations with algebraic geometry via the Euler characteristic, providing a new way to compute master integrals.

Findings

The number of master integrals equals the Euler characteristic of the Lee-Pomeransky polynomial.

Techniques are developed to compute this Euler characteristic in various examples.

Comparison with previous methods shows consistency and potential advantages.

Abstract

We study shift relations between Feynman integrals via the Mellin transform through parametric annihilation operators. These contain the momentum space IBP relations, which are well-known in the physics literature. Applying a result of Loeser and Sabbah, we conclude that the number of master integrals is computed by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate techniques to compute this Euler characteristic in various examples and compare it with numbers of master integrals obtained in previous works.