The number of master integrals as Euler characteristic

TL;DR

This paper introduces a parametric method to derive shift relations between Feynman integrals, revealing that the number of master integrals equals the Euler characteristic of a specific polynomial, thus connecting algebraic geometry with quantum field theory.

Contribution

It presents a novel approach using parametric annihilators of the Lee-Pomeransky polynomial to determine the number of master integrals as an Euler characteristic.

Findings

The number of master integrals equals the Euler characteristic of the polynomial.

The approach generates all shift relations between Feynman integrals.

Feynman integrals form a vector space with a dimension given by this Euler characteristic.

Abstract

We give a brief introduction to a parametric approach for the derivation of shift relations between Feynman integrals and a result on the number of master integrals. The shift relations are obtained from parametric annihilators of the Lee-Pomeransky polynomial $\mathcal{G}$. By identification of Feynman integrals as multi-dimensional Mellin transforms, we show that this approach generates every shift relation. Feynman integrals of a given family form a vector space, whose finite dimension is naturally interpreted as the number of master integrals. This number is an Euler characteristic of the polynomial $\mathcal{G}$.