Critical points and number of master integrals

TL;DR

This paper links the number of master integrals in multiloop Feynman diagrams to the critical points of specific polynomials, providing a topological method and a computational tool for counting them.

Contribution

It introduces a topological approach to determine the number of master integrals using critical points of polynomials and provides a Mathematica package for automation.

Findings

Number of master integrals equals the sum of Milnor numbers of critical points.

Critical points of Symanzik polynomials determine master integral count.

Mathematica package Mint automates the counting process.

Abstract

We consider the question about the number of master integrals for a multiloop Feynman diagram. We show that, for a given set of denominators, this number is totally determined by the critical points of the polynomials entering either of the two representations: the parametric representation and the Baikov representation. In particular, for the parametric representation the corresponding polynomial is just the sum of Symanzik polynomials. The relevant topological invariant is the sum of the Milnor numbers of the proper critical points. We present a Mathematica package Mint to automatize the counting of the master integrals.