The number of master integrals is finite

TL;DR

This paper proves that for any fixed Feynman graph, the process of reducing integrals with various propagator powers always results in a finite set of master integrals, confirming a long-standing empirical observation.

Contribution

The paper provides a rigorous proof that the number of master integrals for a fixed Feynman graph is always finite, establishing a fundamental theoretical result.

Findings

Number of master integrals is finite for any fixed Feynman graph.

The finiteness is a proven theoretical fact, not just empirical.

Supports the validity of reduction procedures in quantum field theory.

Abstract

For a fixed Feynman graph one can consider Feynman integrals with all possible powers of propagators and try to reduce them, by linear relations, to a finite subset of integrals, the so-called master integrals. Up to now, there are numerous examples of reduction procedures resulting in a finite number of master integrals for various families of Feynman integrals. However, up to now it was just an empirical fact that the reduction procedure results in a finite number of irreducible integrals. It this paper we prove that the number of master integrals is always finite.