Periods and Feynman integrals

TL;DR

The paper proves that, for scalar multi-loop Feynman integrals in the Euclidean region with rational ratios of invariants and masses, every coefficient in the Laurent expansion in $\varepsilon$ is a period. It achieves a constructive proof using iterated sector decomposition to reduce tensor integrals to scalar ones and to monomialise polynomials, linking Feynman integral structure to algebraic geometry through $\mathcal U$ and $\mathcal F$ and Hironaka’s polyhedra game. The result extends known periodicity phenomena from Igusa zeta functions to general multi-loop integrals and constrains the types of special numbers that can appear in perturbative QFT calculations. This deepens the connection between quantum field theory and the theory of periods via a geometric, algorithmic framework.

Abstract

We consider multi-loop integrals in dimensional regularisation and the corresponding Laurent series. We study the integral in the Euclidean region and where all ratios of invariants and masses have rational values. We prove that in this case all coefficients of the Laurent series are periods.

Figures (1)

• Figure 1: An example of a two-loop Feynman graph with three external legs.