On the periods of some Feynman integrals

TL;DR

This paper investigates the conditions under which massless phi^4 theory Feynman integrals evaluate to multiple zeta values and have mixed Tate motives, providing geometric criteria applicable to many graphs.

Contribution

It introduces geometric and combinatorial criteria for Feynman graphs to determine when their integrals evaluate to multiple zeta values and have mixed Tate motives, extending previous known cases.

Findings

Criteria hold for infinite classes of graphs including all previously known cases.

Calabi-Yau varieties emerge where criteria fail, indicating more complex motives.

Provides a unified geometric framework for understanding Feynman integral evaluations.

Abstract

We study the related questions: (i) when Feynman amplitudes in massless $\phi^4$ theory evaluate to multiple zeta values, and (ii) when their underlying motives are mixed Tate. More generally, by considering configurations of singular hypersurfaces which fiber linearly over each other, we deduce sufficient geometric and combinatorial criteria on Feynman graphs for both (i) and (ii) to hold. These criteria hold for some infinite classes of graphs which essentially contain all cases previously known to physicists. Calabi-Yau varieties appear at the point where these criteria fail.