On the periods of some Feynman integrals

TL;DR

This work develops an intricate bridge between Feynman integral periods in massless $\phi^4$ theory and the geometry of graph hypersurfaces. By introducing and exploiting the framework of linearly reducible hypersurfaces, Dodgson polynomial identities, and Landau varieties, the authors establish conditions under which graph motives are mixed Tate and residues reduce to multiple zeta values. They show that graphs with vertex width at most 3 are of matrix type and hence linearly reducible, enabling a constructive route to compute periods via moduli spaces $\mathfrak{M}_{0,n}$ and the bar construction, with outcomes in the ring of multiple zetas. The approach also identifies obstructions (e.g., the five-invariant) that yield non-Tate phenomena and connections to Calabi–Yau varieties, illuminating the boundary between Tate and non-Tate behavior in Feynman integral physics. Overall, the paper provides a comprehensive geometric-combinatorial program for understanding when Feynman amplitudes lie in the realm of multiple zeta values and mixed Tate motives, with concrete algorithms for a broad family of graphs.

Abstract

We study the related questions: (i) when Feynman amplitudes in massless $φ^4$ 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.

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Theorems & Definitions (196)

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