Feynman motives and deletion-contraction relations

TL;DR

This paper establishes a deletion-contraction formula for motivic Feynman rules, providing explicit recursions and connecting them to Tutte polynomial relations, with implications for the category of mixed motives.

Contribution

It introduces a deletion-contraction relation for motivic Feynman rules and links it to Tutte polynomial recursions, extending to the category of mixed motives.

Findings

Derived explicit recursions for motivic Feynman rules.

Compared motivic rules with Tutte polynomial relations.

Lifted deletion-contraction to the category of mixed motives.

Abstract

We prove a deletion-contraction formula for motivic Feynman rules given by the classes of the affine graph hypersurface complement in the Grothendieck ring of varieties. We derive explicit recursions and generating series for these motivic Feynman rules under the operation of multiplying edges in a graph and we compare it with similar formulae for the Tutte polynomial of graphs, both being specializations of the same universal recursive relation. We obtain similar recursions for graphs that are chains of polygons and for graphs obtained by replacing an edge by a chain of triangles. We show that the deletion-contraction relation can be lifted to the level of the category of mixed motives in the form of a distinguished triangle, similarly to what happens in categorifications of graph invariants.