Position Space Feynman quadrics and their motives

TL;DR

This paper introduces position space Feynman quadrics related to quantum field theory divergences and proves they are mixed Tate motives for complete graphs, contrasting with the non-mixed Tate nature of graph hypersurfaces.

Contribution

It establishes that position space Feynman quadrics are mixed Tate motives for complete graphs, providing a new perspective on the motivic nature of quantum field theory divergences.

Findings

Feynman quadrics are in the category of mixed Tate motives.

Complete graphs yield mixed Tate Feynman quadrics.

Contrasts with non-mixed Tate graph hypersurfaces.

Abstract

In this note, we introduce and study position space Feynman quadrics that are the loci of divergences of the position space Feynman integrals for Euclidean massless scalar quantum field theories. We prove that the Feynman quadrics define objects in the category of mixed Tate motives for complete graphs with a bound on the number of vertices. This result shows a strong contrast with the graph hypersurfaces approach which produces also non-mixed Tate examples.