Properties of c_2 invariants of Feynman graphs

TL;DR

This paper explores the properties of the c_2 invariant in Feynman graphs, establishing its equivalence in different spaces, its vanishing under subdivergences, and its relation to knot theory identities.

Contribution

It defines the c_2 invariant in momentum space, proves its equivalence with parametric space, and investigates its behavior and relations in Feynman graphs.

Findings

c_2 invariant equals in momentum and parametric space for certain graphs

c_2 invariant vanishes with subdivergences

relation to knot theory identities like the four-term relation

Abstract

The c_2 invariant of a Feynman graph is an arithmetic invariant which detects many properties of the corresponding Feynman integral. In this paper, we define the c_2 invariant in momentum space and prove that it equals the c_2 invariant in parametric space for overall log-divergent graphs. Then we show that the c_2 invariant of a graph vanishes whenever it contains subdivergences. Finally, we investigate how the c_2 invariant relates to identities such as the four-term relation in knot theory.