Properties of $c_2$ invariants of Feynman graphs

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.