Advances in Theoretical and Mathematical Physics

Volume 18 (2014)

Number 2

Properties of $c_2$ invariants of Feynman graphs

Pages: 323 – 362



Francis Brown (Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France)

Oliver Schnetz (Department Mathematik, Universität Erlangen-Nürnberg, Germany)

Karen Yeats (Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada)


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.

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