# Further investigations into the graph theory of $ϕ_{4}$-periods and the $c_{2}$ invariant

### Simone Hu

University of Waterloo, Canada### Oliver Schnetz

Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany### Jim Shaw

University of British Columbia, Vancouver, Canada### Karen Yeats

University of Waterloo, Canada

## Abstract

A Feynman period is a particular residue of a scalar Feynman integral which is both physically and number theoretically interesting. Two ways in which the graph theory of the underlying Feynman graph can illuminate the Feynman period are via graph operations which are period invariant and other graph quantities which predict aspects of the Feynman period, one notable example is known as the $c_{2}$ invariant. We give results and computations in both these directions, proving a new period identity and computing its consequences up to $11$ loops in $ϕ_{4}$-theory, proving a $c_{2}$ invariant identity, and giving the results of a computational investigation of $c_{2}$ invariants at $11$ loops.

## Cite this article

Simone Hu, Oliver Schnetz, Jim Shaw, Karen Yeats, Further investigations into the graph theory of $ϕ_{4}$-periods and the $c_{2}$ invariant. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 9 (2022), no. 3, pp. 473–524

DOI 10.4171/AIHPD/123