Communications in Number Theory and Physics

Volume 11 (2017)

Number 3

The Galois coaction on $\phi^4$ periods

Pages: 657 – 705



Erik Panzer (All Souls College, University of Oxford, United Kingdom)

Oliver Schnetz (Department Mathematik, Emmy-Noether-Zentrum, FAU Erlangen-Nürnberg, Erlangen, Germany)


We report on calculations of Feynman periods of primitive log-divergent $\phi^4$ graphs up to eleven loops. The structure of $\phi^4$ periods is described by a series of conjectures. In particular, we discuss the possibility that $\phi^4$ periods are a comodule under the Galois coaction. Finally, we compare the results with the periods of primitive log-divergent non-$\phi^4$ graphs up to eight loops and find remarkable differences to $\phi^4$ periods. Explicit results for all periods we could compute are provided in ancillary files.

Received 18 March 2016

Accepted 23 August 2016

Published 2 October 2017