The Galois coaction on $\ddot{I}{•}^4$ periods

Abstract

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

extensive data sets for phi^4 periods up to 11 loops (complete up to 7 loops) and non-phi^4 periods up to 8 loops are attached as ancillary files; v2: several clarifications and examples added