The Galois coaction on $\phi^4$ periods

TL;DR

This paper computes and analyzes Feynman periods of primitive log-divergent $$ graphs up to eleven loops, exploring their structure through conjectures and Galois coaction, and comparing with non-$$ graph periods.

Contribution

It provides the first extensive calculations of $$ periods up to eleven loops and investigates their algebraic structure via Galois coaction conjectures.

Findings

$$ periods exhibit a conjectured Galois coaction structure.

Distinct differences are observed between $$ and non-$$ graph periods.

Explicit period data are provided for future research.

Abstract

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.