Recursive computation of Feynman periods

TL;DR

This paper introduces recursive transformation rules for computing Feynman periods, simplifying calculations in quantum field theory and providing extensive results up to seven loops, supporting conjectures about their algebraic structure.

Contribution

It presents a novel recursive method for computing Feynman periods using transformation rules, enabling calculations in arbitrary even dimensions and extensive results in $$ theory.

Findings

Computed all subdivergence-free Feynman periods up to six loops.

Calculated 561 out of 607 Feynman periods at seven loops.

Results support the conjectured coaction structure in quantum field theory.

Abstract

Feynman periods are Feynman integrals that do not depend on external kinematics. Their computation, which is necessary for many applications of quantum field theory, is greatly facilitated by graphical functions or the equivalent conformal four-point integrals. We describe a set of transformation rules that act on such functions and allow their recursive computation in arbitrary even dimensions. As a concrete example we compute all subdivergence-free Feynman periods in $\phi^3$ theory up to six loops and 561 of 607 Feynman periods at seven loops. Our results support the conjectured existence of a coaction structure in quantum field theory and suggest that $\phi^3$ and $\phi^4$ theory share the same number content.