Recursive computation of Feynman periods

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.