Feynman diagrams as a weight system: four-loop test of a four-term relation

TL;DR

This paper tests a four-term relation in four-loop Feynman diagrams within a four-dimensional field theory, confirming a non-trivial knot-related cancellation and suggesting deeper algebraic structures at higher loops.

Contribution

It provides the first four-loop verification of the four-term relation in this context, linking counterterms to knot theory and revealing potential for richer structures at five loops.

Findings

Four-loop test confirms the four-term relation with specific transcendental number cancellations.

Supports the connection between Feynman diagrams, knot theory, and algebraic relations.

Indicates more complex structures may emerge at five loops.

Abstract

At four loops there first occurs a test of the four-term relation derived by the second author in the course of investigating whether counterterms from subdivergence-free diagrams form a weight system. This test relates counterterms in a four-dimensional field theory with Yukawa and $\phi^4$ interactions, where no such relation was previously suspected. Using integration by parts, we reduce each counterterm to massless two-loop two-point integrals. The four-term relation is verified, with $<G_1-G_2+G_3-G_4> = 0 - 3\zeta_3 + 6\zeta_3 - 3\zeta_3 = 0$, demonstrating non-trivial cancellation of the trefoil knot and thus supporting the emerging connection between knots and counterterms, via transcendental numbers assigned by four-dimensional field theories to chord diagrams. Restrictions to scalar couplings and renormalizable interactions are found to be necessary for the existence of a pure four-term relation. Strong indications of richer structure are given at five loops.