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

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 φ4 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 〈G1−G2+G3−G4〉=0−3ζ3+6ζ3−3ζ3=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.