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

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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.