FEYNMAN DIAGRAMS AS A WEIGHT SYSTEM - 4-LOOP TEST OF A 4-TERM RELATION

At four loops there first occurs a test of the four-term relation deri ved by the second author in the course of investigating whether counte rterms from subdivergence-free diagrams form a weight system. This tes t relates counterterms in a four-dimensional field theory with Yukawa and phi(4) interactions, where no such relation was previously suspect ed. Using integration by parts, we reduce each counterterm to massless two-loop two-point integrals. The four-term relation is verified, wit h (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 th us 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 renormalizabl e interactions are found to be necessary for the existence of a pure f our-term relation. Strong indications of richer structure are given at five loops. (C) 1998 Published by Elsevier Science B.V. All rights re served.