Weight Systems from Feynman Diagrams

Abstract

We find that the overall UV divergences of a renormalizable field theory with trivalent vertices fulfil a four-term relation. They thus come close to establish a weight system. This provides a first explanation of the recent successful association of renormalization theory with knot theory.

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

“…In Section 2 we prosecute a successful test in a combined Yukawa and φ 4 theory, at four loops. Sections 3 and 4 confirm the expectations [1] that a pure four-term relation is vitiated by vector couplings, and by non-renormalizable interactions. Section 4 also considers a specific three-term relation, derived in [1].…”

“…Finally, we remark on a specific three-term relation, derived in [1]. It is possible that such relations, called STU relations in the theory of chord diagrams [2], impose even stronger constraints upon the structure of field-theory counterterms.…”

On knots in subdivergent diagrams

“…In recent results, [9,10], results are obtained clarifying the role of the four-term relation in counterterms free of subdivergences. These results still suffer from an exclusion of graphs with subdivergences.…”

On the Hopf algebra structure of perturbative quantum field theories

“…• It is always interesting to understand the primitive elements in a Hopf algebra. Clearly, from the results in [19], there are relations between Feynman diagrams without subdivergences. This should be a major focus of future work.…”

“…With this decomposition, we have completely reduced the calculation of Z-factors (and quantities derived from them, like anomalous dimensions and β-functions) to the calculation of primitive elements. What remains is to classify the latter, which is a non-trivial task in view of the relations amongs them [19].…”