Chord diagrams and BPHZ subtractions

TL;DR

This paper explores the connection between knot theory and renormalization in quantum field theory by translating BPHZ subtraction combinatorics into chord diagram relations, confirming a fundamental link.

Contribution

It introduces a novel combinatorial approach using chord diagrams to represent BPHZ subtractions in ladder graphs within  theory, linking knot theory to renormalization.

Findings

Chord diagrams encode BPHZ subtraction combinatorics.

Resolution of singular crossings confirms the knot-renormalization relationship.

Provides a new perspective on perturbative quantum field theory.

Abstract

The combinatorics of the BPHZ subtraction scheme for a class of ladder graphs for the three point vertex in $\phi^3$ theory is transcribed into certain connectivity relations for marked chord diagrams (knots with transversal intersections). The resolution of the singular crossings using the equivalence relations in these examples provides confirmation of a proposed fundamental relationship between knot theory and renormalization in perturbative quantum field theory.