Contraction of Dirac matrices via chord diagrams

TL;DR

This paper introduces a novel combinatorial approach using chord diagrams to simplify Dirac matrix contractions, leading to more efficient calculations in quantum electrodynamics (QED) Feynman integrals.

Contribution

It develops a new formula for Dirac matrix contractions using chord diagrams, extending previous algorithms and simplifying the tensor structures in QED calculations.

Findings

Derived a simple, general contraction formula for Dirac matrices

Simplified the Schwinger parametric integrand in QED Feynman integrals

Eliminated internal tensor structures in the integrand

Abstract

Chord diagrams and combinatorics of word algebras are used to model products of Dirac matrices, their traces, and contractions. A simple formula for the result of arbitrary contractions is derived, simplifying and extending an old contraction algorithm due to Kahane. This formula is then used to express the Schwinger parametric integrand of a QED Feynman integral in a much simplified form, with the entire internal tensor structure eliminated. Possible next steps for further simplification, including a specific conjecture, are discussed.