Four-point functions and the permutation group S4

TL;DR

This paper introduces a new notation for handling the permutation group S4 to organize tensor structures in four-point functions, aiding calculations in particle physics such as light-by-light scattering.

Contribution

It presents an efficient S4 multiplet notation and demonstrates its usefulness in analyzing four-point functions with gauge bosons, improving the organization of tensor bases.

Findings

Efficient S4 multiplet notation for four-point functions

Application to four-gluon vertex and light-by-light scattering

Enhanced analysis of kinematic regions and singularities

Abstract

Four-point functions are at the heart of many interesting physical processes. A prime example is the light-by-light scattering amplitude, which plays an important role in the calculation of hadronic contributions to the anomalous magnetic moment of the muon. In the calculation of such quantities one faces the challenge of finding a suitable and well-behaved basis of tensor structures in coordinate and/or momentum space. Provided all (or many) of the external legs represent similar particle content, a powerful tool to construct and organize such bases is the permutation group S4. We introduce an efficient notation for dealing with the irreducible multiplets of S4, and we highlight the merits of this treatment by exemplifying four-point functions with gauge-boson legs such as the four-gluon vertex and the light-by-light scattering amplitude. The multiplet analysis is also useful for isolating the important kinematic regions and the dynamical singularity content of such amplitudes. Our analysis serves as a basis for future efficient calculations of these and similar objects.