Classification of four-point local gluon S-matrices

TL;DR

This paper classifies four-point local gluon S-matrices across arbitrary dimensions and gauge groups, providing explicit constructions and Lagrangians, extending previous classifications of photons and gravitons.

Contribution

It introduces a systematic method to classify and construct four-point gluon S-matrices for SO(N) and SU(N) gauge groups in any dimension, including explicit Lagrangians.

Findings

Explicit classification of gluon S-matrices for SO(N) and SU(N)

Construction of local Lagrangians for D≥9

Representation of Yang-Mills four-gluon S-matrix in the basis

Abstract

In this paper, we classify four-point local gluon S-matrices in arbitrary dimensions. This is along the same lines as \cite{Chowdhury:2019kaq} where four-point local photon S-matrices and graviton S-matrices were classified. We do the classification explicitly for gauge groups $SO(N)$ and $SU(N)$ for all $N$ but our method is easily generalizable to other Lie groups. The construction involves combining not-necessarily-permutation-symmetric four-point S-matrices of photons and those of adjoint scalars into permutation symmetric four-point gluon S-matrix. We explicitly list both the components of the construction, i.e permutation symmetric as well as non-symmetric four point S-matrices, for both the photons as well as the adjoint scalars for arbitrary dimensions and for gauge groups $SO(N)$ and $SU(N)$ for all $N$. In this paper, we explicitly list the local Lagrangians that generate the local gluon S-matrices for $D\geq 9$ and present the relevant counting for lower dimensions. Local Lagrangians for gluon S-matrices in lower dimensions can be written down following the same method. We also express the Yang-Mills four gluon S-matrix with gluon exchange in terms of our basis structures.