Counting parity-violating local S-matrices

TL;DR

This paper enumerates parity-violating four-point local S-matrices for photons, gravitons, and gluons using plethystic techniques, revealing dimension-dependent contributions and extending previous counting methods.

Contribution

It generalizes the enumeration of local four-point S-matrices to include parity-violating sectors and evaluates these in various dimensions, providing new insights into their structure.

Findings

Parity-violating contributions are explicitly enumerated in 4 and 6 dimensions.

Parity violation does not contribute to four-particle scattering in dimensions ≥8.

Results align with previous conjectures on S-matrix transformation properties.

Abstract

Four point tree-level local S-matrices form a module over ring of polynomials of mandelstam invariants s, t and u. The module of local analytic S-matrices can be encoded in terms of a partition function which is enumerated using plethystic techniques. In this paper, we enumerate the plethystic contribution to local four point photon, graviton and gluon multi-particle partition functions that encode parity violating $2 \rightarrow 2$ scattering. We generalise the counting problem solved in \cite{ Chowdhury:2019kaq, Chowdhury:2020ddc} to project out parity violating sectors, a subtle task in even dimensions \cite{Henning:2017fpj}. We explicitly enumerate the parity odd contributions to the multi-letter partition function for gauge fields, gravitons and gluons and evaluate the resulting parity violating partition functions in $D=4,6$. We also perform a large $D$ analysis to show that parity violating local interactions do not contribute to four particle scattering in higher dimensions ($D\geq 8$). Our computations and observations for photons, gravitons and gluons agree with the transformation properties of these S-matrices previously conjectured in \cite{Chowdhury:2019kaq, Chowdhury:2020ddc}