# Standard Model Plethystics

## TL;DR

This paper analyzes the vacuum geometry of the MSSM using the Plethystic Programme, computing the Hilbert series to understand the structure of gauge invariant operators.

## Contribution

It introduces a computational approach to derive the Hilbert series of MSSM invariants, including a fully refined series as a sum of rational functions.

## Key findings

- Explicit Hilbert series with 1422 rational functions
- Identification of weights for unrefining the series
- Determination of vacuum moduli space dimension and degree

## Abstract

We study the vacuum geometry prescribed by the gauge invariant operators of the MSSM via the Plethystic Programme. This is achieved by using several tricks to perform the highly computationally challenging Molien-Weyl integral, from which we extract the Hilbert series, encoding the invariants of the geometry at all degrees. The fully refined Hilbert series is presented as the explicit sum of 1422 rational functions. We found a good choice of weights to unrefine the Hilbert series into a rational function of a single variable, from which we can read off the dimension and the degree of the vacuum moduli space of the MSSM gauge invariants. All data in Mathematica format are also presented.