Hilbert Series for Constructing Lagrangians: expanding the phenomenologist's toolbox

TL;DR

This paper introduces the Hilbert series technique as an efficient and reliable method for constructing and counting group-invariant Lagrangian operators, especially in complex scenarios where traditional methods are challenging.

Contribution

It extends the application of Hilbert series to a broader audience for constructing Lagrangians, providing a quick counting tool and cross-check for invariants without derivatives.

Findings

Hilbert series efficiently counts operators of given mass dimension.

The method serves as a cross check for traditional group theoretical techniques.

Practical examples demonstrate the technique's effectiveness in complex cases.

Abstract

This note presents the Hilbert series technique to a wider audience in the context of constructing group-invariant Lagrangians. This technique provides a fast way to calculate the number of operators of a specified mass dimension for a given field content, and is a useful cross check on more well-known group theoretical methods. In addition, at least when restricted to invariants without derivatives, the Hilbert series technique supplies a robust way of counting invariants in scenarios which, due to the large number of fields involved or to high dimensional group representations, are intractable by traditional methods. We work out several practical examples.