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

TL;DR

The paper demonstrates how the Hilbert series, implemented via the plethystic exponential and Haar projection, yields a compact generating function $H$ that counts independent, gauge-invariant operators in Lagrangians. It provides concrete SMEFT examples, including dim-6 baryon-number-violating operators and the dim-7 $LLL\overline{e}H$ sector, confirming operator counts (e.g., 57 independent structures in certain classes) and exposing flavor-relations via Fierz identities. Extending to BSM, it shows counting invariants in an extended Higgs sector with a scalar quadruplet yields precise dim-4 and dim-6 operator counts and can reveal missing structures. The discussion of derivatives and EOM outlines a path to include dynamics in the Hilbert series framework, highlighting both current capabilities and future work toward a complete SMEFT Hilbert series. Overall, the method offers a fast, automatable cross-check and counting tool for constructing complete, symmetry-respecting Lagrangians in complex operator spaces.

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.