Counting operators in Effective Field Theories

TL;DR

This paper develops a systematic Hilbert series method for counting independent operators in Effective Field Theories, including detailed group theory background and applications to scalar and electromagnetic fields.

Contribution

It introduces a comprehensive Hilbert series framework with detailed mathematical foundations for operator counting in EFTs, extending previous results.

Findings

Successfully applies the formalism to scalar and electromagnetic fields

Reproduces known operator counts and extends them

Provides a detailed, accessible introduction to the mathematical tools

Abstract

We systematically develop the Hilbert series technique for counting independent operators in Effective Field Theories. In the hope of providing more approachable entry point to the subject we include a detailed introduction of all necessary group theoretic tools (in a rather mathematical definition-theorem-proof style). Finally, we apply the formalism in the case of a single scalar field and also electromagnetic field, partly reproducing and partly extending known results.