# Hilbert Series and Operator Basis for NRQED and NRQCD/HQET

## TL;DR

This paper employs a Hilbert series approach to systematically construct operator bases in nonrelativistic effective theories like NRQED and NRQCD/HQET, providing new insights and discrepancies at higher dimensions.

## Contribution

It introduces a Hilbert series method to derive operator bases in NRQED and NRQCD/HQET, extending previous work and identifying discrepancies at higher mass dimensions.

## Key findings

- Operator bases constructed up to dimension 8.
- Results agree with existing literature for NRQED at these dimensions.
- Discrepancies found in NRQCD/HQET at dimensions 7 and 8.

## Abstract

We use a Hilbert series to construct an operator basis in the $1/m$ expansion of a theory with a nonrelativistic heavy fermion in an electromagnetic (NRQED) or color gauge field (NRQCD/HQET). We present a list of effective operators with mass dimension $d\leq 8$. Comparing to the current literature, our results for NRQED agree for $d\leq 8$, but there are some discrepancies in NRQCD/HQET at $d=7$ and 8.