Hilbert Series and Operator Basis for NRQED and NRQCD/HQET

TL;DR

The paper develops a Hilbert-series approach to construct a complete operator basis for a single heavy fermion in NRQED and NRQCD/HQET, organized in a 1/m expansion with operators up to mass dimension d ≤ 8. It uses plethystic exponentials and Molien-Weyl integrals to enumerate invariants under the relevant symmetry groups, systematically removing redundancies from equations of motion and integration by parts. NRQED operators up to d ≤ 8 agree with established results, while NRQCD/HQET shows historical discrepancies at d = 7–8 that are clarified in light of newer analyses and considerations of color contractions; a note highlights updated concordance with recent work for d ≤ 8. The work also discusses automation of parity and time-reversal constraints (the latter automated in the abelian NRQED case but handled manually for NRQCD/HQET) and points toward future reformulations in terms of non-relativistic conformal symmetry and reparameterization invariance.

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.

Figures (1)

• Figure 1: The total number of rotationally-invariant operators in NRQED, which are even under parity and time reversal, as a function of the operator dimension $d$. Explicit form of the operators for $d=5,6,7,8$ can be found in Table \ref{['NRQED']}.