Reparameterization Invariant Operator Basis for NRQED and HQET

We discuss how reparameterization invariance connects a rotationally-invariant heavy particle effective theory with a single fermion to a Lorentz-invariant theory. Using Hilbert-series methods, a Lorentz-invariant operator basis is tabulated, up to and including operators of order $1\mathrm{/}{M}^4$, when the fermion couples to an external $U(1)$ or $SU(3)$ gauge interaction.