Reparameterization Invariance Revisited

Reparameterization invariance, a symmetry of heavy quark effective theory, appears in different forms in the literature. The most commonly cited forms of the reparameterization transformation are shown to induce the same constraints on operators that do not vanish under the equation of motion to order $1/m^2$, and to be related by a redefinition of the heavy quark field. We give a new, very straightforward proof that that the reparameterization invariance constraints apply to all orders in $\alpha_s$ under matching to full QCD and renormalization-group running, at least up to and including $O(1/m^2)$.