Reparametrization invariance to all orders in heavy quark effective theory

Heavy quark effective theory splits heavy quark momentum into a large fixed momentum and a variable residual momentum $p_{\mu }{=m}_Q{v}_{\mu }{+k}_{\mu }.$ It thereby suffers a redundancy of description corresponding to small changes in the choice of the fixed velocity $v_{\mu }.$ The fact that full QCD is manifestly $v$ independent should lead to a nontrivial constraint on the form of the effective theory, known as reparametrization invariance. For spin-$\frac{1}{2}$ quarks, the precise form of the constraint and its solution at the level of the effective Lagrangian have proven to be rather subtle, and the original proposal by Luke and Manohar has been questioned. In this paper I employ a version of heavy quark effective theory containing the “antiparticle” field as a nonpropagating auxiliary field, which greatly simplifies keeping track of $v$ dependence. This permits a very simple derivation of reparametrization invariance from first principles. The auxiliary field can also be integrated out to return to the standard formulation of the effective theory, but with the effective Lagrangian now satisfying the full reparametrization constraint. I compare this result with earlier proposals.