Nonperturbative evaluation of the physical classical velocity in the lattice heavy quark effective theory

In the lattice formulation of heavy quark effective theory, the value of the “classical velocity” $v,$ as defined through the separation of the four-momentum of a heavy quark into a part proportional to the heavy quark mass and a residual part that remains finite in the heavy quark limit $\mathrm{}(P=Mv+p),$ is different from its value as it appears in the bare heavy quark propagator $\left[S^{-1}\mathrm{}\right(p)=v\cdot p].$ The origin of the difference, which is effectively a lattice-induced renormalization, is the reduction of Lorentz [or O(4)] invariance to (hyper)cubic invariance. The renormalization is finite and depends specifically on the form of the discretization of the reduced heavy quark Dirac equation. For the forward time, centered space discretization, we compute this renormalization nonperturbatively, using an ensemble of lattices at $\beta =6.1\mathrm{}$ provided by the Fermilab ACP-MAPS Collaboration. The calculation makes crucial use of a variationally optimized smeared operator for creating composite heavy-light mesons. It has the property that its propagator achieves an asymptotic plateau in just a few Euclidean time steps. For comparison, we also compute the shift perturbatively, to one loop in lattice perturbation theory. The nonperturbative calculation of the leading multiplicative shift in the classical velocity is considerably different from the one-loop estimate and indicates that for the above parameters $\overrightarrow{v}$ is reduced by about 10–13 %.