Heavy-quark renormalization parameters in nonrelativistic QCD

We calculate the self-energy of the heavy quark to $O\left({\alpha }_s\right)$ in nonrelativistic QCD (NRQCD) on the lattice. We use the lowest-order (in $\frac{1}{M}$) NRQCD action which includes the time derivative and kinetic energy terms, but we ignore spin effects. Results are given for the wave-function renormalization, mass renormalization, and energy shift. Our results agree with the lowest-order static effective field theory as $\mathrm{Ma}\to \infty$, where $\mathrm{Ma}$ is the quark mass in lattice units. At values of $\mathrm{Ma}$ appropriate to $b$- and $c$-quark calculations, the differences are significant. We also consider $\frac{1}{\mathrm{Ma}}$ corrections to the static theory arising from the kinetic term.