Tadpole renormalization and relativistic corrections in lattice NRQCD

We make a detailed comparison of two tadpole renormalization schemes in the context of the quarkonium hyperfine splittings in lattice NRQCD. We renormalize improved gauge-field and NRQCD actions using the mean-link $u_{0,L}$ in the Landau gauge, and using the fourth root of the average plaquette $u_{0,P}.$ Simulations are done for the three quarkonium systems $c\overline{c},$ $b\overline{c},$ and $b\overline{b}.$ The hyperfine splittings are computed both at leading $\left[{O(M}_Q{v}^4\right)]$ and at next-to-leading $\left[{O(M}_Q{v}^6\right)]$ order in the relativistic expansion, where $M_Q$ is the renormalized quark mass, and $v^2$ is the mean-squared velocity. Results are obtained at a large number of lattice spacings, in the range of about 0.14–0.38 fm. A number of features emerge, all of which favor tadpole renormalization using $u_{0,L}.$ This includes a much better scaling behavior of the hyperfine splittings in the three quarkonium systems when $u_{0,L}$ is used. We also find that relativistic corrections to the spin splittings are smaller when $u_{0,L}$ is used, particularly for the $c\overline{c}$ and $b\overline{c}$ systems. We also see signs of a breakdown in the NRQCD expansion when the bare quark mass falls below about 1 in lattice units. Simulations with $u_{0,L}$ also appear to be better behaved in this context: the bare quark masses turn out to be larger when $u_{0,L}$ is used, compared to when $u_{0,P}$ is used on lattices with comparable spacings. These results also demonstrate the need to go beyond tree-level tadpole improvement for precision simulations.