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 u0,L in the Landau gauge, and using the fourth root of the average plaquette u0,P. Simulations are done for the three quarkonium systems cc̄, bc̄, and bb̄. The hyperfine splittings are computed both at leading [O(MQv⁴)] and at next-to-leading [O(MQv⁶)] order in the relativistic expansion, where MQ is the renormalized quark mass, and v² 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 u0,L. This includes a much better scaling behavior of the hyperfine splittings in the three quarkonium systems when u0,L is used. We also find that relativistic corrections to the spin splittings are smaller when u0,L is used, particularly for the cc̄ and bc̄ 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 u0,L also appear to be better behaved in this context: the bare quark masses turn out to be larger when u0,L is used, compared to when u0,P is used on lattices with comparable spacings. These results also demonstrate the need to go beyond tree-level tadpole improvement for precision simulations.