Quarkonium spin structure in lattice NRQCD

Numerical simulations of the quarkonium spin splittings are done in the framework of lattice nonrelativistic quantum chromodynamics. At leading order in the velocity expansion the spin splittings are of ${O(M}_Q{v}^4)$, where $M_Q$ is the renormalized quark mass and $v^2$ is the mean squared quark velocity ${(v}_{\psi }^2\approx 0.3$ and $v_{{\rm Y}}^2\approx 0.1)$. A systematic analysis is done of all next-to-leading order corrections. This includes the addition of ${O(M}_Q{v}^6)$ relativistic interactions, and the removal of ${O(a}^2{M}_Q{v}^4)$ discretization errors in the leading-order interactions. Simulations are done for both $S$- and $P$-wave mesons, with a variety of heavy quark actions and over a wide range of lattice spacings. Two prescriptions for the tadpole improvement of the action are also studied in detail: one using the measured value of the average plaquette, the other using the mean link measured in the Landau gauge. Next-to-leading order interactions result in a very large reduction in the charmonium splittings, down by about 60% from their values at leading order. There are further indications that the velocity expansion may be poorly convergent for charmonium. Preliminary results show a small correction to the hyperfine splitting in the ${\rm Y}$ system.