We give results for the Upsilon spectrum from lattice QCD using an improved version of the nonrelativistic QCD action for $b$ quarks which includes radiative corrections to kinetic terms at $\mathcal{O}(v^4)$ in the velocity expansion. We also include for the first time the effect of up, down, strange and charm quarks in the sea using “second generation” gluon field configurations from the MILC Collaboration. Using the ${\rm Y}$ $2S-1S$ splitting to determine the lattice spacing, we are able to obtain the $1P-1\overline{S}$ splitting to 1.4% and the $3S-1S$ splitting to 2.4%. Our improved result for $M({\rm Y})-M({\eta }_b)$ is 70(9) MeV and we predict $M({{\rm Y}}^{\prime })-M({\eta }_b^{\prime })=35(3)\mathrm{MeV}$. We also calculate $\pi$, $K$ and ${\eta }_s$ correlators using the highly improved staggered quark action and perform a chiral and continuum extrapolation to give values for $M_{{\eta }_s}(0.6893(12)\mathrm{GeV})$ and $f_{{\eta }_s}(0.1819(5)\mathrm{GeV})$ that allow us to tune the strange quark mass as well as providing an independent and consistent determination of the lattice spacing. Combining the nonrelativistic QCD and highly improved staggered quark analyses gives $m_b/m_s=54.7(2.5)$ and a value for the heavy quark potential parameter of $r_1=0.3209(26)\mathrm{fm}$.

• Received 1 November 2011

DOI:https://doi.org/10.1103/PhysRevD.85.054509