We calculate the masses of bottom mesons using an improved relativistic action for the $b$-quarks and the RBC/UKQCD Iwasaki gauge configurations with $2+1$ flavors of dynamical domain-wall light quarks. We analyze configurations with two lattice spacings: $a^{-1}=1.729\mathrm{GeV}$ ($a\approx 0.11\mathrm{fm}$) and $a^{-1}=2.281\mathrm{GeV}$ ($a\approx 0.086\mathrm{fm}$). We use an anisotropic, clover-improved Wilson action for the $b$-quark, and tune the three parameters of the action nonperturbatively such that they reproduce the experimental values of the $B_s$ and $B_s^{*}$ heavy-light meson states. The masses and mass splittings of the low-lying bottomonium states (such as the ${\eta }_b$ and ${\rm Y}$) can then be computed with no additional inputs, and comparison between these predictions and experiment provides a test of the validity of our method. We obtain bottomonium masses with total uncertainties of $\sim 0.5–0.6\%$ and fine-structure splittings with uncertainties of $\sim 45–55\%$: for all cases we find good agreement with experiment. The parameters of the relativistic heavy-quark action tuned for $b$-quarks presented in this work can be used for precise calculations of weak matrix elements such as $B$-meson decay constants and mixing parameters with lattice discretization errors that are of the same size as in light pseudoscalar meson quantities. This general method can also be used for charmed meson masses and matrix elements if the parameters of the heavy-quark action are appropriately tuned.

• Received 21 June 2012

DOI:https://doi.org/10.1103/PhysRevD.86.116003