High-precision determination of the light-quark masses from realistic lattice QCD

Three-flavor lattice QCD simulations and two-loop perturbation theory are used to make the most precise determination to date of the strange-, up-, and down-quark masses, $m_s$, $m_u$, and $m_d$, respectively. Perturbative matching is required in order to connect the lattice-regularized bare-quark masses to the masses as defined in the $\overline{\mathrm{MS}}$ scheme, and this is done here for the first time at next-to-next-to leading (or two-loop) order. The bare-quark masses required as input come from simulations by the MILC collaboration using so-called staggered quarks, with three flavors of light quarks in the Dirac sea; these simulations were previously analyzed in a joint study by the HPQCD and MILC collaborations, using degenerate $u$ and $d$ quarks, with masses as low as $m_s/8$, and two values of the lattice spacing, with chiral extrapolation/interpolation to the physical masses. With the new perturbation theory presented here, the resulting $\overline{\mathrm{MS}}$ masses are $m_s^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=87(0)(4)(4)(0)\mathrm{MeV}$, and ${\widehat{m}}^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=3.2(0)(2)(2)(0)\mathrm{MeV}$, where $\widehat{m}=\frac{1}{2}(m_u+m_d)$ is the average of the $u$ and $d$ masses. The respective uncertainties are from statistics, simulation systematics, perturbation theory, and electromagnetic/isospin effects. The perturbative errors are about a factor of 2 smaller than in an earlier study using only one-loop perturbation theory. Using a recent determination of the ratio $m_u/m_d=0.43(0)(1)(0)(8)$ due to the MILC collaboration, these results also imply $m_u^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=1.9(0)(1)(1)(2)\mathrm{MeV}$ and $m_d^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=4.4(0)(2)(2)(2)\mathrm{MeV}$. A technique for estimating the next order in the perturbative expansion is also presented, which uses input from simulations at more than one lattice spacing; this method is used here in the estimate of the systematic uncertainties.