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 \msbar 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 of a highly-efficient formalism (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 $m_s\mathrm{/}8$, and two values of the lattice spacing, with chiral extrapolation/interpolation to the physical masses. With the new perturbation theory presented here, the resulting \msbar
masses are $m_s^{\text{msbar}}(2GeV)=87(0)(4)(4)(0)$ MeV, and $\hat m^\msbar(2 {GeV}) = 3.2(0)(2)(2)(0)$MeV,where$\hat m = \sfrac12 (m_u + m_d)$ 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 two smaller than in an earlier study using only one-loop perturbation theory. Using a recent determination of the ratio $m_u\mathrm{/}{m}_d=0.43(0)(1)(0)(8)$ due to the MILC collaboration, these results also imply $m^\msbar_u(2 {GeV}) = 1.9(0)(1)(1)(2)$MeVand$m^\msbar_d(2 {GeV}) = 4.4(0)(2)(2)(2)$ 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.