Light quark masses from lattice QCD

We present estimates of the masses of light quarks using lattice data. Our main results are based on a global analysis of all the published data for Wilson, Sheikholeslami-Wohlert (clover), and staggered fermions, both in the quenched approximation and with $n_f=2\mathrm{}$ dynamical flavors. We find that the values of masses with the various formulations agree after extrapolation to the continuum limit for the $n_f=0\mathrm{}$ theory. Our best estimates, in the $\overline{\mathrm{MS}}$ scheme at $\mu =2\mathrm{}\mathrm{GeV},$ are $\overline{m}=3.4\mathrm{}\pm 0.4\pm 0.3\mathrm{MeV}$ and $m_s=100\mathrm{}\pm 21\pm 10\mathrm{MeV}$ in the quenched approximation. The $n_f=2\mathrm{}$ results, $\overline{m}=2.7\mathrm{}\pm 0.3\pm 0.3\mathrm{MeV}$ and $m_s=68\mathrm{}\pm 12\pm 7\mathrm{MeV},$ are preliminary. (A linear extrapolation in $n_f$ would further reduce these estimates for the physical case of three dynamical flavors.) These estimates are smaller than phenomenological estimates based on sum rules, but maintain the ratios predicted by chiral perturbation theory. The new results have a significant impact on the extraction of ${\epsilon }^{\prime }/\epsilon$ from the standard model. Using the same lattice data we estimate the quark condensate using the Gell-Mann–Oakes–Renner relation. Again the three formulations give consistent results after extrapolation to $a=0,$ and the value turns out to be correspondingly larger, roughly preserving $m_s〈\overline{\psi }\psi 〉.$