Determination of light quark masses from the electromagnetic splitting of pseudoscalar meson masses computed with two flavors of domain wall fermions

We determine the light quark masses from lattice QCD simulations incorporating the electromagnetic interaction of valence quarks, using the splittings of charged and neutral pseudoscalar meson masses as inputs. The meson masses are calculated on lattice QCD configurations generated by the RBC Collaboration for two flavors of dynamical domain-wall fermions, which are combined with QED configurations generated via quenched noncompact lattice QED. The electromagnetic part of the pion mass splitting is found to be $m_{{\pi }^{+}}-m_{{\pi }^0}=4.12(21)\mathrm{MeV}$, where only the statistical error is quoted, and similarly for the kaon, 1.443(55) MeV. Our results for the light quark masses are $m_u^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=3.02(27)(19)\mathrm{MeV}$, $m_d^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=5.49(20)(34)\mathrm{MeV}$, and $m_s^{\overline{\mathrm{MS}}}(2\mathrm{GeV})=119.5(56)(74)\mathrm{MeV}$, where the first error is statistical and the second reflects the uncertainty in our nonperturbative renormalization procedure. By averaging over $\pm e$ to cancel $\mathcal{O}(e)$ noise exactly on each combined gauge field configuration, we are able to work at physical $\alpha =1/137$ and obtain very small statistical errors. In our calculation, several sources of systematic error remain, including finite volume, nonzero lattice spacing, chiral extrapolation, quenched QED, and quenched strange quark, which may be more significant than the errors quoted above. We discuss these systematic errors and how to reduce or eliminate them.