We have simulated QCD using 2 þ 1 flavors of domain wall quarks and the Iwasaki gauge action on a ð2:74 fmÞ 3 volume with an inverse lattice scale of a À1 ¼ 1:729ð28Þ GeV. The up and down (light) quarks are degenerate in our calculations and we have used four values for the ratio of light quark masses to the strange (heavy) quark mass in our simulations: 0.217, 0.350, 0.617, and 0.884. We have measured pseudoscalar meson masses and decay constants, the kaon bag parameter B K , and vector meson couplings. We have used SU(2) chiral perturbation theory, which assumes only the up and down quark masses are small, and SU(3) chiral perturbation theory to extrapolate to the physical values for the light quark masses. While next-to-leading order formulas from both approaches fit our data for light quarks, we find the higher-order corrections for SU(3) very large, making such fits unreliable. We also find that SU(3) does not fit our data when the quark masses are near the physical strange quark mass. Thus, we rely on SU(2) chiral perturbation theory for accurate results. We use the masses of the baryon, and the and K mesons to set the lattice scale and determine the quark masses. We then find f ¼ 124:1ð3:6Þ stat  ð6:9Þ syst MeV, f K ¼ 149:6ð3:6Þ stat ð6:3Þ syst MeV, and f K =f ¼ 1:205ð0:018Þ stat ð0:062Þ syst . Using nonperturbative renormalization to relate lattice regularized quark masses to regularization independent momentum scheme masses, and perturbation theory to relate these to MS, we find m MS ud ð2 GeVÞ ¼ 3:72ð0:16Þ stat ð0:33Þ ren ð0:18Þ syst MeV, m MS s ð2 GeVÞ ¼ 107:3ð4:4Þ stat ð9:7Þ ren ð4:9Þ syst MeV, and mud : ms ¼ 1:28:8ð0:4Þ stat ð1:6Þ syst . For the kaon bag parameter, we find B MS K ð2 GeVÞ ¼ 0:524ð0:010Þ stat ð0:013Þ ren  ð0:025Þ syst . Finally, for the ratios of the couplings of the vector mesons to the vector and tensor currents (f V and f T V , respectively) in the MS scheme at 2 GeV we obtain f T =f ¼ 0:687ð27Þ; f T K à =f K à ¼ 0:712ð12Þ, and f T =f ¼ 0:750ð8Þ.

We present physical results obtained from simulations using 2+1 flavors of domain wall quarks and the Iwasaki gauge action at two values of the lattice spacing a, (a −1 = 1.73 (3) GeV and a −1 = 2.28 (3) GeV). On the coarser lattice, with 24 3 × 64 × 16 points (where the 16 corresponds to L s , the extent of the 5 th dimension inherent in the domain wall fermion (DWF) formulation

Hadronic matrix elements of operators relevant to nucleon decay in grand unified theories are calculated numerically using lattice QCD. In this context, the domain-wall fermion formulation, combined with non-perturbative renormalization, is used for the first time. These techniques bring reduction of a large fraction of the systematic error from the finite lattice spacing. Our main effort is devoted to a calculation performed in the quenched approximation, where the direct calculation of the nucleon to pseudoscalar matrix elements, as well as the indirect estimate of them from the nucleon to vacuum matrix elements, are performed. First results, using two flavors of dynamical domain-wall quarks for the nucleon to vacuum matrix elements are also presented to address the systematic error of quenching, which appears to be small compared to the other errors. Our results suggest that the representative value for the low energy constants from the nucleon to vacuum matrix elements are given as |α| ≃ |β| ≃ 0.01 GeV 3 . For a more reliable estimate of the physical low energy matrix elements, it is better to use the relevant form factors calculated in the direct method. The direct method tends to give smaller value of the form factors, compared to the indirect one, thus enhancing the proton life-time; indeed for the π 0 final state the difference between the two methods is quite appreciable.

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