Physical results from<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:math>flavor domain wall QCD and SU(2) chiral perturbation theory

We have simulated QCD using $2+1$ flavors of domain wall quarks and the Iwasaki gauge action on a $(2.74\text{ }\text{ }\mathrm{fm}{)}^{3}$ volume with an inverse lattice scale of ${a}^{\ensuremath{-}1}=1.729(28)\text{ }\text{ }\mathrm{GeV}$. 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 $\ensuremath{\Omega}$ baryon, and the $\ensuremath{\pi}$ and $K$ mesons to set the lattice scale and determine the quark masses. We then find ${f}_{\ensuremath{\pi}}=124.1(3.6{)}_{\mathrm{stat}}(6.9{)}_{\mathrm{syst}}\text{ }\text{ }\mathrm{MeV}$, ${f}_{K}=149.6(3.6{)}_{\mathrm{stat}}(6.3{)}_{\mathrm{syst}}\text{ }\text{ }\mathrm{MeV}$, and ${f}_{K}/{f}_{\ensuremath{\pi}}=1.205(0.018{)}_{\mathrm{stat}}(0.062{)}_{\mathrm{syst}}$. Using nonperturbative renormalization to relate lattice regularized quark masses to regularization independent momentum scheme masses, and perturbation theory to relate these to $\overline{\mathrm{MS}}$, we find ${m}_{ud}^{\overline{\mathrm{MS}}}(2\text{ }\text{ }\mathrm{GeV})=3.72(0.16{)}_{\mathrm{stat}}(0.33{)}_{\mathrm{ren}}(0.18{)}_{\mathrm{syst}}\text{ }\text{ }\mathrm{MeV}$, ${m}_{s}^{\overline{\mathrm{MS}}}(2\text{ }\text{ }\mathrm{GeV})=107.3(4.4{)}_{\mathrm{stat}}(9.7{)}_{\mathrm{ren}}(4.9{)}_{\mathrm{syst}}\text{ }\text{ }\mathrm{MeV}$, and ${\stackrel{\texttildelow{}}{m}}_{ud}\ensuremath{\mathbin:}{\stackrel{\texttildelow{}}{m}}_{s}=1\ensuremath{\mathbin:}28.8(0.4{)}_{\mathrm{stat}}(1.6{)}_{\mathrm{syst}}$. For the kaon bag parameter, we find ${B}_{K}^{\overline{\mathrm{MS}}}(2\text{ }\text{ }\mathrm{GeV})=0.524(0.010{)}_{\mathrm{stat}}(0.013{)}_{\mathrm{ren}}(0.025{)}_{\mathrm{syst}}$. Finally, for the ratios of the couplings of the vector mesons to the vector and tensor currents (${f}_{V}$ and ${f}_{V}^{T}$, respectively) in the $\overline{\mathrm{MS}}$ scheme at 2 GeV we obtain ${f}_{\ensuremath{\rho}}^{T}/{f}_{\ensuremath{\rho}}=0.687(27)$; ${f}_{{K}^{*}}^{T}/{f}_{{K}^{*}}=0.712(12)$, and ${f}_{\ensuremath{\phi}}^{T}/{f}_{\ensuremath{\phi}}=0.750(8)$.