Continuum limit physics 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

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 $a$, [${a}^{\ensuremath{-}1}=1.73(3)\text{ }\text{ }\mathrm{GeV}$ and ${a}^{\ensuremath{-}1}=2.28(3)\text{ }\text{ }\mathrm{GeV}$]. On the coarser lattice, with ${24}^{3}\ifmmode\times\else\texttimes\fi{}64\ifmmode\times\else\texttimes\fi{}16$ points (where the 16 corresponds to ${L}_{s}$, the extent of the 5th dimension inherent in the domain wall fermion formulation of QCD), the analysis of C. Allton et al. (RBC-UKQCD Collaboration), Phys. Rev. D 78 is extended to approximately twice the number of configurations. The ensembles on the finer ${32}^{3}\ifmmode\times\else\texttimes\fi{}64\ifmmode\times\else\texttimes\fi{}16$ lattice are new. We explain in detail how we use lattice data obtained at several values of the lattice spacing and for a range of quark masses in combined continuum-chiral fits in order to obtain results in the continuum limit and at physical quark masses. We implement this procedure for our data at two lattice spacings and with unitary pion masses in the approximate range 290--420 MeV (225--420 MeV for partially quenched pions). We use the masses of the $\ensuremath{\pi}$ and $K$ mesons and the $\ensuremath{\Omega}$ baryon to determine the physical quark masses and the values of the lattice spacing. While our data in the mass ranges above are consistent with the predictions of next-to-leading order SU(2) chiral perturbation theory, they are also consistent with a simple analytic ansatz leading to an inherent uncertainty in how best to perform the chiral extrapolation that we are reluctant to reduce with model-dependent assumptions about higher order corrections. In some cases, particularly for ${f}_{\ensuremath{\pi}}$, the pion leptonic decay constant, the uncertainty in the chiral extrapolation dominates the systematic error. Our main results include ${f}_{\ensuremath{\pi}}=124(2{)}_{\mathrm{stat}}(5{)}_{\mathrm{syst}}\text{ }\text{ }\mathrm{MeV}$, ${f}_{K}/{f}_{\ensuremath{\pi}}=1.204(7)(25)$ where ${f}_{K}$ is the kaon decay constant, ${m}_{s}^{\overline{\mathrm{MS}}}(2\text{ }\text{ }\mathrm{GeV})=(96.2\ifmmode\pm\else\textpm\fi{}2.7)\text{ }\text{ }\mathrm{MeV}$ and ${m}_{ud}^{\overline{\mathrm{MS}}}(2\text{ }\text{ }\mathrm{GeV})=(3.59\ifmmode\pm\else\textpm\fi{}0.21)\text{ }\text{ }\mathrm{MeV}$ (${m}_{s}/{m}_{ud}=26.8\ifmmode\pm\else\textpm\fi{}1.4$) where ${m}_{s}$ and ${m}_{ud}$ are the mass of the strange quark and the average of the up and down quark masses, respectively, $[{\ensuremath{\Sigma}}^{\overline{\mathrm{MS}}}(2\text{ }\text{ }\mathrm{GeV}){]}^{1/3}=256(6)\text{ }\text{ }\mathrm{MeV}$, where $\ensuremath{\Sigma}$ is the chiral condensate, the Sommer scale ${r}_{0}=0.487(9)\text{ }\text{ }\mathrm{fm}$ and ${r}_{1}=0.333(9)\text{ }\text{ }\mathrm{fm}$.