Light pseudoscalar decay constants, quark masses, and low energy constants from three-flavor lattice QCD

As part of our program of lattice simulations of three-flavor QCD with improved staggered quarks, we have calculated pseudoscalar meson masses and decay constants for a range of valence quark masses and sea-quark masses on lattices with lattice spacings of about 0.125 and 0.09 fm. We fit the lattice data to forms computed with “staggered chiral perturbation theory.” Our results provide a sensitive test of the lattice simulations, and especially of the chiral behavior, including the effects of chiral logarithms. We find: $f_{\pi }=129.5\pm 0.9\pm 3.5\mathrm{M}\mathrm{e}\mathrm{V}$, $f_K=156.6\pm 1.0\pm 3.6\mathrm{M}\mathrm{e}\mathrm{V}$, and $f_K/f_{\pi }=1.210(4)(13)$, where the errors are statistical and systematic. Following a recent paper by Marciano, our value of $f_K/f_{\pi }$ implies $|V_{us}|=0.2219(26)$. Further, we obtain $m_u/m_d=0.43(0)(1)(8)$, where the errors are from statistics, simulation systematics, and electromagnetic effects, respectively. The partially quenched data can also be used to determine several of the constants of the low energy chiral effective Lagrangian: In particular, we find $2L_8-L_5=-0.2(1)(2)\times {10}^{-3}$ at chiral scale $m_{\eta }$, where the errors are statistical and systematic. This provides an alternative (though not independent) way of estimating $m_u$; the value of $2L_8-L_5$ is far outside the range that would allow the up quark to be massless. Results for $m_s^{\overline{\mathrm{M}\mathrm{S}}}$, ${\widehat{m}}^{\overline{\mathrm{M}\mathrm{S}}}$, and $m_s/\widehat{m}$ can be obtained from the same lattice data and chiral fits, and have been presented previously in joint work with the HPQCD and UKQCD collaborations. Using the perturbative mass renormalization reported in that work, we obtain $m_u^{\overline{\mathrm{M}\mathrm{S}}}=1.7(0)(1)(2)(2)\mathrm{M}\mathrm{e}\mathrm{V}$ and $m_d^{\overline{\mathrm{M}\mathrm{S}}}=3.9(0)(1)(4)(2)\mathrm{M}\mathrm{e}\mathrm{V}$ at scale 2 GeV, with errors from statistics, simulation, perturbation theory, and electromagnetic effects, respectively.