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

TL;DR

This study reports a three-flavor lattice QCD calculation using improved staggered quarks to determine $f_c$, $f_c$, their ratio, light quark masses, and Gasser-Leutwyler constants $L_i$ by fitting to staggered chiral perturbation theory forms that incorporate taste violations. The methodology combines precise correlator analyses with random-wall sources, continuum and chiral extrapolations, and perturbative mass renormalization, all anchored by the scale $r_1$ and finite-volume considerations. Key results include $f_c=129.5(0.9)(3.5)$ MeV, $f_K=156.6(1.0)(3.6)$ MeV, $f_K/f_c=1.210(4)(13)$, and $m_u/m_d=0.43(0)(1)(8)$, with corresponding $ar{ ext{MS}}$ masses and $2L_8-L_5=-0.2(1)(2) imes10^{-3}$. The work provides a precise determination of $|V_{us}|=0.2219(26)$ via $f_K/f_c$, emphasizes the necessity of S$b5$PT to account for taste-violating lattice artifacts, and outlines avenues to reduce systematic uncertainties through lighter sea-quark masses and finer lattice spacings.

Abstract

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 fm 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_π=129.5(0.9)(3.5)MeV, f_K=156.6(1.0)(3.6)MeV, and f_K/f_π=1.210(4)(13), where the errors are statistical and systematic. Following a recent paper by Marciano, our value of f_K/f_πimplies |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 data can also be used to determine several of the constants of the low energy effective Lagrangian: in particular we find 2L_8-L_5=-0.2(1)(2) 10^{-3} at chiral scale m_η. This provides an alternative (though not independent) way of estimating m_u; 2L_8-L_5 is far outside the range that would allow m_u=0. Results for m_s^\msbar, \hat m^\msbar, and m_s/\hat 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^\msbar=1.7(0)(1)(2)(2)MeV and m_d^\msbar=3.9(0)(1)(4)(2)MeV at scale 2 GeV, with errors from statistics, simulation, perturbation theory, and electromagnetic effects, respectively.

Figures (22)

• Figure 1: Pion masses with random-wall and Coulomb-wall sources and point and Coulomb-wall sinks from the coarse set with sea quark lattice masses 0.01,0.05 (see Table \ref{['RUNTABLE']}). The red crosses are random-wall source and Coulomb-wall sink, and the green octagons are Coulomb-wall source and point sink (summed over spatial sites to project out the zero momentum states). The blue bursts are from a random-wall source and point sink, and the magenta squares have a Coulomb-wall source and sink. The lower set of "WW" points include an excited state in the fit. The symbol size is proportional to the confidence level of the fit, with the symbol size in the labels corresponding to 50%.
• Figure 2: Same as Fig. \ref{['masses_fig']} but for pion propagator amplitudes. The lower set of "WW" points again include an excited state in the fit. The "PW" symbols have been displaced slightly to the right to separate them from the "WP" points.
• Figure 3: Ratio of pion propagators. Here $P_{WP}$ is the Coulomb wall source and point sink pion propagator, etc. The point source was implemented with a random wall as discussed in the text.
• Figure 4: Pion masses (red octagons) and amplitudes (blue crosses) as a function of the minimum time distance in the fit, from the fine set with sea quark lattice masses 0.0062,0.031 (see Table \ref{['RUNTABLE']}). The amplitudes have been multiplied by 175.
• Figure 5: Pseudoscalar masses with $a \approx 0.125$ fm. The horizontal axis is the sum of the valence quark mass (in units of $r_1$). For each set of values of $m_{\rm sea}$, the first symbol shows "pion" points with $m_x=m_y$; while the second shows "kaon" points with $m_y=m'_s$. Bursts are pion points with valence masses equal to sea quark masses.