$2+1$ flavor domain wall QCD on a $(2\mathrm{fm}{)}^3$ lattice: Light meson spectroscopy with $L_s=16$

We present results for light meson masses and pseudoscalar decay constants from the first of a series of lattice calculations with $2+1$ dynamical flavors of domain wall fermions and the Iwasaki gauge action. The work reported here was done at a fixed lattice spacing of about 0.12 fm on a ${16}^3\times 32$ lattice, which amounts to a spatial volume of $(2\mathrm{fm}{)}^3$ in physical units. The number of sites in the fifth dimension is 16, which gives $m_{\mathrm{res}}=0.00308(4)$ in these simulations. Three values of input light sea quark masses, $m_l^{\mathrm{sea}}\approx 0.85m_s$, $0.59m_s$ and $0.33m_s$ were used to allow for extrapolations to the physical light quark limit, while the heavier sea quark mass was fixed to approximately the physical strange quark mass $m_s$. The exact rational hybrid Monte Carlo algorithm was used to evaluate the fractional powers of the fermion determinants in the ensemble generation. We have found that $f_{\pi }=127(4)\mathrm{MeV}$, $f_K=157(5)\mathrm{MeV}$ and $f_K/f_{\pi }=1.24(2)$, where the errors are statistical only, which are in good agreement with the experimental values.

Figure 1

Evolution history of the plaquette for all three ensembles, measured every MD time unit. The straight lines are the averages from trajectory 2000 to 4000 for each ensemble.

Figure 2

Evolution history of the quark condensate $⟨\overline{q}q⟩$ for the first 3000 molecular dynamics time units at three dynamical points: $m_{\mathrm{val}}=m_l^{\mathrm{sea}}=0.01$ (top), $m_{\mathrm{val}}=m_l^{\mathrm{sea}}=0.02$ (middle) and $m_{\mathrm{val}}=m_l^{\mathrm{sea}}=0.03$ (bottom). The measurements were taken every 5 molecular dynamics time units.

Figure 3

Integrated autocorrelation time for the pseudoscalar two-point correlation function at time slice 12 with $m_l^{\mathrm{sea}}=m_{\mathrm{val}}=0.01$.

Figure 4

The topological charge as measured by gluonic fields on the three ensembles. The left three panels are the topological charge as a function of the molecular dynamics time. Each of the three graphs on the right represents the normalized histogram of the corresponding topological charge evolution on the left.

Figure 5

Variation of the pseudoscalar masses obtained from the correlated fits and the corresponding ${\chi }^2/\mathrm{dof}$ with respect to $t_{\min }$ for each of the dynamical points.

Figure 6

Pseudoscalar effective masses from WL-WL, LL-LL and GL-GL correlation functions for the three ensembles. The valence quarks are degenerate, with the masses equal to the light sea quark mass of that ensemble.

Figure 7

Vector effective masses from WL-WL and GL-GL correlation functions for the three ensembles. The valence quarks are degenerate, with the masses equal to the light sea quark mass of that ensemble.

Figure 8

The residual mass determined from the WL-WL correlators. The solid line is a linear fit to the dynamical points with $m_{\mathrm{val}}=m_l^{\mathrm{sea}}$.

Figure 9

Linear extrapolations of the vector mass using Eq. (14). Here the circles are partially quenched data points from the $m_l^{\mathrm{sea}}=0.01$ ensemble, the squares are the unitary points with $m_1=m_2=m_l^{\mathrm{sea}}$ and the diamond is the value of $m_V$ in the chiral limit.

Figure 10

Linear extrapolation of $a^{-1}$ versus $m_l^{\mathrm{sea}}$ from the method of planes.

Figure 11

Linear extrapolation of the pseudoscalar meson mass through the light dynamical points. The crosses are the physical $\pi$ and $K$ points from which we obtain the values of $\overline{m}$ and $m_s$.

Figure 12

Linear extrapolation of the pseudoscalar meson decay constant through the light dynamical points. The diamonds are the values of $f_{\pi }$ and $f_K$ obtained from $f_P^{(2)}$.

Figure 13

The comparison of the topological charge evolution and corresponding histogram measured from the two portions of the $m_l^{\mathrm{sea}}=0.03$ ensemble, in which different algorithms (RHMC I and RHMC II) were used to evaluate Eq. (1). The dashed line indicates where the algorithm was changed.

Figure 14

The comparison of the effective masses for the pseudoscalar and vector WL-WL correlation functions at the unitary point from the two portions of the $m_l^{\mathrm{sea}}=0.03$ ensemble (RHMC I and RHMC II).