$2+1$ flavor lattice QCD toward the physical point

We present the first results of the PACS-CS project which aims to simulate $2+1$ flavor lattice QCD on the physical point with the nonperturbatively $O(a)$-improved Wilson quark action and the Iwasaki gauge action. Numerical simulations are carried out at $\beta =1.9$, corresponding to the lattice spacing of $a=0.0907(13)\mathrm{fm}$, on a ${32}^3\times 64$ lattice with the use of the domain-decomposed HMC algorithm to reduce the up-down quark mass. Further algorithmic improvements make possible the simulation whose up-down quark mass is as light as the physical value. The resulting pseudoscalar meson masses range from 702 MeV down to 156 MeV, which clearly exhibit the presence of chiral logarithms. An analysis of the pseudoscalar meson sector with SU(3) chiral perturbation theory reveals that the next-to-leading order corrections are large at the physical strange quark mass. In order to estimate the physical up-down quark mass, we employ the SU(2) chiral analysis expanding the strange quark contributions analytically around the physical strange quark mass. The SU(2) low energy constants ${\overline{l}}_3$ and ${\overline{l}}_4$ are comparable with the recent estimates by other lattice QCD calculations. We determine the physical point together with the lattice spacing employing $m_{\pi }$, $m_K$ and $m_{\Omega }$ as input. The hadron spectrum extrapolated to the physical point shows an agreement with the experimental values at a few % level of statistical errors, albeit there remain possible cutoff effects. We also find that our results of $f_{\pi }$, $f_K$ and their ratio, where renormalization is carries out perturbatively at one loop, are compatible with the experimental values. For the physical quark masses we obtain $m_{ud}^{\overline{\mathrm{MS}}}$ and $m_s^{\overline{\mathrm{MS}}}$ extracted from the axial-vector Ward-Takahashi identity with the perturbative renormalization factors. We also briefly discuss the results for the static quark potential.

Figure 1

Simulation cost at $({\kappa }_{ud},{\kappa }_s)=(0.13770,0.13640)$ by DDHMC (blue open circle) and $({\kappa }_{ud},{\kappa }_s)=(0.13781,0.13640)$ by MPDDHMC (blue closed circle) for 10000 trajectories. Solid line indicates the cost estimate of $N_f=2+1$ QCD simulations with the HMC algorithm at $a=0.1\mathrm{fm}$ with $L=3\mathrm{fm}$ for 100 independent configurations. Vertical line denotes the physical point.

Figure 2

Plaquette history (left) and normalized autocorrelation function (right) for $({\kappa }_{ud},{\kappa }_s)=(0.13727,0.13640)$. Horizontal lines in the left denote the average value of the plaquette with one standard deviation error band.

Figure 3

Effective masses for the mesons (left) and the baryons (right) at $({\kappa }_{ud},{\kappa }_s)=(0.13754,0.13640)$. Horizontal lines represent the fitting results with 1 standard deviation error band.

Figure 4

Same as Fig. 3 for $({\kappa }_{ud},{\kappa }_s)=(0.13754,0.13660)$.

Figure 5

Same as Fig. 3 for $({\kappa }_{ud},{\kappa }_s)=(0.13770,0.13640)$.

Figure 6

Same as Fig. 3 for $({\kappa }_{ud},{\kappa }_s)=(0.13781,0.13640)$.

Figure 7

Noise-to-signal ratio for the $\pi$, $\rho$ and nucleon propagators. The data are normalized by the value at $t=10$. See text for red and bule lines.

Figure 8

Binsize dependence of the magnitude of error for $m_{\pi }$ (left) and $m_{{\eta }_{\mathrm{ss}}}$ (right) at ${\kappa }_{ud}\ge 0.13754$.

Figure 9

Effective masses for the $\pi$ (left) and the nucleon (right) at $({\kappa }_{ud},{\kappa }_s)=(0.13700,0.13640)$. Black and red symbols denote the PACS-CS and the CP-PACS/JLQCD results, respectively.

Figure 10

Comparison of the PACS-CS (red) and the CP-PACS/JLQCD (black) results for $m_{\pi }^2/m_{ud}^{\mathrm{AWI}}$ (left) and $f_K/f_{\pi }$ (right) as a function of $m_{ud}^{\mathrm{AWI}}$. Vertical line denotes the physical point and star symbol represents the experimental value.

Figure 11

Comparison of the results for ${\overline{l}}_3$ and ${\overline{l}}_4$. Black symbols denote the phenomenological estimates. Blue ones are for 2 flavor lattice results. Red closed (open) symbols represent the results for the SU(3) (SU(2)) ChPT analyses in the $2+1$ flavor dynamical simulations. See text and Table 8 for details.

Figure 12

SU(3) ChPT fit for $m_{\pi }^2/m_{ud}^{\mathrm{AWI}}$ (left) and $2m_K^2/(m_{ud}^{\mathrm{AWI}}+m_s^{\mathrm{AWI}})$ (right).

Figure 13

SU(3) ChPT fit for $f_{\pi }$ (left) and $f_K$ (right).

Figure 14

Ratio of the NLO contribution to the LO one in the SU(3) ChPT fit for $m_{\pi }^2/m_{ud}^{\mathrm{AWI}}$ (left) and $2m_K^2/(m_{ud}^{\mathrm{AWI}}+m_s)$ (right).

Figure 15

Ratio of the NLO contribution to the LO one in the SU(3) ChPT fit for $f_{\pi }$ (left) and $f_K$ (right).

Figure 16

SU(2) ChPT fit for $m_{\pi }^2/m_{ud}^{\mathrm{AWI}}$.

Figure 17

SU(2) ChPT fit for $f_{\pi }$ (left) and $f_K$ (right).

Figure 18

Ratio of the NLO contribution to the LO one in the SU(2) ChPT fit for $m_{\pi }^2/m_{ud}^{\mathrm{AWI}}$, $f_{\pi }$ and $f_K$.

Figure 19

$|R_X|$ ($R_{m_{\mathrm{PS}}}>0$ and $R_{f_{\mathrm{PS}}}<0$) for $X=m_{\pi }$, $m_K$, $f_{\pi }$, and $f_K$ with $L=2.9\mathrm{fm}$ as a function of $m_{\pi }$ at the physical strange quark mass based on the NLO SU(3) ChPT. Dotted vertical line denotes the physical point and the solid ones are for our simulation points. Orange vertical line represents the pion mass with $m_{\pi }L=2$.

Figure 20

$|R_X^{\prime }|$ ($R_{m_{\mathrm{PS}}}^{\prime }>0$ and $R_{f_{\mathrm{PS}}}^{\prime }<0$) for $X=m_{\pi }$, $f_{\pi }$ with $L=2.9\mathrm{fm}$ as a function of $m_{\pi }$ based on the NLO SU(2) ChPT. Dotted vertical line denotes the physical point and the solid ones are for our simulation points. Orange vertical line represents the pion mass with $m_{\pi }L=2$.

Figure 21

Linear chiral extrapolation for the vector meson masses in lattice units. Red triangles represent the fit results at the measured values of $m_{ud}^{\mathrm{AWI}}$. Red star symbol denotes the extrapolated value at the physical point. Experimental value in lattice units is also plotted at $m_{ud}^{\mathrm{AWI}}=0$ for comparison.

Figure 22

Same as Fig. 21 for the octet baryons.

Figure 23

Same as Fig. 21 for the decuplet baryons.

Figure 24

Light hadron spectrum extrapolated to the physical point using $m_{\pi }$, $m_K$ and $m_{\Omega }$ as input. Horizontal bars denote the experimental values.

Figure 25

Effective potential $V_{\mathrm{eff}}(r,t)$ with $r=4$, 8, 12 at ${\kappa }_{ud}=0.13770$ as a representative case.

Figure 26

Static quark potential $V(r)$ at ${\kappa }_{ud}=0.13770$ as a representative case. Solid line denote the fit result with Eq. (79).

Figure 27

Linear chiral extrapolation for $1/r_0$ at the physical point. Red triangles denote the fit results at the measured values of $m_{ud}^{\mathrm{AWI}}$.