Equation of state in $2\mathbf{+}1$ flavor QCD with improved Wilson quarks by the fixed scale approach

We study the equation of state in $2+1$ flavor QCD with nonperturbatively improved Wilson quarks coupled with the renormalization group-improved Iwasaki glue. We apply the $T$-integration method to nonperturbatively calculate the equation of state by the fixed-scale approach. With the fixed-scale approach, we can purely vary the temperature on a line of constant physics without changing the system size and renormalization constants. Unlike the conventional fixed-$N_t$ approach, it is easy to keep scaling violations small at low temperature in the fixed-scale approach. We study $2+1$ flavor QCD at light quark mass corresponding to $m_{\pi }/m_{\rho }\simeq 0.63$, while the strange quark mass is chosen around the physical point. Although the light quark masses are still heavier than the physical values, our equation of state is roughly consistent with recent results with highly improved staggered quarks at large $N_t$.

Figure 1

Temperature vs lattice spacing at each $N_t$. The horizontal dashed line at $a\simeq 0.07\mathrm{fm}$ represents the lattice spacing in this study. The vertical shaded line represents the approximate location of the pseudocritical temperature at our quark masses.

Figure 2

Time history of the Polyakov loop measured on finite-temperature lattices. The horizontal axis is the trajectory length.

Figure 3

Polyakov loop expectation value and its susceptibility as functions of $T$. The left panel shows the bare results; the right panel shows renormalized results using the renormalization scheme of 27. ${\chi }_L$ is multiplied by 2 and ${\chi }_{L_{\mathrm{ren}}}$ is multiplied by 0.004 to fit into the same scale. Also shown in the right panel are the results of $⟨L_{\mathrm{ren}}⟩$ from the p4 staggered quark action obtained at $m_{ud}^{\mathrm{bare}}/m_s^{\mathrm{bare}}=0.05$ in the fixed $N_t$ approach at $N_t=8$ [29], where the horizontal axis is rescaled using $r_0=0.5\mathrm{fm}$.

Figure 4

The global fit for coupling parameters, $\beta$, as a function of $m_{\rho }a$. Square symbols show coupling parameters in the $\mathrm{CP}\mathrm{\text{−}}\mathrm{PACS}/\mathrm{JLQCD}$ study. The solid lines show the global fit results for each simulation point with corresponding $m_{\rho }/m_{\pi }$ and $m_{{\eta }_{ss}}/m_{\varphi }$. To avoid a plot that is too busy, only half of the data points are shown (${\kappa }_s=0.1371$, 0.1358, and 0.1351 at $\beta =1.83$, 1.90, and 2.05, respectively).

Figure 5

The same as Fig. 4 but for ${\kappa }_{ud}$ and ${\kappa }_s$.

Figure 6

Beta functions on our LCP, $m_{\pi }/m_{\rho }=0.6337$ and $m_{{\eta }_{ss}}/m_{\varphi }=0.7377$, as functions of $\beta$. The scale setting is made with $a{m}_{\rho }$. Beta functions for ${\kappa }_{ud}$ and ${\kappa }_s$ are magnified by a factor of 100. Horizontal and vertical bars at each data point represent statistical errors.

Figure 7

Beta functions at light quark masses corresponding to $m_{\pi }/m_{\rho }\simeq 0.5$, 0.6, and 0.7, with the $s$-quark mass $m_{{\eta }_{ss}}/m_{\varphi }\simeq 0.7$, as functions of $\beta$. The simulation point of this study is marked by an open square. Horizontal and vertical bars at each data point represent statistical errors.

Figure 8

Trace anomaly $(ϵ-3p)/T^4$, energy density $ϵ/T^4$, and pressure $3p/T^4$ in $2+1$ flavor QCD. The thin and thick vertical bars represent statistic and systematic errors, respectively. The curves are drawn by the Akima spline interpolation.

Figure 9

Straight line, cubic spline, and Akima spline interpolations for the trace anomaly and the pressure.