Critical endline of the finite temperature phase transition for $2+1$ flavor QCD around the SU(3)-flavor symmetric point

We investigate the critical endline of the finite temperature phase transition of QCD around the SU(3)-flavor symmetric point at zero chemical potential. We employ the renormalization-group improved Iwasaki gauge action and nonperturbatively $O(a)$-improved Wilson-clover fermion action. The critical endline is determined by using the intersection point of kurtosis, employing the multiparameter, multiensemble reweighting method to calculate observables off the SU(3)-symmetric point, at the temporal size $N_{\mathrm{T}}=6$ and lattice spacing as low as $a\approx 0.19\mathrm{fm}$. We confirm that the slope of the critical endline takes the value of $-2$, and find that the second derivative is positive, at the SU(3)-flavor symmetric point on the Columbia plot parametrized with the strange quark mass $m_{\mathrm{s}}$ and degenerated up-down quark mass $m_{\mathrm{l}}$.

Figure 1

$\mathrm{\Sigma }$ (top left), $\chi$ (top right), $S$ (bottom left), and $K$ (bottom right) as a function of ${\kappa }_{\mathrm{l}}$ obtained by the reweighting of $N_{\mathrm{f}}=3$ data in Table 1 (curves) and those at $\beta =1.73$, ${\kappa }_{\mathrm{s}}=0.13950$, $N_{\mathrm{L}}=12$ obtained by the direct $N_{\mathrm{f}}=2+1$ simulations (square data points).

Figure 2

$\chi$ and $K$ for ${\kappa }_{\mathrm{s}}=0.13910$, 0.14070, and 0.14170 as a function of ${\kappa }_{\mathrm{l}}$ at $\beta =1.715$ (upper) and 1.73 (lower).

Figure 3

Phase diagram in bare parameter space of ($\beta$, ${\kappa }_{\mathrm{l}}$). Each line represents a transition line with the quadratic form in Eq. (5) for each value of ${\kappa }_{\mathrm{s}}$. Open symbols denote transition points at $\beta =1.715$, 1.73, and 1.75. Filled symbols denote critical end points. Note that the values of $\beta$ at critical end points are almost constant for $0.1391\le {\kappa }_{\mathrm{s}}\le 0.1417$.

Figure 4

Typical kurtosis intersection plots for ${\kappa }_{\mathrm{s}}=0.13910$, 0.14070, and 0.14170 from left to right, and the linear functional form with two $\beta$ points (top), linear with three points (middle), and quadratic with three points (bottom).

Figure 5

Results for the critical endpoints in the plane of $1/{\kappa }_{\mathrm{l}}$ and $1/{\kappa }_{\mathrm{s}}$ (black open circles). Also shown (red squares) are the points where zero temperature simulations are carried out to calculate hadron masses. An SU(3)-symmetric line is drawn in pink, while for the green and blue lines the sum of three quark masses is constant.

Figure 6

CEL in the plane of $(\sqrt{t_0}{m}_{\pi }{)}^2$ and $(\sqrt{t_0}{m}_{{\eta }_{\mathrm{s}}}{)}^2$ and the slope along CEL calculated by the fit in Eq. (8). The pink line denotes the line of SU(3) symmetry ($N_{\mathrm{f}}=3$).