Nature of the phase transition for finite temperature $N_{\mathrm{f}}=3$ QCD with nonperturbatively $\mathrm{O}(a)$ improved Wilson fermions at $N_{\mathrm{t}}=12$

We study the nature of the finite temperature phase transition for three-flavor QCD. In particular we investigate the location of the critical endpoint along the three flavor symmetric line in the light quark mass region of the Columbia plot. In the study, the Iwasaki gauge action and the nonperturvatively O($a$) improved Wilson-Clover fermion action are employed. We newly generate data at $N_{\mathrm{t}}=12$ and set an upper bound of the critical pseudoscalar meson mass in the continuum limit $m_{\mathrm{PS},\mathrm{E}}\lesssim 110\mathrm{MeV}$.

Figure 1

Columbia plot for $N_{\mathrm{f}}=2+1$ QCD at zero density.

Figure 2

An illustration of the kurtosis intersection analysis. $K_t$ denotes the kurtosis value for the phase transition and $E$ indicates the critical endpoint, where $K_t$ is independent of the volume.

Figure 3

The susceptibility (upper half) and kurtosis (lower half) of the chiral condensate as a function $\kappa$ for several spatial sizes, $N_{\mathrm{s}}=16–32$. The left (right) panel is for $\beta =1.80$ ($\beta =1.81$). The raw data points (as symbols) as well as the multiensemble reweighting results (1-$\sigma$ band) are plotted.

Figure 4

Phase diagram for the bare parameter space $(\beta ,\kappa )$ at $N_{\mathrm{t}}=12$ together with $N_{\mathrm{t}}=6$, 8, and 10 results in Ref. [34]. The open symbols represent a transition point (TP) while the filled symbols are the critical endpoints (CEP) determined by the kurtosis intersection with a correction term. On the transition line, the left (right) hand side of an critical endpoint is the first order phase transition (crossover) side. The phase transition line for each $N_{\mathrm{t}}$ is a polynomial interpolation. For $N_{\mathrm{t}}=12$ the interpolation formula is given in Eq. (2). In the plot, ${\kappa }_{\mathrm{c}}$ is the pseudoscalar massless point with $N_{\mathrm{f}}=3$ which is determined by zero temperature simulations as shown in Table 7 in Appendix.

Figure 5

Kurtosis intersection for the chiral condensate at $N_{\mathrm{t}}=12$. The left panel contains all data points of $N_{\mathrm{s}}=16–32$. The right panel includes larger volumes $N_{\mathrm{s}}\ge 24$ together with the fitting function in Eq. (3) which assumes the 3D ${\mathrm{Z}}_2$ universality class and contains a correction term. The black pentagon represents the resulting critical value of $\beta$.

Figure 6

The volume scaling of susceptibility peak height for $N_{\mathrm{t}}=12$. Both axes are scaled logarithmically. The filled symbols are included in the fit but open ones are not.

Figure 7

Exponent of the susceptibility peak height along the transition line projected on $\beta$ value for $N_{\mathrm{t}}=4$, 6, 8, 10, and 12. The line connecting the data points is to guide readers’ eyes. The point where the line for each $N_{\mathrm{t}}$ intersects the (green) horizontal line is an estimate of the critical point assuming the ${\mathrm{Z}}_2$ universality class. On the other hand, the shaded areas represent the critical $\beta$ determined by the kurtosis intersection analysis.

Figure 8

The hadronic quantities in lattice units $(a{m}_{\mathrm{PS}}{)}^2$, $\sqrt{t_0}/a$ as a function of $\kappa$ at $\beta =1.80$ (left) and 1.81 (right). The vertical blue line shows the location of the transition point for $\kappa$ at the corresponding $\beta$ with $N_{\mathrm{t}}=12$.

Figure 9

$\sqrt{t_0}T$, $m_{\mathrm{PS}}/T$, and $\sqrt{t_0}{m}_{\mathrm{PS}}$ along the transition line projected on $\beta$ for $N_{\mathrm{t}}=10$ (left) and 12 (right). The vertical red line shows the location of the critical value of $\beta$ determined by the kurtosis intersection analysis.

Figure 10

Continuum extrapolation of the critical endpoint. In upper panels, the left one is for $\sqrt{t_0}{m}_{\mathrm{PS},\mathrm{E}}$ vs $1/N_{\mathrm{t}}^2$ while the right one is for $\sqrt{t_0}{m}_{\mathrm{PS},\mathrm{E}}$ vs $1/N_{\mathrm{t}}$. In the lower panel $m_{\mathrm{PS},\mathrm{E}}/T_{\mathrm{E}}$ (left) and $\sqrt{t_0}{T}_{\mathrm{E}}$ (right) are plotted as a function of $1/N_{\mathrm{t}}^2$.

Figure 11

Columbia-like plot with axes $m_{\pi }^2$ and $m_{{\eta }_s}^2$ in physical units. The blue symbol denotes the upper bound obtained in this work $N_{\mathrm{t}}\le 12$ while the red one is given in our previous study $N_{\mathrm{t}}\le 10$ [34]. The physical point is also shown just for a reference.