Critical endpoint of the finite temperature phase transition for three flavor QCD

We investigate the critical endpoint of finite temperature phase transition of $N_f=3$ QCD at zero chemical potential. We employ the renormalization-group improved Iwasaki gauge action and nonperturbatively $O(a)$-improved Wilson-clover fermion action. The critical endpoint is determined by using the intersection point of kurtosis for the temporal size $N_t=4$, 6, 8. Spatial sizes of $N_l=6–16$ ($N_t=4$), 10–24 ($N_t=6$), and 12–24 ($N_t=8$) are employed. We find that $N_t=4$ is out of the scaling region. Using results for $N_t=6$ and 8 and making linear extrapolations in $1/N_t^2$, we obtain $\sqrt{t_0}{T}_{\mathrm{E}}=0.0988(14)(9)$, $\sqrt{t_0}{m}_{\mathrm{PS},\mathrm{E}}=0.2274(52)(108)$, and $m_{\mathrm{PS},\mathrm{E}}/T_{\mathrm{E}}=2.302(62)(13)$, where the first error is statistical error, the second error is systematic error, and $m_{\mathrm{PS}}$ is the pseudoscalar meson mass. If one uses $1/\sqrt{t_0}=1.347(30)\mathrm{GeV}$, as reported by Borsanyi et al., one finds $T_{\mathrm{E}}=133(2)(1)(3)\mathrm{MeV}$, $m_{\mathrm{PS},\mathrm{E}}=306(7)(14)(7)\mathrm{MeV}$, and $m_{\mathrm{PS},\mathrm{E}}/m_{\mathrm{PS}}^{\text{phys},\text{sym}}=0.745(17)(35)(17)$, where the third error comes from the error of $\sqrt{t_0}$ and $m_{\mathrm{PS}}^{\text{phys},\text{sym}}=\sqrt{(m_{\pi }^2+2m_K^2)/3}$. Our current estimation of $\sqrt{t_0}{m}_{\mathrm{PS},\mathrm{E}}$ in the continuum limit is about 25% smaller than the SU(3) symmetric point.

Figure 1

Schematic picture of a method to determine the critical endpoint.

Figure 2

(Left panel) $(a{m}_{\mathrm{PS}}{)}^2$ vs $1/\kappa$. (Right panel) $\sqrt{t_0}/a$ vs $1/\kappa$.

Figure 3

Kurtosis intersection for $G,P,L$ on smaller lattices (top-left panel), larger lattices (top-right panel), $\mathrm{\Sigma }$ on smaller lattices (bottom-left panel), and larger lattices (bottom-right panel) at $N_t=4$.

Figure 4

$K_{\mathrm{t}}$, $\sqrt{t_0}{m}_{\mathrm{PS},\mathrm{t}}$, $\sqrt{t_0}{T}_{\mathrm{t}}$ vs $\beta$ at $N_t=4$ (top-left panel), $N_t=6$ (top-right panel), $N_t=8$ (bottom panel) determined by the quark observable $\mathrm{\Sigma }$.

Figure 5

$K_{\mathrm{t}}$, $\sqrt{t_0}{m}_{\mathrm{PS},\mathrm{t}}$, $\sqrt{t_0}{T}_{\mathrm{t}}$ vs $\beta$ at $N_t=4$ (top-left panel), $N_t=6$ (top-right panel), and $N_t=8$ (bottom panel) determined by the gluon observables $G,P,L$.

Figure 6

Susceptibility exponent $b$ as a function of $\beta$. The horizontal line corresponds to the value $\gamma /\nu =1.964$ for Z(2) universality in three dimensions. Colored blocks superimposed are estimates of the critical endpoint ${\beta }_E$ obtained in the kurtosis intersection analysis.

Figure 7

Phase diagram of $N_f=3$ lattice QCD on the $(\beta ,\kappa )$ plane. An estimation of the line of first order phase transition ending at a critical point marked by full circles is shown for $N_t=4$, 6, 8, together with the line of ${\kappa }_c$ where pion mass vanishes.

Figure 8

Continuum extrapolation of $\sqrt{t_0}{m}_{\mathrm{PS},\mathrm{E}}$ and $\sqrt{t_0}{T}_{\mathrm{E}}$ as a function of $1/N_t^2$.

Figure 9

Susceptibility and kurtosis as functions of $\kappa$ at $\beta =1.60$ and $N_t=4$, together with quadratic fits. Gluon observable $P$ (top-left panel), $G$ (top-right panel), $L$ (bottom-right panel), and quark observable $\mathrm{\Sigma }$ (bottom-left panel) are shown.

Figure 10

Same as Fig. 9, but at $\beta =1.65$ and $N_t=4$.

Figure 11

Susceptibility and kurtosis as functions of $\kappa$ at $\beta =1.715$ and $N_t=6$, together with quadratic fits. Gluon observables $P$ (top-left panel), $G$ (top-right panel), and $L$ (bottom-right panel) and quark observable $\mathrm{\Sigma }$ (bottom-left panel) are shown.

Figure 12

Same as Fig. 11, but at $\beta =1.73$ and $N_t=6$.

Figure 13

Susceptibility and kurtosis as functions of $\beta$ at $\kappa =0.14024$ and $N_t=8$, with curves showing reweighted estimates. Gluon observables $P$ (top-left panel), $G$ (top-right panel), and $L$ (bottom-right panel) and quark observable $\mathrm{\Sigma }$ (bottom-left panel) are shown.

Figure 14

Same as Fig. 13, but at $\kappa =0.13995$ and $N_t=8$.