Complex saddle points and disorder lines in QCD at finite temperature and density

The properties and consequences of complex saddle points are explored in phenomenological models of QCD at nonzero temperature and density. Such saddle points are a consequence of the sign problem and should be considered in both theoretical calculations and lattice simulations. Although saddle points in finite-density QCD are typically in the complex plane, they are constrained by a symmetry that simplifies analysis. We model the effective potential for Polyakov loops using two different potential terms for confinement effects and consider three different cases for quarks: very heavy quarks, massless quarks without modeling of chiral symmetry breaking effects, and light quarks with both deconfinement and chiral symmetry restoration effects included in a pair of Polyakov-Nambu-Jona Lasinio models. In all cases, we find that a single dominant complex saddle point is required for a consistent description of the model. This saddle point is generally not far from the real axis; the most easily noticed effect is a difference between the Polyakov loop expectation values $⟨{\mathrm{Tr}}_{F}P⟩$ and $⟨{\mathrm{Tr}}_F{P}^{†}⟩$, and that is confined to a small region in the $\mu -T$ plane. In all but one case, a disorder line is found in the region of critical and/or crossover behavior. The disorder line marks the boundary between exponential decay and sinusoidally modulated exponential decay of correlation functions. Disorder line effects are potentially observable in both simulation and experiment. Precision simulations of QCD in the $\mu -T$ plane have the potential to clearly discriminate between different models of confinement.

Figure 1

$⟨{\mathrm{Tr}}_{F}P⟩$ as a function of $T$ for pure $SU(3)$ with Model A and Model B for confinement effects.

Figure 2

$⟨{\mathrm{Tr}}_{F}P⟩$ (solid line) and $⟨{\mathrm{Tr}}_F{P}^{†}⟩$ (dashed line) as a function of $T$ for $\mu =1000$, 1400, and 1800 MeV for heavy quarks using Model A for confinement effects. The Polyakov loops are normalized to one as the temperature becomes large.

Figure 3

$⟨{\mathrm{Tr}}_{F}P⟩$ (solid line) and $⟨{\mathrm{Tr}}_F{P}^{†}⟩$ (dashed line) as a function of $T$ for $\mu =1000$, 1400, and 1800 MeV for heavy quarks using Model B for confinement effects. The Polyakov loops are normalized to one as the temperature becomes large.

Figure 4

Contour plot of $\psi$ in the $\mu -T$ plane for heavy quarks ($m=2000\mathrm{MeV}$) using Model A for confinement effects. The region where ${\kappa }_I\ne 0$ is shaded.

Figure 5

Contour plot of ${\kappa }_I$ in the $\mu -T$ plane for heavy quarks ($m=2000\mathrm{MeV}$) using Model A for confinement effects. Contours are given in MeV with ${\alpha }_S$ set to one. The region where ${\kappa }_I\ne 0$ is shaded.

Figure 6

The shaded region indicates where ${\kappa }_I\ne 0$ for heavy quarks ($m=2000\mathrm{MeV}$) using Model A for confinement effects. The boundary of this region is also shown using an approximation appropriate for very heavy quarks ($\beta m\gg 1$) as well as for massless quarks, appropriate when $\beta m\ll 1$.

Figure 7

Contour plot of $\psi$ in the $\mu -T$ plane for heavy quarks ($m=2000\mathrm{MeV}$) using Model B for confinement effects. The region where ${\kappa }_I\ne 0$ is shaded.

Figure 8

Contour plot of ${\kappa }_I$ in the $\mu -T$ plane for heavy quarks ($m=2000\mathrm{MeV}$) using Model B for confinement effects. Contours are given in MeV with ${\alpha }_S$ set to one. The region where ${\kappa }_I\ne 0$ is shaded.

Figure 9

A comparison of the regions where ${\kappa }_I\ne 0$ for heavy quarks with Model A and Model B along with the corresponding disorder lines.

Figure 10

Contour plot of $\psi$ in the $\mu -T$ plane for Model A with massless quarks, showing where ${\mathrm{Tr}}_{F}P$ is most different from ${\mathrm{Tr}}_F{P}^{†}$. The region where ${\kappa }_I\ne 0$ is shaded.

Figure 11

Contour plot of ${\kappa }_I$ in the $\mu -T$ plane for Model A with massless quarks. Contours are given in MeV, with ${\alpha }_s$ set to one. The region where ${\kappa }_I\ne 0$ is shaded.

Figure 12

Contour plot of $\psi$ in the $\mu -T$ plane for Model B with massless quarks, showing where ${\mathrm{Tr}}_{F}P$ is most different from ${\mathrm{Tr}}_F{P}^{†}$.

Figure 13

The constituent mass $m$ (dotted line), $⟨{\mathrm{Tr}}_{F}P⟩$ (solid line), and $⟨{\mathrm{Tr}}_F{P}^{†}⟩$ (dashed line) as a function of $\mu$ for $T=50$, 150, and 210 MeV for a PNJL model using Model A for confinement effects. The constituent mass $m$ is normalized to one at $T=0$, and the Polyakov loops are normalized to one as the temperature becomes large.

Figure 14

The constituent mass $m$ (dotted line), $⟨{\mathrm{Tr}}_{F}P⟩$ (solid line), and $⟨{\mathrm{Tr}}_F{P}^{†}⟩$ (dashed line) as a function of $T$ for $\mu =0$, 250, and 325 MeV for a PNJL model using Model A for confinement effects. The constituent mass $m$ is normalized to one at $T=0$, and the Polyakov loops are normalized to one as the temperature becomes large.

Figure 15

The constituent mass $m$ (dotted line), $⟨{\mathrm{Tr}}_{F}P⟩$ (solid line), and $⟨{\mathrm{Tr}}_F{P}^{†}⟩$ (dashed line) as a function of $\mu$ for $T=50$, 150, and 210 MeV for a PNJL model using Model B for confinement effects. The constituent mass $m$ is normalized to one at $T=0$, and the Polyakov loops are normalized to one in the limit as the temperature becomes large.

Figure 16

The constituent mass $m$ (dotted line), $⟨{\mathrm{Tr}}_{F}P⟩$ (solid line), and $⟨{\mathrm{Tr}}_F{P}^{†}⟩$ (dashed line) as a function of $T$ for $\mu =0$, 250, and 325 MeV for a PNJL model using Model B for confinement effects. The constituent mass $m$ is normalized to one at $T=0$, and the Polyakov loops are normalized to one in the limit as the temperature becomes large.

Figure 17

Contour plot of $\psi$ in the $\mu -T$ plane for a PNJL model using Model A for confinement effects. The region where ${\kappa }_I\ne 0$ is shaded. The critical line and its end point are also shown.

Figure 18

Contour plot of ${\kappa }_I$ in the $\mu -T$ plane for a PNJL model using Model A for confinement effects. Contours are given in MeV, with ${\alpha }_s$ set to one. The region where ${\kappa }_I\ne 0$ is shaded. The critical line and its end point are also shown.

Figure 19

Contour plot of $\psi$ in the $\mu -T$ plane for a PNJL model using Model B for confinement effects. The region where ${\kappa }_I\ne 0$ is shaded. The critical line and its end point are also shown.

Figure 20

Contour plot of ${\kappa }_I$ in the $\mu -T$ plane for a PNJL model using Model B for confinement effects. Contours are given in MeV, with ${\alpha }_s$ set to one. The region where ${\kappa }_I\ne 0$ is shaded. The critical line and its end point are also shown.