A model for QCD at high density and large quark mass

We study the high density region of QCD within an effective model obtained in the frame of the hopping parameter expansion and choosing Polyakov-type loops as the main dynamical variables representing the fermionic matter. To get a first idea of the phase structure, the model is analyzed in strong coupling expansion and using a mean field approximation. In numerical simulations, the model still shows the so-called sign problem, a difficulty peculiar to nonzero chemical potential, but it permits the development of algorithms which ensure a good overlap of the Monte Carlo ensemble with the true one. We review the main features of the model and present calculations concerning the dependence of various observables on the chemical potential and on the temperature, in particular, of the charge density and the diquark susceptibility, which may be used to characterize the various phases expected at high baryonic density. We obtain in this way information about the phase structure of the model and the corresponding phase transitions and crossover regions, which can be considered as hints for the behavior of nonzero density QCD.

Figure 1

Periodic lattice, loops, temporal gauge. In the maximal temporal gauge also the links of the basis line are fixed to 1 up to the rightmost one.

Figure 2

Comparison with strong coupling at $\beta =3$ (upper plot) and $\beta =5$ (lower plot), $4^4$ lattice. Full symbols denote $\mathrm{Re}P$, empty symbols $\mathrm{Re}{P}^{*}$, the lines show the corresponding strong coupling results.

Figure 3

Comparison with strong coupling, $\beta =5.5$ (upper plot) and $\beta =5.6$ (lower plot), $6^4$ lattice. Symbols as in Fig. 2.

Figure 4

Mean field phase diagram (abscissa $\mu$, ordinate $\gamma =N_{\tau }aT$).

Figure 5

Tentative phase diagram in $T$ and $\mu$ for various $\kappa$.

Figure 6

Fixed mass plane phase diagram; dotted arrows indicate sequences of runs.

Figure 7

Paths contributing to quark and diquark “propagators.”

Figure 8

Data taken in the plane $\beta$ vs $\mu$ for fixed $\kappa =0.12$.

Figure 9

Baryonic density versus $\beta$ at fixed $\mu$.

Figure 10

Baryonic density versus $\mu$ at fixed $\beta$.

Figure 11

Landscape of the baryonic density. The color scale (right) is based on ${log}_{10}(n_B)$.

Figure 12

Landscape of the baryon density susceptibility. The color scale (right) is based on ${log}_{10}({\chi }_{n_B})$.

Figure 13

Polyakov loop susceptibility versus $\beta$ at fixed $\mu$.

Figure 14

Polyakov loop susceptibility versus $\mu$ at fixed $\beta$.

Figure 15

Landscape of the Polyakov loop susceptibility. The color scale (right) is based on ${log}_{10}({\chi }_P)$.

Figure 16

3D view of Fig. 15.

Figure 17

Phase diagram in the $\beta (\mathrm{\text{or}}T/T_c)\mathrm{\text{−}}{\mu }_{\mathrm{phys}}/T_c$ QCD plane. The dotted straight lines correspond to constant $\mu$, the dashed lines to constant $\beta$. The blobs, shadowing, and other features are explained in the text.

Figure 18

Polyakov loop “histogram” $H_{\Delta }(x,y)$ of Eq. (4.9) versus $\mu$ at $\beta =5.65$.

Figure 19

Polyakov loop “histogram” $H_{\Delta }(x,y)$ of Eq. (4.9) versus $\beta$ at $\mu =0.70$.

Figure 20

Real part of the Polyakov loop “distribution” $T_{\Delta }(x,y)$ of Eq. (4.10) versus $\mu$ at $\beta =5.65$ fixed.

Figure 21

Real part of the Polyakov loop “distribution” $T_{\Delta }(x,y)$ of Eq. (4.10) versus $\beta$ at $\mu =0.70$ fixed.

Figure 22

Imaginary part of the Polyakov loop “distribution” $T_{\Delta }(x,y)$ of Eq. (4.10) versus $\mu$ at $\beta =5.65$ fixed.

Figure 23

Imaginary part of the Polyakov loop “distribution” $T_{\Delta }(x,y)$ of Eq. (4.10) versus $\beta$ at $\mu =0.70$ fixed.

Figure 24

Plaquette averages versus $\mu$ at fixed $\beta =5.65$.

Figure 25

Plaquette averages versus $\beta$ at fixed $\mu =0.70$.

Figure 26

Topological susceptibility average versus $\mu$ at fixed $\beta =5.65$.

Figure 27

Topological susceptibility average versus $\beta$ at fixed $\mu =0.70$.

Figure 28

Diquark susceptibility average versus $\mu$ at fixed $\beta =5.65$.

Figure 29

Diquark susceptibility average versus $\beta$ at fixed $\mu =0.70$.