Matrix model for deconfinement in a $SU(N_c)$ gauge theory in $2+1$ dimensions

We use an effective matrix model to study deconfinement in a pure $SU(N_c)$ gauge theory, without quarks, in $d=2+1$ dimensions. Expanding about a constant background ${\mathcal{A}}_0$ field, we construct an effective potential for the eigenvalues of the thermal Wilson line for general $N_c$ and in the large-$N_c$ limit. The numerical results are presented using one, two, and four free parameters, which are determined by fitting directly to the lattice data for the pressure. The matrix model shows a good agreement with numerical lattice simulations for the pressure and the interaction measure, starting from the perturbative limit up to the critical temperature. For the pressure, the details of ${\mathcal{A}}_0$-dependent nonperturbative terms are relevant only in a narrow transition region below $\sim 1.2T_d$. This is also the range where the Polyakov loop deviates notably from one. In accordance with the lattice results we find that, up to a trivial factor ${N_c}^2-1$, there is only a mild dependence on the number of colors.

Figure 1

Boundaries for values of the Polyakov loop for $N_c=3$ (left top), 4 (right top), 5 (left bottom), 6 (right bottom). The dashed lines indicate the region between the perturbative vacuum at $l=1$ and the confining vacuum at $l=0$.

Figure 2

The perturbative potential ${\tilde{V}}_{\mathrm{pt}}$ (solid line), the Vandermonde nonperturbative term ${\check{V}}_{\mathrm{Vdm}}$ (dashed line), and the linear nonperturbative term ${\tilde{V}}_{\text{lin}}$ (dotted line) as a function of $q_{ij}$.

Figure 3

The Polyakov loop as a function of $s$ in the uniform eigenvalue ansatz for $N_c=3$ (left top), $N_c=4$ (right top), $N_c=5$ (left bottom), $N_c=6$ (right bottom).

Figure 4

The perturbative potential $V_{\mathrm{pt}}(s)$ as a function of $s$ in the uniform eigenvalue ansatz. The plots are shown for $N_c=2$ (dashed line), $N_c=3$ (dashed-dotted line), $N_c=4$ (dotted line), and in the large-$N_c$ limit (solid line).

Figure 5

The Vandermonde term $V_{\mathrm{Vdm}}(s)$ as a function of $s$ for $N_c=2$ (dashed line), $N_c=3$ (dashed-dotted line), $N_c=4$ (dotted line), and in the large-$N_c$ limit (solid line).

Figure 6

The linear term $V_{\text{lin}}(s)$ as a function of $s$ for $N_c=2$ (dashed line), $N_c=3$ (dashed-dotted line), $N_c=4$ (dotted line), and in the large-$N_c$ limit (solid line).

Figure 7

The effective potential $V_{\text{eff}}(s)$ for $N_c=6$, and its derivative $V_{\text{eff}}^{\prime }(s)$ as a function of the variable $s$. The plots are obtained by using the Vandermonde term for three different values of $a:a<a_d$ (dashed line) represents the semi-QGP, at $a=a_d$ (solid line) the phase transition to confinement takes place, and for $a>a_d$ (dotted line) the system is in the confined phase.

Figure 8

Left panel: the minimum of the effective potential as a function of $a$, $s_{\min }(a)$, for the linear term (solid line), and the Vandermonde determinant (dashed line). Right panel: the potential at the minimum as a function of $a$, $V_{\min }(a)$. In the upper panels we show the results for $N_c=2$, and in the lower panels for $N_c=3$.

Figure 9

Left panel: the minimum of the effective potential as a function of $a$, $s_{\min }(a)$, for the linear term (solid line), and the Vandermonde determinant (dashed line). Right panel: the potential at the minimum as a function of $a$, $V_{\min }(a)$. In the upper panels we show the results for $N_c=4$, and in the lower panels for $N_c=5$.

Figure 10

Left panel: the minimum of the effective potential for $N_c=6$ as a function of $a$, $s_{\min }(a)$, for the linear term (solid line), and the Vandermonde determinant (dashed line). Right panel: the potential at the minimum as a function of $a$, $V_{\min }(a)$.

Figure 11

Lattice data of Ref. [1] for $SU(2{)}_c$ close to the critical temperature. Left panel: the pressure with the tangent to the inflection point. Right panel: the susceptibility ${\chi }_p$ obtained from the lattice pressure. The critical temperature $T_d$ is denoted by a square, the intercept temperature $T_i$ by a circle, and the inflection point $T_p$ by a diamond.

Figure 12

Pressure (upper panels) and interaction measure (lower panels) scaled by ${N_c}^2-1=3$ for $SU(N_c=2)$. Lattice data, taken from Ref. [1], are denoted by dots. For the interaction measure we also show the lattice error bars, while for the pressure the error bars are smaller than the symbols. The results of the matrix model are obtained by using the Vandermonde determinant (left panel), and the linear term (right panel) within the one-parameter model (dashed line), two-parameter model (solid line), and four-parameter fit (dotted line). The horizontal lines represent the perturbative limit of the pressure $c=\zeta (3)/2\pi$.

Figure 13

Pressure (upper panels) and interaction measure (lower panels) scaled by ${N_c}^2-1=8$ for $SU(N_c=3)$. Lattice data are denoted by dots. The results of the matrix model are obtained by using the Vandermonde determinant (left panel), and the linear term (right panel) within the one-parameter model (dashed line), two-parameter model (solid line), and four-parameter fit (dotted line). The horizontal lines represent the Stefan-Boltzmann limit of the pressure $c=\zeta (3)/2\pi$.

Figure 14

Pressure (upper panels) and interaction measure (lower panels) scaled by ${N_c}^2-1=15$ for $SU(N_c=4)$. Lattice data [1] are denoted by dots. The results of the matrix model are obtained by using the Vandermonde determinant (left panel), and the linear term (right panel) within the one-parameter model (dashed line), two-parameter model (solid line), and four-parameter fit (dotted line). The horizontal lines represent the perturbative limit of the pressure $c=\zeta (3)/2\pi$.

Figure 15

Pressure (upper panels) and interaction measure (lower panels) scaled by ${N_c}^2-1=24$ for $SU(N_c=5)$. Lattice data [1] are denoted by dots. The results of the matrix model are obtained by using the Vandermonde determinant (left panel), and the linear term (right panel) within the one-parameter model (dashed line), two-parameter model (solid line), and four-parameter fit (dotted line). The horizontal lines represent the perturbative limit of the pressure $c=\zeta (3)/2\pi$.

Figure 16

Pressure (upper panels) and interaction measure (lower panels) scaled by ${N_c}^2-1=35$ for $SU(N_c=6)$. Lattice data [1] are denoted by dots. The results of the matrix model are obtained by using the Vandermonde determinant (left panel), and the linear term (right panel) within the one-parameter model (dashed line), two-parameter model (solid line), and four-parameter fit (dotted line). The horizontal lines represent the perturbative limit of the pressure $c=\zeta (3)/2\pi$.

Figure 17

The Polyakov loop for $SU(2{)}_c$ (upper panels) and $SU(3{)}_c$ (lower panels). The results are obtained for the Vandermonde determinant (left panel), and the linear term (right panel) in the one-parameter model (dashed line), two-parameter model (solid line), and four-parameter fit (dotted line).

Figure 18

The Polyakov loop for $SU(4{)}_c$ (upper panels) and $SU(5{)}_c$ (lower panels). The results are obtained for the Vandermonde determinant (left panel), and the linear term (right panel) in the one-parameter model (dashed line), two-parameter model (solid line), and four-parameter fit (dotted line).

Figure 19

The Polyakov loop for $SU(6{)}_c$ obtained for the Vandermonde determinant (left panel), and the linear term (right panel) in the one-parameter model (dashed line), two-parameter model (solid line), and four-parameter fit (dotted line).

Figure 20

Shift in the critical temperature, $T_d-T_i$, as a function of $N_c$, where $T_i$ is the estimated temperature, at which the pressure vanishes. We also show three different fits: $1/(N_c-1)$ (dashed line), $1/({N_c}^2-1)$ (solid line), and $1/({N_c}^3-1)$ (dotted line).

Figure 21

The perturbative potential $V_{\mathrm{pt}}(s)$ (solid), the Vandermonde term $V_{\mathrm{Vdm}}(s)$ (dashed), and the linear term $V_{\text{lin}}(s)$ (dotted) as a function of $s$ in the uniform eigenvalue ansatz, for $N_c=3$ (left top), $N_c=4$ (right top), $N_c=5$ (left bottom), $N_c=6$ (right bottom).

Figure 22

The effective potential $V_{\text{eff}}(s)$ for $N_c=6$ using the Vandermonde term $V_{\mathrm{Vdm}}(s)$, in the semi-QGP (dashed), at the phase transition (solid), and in the confined phase (dotted).

Figure 23

Density plot of the perturbative potential $V_{\mathrm{pt}}[\mathbf{q}(s,s_1)]$ for $N_c=3$. The $s$ axis points in the direction of the uniform eigenvalue ansatz, while the $s_1$ axis points in the transverse direction.

Figure 24

Density plot of the perturbative potential for $N_c=4$. The potential is plotted for different slices in the $s_{1,}{s}_2$ plane perpendicular to ${\mathbf{q}}_c$, at $s=0$ (a), $s=1.35$ (b) and $s=2$ (c).

Figure 25

Density plot of the perturbative potential for $N_c=4$. The potential is plotted for different $s,s_1$ slices, varying the distance in the perpendicular direction: $s_2=0$ (a), $s_2=0.5$ (b), $s_2=1$ (c), $s_2=1.5$ (d). (To be continued in Fig. 26.)

Figure 26

Continuation of Fig. 25 for $s_2=2$ (a), $s_2=2.5$ (b), $s_2=3$ (c), $s_2=3.5$ (d).