Polyakov loop models in the large $N$ limit: Correlation function and screening masses

We explore the ’t Hooft-Veneziano limit of the Polyakov loop models at finite baryon chemical potential. Using methods developed by us earlier we calculate the two- and $N$-point correlation functions of the Polyakov loops. This gives a possibility to compute the various potentials in the confinement phase and to derive the screening masses outside the confinement region. In particular, we establish the existence of complex masses and an oscillating decay of correlations in a certain range of parameters. Furthermore, it is shown that the calculation of the $N$-point correlation function in the confinement phase reduces to the geometric median problem. This leads to a large $N$ analog of the $Y$ law for the baryon potential.

Figure 1

Cross sections of the phase diagram of the SU(N) model. Left: $\mu –\alpha$ coordinates at fixed $\beta d=0.1$. Right: $\mu –\beta d$ coordinates at fixed $\alpha =0.2$. The blue curve shows the critical line of the third order phase transition. See text for details.

Figure 2

Oscillating decay of the correlation function with a distance in region II of the phase diagram. Left: $m_i/m_r\approx 7$, $\alpha =0.05$, $\beta d=0.1$, $\mu =2.595$. Right: $m_i/m_r\approx 30$, $\alpha =0.01$, $\beta d=0.1$, $\mu =4.1935$.

Figure 3

Real (red) and imaginary (blue) parts of the screening masses vs $\mu$ at fixed $\beta d=0.1$ and fixed $\alpha =0.7$ (top left), $\alpha =0.2$ (top right), $\alpha =0.1$ (bottom left), $\alpha =0.05$ (bottom right). See text for details.