Phenomenological equations of state for the quark-gluon plasma

Two phenomenological models describing an $\mathrm{SU}(N)$ quark-gluon plasma are presented. The first is obtained from high temperature expansions of the free energy of a massive gluon, while the second is derived by demanding color neutrality over a certain length scale. Each model has a single free parameter, exhibits behavior similar to lattice simulations over the range ${T}_{d}\ensuremath{-}{5T}_{d},$ and has the correct blackbody behavior for large temperatures. The $N=2$ deconfinement transition is second order in both models, while $N=3, 4,$ and $5$ are first order. Both models appear to have a smooth large-N limit. For $Ng~4,$ it is shown that the trace of the Polyakov loop is insufficient to characterize the phase structure; the free energy is best described using the eigenvalues of the Polyakov loop. In both models, the confined phase is characterized by a mutual repulsion of Polyakov loop eigenvalues that makes the Polyakov loop expectation value zero. In the deconfined phase, the rotation of the eigenvalues in the complex plane towards $1$ is responsible for the approach to the blackbody limit over the range ${T}_{d}\ensuremath{-}{5T}_{d}.$ The addition of massless quarks in $\mathrm{SU}(3)$ breaks $Z(3)$ symmetry weakly and eliminates the deconfining phase transition. In contrast, a first-order phase transition persists with sufficiently heavy quarks.