Effective matrix model for deconfinement in pure gauge theories

TL;DR

The authors present a two-parameter SU(N) matrix model for deconfinement in pure gauge theories, combining a one-loop perturbative potential with a tunable nonperturbative sector to fit lattice data for the pressure and latent heat in the semi-QGP window. By exploiting a uniform eigenvalue ansatz and Weyl-group invariance, they obtain analytic control for arbitrary N and derive both order-disorder and order-order interface tensions, providing a natural explanation for first-order transitions at N≥3. The model also extends to centerless groups like G(2), illustrating how maximal eigenvalue repulsion can drive confinement without center symmetry, though matching lattice Polyakov-loop data remains challenging. Overall, the framework yields good agreement with several observables, explains the generic first-order nature of deconfinement for N≥3, and points to refinements needed to capture the full renormalized Polyakov-loop behavior and QCD with dynamical quarks.

Abstract

We construct matrix models for the deconfining phase transition in SU(N) gauge theories, without dynamical quarks, at a nonzero temperature T. We generalize models with zero and one free parameter to study a model with two free parameters: besides perturbative terms ~T^4, we introduce terms ~T^2 and ~T^0. The two N-dependent parameters are determined by fitting to data from numerical simulations on the lattice for the pressure, including the latent heat. Good agreement is found for the pressure in the semi-quark gluon plasma (QGP), which is the region from Tc, the critical temperature, to about ~4 Tc. Above ~1.2 Tc, the pressure is a sum of a perturbative term, ~ +T^4, and a simple non-perturbative term, essentially just a constant times ~ -Tc^2 T^2. For the pressure, the details of the matrix model only enter within a very narrow window, from Tc to ~1.2 Tc, whose width does not change significantly with N. Without further adjustment, the model also agrees well with lattice data for the 't Hooft loop. This is notable, because in contrast to the pressure, the 't Hooft loop is sensitive to the details of the matrix model over the entire semi-QGP. For the (renormalized) Polyakov loop, though, our results disagree sharply with those from the lattice. Matrix models provide a natural and generic explanation for why the deconfining phase transition in SU(N) gauge theories is of first order not just for three, but also for four or more colors. Lastly, we consider gauge theories where there is no strict order parameter for deconfinement, such as for a G(2) gauge group. To agree with lattice measurements, in the G(2) matrix model it is essential to add terms which generate complete eigenvalue repulsion in the confining phase.

Figures (26)

• Figure 1: Plot of the trace anomaly divided by $T^2$, $(e-3p)/(8 T^2 T_c^2)$, from the data of Umeda et al., Ref. Umeda:2008bd.
• Figure 2: The Weyl chamber for three (a) and four (b) colors. See also fig. (\ref{['fig:su4realtraceplane']}).
• Figure 3: A comparison of the interaction measure, $(e-3p)/(8 T^4)$ for three colors in the models with zero Meisinger:2001cq, one Dumitru:2010mj, and two parameters, versus lattice measurements.
• Figure 4: Thermodynamics of SU(3): pressure $p/T^4$, energy density $e/(3 T^4)$, and the interaction measure times $T^2/T_c^2$, $\widetilde{\Delta}$ in Eq. (\ref{['rescaled_conf_anomaly']}). All quantities are also scaled by $1/8$.
• Figure 5: Thermodynamics of SU(4) under the uniform eigenvalue ansatz: pressure $p/T^4$, energy density $e/(3 T^4)$, and the interaction measure times $T^2/T_c^2$, $\widetilde{\Delta}$ in Eq. (\ref{['rescaled_conf_anomaly']}). All quantities are also scaled by $1/15$.