A Phenomenological Treatment of Chiral Symmetry Restoration and Deconfinement

A phenomenological expression for the thermodynamic potential of gluons and quarks is constructed which incorporates the features of deconfinement and chiral symmetry restoration known from lattice simulations. The thermodynamic potential is a function of the Polyakov loop and chiral condensate expectation values. The gluonic sector uses a successful model for pure (SU(N_c)) gauge theories in which the Polyakov loop eigenvalues are the fundamental order parameters for deconfinement. The quark sector is given by a Nambu-Jona-Lasinio model in which a constant background (A_0) field couples the chiral condensate to the Polyakov loop. We consider the case of (N_f = 2) in detail. For two massless quarks, we find a second order chiral phase transition. Confinement effects push the transition to higher temperatures, but the entropy associated with the gluonic sector acts in the opposite direction. For light mass quarks, only a rapid crossover occurs. For sufficiently heavy quarks, a first order deconfinement transition emerges. This simplest model has one adjustable parameter, which can be set from the chiral transition temperature for light quarks. It predicts all thermodynamic quantities as well as the behavior of the chiral condensate and the Polyakov loop over a wide range of temperatures.Comment: 3 pages, 4 eps figures, Lattice 2002 conference contribution, Lattice2002(nonzerot

PT symmetry and large-N models

Recently developed methods for PT-symmetric models can be applied to quantum-mechanical matrix and vector models. In matrix models, the calculation of all singlet wave functions can be reduced to the solution a one-dimensional PT-symmetric model. The large-N limit of a wide class of matrix models exists, and properties of the lowest-lying singlet state can be computed using WKB. For models with cubic and quartic interactions, the ground state energy appears to show rapid convergence to the large-N limit. For the special case of a quartic model, we find explicitly an isospectral Hermitian matrix model. The Hermitian form for a vector model with O(N) symmetry can also be found, and shows many unusual features. The effective potential obtained in the large-N limit of the Hermitian form is shown to be identical to the form obtained from the original PT-symmetric model using familiar constraint field methods. The analogous constraint field prescription in four dimensions suggests that PT-symmetric scalar field theories are asymptotically free.Comment: 15 pages, to be published in J. Phys. A special issue on Pseudo Hermitian Hamiltonians in Quantum Physic

Landau-Ginsberg Theory of Quark Confinement

We describe the SU(3) deconfinement transition using Landau-Ginsberg theory. Drawing on perturbation theory and symmetry principles, we construct the free energy as a function of temperature and the Polyakov loop. Once the two adjustable parameters of the model are fixed, the pressure p, energy epsilon and Polyakov loop expectation value P_F are calculable functions of temperature. An excellent fit to the continuum extrapolation of lattice thermodynamics data can be achieved. In an extended form of the model, the glueball potential is responsible for breaking scale invariance at low temperatures. Three parameters are required, but the glueball mass and the gluon condensate are calculable functions of temperature, along with p, epsilon and P_F.Comment: Lattice99(Finite Temperature and Density) <= added keywords only change in revised version, sorry; 3 pages, LaTeX with espcrc2.sty and epsf.tex. Talk presented at Lattice99, Pisa, 29 June - 3 July 1999, to appear in Nucl. Phys. B (Proc.Suppl.

Finite Temperature Quark Confinement

Confinement may be more easily demonstrated at finite temperature using the Polyakov loop than at zero temperature using the Wilson loop. A natural mechanism for confinement can arise via the coupling of the adjoint Polyakov loop to F_{mu nu}^2. We demonstrate this mechanism with a one-loop calculation of the effective potential for SU(2) gluons in a background field consisting of a non-zero color magnetic field and a non-trivial Polyakov loop. The color magnetic field drives the Polyakov loop to non-trivial behavior, and the Polyakov loop can remove the well-known tachyonic mode associated with the Saviddy vacuum. Minimizing the real part of the effective potential leads to confinement, as determined by the Polyakov loop. Unfortunately, we cannot arrange for simultaneous stability and confinement for this simple class of field configurations. We show for a large class of abelian background fields that at one loop tachyonic modes are necessary for confinement.Comment: 15 pages, 7 figures, RevTe

Polyakov Loops, Z(N) Symmetry, and Sine-Law Scaling

We construct an effective action for Polyakov loops using the eigenvalues of the Polyakov loops as the fundamental variables. We assume Z(N) symmetry in the confined phase, a finite difference in energy densities between the confined and deconfined phases as $T\to 0$, and a smooth connection to perturbation theory for large $T$. The low-temperature phase consists of $N-1$ independent fields fluctuating around an explicitly Z(N) symmetric background. In the low-temperature phase, the effective action yields non-zero string tensions for all representations with non-trivial $N$-ality. Mixing occurs naturally between representations of the same $N$-ality. Sine-law scaling emerges as a special case, associated with nearest-neighbor interactions between Polyakov loop eigenvalues.Comment: Talk presented at Lattice2004(topology), Fermilab, June 21-26, 2004, 3 page

The sign problem and Abelian lattice duality

For a large class of Abelian lattice models with sign problems, including the case of non-zero chemical potential, duality maps models with complex actions into dual models with real actions. For extended regions of parameter space, calculable for each model, duality resolves the sign problem for both analytic methods and computer simulations. Explicit duality relations are given for models for spin and gauge models based on Z(N) and U(1) symmetry groups. The dual forms are generalizations of the Z(N) chiral clock model and the lattice Frenkel-Kontorova model, respectively. From these equivalences, rich sets of spatially-modulated phases are found in the strong-coupling region of the original models.Comment: Latex, 7 pages, 1 figure. Presented at the 31st International Symposium on Lattice Field Theory (Lattice 2013), 29 July - 3 August 2013, Mainz, German