Confinement at Weak Coupling

The free energy of U(N) and SU(N) gauge theory was recently found to be of order N^0 to all orders of a perturbative expansion about a center-symmetric orbit of vanishing curvature. Here I consider extended models for which this expansion is perturbatively stable. The extreme case of an SU(2) gauge theory whose configuration space is restricted to center-symmetric orbits has recently been investigated on the lattice hep-lat/0509156. In extension of my talk, a discussion and possible interpretation of the observed finite temperature phase transition is given. The transfer matrix of constrained SU(N) lattice gauge theory is constructed for any finite temperature.Comment: 8 pages, no figures, updated talk given at LC2005 in Cairns, Australi

The Finite Temperature SU(2) Savvidy Model with a Non-trivial Polyakov Loop

We calculate the complete one-loop effective potential for SU(2) gauge bosons at temperature T as a function of two variables: phi, the angle associated with a non-trivial Polyakov loop, and H, a constant background chromomagnetic field. Using techniques broadly applicable to finite temperature field theories, we develop both low and high temperature expansions. At low temperatures, the real part of the effective potential V_R indicates a rich phase structure, with a discontinuous alternation between confined (phi=pi) and deconfined phases (phi=0). The background field H moves slowly upward from its zero-temperature value as T increases, in such a way that sqrt(gH)/(pi T) is approximately an integer. Beyond a certain temperature on the order of sqrt(gH), the deconfined phase is always preferred. At high temperatures, where asymptotic freedom applies, the deconfined phase phi=0 is always preferred, and sqrt(gH) is of order g^2(T)T. The imaginary part of the effective potential is non-zero at the global minimum of V_R for all temperatures. A non-perturbative magnetic screening mass of the form M_m = cg^2(T)T with a sufficiently large coefficient c removes this instability at high temperature, leading to a stable high-temperature phase with phi=0 and H=0, characteristic of a weakly-interacting gas of gauge particles. The value of M_m obtained is comparable with lattice estimates.Comment: 28 pages, 5 eps figures; RevTeX 3 with graphic

Fluctuations and the QCD phase diagram

In this contribution the role of quantum fluctuations for the QCD phase diagram is discussed. This concerns in particular the importance of the matter back-reaction to the gluonic sector. The impact of these fluctuations on the location of the confinement/deconfinement and the chiral transition lines as well as their interrelation are investigated. Consequences of our findings for the size of a possible quarkyonic phase and location of a critical endpoint in the phase diagram are drawn.Comment: 7 pages, 3 figures, to appear in Physics of Atomic Nucle

Phenomenological Equations of State for the Quark-Gluon Plasma

Two phenomenological models describing an 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 - 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 N >= 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 - 5T_d. The addition of massless quarks in 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.Comment: 22 pages, RevTeX, 9 eps file

Calculation of the Hidden Symmetry Operator in PT-Symmetric Quantum Mechanics

In a recent paper it was shown that if a Hamiltonian H has an unbroken PT symmetry, then it also possesses a hidden symmetry represented by the linear operator C. The operator C commutes with both H and PT. The inner product with respect to CPT is associated with a positive norm and the quantum theory built on the associated Hilbert space is unitary. In this paper it is shown how to construct the operator C for the non-Hermitian PT-symmetric Hamiltonian $H={1\over2}p^2+{1\over2}x^2 +i\epsilon x^3$ using perturbative techniques. It is also shown how to construct the operator C for $H={1\over2}p^2+{1\over2}x^2-\epsilon x^4$ using nonperturbative methods

Dirac eigenvalues and eigenvectors at finite temperature

We investigate the eigenvalues and eigenvectors of the staggered Dirac operator in the vicinity of the chiral phase transition of quenched SU(3) lattice gauge theory. We consider both the global features of the spectrum and the local correlations. In the chirally symmetric phase, the local correlations in the bulk of the spectrum are still described by random matrix theory, and we investigate the dependence of the bulk Thouless energy on the simulation parameters. At and above the critical point, the properties of the low-lying Dirac eigenvalues depend on the $Z_3$-phase of the Polyakov loop. In the real phase, they are no longer described by chiral random matrix theory. We also investigate the localization properties of the Dirac eigenvectors in the different $Z_3$-phases.Comment: Lattice 2000 (Finite Temperature), 5 page

Phase diagram and critical properties within an effective model of QCD: the Nambu-Jona-Lasinio model coupled to the Polyakov loop

We investigate the phase diagram of the so-called Polyakov--Nambu--Jona-Lasinio model at finite temperature and non-zero chemical potential with three quark flavors. Chiral and deconfinement phase transitions are discussed and the relevant order-like parameters are analyzed. The results are compared with simple thermodynamic expectations and lattice data. We present the phase diagram in the ($T,\,\mu_B)$ plane, paying special attention to the critical end point: as the strength of the flavor-mixing interaction becomes weaker, the critical end point moves to low temperatures and can even disappear.Comment: 46 pages, 11 figures, 3 table

Effects of mesonic correlations in the QCD phase transition

The finite temperature phase transition of strongly interacting matter is studied within a nonlocal chiral quark model of the NJL type coupled to a Polyakov loop. In contrast to previous investigations which were restricted to the mean-field approximation, mesonic correlations are included by evaluating the quark-antiquark ring sum. For physical pion masses, we find that the pions dominate the pressure below the phase transition, whereas above T_c the pressure is well described by the mean-field approximation result. For large pion masses, as realized in lattice simulations, the meson effects are suppressed.Comment: 11 pages, 4 figures; version accepted for publication in Yad. Fiz., text extended, 1 figure adde

Calorons and localization of quark eigenvectors in lattice QCD

We analyze the localization properties for eigenvectors of the Dirac operator in quenched lattice QCD in the vicinity of the deconfinement phase transition. Studying the characteristic differences between the Z_3 sectors above the critical temperature T_c, we find indications for the presence of calorons.Comment: 4 pages, 4 figure