Renormalized Polyakov Loops, Matrix Models and the Gross-Witten Point

The values of renormalized Polyakov loops in the three lowest representations of SU(3) were measured numerically on the lattice. We find that in magnitude, condensates respect the large-N property of factorization. In several ways, the deconfining phase transition for N=3 appears to be like that in the N=infinity matrix model of Gross and Witten. Surprisingly, we find that the values of the renormalized triplet loop are described by an SU(3) matrix model, with an effective action dominated by the triplet loop. Future numerical simulations with a larger number of colors should be able to show whether or not the deconfining phase transition is close to the Gross-Witten point.Comment: 9 pages, 3 figures, Combined contribution to proceedings of Strong and Electroweak Matter 2004 (SEWM 2004), Helsinki, Finland, 16-19 June 200

Coherent Topological Charge Structure in CPN−1CP^{N-1} Models and QCD

In an effort to clarify the significance of the recent observation of long-range topological charge coherence in QCD gauge configurations, we study the local topological charge distributions in two-dimensional $CP^{N-1}$ sigma models, using the overlap Dirac operator to construct the lattice topological charge. We find long-range sign coherence of topological charge along extended one-dimensional structures in two-dimensional spacetime. We discuss the connection between the long range topological structure found in $CP^{N-1}$ and the observed sign coherence along three-dimensional sheets in four-dimensional QCD gauge configurations. In both cases, coherent regions of topological charge form along membrane-like surfaces of codimension one. We show that the Monte Carlo results, for both two-dimensional and four-dimensional gauge theory, support a view of topological charge fluctuations suggested by Luscher and Witten. In this framework, the observed membranes are associated with boundaries between ``k-vacua,'' characterized by an effective local value of $\theta$ which jumps by $\pm 2\pi$ across the boundary.Comment: 26 page

Influence of the U(1)_A Anomaly on the QCD Phase Transition

The SU(3)_{r} \times SU(3)_{\ell} linear sigma model is used to study the chiral symmetry restoring phase transition of QCD at nonzero temperature. The line of second order phase transitions separating the first order and smooth crossover regions is located in the plane of the strange and nonstrange quark masses. It is found that if the U(1)_{A} symmetry is explicitly broken by the U(1)_{A} anomaly then there is a smooth crossover to the chirally symmetric phase for physical values of the quark masses. If the U(1)_{A} anomaly is absent, then there is a phase transition provided that the \sigma meson mass is at least 600 MeV. In both cases, the region of first order phase transitions in the quark mass plane is enlarged as the mass of the \sigma meson is increased.Comment: 5 pages, 3 figures, Revtex, discussion extended and references added. To appear in PR

The O(N) Model at Finite Temperature: Renormalization of the Gap Equations in Hartree and Large-N Approximation

The temperature dependence of the sigma meson and pion masses is studied in the framework of the O(N) model. The Cornwall-Jackiw-Tomboulis formalism is applied to derive gap equations for the masses in the Hartree and large-N approximations. Renormalization of the gap equations is carried out within the cut-off and counter-term renormalization schemes. A consistent renormalization of the gap equations within the cut-off scheme is found to be possible only in the large-N approximation and for a finite value of the cut-off. On the other hand, the counter-term scheme allows for a consistent renormalization of both the large-N and Hartree approximations. In these approximations, the meson masses at a given nonzero temperature depend in general on the choice of the cut-off or renormalization scale. As an application, we also discuss the in-medium on-shell decay widths for sigma mesons and pions at rest.Comment: 21 pages, 6 figures, typos corrected and refs. added, accepted in Journal of Physics

Mesoscopic QCD and the Theta Vacua

The partition function of QCD is analyzed for an arbitrary number of flavors, N_f, and arbitrary quark masses including the contributions from all topological sectors in the Leutwyler--Smilga regime. For given N_f and arbitrary vacuum angle, \theta, the partition function can be reduced to N_f-2 angular integrations of single Bessel functions. For two and three flavors, the \theta dependence of the QCD vacuum is studied in detail. For N_f= 2 and 3, the chiral condensate decreases monotonically as \theta increases from zero to \pi and the chiral condensate develops a cusp at \theta=\pi for degenerate quark masses in the macroscopic limit. We find a discontinuity at \theta=\pi in the first derivative of the energy density with respect to \theta for degenerate quark masses. This corresponds to the first--order phase transition in which CP is spontaneously broken, known as Dashen's phenomena.Comment: 31 pages, revtex, 10 figures, final version to appear in Nucl. Phys.

Subprocess Size in Hard Exclusive Scattering

The interaction region of hard exclusive hadron scattering can have a large transverse size due to endpoint contributions, where one parton carries most of the hadron momentum. The endpoint region is enhanced and can dominate in processes involving multiple scattering and quark helicity flip. The endpoint Fock states have perturbatively short lifetimes and scatter softly in the target. We give plausible arguments that endpoint contributions can explain the apparent absence of color transparency in fixed angle exclusive scattering and the dimensional scaling of transverse rho photoproduction at high momentum transfer, which requires quark helicity flip. We also present a quantitative estimate of Sudakov effects.Comment: 16 pages, 4 figures, JHEP style; v2: quantitative estimate of Sudakov effects and more detailed discussion of endpoint behaviour of meson distribution amplitude added, few other clarifications, version to appear in Phys. Rev.

Deconfining Phase Transition as a Matrix Model of Renormalized Polyakov Loops

We discuss how to extract renormalized from bare Polyakov loops in SU(N) lattice gauge theories at nonzero temperature in four spacetime dimensions. Single loops in an irreducible representation are multiplicatively renormalized without mixing, through a renormalization constant which depends upon both representation and temperature. The values of renormalized loops in the four lowest representations of SU(3) were measured numerically on small, coarse lattices. We find that in magnitude, condensates for the sextet and octet loops are approximately the square of the triplet loop. This agrees with a large $N$ expansion, where factorization implies that the expectation values of loops in adjoint and higher representations are just powers of fundamental and anti-fundamental loops. For three colors, numerically the corrections to the large $N$ relations are greatest for the sextet loop, $\leq 25$; these represent corrections of $\sim 1/N$ for N=3. The values of the renormalized triplet loop can be described by an SU(3) matrix model, with an effective action dominated by the triplet loop. In several ways, the deconfining phase transition for N=3 appears to be like that in the $N=\infty$ matrix model of Gross and Witten.Comment: 24 pages, 7 figures; v2, 27 pages, 12 figures, extended discussion for clarity, results unchange