Chiral symmetry breaking in hot matter

This series of three lectures covers (a) a basic introduction to symmetry breaking in general and chiral symmetry breaking in QCD, (b) an overview of the present status of lattice data and the knowlegde that we have at finite temperature from chiral perturbation theory. (c) Results obtained from the Nambu--Jona-Lasinio model describing static mesonic properties are discussed as well as the bulk thermodynamic quantities. Divergences that are observed in the elastic quark-antiquark scattering cross-section, reminiscent of the phenomenon of critical opalescence in light scattering, is also discussed. (d) Finally, we deal with the realm of systems out of equilibrium, and examine the effects of a medium dependent condensate in a system of interacting quarks.Comment: 62 LaTex pages, incorporating 23 figures. Lectures given at the eleventh Chris-Engelbrecht Summer School in Theoretical Physics, 4-13 February, 1998, to be published by Springer Verla

Transition rates for q q bar -> pi pi pi in a chiral model

We investigate the nature of transition rates for the hadronization process of q q bar -> Pi Pi Pi as opposed to the transition rates for q q bar -> Pi Pi, within the Nambu--Jona-Lasinio model that has manifest chiral symmetry. Feynman diagrams appropriate to this process can be classified according to the expansion in the inverse number of colors 1/N_c. Two of these types of graphs are seen to be either 's-like' or 't-like' in nature. A further graph that contains both s-like and t-like elements, and which is denoted as st-like, is also present. To describe such a process with two incoming and three outgoing particles, it is necessary to extend the number of Mandelstam invariants. It is convenient to introduce seven such variables, of which only five are independent. The cross section for two incoming particles to three outgoing particles is then reexpressed in integral form in terms of these invariants. As a function of sqrt s, the final expression is reduced to an integral over the four remaining invariants. The limits of integration, which are now non-trivial, are also discussed. The transition rate for the explicit case of u u bar -> Pi^+ Pi^- Pi^0, is evaluated numerically, assuming non-chiral pions, m_Pi = 135MeV. The rate for three pion production is found to be of the same order of magnitude as for two pion production, making this a non-negligible contribution to hadronization.Comment: 33 LaTex pages plus 1 postscript file containing 12 figures. To be published in Nuclear Physics

Confinement in the Coulomb Gauge Model

The Coulomb gauge model of QCD is studied with the introduction of a confining potential into the scalar part of the vector potential. Using a Green function formalism, we derive the self-energy for this model, which has both scalar and vector parts, $\Sigma^S(p)$ and $\Sigma^V(p)$. A rotation of these variables leads to the so-called gap and energy equations. We then analyse the divergence structure of these equations. As this depends explicitly on the form of potential, we give as examples both the linear plus Coulomb and quadratically confining potentials. The nature of the confining single particle Green function is investigated, and shown to be divergent due to the infrared singularities caused by the confining potential. Solutions to the gap equation for the simpler case of quadratic confinement are found both semi-analytically and numerically. At finite temperatures, the coupled set of equations are solved numerically in two decoupling approximations. Although chiral symmetry is found only to be exactly restored as $T\to\infty$, the chiral condensate displays a steep drop over a somewhat small temperature range.Comment: 31 pages Revtex, 2 PS files containing 11 figures, accepted for publication in Annals Of Physics (NY

Comment on ``Relativistic kinetic equations for electromagnetic, scalar and pseudoscalar interactions''

It is found that the extra quantum constraints to the spinor components of the equal-time Wigner function given in a recent paper by Zhuang and Heinz should vanish identically. We point out here the origin of the error and give an interpretation of the result. However, the principal idea of obtaining a complete equal-time transport theory by energy averaging the covariant theory remains valid. The classical transport equation for the spin density is also found to be incorrect. We give here the correct form of that equation and discuss briefly its structure.Comment: 5 pages LaTe

Mass Generation in the Supersymmetric Nambu--Jona--Lasinio Model in an External Magnetic Field

The mass generation in the (3+1)-dimensional supersymmetric Nambu-Jona-Lasinio model in a constant magnetic field is studied. It is shown that the external magnetic field catalyzes chiral symmetry breaking.Comment: LaTeX file, 6 pages. Talk given at the International Seminar dedicated to the memory of Dmitrij Volkov "Supersymmetry and Quantum Field Theory", Kharkov, Ukraine, January 5-7, 199

Derivation of transport equations for a strongly interacting Lagrangian in powers of ℏ\hbar and 1/Nc1/N_c

Transport theory for an interacting fermionic system is reviewed and applied to the chiral Lagrangian of the Nambu-Jona-Lasinio model. Two expansions must be applied: an expansion in the inverse number of colors, $1/N_c$, due to the nature of the strong coupling theory, and a semiclassical expansion, in powers of $\hbar$. The quasiparticle approximation is implemented at an early stage, and spin effects are omitted. The self-energy is evaluated, self-consistently only in the Hartree approximation, and semi-perturbatively in the collision integral. In the Hartree approximation, $O((1/N_c)^0)$, the Vlasov equation is recovered to $O(\hbar^1)$, together with an on-mass shell constraint equation, that is automatically fulfilled by the quasiparticle ansatz. The expressions for the self-energy to order $O((1/N_c))$ lead to the collision term. Here one sees explicitly that particle-antiparticle creation and annihilation processes are suppressed that would otherwise be present, should an off-shell energy spectral function be admitted. A clear identification of the $s$, $t$ and $u$ channel scattering processes in connection with the self-energy graphs is made and the origin of the mixed terms is made evident. Finally, after ordering according to powers in $\hbar$, a Boltzmann-like form for the collision integral is obtained

Responses of quark condensates to the chemical potential

The responses of quark condensates to the chemical potential, as a function of temperature T and chemical potential \mu, are calculated within the Nambu--Jona-Lasinio (NJL) model. We compare our results with those from the recent lattice QCD simulations [QCD-TARO Collaboration, Nucl. Phys. B (Proc. Suppl.) 106, 462 (2002)]. The NJL model and lattice calculations show qualitatively similar behavior, and they will be complimentary ways to study hadrons at finite density. The behavior above T_c requires more elaborated analyses.Comment: 3 pages, 2 figs, based on a contribution to the Prof. Osamu Miyamura memorial symposium, Hiroshima University, Nov. 16-17, 2001; slightly revised, accepted for publication in Physical Review

A Study of the Nambu--Jona-Lasinio Model on the Lattice

We present our full analysis of the two flavor Nambu--Jona-Lasinio model with $SU(2) \times SU(2)$ chiral symmetry on the four--dimensional hypercubic lattice with naive and Wilson fermions. We find that this model is an excellent toy field theory to investigate issues related to lattice QCD. We use the large $N$ approximation to leading order in $1/N$ to obtain non perturbative analytical results over almost the whole parameter range. By using numerical simulations we estimate that the size of the $1/N$ corrections for most of the quantities we consider are small and in this way we strengthen the validity of the leading order large $N$ calculations. We obtain results regarding the approach to the continuum chiral limit, the effects of the zero momentum fermionic modes on finite lattices and the scalar and pseudoscalar spectrum. Note: The full ps file of this preprint is also available via anonymous ftp to ftp.scri.fsu.edu. To get the ps file, ftp to this address and use for username "anonymous" and for password your complete E-mail address. The file is in the directory pub/vranas (to go to that directory type: cd pub/vranas) and is called NJL_long.ps (to get it type: get NJL_long.ps)Comment: 35 pages, LaTex file. (Added section with title: "The zero pion mass line on a finite lattice at large $N$".

Density Effect on Hadronization of a Quark Plasma

The hadronization cross section in a quark plasma at finite temperature and density is calculated in the framework of Nambu--Jona-lasinio model with explicit chiral symmetry breaking. In apposition to the familiar temperature effect, the quark plasma at high density begins to hadronize suddenly. It leads to a sudden and strong increase of final state pions in relativistic heavy ion collisions which may be considered as a clear signature of chiral symmetry restoration.Comment: Latex2e, 11 pages, 7 Postscript figures, submitted to Phys. Rev.