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, 7 S ( p) and 7 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 Γ , the chiral condensate displays a steep drop over a somewhat small temperature range.
1997 Academic Press
I. INTRODUCTION
From a phenomenological and experimental viewpoint, two aspects of quantum chromodynamics (QCD) are believed to be fundamental, (a) that chiral symmetry is spontaneously broken in the ground state and (b) that partons are confined. Both chiral symmetry and the phenomenon of confinement underly a common speculation, viz. that each is believed to give rise to a phase transition at a characteristic temperature or density T / (T c ) and \ / ( \ c ) respectively. As a side remark, one may note that these two phase transitions have a very different character the chiral order parameter is zero in the ordered phase, and non-zero in the broken phase, in character with the notions of condensed matter physics, while the confinement transition has an inverted structure: the non-zero value of the order parameter, or Polyakov line in this case, usually characterizes the deconfined or ordered phase. The current belief, stemming from lattice gauge simulations, is that, for QCD including quarks, the critical temperatures coincide [1]. This feature is not understood on a firm basis, and is merely an empirical observation.
Phenomenological studies over the last decade have predominantly investigated the chiral symmetry aspect of QCD. For example, the Nambu Jona Lasinio (NJL) model [2, 3] has enjoyed tremendous success, due to its mathematical tractability and ease of use. Calculations of the low-energy mesonic and baryonic sectors demonstrate the well-known fact that chiral symmetry is broken at T=0 in nature.