M ar 2 00 4 Temperature Dependent Dyson - Schwinger Equation In QCD With Linear Confinement : The Critical Point

The age-old problem of the quark mass function in QCD is examined afresh with special reference to the infrared behavour of the gluon propagator with small mass (m g). Linear confinement in the instantaneous limit is ensured by a non-perturbative term proportional to k −4 , obtained via two differentiations: µ 2 ∂ 2 m (m 2 g + k 2) −1 , µ being another scale (to be determined). With this input, the Dyson-Schwinger Equation (DSE) (keeping rigorous track of gauge invariance, reinforced by the Landau gauge), is solved for the mass function Σ(p) with a new sub-tractive renormalization approach which leads to a simplified differential equation for the DSE. The latter is satisfied exactly in the large p 2 limit by the ansatz Σ(p) = Σ)0/(1 + βp 2). The limit p 2 = 0 determines Σ 0 .A third limit p 2 = −m 2 0 formally defines the dynamical mass via the equation Σ(im 0) = +m 0 , together with a mass renormalization factor Z m. After two approximate checks (f π = 110M eV and < q ¯ q >= (230M eV) 3 , the formalism is extended to T-dependent DSE in the " real time " formalism, when m 0 , m g become m t , m gt , as functions of T. Taking account of the temperature dependence of the IR part of the gluon propagator, the " critical " index γ works out at 1/3, and the critical temperature T c = 150M eV .