X iv : h ep - p h / 04 03 15 3 v 3 2 1 A pr 2 00 4 Temperature Dependent Dyson - Schwinger Equation In QCD With Linear Confinement : The Critical Point

The quark mass problem is revisited, regarding the infrared behaviour of the gluon propagator with small mass (m g). Linear confinement in the instantaneous limit is ensured by a non-perturbative term 2µ 2 /(k 2 + m 2 g) −2 , µ being another scale (to be determined). Now, we propose a new (differential) form of the Dyson-Schwinger Equation (DSE) for solving the mass function Σ(p), based on a subtractive form of Renormalization in QCD. Consequently, the differential operator applied on both sides of the DSE, is to lower the degree of divergence w.r.t. the integration variable k µ , on the RHS, by T W O units. This warrants Σ(p − k) ≈ Σ(p) inside the integral since its k-dependence is no longer sensitive to the principal term (p − k) 2 in the quark propagator. This simplified DSE (preserving gauge invari-ance, reinforced by the Landau gauge) is satisfied exactly in the large p 2 limit by 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 checks (f π = 103M eV and < q ¯ q >= (223M eV) 3 , the T-dependent DSE is used in the " real time " formalism, when m 0 , m g become m t , m gt , as functions of T. The " critical " index γ works out at 1/3, and the critical temperature T c = 150M eV , while f π (T) vanishes at the critical point.