ar X iv : h ep - p h / 04 03 15 3 v 5 2 3 A ug 2 00 4 T - Dependent Dyson - Schwinger Equation In IR Regime Of QCD : The Critical Point

The quark mass function Σ(p) in QCD is revisited, using a gluon propagator in the form 1/(k 2 + m 2 g) plus 2µ 2 /(k 2 + m 2 g) 2 , where the second (IR) term gives linear confinement for m g = 0 in the instantaneous limit, µ being another scale. To find Σ(p) we propose a new (differential) form of the Dyson-Schwinger Equation (DSE) for Σ(p), based on an infinitesimal subtractive Renormalization via a differential operator which lowers the degree of divergence in integration on the RHS, by T W O units. This warrants Σ(p − k) ≈ Σ(p) in the in-tegrand since its k-dependence is no longer sensitive to the principal term (p − k) 2 in the quark propagator. The simplified DSE (which incorporates WT identity in the Landau gauge) is satisfied for large p 2 by Σ(p) = Σ(0)/(1 + βp 2), except for Log factors. The limit p 2 = 0 determines Σ 0 .A third limit p 2 = −m 2 0 defines the dynamical mass m 0 via Σ(im 0) = +m 0. After two checks (f π = 93 ± 1M eV and < q ¯ q >= (280 ± 5M eV) 3), for 1.5 < β < 2 with Σ 0 = 300M eV , the T-dependent DSE is used in the real time formalism to determine the " critical " index γ = 1/3 analytically, with the IR term partly serving for the H field. We find T c = 180 ± 20M eV and check the vanishing of f π and < q ¯ q > at T c .