T-dependent Dyson-Schwinger equation in IR regime of QCD:: the critical point

The quark mass function Sigma(p) in QCD is revisited, using a gluon propagator in the form 1/(k(2) + m(g)(2)) plus 2mu(2)/(k(2) + m(g)(2))(2), where the second (IR) term gives linear confinement for m(g) = 0 in the instantaneous limit, mu being another scale. To find Sigma(p) we propose a new (differential) form of the Dyson - Schwinger equation (DSE) for Sigma(p), based on an infinitesimal subtractive renormalization via a differential operator which lowers the degree of divergence in integration on the RHS, by two units. This warrants Sigma(p - k) approximate to Sigma(p) in the integrand since its k-dependence is no longer sensitive to the principal term (p-k)(2) in the quark propagator. The simplified DSE ( which incorporates the Ward - Takahashi (WT) identity in the Landau gauge) is satisfied for large p(2) by Sigma(p) = Sigma(0)/(1+betap(2)), except for Log factors. The limit p(2) = 0 determines Sigma(0). A third limit, p(2) = - m(0)(2), defines the dynamical mass m(0) via Sigma(im(0)) = + m(0). After two checks (f(pi) = 93 +/- 1 MeV and [q (q) over bar] = ( 280 +/- 5MeV)(3)), for 1.5 < beta < 2 with Sigma(0) = 300 MeV, the T-dependent DSE is used in the real time formalism to determine the "critical" index gamma = 1/3 analytically, with the IR term partly serving as the H-field. We find T-c = 180 +/- 20 MeV and check the vanishing of f(pi) and [q (q) over bar] at T-c.