T-dependent Dyson-Schwinger equation in IR regime of QCD: the critical point
A. N. Mitra, W-Y. P. Hwang
European Physical Journal C: Particles and Fields
The quark mass function Σ(p) in QCD is revisited, using a gluon propagator in the form 1/(k^2 + m_g^2) plus 2μ^2/ (k^2 + m_g^2)^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 TWO
... its. This warrants Σ(p-k)≈Σ(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 WT identity in the Landau gauge) is satisfied for large p^2 by Σ(p) = Σ(0)/(1 + β p^2), except for Log factors. The limit p^2 =0 determines Σ_0.A third limit p^2 = -m_0^2 defines the dynamical mass m_0 via Σ(im_0) = + m_0. After two checks (f_π = 93± 1 MeV and = (280 ± 5 MeV)^3), for 1.5<β<2 with Σ_0=300 MeV, 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 ± 20 MeV and check the vanishing of f_π and at T_c. PACS: 24.85.+p; 12.38.Lg; 12.38.Aw.