T-Dependent Dyson- Schwinger Equation In IR Regime Of QCD: The Critical Point

Abstract

The quark mass function $\Sigma(p)$ in QCD is revisited, using a gluon propagator in the form $1/(k^2 + m_g^2)$ plus $2\mu^2/ (k^2 + m_g^2)^2$, 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 $\Sigma(p-k)\approx \Sigma(p)$ 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 WT identity in the Landau gauge) is satisfied for large $p^2$ by $\Sigma(p)$ = $\Sigma(0)/(1 + \beta p^2)$, 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\pm 1 MeV $ and $ <q{\bar q}>$= $(280 \pm 5 MeV)^3$), 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 for the $H$ field. We find $T_c = 180 \pm 20 MeV$ and check the vanishing of $f_\pi$ and $<q{\bar q}>$ at $T_c$. PACS: 24.85.+p; 12.38.Lg; 12.38.Aw.