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 $\Sigma (i{m}_0)=+m_0$. After two checks ($f_{\pi }=93\pm 1MeV$ and $<q\bar{q}>$= $(280\pm 5MeV{)}^3$), for $1.5<\beta <2$ with ${\Sigma }_0=300MeV$, 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 $T_c=180\pm 20MeV$ and check the vanishing of $f_{\pi }$ and $<q\bar{q}>$ at $T_c$. PACS: 24.85.+p; 12.38.Lg; 12.38.Aw.

Comments: 25pp on dvi, LaTex File