Minimum Of QCD Effective Action As Test Of QCD Confinement Parameter

Abstract

A new approach to the non-perturbative regime of QCD is proposed by introducing a (non-hermitian) field $B$ related to the usual gluon field $A$ by $B_\mu$ = $(1+ \sigma \partial_m) A_\mu$ where $m$ goes to zero after differentiation, and $\sigma$ is a parameter which `runs' with momentum ($k$). An exact treatment yields a structure $[1/k^2 + 2\mu^2/k^4]$ for the gluon propagator, where $\sigma^2$ =$\pi k^4/9\mu^2\alpha_s$, showing $linear$ confinement in the instantaneous limit. This propagator was recently employed to evaluate some basic condensates and their temperature dependence (in the cosmological context), which were all reproduced for $\mu = 1GeV$ (termed the `confinement scale parameter'), in association with the QCD scale parameter $\Lambda_{qcd}= 200 MeV$ [hep-ph/0109278]. This paper seeks to provide a formal basis for the ratio $\Lambda_{qcd} / \mu$ by employing the minimality condition for the $integrated$ effective action $\Gamma$, up to the 2-loop level, using the Cornwall-Jackiw formalism for composite operators. To that end the mass function $m(p)$, determined via the Schwinger-Dyson equation (as a zero of the functional derivative of $\Gamma$ w.r.t $S_F'$), acts as a feeder, and the stationarity condition on $\Gamma$ as function of $\mu$ and $\alpha_s(\mu)$ gives the ratio $\Lambda /\mu$ = 0.246, in fair accord with the value 0.20 given above. Inclusion of a two-loop $\Gamma$ is crucial for the agreement. \\ Keywords: confinement scale, QCD effective action, minimality condition, $B$-field.