The Vacuum Energy Density of QCD with n_f=3 Quark Flavors

Abstract

An equivariant BRST-construction is used to define the continuum SU(3) gauge theory on a finite torus. I corroborate previous results using renormalization group techniques by explicitly computing the measure on the moduli-space of the model with 3 quark flavors to two loops. I find that the correction to the maximum of the one-loop effective action is indeed of order $g^2$ in the critical covariant gauge. The leading logarithmic corrections from higher loops are also shown to be suppressed by at least one order of $g^2$. I therefore am able to relate the expectation value of the moduli to the asymptotic scale parameter of the modified minimal subtraction scheme. An immediate consequence is the determination of the non-perturbative proportionality constant in the relation between the vacuum expectation value of the trace of the energy momentum tensor and $\Lambda_{QCD}$ for the modified minimal subtraction scheme with three quark flavors. The result compares favorably with phenomenological estimates of the gluon condensate from QCD sumrules for the charmonium system and $\Lambda_{QCD}$ from $\tau$-decay.