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We derive some exact bounds on the free energy W(J) in an SU(N) gauge theory,where J_mu^b is a source for the gluon field A_mu^b in the minimal Landaugauge, and W(J) is the generating functional of connected correlators, exp W(J)= <exp(J, A)>. We also provide asymptotic expressions for the free energy W(J)at large J and for the quantum effective action Gamma(A) at large A. Wespecialize to a source J(x)=h cos(kx) of definite momentum k and sourcestrength h, and study the gluon propagator D(k,h) in the presence of thissource. Among other relations, we prove int_0^inf dh D(k,h)<=2^1/2 k, whichimplies lim_(k->0) D(k,h) = 0, for all positive h>0. Thus the system does notrespond to a static color probe, no matter how strong. Recent lattice data inminimal Landau gauge in d =3 and 4 dimensions at h=0 indicate that the gluonpropagator in the minimum Landau gauge is finite, lim_(k->0) D(k,0)>0. Thusthese lattice data imply a jump in the value of D(k,h) at h=0 and k=0, and thevalue of D(k,h) at this point depends on the order of limits. We also presentnumerical evaluations of the free energy W(k,h) and the gluon propagator D(k,h)for the case of SU(2) Yang-Mills theory in various dimensions which support allof these findings.

Analytic and numerical study of the free energy in gauge theory