  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 Landau
gauge, 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. We
specialize to a source J(x)=h cos(kx) of definite momentum k and source
strength h, and study the gluon propagator D(k,h) in the presence of this
source. Among other relations, we prove int_0^inf dh D(k,h)<=2^1/2 k, which
implies lim_(k->0) D(k,h) = 0, for all positive h>0. Thus the system does not
respond to a static color probe, no matter how strong. Recent lattice data in
minimal Landau gauge in d =3 and 4 dimensions at h=0 indicate that the gluon
propagator in the minimum Landau gauge is finite, lim_(k->0) D(k,0)>0. Thus
these lattice data imply a jump in the value of D(k,h) at h=0 and k=0, and the
value of D(k,h) at this point depends on the order of limits. We also present
numerical 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 all
of these findings.
