Gluon propagator in an external field; what happens when the field is removed?

We exhibit some general bounds on the free energy W(J) in an SU(N) gauge theory, where J^b_μ is a source for the gluon field A^b_μ in the minimal Landau gauge, and W(J) is the generating functional of connected correlators, expW(J) = (exp(J,A)). We then specialize to a source J(x) = hcos(k-x) 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 f^∞₀ dh D(k,h) ≤ V2k, which implies lim_k→0 D(k, h) = 0, for all positive h>0. This means that the system does not respond to a static color probe, no matter how strong. 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 dimensions 2, 3 and 4 which are consistent with these findings, and we compare with recent lattice calculations at h = 0 which indicate that the gluon propagator in the minimum Landau gauge is finite, lim_k→0 D(k, 0) > 0. These lattice data together with our analytic results 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.