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 $\mathrm{\Gamma }(A)$ at large $A$. We specialize to a source $J(x)=h\cos (k·x)$ 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 ${\int }_0^{\infty }dhD(k,h)\le \sqrt{2}k$, which implies $\underset{k\to 0}{\lim }D(k,h)=0$, 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, ${\lim }_{k\to 0}D(k,0)>0$. 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.

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