Analytic and numerical study of the free energy in gauge theory

Abstract

We derive some exact bounds on the free energy $W(J)$ in an $SU(N)$ gauge theory, where ${J}_{\ensuremath{\mu}}^{b}$ is a source for the gluon field ${A}_{\ensuremath{\mu}}^{b}$ in the minimal Landau gauge, and $W(J)$ is the generating functional of connected correlators, $\mathrm{exp}\text{ }W(J)=⟨\mathrm{exp}(J,A)⟩$. We also provide asymptotic expressions for the free energy $W(J)$ at large $J$ and for the quantum effective action $\mathrm{\ensuremath{\Gamma}}(A)$ at large $A$. We specialize to a source $J(x)=h\text{ }\mathrm{cos}(k\ifmmode\cdot\else\textperiodcentered\fi{}x)$ 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 ${\ensuremath{\int}}_{0}^{\ensuremath{\infty}}dhD(k,h)\ensuremath{\le}\sqrt{2}k$, which implies $\underset{k\ensuremath{\rightarrow}0}{\mathrm{lim}}D(k,h)=0$, 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, ${\mathrm{lim}}_{k\ensuremath{\rightarrow}0}D(k,0)>0$. 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.