Exact bounds on the free energy in QCD

Abstract

We consider the free energy $W[J]=W_k(H)$ of QCD coupled to an external source $J_{\mu }^b(x)=H_{\mu }^b\cos (k\cdot x)$, where $H_{\mu }^b$ is, by analogy with spin models, an external "magnetic" field with a color index that is modulated by a plane wave. We report an optimal bound on $W_k(H)$ and an exact asymptotic expression for $W_k(H)$ at large $H$. They imply confinement of color in the sense that the free energy per unit volume $W_k(H)/V$ and the average magnetization $m(k,H)=\frac{1}{V}\frac{\text{p}{W}_k(H)}{\text{p}H}$ vanish in the limit of constant external field $k\to 0$. Recent lattice data indicate a gluon propagator $D(k)$ which is non-zero, $D(0)\ne 0$, at $k=0$. This would imply a non-analyticity in $W_k(H)$ at $k=0$. We also give some general properties of the free energy $W(J)$ for arbitrary $J(x)$. Finally we present a model that is consistent with the new results and exhibits (non)-analytic behavior. Direct numerical tests of the bounds are proposed.

Comments: 7 pages, International Workshop on QCD Green's Functions, Confinement and Phenomenology, September 05-09, 2011, Trento Italy