Exact bounds on the free energy in QCD

TL;DR

This paper derives exact bounds and asymptotic behavior for the free energy in QCD under external sources, providing insights into color confinement and discussing implications of lattice data on gluon propagators.

Contribution

It presents an optimal bound and exact asymptotics for the QCD free energy with external sources, linking to confinement and non-analyticity at zero momentum.

Findings

Bound on free energy $W_k(H)$ established

Asymptotic expression for large $H$ derived

Implications for gluon propagator and confinement discussed

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 $m(k, H) ={1 \over V} {\p W_k(H) \over \p H}$ vanish in the limit of constant external field $k \to 0$. Recent lattice data indicate a gluon propagator $D(k)$ which is non-zero, $D(0) \neq 0$, 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.