An improved model of color confinement

TL;DR

This paper establishes bounds and asymptotic behavior for the free energy in QCD with an external source, providing insights into color confinement and reconciling theoretical models with lattice data indicating a non-zero gluon propagator at zero momentum.

Contribution

It introduces an optimal bound and exact asymptotic form for the free energy in QCD with external sources, and proposes a model consistent with recent lattice findings.

Findings

Color confinement implied by vanishing free energy and magnetization at zero external field.

A model consistent with non-zero gluon propagator at zero momentum.

Proposes numerical tests for the derived bounds.

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 present a model that is consistent with the new results and exhibits (non)-analytic behavior. Direct numerical tests of the bounds are proposed.