No confinement without Coulomb confinement

TL;DR

The paper demonstrates that if the physical potential between a quark-antiquark pair is confining, then the color-Coulomb potential must also be confining, establishing a fundamental link between these two potentials in SU(N) gauge theory.

Contribution

It proves that confinement of the physical potential implies confinement of the Coulomb potential in Coulomb gauge, revealing a key relationship between these potentials.

Findings

If $V_D(R)$ is confining, then $V_{ m coul}(R)$ is also confining.

The inequality $V_D(R) leq - C_D V_{ m coul}(R)$ holds asymptotically.

Confinement in the physical potential implies confinement in the Coulomb potential.

Abstract

We compare the physical potential $V_D(R)$ of an external quark-antiquark pair in the representation $D$ of SU(N), to the color-Coulomb potential $V_{\rm coul}(R)$ which is the instantaneous part of the 44-component of the gluon propagator in Coulomb gauge, $D_{44}(\vx,t) = V_{\rm coul}(|\vx|) \delta(t)$ + (non-instantaneous). We show that if $V_D(R)$ is confining, $\lim_{R \to \infty}V_D(R) = + \infty$, then the inequality $V_D(R) \leq - C_D V_{\rm coul}(R)$ holds asymptotically at large $R$, where $C_D > 0$ is the Casimir in the representation $D$. This implies that $ - V_{\rm coul}(R)$ is also confining.