No confinement without Coulomb confinement

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 $V_{\mathrm{coul}}(R)$ which is the instantaneous part of the 44-component of the gluon propagator in Coulomb gauge, $D_{44}(\text{vx},t)=V_{\mathrm{coul}}(|\text{vx}|)\delta (t)$ + (non-instantaneous). We show that if $V_D(R)$ is confining, ${\lim }_{R\to \infty }{V}_D(R)=+\infty$, then the inequality $V_D(R)\le -C_D{V}_{\mathrm{coul}}(R)$ holds asymptotically at large $R$, where $C_D>0$ is the Casimir in the representation $D$. This implies that $-V_{\mathrm{coul}}(R)$ is also confining.

Comments: 9 pages