No Confinement without Coulomb Confinement

We compare the physical potential $V_D(R)$ of an external quark-antiquark pair in the representation $D$ of $\mathrm{S}\mathrm{U}(N)$, to the color-Coulomb potential $V_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{l}}(R)$ which is the instantaneous part of the 44-component: of the gluon propagator in Coulomb-gauge $D_{44}(\overrightarrow{x},t)=V_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{l}}(|\overrightarrow{x}|)\delta (t)+(\mathrm{\text{noninstantaneous}})$. We show that if $V_D(R)$ is confining, $\underset{R\to \infty }{lim}{V}_D(R)=+\infty$, as is believed to hold in the absence of dynamical quarks, then the inequality $V_D(R)\le -C_D{V}_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{l}}(R)$ holds asymptotically at large $R$, where $C_D>0$ is the Casimir in the representation $D$. This implies that $-V_{\mathrm{c}\mathrm{o}\mathrm{u}\mathrm{l}}(R)$ is also confining.