No confinement without Coulomb confinement

We compare the physical potential VD(R) of an external quark-antiquark pair in the representation D of SU(N), to the color-Coulomb potential Vcoul(R) which is the instantaneous part of the 44-component: of the gluon propagator in Coulomb-gauge D44(x,t)=Vcoul(|x|)δ(t)+(noninstantaneous). We show that if VD(R) is confining, limR→∞VD(R)=+∞, as is believed to hold in the absence of dynamical quarks, then the inequality VD(R)≤-CDVcoul(R) holds asymptotically at large R, where CD>0 is the Casimir in the representation D. This implies that -Vcoul(R) is also confining