Coulomb Potential Is Not a Part of The QCD Potential

Abstract

The Coulomb plus linear potential is widely used in QCD. However, in this paper we show that the Coulomb potential of the form $\frac{1}{r}$ is not a part of the QCD potential. This is because the form $\frac{g^2}{r}$ is for abelian theory (not QCD) and the form $\frac{g^2(\mu)}{r}$ in QCD at short distance is not of the Coulomb form $\frac{1}{r}$ because $g(\mu)$ depends on the mass/length scale $\mu$. Similarly at long distance the QCD potential corresponds to the potential in the classical Yang-Mills theory which does not have the Coulomb form $\frac{1}{r}$ because the fundamental color charge of the quark is time dependent in the classical Yang-Mills theory. This is unlike the QED potential which reduces to Coulomb potential at long distance.