Gauge non-invariance of quark-quark interactions

We discuss the gauge invariance of quark-quark potential in QCD in the perturbative treatment, and show that the gauge dependence of the quark-gluon interaction cannot be eliminated. Therefore, the quark-quark potential cannot properly be written in terms of physical variables in the perturbation theory. This is in contrast to the QED case in which the gauge invariance of the electron-photon interaction is guaranteed by the current conservation. As an example, we show that the different choices of the gauge fixing give different Coulomb interactions when the fermion current is not conserved. Also, we examine a possible existence of $QG-$state (one quark$-$one gluon) corresponding to the conserved color current, which should have a fractional electric charge.