Path Integration in QCD with Arbitrary Space-Dependent Static Color Potential

We perform path integral for a quark (antiquark) in the presence of an arbitrary space-dependent static color potential A^a_0(x)(=-\int dx E^a(x)) with arbitrary color index a=1,2,...8 in SU(3) and obtain an exact non-perturbative expression for the generating functional. We show that such a path integration is possible even if one can not solve the Dirac equation in the presence of arbitrary space-dependent potential. It may be possible to further explore this path integral technique to study non-perturbative bound state formation.Comment: 11 pages latex, typos corrected, Accepted for Publication in Journal of High Energy Physic

General Form of Color Charge of the Quark

In Maxwell theory the constant electric charge e of the electron is consistent with the continuity equation $\partial_\mu j^\mu(x)=0$ where $j^\mu(x)$ is the current density of the electron where the repeated indices $\mu=0,1,2,3$ are summed. However, in Yang-Mills theory the Yang-Mills color current density $j^{\mu a}(x)$ of the quark satisfies the equation $D_\mu[A]j^{\mu a}(x)=0$ which is not a continuity equation ($\partial_\mu j^{\mu a}(x)\neq 0$) which implies that the color charge of the quark is not constant where a=1,2,...,8 are the color indices. Since the charge of a point particle is obtained from the zero ($\mu =0$) component of a corresponding current density by integrating over the entire (physically) allowed volume, the color charge $q^a(t)$ of the quark in Yang-Mills theory is time dependent. In this paper we derive the general form of eight time dependent fundamental color charges $q^a(t)$ of the quark in Yang-Mills theory in SU(3) where a=1,2,...,8.Comment: 52 pages latex, final version, accepted for publication in Eur. Phys. J. C. arXiv admin note: substantial text overlap with arXiv:1201.266