In this paper we derive a gauge fixing identity by varying the covariant gauge fixing term in Z[A,J,η,ηˉ] in the background field method of QCD in pure gauge. Using this gauge fixing identity we establish a relation between Z[J,η,ηˉ]
Nearby in the stack
in QCD and
Z[A,J,η,ηˉ]
in background field method of QCD in pure gauge. We show the validity of this gauge fixing identity in general non-covariant and general Coulomb gauge fixings respectively. This gauge fixing identity is used to prove factorization theorem in QCD at high energy colliders and in non-equilibrium QCD at high energy heavy-ion colliders.