  By using path integral formulation of QCD and QED we prove that the
factorization theorem is valid for light-like Wilson line but is not valid for
non-light-like Wilson line. This conclusion is shown to be consistent with Ward
identity and Grammer-Yennie approximation. Hence we find that the factorization
theorem is violated in all the previous studies which used the non-light-like
Wilson line in the definition of the (transverse momentum dependent) parton
distribution function and fragmentation function at high energy colliders.
