  We demonstrate that the recently proposed soft gluon factorization (SGF) is
equivalent to the nonrelativistic QCD (NRQCD) factorization for heavy
quarkonium production or decay, which means that for any given process these
two factorization theories are either both valid or both violated. We use two
methods to achieve this conclusion. In the first method, we apply the two
factorization theories to the physical process $J/\psi \to e^+e^-$. Our
explicit calculation shows that both SGF and NRQCD can correctly reproduce low
energy physics of full QCD, and thus the two factorizations are equivalent. In
the second method, by using equations of motion we successfully deduce SGF from
NRQCD effective field theory. By identifying SGF with NRQCD factorization, we
establish relations between the two factorization theories and prove the
generalized Gremm-Kapustin relations as a by product. Comparing with the NRQCD
factorization, the advantage of SGF is that it resums the series of
relativistic corrections originated from kinematic effects to all powers, which
gives rise to a better convergence in relativistic expansion.
