  We complete the procedure of extracting parton distribution functions (PDFs)
using large momentum effective theory (LaMET) at leading power accuracy in the
hadron momentum. We derive a general factorization formula for the quasi PDFs
in the presence of mixing, and give the corresponding hard matching kernel at
$\mathcal O(\alpha_s)$, both for the unpolarized and for the polarized quark
and gluon quasi-PDFs. Our calculation is performed in a
regularization-independent momentum subtraction scheme. The results allow us to
match the nonperturbatively renormalized quasi-PDFs to normal PDFs in the
presence of mixing, and therefore can be used to extract flavor-singlet quark
PDFs as well as gluon PDFs from lattice simulations.
