  Quasi-PDFs provide a path toward an ab initio calculation of parton
distribution functions (PDFs) using lattice QCD. One of the problems faced in
calculations of quasi-PDFs is the renormalization of a nonlocal operator. By
introducing an auxiliary field, we can replace the nonlocal operator with a
pair of local operators in an extended theory. On the lattice, this is closely
related to the static quark theory. In this approach, we show how to understand
the pattern of mixing that is allowed by chiral symmetry breaking, and obtain a
master formula for renormalizing the nonlocal operator that depends on three
parameters. We present an approach for nonperturbatively determining these
parameters and use perturbation theory to convert to the MS-bar scheme.
Renormalization parameters are obtained for two lattice spacings using Wilson
twisted mass fermions and for different discretizations of the Wilson line in
the nonlocal operator. Using these parameters we show the effect of
renormalization on nucleon matrix elements with pion mass approximately 370
MeV, and compare renormalized results for the two lattice spacings. The
renormalized matrix elements are consistent among the different Wilson line
discretizations and lattice spacings.
