  Nonlocal quark bilinear operators connected by link paths are used for
studying parton distribution functions (PDFs) and transverse momentum-dependent
PDFs of hadrons using lattice QCD. The nonlocality makes it difficult to
understand the renormalization and improvement of these operators using
standard methods. In previous work, we showed that by introducing an auxiliary
field on the lattice, one can understand an on-axis Wilson-line operator as the
product of two local operators in an extended theory. In this paper, we provide
details about the calculation in perturbation theory of the factor for
conversion from our lattice-suitable renormalization scheme to the MS-bar
scheme. Extending our work, we study Symanzik improvement of the extended
theory to understand the pattern of discretization effects linear in the
lattice spacing, $a$, which are present even if the lattice fermion action
exactly preserves chiral symmetry. This provides a prospect for an eventual
$O(a)$ improvement of lattice calculations of PDFs. We also generalize our
approach to apply to Wilson lines along lattice diagonals and to
piecewise-straight link paths.
