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Nonlocal quark bilinear operators connected by link paths are used forstudying parton distribution functions (PDFs) and transverse momentum-dependentPDFs of hadrons using lattice QCD. The nonlocality makes it difficult tounderstand the renormalization and improvement of these operators usingstandard methods. In previous work, we showed that by introducing an auxiliaryfield on the lattice, one can understand an on-axis Wilson-line operator as theproduct of two local operators in an extended theory. In this paper, we providedetails about the calculation in perturbation theory of the factor forconversion from our lattice-suitable renormalization scheme to the MS-barscheme. Extending our work, we study Symanzik improvement of the extendedtheory to understand the pattern of discretization effects linear in thelattice spacing, $a$, which are present even if the lattice fermion actionexactly preserves chiral symmetry. This provides a prospect for an eventual$O(a)$ improvement of lattice calculations of PDFs. We also generalize ourapproach to apply to Wilson lines along lattice diagonals and topiecewise-straight link paths.

Improvement, generalization, and scheme conversion of Wilson-line operators on the lattice in the auxiliary field approach