  In this paper we present one-loop results for the renormalization of nonlocal
quark bilinear operators, containing a staple-shaped Wilson line, in both
continuum and lattice regularizations. The continuum calculations were
performed in dimensional regularization, and the lattice calculations for the
Wilson/clover fermion action and for a variety of Symanzik-improved gauge
actions. We extract the strength of the one-loop linear and logarithmic
divergences (including cusp divergences), which appear in such nonlocal
operators; we identify the mixing pairs which occur among some of these
operators on the lattice, and we calculate the corresponding mixing
coefficients. We also provide the appropriate RI'-like scheme, which
disentangles this mixing nonperturbatively from lattice simulation data, as
well as the one-loop expressions of the conversion factors, which turn the
lattice data to the MS-bar scheme. Our results can be immediately used for
improving recent nonperturbative investigations of transverse
momentum-dependent distribution functions (TMDs) on the lattice. Finally,
extending our perturbative study to general Wilson-line lattice operators with
n cusps, we present results for their renormalization factors, including
identification of mixing and determination of the corresponding mixing
coefficients, based on our results for the staple operators.
