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In this work, we study the renormalization of nonlocal quark bilinearoperators containing an asymmetric staple-shaped Wilson line at the one-looplevel in both lattice and continuum perturbation theory. These operators enterthe first-principle calculation of transverse momentum-dependent partondistribution functions (TMDPDFs) in lattice QCD using the formulation of LargeMomentum Effective Theory. We provide appropriate RI$'$-type conditions thataddress the power and logarithmic divergences, as well as the mixing amongstaple operators of different Dirac structures, using a number of differentpossible projectors. A variant of RI$'$, including calculations of rectangularWilson loops, which cancel the pinch-pole singularities of the staple operatorsat infinite length and reduce residual power divergences, is also employed. Wecalculate at one-loop order the conversion matrix, which relates thequasi-TMDPDFs in the RI$'$-type schemes to the reference scheme $\overline{\rmMS}$ for arbitrary values of the renormalization momentum scale and of thedimensions of the staple.

Renormalization of asymmetric staple-shaped Wilson-line operators in lattice and continuum perturbation theory