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We present one-loop perturbative results of the renormalization functions fora complete set of nonlocal quark bilinear operators containing an asymmetricstaple-shaped Wilson line, using a family of improved lattice actions. Thisstudy is relevant for the nonperturbative investigations regarding therenormalization of the unpolarized, helicity and transversitytransverse-momentum dependent parton distribution functions (TMDPDFs) inlattice QCD. We employ a number of different versions ofregularization-independent (RI$'$) renormalization prescriptions which addressthe power and logarithmic divergences of such nonlocal operators, thepinch-pole singularities at infinite Wilson-line lengths, as well as the mixingamong operators of different Dirac structures, as dictated by discretesymmetries. All cancelations of divergences and admixtures are confirmed by ourresults at one-loop level. We compare all the different prescriptions and weprovide the conversion matrices at one-loop order which relate the matrixelements of the staple operators in RI$'$ to the reference scheme$\overline{\rm MS}$.

Perturbative study of renormalization and mixing for asymmetric staple-shaped Wilson-line operators on the lattice