Wilson lines in the operator definition of TMDs: spin degrees of freedom and renormalization

A generalized idea of gauge invariance, that embodies into the Wilson lines the spin-dependent Pauli term $\sim F^{\mu \nu }[{\gamma }_{\mu },{\gamma }_{\nu }]$, is applied to set up a new framework for the operator definition of transverse-momentum-dependent parton densities (TMDs). We show that such a treatment of gauge invariance is justified, since it does not change the leading-twist behavior of the TMDs, albeit it contributes to their twist-three properties, in particular, to their anomalous dimensions. We discuss other consequences of this generalization and its possible applications to lattice simulations of the TMDs.