  We propose a new definition of a transverse-momentum-dependent (TMD) wave
function with simpler soft subtraction for $k_T$ factorization of hard
exclusive processes. The un-subtracted wave function involves two pieces of
non-light-like Wilson links oriented in different directions, so that the
rapidity singularity appearing in usual $k_T$ factorization is regularized, and
the pinched singularity from Wilson-link self-energy corrections is alleviated
to a logarithmic one. In particular no soft function is needed, when the two
pieces of Wilson links are orthogonal to each other. We show explicitly at
one-loop level that the simpler definition with the non-dipolar Wilson links
exhibits the same infrared behavior as the one with the dipolar Wilson links
and complicated soft subtraction. It is pointed out that both definitions
reduce to the naive TMD wave function as the non-light-like Wilson links
approach to the light cone. Their equivalence is then extended to all orders by
considering the evolution in the Wilson-link rapidity.

High Energy Physics - Phenomenology;High Energy Physics - Lattice;Nuclear Theory