  At hadron colliders, the differential cross section for $W$ production can be
factorized and it is sensitive transverse momentum dependent distributions
(TMD) for low boson transverse momentum. While, often, the corresponding
non-perturbative QCD contributions are extrapolated from $Z$ boson production,
here we use an existing extraction (based on the code Artemide) of TMD which
includes data coming from Drell-Yan and semi-inclusive deep inelastic
scattering, to provide checks and predictions for the $W$ case. Including
fiducial cuts with different configurations and kinematical power corrections,
we consider transverse momentum dependent cross sections within several
intervals of the vector boson transverse mass. We perform the same study for
the $p_T^{W^-}/p_T^{W^+}$ and $p_T^Z/p_T^W$ distributions. We compare our
predictions with recent extractions of these quantities at ATLAS and CMS and
results from TeVatron. The results encourage a broader experimental and
phenomenological work, and a deeper study of TMD for the $W$ case.
