  Recent BELLE measurements provide the cross section for single hadron
production in $e^+e^-$ annihilations, differential in thrust and in the hadron
transverse momentum with respect to the thrust axis. Universality breaking
effects due to process-dependent soft factors make it very difficult to relate
this cross sections to those corresponding to hadron-pair production in
$e^+e^-$ annihilations, where transverse momentum dependent (TMD) factorization
can be applied. The correspondence between these two cross sections is examined
in the framework of the Collins-Soper-Sterman factorization, in the collinear
as well as in the TMD approach. We propose a scheme that allows to relate the
TMD parton densities defined in 1-hadron and in 2-hadron processes, neatly
separating, within the soft and collinear parts, the non-perturbative terms
from the contributions that can be calculated perturbatively. The
regularization of rapidity divergences introduces cut-offs, the arbitrariness
of which will be properly reabsorbed by means of a mechanism closely
reminiscent of a gauge transformation. In this way, we restore the possibility
to perform global phenomenological studies of TMD physics, simultaneously
analyzing data belonging to different hadron classes.
