  It is shown that transverse momentum dependent azimuthal spin asymmetries
suffer from suppression due to Sudakov factors, in the region where the
transverse momentum is much smaller than the large energy scale Q^2. The size
and Q^2 dependence of this suppression are studied numerically for two such
asymmetries, both arising due to the Collins effect. General features are
discussed of how the fall-off with Q^2 is affected by the nonperturbative
Sudakov factor and by the transverse momentum weights and angular dependences
that appear in different asymmetries. For a subset of asymmetries the
asymptotic Q^2 behavior is calculated analytically, providing an upper bound
for the decrease with energy of other asymmetries. The effect of Sudakov
factors on the transverse momentum distributions is found to be very
significant already at present-day collider energies. Therefore, it is
essential to take into account Sudakov factors in transverse momentum dependent
azimuthal spin asymmetries.
