  We study factorization in single transverse spin asymmetries for dijet
production in proton-proton collisions, by considering soft gluon radiation at
one-loop order. We show that the associated transverse momentum dependent (TMD)
factorization is valid at the leading logarithmic level. At
next-to-leading-logarithmic (NLL) accuracy, however, we find that soft gluon
radiation generates terms in the single transverse spin dependent cross section
that differ from those known for the unpolarized case. As a consequence, these
terms cannot be organized in terms of a spin independent soft factor in the
factorization formula. We present leading logarithmic predictions for the
single transverse spin dijet asymmetry for $pp$ collisions at RHIC, based on
quark Sivers functions constrained by semi-inclusive deep inelastic scattering
data. We hope that our results will contribute to a better understanding of TMD
factorization breaking effects at NLL accuracy and beyond.
