TMD factorization and evolution at large $b_T$

Abstract: In using transverse-momentum-dependent (TMD) parton densities and fragmentation functions, important non-perturbative information is at large transverse position $b_T$. This concerns both the TMD functions and their evolution. Fits to high energy data tend to predict too rapid evolution when extrapolated to low energies where larger values of $b_T$ dominate. I summarize a new analysis of the issues. It results in a proposal for much weaker $b_T$ dependence at large $b_T$ for the evolution kernel, while preserving the accuracy of the existing fits. The results are particularly important for using transverse-spin-dependent functions like the Sivers function.