Limits on TMD Evolution From Semi-Inclusive Deep Inelastic Scattering at Moderate $Q$

In the QCD evolution of transverse momentum dependent parton distribution and fragmentation functions, the Collins-Soper evolution kernel includes both a perturbative short-distance contribution as well as a large-distance non-perturbative, but strongly universal, contribution. In the past, global fits, based mainly on larger $Q$ Drell-Yan-like processes, have found substantial contributions from non-perturbative regions in the Collins-Soper evolution kernel. In this article, we investigate semi-inclusive deep inelastic scattering measurements in the region of relatively small $Q$, of the order of a few GeV, where sensitivity to non-perturbative transverse momentum dependence may become more important or even dominate the evolution. Using recently available deep inelastic scattering data from the COMPASS experiment, we provide estimates of the regions of coordinate space that dominate in TMD processes when the hard scale is of the order of only a few GeV. We find that distance scales that are much larger than those commonly probed in large $Q$ measurements become important, suggesting that the details of non-perturbative effects in TMD evolution are especially significant in the region of intermediate $Q$. We highlight the strongly universal nature of the non-perturbative component of evolution, and its potential to be tightly constrained by fits from a wide variety of observables that include both large and moderate $Q$. On this basis, we recommend detailed treatments of the non-perturbative component of the Collins-Soper evolution kernel for future TMD studies.