A Unified Treatment of the QCD Evolution of All (Un-)Polarized TMD Functions: Collins Function as a Study Case

By considering semi-inclusive deep-inelastic scattering and the (complementary) $q_T$-spectrum for Drell-Yan lepton pair production we derive the QCD evolution for all the leading-twist transverse momentum dependent distribution and fragmentation functions. We argue that all of those functions evolve with $Q^2$ following a \emph{single} evolution kernel. This kernel is independent of the underlying kinematics and it is also spin-independent. Those features hold, in impact parameter space, to all values of $b_T$. The evolution kernel presented has all of its large logarithms resummed up next-to-next-to leading logarithmic accuracy, which is the highest possible accuracy given existing perturbative calculations. As a study case we apply this kernel to investigate the evolution of Collins function, one of the ingredients that have attracted recently much attention within the phenomenological studies of spin asymmetries. Our analysis can be readily implemented to revisit previously obtained fits that involve data at different scales for other spin-dependent functions. Such improved fits are important to get better predictions --with the correct evolution kernel-- for certain upcoming experiments aiming to measure Sivers function, Collins function, transversity and other spin-dependent functions as well.