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All (un)polarized transverse momentum dependent functions (TMDs), bothdistribution and fragmentation functions, are defined with the same universalsoft function, which cancels spurious rapidity divergences within an individualTMD and renders them well-defined hadronic quantities. Moreover, it isindependent of the kinematics, whether it is Drell-Yan, deep inelasticscattering or $e^+e^-\rightarrow 2$ hadrons. In this paper we provide this softfunction at next-to-next-to-leading order (NNLO), necessary for the calculationof all TMDs at the same order, and to perform the resummation of largelogarithms at next-to-next-to-next-to-leading-logarithmic accuracy. From theresults we obtain the $D$ function at NNLO, which governs the evolution of allTMDs. This work represents the first independent and direct calculation of thisquantity. Given the all order relation through a Casimir scaling between thesoft function relevant for gluon TMDs and the one for quark TMDs, we alsoobtain the first at NNLO. The used regularization method to deal with therapidity divergences is discussed as well.

The Universal Transverse Momentum Dependent Soft Function at NNLO