The Transverse Momentum Dependent Fragmentation Function at NNLO

Abstract: We calculate the unpolarized transverse momentum dependent fragmentation function (TMDFF) at next-to-next-to-leading order (NNLO), evaluating separately TMD soft factor and TMD collinear correlator. For the first time the cancellation of spurious rapidity divergences in a properly defined individual TMD beyond the first non-trivial order is shown. This represents a strong check of the given TMD definition. We extract the matching coefficient necessary to perform the transverse momentum resummation at next-to-next-to-next-to-leading-logarithmic accuracy. The universal character of the soft function, which enters the definition of all (un)polarized TMD distribution/fragmentation functions, facilitates the future calculation of all the other TMDs and their coefficients at NNLO, pushing forward the accuracy of theoretical predictions for the current and next generation of high energy colliders.