We compute the transverse momentum dependent (TMD) soft function for the production of a color-neutral nal state at the LHC within the rapidity renormalization group (RRG) framework to next-to-next-to-leading order (NNLO). We use this result to extract the universal renormalized TMD beam functions (aka TMDPDFs) in the same scheme and at the same order from known results in another scheme. We derive recurrence relations for the logarithmic structure of the soft and beam functions, which we use to cross check our calculation. We also explicitly con rm the non-Abelian exponentiation of the TMD soft function in the RRG framework at two loops. Our results provide the ingredients for resummed predictions of p?-di erential cross sections at NNLL0 in the RRG formalism. The RRG provides a systematic framework to resum large (rapidity) logarithms through (R)RG evolution and assess the associated perturbative uncertainties.
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