| Summary: | Abstract The singular behaviour of QCD squared amplitudes in the collinear limit is factorized and controlled by splitting kernels with a process-independent structure. We use these kernels to define collinear functions that can be used in QCD resummation formulae of hard-scattering observables. Different collinear functions are obtained by integrating the splitting kernels over different phase-space regions that depend on the hard-scattering observables of interest. The collinear functions depend on an auxiliary vector n μ that can be either light-like (n 2 = 0) or time-like (n 2 > 0). In the case of transverse-momentum dependent (TMD) collinear functions, we show that the use of a time-like auxiliary vector avoids the rapidity divergences, which are instead present if n 2 = 0. The perturbative computation of the collinear functions lead to infrared (IR) divergences that can be properly factorized with respect to IR finite functions that embody the logarithmically-enhanced collinear contributions to hard-scattering cross sections. We evaluate various collinear functions and their n μ dependence at O $$ \mathcal{O} $$ (α S). We compute the azimuthal-correlation component of the TMD collinear functions at O α S 2 $$ \mathcal{O}\left({\alpha}_{\textrm{S}}^2\right) $$ , and we present the results of the O α S 2 $$ \mathcal{O}\left({\alpha}_{\textrm{S}}^2\right) $$ contribution of linearly-polarized gluons to transverse-momentum resummation formulae. Beyond O α S 2 $$ \mathcal{O}\left({\alpha}_{\textrm{S}}^2\right) $$ the collinear functions of initial-state colliding partons are process dependent, as a consequence of the violation of strict collinear factorization of QCD squared amplitudes.
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