Collinear functions for QCD resummations

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 $n^{\mu }$ 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 $n^{\mu }$ dependence at $\mathcal{O}({\alpha }_{\mathrm{S}})$. We compute the azimuthal-correlation component of the TMD collinear functions at $\mathcal{O}({\alpha }_{\mathrm{S}}^2)$, and we present the results of the $\mathcal{O}({\alpha }_{\mathrm{S}}^2)$ contribution of linearly-polarized gluons to transverse-momentum resummation formulae. Beyond $\mathcal{O}({\alpha }_{\mathrm{S}}^2)$ 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.

55 pages; some comments added, results unchanged

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• Angular integrals with three denominators via IBP, mass reduction, dimensional shift, and differential equations
• The $q_{\mathrm{T}}$ and $\Delta {\varphi }_{t\bar{t}}$ spectra in top-antitop hadroproduction at NNLL+NNLO: the interplay of soft-collinear resummation and Coulomb singularities