Transverse momentum dependent PDFs at N<sup>3</sup>LO

Abstract

We compute the quark and gluon transverse momentum dependent parton distribution functions at next-to-next-to-next-to-leading order (N<jats:sup>3</jats:sup>LO) in perturbative QCD. Our calculation is based on an expansion of the differential Drell-Yan and gluon fusion Higgs production cross sections about their collinear limit. This method allows us to employ cutting edge multiloop techniques for the computation of cross sections to extract these universal building blocks of the collinear limit of QCD. The corresponding perturbative matching kernels for all channels are expressed in terms of simple harmonic polylogarithms up to weight five. As a byproduct, we confirm a previous computation of the soft function for transverse momentum factorization at N<jats:sup>3</jats:sup>LO. Our results are the last missing ingredient to extend the <jats:italic>q</jats:italic><jats:sub><jats:italic>T</jats:italic></jats:sub> subtraction methods to N<jats:sup>3</jats:sup>LO and to obtain resummed <jats:italic>q</jats:italic><jats:sub><jats:italic>T</jats:italic></jats:sub> spectra at N<jats:sup>3</jats:sup>LL' accuracy both for gluon as well as for quark initiated processes.