Transverse momentum dependent PDFs at N3LO

Abstract

We compute the quark and gluon transverse momentum dependent parton distribution functions at next-to-next-to-next-to-leading order (N3LO) 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 N3LO. Our results are the last missing ingredient to extend the qT subtraction methods to N3LO and to obtain resummed qT spectra at N3LL′ accuracy both for gluon as well as for quark initiated processes.

Calculation of transverse momentum dependent distributions beyond the leading power

“…The leading term of this expression is well-known. Moreover, the perturbative corrections for the leading term are know up to α 3 s -order [22,23]. Peculiarly, the contributions proportional to the twist-3 are absent at all powers.…”

Collinear expansion for color singlet cross sections

“…Since the beam function counter term Z i B gives rise to the renormalization group equation of the beam function, in practice one can either predict Z i B from the RGEs and check that this cancels all poles in , or determine Z i B by absorbing all poles remaining after QCD UV and IR subtraction and verify that it reproduces the RGE dictated by the EFT. For the T N and q T beam functions, this is discussed in more detail in our companion papers [105,106].…”

“…These results are necessary input for the calculation of aforementioned beam functions at N 3 LO in QCD, where much progress has been already made for the quark T N beam function [101][102][103], and which has already been achieved for the quark q T beam function and TMDPDF [104]. In our companion papers [105,106], we complete this task by computing the q T and T 0 beam functions in all channels at N 3 LO based on the methods outlined in this article.…”

“…The all-order structure of the beam function counter termZ i B can be predicted from the beam function RGE, which we show in detail in ref. [105].…”

Quarkonium TMD fragmentation functions in NRQCD

“…The factorization of the TMD cross section for the light-quark fragmentation mechanism is structurally very similar to the conventional SIDIS process [29][30][31][32], where the rapidity scale ζ and the renormalization scale µ are responsible of the evolution of the TMD matrix elements in initial and final states [33]. This process has been extensively studied in the literature and we now have high-order QCD calculations for hard factors [34][35][36], evolution kernel [37][38][39] and TMDPDFs [40][41][42][43][44][45][46]. It is also possible then to use extraction of TMDs [47][48][49] to perform a phenomenological analysis.…”