Abstract
The transverse momentum dependent parton distribution/fragmentation functions (TMDs) are essential in the factorization of a number of processes like Drell-Yan scattering, vector boson production, semi-inclusive deep inelastic scattering, etc. We provide a comprehensive study of unpolarized TMDs at next-to-next-to-leading order, which includes an explicit calculation of these TMDs and an extraction of their matching coefficients onto their integrated analogues, for all flavor combinations. The obtained matching coefficients are important for any kind of phenomenology involving TMDs. In the present study each individual TMD is calculated without any reference to a specific process. We recover the known results for parton distribution functions and provide new results for the fragmentation functions. The results for the gluon transverse momentum dependent fragmentation functions are presented for the first time at one and two loops. We also discuss the structure of singularities of TMD operators and TMD matrix elements, crossing relations between TMD parton distribution functions and TMD fragmentation functions, and renormalization group equations. In addition, we consider the behavior of the matching coefficients at threshold and make a conjecture on their structure to all orders in perturbation theory.
Highlights
The transverse momentum dependent parton distribution and fragmentation functions (TMDs) play a central role in our understanding of QCD dynamics in multi-differential cross sections and spin physics
We provide a comprehensive study of unpolarized transverse momentum dependent parton distribution/fragmentation functions (TMDs) at next-to-next-to-leading order, which includes an explicit calculation of these TMDs and an extraction of their matching coefficients onto their integrated analogues, for all flavor combinations
In this work we focus on unpolarized TMDs, which have received much attention recently, being the simplest functions and for which the relevant factorization theorems have been explicitly checked at next-to-leading order (NLO), with various quantum numbers, by several groups
Summary
The transverse momentum dependent parton distribution and fragmentation functions (TMDs) play a central role in our understanding of QCD dynamics in multi-differential cross sections and spin physics. The TMD factorization theorem offers a strategy to remove the rapidity divergences in order to achieve a well-defined TMDs. Recently our group has provided a direct calculation of an individual TMD at NNLO, namely the unpolarized quark transverse momentum dependent fragmentation function (TMDFF) [21], and a complete study of the structure of rapidity divergences at the same order in the soft function [22] (see [20, 23]). Some properties of the TMDPDFs, like their matching onto integrated parton distribution functions (PDFs) can be found in previous works [15,16,17,18,19], where they were obtained by decomposing the product of two TMDs and did not use the fact that each TMD is per se calculable In other words, these calculations did not fully exploit the results of the TMD factorization theorem of [1,2,3,4]. The set of appendices includes several necessary definitions, some intermediate expressions and side results which were used in the paper
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