Abstract
In this paper we calculate analytically the perturbative matching coefficients for unpolarized quark and gluon Transverse-Momentum-Dependent (TMD) Parton Distribution Functions (PDFs) and Fragmentation Functions (FFs) through Next-to-Next-to-Next-to-Leading Order (N3LO) in QCD. The N3LO TMD PDFs are calculated by solving a system of differential equation of Feynman and phase space integrals. The TMD FFs are obtained by analytic continuation from space-like quantities to time-like quantities, taking into account the probability interpretation of TMD PDFs and FFs properly. The coefficient functions for TMD FFs exhibit double logarithmic enhancement at small momentum fraction z. We resum such logarithmic terms to the third order in the expansion of αs. Our results constitute important ingredients for precision determination of TMD PDFs and FFs in current and future experiments.
Highlights
Kinemtical limit [16, 17]
In summary we presented calculations for the unpolarized quark and gluon TMD Parton Distribution Functions (PDFs) and Fragmentation Functions (FFs) at N3LO in QCD
Unpolarized quark and gluon TMD PDFs have been calculated in [48] recently using an independent method, whose results are in full agreement with ours
Summary
We begin with the bare TMD PDFs for unpolarized quark and gluon, which can be defined in terms of SCET [58,59,60,61,62] collinear fields. And Aan,⊥μ is the gauge invariant collinear gluon field with color index a and Lorentz index μ. The usual bare partonic collinear PDFs are just φbi/ajre(x) = δijδ(1 − x), . One can perform a further decomposition into two independent Lorentz structures in d = 4 − 2 dimension, Igbiare,μν (ξ, b⊥). Where we have defined two scalar form factors, Igbiare, the unpolarized gluon coefficient functions, and Igbiare, the linearly-polarized gluon coefficient functions. We focus on unpolarized TMD distributions for the current paper, and the results for the linearly-polarized gluon distribution are left for future work
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