Abstract
In this paper we consider the parametrizations of gluon transverse momentum dependent (TMD) correlators in terms of TMD parton distribution functions (PDFs). These functions, referred to as TMDs, are defined as the Fourier transforms of hadronic matrix elements of nonlocal combinations of gluon fields. The nonlocality is bridged by gauge links, which have characteristic paths (future or past pointing), giving rise to a process dependence that breaks universality. For gluons, the specific correlator with one future and one past pointing gauge link is, in the limit of small $x$, related to a correlator of a single Wilson loop. We present the parametrization of Wilson loop correlators in terms of Wilson loop TMDs and discuss the relation between these functions and the small-$x$ `dipole' gluon TMDs. This analysis shows which gluon TMDs are leading or suppressed in the small-$x$ limit. We discuss hadronic targets that are unpolarized, vector polarized (relevant for spin-$1/2$ and spin-$1$ hadrons), and tensor polarized (relevant for spin-$1$ hadrons). The latter are of interest for studies with a future Electron-Ion Collider with polarized deuterons.
Highlights
In this paper we consider the parametrizations of gluon transverse momentum dependent (TMD) correlators in terms of TMD parton distribution functions (PDFs)
The starting point for the gluon TMD correlators are Fourier transforms of hadronic matrix elements of field strength tensors connected by Wilson lines or gauge links [1,2,3,4,5,6] that bridge the nonlocality of the field operators, ensuring color gauge invariance
We present the parametrizations of the gluon-gluon and Wilson loop TMD correlators in terms of TMDs of definite rank for unpolarized, vector polarized, and tensor polarized hadrons, the latter being considered here for the first time
Summary
In 2001, Mulders and Rodrigues [7] presented the first parametrization of the gluon-gluon light-front correlator in terms of TMDs considering both unpolarized and vector polarized hadrons. The light-front correlators are expanded in a Lorentz basis of completely symmetric traceless tensors built from the partonic momentum kT (see appendix C.1 for the definitions of the relevant symmetric traceless tensors), and are expressed in terms of TMDs. a more systematic way of naming the various TMDs is introduced, keeping and extending the notation proposed in ref. Which become the standard time and spatial z-directions in the hadron rest frame They are useful since the spin vector and tensor only contain the spacelike combination z: Sμ = SL zμ + STμ, zμzν.
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