A spectator-model way to transverse-momentum-dependent gluon distribution functions

We present exploratory analyses of the 3D gluon content of the proton via a study of unpolarized and polarized gluon TMDs at twist-2, calculated in a spectator model for the parent nucleon. Our approach embodies a flexible parametrization for the spectator-mass function, suited to describe both moderate and small-$x$ effects. All these studies can serve as a useful guidance in the investigation of the gluon dynamics inside nucleons and nuclei, which constitutes one of the major goals of new-generation colliding machines, as the EIC, the HL-LHC, NICA, and the FPF.


Introduction
The study of the proton content via transverse-momentum-dependent (TMD) parton distribution functions represents a challenging line of research plans at current and new-generation colliding machines. While in the last years the investigation of the quark-TMD field has reached important milestones, from the deep knowledge of formal properties to the more and more accurate extraction of quark densities from global fits, the gluon-TMD sector still represents a largely unexplored territory. A first classification of unpolarized and polarized gluon TMD distributions was first made in Ref.
[1] and subsequently extended in Refs. [2][3][4]. Recent phenomenological analyses on gluon TMDs can be found in Refs. [5][6][7][8][9]. A major difficulty that emerges in formal studies of gluon TMDs is their process dependence. Different kinds of reactions are sensitive to distinct gauge-link structures, and this leads to a more intricate modified universality with respect to what we observe for quark TMDs. Two main gauge links can be identified. They have been classified in the context of small-x analyses as Weiszäcker-Williams and dipole TMDs [10]. They are strictly related to gluon correlators where for T -odd TMDs the f abc and d abc QCD color structures respectively emerge. Therefore, they are also known among the TMD community as f -type and d-type gluon TMDs. At low-x values and large transverse momenta, the gluon content of the proton is described by the so-called unintegrated gluon distribution (UGD), whose evolution is governed by the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation [11,12] (for recent applications see Refs. [13][14][15][16][17][18][19][20][21][22][23]). Its relation to the low-x limit of gluon TMDs and, more in general, to the Collins-Soper-Sterman (CSS) evolution [24,25] has been investigated in Refs. [10] and [26,27], respectively. In this work we present a study on leading-twist T -even gluon TMDs calculated in a spectator model for the parent proton. Our framework is suited to analyses both in moderate and small-x ranges.

TMD gluon distribution functions
According to the spectator-model approximation, the proton can emit a gluon with longitudinalmomentum fraction x and transverse momentum p T , and the remainders are treated as an effective colored particle with mass M X and possessing the quantum numbers of a fermion, that we call spectator. The nucleon-gluon-spectator coupling is encoded in a effective vertex that contains two form factors, chosen as dipolar functions of p 2 T . The main advantage of using dipolar form factors consists in the possibility of cancelling gluon-propagator singularities, quenching the effects of large transverse momenta where a pure TMD description is not anymore adequate, and removing logarithmic divergences emerging in p T -integrated densities.
In Ref.
[28] a pioneering study on quark TMDs was proposed, by considering different di-quark spectator polarization states and nucleon-parton-spectator form factors. In Ref. [29] the weight of azimuthal asymmetries was assessed.
In the present study we present our calculation in the spectator model of T -even gluon TMDs at twist-2. We improved the genuine spectator-model approach by allowing the spectator mass, M X , to be in a range of values weighed by the following 7-parameter spectral function (1) The expression for a given TMD reads withF g the corresponding TMD obtained in a pure spectator-model calculation. Model parameters were fitted to simultaneously reproduce the gluon unpolarized ( f g 1 (x)) and helicity (g g 1 (x)) collinear parton distribution functions (PDFs), obtained in global fits at the initial scale Q 0 = 1.64 GeV (see Fig. 1). We performed our fit by making use of the so-called bootstrap method. We created N replicas of the central value of the NNPDF parametrization by randomly varying it with a Gaussian noise that keeps the same variance of the original parametrization uncertainty. We fitted each replica separately and we obtained N -dimensional vector for each parameter of the model. A complete description of our model together all technical details of our fit procedure can be found in Ref. [30] (see also Refs. [31][32][33][34][35][36]). We show in Fig. 2  dashed borders stand for the NNPDF3.1x [37] and the NNPDFpol1.1 [38] parametrizations. Blue curves depict the 100 replicas for our integrated TMDs. Red curve for the most representative replica #11.

Conclusion
We presented a model dependent calculation of all twist-2 T -even gluon TMDs based on the assumption that what remains of the proton after gluon emission can be described as an effective spin-1/2 spectator particle. We improved the genuine spectator-model description by weighing its mass via a versatile spectral function. We fitted model parameters to reproduce the x-shape of collinear unpolarized and helicity gluon PDFs that were extracted from global fits. At the current level, our model does not incorporate any gauge-link dependence, and the extension to twist-2 T -odd gluon TMD distributions is underway. Another intriguing perspective is represented by encoding in the description of the unpolarized gluon TMD genuine small-x effect from the BFKL resummation [11,12]. Exploratory studies on gluon-TMD phenomenology via our model can represent a useful guidance in accessing the proton content at new-generation colliding machines, as the Electron-Ion Collider (EIC) [39], the High-Luminosity Large Hadron Collider (HL-LHC) [40], NICA [41], and the Forward Physics Facility [42,43].