Towards leading-twist $T$-odd TMD gluon distributions

We present exploratory studies of the 3D proton tomography through polarized $T$-odd gluon TMDs at leading twist, obtained in a spectator-model framework. We embody in our approach a flexible parameterization for the spectator-mass spectral function, suited to catch both small- and moderate-$x$ effects. All these studies are relevant to unveil the gluon dynamics inside hadrons, which represents a core research line of studies at new-generation colliders, such as the Electron-Ion Collider, NICA-SPD, the High-Luminosity LHC, and the Forward Physics Facility.


Introduction
One of the ultimate goals of frontier researches in particle physics is unraveling the inner structure of nucleons in terms of the distribution of their constituents. The collinear factorization is a wellestablished formalism that has collected many successes since the advent of the parton model. A key role in the description of high-energy hadronic and lepto-hadronic collisions is played by the onedimensional parton distribution functions (PDFs). However, there are fundamental questions about the deep nature of strong interactions that are still open and whose answers go beyond the reach of a pure collinear description. As an example, unveiling the origin of proton mass and spin requires a viewpoint stretched to a three-dimensional, tomographic description, which is naturally provided by the so-called transverse-momentum-dependent (TMD) factorization.
A striking difference between TMD and collinear densities is represented by the gauge-link sensitivity. In particular, the fact that TMDs are sensitive to the transverse components of the gauge link makes them process dependent (see Refs. [13][14][15]). Quark TMDs depend on processes through the [+] and [−] staple links, which determine the direction of future-and past-pointing Wilson lines, respectively. The gluon TMDs have a more complicated gauge-link dependence, since they are sensitive on combinations of staple links. This fact leads to a more diversified kind of modified universality. Two major gluon gauge links emerge: the f -type and the d-type ones. They are also known in the context of small-x studies as Weiszäcker-Williams and dipole structures, respectively. The antisymmetric f abc QCD color structure is part of the f -type T -odd gluon-TMD correlator, whereas the symmetric d abc structure appears in the d-type T -odd one. This brings to a dependence of f -type gluon TMDs on the [±, ±] gauge-link combinations, while d-type gluon TMDs are characterized by the [±, ∓] ones. More intricate, box-loop gauge links appear in processes where multiple color exchanges connect both initial and final state states [16], thus leading however to a violation of the TMD factorization [17].
A spectator-model calculation of quark TMDs in the proton was done in Refs. [43,44]. A comprehensive framework was recently built [45] (see also Refs. [46][47][48][49]) for all the T -even gluon TMDs at twist-2 by defining an enhanced spectator model for the parent proton to effectively catch effects coming from the high energy resummation.
In this work we report a preliminary study on the T -odd gluon TMDs, the f -type Sivers and linearity functions, which are connected to relevant single-spin asymmetries arising from the distribution of unpolarized and linearly-polarized gluons inside a transversely polarized proton.

T-odd gluon TMDs in a spectator model
The spectator-model framework is based on a simple and intuitive assumption, namely that the incoming proton with mass M and four-momentum P emits a gluon having longitudinal fraction x, four-momentum p, and transverse momentum p T , and the remainders are effectively treated as an onshell spectator particle with mass M X and spin-1/2. The nucleon-gluon-spectator vertex is modeled as follows the τ 1 and τ 2 functions being dipolar form factors in p 2 T . A dipolar choice for the couplings is useful to remove gluon-propagator divergences, suppress large-p T effects which are beyond the reach of a pure TMD description, and dampen logarithmic singularities coming from p T -integrated distributions. All the unpolarized and polarized spectator-model T -even gluon TMDs at twist-2 in the proton were obtained in [45]. In that work the naive spectator-model approach was improved by allowing the spectator mass M X to spread over a continuous range of values via a flexible spectral function suited to capture both small-and moderate-x effects (see Eqs. (16) and (17) of Ref. [45]). The model parameters encoded in the definition of the spectral function and in the spectator-model correlator were determined through a simultaneous fit of the unpolarized and helicity gluon TMD densities, f g 1 and g g 1 , to the corresponding collinear PDF distributions obtained from NNPDF [50,51] at the initial scale Q 0 = 1.64 GeV. The size of the statistical uncertainty was assessed by means of the bootstrap method.
Since the tree-level approximation for the gluon correlator does not account for the gauge link, our T -even TMD distributions turn out to be process-independent. In order to generate T -odd structures in the gluon correlator, we need to go beyond the tree level and include its interference with a distinct channel. Similarly to the quark TMD case, we have considered the one-gluon exchange in eikonal approximation. This diagram corresponds to the truncation at first order of the whole gauge-link operator. The main effect of this procedure is that the obtained T -odd functions become sensitive to gauge links, and thus process dependent. For the given f -type gauge link, two Sivers TMDs ( f ⊥ 1T ) and two linearity TMDs (h 1 ) are obtained by suitably projecting the transverse part of the corresponding gluon correlator. For each pair, the two partners are connected by the following modified-universality relation In our preliminary analysis we have employed a simplified expression for the nucleon-gluon-spectator vertex, with the τ 2 form factor in Eq. (1) set to zero. For the sake of consistency, we have fitted the model parameters to NNPDF parametrizations by using the simplified expression for the vertex.
In upper panels of Fig. 1 we present the transverse-momentum dependence of the p T -weighted [+, +] Sivers function for two representative values of the longitudinal fraction, x = 10 −1 and x = 10 −3 , and at the initial scale Q 0 = 1.64 GeV. Corresponding results for the [+, +] linearity function are given in ower panels. By inspecting our plots, it emerges that both the distributions have a non-Gaussian pattern in p 2 T , with a large flattening tail at large p 2 T -values and a small but nonzero value when p 2 T → 0, which suggests that in this limit both TMDs diverge at most as 1/|p T |. At variance with the T -even unpolarized and the Boer-Mulders gluon functions (see Fig. (4) of Ref. [45]), the bulk of our f -type T -odd functions increases when x grows. This suggests that transverse single-spin asymmetries could be less manifest in the low-x regime. We remark, however, that our results could change even radically when the full-vertex calculation will become available.

Conclusions and prospects
We have enhanced our spectator-model framework by performing preliminary calculation of two f -type T -odd gluon TMDs: the Sivers and the linarity functions. The full calculation of all the Todd gluon TMDs, including the d-type ones is underway. They can serve as a useful guidance to shed light on gluon-TMD dynamics at new-generation particle colliders and experiments, such as the Electron-Ion Collider (EIC) [52], NICA-SPD [53], the High-Luminosity Large Hadron Collider (HL-LHC) [54], and the Forward Physics Facility (FPF) [55].