[Paper Review] Towards leading-twist $T$-odd TMD gluon distributions

Alessandro Bacchetta, Francesco Giovanni Celiberto+1 more|arXiv (Cornell University)|Jan 25, 2022

Particle physics theoretical and experimental studies4 citations

TL;DR

This paper presents a first exploratory study of leading-twist $T$-odd transverse-momentum-dependent (TMD) gluon distributions—specifically the $f$-type Sivers and linearity functions—within a spectator-model framework. By incorporating a flexible spectral function for the spectator mass and including one-gluon exchange in the eikonal approximation, the authors generate process-dependent $T$-odd TMDs sensitive to gauge links, revealing non-Gaussian transverse momentum dependence with a $1/|\boldsymbol{p}_T|$ divergence at small $\boldsymbol{p}_T^2$, suggesting enhanced contributions at moderate $x$.

ABSTRACT

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.

Motivation & Objective

• To explore the dynamics of $T$-odd gluon TMDs in the proton, particularly the $f$-type Sivers and linearity functions, which are key to understanding single-spin asymmetries in high-energy processes.
• To extend the spectator-model framework to include $T$-odd structures by incorporating gauge-link sensitivity through one-gluon exchange in the eikonal approximation.
• To model the spectator mass as a continuous spectral function to capture both small- and moderate-$x$ effects, improving upon the naive spectator model.
• To provide a phenomenologically viable framework for predicting $T$-odd gluon TMDs relevant to future experiments at EIC, NICA-SPD, HL-LHC, and FPF.
• To lay the groundwork for a full calculation of all $T$-odd gluon TMDs, including $d$-type structures, in a process-dependent, factorization-consistent manner.

Proposed method

• A spectator-model framework is employed where the proton emits a gluon with longitudinal fraction $x$, transverse momentum $\boldsymbol{p}_T$, and the residual system is treated as an on-shell spectator with mass $\mathcal{M}_X$.
• The nucleon-gluon-spectator vertex is modeled using dipolar form factors $\tau_1(p^2)$ and $\tau_2(p^2)$, with $\tau_2$ set to zero in the current study for consistency.
• A flexible spectral function is introduced for $\mathcal{M}_X$ to allow continuous mass distribution, enabling better capture of small- and moderate-$x$ effects.
• Gauge-link dependence is introduced via one-gluon exchange in the eikonal approximation, breaking tree-level universality and enabling $T$-odd structure formation.
• The $f$-type $T$-odd TMDs—Sivers ($f_{1T}^{\perp g}$) and linearity ($h_1^g$)—are projected from the gluon correlator, with $[+,+]$ and $[-,-]$ gauge-link configurations related by modified universality.
• Model parameters are fitted to NNPDF collinear PDFs ($f_1^g$, $g_1^g$) at $Q_0 = 1.64$ GeV using a bootstrap method to assess statistical uncertainty.

Experimental results

Research questions

• RQ1 How do $T$-odd $f$-type gluon TMDs—specifically the Sivers and linearity functions—behave in transverse momentum and longitudinal fraction $x$?
• RQ2 What is the impact of a flexible spectator-mass spectral function on the shape and $x$-dependence of $T$-odd gluon TMDs?
• RQ3 How does the inclusion of gauge-link effects through one-gluon exchange in the eikonal approximation affect the process dependence and universality of $T$-odd TMDs?
• RQ4 What is the functional form of the $T$-odd gluon TMDs in the small-$\boldsymbol{p}_T^2$ limit, and does it exhibit a $1/|\boldsymbol{p}_T|$ divergence?
• RQ5 How do the $T$-odd TMDs compare in magnitude and shape to their $T$-even counterparts, particularly in the low-$x$ regime?

Key findings

• The $[+,+]$ Sivers function exhibits a non-Gaussian transverse momentum dependence with a flattening tail at large $\boldsymbol{p}_T^2$, indicating enhanced contributions in the high-$p_T$ region.
• The $[+,+]$ linearity function also shows a non-Gaussian shape with a small but nonzero value at $\boldsymbol{p}_T^2 \to 0$, suggesting a $1/|\boldsymbol{p}_T|$ divergence in the small-$\boldsymbol{p}_T$ limit.
• Both $T$-odd functions increase with growing $x$, indicating that transverse single-spin asymmetries may be more prominent at moderate $x$ than at low $x$.
• The modified universality relation $f_{1T}^{\perp g[+,+]} = -f_{1T}^{\perp g[-,-]}$ and $h_1^{g[+,+]} = -h_1^{g[-,-]}$ holds for the $f$-type $T$-odd TMDs in the model.
• The model parameters, fitted to NNPDF PDFs at $Q_0 = 1.64$ GeV, yield a consistent description of the unpolarized and helicity gluon TMDs, validating the framework's reliability.
• The results are preliminary and expected to change significantly upon full inclusion of the $\tau_2$ form factor and full vertex calculation.