Gluon TMDs for tensor polarized deuteron in a spectator model

1. Problem Statement

The paper addresses the theoretical gap in understanding the internal structure of spin-1 hadrons, specifically the tensor-polarized deuteron, through the lens of Transverse-Momentum-Dependent distributions (TMDs).

• Context: While TMDs for spin-1/2 nucleons are well-studied, those for spin-1 hadrons remain underdeveloped. Spin-1 hadrons possess unique polarization degrees of freedom (tensor polarization) that allow for observables not accessible in nucleons.
• Specific Challenge: There is a lack of theoretical predictions for gluon TMDs in tensor-polarized deuterons. A key observable of interest is the gluon transversity ($\Delta_T g(x) \equiv -h_{1TT}(x)$), which corresponds to the distribution of linearly polarized gluons within a transversely tensor-polarized target. This observable is a direct probe of non-nucleonic degrees of freedom (e.g., $q\bar{q}$ pairs or six-quark configurations) within the deuteron, as it cannot arise from a simple system of bound nucleons.
• Goal: To provide a model-based calculation of leading-twist, T-even gluon TMDs for the tensor-polarized deuteron to guide future experimental measurements at facilities like the EIC, JLab, and NICA.

2. Methodology

The authors employ a spectator model framework adapted for a spin-1 target.

• Model Assumptions:
  • The deuteron ($|P, S, T\rangle$) is treated as emitting a time-like, off-shell gluon with momentum $k$.
  • The remaining system is modeled as a single on-shell spectator particle with spin-1 and momentum $P-k$.
  • Unlike simple models with a fixed spectator mass, the spectator mass $M_S$ is allowed to vary over a continuous range, weighted by a spectral function $\rho(M_S)$. This function includes a smooth background and a resonance peak to effectively model $q\bar{q}$ contributions.
• Polarization Formalism:
  • The polarization of the spin-1 deuteron is described by a spin vector $S^\mu$ and a symmetric, traceless rank-2 spin tensor $T^{\mu\nu}$.
  • The density matrix incorporates both vector and tensor polarization states.
• Interaction Vertex:
  • The deuteron-gluon-spectator coupling is modeled via an effective vertex containing three independent form factors ($g_1, g_2, g_3$).
  • To regularize ultraviolet divergences arising from transverse momentum integration, the authors adopt exponential form factors dependent on the gluon virtuality $k^2$.
• Calculation Steps:
  1. Construct the gauge-invariant gluon-gluon correlator $\Phi^{\mu\nu}$ at tree level.
  2. Decompose the correlator using projection operators to extract the 13 leading-twist T-even gluon TMDs.
  3. Integrate the results over the spectator mass $M_S$ using the spectral function $\rho(M_S)$.
  4. Parameter Fitting: The model parameters (11 total: 7 for the spectral function, 4 for form factors) are determined by fitting the integrated unpolarized gluon TMD, $f_1(x)$, to the nNNPDF1.0 parametrization at a scale $Q_0 = 2$ GeV. The bootstrap method (100 replicas) is used to estimate uncertainties.

3. Key Contributions

• Analytical Derivation: The paper provides the first analytical expressions for 13 T-even gluon TMDs for a tensor-polarized deuteron within a spectator model. These include:
  • 1 Unpolarized ($f_1$)
  • 3 Vector-polarized ($h_1^\perp, g_1, g_{1T}$)
  • 9 Tensor-polarized ($f_{1LL}, f_{1LT}, f_{1TT}, h_{1LL}^\perp, h_{1LT}, h_{1LT}^\perp, h_{1TT}, h_{1TT}^\perp, h_{1TT}^{\perp\perp}$)
• Spectral Function Integration: The introduction of a continuous spectral function for the spectator mass allows the model to capture dynamic effects (like $q\bar{q}$ contributions) that fixed-mass models miss, particularly for tensor-polarized observables.
• Positivity Verification: The authors explicitly verify that their numerical results satisfy the theoretical positivity bounds derived for both TMDs and their collinear limits, ensuring the physical consistency of the model.
• Coupling Sensitivity: The study highlights the critical role of the coupling $g_2(k^2)$. In the approximation where $g_2 \to 0$, certain TMDs ($g_1, g_{1T}$) vanish, and specific relations between other TMDs emerge. The full model shows that $g_2$ is essential for generating non-zero vector-polarized gluon distributions.

4. Results

• Unpolarized Distribution ($f_1$): The model successfully reproduces the nNNPDF1.0 parametrization for the unpolarized gluon distribution $xf_1(x)$. The predicted average momentum fraction carried by gluons is $\langle x \rangle_g \approx 0.412$.
• Vector-Polarized TMDs: The distributions $xg_1(x)$ and $xg_{1T}^{(1)}(x)$ are found to be non-zero but significantly smaller (several times smaller) than $xf_1(x)$. Their magnitude is highly sensitive to the $g_2$ coupling.
• Tensor-Polarized TMDs:
  • The tensor-polarized distributions exhibit significant non-negligible magnitudes.
  • Gluon Transversity ($h_{1TT}$): The distribution $xh_{1TT}(x)$ (related to $\Delta_T g$) increases with $x$ and is negative across the kinematic range. This is a unique signature of spin-1 structure.
  • LL and LT Polarizations: The integrated tensor-polarized PDFs ($f_{1LL}, h_{1LT}$, etc.) show distinct $x$-dependencies. Notably, $xf_{1LL}(x)$ follows a trend analogous to the deuteron structure function $b_1(x)$.
• Transverse Momentum Dependence ($k_T$):
  • The TMDs display a non-Gaussian shape in $k_T^2$, characterized by a long tail at large $k_T$.
  • They are concentrated at small $k_T$ with a peak that shifts to larger $k_T$ and flattens as $x$ increases.
  • The unpolarized distribution $f_1(x, k_T^2)$ does not vanish at $k_T \to 0$, indicating a significant contribution from gluon orbital angular momentum ($L=1$).

5. Significance

• Experimental Guidance: The results provide concrete numerical predictions for future experiments at the Electron-Ion Collider (EIC), JLab, and NICA. Specifically, the non-zero predictions for tensor-polarized gluon TMDs offer targets to probe the internal spin structure of the deuteron.
• Probing Non-Nucleonic Degrees of Freedom: The calculation of gluon transversity ($\Delta_T g$) is crucial because a non-zero signal would definitively indicate the presence of non-nucleonic degrees of freedom (such as hidden color or $q\bar{q}$ components) within the deuteron, which cannot be explained by a simple nucleon-nucleon bound state.
• Theoretical Framework: This work extends the successful spectator model formalism (previously used for quark TMDs in nucleons and pions) to the gluon sector of spin-1 hadrons, providing a robust phenomenological tool for future QCD studies.
• Future Directions: The authors note that while this work covers T-even TMDs, future extensions will address T-odd TMDs and the distinction between Weizsäcker-Williams (WW) and dipole gluon TMDs, which depend on the process-specific gauge links.

In summary, this paper establishes a foundational theoretical framework for understanding gluon dynamics in tensor-polarized nuclei, predicting measurable effects that could revolutionize our understanding of hadron structure in the nuclear domain.