Phenomenological studies of exclusive heavy-quarkonium electroproduction at NLO

1. Problem Statement

The paper addresses the theoretical description of exclusive heavy vector meson electroproduction ($\gamma^* p \to V p$, where $V = J/\psi$ or $\Upsilon$) in electron-proton ($ep$) collisions. While these processes are crucial probes for understanding the small-$x$ behavior of the gluon Parton Distribution Function (PDF) and saturation physics, previous theoretical descriptions often relied on Leading Order (LO) approximations or lacked a complete Next-to-Leading Order (NLO) treatment within the Collinear Factorisation (CF) framework combined with Non-Relativistic QCD (NRQCD).

Key challenges include:

• The need to confront NLO predictions with existing HERA data across a wide range of photon virtualities ($Q^2$) and center-of-mass energies ($W$).
• Preparing robust theoretical benchmarks for the upcoming Electron-Ion Collider (EIC), which will extend measurements to higher $Q^2$ and allow for the first measurements of $\Upsilon$ electroproduction with off-shell kinematics.
• Investigating the necessity of resumming logarithmically enhanced terms (specifically double logarithms $\ln^2(Q^2/m_Q^2)$) that arise in the high-$Q^2$ regime within the Fixed-Flavour-Number Scheme (FFNS).

2. Methodology

The authors build upon their previous derivation of NLO coefficient functions [17] to perform a comprehensive phenomenological analysis.

• Theoretical Framework:

  • Factorisation: The process is calculated within the collinear factorisation framework, convolving Generalised Parton Distributions (GPDs) with perturbative coefficient functions.
  • NRQCD: The transition from the open heavy quark-antiquark pair ($Q\bar{Q}$) to the bound vector meson is handled via the color-singlet NRQCD matrix element $\langle O_1 \rangle_V$.
  • Amplitude Calculation: The authors compute the real and imaginary parts of the coefficient functions directly using integral reduction techniques (QGRAF for diagrams, FORM for algebraic manipulation, Apart for partial fractioning, and Reduze 2 for IBP reduction to master integrals).
  • Renormalisation: The calculation employs the on-shell scheme for heavy quark mass and wavefunction renormalisation, and the $\overline{\text{MS}}$ scheme for the strong coupling constant (with heavy quark loops subtracted at zero momentum). Mass factorisation absorbs remaining collinear singularities into the GPDs.

• GPD Construction:

  • Since experimental data for exclusive production primarily probes the charge-conjugation-even quark GPD, the authors construct the quark-singlet and gluon GPDs using the Shuvaev transform. This method relates small-skewness GPDs to forward PDFs (CT18ANLO) via conformal moments.
  • The transform is applied in the region $|X| > \xi$ (DGLAP region) to avoid singularities associated with analytic continuation in the complex $N$-plane.

• Kinematics and Observables:

  • The study calculates the $t$-integrated cross-section $\sigma(\gamma^* p \to V p)$ and the ratio of longitudinal to transverse cross-sections $R = \sigma_L / \sigma_T$.
  • Scale variations ($\mu_F, \mu_R$) are performed around the central scale $\mu = \sqrt{Q^2 + 4m_c^2}$ to estimate theoretical uncertainties.
  • The $t$-dependence is modeled via an exponential slope $B(W)$, parameterized based on HERA fits and Pomeron trajectory models.

3. Key Contributions

• First NLO Phenomenological Application: This is the first detailed phenomenological study applying the complete NLO coefficient functions for exclusive heavy vector meson electroproduction to experimental data.
• EIC Predictions: The paper provides quantitative predictions for the EIC, including event rates and kinematic coverage for both $J/\psi$ and $\Upsilon$ production, highlighting the collider's potential to probe gluon PDFs at intermediate resolution scales.
• Analysis of Logarithmic Enhancements: The authors identify and analyze the presence of double-logarithmic terms ($\propto \ln^2(Q^2/m_Q^2)$) in the NLO gluon coefficient functions. They discuss the potential need for resummation using the Infrared Evolution Equation (IREE) matched with DGLAP, though they find current data does not yet demand it.
• Resolution of Scale Dependence: The study demonstrates that while individual cross-sections show significant scale dependence, the ratio $R = \sigma_L/\sigma_T$ exhibits much milder scale dependence, making it a robust observable for testing QCD dynamics.

4. Results

• Comparison with HERA Data:

  • The NLO predictions for $J/\psi$ electroproduction show good agreement with HERA data (H1 and ZEUS) across a broad range of $Q^2$ (up to $\sim 100 \text{ GeV}^2$) and $W$.
  • The NLO framework successfully describes the data where LO calculations often struggle, particularly regarding the scale stability and the magnitude of the cross-section.
  • The ratio $R = \sigma_L/\sigma_T$ is well-described by NLO, with the longitudinal component becoming dominant at higher $Q^2$ due to the coupling to small-size $c\bar{c}$ pairs.

• PDF Sensitivity:

  • At low $Q^2$, the cross-section is sensitive to the choice of input PDFs (CT18A, MSHT20, NNPDF3.1) due to the interplay between quark and gluon contributions.
  • At high $Q^2$ ($Q^2 \gg m_c^2$), the gluon contribution dominates, and the predictions become largely independent of the specific PDF set choice, with uncertainties driven primarily by the gluon PDF magnitude.

• $\Upsilon$ Production:

  • Predictions for $\Upsilon$ electroproduction indicate significantly lower event rates compared to $J/\psi$ due to the heavier mass ($M_\Upsilon$).
  • Even at the EIC, $\Upsilon$ electroproduction will be limited to the photoproduction bin and low $x$ regions, offering a more restricted kinematic lever arm for studying scale evolution compared to $J/\psi$.

• Perturbative Convergence:

  • For $Q^2 \gg m_c^2$, the perturbative series converges well (LO and NLO are in line).
  • For $Q^2 \lesssim m_c^2$ (photoproduction limit), convergence deteriorates, and the NLO amplitude exhibits a sign change in its imaginary part relative to LO, a known issue in CF for exclusive photoproduction.

• Logarithmic Resummation:

  • While double-logarithmic terms ($\ln^2(Q^2/m_Q^2)$) are present and grow with $Q^2$, the current agreement between NLO CF predictions and HERA data up to the highest measured virtualities suggests that resummation is not yet strictly necessary. However, the EIC will provide the statistical precision required to determine if resummation becomes relevant at higher scales.

5. Significance

This work solidifies the NLO Collinear Factorisation framework as a robust tool for describing exclusive heavy quarkonium production. Its significance lies in:

1. Validating Theory: Confirming that NLO calculations can accurately describe existing HERA data, thereby increasing confidence in theoretical extrapolations.
2. Guiding Future Experiments: Providing essential benchmarks for the EIC, helping experimentalists optimize their strategies for measuring gluon PDFs and testing saturation physics.
3. Identifying Future Theoretical Needs: Highlighting the specific kinematic regimes (high $Q^2$) where current fixed-order calculations may eventually require resummation of logarithmic terms, setting the stage for future theoretical developments involving IREE-DGLAP matching.
4. Precision QCD: Enabling more precise extractions of the gluon distribution function at intermediate resolution scales, deepening the understanding of nucleon structure and strong interaction dynamics.