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Observables in exclusive heavy quarkonium electroproduction at NLO and prospects for the EIC

This paper presents next-to-leading order perturbative QCD predictions for exclusive heavy-quarkonium electroproduction at HERA to establish benchmarks for future Electron-Ion Collider measurements, while advocating for a dedicated resummation framework to address challenges at large photon virtualities.

Original authors: Chris A. Flett

Published 2026-09-14

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Chris A. Flett

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep inside the heart of every atom, protons and neutrons are not solid, featureless spheres. They are bustling cities of energy, filled with a seething cloud of particles called gluons that bind the fundamental building blocks of matter together. Understanding how these gluons are arranged, especially when they are packed tightly together or moving at extreme speeds, is one of the great challenges of modern physics. For decades, scientists have used high-energy collisions to probe this invisible world, much like shining a flashlight into a dark room to see what is hidden in the shadows. A particularly powerful way to do this is by firing electrons at protons and watching how the electrons scatter. When an electron hits a proton, it can emit a burst of energy that briefly transforms into a heavy particle, which then interacts with the proton's internal gluons before emerging as a new, distinct particle. By studying these rare events, physicists can map the density and behavior of the gluons inside the proton, revealing the structure of matter itself.

A recent study by physicist Chris Flett takes a closer look at these rare events, specifically focusing on a type of heavy particle known as the J/psi meson. This particle is made of a heavy quark and its antimatter partner, and it serves as a very clean probe for the gluons inside a proton. Chris Flett performed detailed calculations to predict exactly what should happen when an electron collides with a proton to create this meson, using the most advanced mathematical tools available in quantum physics. They compared their predictions against data collected by the HERA accelerator in Germany, which operated for many years, and looked ahead to what the future Electron-Ion Collider will be able to measure. The goal was to see if the current theories hold up under the most precise scrutiny and to identify where our understanding might need to be refined.

The researchers found that their calculations, which included complex corrections to the basic theory, matched the existing experimental data from HERA very well. They looked at how often these heavy particles are produced at different energy levels and found that the predictions remained stable and reliable across a wide range of conditions. One of the most interesting discoveries was how the orientation of the produced particle changes as the energy of the collision increases. At lower energies, the particles emerge in a mix of orientations, but as the energy rises, they become almost perfectly aligned in a specific direction. This shift was predicted by the theory and confirmed by the data, showing that the underlying physics is well understood in this high-energy regime.

However, the study also highlighted a specific area where the current theory becomes difficult to manage. When the energy of the collision becomes extremely high, certain mathematical terms in the equations begin to grow very large, making the predictions less precise and harder to trust. The researchers showed that while the standard calculations work well for the energies currently available, they will struggle to provide accurate answers for the much higher energies that the future Electron-Ion Collider will reach. To solve this, they argue that scientists need to develop a new, specialized method to handle these large terms, essentially reorganizing the math to keep the predictions reliable even at the highest possible energies.

The work serves as a crucial bridge between the past achievements of the HERA accelerator and the upcoming capabilities of the Electron-Ion Collider. By confirming that the current theories work well for the data we have, the study gives scientists confidence in their models. At the same time, by pinpointing exactly where the models start to break down at very high energies, it provides a clear roadmap for the next generation of research. The Electron-Ion Collider will be able to collect far more data than ever before, allowing physicists to test these refined theories with unprecedented precision. This will not only confirm the behavior of gluons in protons but also extend these studies to heavier atomic nuclei, potentially revealing how gluons behave when they are packed even more densely, a state of matter that existed just moments after the Big Bang.

Technical Summary: Observables in Exclusive Heavy Quarkonium Electroproduction at NLO and Prospects for the EIC

Problem Statement
The paper addresses the theoretical description of exclusive heavy-quarkonium electroproduction ( $e^-p \to e^-Vp$ , where $V = J/\psi, \psi', \Upsilon$ ) in the context of Next-to-Leading Order (NLO) perturbative QCD (pQCD). While Leading Order (LO) calculations have established the sensitivity of these processes to the small- $x$ gluon density, precise phenomenological predictions at NLO are required to interpret existing HERA data and prepare for the upcoming Electron-Ion Collider (EIC). A specific challenge lies in the factorization of transversely polarized photons, for which no general proof exists, and the handling of large logarithmic corrections ( $\ln(1/\xi)$ and $\ln(Q^2/m_c^2)$ ) that arise at high energies and large photon virtualities ( $Q^2$ ).

Methodology
The author performs a phenomenological study of the hard-scattering subprocess $\gamma^* p \to V p$ within the collinear factorization framework.

Formalism: The scattering amplitudes are expressed as convolutions of Generalized Parton Distributions (GPDs) and NLO coefficient functions. The study adopts the static approximation for the heavy quark-antiquark pair ( $m_Q = M_V/2$ ).
GPD Construction: Since direct GPD data is limited, the author utilizes the Shuvaev transform to relate GPDs to standard forward Parton Distribution Functions (PDFs) at small $x$ and skewness $\xi$ . The input PDFs are taken from the CT18ANLO set.
Kinematics and Scales: Calculations are performed for various scales $\mu_F = \mu_R = \zeta \mu$ , where $\mu = \sqrt{Q^2 + 4m_c^2}$ and $\zeta \in \{1/2, 1, 2\}$ . The analysis covers a range of photon virtualities ( $Q^2$ ) and center-of-mass energies ( $W$ ).
Cross-Section Integration: Differential cross-sections at $t=0$ are calculated using helicity amplitudes. Integrated cross-sections are derived by assuming an exponential $t$ -dependence parametrized by a Regge form fitted to elastic $J/\psi$ and diffractive $pp$ data.

Key Contributions and Results

NLO Amplitude Stability: The author presents the imaginary parts of helicity amplitudes ( $M_{\pm\pm}$ and $M_{00}$ ) at LO and NLO. They observe that while LO and NLO predictions differ significantly at low $Q^2$ (including sign changes in the transverse amplitude for specific scale choices), the predictions converge as $Q^2$ increases, indicating good perturbative stability.
High-Energy Limit and Logarithms: In the high-energy limit ( $W^2 \gg Q^2 + 4m_c^2$ ), the NLO correction exhibits a large double-logarithmic enhancement $\sim \alpha_s \ln(1/\xi) \ln(\bar{Q}^2/\mu_F^2)$ . The author demonstrates that choosing the factorization scale $\mu_F^2 = \bar{Q}^2 = (Q^2 + 4m_c^2)/4$ effectively eliminates this enhancement, thereby resumming the associated family of terms $(\alpha_s \ln(1/\xi) \ln(\bar{Q}^2/\mu_F^2))^n$ .
Comparison with HERA Data:
- Cross-Sections: Predictions for the exclusive $J/\psi$ cross-section as a function of $W$ (at fixed $Q^2$ ) and $Q^2$ (at fixed $W$ ) show favorable agreement with HERA data (H1 and ZEUS). The scale variation bands at NLO are generally consistent with data, though the lower boundary of the band (corresponding to the resummed scale) exhibits a distinct energy dependence that avoids broadening at large $W$ .
- Polarization: The ratio of longitudinal to transverse cross-sections ( $R = \sigma_L/\sigma_T$ ) and the resulting polarization parameter $\lambda_\theta$ are calculated. The results show that $\lambda_\theta \to -1$ at large $Q^2$ , indicating a dominance of longitudinally polarized production, which aligns with experimental observations. Notably, the factorization-scale uncertainty largely cancels in the ratio $R$ .
EIC Prospects: The paper projects that the EIC will provide significantly higher statistics, particularly in the $Q^2$ region where current data is sparse. This will allow for a more precise determination of the $Q^2$ dependence and clarification of discrepancies between existing HERA datasets.

Significance and Future Directions
The paper establishes that NLO pQCD calculations, when combined with the Shuvaev transform and appropriate scale choices, provide a robust framework for describing exclusive heavy quarkonium production. The work validates the use of these calculations as benchmarks for the EIC.

Crucially, the author identifies a limitation in the fixed-order approach at large $Q^2$ : the gluon coefficient functions contain explicit $\ln^2(Q^2/m_c^2)$ terms, leading to parametric enhancements of NLO corrections relative to LO. Consequently, the paper advocates for the development of a dedicated and systematic resummation framework to handle these large logarithms. The author notes that an approach matching the Infrared Evolution Equation (IREE) with DGLAP evolution has been proposed in prior work but remains to be implemented for this specific process.

Technical Summary: Observables in Exclusive Heavy Quarkonium Electroproduction at NLO and Prospects for the EIC

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