⚛️ phenomenology

Inclusive productions of $J/\psi+\eta_c$ in $e^{+}e^{-}$ annihilation at Belle

This paper presents a next-to-leading order NRQCD analysis of inclusive $J/\psi+\eta_c$ production in $e^+e^-$ annihilation, demonstrating that color-octet mechanisms and feed-down effects significantly enhance cross sections at $\Upsilon(4S)$ energies and offering a promising avenue for observation using Belle's detector capabilities.

Original authors: Zhan Sun, Shu-Jian Qi, Ying-Zhao Jiang

Published 2026-09-10

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CC BY 4.0

Original authors: Zhan Sun, Shu-Jian Qi, Ying-Zhao Jiang

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

In the subatomic world, matter is built from a handful of fundamental particles, among them the quark. Quarks rarely travel alone; they are bound together by the strong force, the most powerful interaction in nature, to form composite particles called hadrons. When a heavy quark and its own antiparticle, an antiquark, bind together, they create a specific type of hadron known as a quarkonium. These particles are like tiny, short-lived laboratories where physicists can test the rules of quantum mechanics. For decades, the leading theory for how these heavy particles form has been a framework called non-relativistic quantum chromodynamics, or NRQCD. This theory suggests that when quarks bind, they can do so in different "colors" of charge, much like how a painter might mix primary colors to get a final shade. The theory predicts that while the most obvious way for them to bind is through a direct, color-neutral connection, there are also more complex, indirect pathways involving colored intermediates. Checking whether these complex pathways actually happen, especially at lower energy levels, is crucial because if the theory fails here, our understanding of how matter is constructed might need a major revision.

A team of researchers at Guizhou Minzu University has taken a fresh look at this problem by simulating a specific collision event: an electron smashing into a positron to produce a pair of charmonium particles, specifically a J/ψ and an ηc, along with other debris. While previous studies had looked at similar events, they often stopped at the simplest level of calculation or ignored the complex, colored pathways. The researchers in this study went much further, performing a highly detailed calculation that included the next level of complexity in the math, known as next-to-leading order. This allowed them to see how the probability of these particles forming changes when you account for the messy, real-world interactions of the strong force, as well as the subtle effects of the electromagnetic force. They ran these simulations across a range of energies, focusing on the conditions found at the Belle experiment in Japan, where such collisions are actually recorded.

The results of this detailed simulation reveal that the complex, colored pathways play a much more significant role than previously thought in this low-energy environment. When the researchers included these indirect formation routes, the predicted number of J/ψ and ηc pairs produced increased by about 20 to 30 percent at the energy level of the Υ(4S) resonance, a specific energy state where the Belle experiment operates. This is a substantial shift. More importantly, the way the production rate changes as the energy of the collision increases is different depending on whether you include these complex pathways or not. The simple, direct formation routes predict a rapid drop-off in production as energy rises, but the inclusion of the complex routes slows this drop significantly. This difference in behavior acts like a fingerprint; if future experiments measure the rate of these particles at different energies, they can see which prediction matches reality and determine if the complex theory holds up.

The study also found that the electromagnetic force, often considered a minor player in these heavy-particle interactions, actually contributes a noticeable amount to the total production rate. At the energies studied, the interference between the strong force and the electromagnetic force adds another 10 to 20 percent to the expected number of events. This contribution grows as the collision energy increases, becoming a major factor that cannot be ignored. Furthermore, the researchers considered that some of the J/ψ particles observed might not be formed directly but could instead come from the decay of a heavier, excited cousin called the ψ(2S). When they added this "feed-down" effect to their calculations, the total predicted number of events increased by another 40 percent at the Υ(4S) energy level.

The researchers concluded that this process is not just a theoretical exercise but is within reach of current experimental capabilities. Using the specific detection methods employed by the Belle collaboration, which can identify the J/ψ particle by its decay into two muons and the ηc through six different decay patterns, the team estimated that the experiment could collect enough data to observe this phenomenon. At the Υ(4S) resonance, they predict that with the high amount of data expected from future runs, hundreds to perhaps a thousand events could be recorded. This potential for observation turns the theoretical prediction into a testable reality. If the experimental data matches the researchers' predictions, it will provide strong evidence that the complex, colored pathways are indeed active and necessary for describing how heavy particles form, even at the relatively low energies of modern particle colliders. If the data does not match, it could signal that the current theory of how quarks bind needs to be fundamentally rethought.

Technical Summary of "Inclusive productions of J/ψ + ηc in e+e− annihilation at Belle"

Problem Statement
The validity of the Nonrelativistic QCD (NRQCD) factorization framework at low energies remains a subject of debate. While NRQCD successfully describes high transverse-momentum ( $p_T$ ) charmonium production in hadroproduction, recent studies indicate discrepancies between NRQCD predictions and experimental data regarding integrated $p_T$ cross sections at colliders like RHIC and LHC. Specifically, the energy dependence observed experimentally often aligns with Leading Order (LO) Color-Singlet (CS) predictions, whereas the inclusion of QCD corrections and Color-Octet (CO) mechanisms can lead to unphysical behaviors or fail to dominate as expected. Electron-positron ( $e^+e^-$ ) annihilation at B-factories (centered around the $\Upsilon(4S)$ resonance) offers a cleaner environment with reduced backgrounds, making it an ideal testing ground for NRQCD validity in the low-energy regime. While exclusive double-charmonium production ( $e^+e^- \to J/\psi + J/\psi$ ) has shown theoretical suppression of CO contributions, the role of the CO mechanism in inclusive double-charmonium production ( $e^+e^- \to J/\psi + \eta_c + X$ ) remains theoretically incomplete. Previous studies were restricted to LO CS contributions or omitted relevant CO terms, leaving a gap in understanding how CO mechanisms influence the energy dependence of the total cross section.

Methodology
The authors perform a systematic Next-to-Leading Order (NLO) analysis of the inclusive process $e^+e^- \to J/\psi + \eta_c + X$ within the NRQCD factorization framework. The calculation involves the following key components:

Factorization: The cross section is factorized into perturbatively calculable Short-Distance Coefficients (SDCs) and non-perturbative Long-Distance Matrix Elements (LDMEs).
Process Classification: The contributing amplitudes are categorized based on the color structure of the intermediate $c\bar{c}$ $c \overset{c}{ˉ}$ pairs:
- CS+CS: $e^+e^- \to c\bar{c}[^3S_1^{[1]}] + c\bar{c}[^1S_0^{[1]}]$ . This channel includes both conventional QCD diagrams and QED contributions (specifically single-photon fragmentation, SPF).
- CS+CO: Processes involving one CS and one CO state (e.g., $c\bar{c}[^3S_1^{[1]}] + c\bar{c}[^3S_1^{[8]}]$ ). These include channels with soft gluon emissions that induce infrared (IR) divergences, which are handled via LDME renormalization.
- CO+CO: Processes involving two CO states. The authors focus on channels expected to yield sizable contributions, particularly those involving the $^3S_1^{[8]}$ single-gluon fragmentation (SGF) topology.
Radiative Corrections: The calculation includes NLO QCD corrections for all relevant channels. For the CS+CS channel, QED corrections at $\mathcal{O}(\alpha^2)$ and their interference with QCD at $\mathcal{O}(\alpha^3)$ are included.
Renormalization: Ultraviolet (UV) and IR divergences are regularized using dimensional regularization. Renormalization is performed in the on-mass-shell (OS) scheme for the charm quark mass and field, and the $\overline{\text{MS}}$ scheme for the QCD coupling and gluon field.
Inputs: The study utilizes four representative sets of $J/\psi$ -related LDMEs from existing literature (Refs. [101–104]). $\eta_c$ LDMEs are derived from $J/\psi$ matrix elements using heavy-quark spin symmetry. Calculations are performed at $\sqrt{s} = 10.52$ GeV ( $\Upsilon(4S)$ ), $\Upsilon(5S)$ , and extended up to 20 GeV.

Key Contributions and Results

Enhancement by QCD Corrections: QCD corrections significantly enhance the LO results for both CS and CO channels. For the pure CS channel, NLO corrections increase the cross section by approximately a factor of two. For CO channels, the enhancement is even more pronounced, increasing LO results by roughly a factor of three.
Role of QED Diagrams: The interference between QED and QCD diagrams in the CS channel contributes an additional 10%–20% to the NLO predictions at $\sqrt{s} = \Upsilon(4S)$ . This contribution is non-negligible and grows rapidly with collision energy, becoming comparable to NNLO corrections at $\sqrt{s} = 20$ GeV.
Impact of Color-Octet Mechanisms: The inclusion of CO terms increases the inclusive cross section by 20%–30% at $\Upsilon(4S)$ $Υ (4 S)$ , but this enhancement is specific to the results obtained using LDME sets 2, 3, and 4. For LDME set 1, the NRQCD prediction remains close to the CS prediction.
- Using "Set 1" LDMEs, the NRQCD prediction is close to the CS prediction, exhibiting a similar energy dependence ( $\sim s^{-4}$ ).
- Using "Sets 2, 3, and 4," the NRQCD predictions are significantly larger (20–30% higher at $\Upsilon(4S)$ , up to 3–4 times higher at 20 GeV) and display a distinct, softer energy dependence ( $\sim s^{-3}$ ) compared to the CS prediction.
Feed-down Contributions: The study estimates that feed-down from $\psi(2S)$ production ( $e^+e^- \to \psi(2S) + \eta_c + X$ ) contributes approximately 6.5 fb to the inclusive $J/\psi + \eta_c$ cross section at $\Upsilon(4S)$ , amplifying the total NRQCD prediction by about 40%.
Experimental Feasibility: Based on the Belle Collaboration's reconstruction strategy for $J/\psi \to \mu^+\mu^-$ and six $\eta_c$ decay channels, the authors estimate that with the high integrated luminosity expected at SuperKEKB (50 ab $^{-1}$ ), several hundred to a thousand events could be accumulated near the $\Upsilon(4S)$ resonance, making the process observable.

Significance and Claims
The paper claims that the inclusive $e^+e^- \to J/\psi + \eta_c + X$ process offers a sensitive test of the NRQCD factorization framework at low energies. The distinct energy dependence of the NRQCD prediction (when using certain LDME sets) compared to the pure CS prediction provides a unique probe to distinguish between these mechanisms. The authors assert that the CO mechanism plays a significant role, enhancing the cross section and altering the energy scaling behavior, which contradicts the expectation that CO contributions are negligible at low energies. Furthermore, the work highlights the theoretical importance of including QED interference terms and feed-down contributions for accurate phenomenological predictions. The study concludes that the process is experimentally viable for observation at Belle, offering a potential pathway to resolve the ongoing debate regarding the validity of NRQCD in the low-energy domain.

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