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NNLO QCD corrections to direct $J/ψ$ production with open charm at B factories

This paper calculates the next-to-next-to-leading-order QCD corrections to direct $J/\psi$ production with open charm at B factories, finding that while the improved perturbative convergence and momentum spectrum bring the theoretical prediction closer to Belle data, the angular distribution discrepancy highlights the need for more precise measurements at Belle II.

Original authors: Xiangyu Chang, Chao Zeng, Bin Gong, Jian-Xiong Wang, Cong Li

Published 2026-10-30

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CC BY 4.0

Original authors: Xiangyu Chang, Chao Zeng, Bin Gong, Jian-Xiong Wang, Cong Li

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, particles called quarks are never found alone. They are permanently bound together by a powerful force, much like how a magnet's pull keeps two pieces of iron stuck together, though this force is far more complex and operates at a scale billions of times smaller. When a heavy quark and its antimatter partner, an antiquark, bind together, they form a particle known as quarkonium. One of the most famous examples is the J/psi particle, a heavy, short-lived object that physicists create in particle accelerators to test the fundamental rules of nature. These rules are described by a theory called Quantum Chromodynamics, or QCD, which governs how quarks interact. While scientists have long been able to predict how these particles behave in simple scenarios, the calculations become incredibly difficult when more particles are involved, requiring a level of mathematical precision that pushes the limits of modern computing.

A team of researchers at the Institute of High Energy Physics in China has taken a significant step forward in this field by performing the most precise calculation yet for a specific process: the creation of a J/psi particle alongside a pair of open charm quarks. This event occurs when an electron and a positron collide at high speeds, annihilating each other to produce a burst of energy that materializes into new particles. The researchers focused on a scenario where the J/psi is produced directly, rather than as a byproduct of a heavier particle's decay, and where it is accompanied by a charm quark and its antiquark. This specific combination of particles had been studied before, but only with approximations that left significant gaps in understanding. By calculating the effects of the strong force to a new, higher level of accuracy, the team was able to refine their predictions and compare them directly with experimental data collected by the Belle collaboration at a particle accelerator in Japan.

The researchers used a sophisticated mathematical technique to handle the complex interactions that occur during these collisions. In these high-energy events, particles can emit other particles, such as gluons, which act as the carriers of the strong force. When calculating the probability of a specific outcome, physicists must account for every possible way these emissions can happen. The team's new calculation included contributions from processes where two extra particles are emitted, a level of detail that had not been fully worked out for this specific reaction before. This allowed them to produce a more reliable prediction for how often this reaction happens and what the resulting particles look like in terms of their speed and direction.

When the team applied their new, highly precise calculations to the conditions of the Belle experiment, they found that the total rate of production was about eighteen percent higher than the previous best estimates. This increase brought their theoretical prediction closer to the actual number of events observed by the Belle detectors, suggesting that the theory is on the right track. However, the story was not entirely one of perfect agreement. While the new calculations improved the prediction for the speed of the J/psi particles, making the peak of the speed distribution match the experimental data better, they made the prediction for the direction of the particles worse. The theory now predicts that the J/psi particles are more likely to fly forward or backward along the path of the collision, whereas the experimental data suggests they are more likely to be produced at angles in the middle.

This mismatch in the direction of the particles is a crucial finding. It indicates that while the team has successfully accounted for many of the complex interactions, there are still missing pieces in the puzzle. The experimental data includes not only the direct production of the J/psi but also contributions from heavier particles that decay into it, as well as events where two J/psi particles are created at once. The researchers suspect that the difference in the angular shape might be resolved by better understanding how these other processes contribute to the final count. They propose that future experiments at the upgraded Belle II facility, which will collect much more data, are needed to separate these different sources of J/psi particles. Only by isolating the direct production events from the others can physicists determine if the theory needs further adjustment or if the current discrepancy is simply due to the mixing of different physical processes in the detector.

The work represents a major achievement in computational physics, demonstrating that even the most intricate calculations involving heavy particles can be tamed with the right methods. By pushing the precision of their predictions, the team has provided a sharper tool for experimentalists to use. The fact that their calculation brings the predicted speed of the particles closer to reality gives confidence in the underlying theory, while the remaining disagreement in direction points the way toward new discoveries. It is a reminder that in the quest to understand the fundamental building blocks of the universe, progress often comes in the form of refining our questions as much as answering them. The path forward lies in gathering more precise data to see if the theory can eventually align perfectly with the reality observed in the laboratory.

Technical Summary: NNLO QCD Corrections to Direct $J/\psi$ Production with Open Charm at B Factories

Problem Statement
The production of charmonium ( $J/\psi$ ) in association with an open charm pair ( $c\bar{c}$ ) in electron-positron annihilation serves as a critical test of heavy-quarkonium production mechanisms within Non-Relativistic QCD (NRQCD). While Next-to-Leading Order (NLO) QCD corrections were established over a decade ago, they revealed a significant enhancement (factor of $\sim 1.7\text{--}1.8$ ) over Leading Order (LO) predictions. However, discrepancies remained between NLO theoretical predictions and experimental data from Belle, particularly regarding the production-angle distribution. The large NLO correction necessitated a test of perturbative stability at the Next-to-Next-to-Leading Order (NNLO) to determine if the series converges and to refine differential predictions for total rates, momentum spectra, and angular distributions.

Methodology
The authors perform the first NNLO QCD calculation for direct colour-singlet $J/\psi$ production with an open charm pair via a virtual photon ( $e^+e^- \to \gamma^* \to J/\psi + c\bar{c} + X$ ). The calculation employs the nested soft-collinear subtraction scheme adapted for massive partons.

Theoretical Framework: The process is defined within NRQCD factorisation, where the cross-section is separated into perturbative short-distance coefficients and a non-perturbative colour-singlet matrix element $\langle O_1 \rangle_{J/\psi}$ . The calculation includes contributions up to $\mathcal{O}(\alpha_s^2)$ relative to the LO term.
Subtraction Scheme: To handle infrared (IR) singularities arising from soft and collinear emissions in the double-real ( $RR$ $R R$ ), real-virtual ( $RV$ $R V$ ), and double-virtual ( $VV$ $V V$ ) channels, the authors utilize nested soft-collinear subtraction.
- Real Emissions: The double-real channels involve two gluons ($gg$) or a light quark pair ( $q\bar{q}$ ). The heavy-quark mass regulates collinear singularities involving the open charm, but overlapping soft and light-parton collinear limits remain. The authors subtract soft limits first, followed by collinear limits, using local counterterms.
- Virtual Corrections: The two-loop virtual amplitudes are reduced to master integrals using Kira and evaluated with AMFlow. The infrared poles of the virtual terms are cancelled analytically against the integrated counterterms derived from the real-emission sectors.
- Phase Space: Mappings are constructed to keep the measured $J/\psi$ momentum fixed between real terms and their counterterms, preserving differential information.
Numerical Implementation: The calculation is performed at $\sqrt{s} = 10.6$ GeV with a charm mass $m_c = 1.5$ GeV. The renormalisation scale $\mu_R$ is varied between $m_c$ and $4m_c$ . The authors approximate the finite virtual terms at $\sqrt{s}=10.6$ GeV using results calculated at $\sqrt{s}=10.5$ GeV to simplify the two-loop integration-by-parts reduction, a method validated by NLO benchmarks.

Key Contributions

First NNLO Calculation: This work presents the first NNLO QCD prediction for the inclusive process $e^+e^- \to J/\psi + c\bar{c} + X$ in the colour-singlet channel.
Implementation of Massive Subtraction: The authors successfully apply the nested soft-collinear subtraction method to a process with a three-body Born final state containing massive coloured particles, demonstrating the cancellation of IR poles across different final-state multiplicities.
Differential Predictions: Beyond total cross-sections, the paper provides predictions for the production-angle distribution and the momentum spectrum, including their scale uncertainties.
Mass Extrapolation and Feed-down: Recognizing that the calculation is for $m_c=1.5$ GeV, the authors estimate the impact of a lower mass ( $m_c=1.4$ GeV) using NLO mass dependence ratios. They also incorporate an estimated $\psi(2S)$ feed-down contribution to compare with Belle's inclusive prompt charm-associated data.

Results

Total Cross Section: At the central scale $\mu_R = 2m_c$ , the approximate NNLO cross section is 0.324 pb (with scale uncertainties $^{+0.041}_{-0.066}$ pb). This represents an 18.4% increase over the NLO result.
Perturbative Convergence: The relative scale variation decreases significantly from NLO ( $^{+48.9\%}_{-29.3\%}$ ) to NNLO ( $^{+12.7\%}_{-20.3\%}$ ). The NNLO correction is less than half the size of the NLO correction, indicating good perturbative convergence.
Mass and Feed-down Effects: Extrapolating to $m_c = 1.4$ GeV and adding the estimated $\psi(2S)$ feed-down (29% of the direct rate) raises the predicted rate to approximately 0.59 pb (at $\mu_R=3$ GeV) or 0.66 pb (at $\mu_R=1.65$ GeV). This brings the theoretical rate closer to Belle's measured prompt charm-associated cross section of $0.74 \pm 0.08 \text{ (stat.)} ^{+0.09}_{-0.08} \text{ (syst.)}$ pb.
Momentum Distribution: The NNLO correction raises the absolute momentum spectrum near its peak, improving agreement with Belle's prompt open-charm data compared to NLO. However, the predicted shape still yields a larger fraction of high-momentum $J/\psi$ mesons than observed, likely due to unmodeled feed-down kinematics.
Angular Distribution: The NNLO correction increases the angular coefficient $\alpha_\theta$ from 0.505 (NLO) to 0.554 (NNLO), enhancing forward and backward production. This shifts the prediction further away from Belle's measured value ( $\alpha_\theta = -0.26$ ), which favours central production. The scale uncertainty on $\alpha_\theta$ also increases at NNLO.

Significance and Claims
The paper claims that the NNLO calculation demonstrates good perturbative convergence for the total cross section, with reduced scale dependence compared to NLO. The authors state that while the momentum spectrum prediction improves with NNLO corrections, the angular shape discrepancy with experimental data persists and is exacerbated at NNLO.

The authors conclude that this difference in angular shape motivates more precise measurements at Belle II that can separate direct open-charm production from feed-down and double-charmonium contributions. They do not claim to resolve the angular discrepancy with current data but rather highlight the need for differential measurements of specific components to fully test the NRQCD colour-singlet mechanism at this order. The work establishes a baseline for future phenomenological studies of quarkonium production in $e^+e^-$ collisions at NNLO accuracy.

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