⚛️ phenomenology

Common origin of the Belle-BESIII line-shape difference of $\Lambda_c^+\bar{\Lambda}_c^-$ , $D_s D_{s1}$ and $D_s D_{s2}^*$ from $e^+ e^-$ annihilation

This paper proposes that radiative production of higher charmonia, specifically $\eta_c(6S)$ and $\chi_{c2}(5P)$ , provides a common explanation for the longstanding discrepancies between Belle and BESIII measurements of line shapes in $e^+e^-$ annihilation to $\Lambda_c^+\bar{\Lambda}_c^-$ , $D_sD_{s1}$ , and $D_sD_{s2}^*$ final states.

Original authors: Cheng-Xi Liu, Tian-Le Gao, Cheng-Ping Shen, Xiang Liu

Published 2026-10-20

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CC0 1.0

Original authors: Cheng-Xi Liu, Tian-Le Gao, Cheng-Ping Shen, Xiang Liu

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 known as charmonia act as a unique laboratory for understanding how matter holds together. These are not ordinary atoms but rather tightly bound pairs of a heavy charm quark and its antimatter twin, orbiting one another. Physicists study them to map out the invisible forces that govern the strong interaction, the glue that binds the very core of all visible matter. When scientists collide electrons and positrons at high speeds, they can create these charmonium states, which then decay into lighter particles. By carefully measuring the energy and mass of the resulting debris, researchers expect to see smooth patterns or distinct peaks that reveal the existence of specific, predicted particles. However, the universe sometimes presents a puzzle: two different teams of scientists, using different machines, can look at the same collision data and see completely different pictures. One team might spot a dramatic spike in the data, a sign of a new particle, while the other team sees nothing but a flat line. This discrepancy does not just represent a minor error; it suggests that something fundamental about how the data is collected or interpreted is being missed.

For over a decade, a persistent mystery has lingered in the field of particle physics involving the production of charmed particles. Two major experiments, one at the Belle facility in Japan and another at BESIII in China, have been measuring how often certain pairs of particles appear when electrons and positrons collide. Specifically, they looked at the creation of a pair of charmed baryons, as well as two different types of charmed meson pairs. The Belle experiment, which operates at a much higher collision energy, reported seeing a significant bump or enhancement in the data around an energy of 4.63 billion electron volts. This bump suggested the presence of a new, heavier charmonium state. In contrast, the BESIII experiment, which scans energies more directly in that specific range, found no such bump. The data from BESIII was smooth and flat, showing no sign of the extra particle that Belle seemed to see. This disagreement has remained unresolved, with both teams rigorously checking their equipment and methods, yet failing to find a reason for the difference. The question became: is there a hidden particle that only one machine can see, or is there a subtle flaw in how the data is being processed?

A new study proposes a solution that does not require a new particle to exist in the way previously thought, but rather suggests that a known type of particle production was being misidentified. The researchers suggest that the difference arises from how the two experiments handle a specific process called initial-state radiation. In the Belle experiment, the incoming particles sometimes emit a photon, a particle of light, before they collide. This emission lowers the energy of the actual collision, allowing the creation of lighter particles even though the machine is running at a very high energy. The experiment then reconstructs the event by looking for the resulting particles and the emitted photon. The problem, the authors argue, is that a different process can mimic this signature. A heavier, higher-energy charmonium particle could be created directly at the full collision energy, which then emits a photon and decays into the same pair of particles. Because the final arrangement of particles looks identical to the radiation process, these events get mixed into the same data sample.

The crucial insight is how the scientists calculate the probability of these events occurring. When the Belle team analyzes their data, they assume all the events in their sample come from the initial-state radiation process. They use a specific mathematical factor to convert the number of events they see into a measure of how likely the reaction is to happen. However, if a significant portion of those events actually came from the direct production of a heavier charmonium state, this calculation becomes distorted. The factor used for the radiation process is much larger than the factor for the direct process. Consequently, when the researchers apply the radiation factor to these "misidentified" direct events, the calculated likelihood of the event skyrockets. This artificial inflation creates the appearance of a large bump in the data, exactly where the discrepancy was observed. The BESIII experiment, which operates at lower energies where this direct production of heavy states is impossible, does not suffer from this confusion and therefore sees a smooth, flat line.

To test this idea, the researchers built a model that includes these hidden contributions. They took the smooth data from BESIII as the true baseline for what the reaction should look like. Then, they added the potential signal from a higher-energy charmonium state to the Belle data. For the baryon pair, they found that a specific type of particle, a pseudoscalar charmonium state known as eta_c(6S), could perfectly explain the extra bump seen by Belle. For the two meson pairs, a different type of particle, a tensor charmonium state called chi_c2(5P), provided the necessary explanation. When these contributions were included, the model successfully reproduced the sharp peak in the Belle data while remaining consistent with the flat line from BESIII. The study suggests that the "missing" particle is not a mystery of nature, but a result of a complex interplay between two different production mechanisms that were previously treated as one.

The findings offer a unified explanation for three separate discrepancies that had puzzled the community for years. By recognizing that the higher-energy machine was inadvertently capturing events from a different production channel, the researchers showed that the data from both experiments can be reconciled without inventing new physics. The study does not claim to have discovered a new particle in the traditional sense of finding something entirely unknown; rather, it suggests that the existing data contains a signal from a higher-energy state that was being misinterpreted. The authors note that this mechanism offers a new way to search for other missing heavy charmonium states at the Belle II facility. If their hypothesis is correct, future experiments could look for these specific particles by adjusting how they analyze the data, potentially revealing a family of heavy particles that have been hiding in plain sight, obscured by the very method used to find them. The work serves as a reminder that in the quest to understand the fundamental building blocks of the universe, the way we look at the data can be just as important as the data itself.

Technical Summary: Common Origin of the Belle-BESIII Line-Shape Difference in $e^+e^- \to \Lambda_c^+\bar{\Lambda}_c^-$ , $D_s D_{s1}$ , and $D_s D_{s2}^*$

Problem Statement
The paper addresses a longstanding discrepancy between the Belle and BESIII collaborations regarding the line shapes of specific hadronic channels produced in electron-positron annihilation. Specifically, Belle observed significant enhancements in the invariant mass spectra near 4.63 GeV for the baryonic channel $\Lambda_c^+\bar{\Lambda}_c^-$ and the mesonic channels $D_s D_{s1}(2536)$ and $D_s D_{s2}^*(2573)$ . In contrast, BESIII measurements in the same energy range showed no comparable structures or significantly weaker signals. This "Y problem" has persisted despite mutual verification of analysis methods by both collaborations. The authors posit that an underlying dynamical mechanism, previously unaccounted for in the interpretation of Belle's data, is responsible for these differences.

Methodology
The authors propose a mechanism based on the radiative production of higher charmonia at the Belle collision energy ( $\sqrt{s_0} \approx 10.58$ GeV), which mimics the signature of Initial-State Radiation (ISR) events.

Production Mechanism:
- BESIII: Measures direct $e^+e^- \to \gamma^* \to h_1 h_2$ annihilation at selected center-of-mass energies near the resonance mass.
- Belle: Primarily measures ISR processes ( $e^+e^- \to \gamma_{ISR} + \gamma^* \to \gamma_{ISR} + h_1 h_2$ ).
- Proposed Contribution: The authors suggest that at Belle's higher energy, resonances $R_i$ (higher charmonia with $C=+1$ ) are produced radiatively via $e^+e^- \to \gamma_{rad} R_i$ , followed by $R_i \to h_1 h_2$ . These events yield the same final state ( $\gamma h_1 h_2$ ) as ISR events.
- Reconstruction: Belle's reconstruction algorithms for these channels rely on detecting an energetic photon and constraining the recoil mass. Since the kinematics of a radiative photon ( $\gamma_{rad}$ ) and an ISR photon ( $\gamma_{ISR}$ ) can be indistinguishable within the detector's acceptance and angular cuts, radiative events are inadvertently included in the ISR samples.
Cross-Section Extraction and Normalization:
- A critical factor is the luminosity normalization. In ISR analyses, the effective luminosity ( $L_{ISR}$ ) is calculated by integrating the ISR radiator function over the mass interval, which is significantly smaller than the total integrated luminosity ( $L_B$ ) collected by Belle.
- If radiative events (which scale with $L_B$ ) are treated as ISR events (scaled by $L_{ISR}$ ), their contribution to the extracted cross-section is artificially enhanced by the factor $L_B / L_{ISR,j}$ . The authors demonstrate that this factor is approximately $10^4$ , allowing a modest radiative production cross-section to appear as a large enhancement in the measured spectrum.
Theoretical Modeling:
- The total cross-section is modeled as the sum of a vector baseline (derived from BESIII data and theoretical form factors) and an additional radiative contribution.
- The radiative contribution is parameterized using a relativistic Breit-Wigner amplitude.
- For the $\Lambda_c^+\bar{\Lambda}_c^-$ channel, the authors test the contribution of a pseudoscalar charmonium $\eta_c(6S)$ .
- For the $D_s D_{s1}$ and $D_s D_{s2}^*$ channels, they investigate higher charmonium candidates with $C=+1$ (specifically $\chi_{cJ}$ states) that can decay into these pairs via allowed orbital angular momenta.
- Coupled-channel effects and open-charm decay widths are calculated using the Godfrey-Isgur Hamiltonian and the Quark Pair Creation (QPC) model to determine resonance masses and decay properties.

Key Contributions and Results

Unified Explanation: The paper provides a common dynamical explanation for the discrepancies in all three channels ( $\Lambda_c^+\bar{\Lambda}_c^-$ , $D_s D_{s1}$ , $D_s D_{s2}^*$ ) by attributing the Belle enhancements to misidentified radiative production of higher charmonia.
$\Lambda_c^+\bar{\Lambda}_c^-$ Channel:
- The inclusion of an $\eta_c(6S)$ contribution significantly improves the fit to the Belle data relative to the BESIII baseline.
- A fixed-input fit using parameters from previous studies ( $M_R = 4629$ MeV, $\Gamma_R = 24.2$ MeV) yields a $\chi^2$ of 6.81 (down from 17.12 for the baseline alone).
- A free fit yields $M_R = 4640.1^{+6.0}_{-13.0}$ MeV and $\Gamma_R = 29^{+34}_{-23}$ MeV, with a significance of $2.34\sigma$ (accounting for the look-elsewhere effect).
Meson Channels ( $D_s D_{s1}$ and $D_s D_{s2}^*$ ):
- The authors identify that higher charmonia lying above the thresholds can couple to these channels.
- Specifically, they propose that a $\chi_{c2}(5P)$ contribution can describe the enhancements observed in both meson channels, while $\chi_{c0}(5P)$ and $\chi_{c1}(5P)$ are also considered as candidates based on quantum number constraints.
- The calculated partial widths for these states (e.g., $\chi_{c2}(5P) \to D_s \bar{D}_{s1}$ ) are small but sufficient to generate the observed effects when scaled by the luminosity factor.
Quantitative Impact: The analysis shows that the "missing" physics is not a new exotic state in the traditional sense, but rather a standard higher charmonium state whose production is enhanced in the Belle dataset due to the specific normalization procedure used for ISR samples.

Significance and Claims
The paper claims to resolve the "Belle-BESIII difference" puzzle by identifying a specific experimental artifact: the contamination of ISR samples by radiative production events, amplified by the luminosity normalization factor.

Mechanism Validation: The work suggests that the observed enhancements are not necessarily new resonances or exotic hadrons, but rather known (or predicted) higher charmonia ( $\eta_c(6S)$ , $\chi_{c2}(5P)$ ) produced via radiative transitions.
Search Strategy: As a secondary outcome, the authors suggest that this mechanism offers a pathway to search for missing higher charmonia at Belle II. By analyzing the specific kinematic distributions and angular correlations, future experiments could isolate these radiative contributions to confirm the existence of these higher charmonium states.
Modesty: The authors acknowledge that the significance of the $\eta_c(6S)$ signal in the free fit is moderate ( $2.34\sigma$ ) and that the mechanism relies on the assumption that radiative events enter the ISR samples without being rejected by kinematic cuts. They frame the result as a "possible common explanation" that warrants further experimental scrutiny rather than a definitive proof of specific resonance properties.

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