⚛️ phenomenology

On the failure of the $12\%$ rule in $J/\psi$ and $\psi(2S)$ decays

This paper demonstrates that the failure of the $12\%$ rule in $J/\psi$ and $\psi(2S)$ decays, particularly in the $\rho\pi$ channel, can be explained by an effective chiral framework incorporating $\mathrm{SU}(3)$ flavor breaking and electromagnetic effects, which renders the previously proposed vector glueball mixing scenario indistinguishable and suggests the discrepancy arises from additional contributions to $J/\psi$ amplitudes rather than a specific suppression in $\psi(2S)$ .

Original authors: Arthur Vereijken, Francesco Giacosa

Published 2026-09-22

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CC BY 4.0

Original authors: Arthur Vereijken, Francesco Giacosa

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 known as charmonia are like heavy, short-lived atoms made of a charm quark and its antimatter twin. These particles are unstable and quickly break apart into lighter, more common particles. Physicists have long observed two specific versions of these particles: one is the ground state, a heavy configuration, while the other is an excited version, carrying more energy and sitting in a higher state of existence. For decades, a simple rule of thumb has guided expectations about how these two versions decay. If the ground state breaks apart into a certain set of light particles with a specific frequency, the excited state should do the same, just at a rate roughly thirteen percent as often. This expectation, known as the twelve percent rule, relies on the idea that the internal structure of the excited state is just a larger, more energetic version of the ground state, sharing the same fundamental decay habits.

However, nature has a habit of defying simple rules. In one particular decay path, where the particles transform into a rho meson and a pion, the excited state behaves strangely. Instead of decaying at the predicted rate, it appears to be almost completely silent, while the ground state decays with surprising vigor. This discrepancy, known as the rho-pion puzzle, has puzzled scientists for decades, leading to various theories about hidden forces or exotic particles that might be interfering with the process. Researchers Arthur Vereijken and Francesco Giacosa have now revisited this mystery using a detailed mathematical framework that treats the strong nuclear force and electromagnetic interactions as a unified system. Their work suggests that the problem does not lie with the excited state behaving poorly, but rather with the ground state behaving unexpectedly well.

The researchers approached the problem by building a model that describes how these heavy particles transform into lighter ones, accounting for both the powerful strong force that binds quarks together and the electromagnetic force that governs how charged particles interact. They began by testing this model against the data for the excited state, the psi(2S). With just three adjustable numbers to fine-tune their calculations, they found that the model could reproduce the experimental data with remarkable accuracy. The excited state behaved exactly as the standard rules of physics predicted, decaying into various combinations of light particles at rates that matched observations almost perfectly. This success implied that the excited state was not the source of the anomaly; it was behaving normally, following the established laws of particle decay without needing any exotic explanations or hidden mechanisms.

When the team applied the same model to the ground state, the J/psi, the picture changed dramatically. The simple version of their model, which worked so well for the excited state, failed to match the experimental data for the ground state. The calculated decay rates were far off from what was actually measured in the laboratory. To fix this, the researchers had to introduce additional complexities into their model. They needed to account for a subtle breaking of symmetry in how the particles interact with different types of quarks, and they had to include a significant contribution from electromagnetic interactions that was much larger than standard theory would suggest. Once these extra ingredients were added, the model finally aligned with the data, producing a fit that was statistically acceptable.

The most striking finding emerged from this comparison. The extra electromagnetic contribution required to make the ground state model work was surprisingly large, comparable in strength to the primary strong force interactions. This suggests that the ground state is not simply a passive victim of a suppression mechanism affecting the excited state. Instead, the ground state appears to be receiving an unexpected boost in its ability to decay into these specific light particles. The researchers explored whether this boost could be explained by the ground state mixing with a hypothetical, invisible particle made entirely of gluons, known as a vector glueball. They found that their mathematical results were effectively identical whether they assumed this mixing existed or not. The data simply could not distinguish between the two scenarios.

Ultimately, the study points toward a solution where the ground state is the active agent of change. The evidence suggests that the rho-pion puzzle is not caused by the excited state being mysteriously suppressed, but rather by the ground state being unexpectedly enhanced. The ground state seems to have access to additional pathways or dynamics that allow it to decay into rho and pion pairs much more frequently than the standard rules would allow. While the possibility of a hidden glueball component remains a valid mathematical interpretation, the researchers conclude that the most direct reading of the data is that the ground state possesses unique, enhanced decay characteristics. This insight shifts the focus of the mystery from what is missing in the excited state to what is present and amplified in the ground state, offering a clearer, if still complex, picture of how these heavy particles break apart.

Technical Summary: On the failure of the 12% rule in J/ψ and ψ(2S) decays

Problem Statement
The "12% rule" (or 13% rule) posits that the ratio of hadronic decay widths between the $\psi(2S)$ and $J/\psi$ should be approximately constant across different final states, determined primarily by the ratio of their wave functions at the origin. While this rule holds for many baryon-antibaryon channels, it is severely violated in several vector-pseudoscalar ( $VP$ ) meson channels. The most prominent violation occurs in the $\rho\pi$ channel, a phenomenon known as the " $\rho\pi$ puzzle." In this channel, the $J/\psi \to \rho\pi$ decay is substantial, whereas the $\psi(2S) \to \rho\pi$ decay is heavily suppressed, deviating significantly from the expected ratio. Previous explanations have ranged from $J/\psi$ mixing with a vector glueball to intrinsic charm components in light mesons or destructive interference mechanisms specific to the $\psi(2S)$ .

Methodology
The authors investigate these decays using an effective chiral framework based on the extended Linear Sigma Model (eLSM), incorporating both strong and electromagnetic interactions. Electromagnetic interactions are implemented via Vector Meson Dominance (VMD). The analysis proceeds in two stages:

Minimal Model: The authors first construct a minimal model using leading-order strong interactions (single and double trace terms in the chiral Lagrangian) and leading-order electromagnetic contributions derived from VMD. This model is applied to the $\psi(2S)$ data first, as it is expected to behave more "normally" than the $J/\psi$ .
Extended Model for $J/\psi$ : Recognizing that the minimal model fails to describe $J/\psi$ $J / ψ$ data, the authors extend the framework by introducing:
- SU(3) Flavor Breaking: A flavor-breaking matrix is applied to the leading-order strong interaction to account for strange quark mass effects.
- Anomalous Couplings: An additional interaction term is introduced, motivated by an anomalous $U(1)_A$ breaking term, which allows for specific electromagnetic contributions.
- Relaxed VMD Constraints: The strict VMD relation between the strong and electromagnetic couplings for the $J/\psi$ is relaxed by introducing an independent electromagnetic coupling parameter ( $g_{J,em}$ ).

The fits are performed against experimental branching ratios for various $VP$ channels (e.g., $\rho\pi$ , $\omega\eta$ , $K\bar{K}^*$ , $\phi\eta$ ) using a $\chi^2$ minimization procedure.

Key Results

$\psi(2S)$ Decays: The minimal model, containing only three free parameters ( $g_{\psi}$ , $g_{2\psi}$ , and $g_{\psi\gamma}$ ), describes the $\psi(2S)$ data well, yielding a reduced chi-squared of $\chi^2_{red} \simeq 1.12$ . The model successfully predicts branching ratios for unmeasured channels (e.g., $\phi\pi^0$ , $\omega\eta$ ) and suggests that the $\psi(2S)$ behavior is consistent with standard strong and electromagnetic dynamics without requiring exotic suppression mechanisms.
$J/\psi$ Decays: The minimal model fails significantly for the $J/\psi$ ( $\chi^2_{red} \simeq 16.5$ ). To achieve a fair agreement with data ( $\chi^2_{red} \simeq 1.97$ ), the model requires the inclusion of SU(3) flavor breaking and, crucially, an independent electromagnetic coupling ( $g_{J,em}$ ) that is substantially larger than the strong coupling.
Coupling Ratios: The ratios of fitted couplings between $\psi(2S)$ and $J/\psi$ deviate significantly from the values expected under the 12% rule (off by factors of roughly 6 for strong channels and 2 for photon couplings), indicating that the rule is not merely a phase-space effect but is fundamentally broken in these channels.
Vector Glueball Equivalence: The authors demonstrate that their extended fit, which introduces an anomalous electromagnetic contribution to the $J/\psi$ , is mathematically equivalent ("de facto") to a model where the $J/\psi$ mixes with a vector glueball. Consequently, the data cannot distinguish whether the observed enhancement in $J/\psi$ decays arises from glueball mixing or other dynamical enhancements specific to the $J/\psi$ .

Significance and Conclusions
The paper concludes that the $\rho\pi$ puzzle is likely not caused by a suppression mechanism specific to the $\psi(2S)$ , as the $\psi(2S)$ data is well-described by a minimal effective chiral model. Instead, the discrepancy arises from an unexpected enhancement of the $J/\psi \to VP$ amplitudes. This enhancement requires additional dynamics beyond the standard strong and electromagnetic contributions, such as SU(3) flavor breaking and significant electromagnetic interference.

The authors emphasize that while their results are consistent with the hypothesis of $J/\psi$ -vector glueball mixing, the current data does not allow for a definitive confirmation of the glueball's presence. The "solution" to the puzzle, therefore, points toward an enhancement of the $J/\psi$ decay amplitudes rather than a suppression of the $\psi(2S)$ . The study underscores the limitations of the 12% rule in vector-pseudoscalar channels and highlights the necessity of including electromagnetic effects and flavor symmetry breaking in effective models of charmonium decays.

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