Radiative charmonium decays in a contact-interaction model with dynamical quark anomalous magnetic moment

This paper utilizes a contact-interaction model incorporating valence-quark anomalous magnetic moments to demonstrate that while the framework aligns with lattice QCD estimates and the BESIII 2026 measurement of the $\eta_c \to \gamma\gamma$ decay width, it cannot accommodate the significantly larger central value reported by BESIII in 2024.

The Gist

The Big Picture: A Puzzle with Two Different Pictures

Imagine the world of tiny particles (subatomic physics) as a giant jigsaw puzzle. Scientists have been trying to fit a specific piece called the $\eta_c$ meson (a heavy particle made of a charm quark and an anti-charm quark) into the picture.

Recently, the BESIII experiment (a team of scientists in China) took two photos of this piece.

1. The 2024 Photo: This picture showed the piece behaving in a very strange, energetic way. It was much "brighter" (had a higher decay rate) than almost any other theory or previous measurement predicted. It was like seeing a car engine revving so loud it seemed impossible.
2. The 2026 Photo: A few months later, the same team took another picture. This one looked much more normal. It fit perfectly with what everyone else expected and with the "world average" of how this particle usually behaves.

This created a mystery: Which photo is right? Is the particle actually super-energetic, or was the first photo a fluke?

The Scientists' Approach: Adding a "Hidden Gear"

The authors of this paper wanted to solve the mystery using a specific theoretical model (a set of mathematical rules) called a Contact Interaction (CI) model. Think of this model as a simulation of how these particles interact.

For a long time, this simulation had a blind spot. It treated the quarks (the building blocks inside the particle) like simple, smooth marbles. However, the authors knew that in the real world, quarks have a "spin" and a magnetic nature, similar to a tiny bar magnet. This is called an Anomalous Magnetic Moment (AMM).

• The Analogy: Imagine you are trying to predict how a spinning top moves. If you ignore the fact that the top is slightly magnetic and interacts with the table's magnetic field, your prediction will be off.
• The Fix: The authors updated their simulation to include this "magnetic gear" (the AMM). They wanted to see if adding this extra detail would make the simulation match the weird 2024 photo, or if it would still look like the normal 2026 photo.

What They Found

The researchers ran their updated simulation with the new "magnetic gear" included. Here is what happened:

1. The Gear Helped, But Not Enough: Adding the magnetic effect did make the particle behave a little more energetically, just as they hoped. It brought the theoretical prediction closer to the experimental data.
2. The 2026 Photo Wins: The updated simulation matched the 2026 result perfectly. It also matched the "world average" and other high-tech computer simulations (called Lattice QCD).
3. The 2024 Photo is Still Too Loud: Even with the new magnetic gear, the simulation could not reach the high energy levels shown in the 2024 result. The 2024 measurement is still "too loud" for their model to explain, even when they tweaked all the knobs and dials to the maximum reasonable settings.

The Conclusion: A Call for a Second Look

The authors conclude that their model, which is very careful about preserving the fundamental laws of physics (symmetries), naturally supports the 2026 measurement.

They are not saying the 2024 measurement is definitely wrong, but they are saying:

• Our current best understanding of how these particles work (including their magnetic quirks) cannot explain the 2024 result.
• The 2026 result fits our understanding perfectly.
• Therefore, the 2024 result might need to be checked again by the experimentalists to see if there was a mistake or if there is some other new physics we haven't discovered yet.

In short: The scientists added a missing piece of physics to their theory. It fixed the problem for the "normal" 2026 data, but the "strange" 2024 data remains an outlier that doesn't fit the current picture of the universe.

Technical Summary

1. Problem Statement

The paper addresses a significant discrepancy in the experimental measurement of the two-photon decay width of the $\eta_c$ meson ($\Gamma_{\eta_c \to \gamma\gamma}$).

• The Conflict: The BESIII Collaboration reported two conflicting measurements: a 2024 result significantly larger (by $>3\sigma$) than the world average and most theoretical predictions, and a subsequent 2026 result aligning with conventional expectations.
• Theoretical Gap: Standard theoretical approaches (Lattice QCD, non-relativistic QCD, and standard Continuum Schwinger Methods) generally favor the lower values but struggle to explain the high 2024 value.
• Missing Physics: Conventional treatments often neglect the valence-quark anomalous magnetic moment (AMM), a non-perturbative effect arising from dynamical chiral symmetry breaking (DCSB). While significant for light quarks, its potential impact on the charm sector (charmonium) remains under-explored.

2. Methodology

The authors employ a Symmetry-Preserving Contact Interaction (CI) model within the framework of Continuum Schwinger Methods (CSMs).

• Model Framework:
  • Gluon Propagator: Modeled as a constant (contact interaction) $D_{\mu\nu} \sim \delta_{\mu\nu}/m_G^2$, where $m_G$ is an effective gluon mass scale.
  • Truncation Scheme: They utilize a Modified Rainbow-Ladder (MRL) truncation. Unlike the standard Rainbow-Ladder (RL) approximation, MRL dynamically generates an AMM term in the quark-photon vertex (QPV) through a parameter $\xi$.
  • Quark-Photon Vertex (QPV): The vertex is decomposed into longitudinal and transverse components. The transverse component $f_A(Q^2)$, which vanishes in the RL limit ($\xi \to 0$), encodes the dynamical AMM.
• Processes Studied:
  • $\eta_c \to \gamma\gamma$ (two-photon decay).
  • $J/\psi \to \gamma\eta_c$ (radiative decay).
• Parameter Fixing:
  • The model parameters (current quark mass $m_c$, effective gluon mass $m_G$, and UV cutoff $\Lambda_{uv}$) are fixed by reproducing the experimental masses of the ground-state $\eta_c$ and $J/\psi$.
  • The AMM strength parameter $\xi$ is varied (0 to 0.151) to assess its impact. The value $\xi=0.151$ is fixed based on light-quark sector constraints.
  • The UV cutoff range is set to $\Lambda_{uv} = 2.0 - 2.5$ GeV, appropriate for the heavier charm quark system.
• Calculations:
  • Bethe-Salpeter Equations (BSE) are solved for meson amplitudes.
  • Decay widths are calculated using the Impulse Approximation, where the external photon couples to a single valence quark.
  • Regularization is handled via a symmetry-preserving scheme to manage divergences and ensure the Ward-Green-Takahashi identity is satisfied.

3. Key Contributions

• Systematic Inclusion of AMM: This work is the first to systematically incorporate the dynamical quark AMM into charmonium radiative decays within a CI model. Previous CI studies of heavy quarkonia neglected this effect.
• Bridging Soft and Hard Physics: The study investigates the transition between non-perturbative (DCSB-driven) and perturbative regimes. The charm quark serves as an ideal probe, as it is heavy enough to be semi-perturbative but light enough for DCSB effects to remain relevant.
• Resolution of Experimental Tension: The authors provide a theoretical framework to test whether the inclusion of AMM can reconcile the theoretical predictions with the anomalous 2024 BESIII data.

4. Results

• Impact of AMM on Decay Widths:
  • Including the AMM (MRL truncation) enhances the decay widths compared to the standard RL truncation.
  • $\eta_c \to \gamma\gamma$: The predicted width increases from $\approx 6.23$ keV (RL) to $\approx 7.48$ keV (MRL).
  • $J/\psi \to \gamma\eta_c$: The width increases from $\approx 2.01$ keV (RL) to $\approx 2.11$ keV (MRL).
• Comparison with Data:
  • 2026 BESIII & World Average: The MRL results (with AMM) show excellent agreement with the 2026 BESIII measurement and the world average ($\approx 5.1$ keV).
  • 2024 BESIII: Even with the maximum reasonable variation of the AMM parameter $\xi$ and the UV cutoff, the theoretical prediction fails to reach the central value of the 2024 BESIII measurement. The 2024 value lies significantly above the range accommodated by the CI+AMM framework.
• Form Factors:
  • The transition form factors (TFF) for $\eta_c \to \gamma^*\gamma$ and $J/\psi \to \gamma^*\eta_c$ were calculated.
  • The interaction radii were extracted: $r_{\eta_c\gamma} \approx 0.133$ fm and $r_{J/\psi\to\eta_c} \approx 0.28$ fm. These are smaller than momentum-dependent model predictions but preserve the relative hierarchy between light and heavy quark systems.

5. Significance and Conclusion

• Validation of 2026 Measurement: The study supports the 2026 BESIII result and the world average, suggesting that standard non-perturbative QCD effects (including AMM) are sufficient to describe the decay without invoking new physics or extreme parameter tuning.
• Implication for 2024 Measurement: The inability of the AMM-enhanced framework to explain the 2024 central value suggests that the 2024 measurement may be an outlier or requires further experimental scrutiny. It indicates that the discrepancy is not merely due to missing AMM effects in standard models.
• Theoretical Insight: The work demonstrates that while quark AMM is a crucial non-perturbative correction that improves agreement with lattice QCD and phenomenological expectations, it is insufficient on its own to explain the highest experimental claims for $\eta_c \to \gamma\gamma$.
• Future Directions: The authors call for independent experimental verification of the 2024 result and more refined theoretical analyses (e.g., momentum-dependent interactions with explicit AMM identification) to fully resolve the tension.

In summary, the paper utilizes a symmetry-preserving contact interaction model to demonstrate that dynamical quark anomalous magnetic moments enhance charmonium radiative decay widths, bringing theory into alignment with the 2026 BESIII data and lattice QCD, but failing to accommodate the anomalously high 2024 BESIII measurement.