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Radiative decays of the $h_c$ meson from lattice QCD

This paper presents the first lattice QCD calculation of the radiative decays $h_c \to \gamma \eta$ and $h_c \to \gamma \eta'$ , utilizing novel techniques to separate form factors and revealing a decay rate suppression consistent with previous $J/\psi$ results, while also providing theoretical predictions for the unmeasured Dalitz decays $h_c \to \ell^+ \ell^- \eta^{(\prime)}$ .

Original authors: Mischa Batelaan, Jozef J. Dudek, Robert G. Edwards

Published 2026-07-29

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Mischa Batelaan, Jozef J. Dudek, Robert G. Edwards

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

Imagine the universe is a giant, cosmic LEGO set, but instead of plastic bricks, everything is built from tiny, fundamental particles called quarks. These quarks are glued together by an invisible, super-strong force called the strong nuclear force, creating heavier particles known as "hadrons." Among these, there's a special family called "charmonium," which are like heavy-duty couples made of a charm quark and its antimatter twin, the anti-charm quark, dancing around each other. Scientists are obsessed with watching these couples break up or change partners because the way they do it—specifically, how they flash a burst of light (a photon) to transform—acts like a high-speed camera. This "radiative decay" is a clean, theoretical window that lets us see the hidden rules of the strong force, helping us understand why the universe is built the way it is.

In this new study, a team of researchers used a massive digital simulation called "Lattice QCD" to watch a specific, tricky character in this cosmic dance: the $h_c$ meson. Think of the $h_c$ as a slightly excited, heavier version of the famous $J/\psi$ particle. The scientists wanted to see what happens when this $h_c$ meson decides to slow down and emit a photon to turn into a lighter, ghostly particle called an $\eta$ or an $\eta'$ (eta-prime). It's like watching a heavy, glowing balloon pop and release a spark that turns into a smaller, different balloon. By simulating this on a supercomputer, they could measure the exact "shape" of the light emitted, which tells us how the quarks are moving and interacting inside.

The team's main job was to figure out the details of this light emission. In the real world, we can't always see the "longitudinal" part of the light (the part that pushes forward) versus the "electric dipole" part (the part that wiggles sideways) easily. But in their computer simulation, they could separate these two distinct "flavors" of light emission. They found that the $h_c$ meson does indeed emit light to become an $\eta$ or $\eta'$ , and they mapped out exactly how this process changes as the energy of the light varies. They even looked at a slightly heavier, excited cousin of the $J/\psi$ called the $\psi'$ , but in their current simulation, the signal for that particle was so faint it looked like zero—like trying to hear a whisper in a hurricane.

Here is the twist, though: when the team compared their computer results to real-world experiments done by the BESIII detector, they found a mismatch. The light emission rates they calculated were significantly lower than what scientists actually see in the lab. It's as if their digital simulation predicted a dimmer spark than the real one. The authors suggest this isn't a mistake in their math, but rather a hint that the "glue" holding the $\eta$ and $\eta'$ particles together behaves differently in their specific simulation environment than it does in reality. They suspect the issue might be related to how the simulation handles the "topology" (the shape and twists) of the invisible force fields, which is a known tricky spot in these kinds of calculations.

Despite the mismatch, the study is a huge success in method. It's the first time anyone has successfully simulated these specific decays for the $h_c$ meson, separating the different types of light emission and even predicting how these particles would decay into pairs of electrons and positrons (a process called a Dalitz decay) that hasn't been measured yet. The researchers conclude that while their numbers are a bit off from the real world, their technique works beautifully. They believe that if we can fix how the simulation handles the complex "twists" in the force fields, we can use these heavy quark couples as a perfect factory to discover new, exotic particles in the future. For now, they've proven that we can simulate these complex quantum dances with high precision, even if the music isn't quite in tune with the real world yet.

Technical Summary: Radiative Decays of the $h_c$ Meson from Lattice QCD

Problem Statement
This work addresses the radiative decays of the $J^{PC} = 1^{+-}$ charmonium state, $h_c$ , into light pseudoscalar mesons ( $\eta$ and $\eta'$ ) and a photon ( $h_c \to \gamma\eta^{(\prime)}$ ). While the related decays of the vector meson $J/\psi \to \gamma\eta^{(\prime)}$ were previously studied by the Hadron Spectrum Collaboration, the $h_c$ presents distinct challenges. Unlike the $J/\psi$ , the $h_c$ is an excited state in the lattice spectrum for many kinematic configurations, often appearing alongside the $J/\psi$ in the same irreducible representations (irreps). Furthermore, the transition between an axial-vector meson ( $h_c$ ) and a pseudoscalar meson involves two independent form-factors (electric dipole and longitudinal), whereas the $J/\psi$ transition is described by a single magnetic dipole amplitude. The study aims to compute these form-factors on three-flavor lattices with a pion mass of $m_\pi \sim 391$ MeV to investigate the suppression of these rates relative to experimental data, a discrepancy previously observed in $J/\psi$ decays.

Methodology
The calculation utilizes anisotropic lattice QCD with improved vector currents. Key methodological developments include:

Accessing Excited States: To isolate the $h_c$ from the $J/\psi$ ground state and other excited states, the authors employ optimized operators constructed via the diagonalization of a large basis of fermion bilinear operators. This allows for the reliable projection of the $h_c$ even when it is not the ground state in specific moving-frame irreps.
Multipole Separation: The transition matrix element for $h_c \to \gamma\eta^{(\prime)}$ is decomposed into an electric dipole form-factor ( $E_1$ ) and a longitudinal form-factor ( $C_1$ ). To separate these, the authors compute three-point correlation functions for various kinematic configurations (different momenta for the source, sink, and current) and helicities. By forming an overconstrained linear system at fixed photon virtuality ( $Q^2$ ), they solve for the independent form-factors.
Simultaneous Fitting: To maximize the utility of data computed at multiple source-sink separations ( $\Delta t/at = \{16, 20, 24, 28\}$ ), a new simultaneous fitting procedure is introduced. This method fits the time-dependence of correlation functions across all $\Delta t$ values simultaneously, varying time-window cuts and model parameterizations (constants, source/sink exponentials). The results are averaged using the Akaike Information Criterion (AIC) to extract a single matrix element value with conservatively estimated uncertainties.
Kinematic Coverage: Due to the heavier mass of the $h_c$ compared to the $J/\psi$ , the study requires higher momenta for the $\eta$ and $\eta'$ mesons to access the region near real photons ( $Q^2 \approx 0$ ). The analysis covers $\eta$ and $\eta'$ momenta up to $|\mathbf{n}|^2 = 9$ .

Key Contributions and Results

First Lattice Calculation: This paper presents the first lattice QCD calculation of the multipole transition form-factors for $h_c \to \gamma\eta$ and $h_c \to \gamma\eta'$ .
Form-Factor Extraction: The authors successfully extracted the electric dipole ( $E_1$ ) and longitudinal ( $C_1$ ) form-factors as functions of timelike $Q^2$ . The data is well-described by dipole, exponential, and conformally mapped polynomial parameterizations.
Decay Rates: The real-photon transition rates are determined by $E_1(0)$ $E_{1} (0)$ . The calculated values are:
- $|E_1^{h_c \to \gamma\eta}(0)| = 0.0195(19) \text{ GeV}$
- $|E_1^{h_c \to \gamma\eta'}(0)| = 0.0556(30) \text{ GeV}$
Comparison to Experiment: Similar to the prior $J/\psi$ calculation, the computed decay rates for $h_c \to \gamma\eta^{(\prime)}$ are significantly suppressed relative to experimental measurements from BESIII.
Ratios and Scaling:
- The ratio of $\eta'$ to $\eta$ production ( $\sim 2.9$ ) is consistent with expectations based on the $SU(3)$ flavor singlet dominance of the $\eta'$ in gluonic intermediate states.
- A dimensionless ratio comparing the electric dipole strength in $h_c$ decays to the magnetic dipole strength in $J/\psi$ decays shows approximate agreement with experimental values, suggesting the discrepancy may lie in the properties of the $\eta$ and $\eta'$ mesons on these specific lattices rather than the charmonium annihilation process itself.
Dalitz Decays: The timelike dependence of both form-factors is provided, which is necessary for predicting the unmeasured Dalitz decays $h_c \to \ell^+\ell^-\eta^{(\prime)}$ . The analysis indicates that while the electric dipole dominates, the longitudinal form-factor is non-negligible.
$\psi'$ Transitions: The study also attempted to compute transitions for the excited vector state $\psi' \to \gamma\eta^{(\prime)}$ . The results are statistically compatible with zero on the current lattices, indicating that higher statistics or improved techniques are required for this process.

Significance and Claims
The authors claim that this work demonstrates the effectiveness of optimized operator technology and simultaneous fitting techniques in extracting signals from excited charmonium states and separating multiple form-factors. The primary significance lies in confirming that the suppression of radiative decay rates observed in $J/\psi \to \gamma\eta^{(\prime)}$ persists in the $h_c$ channel.

The paper modestly concludes that the origin of the discrepancy with experiment is likely tied to the specific properties of the $\eta$ and $\eta'$ mesons on the lattices used, potentially related to the sampling of gauge-field topology and the $U(1)$ axial anomaly, rather than a failure of the charmonium annihilation mechanism. The authors suggest that future work should focus on lattices with better topological sampling and extend these techniques to multi-hadron production channels (e.g., $J/\psi \to \gamma\pi\pi$ ) to further explore light meson resonance factories.

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