Extraction of charmonium branching fractions from $J/\psi\to\gamma\eta_c$ radiative decays

This paper proposes a theoretically grounded method for extracting charmonium branching fractions from $J/\psi\to\gamma\eta_c$ radiative decays that resolves tensions between experimental data and theoretical predictions by eliminating the need for empirical damping functions in photon line shape analysis.

The Gist

The Big Picture: A Mismatch in the Physics World

Imagine you are trying to weigh a specific type of fruit (let's call it a "charmonium" fruit) by looking at how much light it reflects. Scientists have been doing this for decades. However, there is a confusing disagreement:

• The Theorists (people who use math to predict how heavy the fruit should be) say it weighs one thing.
• The Experimentalists (people who actually measure the fruit) say it weighs something lighter.

The Particle Data Group (PDG), which acts like the "official referee" for physics, has been averaging these measurements. But their average is lower than what the math predicts. This paper suggests the referee might be using a broken scale.

The Problem: The "Fuzzy" Scale

To measure the fruit, scientists look at a "spectrum," which is like a graph showing how much light is emitted at different energies. The signal they are looking for is a sharp peak (the fruit), but the graph has a long, messy "tail" that stretches out far away from the peak.

The Old Way (The Broken Scale):
In the past, when scientists tried to count the fruit, they had to deal with this messy tail. Since the math said the tail should go on forever (making the total count infinite), they invented a "cutoff."

• The Analogy: Imagine you are counting apples in a basket, but there are some stray apples rolling off the edge of the table. To get a number, the old method said, "Let's just pretend the apples stop rolling after 5 feet." They used a made-up "damping function" (a mathematical filter) to cut off the tail.
• The Flaw: The problem is that where you cut off the tail is arbitrary. If you cut it at 5 feet, you get one number. If you cut it at 6 feet, you get a different number. This introduced a "fudge factor" into the results, making the measurements unreliable and inconsistent with the math.

The New Solution: A Sharper Lens

The authors of this paper propose a new way to look at the data that doesn't require cutting off the tail.

The New Method:
Instead of trying to count every apple in the basket (including the ones rolling off the table), they realized they only need to look at the very center of the pile.

• The Analogy: Think of the signal as a mountain. The old method tried to measure the volume of the whole mountain, including the tiny, endless foothills, so they had to draw a line in the sand to say "stop here."
• The New Approach: The authors say, "We don't need to measure the whole mountain. We just need to measure the height of the peak."
• Why it works: The height of the peak is a fixed, clear number. It doesn't depend on where you draw a line in the sand. By using a specific mathematical formula that focuses only on the peak's height, they can calculate the number of events without needing any arbitrary "cutoffs" or "damping functions."

What They Found

When the authors applied this new "peak-height" method to old data from experiments like CLEO and BESIII:

1. The Numbers Changed: The calculated "weight" (branching fraction) of the particle became larger.
2. The Disagreement Vanished: This new, larger number matches perfectly with what the theorists predicted using advanced supercomputer simulations (Lattice QCD).
3. The "Referee" Update: When they fed this new number back into the PDG's official calculations, the tension disappeared. The experimental data and the theoretical predictions finally agreed.

The Takeaway

The paper claims that the long-standing disagreement between theory and experiment wasn't because the laws of physics were wrong or because the particles were behaving strangely. It was simply because the scientists were using a messy, arbitrary method to count the data.

By switching to a cleaner, more precise method that focuses on the "peak" of the signal rather than the messy "tail," they resolved the conflict. The universe is consistent; we just needed a better way to read the ruler.

In short: They fixed a measurement error caused by an arbitrary "cut-off" rule, and suddenly, the experimental data and theoretical predictions finally agreed with each other.

Technical Summary

Technical Summary: Extraction of Charmonium Branching Fractions from $J/\psi \to \gamma \eta_c$ Radiative Decays

Problem Statement
The paper addresses a persistent tension between theoretical predictions and experimental values regarding the partial decay width $\Gamma(J/\psi \to \gamma \eta_c)$ and the associated branching fraction $B(J/\psi \to \gamma \eta_c)$. The Particle Data Group (PDG) quotes an average branching fraction of $(1.7 \pm 0.4)\%$, which is lower than values predicted by weakly-coupled potential nonrelativistic QCD (pNRQCD) and recent high-precision lattice QCD calculations. This discrepancy is critical because $B(J/\psi \to \gamma \eta_c)$ serves as a normalization factor for extracting numerous $\eta_c$ branching fractions and the $\eta_c$ total decay width.

The authors identify the root of this inconsistency in the experimental methodology used to extract the signal yield from the photon energy spectrum. Current analyses (Crystal Ball, CLEO, KEDR, BESIII) model the spectrum using a relativistic Breit–Wigner distribution modified by an $E_\gamma^3$ factor. While the $E_\gamma^3$ factor is theoretically motivated by Low's theorem and the Ore–Powell spectrum, it induces a power-law tail at high energies. Integrating this line shape to determine the signal yield results in a power divergence, necessitating the introduction of ad hoc, empirically motivated damping functions (or cutoffs) to suppress the tail. The authors argue that these damping functions lack theoretical justification, introduce unphysical cutoff dependencies, and create an inherent ambiguity in the determination of the branching fraction.

Methodology
The paper proposes a theoretically grounded alternative for extracting the $J/\psi \to \gamma \eta_c$ signal yield that eliminates the need for arbitrary damping functions.

1. Theoretical Framework: The authors utilize the pNRQCD framework to describe the magnetic dipole (M1) transition. They establish that the differential decay width for the inclusive process $J/\psi \to \gamma X$ near the $\eta_c$ peak is given by a nonrelativistic Breit–Wigner distribution multiplied by $E_\gamma^3$.
2. Signal Extraction Prescription: Instead of integrating the spectrum over a cutoff-dependent range, the authors propose extracting the number of signal events ($N_{\eta_c}$) directly from the peak strength of the photon energy spectrum. By applying a specific weighting function $w(E_\gamma)$ proportional to a delta function at the hyperfine splitting energy ($\Delta E_{\text{HFS}}$), they derive a formula (Eq. 9) where the yield is proportional to the differential event rate evaluated exactly at the peak energy:
   $$N_{\eta_c} = \frac{\pi \Gamma_{\eta_c}}{2} \left. \frac{dN_\gamma}{dE_\gamma} \right|_{E_\gamma = \Delta E_{\text{HFS}}}$$
   This approach is model-independent within the leading-order (LO) nonrelativistic expansion and consistent with perturbation theory, as it avoids integrating over the divergent high-energy tail.
3. Reanalysis of Experimental Data: The authors re-evaluate existing experimental data from CLEO and BESIII using their proposed line shape and extraction prescription. They also perform a "profile scan" of the PDG multiparticle fit, fixing $B(J/\psi \to \gamma \eta_c)$ to various values to observe the resulting impact on the extracted $\Gamma(\eta_c \to \gamma \gamma)$ and compare these results with lattice QCD determinations.

Key Contributions and Results

• Resolution of Divergence: The proposed method provides a cutoff-independent extraction of the branching fraction, removing the ambiguity associated with the choice of damping functions in previous analyses.
• Revised Branching Fractions:
  • CLEO: Re-evaluating CLEO data yields $B(J/\psi \to \gamma \eta_c) = (2.01 \pm 0.13_{\text{stat}} \pm 0.27_{\text{syst}})\%$. This is consistent with the original CLEO value but derived without ad hoc damping.
  • BESIII: Re-evaluating BESIII data for exclusive decays ($J/\psi \to \gamma \eta_c \to \gamma \gamma \gamma$ and $\gamma p \bar{p}$) leads to a combined branching fraction of $B(J/\psi \to \gamma \eta_c) = (2.50 \pm 0.07_{\text{stat}} \pm 0.22_{\text{syst}})\%$. This represents a significant increase over the value reported by BESIII using standard damping functions.
• Compatibility with Lattice QCD: The profile scan of the PDG multiparticle fit demonstrates that larger values of $B(J/\psi \to \gamma \eta_c)$ (specifically in the range of 2.0%–2.5%) render the extracted values of $\Gamma(\eta_c \to \gamma \gamma)$ and $\Gamma(J/\psi \to \gamma \eta_c)$ compatible with recent lattice QCD calculations (e.g., HPQCD, MFLWZ, MLWY).
• Consistency Check: The revised extraction of $B(\eta_c \to \gamma \gamma)$ from the BESIII data, when combined with the new $B(J/\psi \to \gamma \eta_c)$, yields a value consistent with recent direct measurements and lattice predictions, resolving the previous tension.

Significance
The paper claims that the observed tension between the PDG average and theoretical/lattice predictions is not due to a failure of QCD or the need for exotic theoretical explanations, but rather stems from an inconsistency in the experimental extraction methodology. By adopting the proposed peak-strength extraction method, the authors demonstrate that:

1. The branching fraction $B(J/\psi \to \gamma \eta_c)$ can be determined in a model-independent manner consistent with LO perturbation theory.
2. The resulting values for $B(J/\psi \to \gamma \eta_c)$ and the derived $B(\eta_c \to \gamma \gamma)$ are fully compatible with lattice QCD determinations.
3. The use of empirical damping functions in previous analyses introduced unphysical biases that suppressed the extracted branching fractions, leading to the discrepancy with theory.

The authors conclude that their approach obviates the need for ad hoc modifications to the photon line shape and provides a robust pathway for reconciling experimental data with theoretical expectations in charmonium physics.