Study of the Magnetic Dipole Transition of $J/ψ\toγη_c$ via $η_c\to p\bar{p}$

Using a large sample of $J/\psi$ events collected by the BESIII detector, this study presents the first amplitude analysis of $J/\psi\to\gamma p\bar{p}$ in the $\eta_c$ mass region to precisely determine the product branching fraction and subsequently derive the individual branching fractions for $J/\psi\to\gamma\eta_c$ and $\eta_c\to\gamma\gamma$, which are found to be consistent with recent lattice quantum chromodynamics calculations.

The Gist

Imagine the universe as a giant, cosmic dance floor. On this floor, particles are the dancers, and the rules of their movement are written in a complex book called Quantum Chromodynamics (QCD). Sometimes, the dancers move in ways that are easy to predict, but other times, they do something mysterious that breaks the rules we thought we understood.

This paper is a report from a team of scientists (the BESIII Collaboration) who watched a very specific, rare dance move to see if they could solve a long-standing mystery in physics.

The Mystery: The "Heavy" vs. "Light" Dance

In the world of subatomic particles, there are "heavy" dancers made of charm quarks. Two of the most famous are:

1. The J/ψ (J-Psi): A heavy, energetic dancer.
2. The ηc (Eta-c): A slightly lighter, calmer version of the same dancer.

Physics theory predicted that when the heavy J/ψ slows down and turns into the lighter ηc, it should release a burst of energy in the form of a photon (a particle of light). This is called a Magnetic Dipole (M1) transition.

The Problem: For decades, scientists calculated how often this should happen. The math said it should happen a lot. But when experiments actually looked, they saw it happening much less often—about half as much as the theory predicted. It was like a recipe saying you need 2 cups of flour, but every time you bake the cake, you only get half a cup. Something was missing.

The Experiment: Catching the Ghost

To solve this, the scientists used a massive particle accelerator (BEPCII) in Beijing to create billions of J/ψ particles. They acted like a high-speed camera, filming what happened when these particles decayed.

They were looking for a very specific sequence:

1. A J/ψ appears.
2. It transforms into an ηc and a photon (light).
3. The ηc immediately splits apart into a proton and an antiproton (a matter-antimatter pair).

The Challenge: This is like trying to find a specific needle in a haystack, where the needle is made of light and the hay is made of billions of other particles. The scientists had to filter out "noise" (other random particle collisions) to find the few thousand events where this specific dance occurred.

The New Technique: The "Amplitude Analysis"

In the past, scientists looked at this data using a simple, one-dimensional map. They just counted how many times the particles appeared. It was like trying to understand a 3D sculpture by only looking at its shadow on the wall.

In this paper, the team used a new, advanced technique called Amplitude Analysis.

• The Analogy: Imagine trying to figure out how a complex machine works. Instead of just counting how many times it clicks, you listen to the rhythm, the pitch, and the timing of every single sound it makes.
• By analyzing the "shape" and "direction" of the particles' movement in 3D space, the scientists could separate the signal (the real dance) from the background noise much more clearly than ever before.

The Results: Solving the Puzzle

The team analyzed over 10 billion J/ψ events. Here is what they found:

1. Precision Measurement: They measured the frequency of this specific dance with incredible accuracy—ten times better than previous attempts.
2. The "Magic" Number: They calculated the probability of the J/ψ turning into an ηc and a photon.
3. The Big Reveal: Their new number matches the latest, most advanced computer simulations (called Lattice QCD).

Why is this a big deal?
The old experimental numbers were low, and the old theory numbers were high. The new, ultra-precise measurement shows that the theory was actually right all along; the previous experiments just weren't sensitive enough to see the full picture. The "missing" half of the cake was there; they just couldn't see it clearly until now.

The Takeaway

This paper is a triumph of precision. By using a "super-microscope" (the BESIII detector) and a "3D map" (Amplitude Analysis), the scientists proved that our understanding of how heavy particles interact is correct.

In simple terms: They finally found the missing piece of the puzzle, proving that the universe's rulebook for how heavy particles dance is correct, and we just needed better eyes to see it. This helps physicists trust their theories and move on to solving even bigger mysteries about the universe.

Technical Summary

1. Problem Statement

The magnetic dipole (M1) transition between the lowest-lying charmonium states, $J/\psi \to \gamma\eta_c$, serves as a critical laboratory for testing Quantum Chromodynamics (QCD) in both perturbative and non-perturbative regimes.

• Theoretical Discrepancy: Non-relativistic predictions for the transition width are significantly larger (by a factor of 2–3) than experimental results. While recent Lattice QCD (LQCD) calculations predict values closer to the experimental range, they remain systematically higher (by $\sim$factor of 2) than the Particle Data Group (PDG) average.
• Experimental Limitations: Previous measurements of the branching fraction (BF) $B(J/\psi \to \gamma\eta_c)$ relied on inclusive hadronic decays or one-dimensional fits that often ignored interference between the $\eta_c$ resonance and non-resonant (NR) amplitudes. This led to potential biases and large uncertainties.
• Recent Tension: A recent BESIII measurement of $B(\eta_c \to \gamma\gamma)$ using the PDG value for $B(J/\psi \to \gamma\eta_c)$ resulted in a value differing from LQCD predictions by more than $3\sigma$. Independent, high-precision measurements of the $J/\psi \to \gamma\eta_c$ transition are required to resolve these inconsistencies.

2. Methodology

The study utilizes a dataset of $(10.087 \pm 0.044) \times 10^9$ $J/\psi$ events collected by the BESIII detector at the BEPCII collider.

• Reaction Channel: The analysis focuses on the decay chain $J/\psi \to \gamma\eta_c$, followed by $\eta_c \to p\bar{p}$. The invariant mass of the $p\bar{p}$ pair ($M_{p\bar{p}}$) is restricted to the $\eta_c$ mass region $[2.70, 3.05] \text{ GeV}/c^2$.
• Event Selection:
  • Candidates require two charged tracks (proton/antiproton) and at least one photon.
  • Strict kinematic cuts are applied: polar angle $|\cos\theta| < 0.93$, vertex constraints, and particle identification (PID) based on energy loss and time-of-flight.
  • Photon candidates are selected from the electromagnetic calorimeter (EMC) with energy thresholds (25 MeV barrel, 50 MeV end-cap) and timing cuts to suppress noise.
  • A 4-constraint (4C) kinematic fit is performed assuming $J/\psi \to \gamma p\bar{p}$.
• Background Suppression:
  • Dominant backgrounds include $J/\psi \to p\bar{p}\gamma_{FSR}$ (Final State Radiation) and $J/\psi \to p\bar{p}\pi^0$.
  • Non-resonant contributions from $N^*$ baryons are estimated to be $<1.4\%$ and ignored.
  • Off-peak data ($\sqrt{s} = 3.080$ GeV) is used to estimate non-$J/\psi$ backgrounds ($<0.7\%$).
• Amplitude Analysis:
  • Unlike previous one-dimensional fits, this study performs a full amplitude analysis using covariant tensor amplitudes.
  • The amplitude includes the $\eta_c$ resonance (modeled by a relativistic Breit-Wigner function) and multiple non-resonant (NR) partial waves with different spin-parity ($J^P$) configurations.
  • Damping Factors: Two forms (CLEO-c and KEDR) were tested to suppress the divergent long tail of the $\eta_c$. The KEDR form was selected based on a significantly better log-likelihood value ($\Delta \ln \mathcal{L} \to 4.7\sigma$ improvement).
  • Fit Strategy: A maximum likelihood fit determines complex coupling constants ($\Lambda_i$) and resonance parameters. Five NR components with significance $>3\sigma$ (including $0^-, 1^+, 2^+, 2^-$ waves) were retained.

3. Key Contributions

• First Amplitude Analysis: This is the first amplitude analysis of $J/\psi \to \gamma p\bar{p}$ in the $\eta_c$ mass region, rigorously accounting for the interference between the $\eta_c$ resonance and non-resonant amplitudes.
• Precision Improvement: The precision of the product branching fraction measurement is improved by one order of magnitude compared to previous measurements.
• Resolution of Tensions: By combining the new measurement with previously determined product BFs ($B(\eta_c \to p\bar{p}) \times B(\eta_c \to \gamma\gamma)$ and $B(J/\psi \to \gamma\eta_c) \times B(\eta_c \to \gamma\gamma)$), the authors extract absolute branching fractions with high precision, directly addressing the discrepancy between experimental data and LQCD predictions.

4. Key Results

A. Resonance Parameters of $\eta_c$:

• Mass: $M_{\eta_c} = (2984.55 \pm 0.09_{\text{stat}} \pm 0.77_{\text{syst}}) \text{ MeV}/c^2$
• Width: $\Gamma_{\eta_c} = (29.74 \pm 0.17_{\text{stat}} \pm 0.33_{\text{syst}}) \text{ MeV}$
• These values are in excellent agreement with PDG world averages.

B. Branching Fractions:

1. Product Branching Fraction:
   $$B(J/\psi \to \gamma\eta_c) \times B(\eta_c \to p\bar{p}) = (2.11 \pm 0.02_{\text{stat}} \pm 0.07_{\text{syst}}) \times 10^{-5}$$

2. Derived Absolute Branching Fractions:
   Using the product BFs and assuming time-reversal invariance, the study calculates:

   • $B(J/\psi \to \gamma\eta_c)$: $(2.29 \pm 0.01_{\text{stat}} \pm 0.04_{\text{syst}} \pm 0.18_{\text{opbf}})\%$
   • $B(\eta_c \to \gamma\gamma)$: $(2.28 \pm 0.01_{\text{stat}} \pm 0.04_{\text{syst}} \pm 0.18_{\text{opbf}}) \times 10^{-4}$
   • $B(\eta_c \to p\bar{p})$: $(0.92 \pm 0.01_{\text{stat}} \pm 0.02_{\text{syst}} \pm 0.07) \times 10^{-3}$
     (Note: "opbf" denotes uncertainty from other product branching fractions used in the calculation.)

C. Mass Splitting:
The mass splitting between the $J/\psi$ ($1^3S_1$) and $\eta_c$ ($1^1S_0$) is determined to be $(112.35 \pm 0.77) \text{ MeV}/c^2$, consistent with theoretical calculations.

5. Significance

• Resolution of the "Puzzle": The newly determined $B(J/\psi \to \gamma\eta_c)$ deviates from the PDG global fit by $3\sigma$ but is in excellent agreement with the latest Lattice QCD calculations (e.g., HPQCD, CLQCD). This suggests that previous experimental averages were biased, likely due to the neglect of NR interference in earlier analyses.
• Validation of Theory: The results support the validity of modern LQCD calculations for heavy quarkonium transitions, bridging the gap between perturbative and non-perturbative QCD descriptions.
• Methodological Benchmark: The success of the amplitude analysis demonstrates the necessity of including full interference effects in resonance studies, setting a new standard for precision measurements in charmonium physics.

In conclusion, this work provides the most precise determination of the $J/\psi \to \gamma\eta_c$ transition to date, effectively resolving a long-standing discrepancy between experiment and theory and validating the predictive power of Lattice QCD.