Measurement of the branching fraction of $\eta \to \mu^+ \mu^-$ and search for $\eta \to e^+ e^-$

Using a sample of approximately 10 million $J/\psi$ events collected by the BESIII detector, this study measures the branching fraction of $\eta \to \mu^+ \mu^-$ to be $(5.8 \pm 1.0_{\rm stat} \pm 0.2_{\rm syst}) \times 10^{-6}$ and sets an improved 90% confidence level upper limit of $2.2 \times 10^{-7}$for the$\eta \to e^+ e^-$ decay.

The Gist

Imagine the subatomic world as a bustling, chaotic city where tiny particles are the citizens. Among these citizens is a very shy, short-lived character named the Eta meson ($\eta$). It's like a ghost that appears for a split second and then vanishes. Usually, it disappears by turning into other, more common particles (like pions or photons).

But physicists have a burning question: Can this ghost ever turn into a pair of "lepton twins"? Specifically, can it turn into a pair of muons ($\mu^+\mu^-$) or a pair of electrons ($e^+e^-$)?

This paper is the report from the BESIII Collaboration, a massive team of scientists working at a giant particle accelerator in Beijing (the BEPCII). They acted like high-tech detectives, sifting through billions of cosmic "crime scenes" to find these rare transformations.

Here is the story of their investigation, broken down into simple concepts:

1. The Setup: Catching a Ghost in a Crowd

The Eta meson is hard to study because it doesn't just hang out alone. To find it, the scientists used a clever trick. They looked at a specific event where a heavy particle called a $J/\psi$ (think of it as a heavy, unstable bouncer) decays.

When this bouncer decays, it often shoots out a photon (a particle of light) and a slightly lighter cousin called an Eta-prime ($\eta'$). The Eta-prime is unstable and quickly breaks apart into two pions and our shy Eta meson.

• The Chain Reaction: $J/\psi \to \text{Photon} + \eta' \to \text{Photon} + \text{Pions} + \eta$.

Once the Eta meson is created in this chain, the scientists waited to see what it would do next. Would it do the usual thing, or would it do the rare thing and turn into a pair of muons or electrons?

2. The Muon Hunt: Finding the "Heavy Twins"

First, they looked for the Eta $\to$ Muon + Antimuon decay.

• The Analogy: Imagine you are looking for a specific type of rare bird in a forest. Most birds in the forest are sparrows (common particles). You are looking for a pair of golden eagles (muons) that only appear if a specific tree (the Eta) falls.
• The Challenge: The forest is noisy. There are many other birds that look almost like golden eagles, or events where two birds happen to fly by at the same time by accident.
• The Result: After analyzing 10 billion of these events (that's a lot of data!), they found 38 clear cases where the Eta meson turned into a muon pair.
• The Measurement: They calculated that this happens about 6 times out of every million Eta decays. This matches perfectly with what the "Standard Model" (the rulebook of physics) predicts. It's like confirming that the golden eagles exist exactly where the map said they would.

3. The Electron Hunt: The "Ghostly" Search

Next, they looked for the Eta $\to$ Electron + Positron decay.

• The Analogy: This is like looking for a pair of invisible fairies. Theoretically, this should happen even less often than the muon pair because electrons are so light. In fact, the rules of physics say this is incredibly difficult to pull off.
• The Challenge: Because it's so rare, and because there are so many other processes that create electron pairs (background noise), finding a signal is like trying to hear a whisper in a hurricane.
• The Result: They looked through their massive dataset and found zero clear cases of this happening.
• The Conclusion: Since they didn't find any, they couldn't give a number. Instead, they set a "speed limit" for how often this could be happening. They said, "If this fairy dance happens, it must be less than 2.2 times out of every 10 million tries."
• Why it matters: This is a huge improvement over previous limits. It's like saying, "We used to think the fairy might appear 7 times out of 10 million, but now we know for sure it's less than 2.2 times." This tightens the net around "New Physics."

4. Why Does This Matter?

You might ask, "So what? We just confirmed some numbers."

Here is the big picture:

• Testing the Rulebook: The Standard Model is our best guess at how the universe works. If the Eta meson turned into muons or electrons more often than predicted, it would be a smoking gun. It would mean there are hidden forces or new particles (like "New Physics") interacting with the Eta that we don't know about yet.
• The Verdict: The muon result fits the rulebook perfectly. The electron result is consistent with the rulebook (because it's so rare, we just haven't seen it yet, but we know it's not happening too often).
• The Future: The fact that they didn't find any "surprises" is actually good news for the rulebook, but it also means we need even bigger, more powerful detectors (like the future "Super Tau-Charm Factory") to catch even rarer events.

Summary

The BESIII team acted like cosmic detectives with a magnifying glass the size of a city. They sifted through 10 billion particle collisions to find a needle in a haystack.

1. They found the muon needle exactly where the map said it would be.
2. They looked for the electron needle, didn't find it, but proved it's even rarer than we thought.

This confirms our current understanding of the universe is solid, but it also sets the stage for the next generation of experiments to push the boundaries even further.

Technical Summary

Here is a detailed technical summary of the paper "Measurement of the branching fraction of $\eta \to \mu^+\mu^-$ and search for $\eta \to e^+e^-$" by the BESIII Collaboration.

1. Problem Statement and Motivation

The decay of the eta meson ($\eta$) into lepton pairs ($\eta \to \ell^+\ell^-$, where $\ell = e$ or $\mu$) is a rare process within the Standard Model (SM). It proceeds via a fourth-order electromagnetic transition dominated by a two-photon intermediate state ($\eta \to \gamma^*\gamma^* \to \ell^+\ell^-$).

• Theoretical Significance: The decay rate is highly sensitive to the $\eta$ transition form factor. The decay to electrons ($\eta \to e^+e^-$) is further suppressed by a helicity factor proportional to the square of the electron mass ($m_e^2$).
• Physics Beyond the Standard Model (BSM): Any deviation between experimental measurements and SM predictions could indicate the presence of hypothetical interactions or new physics.
• Current Status:
  • The world average for $\mathcal{B}(\eta \to \mu^+\mu^-)$ is $(5.7 \pm 0.8) \times 10^{-6}$, based on data from over 30 years ago.
  • The upper limit for $\mathcal{B}(\eta \to e^+e^-)$ was previously $< 7 \times 10^{-7}$ (90% CL).
  • Previous attempts to study these decays via $J/\psi \to \gamma \eta$ suffered from large backgrounds from $J/\psi \to \gamma \ell^+\ell^-$.

2. Methodology

The analysis utilizes a dataset of $(10087 \pm 44) \times 10^6$ $J/\psi$ events collected by the BESIII detector at the BEPCII storage ring.

A. Decay Chain and Tagging Strategy

To suppress backgrounds, the authors employ a novel tagging approach using the radiative decay of the $J/\psi$ to an $\eta'$ meson, followed by the decay of the $\eta'$:
$$J/\psi \to \gamma \eta' \quad \text{with} \quad \eta' \to \pi^+\pi^-\eta \quad \text{and} \quad \eta \to \ell^+\ell^-$$
This chain allows for the reconstruction of the $\eta$ mass via the $\pi^+\pi^-$ system and the lepton pair, significantly reducing background compared to direct $J/\psi \to \gamma \eta$ studies.

B. Event Selection and Reconstruction

• Final State: Events are required to have one photon candidate and four charged tracks ($\pi^+\pi^-\ell^+\ell^-$) with a total charge of zero.
• Kinematic Fits: A four-constraint (4C) kinematic fit is performed imposing energy and momentum conservation for the hypothesis $J/\psi \to \gamma \pi^+\pi^-\ell^+\ell^-$.
• Particle Identification (PID):
  • Muons: Identified using Time-of-Flight (TOF) and specific ionization energy loss ($dE/dx$). Muon counter information was not used due to low muon momentum.
  • Electrons: Identified using TOF and $dE/dx$.
• Background Rejection:
  • For $\eta \to \mu^+\mu^-$, a requirement $\chi^2_{sum}(\gamma\pi^+\pi^-\pi^+\pi^-) > \chi^2_{sum}(\gamma\pi^+\pi^-\mu^+\mu^-)$ rejects the dominant background from $\eta' \to \pi^+\pi^-\pi^+\pi^-$.
  • Inclusive Monte Carlo (MC) samples of 10 billion $J/\psi$ events were used to model and estimate backgrounds.

C. Analysis Techniques

• $\eta \to \mu^+\mu^-$: An extended unbinned maximum likelihood fit is performed on the $\mu^+\mu^-$ invariant mass ($M_{\mu^+\mu^-}$) distribution. The signal shape is derived from MC simulation, while background shapes are modeled using dedicated MC samples for specific channels ($\eta' \to \pi^+\pi^-\pi^+\pi^-$, $\eta' \to \pi^+\pi^-\mu^+\mu^-$, etc.).
• $\eta \to e^+e^-$: A search is conducted in the $e^+e^-$ invariant mass ($M_{e^+e^-}$) distribution. Since no signal events were observed, an upper limit is calculated using the method of Rolke and Lopez (Ref. [36]) at 90% confidence level (CL), accounting for estimated background yields in sideband regions.

3. Key Contributions

1. Novel Tagging Channel: Successfully demonstrated the use of the $J/\psi \to \gamma \eta' (\to \pi^+\pi^-\eta)$ chain to isolate $\eta \to \ell^+\ell^-$ decays, effectively mitigating the large backgrounds that plagued previous $J/\psi \to \gamma \eta$ analyses.
2. High-Precision Measurement: Provided the most precise measurement of $\mathcal{B}(\eta \to \mu^+\mu^-)$ to date, utilizing the largest $J/\psi$ dataset available.
3. Improved Sensitivity: Set a significantly tighter upper limit on the ultra-rare $\eta \to e^+e^-$ decay, improving upon the previous limit by a factor of more than 3.
4. Systematic Control: Detailed evaluation of systematic uncertainties arising from tracking efficiency, PID, photon detection, and kinematic fitting, ensuring the robustness of the results.

4. Results

A. Measurement of $\eta \to \mu^+\mu^-$

• Signal Yield: $37.9 \pm 6.7$ events.
• Statistical Significance: $9.8\sigma$.
• Branching Fraction:
  $$\mathcal{B}(\eta \to \mu^+\mu^-) = (5.8 \pm 1.0_{\text{stat}} \pm 0.2_{\text{syst}}) \times 10^{-6}$$
• Comparison: This result is consistent with the previous world average and theoretical predictions (e.g., Canterbury approximants method predicts $(4.72^{+0.05}_{-0.21}) \times 10^{-6}$ and dispersive representations predict $(4.54 \pm 0.04 \pm 0.02) \times 10^{-6}$). The measured value is within $1.2\sigma$ of the latest theoretical predictions.

B. Search for $\eta \to e^+e^-$

• Observation: No signal events were observed in the signal region ($M_{e^+e^-} \in [0.534, 0.560]$ GeV/$c^2$).
• Background Estimate: Estimated background in the signal region was $N_{bkg} = 1.5$.
• Upper Limit:
  $$\mathcal{B}(\eta \to e^+e^-) < 2.2 \times 10^{-7} \quad (\text{at } 90\% \text{ CL})$$
• Improvement: This improves the previous limit of $7 \times 10^{-7}$ (from the SND experiment) by more than a factor of 3.

5. Significance

• Validation of SM: The measurement of $\mathcal{B}(\eta \to \mu^+\mu^-)$ confirms the Standard Model prediction for this rare electromagnetic decay, constraining potential BSM contributions.
• Constraints on New Physics: The improved upper limit on $\mathcal{B}(\eta \to e^+e^-)$ places stricter constraints on hypothetical interactions (such as axion-like particles or leptoquarks) that could enhance this helicity-suppressed decay.
• Future Outlook: The success of this analysis paves the way for future experiments at facilities like the Super $\tau$-Charm Factory (STCF), JLab Eta Factory (JEF), and REDTOP, which will provide even larger $\eta$ samples to further probe the internal structure of the $\eta$ meson and search for new physics.