Observation of the rare decay $\eta$ $\to$… — Plain-Language Explanation

Imagine the universe is a giant, bustling construction site where tiny building blocks called particles are constantly crashing into each other. At the CERN laboratory in Europe, scientists use a massive machine called the Large Hadron Collider (LHC) to smash protons together at incredible speeds, creating a shower of new, short-lived particles.

This paper is about the CMS Collaboration, a team of thousands of scientists who act like detectives at this construction site. They are looking for a very specific, extremely rare event: a particle called the eta meson (let's call it "Eta") breaking apart in a very unusual way.

The Rare Breakup

Usually, when Eta breaks apart, it follows a predictable pattern, like a toy car rolling down a ramp. But sometimes, it does something weird. In this study, the scientists caught Eta breaking into four pieces: two positive muons, two negative muons, two positive electrons, and two negative electrons (wait, that's too many! Let's correct that: it breaks into two muons and two electrons, one positive and one negative of each).

Think of Eta as a fragile, magical balloon. Usually, when it pops, it releases a specific set of confetti. But this time, the scientists wanted to see if it could pop and release a different mix of confetti: a pair of muons and a pair of electrons. This specific mix had never been seen before in a single event; it was like finding a unicorn in a herd of horses.

The Challenge: Finding a Needle in a Haystack

The problem is that this event is incredibly rare. It's like trying to find one specific grain of sand on a beach, but the beach is constantly being covered by new sand.

To make this harder, the "haystack" is full of other particles that look almost exactly like the ones they are searching for. For example, there's a common event where Eta breaks into two muons and a photon (a particle of light). If that photon hits the detector and turns into an electron-positron pair (which happens sometimes), it looks exactly like the rare event the scientists are hunting for. This is the "fake" signal, or the "resonant background."

The Detective Work: How They Found It

The CMS team used a clever trick to solve this mystery:

The High-Speed Camera: They used a special "trigger" system. Imagine a security camera that usually only records when a car drives by at 100 mph. But for this experiment, they programmed the camera to also record cars driving at 30 mph. This allowed them to catch the slow, rare events that usually get ignored.
The Reference Point: To know how rare their find was, they needed a ruler. They used the "fake" event (Eta breaking into two muons and a photon that turns into electrons) as a reference. They counted how many of these "fake" events happened and compared it to the "real" rare events.
The Filter: They applied strict rules to their data. They looked for events where the four particles (two muons, two electrons) came from the exact same spot (a common vertex) and had the right energy. They also checked that the electrons didn't come from a "conversion" in the wrong place, which helped them separate the real signal from the noise.

The Result: A Unicorn Found!

After analyzing data from 2022 (equivalent to 38 "inverse femtobarns" of collisions—a unit of measurement for how many crashes they watched), they found 127 clear examples of this rare decay.

The Discovery: They confirmed that the decay $\eta \to \mu^+\mu^-e^+e^-$ exists. It's the first time anyone has ever seen it happen.
The Frequency: They calculated that for every million times Eta decays, this specific four-particle breakup happens about 2.4 times.
The Significance: Before this, the best scientists could do was say, "It happens less than 160 times per million." This new result is two orders of magnitude (100 times) more precise than the old limit. It's like going from guessing a coin is "somewhat heavy" to weighing it on a scale that shows it is exactly 5.2 grams.

Why Does This Matter?

The paper explains that this isn't just about finding a rare particle; it's about understanding the "rules of the game" for the universe.

Testing the Theory: The result matches the predictions made by the "Standard Model" (the current best theory of how particles work). It's like checking if a new puzzle piece fits perfectly into the picture we already have.
Magnetic Mystery: The data helps scientists calculate something called the "muon anomalous magnetic moment." Think of a muon as a tiny spinning top. Scientists are trying to measure exactly how fast it spins and wobbles. The way Eta decays helps them understand the "air resistance" (quantum effects) the top feels, which is crucial for solving a major mystery in physics about why muons behave slightly differently than expected.

In Summary

The CMS team successfully caught a "ghost" event that had been hiding in the noise of particle collisions. By using a high-speed trigger and a clever comparison method, they proved that the eta meson can indeed split into two muons and two electrons. This discovery tightens the screws on our understanding of the subatomic world, confirming that our current theories are on the right track, even when dealing with the rarest of events.

1. Problem and Motivation

The paper addresses the first observation of the rare double-Dalitz decay of the eta meson ( $\eta$ ) into a muon pair and an electron pair: $\eta \to \mu^+\mu^-e^+e^-$ .

Scientific Context: The $\eta$ and $\eta'$ mesons are light, neutral, spin-0 particles. Their rare decays into purely leptonic final states are governed by chiral anomalies involving one or two virtual photons.
Theoretical Importance: Measuring these decay rates is crucial for:
- Calculating the light-by-light hadronic contributions to the muon anomalous magnetic moment ( $a_\mu$ ).
- Testing lepton flavor universality.
- Probing Beyond-Standard-Model (BSM) theories that predict deviations in these rates.
Gap in Knowledge: While other modes like $\eta \to \mu^+\mu^-$ , $\eta \to e^+e^-e^+e^-$ , and $\eta \to \mu^+\mu^-\mu^+\mu^-$ have been observed, the $\eta \to \mu^+\mu^-e^+e^-$ channel had not been observed, with a previous upper limit of $1.6 \times 10^{-4}$ (90% CL) set by the WASA experiment.

2. Methodology

Data Sample and Trigger

Data Source: Proton-proton collisions at $\sqrt{s} = 13.6$ TeV collected by the CMS experiment in 2022.
Integrated Luminosity: $38.0 \text{ fb}^{-1}$ .
Trigger Strategy: A specialized high-rate dimuon "data-parking" trigger was employed. Unlike standard triggers that require high transverse momentum ( $p_T$ ), this trigger accepted events with lower-momentum muons ( $p_T > 3$ and $4$ GeV) to capture the low-energy $\eta$ decays. This allowed the storage of events for offline reconstruction when computing resources were available, significantly improving sensitivity.

Event Selection and Reconstruction

Signal Channel ( $\eta \to \mu^+\mu^-e^+e^-$ ):
- Requires an opposite-sign muon pair and an opposite-sign electron pair originating from a common vertex ( $\chi^2$ probability $> 10\%$ ).
- Electrons must pass strict identification criteria (Boosted Decision Tree) but must not be part of a reconstructed photon conversion candidate.
- Electrons must have $p_T > 2$ GeV and specific pseudorapidity ( $|\eta| < 1.44$ or $1.56 < |\eta| < 2.5$ ).
Reference Channel ( $\eta \to \mu^+\mu^-\gamma$ ):
- Used for normalization. The final state is physically identical to the signal if the photon converts to an $e^+e^-$ pair in the detector.
- Selection requires the electrons to be part of a reconstructed conversion candidate.
- This "inverted" selection minimizes the efficiency for the signal mode while maximizing it for the reference mode.

Background Estimation

Dominant Background: The primary background is the resonant process $\eta \to \mu^+\mu^-\gamma$ where the photon converts to an $e^+e^-$ pair in the innermost tracker layers. This mimics the signal topology.
Other Backgrounds: Misidentification of pions as muons or photon conversions to muons are negligible due to low misidentification rates and kinematic shifts in the reconstructed mass.
Simulation: Monte Carlo (MC) simulations using PYTHIA 8.3 (for $\eta \to \mu^+\mu^-\gamma$ $η \to μ^{+} μ^{-} γ$ ) and PLUTO (for $\eta \to \mu^+\mu^-e^+e^-$ $η \to μ^{+} μ^{-} e^{+} e^{-}$ ) were used.
- The $\eta \to \mu^+\mu^-\gamma$ simulation was reweighted to match the experimentally observed dimuon invariant mass distribution (based on Vector Meson Dominance models) rather than a simple phase-space model.
- Corrections were applied for trigger efficiency, track reconstruction, and electron identification scale factors (SF).

Analysis Technique

Branching Fraction Calculation: The branching fraction (BF) was calculated using the ratio method:
$\frac{\mathcal{B}_{2\mu2e}}{\mathcal{B}_{2\mu\gamma}} = \frac{N_{2\mu2e}}{N_{2\mu\gamma}} \times \frac{\epsilon_{2\mu\gamma}}{\epsilon_{2\mu2e}}$
Where $N$ is the yield and $\epsilon$ is the efficiency.
Fitting: An extended maximum likelihood fit was performed on the four-lepton invariant mass spectrum ( $m_{\mu\mu ee}$ $m_{μμ ee}$ ).
- Signal: Modeled with a double-Gaussian function.
- Resonant Background: Modeled with a Breit-Wigner function (centered at 555 MeV due to conversion kinematics).
- Combinatorial Background: Modeled with a threshold function.

3. Key Contributions

First Observation: This is the first experimental observation of the $\eta \to \mu^+\mu^-e^+e^-$ decay.
Trigger Innovation: Demonstrated the efficacy of the "data-parking" technique combined with high-rate dimuon triggers to access rare, low-momentum decay modes at the LHC, improving sensitivity by nearly two orders of magnitude over previous limits.
Normalization Strategy: Successfully utilized the $\eta \to \mu^+\mu^-\gamma$ channel with photon conversion as a reference, effectively canceling many systematic uncertainties related to $\eta$ production cross-sections and detector acceptance.

4. Results

Yields:
- Signal yield ( $N_{2\mu2e}$ ): $127 \pm 13$ events.
- Reference yield ( $N_{2\mu\gamma}$ ): $329 \pm 22$ events.
- Background in signal window: $\approx 4$ resonant background events and $\approx 10$ combinatorial events.
Significance: The likelihood-ratio test against the background-only hypothesis yields a significance far exceeding 5 standard deviations.
Branching Fraction Measurement:
- Ratio: $\mathcal{B}(\eta \to \mu^+\mu^-e^+e^-) / \mathcal{B}(\eta \to \mu^+\mu^-\gamma) = (7.3 \pm 2.2) \times 10^{-3}$ .
- Absolute Branching Fraction:
  $\mathcal{B}(\eta \to \mu^+\mu^-e^+e^-) = (2.4 \pm 0.8) \times 10^{-6}$
  (Uncertainty includes statistical, systematic, and the uncertainty of the reference BF).
Comparison:
- The result is approximately two orders of magnitude smaller than the previous upper limit ( $1.6 \times 10^{-4}$ ).
- The measurement is consistent with recent data-driven theoretical predictions ( $2.3\text{--}2.4 \times 10^{-6}$ ) based on chiral perturbation theory and Vector Meson Dominance.

5. Significance

Validation of Theory: The result validates the data-driven theoretical approaches used to model light-by-light scattering and transition form factors.
Muon g-2: The precise measurement of this decay channel provides essential input for calculating the hadronic light-by-light contribution to the muon anomalous magnetic moment, a key parameter in the search for new physics.
Methodological Benchmark: The analysis sets a new standard for measuring rare, low-mass resonances in high-luminosity hadron collider environments, proving that specialized trigger strategies can overcome the limitations of standard high- $p_T$ triggers.

Observation of the rare decay η\etaη →\to→ μ+μ−\mu^+\mu^-μ+μ−e+^++e−^-−