Measurement of the $η$ transition form factor through $η' \rightarrow π^+π^-η$ decay

Using a large sample of $J/\psi$ events collected by the BESIII experiment, this study extracts the $\eta$ transition form factor slope and branching fractions for $\eta \to \gamma e^+e^-$ and $\eta \to \gamma \mu^+\mu^-$ decays via the $\eta' \to \pi^+\pi^-\eta$ channel, combines these results with previous measurements to improve precision, and sets new upper limits on dark photon production without observing a significant signal.

The Gist

The Big Picture: Peeking Inside a Ghost

Imagine the universe is filled with tiny, invisible building blocks called quarks. These quarks stick together to form particles like protons and neutrons. But they also form "ghosts" called mesons (specifically the eta meson, or $\eta$).

These ghosts are tricky. They don't have a hard shell like a billiard ball; instead, they are fuzzy clouds of energy. Physicists want to know: How is the electric charge spread out inside this fuzzy cloud?

To answer this, they measure something called a Transition Form Factor (TFF). Think of the TFF as a magnifying glass. If you look at the eta meson from far away, it looks like a single point. But if you zoom in (by hitting it with high energy), you start to see its internal structure. The "slope" of this magnifying glass tells us how "squishy" or spread out the charge is inside the particle.

The Experiment: A Cosmic Pinball Machine

The scientists used a giant machine called BESIII (located in Beijing), which is essentially a massive, high-speed pinball machine for particles.

1. The Setup: They smashed electrons and positrons (anti-electrons) together to create a massive amount of $J/\psi$ particles. Think of the $J/\psi$ as a heavy, unstable parent particle that loves to break apart.
2. The Decay Chain: When the $J/\psi$ breaks, it sometimes spits out a photon (a particle of light) and an $\eta'$ meson (a heavier cousin of our target).
   • The $\eta'$ is unstable and immediately breaks apart into two pions (charged particles) and our target, the $\eta$ meson.
   • Finally, the $\eta$ meson decays into a photon and a pair of leptons (either an electron/positron pair or a muon/anti-muon pair).

The Analogy: Imagine a Russian nesting doll.

• You open the outer doll ($J/\psi$) and find a middle doll ($\eta'$).
• You open the middle doll and find the smallest doll ($\eta$).
• You open the smallest doll, and pop! out come a photon and a pair of tiny, fast-moving marbles (the electron or muon pair).

The scientists caught billions of these "pops" (about 10 billion events!) to get a clear picture.

The New Trick: A Better Way to Find the Target

In previous experiments, the scientists tried to find the $\eta$ meson directly. It was like trying to find a specific needle in a haystack while the hay was blowing in the wind. There was too much "noise" (background interference) from other particle collisions.

The Innovation: In this paper, they used a new strategy. They waited for the $\eta$ to be born from the decay of the $\eta'$ (the middle doll).

• Why is this better? The $\eta'$ is a very precise "factory." When it decays, it leaves a very clean signature. It's like switching from looking for a needle in a haystack to looking for a needle that was just dropped by a robot arm in a clean room. This allowed them to filter out the noise much better and get a sharper measurement.

The Results: Measuring the "Squishiness"

By analyzing the energy and angles of the electron/muon pairs, the scientists calculated the slope of the form factor.

• The Result: They found that the charge inside the eta meson is distributed in a specific way, described by a number called $\Lambda^{-2}$.
• The Analogy: Imagine the eta meson is a balloon. If you squeeze it, how much does it resist? The scientists measured exactly how "stiff" or "soft" that balloon is. Their measurement matches what other teams found, but with much higher precision (less error bars).

They also measured Branching Fractions, which is just a fancy way of asking: "Out of 1,000 times this particle decays, how many times does it choose to split into electrons vs. muons?" They found the answers to be very consistent with previous theories.

The Treasure Hunt: Searching for "Dark Photons"

The most exciting part of the hunt is looking for things we don't expect to find. The scientists were also looking for a hypothetical particle called the Dark Photon ($A'$).

• The Metaphor: Imagine you are listening to a radio station. You expect to hear music (standard physics). But you suspect there might be a secret, encrypted channel (Dark Matter) playing a different song.
• The Search: They scanned the data, looking for a tiny "blip" or a specific frequency in the electron pairs that didn't fit the standard music.
• The Outcome: They didn't find the secret channel. No Dark Photon was detected.
• Why is this good? In science, finding nothing is also a discovery! It tells us that if Dark Photons exist, they must be even more elusive or weaker than we thought. The scientists set a "fence" (an upper limit) saying, "If a Dark Photon exists, it cannot be stronger than this line."

Summary

1. Goal: To understand the internal structure of the eta meson.
2. Method: They used a massive particle collider to create billions of events, using a clever new decay chain to get a cleaner signal than before.
3. Discovery: They measured the "shape" of the eta meson's charge distribution with high precision.
4. Bonus: They looked for a mysterious "Dark Photon" but didn't find it, which helps rule out some theories about dark matter.

In short, this paper is a masterclass in precision engineering at the subatomic level, proving that even when we think we know the rules of the universe, there's always room to measure them a little more accurately.

Technical Summary

1. Problem Statement and Motivation

The Transition Form Factor (TFF) of light mesons provides critical insights into their internal structure, specifically the distribution of charge and magnetization among quarks and gluons. Understanding the TFF is fundamental to Quantum Chromodynamics (QCD) and is a necessary input for calculating the hadronic light-by-light scattering contribution to the anomalous magnetic moment of the muon ($a_\mu$).

Prior to this work, measurements of the $\eta$ meson TFF were limited:

• The A2 collaboration measured the TFF via the $\eta \to \gamma e^+e^-$ channel.
• The NA60 collaboration measured it via the $\eta \to \gamma \mu^+\mu^-$ channel.
• A previous BESIII measurement utilized the $J/\psi \to \gamma \eta$ decay channel (referred to as Sample II).

The BESIII collaboration aimed to improve the precision of these measurements and explore a new production mechanism to reduce background and increase signal yield. Specifically, they sought to measure the TFF and the branching fractions (BFs) of the Dalitz decays $\eta \to \gamma l^+l^-$ ($l=e, \mu$) using a novel sample where the $\eta$ is produced via the radiative decay of the $\eta'$ meson.

2. Methodology

Data Samples and Detector

• Data Source: The analysis is based on a sample of $(1.0087 \pm 0.0044) \times 10^{10}$ $J/\psi$ events collected by the BESIII detector at the BEPCII collider between 2009 and 2019.
• Novel Approach (Sample I): Instead of the traditional $J/\psi \to \gamma \eta$ channel, this study focuses on the chain:
  $$J/\psi \to \gamma \eta', \quad \eta' \to \pi^+\pi^-\eta, \quad \eta \to \gamma l^+l^-$$
  This approach offers two main advantages over the previous BESIII method:
  1. Better Mass Resolution: The $\eta'$ mass resolution is superior, leading to reduced background from other $J/\psi$ decays.
  2. Higher Efficiency: Due to the higher branching fraction of the intermediate steps and the reconstruction efficiency of the charged pions, Sample I yields approximately twice the signal events compared to Sample II.

Event Selection and Background Suppression

• Reconstruction: Events are selected requiring two photons (one radiative $\gamma_r$ from $J/\psi$, one from $\eta$ decay) and four charged tracks (two pions from $\eta'$, two leptons from $\eta$).
• Kinematic Fit: A 6-constraint (6C) kinematic fit is applied, enforcing energy-momentum conservation and constraining the masses of the $\eta'$ and $\eta$ to their Particle Data Group (PDG) values.
• Photon Conversion Veto: A major background source for the $e^+e^-$ channel is $\eta \to \gamma\gamma$ where one photon converts to an $e^+e^-$ pair in the detector material. The authors use a Photon Conversion Finder (PCF) to reconstruct the conversion vertex (CV). Events where the CV is displaced from the interaction point (IP) by $2 < R_{xy} < 8$ cm and the angle $\theta_{eg}$ is small are rejected.
• Background Estimation: Backgrounds from non-$J/\psi$ processes are negligible. Backgrounds from other $J/\psi$ decays (e.g., $\eta \to \gamma \pi^+\pi^-$) are estimated using exclusive Monte Carlo (MC) simulations normalized to known branching fractions.

Analysis Techniques

• TFF Extraction: An unbinned maximum likelihood fit is performed on the invariant mass distribution of the lepton pair ($q^2$). The TFF is parameterized using a single-pole approximation:
  $$F(q^2) = \frac{1}{1 - q^2/\Lambda^2}$$
  The slope parameter $\Lambda^{-2}$ is the free parameter extracted from the fit.
• Branching Fraction Calculation: The BFs are calculated by normalizing the signal yield to the total number of $J/\psi$ events and the known branching fractions of the intermediate decays.
• Combination: The results from Sample I (new) and Sample II (previous $J/\psi \to \gamma \eta$ data) are combined using a weighted average that accounts for correlated systematic uncertainties.
• Dark Photon Search: The combined $\eta \to \gamma e^+e^-$ sample is scanned for a narrow resonance peak ($A'$) in the $e^+e^-$ invariant mass spectrum to search for a dark photon.

3. Key Contributions

1. Novel Production Channel: Successfully demonstrated the feasibility and superiority of using the $J/\psi \to \gamma \eta' \to \gamma \pi^+\pi^-\eta$ chain for $\eta$ TFF measurements, achieving higher statistics and better background rejection than the direct $J/\psi \to \gamma \eta$ method.
2. First BESIII Measurement of $\eta \to \gamma \mu^+\mu^-$ TFF: Provided a new determination of the TFF slope for the dimuon channel, which had not been measured by BESIII previously.
3. Precision Combination: Performed a rigorous statistical combination of the new Sample I data with the existing Sample II data, reducing the total uncertainty on the TFF slope and the electron-channel branching fraction.
4. Dark Photon Limits: Set new, competitive upper limits on the branching fraction of $\eta \to \gamma A'$ for dark photon masses between 0.005 and 0.535 GeV/$c^2$.

4. Results

Transition Form Factor Slope ($\Lambda^{-2}$)

The slope of the TFF was measured for both channels using Sample I:

• Di-electron channel ($\eta \to \gamma e^+e^-$):
  $$\Lambda^{-2} = 1.668 \pm 0.093_{\text{stat}} \pm 0.024_{\text{sys}} \, (\text{GeV}/c^2)^{-2}$$
• Di-muon channel ($\eta \to \gamma \mu^+\mu^-$):
  $$\Lambda^{-2} = 1.645 \pm 0.343_{\text{stat}} \pm 0.017_{\text{sys}} \, (\text{GeV}/c^2)^{-2}$$
  (Note: The statistical uncertainty dominates the muon channel due to lower event statistics.)

Combined Result (Sample I + Sample II):
By combining the new data with previous BESIII results ($J/\psi \to \gamma \eta$), the precision for the electron channel was improved:
$$\Lambda^{-2} = 1.707 \pm 0.076_{\text{stat}} \pm 0.029_{\text{sys}} \, (\text{GeV}/c^2)^{-2}$$
This result is consistent with previous BESIII measurements and theoretical predictions (Chiral Perturbation Theory, Pade Approximant), though slightly lower than the A2 collaboration's result (within 2$\sigma$).

Branching Fractions

• $\eta \to \gamma e^+e^-$: $(6.79 \pm 0.04_{\text{stat}} \pm 0.36_{\text{sys}}) \times 10^{-3}$
• $\eta \to \gamma \mu^+\mu^-$: $(2.97 \pm 0.11_{\text{stat}} \pm 0.07_{\text{sys}}) \times 10^{-4}$
• Combined $\eta \to \gamma e^+e^-$: $(6.93 \pm 0.28_{\text{tot}}) \times 10^{-3}$

Dark Photon Search

No significant signal for a dark photon ($A'$) was observed in the $e^+e^-$ invariant mass spectrum.

• Upper Limits: 90% confidence level (CL) upper limits were set on the branching fraction $B(\eta \to \gamma A', A' \to e^+e^-)$ for $A'$ masses ranging from 0.005 to 0.535 GeV/$c^2$.
• These limits improve upon or are competitive with previous constraints from experiments like APEX, BaBar, and NA64 in the relevant mass range.

5. Significance

This work represents a significant advancement in the precision of $\eta$ meson structure studies. By introducing a new production channel ($\eta'$ decay) that offers higher statistics and cleaner backgrounds, the BESIII collaboration has reduced the systematic and statistical uncertainties on the TFF slope. The combined result provides a robust benchmark for theoretical models of light meson structure and contributes essential data for reducing the uncertainty in the Standard Model prediction of the muon anomalous magnetic moment ($g-2$). Furthermore, the stringent limits placed on the dark photon search constrain models of physics beyond the Standard Model involving light hidden sectors.