Search for the electromagnetic Dalitz decays $χ_{cJ}\to e^{+}e^{-}ϕ$

Using a large sample of $\psi(3686)$ events collected by the BESIII detector, this study presents the first search for the rare electromagnetic Dalitz decays $\chi_{cJ} \to e^+e^-\phi$ ($J=0,1,2$), finding no significant signals and establishing upper limits on their branching fractions at the 90% confidence level.

The Gist

The Big Picture: Hunting for a "Ghost" Particle Decay

Imagine you have a very expensive, high-speed camera (the BESIII detector) sitting at a giant particle collider (the BEPCII). This camera has taken millions of photos of a specific type of subatomic particle called the $\psi(3686)$.

In this paper, the scientists are looking for a very specific, rare event: a "ghost" decay. They are trying to catch a heavy particle called $\chi_c$ (a cousin of the $\psi$) as it transforms into three things at once:

1. A $\phi$ particle (a heavy, short-lived particle made of strange quarks).
2. An electron ($e^-$).
3. A positron ($e^+$, the electron's antimatter twin).

This process is called an electromagnetic Dalitz decay. It's like watching a heavy ball break apart into a smaller ball and a pair of tiny sparks.

The Challenge: Finding a Needle in a Haystack

The scientists had a massive dataset: about 2.7 billion $\psi(3686)$ events. That's a lot of data! However, the decay they are looking for is incredibly rare.

• The Analogy: Imagine you have a stadium full of people (the 2.7 billion events). You are looking for one specific person who is wearing a red hat, holding a blue balloon, and juggling three pineapples.
• The Problem: Most people in the stadium are just wearing hats or holding balloons, but not that specific combination. Even worse, there are "fake" people who look almost exactly like your target (background noise), such as someone who dropped a balloon that bounced off a wall and looked like a second balloon.

How They Searched: The "Filter" Process

To find their target, the BESIII team built a series of digital filters (selection criteria):

1. The Trackers: They looked for exactly four charged tracks (two electrons, two kaons from the $\phi$ decay) and one photon.
2. The ID Check: They used the detector's "ID cards" (Particle Identification) to make sure the tracks were actually electrons and kaons, and not pions or protons pretending to be them.
3. The "Ghost" Veto: One of the biggest headaches was "photon conversion." Sometimes, a real photon hits a piece of metal in the detector and splits into an electron-positron pair. This looks exactly like the signal they want, but it's a fake. They built a special algorithm to spot these "ghosts" and throw them out.
4. The Mass Window: They checked the "weight" (invariant mass) of the particles. If the combined weight of the electron-positron pair and the $\phi$ matched the known weight of the $\chi_c$ particle, it was a potential candidate.

The Result: The Great Silence

After running all these filters through their 2.7 billion events, the scientists looked at the final results.

• What they expected: They hoped to see a little "bump" in the data—a cluster of events that didn't fit the background noise.
• What they found: Nothing. Just a flat line. The number of events they saw was consistent with random background noise.

The Verdict: They did not find the $\chi_c \to e^+e^-\phi$ decay.

The Silver Lining: Setting the "Speed Limit"

Even though they didn't find the particle, the paper is still a huge success. In science, "not finding it" is valuable if you can say, "If it were there, it would have to be this rare."

The scientists calculated the Upper Limit. Think of this like a speed limit sign.

• They can't say the decay happens 100% of the time.
• They can't even say it happens 1 in a million times.
• They can confidently say: "If this decay happens, it happens less than 1 time in every 4 million attempts."

They set these "speed limits" (upper limits on the branching fraction) for three different versions of the $\chi_c$ particle ($\chi_{c0}$, $\chi_{c1}$, and $\chi_{c2}$).

Why Does This Matter?

1. First Time Ever: This is the first time anyone has ever looked for this specific type of decay in these specific particles. It's like exploring a new continent; even if you don't find gold, you've mapped the coastline.
2. Testing the Rules: The decay involves a "virtual photon" (a photon that exists for a split second). The way this happens is governed by the laws of Quantum Electrodynamics (QED) and Quantum Chromodynamics (QCD). By setting these strict limits, they are testing if our current theories of how particles interact are correct.
3. Future Hunting: Now that they know the decay is incredibly rare (or doesn't exist), future experiments with even bigger cameras (like a proposed "Super Tau-Charm Factory") will know exactly how sensitive they need to be to finally catch a glimpse of this ghost.

Summary in One Sentence

The BESIII collaboration used a massive dataset to hunt for a rare particle breakup that had never been seen before; while they didn't find it, they successfully proved that if it does exist, it is vanishingly rare, setting the stage for future, more powerful experiments to solve the mystery.

Technical Summary

1. Problem Statement and Motivation

The paper addresses the lack of experimental data regarding electromagnetic (EM) Dalitz decays of excited charmonium states, specifically the P-wave charmonium triplet $\chi_{cJ}$ ($J=0, 1, 2$).

• Physics Context: While EM Dalitz decays of light vector mesons ($\rho, \omega, \phi$) and S-wave charmonium ($J/\psi$) have been extensively studied, analogous transitions for P-wave states remain unexplored.
• Scientific Goal: These decays ($\chi_{cJ} \to \phi \gamma^* \to \phi e^+e^-$) provide critical insights into:
  • The intrinsic structure of mesons and the transition form factors (TFF) describing the EM vertex.
  • The dynamics of low-energy non-perturbative Quantum Chromodynamics (QCD), a regime where theoretical calculations are challenging.
  • Potential deviations from Standard Model predictions, which could hint at new physics (e.g., dark photons).
• Specific Challenge: The branching fraction for this Dalitz decay is expected to be two orders of magnitude lower than the corresponding radiative decay ($\chi_{cJ} \to \gamma \phi$) due to the additional EM vertex, making the search experimentally difficult.

2. Methodology

The analysis utilizes data collected by the BESIII detector at the BEPCII collider.

• Data Sample:
  • Source: $(2.712 \pm 0.014) \times 10^9$ $\psi(3686)$ events.
  • Center-of-mass energy: $\sqrt{s} = 3.686$ GeV.
  • Production mechanism: $\psi(3686) \to \gamma \chi_{cJ}$.
• Signal Topology:
  • The decay chain is $\psi(3686) \to \gamma \chi_{cJ} \to \gamma \phi e^+e^- \to \gamma (K^+K^-) e^+e^-$.
  • Final state requirement: 4 charged tracks ($K^+, K^-, e^+, e^-$) and 1 photon.
• Event Selection & Reconstruction:
  • Tracking & PID: Charged tracks are identified using the Drift Chamber (MDC) and Time-of-Flight (TOF). Electrons are identified via $dE/dx$ and TOF probabilities; Kaons are identified similarly.
  • Photon Detection: Photons are identified as isolated showers in the Electromagnetic Calorimeter (EMC) with energy thresholds (25 MeV barrel, 50 MeV endcap).
  • Kinematic Fit: A 4-constraint (4C) kinematic fit is applied, constraining the total four-momentum to the initial beam energy. Events with $\chi^2_{4C} < 40$ are retained.
  • Particle Identification Cuts: An $E/p$ ratio cut ($0.8 < E/p < 1.1$) is applied to electron candidates to suppress pion misidentification.
• Background Suppression:
  • Photon Conversion: A specific algorithm rejects events where the $e^+e^-$ pair originates from photon conversion in the detector material (beam pipe or MDC wall) by requiring the conversion vertex distance $R_{xy} < 2$ cm.
  • Resonance Veto: Events with invariant masses $M_{\gamma e^+e^-}$ or $M_{e^+e^-}$ consistent with $\eta$ or $\phi$ resonances (excluding the signal $\phi$) are rejected to suppress backgrounds like $\psi(3686) \to \eta \phi$ or $\chi_{cJ} \to \phi \phi$.
  • Mass Windows: Signal regions are defined for the $\phi$ meson ($|M_{K^+K^-} - M_\phi| < 8.5$ MeV/$c^2$) and $\chi_{cJ}$ states (approx. $3\sigma$ width).
• Monte Carlo (MC) Simulation:
  • Signal MC events are generated using a $q^2$-dependent decay amplitude with a Vector Meson Dominance (VMD) model for the Transition Form Factor (TFF).
  • Inclusive MC samples are used to estimate background levels.
• Statistical Analysis:
  • Upper limits are calculated using an unbounded profile likelihood method at 90% Confidence Level (C.L.).
  • Systematic uncertainties are incorporated as nuisance parameters in the likelihood function.

3. Key Contributions

• First Search: This is the first experimental search for the electromagnetic Dalitz transition of P-wave charmonium states ($\chi_{cJ}$) to a light vector meson ($\phi$).
• Comprehensive Background Handling: The analysis implements a rigorous multi-dimensional sideband method to estimate backgrounds from non-$\phi$ and non-$\chi_{cJ}$ components, as well as specific vetoes for photon conversions and other resonant backgrounds.
• Systematic Uncertainty Budget: A detailed breakdown of 13 sources of systematic uncertainty is provided, including tracking, PID, kinematic fitting, TFF modeling, and sideband choices. The total systematic uncertainty ranges from 7.8% to 8.1%.

4. Results

• Observation: No statistically significant signal was observed in the data for any of the three $\chi_{cJ}$ states ($J=0, 1, 2$).
  • Observed events in signal regions: 3 ($\chi_{c0}$), 7 ($\chi_{c1}$), and 3 ($\chi_{c2}$).
  • Estimated background events: $7.1^{+5.1}_{-3.5}$, $0.46^{+3.0}_{-1.0}$, and $0.46^{+3.0}_{-1.0}$, respectively.
• Upper Limits: Upper limits on the branching fractions $\mathcal{B}(\chi_{cJ} \to e^+e^-\phi)$ were set at 90% C.L., excluding contributions from the $\phi$ resonance decaying directly to $e^+e^-$ (i.e., focusing on the Dalitz transition via a virtual photon).
  • $\chi_{c0} \to e^+e^-\phi$: $2.4 \times 10^{-7}$
  • $\chi_{c1} \to e^+e^-\phi$: $6.7 \times 10^{-7}$
  • $\chi_{c2} \to e^+e^-\phi$: $4.1 \times 10^{-7}$

5. Significance

• Benchmark for Theory: These results provide the first experimental constraints on the TFF for P-wave charmonium Dalitz decays. This data is crucial for testing theoretical models (such as VMD and quark triangle loop mechanisms) in the non-perturbative QCD regime.
• New Physics Sensitivity: By establishing these upper limits, the paper sets a baseline for future searches. Significant deviations from these limits in future high-luminosity experiments could indicate the existence of new particles, such as dark photons.
• Future Prospects: The authors note that precise measurements of these branching fractions and TFFs will be feasible at future high-luminosity facilities, such as a super tau-charm factory, where the statistical precision will be significantly improved.