Amplitude analysis of $ψ(3686)\to γK_S^0 K_S^0$

Using a large sample of $\psi(3686)$ events collected by the BESIII detector, this paper presents the first amplitude analysis of the radiative decay $\psi(3686)\to\gamma K_S^0 K_S^0$, identifying well-established $f_0$ and $f_2$ resonances and determining their production ratios relative to $J/\psi$ decays to provide crucial insights into their internal structure and potential glueball mixing.

The Gist

Imagine the universe as a giant, bustling kitchen where the most fundamental ingredients of matter (quarks) are constantly being cooked up into complex dishes called particles. Sometimes, these particles are made of just two quarks (like a simple sandwich), but sometimes, they are made entirely of the "glue" that holds them together. These glue-only particles are called glueballs.

For decades, physicists have been trying to find these glueballs in their kitchen. The problem? They look exactly like the regular "sandwich" particles. It's like trying to find a specific ghost in a crowd of people wearing identical masks.

This paper from the BESIII Collaboration is a major step forward in solving this mystery. Here is what they did, explained simply:

1. The Setup: A Massive Particle Factory

The scientists used a giant machine called the BESIII detector (located in Beijing, China) to smash electrons and positrons together. Think of this like a high-speed car crash, but instead of metal, they smash subatomic particles.

They didn't just crash a few cars; they created 2.7 billion specific events involving a particle called the $\psi(3686)$. This is like having a massive warehouse full of identical boxes, hoping that one of them contains the rare item you are looking for.

2. The Hunt: The "Ghost" in the Machine

When the $\psi(3686)$ particle crashes, it sometimes decays into a photon (a particle of light) and a pair of neutral Kaons ($K^0_S K^0_S$).

• The Analogy: Imagine the $\psi(3686)$ is a magician. When it disappears, it leaves behind a flash of light (the photon) and two identical, invisible ghosts (the Kaons).
• The scientists can't see the ghosts directly, but they can see where they go and how fast they are moving. By measuring the "weight" (mass) of these two ghosts combined, they can figure out what kind of "intermediate" particle existed for a split second before turning into the ghosts.

3. The Method: Tuning a Radio

The scientists needed to figure out exactly what "songs" (resonances) were playing in the background of this data.

• The Old Way: Before this, scientists just looked at the data and tried to fit simple bell curves (like guessing the shape of a hill). This is like trying to identify a symphony by listening to a single, muffled note.
• The New Way (Amplitude Analysis): This paper uses a sophisticated mathematical tool called a K-matrix.
  • The Analogy: Imagine the data is a messy recording of a crowded party. The K-matrix is like a high-tech audio filter that separates the voices. It doesn't just look for one voice; it identifies specific singers (resonances) even if they are singing over each other, overlapping, or whispering.

4. The Discovery: Finding the Singers

By using this advanced filter, the team identified seven distinct "singers" (resonance states) in the data:

• Four "singers" with a spin of 0 (called $f_0$ states).
• Three "singers" with a spin of 2 (called $f_2$ states).

They found that these singers matched known particles perfectly, like $f_0(1710)$ and $f_2(1525)$. But the real goal was to see if any of them were the elusive glueballs.

5. The Big Question: Are They Glueballs?

To figure out if a particle is a "glueball" or a regular "quark sandwich," the scientists compared how often these particles were made in two different types of crashes:

1. The $J/\psi$ crash: A lower-energy crash.
2. The $\psi(3686)$ crash: The high-energy crash from this paper.

• The Analogy: Imagine you have two different ovens. Oven A (low energy) and Oven B (high energy). You bake cookies in both.
  • If a cookie is made of flour (quarks), it might bake differently in Oven A than in Oven B.
  • If a cookie is made of pure fire (glue), it might bake the same way in both ovens, or follow a specific, predictable pattern.

The team calculated the "baking ratios" (how often the particle appeared in one oven vs. the other). They found that the behavior of these particles in the high-energy crash matched the behavior seen in the low-energy crash.

6. The Conclusion: A Clue, Not a Smoking Gun

The paper concludes that:

• They have successfully mapped out the "family tree" of these particles with much higher precision than ever before.
• The behavior of these particles is consistent with what we expect from a mix of regular quark particles and potential glueballs.
• While they haven't definitively caught a "pure glueball" yet, they have provided the crucial experimental clues needed to prove which of these particles is actually made of glue.

In short: This paper is like a high-resolution map of a foggy forest. The scientists didn't find the "monster" (the pure glueball) yet, but they have drawn the map so accurately that we now know exactly where to look next to catch it. They have cleared the fog enough to see the path forward.

Technical Summary

1. Problem and Motivation

• Scientific Context: Quantum Chromodynamics (QCD) predicts the existence of glueballs (bound states of gluons). The lightest glueball is theoretically expected to be a scalar state ($J^{PC} = 0^{++}$) with a mass between 1.45 and 1.75 GeV/$c^2$. However, glueballs are difficult to identify experimentally because they mix with conventional $q\bar{q}$ mesons (specifically the $f_0$ family) that share the same quantum numbers.
• The Gap: While the BESIII collaboration has extensively studied radiative decays of the $J/\psi$ meson (e.g., $J/\psi \to \gamma K^0_S K^0_S$) to probe these states, there has been a lack of comprehensive amplitude analyses for the $\psi(3686)$ (the first radial excitation of the $J/\psi$). Previous studies on $\psi(3686) \to \gamma K^0_S K^0_S$ were limited to simple Breit-Wigner fits on small datasets (CLEO-c).
• Objective: To perform the first full amplitude analysis of $\psi(3686) \to \gamma K^0_S K^0_S$ using a large dataset to extract pole positions and residues of $f_0$ and $f_2$ resonances. This allows for a comparison between $\psi(3686)$ and $J/\psi$ production mechanisms, providing crucial inputs to understand the internal structure (quark vs. glueball content) of these scalar and tensor mesons.

2. Methodology

• Data Sample: The analysis utilizes $(2712 \pm 14) \times 10^6$ $\psi(3686)$ events collected by the BESIII detector at the BEPCII collider.
• Event Selection:
  • Reconstructed via the decay chain $\psi(3686) \to \gamma K^0_S K^0_S$, where $K^0_S \to \pi^+ \pi^-$.
  • Strict kinematic constraints were applied: 4-constraint (4C) kinematic fit, vertex fitting for $K^0_S$, and invariant mass windows.
  • The analysis is restricted to the $K^0_S K^0_S$ invariant mass region $M_{K^0_S K^0_S} < 2.8$ GeV/$c^2$ to suppress backgrounds from $\chi_{c0,2} \to K^0_S K^0_S$.
  • Backgrounds from continuum processes and other $\psi(3686)$ decays were estimated using Monte Carlo (MC) simulations and found to be negligible (<1%).
• Amplitude Analysis Framework:
  • Model: A one-channel K-matrix approach was employed to describe the dynamics of the $K^0_S K^0_S$ system. This is preferred over a simple sum of Breit-Wigner functions to better handle overlapping resonances and unitarity constraints (though coupled-channel effects are neglected in this single-channel approximation).
  • Parameterization: The production amplitude $\psi(3686) \to \gamma f_{0,2}$ is parameterized using the P-vector method. The dynamic part includes the K-matrix ($K$) and the production vector ($P$).
  • Fit Strategy: An event-based maximum likelihood fit was performed. The model included various combinations of $f_0$ (scalar) and $f_2$ (tensor) poles. The significance of adding new poles was determined using the change in log-likelihood ($\Delta \ln \mathcal{L}$) and Wilks' theorem.
  • Residue Calculation: To quantify the production strength, the residues of the poles were calculated in the complex $s$-plane. These were converted from covariant tensor amplitudes to helicity amplitudes (E1, M2, E3 multipoles) to allow for physical interpretation.

3. Key Contributions

• First Amplitude Analysis: This is the first rigorous amplitude analysis of the $\psi(3686) \to \gamma K^0_S K^0_S$ decay channel.
• Model Development: Successfully applied a one-channel K-matrix approach with P-vector parameterization to the $\psi(3686)$ radiative decay, demonstrating its effectiveness in describing the $K^0_S K^0_S$ spectrum.
• Comparative Study: Established a direct comparison between the production of $f_0$ and $f_2$ states in $\psi(3686)$ radiative decays versus $J/\psi$ radiative decays. This comparison is vital for testing theoretical models regarding the glueball content of these mesons, as the ratio of production rates ($\psi(3686)/J/\psi$) is sensitive to the internal structure (e.g., mixing with glueballs).

4. Results

• Pole Positions: The data is well-described by a model containing four $f_0$ poles and three $f_2$ poles. The determined pole positions are consistent with well-established resonance states in the Particle Data Group (PDG) listings:
  • $f_0$ states: $f_0(1370)$, $f_0(1500)$, $f_0(1710)$, and $f_0(2020)$.
  • $f_2$ states: $f_2(1270)$, $f'_2(1525)$, and $f_2(1950)$.
  • Note: The $f_0(1370)$ pole position showed some instability in the fit, and $f_0(2330)$ was not observed.
• Branching Fractions: The total branching fraction for $\psi(3686) \to \gamma K^0_S K^0_S$ (with $M < 2.8$ GeV/$c^2$) was measured as:
  $$\mathcal{B} = (5.91 \pm 0.05 \text{ (stat)} \pm 0.02 \text{ (syst)}) \times 10^{-5}$$
  Individual branching fractions for the resonances were also calculated based on their residues.
• Production Ratios: The ratios of branching fractions between $\psi(3686)$ and $J/\psi$ decays were determined:
  $$R = \frac{\mathcal{B}(\psi(3686) \to \gamma f_{0,2})}{\mathcal{B}(J/\psi \to \gamma f_{0,2})}$$
  These ratios were compared to the theoretical expectation for pure glueball production, $\frac{\mathcal{B}(\psi(3686) \to \gamma gg)}{\mathcal{B}(J/\psi \to \gamma gg)} \approx 11.7\%$.
  • The observed ratios for $f_0(1500)$, $f_0(1710)$, and $f_2(1270)$ are close to this value (around 9-11%), suggesting a significant glueball component or similar production mechanisms.
  • The ratios for $f_0(2020)$ and $f_2(1950)$ are significantly lower (~1.7% and ~2.3%), suggesting different internal structures or suppression mechanisms.
• Multipole Analysis: For the $f_2$ states, the residues were decomposed into E1, M2, and E3 multipoles. The results generally follow the hierarchy $E1 > M2 > E3$ predicted by the conventional quark model, though statistical precision limits strong conclusions on deviations.

5. Significance

• Glueball Identification: By comparing the production rates of $f_0$ and $f_2$ states in $\psi(3686)$ and $J/\psi$ decays, the study provides critical experimental constraints on the glueball mixing hypothesis. The fact that some states (like $f_0(1710)$) follow the expected scaling for glueball production while others do not helps disentangle the quark-gluon composition of these resonances.
• Spectroscopy: The precise determination of pole positions and residues for the $f_0$ and $f_2$ spectrum in the $\psi(3686)$ channel fills a gap in the charmonium spectroscopy database, complementing previous $J/\psi$ studies.
• Methodological Benchmark: The successful application of the K-matrix approach to $\psi(3686)$ radiative decays sets a precedent for future amplitude analyses in similar channels, improving the reliability of resonance parameter extraction in the presence of overlapping states.

In conclusion, this paper establishes a robust baseline for understanding the scalar and tensor meson spectrum in $\psi(3686)$ radiative decays, offering new experimental evidence to refine theoretical models of QCD bound states and glueball candidates.