Simultaneous Dalitz-plot decomposition of the $e^+ e^- \to J/\psi \, \pi \, \pi \, (K \bar{K})$ processes in the 4.13-4.36 GeV region using dispersive final-state interactions

This paper presents a joint dispersive analysis of $e^+e^- \to J/\psi\pi^+\pi^-$ and $J/\psi K^+K^-$ processes in the 4.13–4.36 GeV region, demonstrating that a single set of energy-independent parameters successfully describes the data only when including both resonant structures ($Y(4220)$, $Y(4320)$, $Z_c(3900)$) and a non-resonant production mechanism subject to coupled-channel final-state interactions.

The Gist

Imagine you are a detective trying to solve a mystery at a high-energy particle collider. The "crime scene" is a specific range of energy (between 4.13 and 4.36 GeV) where electrons and positrons smash together. When they collide, they don't just vanish; they transform into a heavy particle called a J/ψ and two lighter particles that can be either pions (like tiny, light marbles) or kaons (slightly heavier marbles).

The mystery is: How exactly do these particles form?

For a long time, scientists thought the answer was simple: The collision creates a "resonance" (a temporary, unstable particle like a Y(4220) or Y(4320)), which then instantly decays into the final pieces. It's like a magician pulling a rabbit out of a hat—the rabbit appears because the magician (the resonance) was there.

However, this new paper by Ermolina, Danilkin, and Vanderhaeghen suggests the story is more complicated. They used a sophisticated mathematical tool called a Dalitz-plot decomposition (think of it as a 3D map that tracks every possible way the particles can fly apart) and a technique called dispersive final-state interactions (a way to account for how the particles bump into and influence each other after they are created).

Here is what they found, explained simply:

1. The "Ghost" in the Machine (Non-Resonant Production)

The authors discovered that the "magic trick" isn't just about the magician (the resonances). There is also a "ghost" in the machine.

• The Analogy: Imagine a band playing a concert. You can hear the specific songs played by the lead singer (the resonances, like Y(4220)). But if you only listen to the lead singer, you miss the background hum and the way the instruments blend together.
• The Finding: The data shows that the particles are also being produced directly, without going through a specific intermediate resonance first. This is called a non-resonant term. It's like a background hum that exists alongside the main songs. If you ignore this background, your description of the concert is wrong.

2. The "Bump and Grind" (Final-State Interactions)

Once the particles are created, they don't just fly away in a straight line. They interact with each other.

• The Analogy: Imagine two dancers (the pions or kaons) who are thrown onto a dance floor. They don't just spin off; they bump into each other, spin around, and change their path based on how they interact.
• The Finding: The paper uses a method called the Omnès representation to mathematically describe this "bumping and grinding." They found that this interaction is crucial. Without accounting for how the particles "rescatter" (bounce off each other) after being created, the math cannot match the experimental data.

3. The "Two-Act Play" (Y(4220) and Y(4320))

The researchers analyzed the data across the whole energy range and found that the story has two main acts, corresponding to two different "resonant structures" (Y(4220) and Y(4320)).

• The Finding: In the lower energy part of the range, the Y(4220) is the star. But as you go higher in energy, the Y(4320) joins the stage. The paper successfully describes the entire performance by combining these two "actors" with the "background hum" (non-resonant production) and the "dance floor interactions" (final-state interactions).

4. What They Measured

By fitting all these pieces together, the team was able to:

• Measure the "ID cards" of the particles: They calculated the precise mass and width (how long they live) for the Zc(3900), Y(4220), and Y(4320). Their numbers match well with previous measurements from the BESIII experiment.
• Map the "Sub-stories": They figured out how much of the total collision energy goes into specific sub-processes, like creating a Zc particle that then turns into a J/ψ and a pion.

The Bottom Line

The main takeaway is that nature is messy. You cannot explain these particle collisions by just pointing to a few "resonant" particles. You must also account for the background production (particles made directly) and the complex interactions between the particles after they are made.

The authors built a single, unified mathematical model that acts like a master key, unlocking the description of both the total energy output and the specific ways the particles fly apart, using just one set of rules that don't change with energy. They proved that a "purely resonant" story (just the magicians) is insufficient; you need the whole cast, including the background actors and the stage dynamics, to tell the truth.

Technical Summary

Technical Summary: Simultaneous Dalitz-plot Decomposition of $e^+e^- \to J/\psi \pi \pi (K \bar{K})$

Problem Statement
The paper addresses the challenge of simultaneously describing the energy-dependent total cross sections and the one-dimensional invariant-mass distributions for the processes $e^+e^- \to J/\psi \pi^+ \pi^-$ and $e^+e^- \to J/\psi K^+ K^-$ in the center-of-mass energy range of 4.13 to 4.36 GeV. Previous analyses have often treated individual invariant-mass distributions or total cross sections separately. A significant difficulty lies in connecting the $Z_c$ structures observed in the $J/\psi \pi$ invariant-mass spectra with the $Y$ vector structures observed in the total cross sections. Furthermore, experimental data suggests that these final states are not produced exclusively through intermediate $Y$ resonances; a non-resonant production mechanism plays a crucial role, particularly in the one-dimensional distributions. Existing purely resonant descriptions have proven insufficient to capture the full dynamics of the BESIII data.

Methodology
The authors construct a joint amplitude framework based on the Dalitz-plot decomposition (DPD) formalism. The key methodological features include:

• Energy Dependence: The $e^+e^-$ energy dependence ($q^2$) is explicitly encoded through the propagators of the $Y(4220)$ and $Y(4320)$ vector resonances, alongside a smooth non-resonant production term. This allows the underlying couplings to be treated as globally constant parameters.
• Final-State Interactions (FSI): The scalar $\pi\pi/K\bar{K}$ final-state interaction is treated dispersively using a coupled-channel Omnès representation. This matrix is constrained by data-driven $N/D$ analyses and Roy/Roy-Steiner analyses of $\pi\pi \to \pi\pi$ and $\pi\pi \to K\bar{K}$ scattering.
• Amplitude Construction:
  • The amplitudes incorporate contributions from $Y(4220)$ and $Y(4320)$ resonances.
  • The $J/\psi \pi^\pm$ subsystems include the $Z_c^\pm(3900)$ contribution.
  • The $\pi^+\pi^-$ subsystem includes scalar ($f_0(500)$, $f_0(980)$) and tensor ($f_2(1270)$) contributions.
  • A non-resonant background term is introduced, which undergoes $\pi\pi/K\bar{K}$ rescattering via the Omnès matrix.
• Fit Strategy: The authors perform two fits:
  1. A minimal fit covering the 4.13–4.23 GeV region, including only $Y(4220)$, $Z_c(3900)$, and the scalar FSI.
  2. A total fit covering the full 4.13–4.36 GeV region, adding the $Y(4320)$ resonance and the $f_2(1270)$ tensor contribution.
     The scalar-isoscalar FSI is fixed by the Omnès input, while resonance masses, widths, and coupling parameters are extracted from the data.

Key Contributions and Results

• Necessity of Non-Resonant Terms: The analysis demonstrates that a purely resonant description is insufficient. A non-resonant scalar production term, followed by $\pi\pi/K\bar{K}$ rescattering, is essential to describe both the total cross sections and the invariant-mass distributions consistently. This term is particularly significant at lower energies within the studied range.
• Resonance Parameters: Within the isobar model, the authors extract effective Breit-Wigner parameters:
  • $Z_c(3900)$: Mass $3888.7 \pm 0.9$ MeV, Width $37.0 \pm 1.5$ MeV. These are consistent with BESIII partial-wave analysis (PWA) results.
  • $Y(4220)$: Mass $4223.6 \pm 0.4$ MeV, Width $37.7 \pm 0.8$ MeV. These are compatible with BESIII determinations from both subprocess and total cross-section analyses.
  • $Y(4320)$: Mass $4283 \pm 2$ MeV, Width $102 \pm 4$ MeV. The fitted mass is closer to the BESIII total cross-section value for the $J/\psi \pi^+ \pi^-$ channel than to the PDG average for $\psi(4360)$, leading the authors to treat it as a phenomenological structure specific to this channel in the current fit.
• Subprocess Cross Sections: The framework allows for the extraction of subprocess cross sections, specifically for $e^+e^- \to \pi^\pm Z_c^\mp(3900) \to J/\psi \pi^+ \pi^-$ and the scalar $J/\psi(\pi^+\pi^-)_S$-wave channel.
• Predictive Capability: The model provides predictions for invariant-mass distributions at energies where measurements are currently unavailable, although the authors note that extrapolation beyond the fitted range requires more detailed modeling of the non-resonant amplitude's $q^2$ dependence.

Significance
The paper claims that its primary significance lies in providing a simultaneous description of total cross sections and invariant-mass distributions using a single set of energy-independent parameters. By explicitly incorporating the energy dependence through $Y$ resonances and a non-resonant mechanism, and by treating the scalar FSI dispersively, the authors resolve the challenge of connecting the $Z_c$ and $Y$ sectors. The work highlights that the dynamics of these exotic candidates cannot be fully understood without accounting for the interplay between resonant production and non-resonant rescattering mechanisms. The authors modestly note that while open-charm loop mechanisms (including triangle singularities) are not explicitly included, their effects are expected to be subdominant and are effectively absorbed into the effective production amplitudes used in the fit.