Study of the $e^{+}e^{-}\to J/\psi\,\pi^{+}\pi^{-}$ lineshape near the $D^{*}\bar{D}+c.c.$ threshold and possible signals for exotic hidden charm states

This paper investigates the $e^{+}e^{-}\to J/\psi\,\pi^{+}\pi^{-}$ lineshape near the $D^{*}\bar{D}$ threshold by incorporating intermediate meson loops and triangle singularities to distinguish between kinematic effects and genuine exotic hidden charm resonances, offering theoretical predictions for identifying states with quantum numbers $(I,J^{P(C)})=(1,1^{-(-)})$ in the $J/\psi\pi$ invariant mass spectrum.

The Gist

Imagine you are a detective trying to solve a mystery in the subatomic world. Your case file involves a specific event: an electron and a positron (matter and antimatter) smashing together to create a "charmed" particle called a J/ψ and two pions (lightweight particles).

The mystery? There is a strange "bump" or enhancement in the data right around a specific energy level (3.9 GeV). Physicists call this the G(3900). The big question is: Is this bump a brand-new, exotic particle hiding in the data, or is it just a trick of the light caused by the way particles bounce around?

Here is a simple breakdown of what the authors, Jun Wang and Qiang Zhao, did to solve this case.

1. The Setting: A Busy Highway

Think of the energy level of 3.9 GeV as a busy highway intersection.

• The Traffic: There are heavy trucks (particles called $D$ and $\bar{D}$) driving on this road.
• The Threshold: Just at 3.9 GeV, there is a specific speed limit where these trucks can just barely start driving. This is called the "threshold."
• The Mystery Bump: When scientists look at the traffic flow (the cross-section), they see a weird spike right at this speed limit. Is it a new type of vehicle (a new particle) parked there? Or is it just a traffic jam caused by the road geometry?

2. The Two Suspects

The paper investigates two main suspects that could be causing this spike:

Suspect A: The "Genuine Resonance" (The New Particle)

This is the idea that there is actually a new, exotic particle sitting at this energy.

• The Analogy: Imagine a new, rare car model (a "tetraquark" or a "hadronic molecule") that is made of four quarks stuck together. It's a unique vehicle that doesn't fit the standard "two-quark" or "three-quark" rulebook. If this car exists, it would naturally cause a spike in traffic because it's a distinct object.

Suspect B: The "Triangle Singularity" (The Kinematic Trick)

This is a purely mathematical effect caused by how particles interact in a loop.

• The Analogy: Imagine three cars driving in a triangle. If they all hit a specific speed and arrive at the same intersection at the exact same time, they create a massive, temporary traffic jam. This isn't because a new car appeared; it's just because the timing and geometry of the road forced them all to pile up at once. In physics, this is called a Triangle Singularity (TS). It looks like a particle, but it's actually just a "ghost" created by the math of the collision.

3. The Investigation: How They Distinguished the Suspects

The authors built a complex computer simulation (a "theoretical framework") to replay the crash millions of times. They tested two scenarios:

• Scenario 1 (The Direct Hit): They assumed the particles just bumped into each other directly (like a contact sport).
• Scenario 2 (The Relay Race): They assumed the particles bounced off an intermediate "resonance" (like a relay runner passing a baton) before creating the final J/ψ and pions.

The Twist:
When they looked at the total amount of traffic (the total cross-section), both scenarios looked almost identical! The simulation couldn't tell the difference between a new particle and a kinematic trick just by counting the total number of crashes. It was like looking at a blurry photo where both suspects look the same.

The Breakthrough:
The authors realized they needed to look at the speed of the individual cars (the invariant mass spectrum of the J/ψ and the pion).

• The Result: When they analyzed the speed of the J/ψ and the pion separately, the two suspects looked very different.
  • If the "Triangle Singularity" (the kinematic trick) is the cause, the speed distribution looks like a specific, sharp "cusp" (a sudden sharp turn).
  • If a "Genuine Resonance" (the new particle) is the cause, the speed distribution looks like a smooth, distinct hill or peak.

4. The Conclusion: What Does It Mean?

The paper concludes that:

1. We can't tell just by looking at the total crash count. The "bump" at 3.9 GeV is ambiguous.
2. We need to look closer. By measuring the specific energy of the J/ψ and pion pair, future experiments can finally tell if the G(3900) is a real, exotic new particle or just a "traffic jam" caused by the laws of physics.
3. The "Triangle Singularity" is powerful. Even if there is no new particle, the math of the triangle loop can create structures that look like particles. This is a crucial warning for experimentalists: don't assume every bump is a new discovery; check the geometry first!

Summary in a Nutshell

The authors are saying: "We found a weird bump in the data. It could be a new exotic particle, or it could just be a mathematical illusion caused by particles looping around each other. If you just count the total crashes, you can't tell the difference. But if you measure the speed of the pieces flying apart, you can solve the mystery. This gives experimentalists a clear roadmap for how to hunt for these exotic particles in the future."

It's a guide on how to stop confusing a mirage (kinematic effect) with a mountain (real particle).

Technical Summary

1. Problem Statement

The paper addresses the nature of the "G(3900)" structure observed in the $e^+e^- \to D\bar{D}$ channel near the $D^*\bar{D} + c.c.$ threshold. While this structure was initially interpreted as a potential P-wave hadronic molecule or a resonance, its nature remains ambiguous.

• The Conflict: Experimental data for $e^+e^- \to J/\psi \pi^+ \pi^-$ from BESIII shows a broad resonant structure near 3.9 GeV, but specific data points (e.g., at $\sqrt{s} = 3.8713$ GeV) are difficult to accommodate with standard resonance fits.
• The Question: Is the observed enhancement near 3.9 GeV a genuine exotic resonance (such as a tetraquark or hadronic molecule with quantum numbers $I=1, J^P=1^-$), or is it a kinematic effect arising from open-charm thresholds and rescattering?
• The Gap: Previous studies focused heavily on two-body scatterings. The effects of three-body kinematics, specifically Triangle Singularities (TS), in the three-body final state $J/\psi \pi^+ \pi^-$ have not been fully disentangled from genuine resonance contributions in this energy region.

2. Methodology

The authors employ a unified theoretical framework combining Vector Meson Dominance (VMD) and effective Lagrangians to model the process $e^+e^- \to J/\psi \pi^+ \pi^-$.

• Theoretical Framework:
  • Effective Lagrangians: The study utilizes heavy quark symmetry and chiral Lagrangians to describe couplings between charmed mesons ($D, D^*$), light pseudoscalars ($\pi$), and vector mesons ($J/\psi, \rho, \omega, \phi$).
  • Amplitude Decomposition: The total amplitude is calculated as a sum of:
    1. Tree-level contributions: Direct coupling of intermediate charmonium states ($J/\psi, \psi(3686), \psi(3770), \dots$) to the final state.
    2. Loop contributions: Intermediate meson loops involving $D^*\bar{D} + c.c.$ rescattering.
• Scenarios Investigated:
  • Scenario I (Contact Interaction): The transition $D\bar{D} \to J/\psi \pi$ occurs via a direct contact vertex (no intermediate resonance in the loop).
  • Scenario II (Intermediate Resonance): The transition $D\bar{D} \to J/\psi \pi$ proceeds via an intermediate vector resonance (simulating a potential G(3900) or P-wave molecule).
• Fitting Strategy:
  • The model is fitted to experimental cross-section data from BESIII ($\sqrt{s} = 3.773 - 4.698$ GeV).
  • Two fitting schemes are used:
    • Fit-1: Includes only known charmonium states (Table I).
    • Fit-2: Includes an additional intermediate state at 3905 MeV (width 346 MeV) to mimic the G(3900) structure.
  • Triangle Singularity (TS): The loop integrals are evaluated using Passarino-Veltman functions (via LoopTools) with a form factor to regulate UV divergence. The TS condition is checked to see if internal particles can go on-shell simultaneously, potentially creating a peak in the invariant mass spectrum.

3. Key Contributions

• Disentanglement of Kinematics vs. Resonance: The paper provides a rigorous method to distinguish between kinematic enhancements (TS) and genuine resonant states by analyzing the $J/\psi \pi$ invariant mass spectrum, rather than just the total cross-section.
• Identification of Exotic Quantum Numbers: The study demonstrates that the triangle mechanism allows access to exotic quantum numbers $(I, J^P(C)) = (1, 1^-( -))$ in the $J/\psi \pi$ spectrum, which are forbidden for conventional charmonium.
• Sensitivity Analysis: It shows that while the total cross-section lineshape is insensitive to the specific mechanism (contact vs. resonance) due to dominant tree-level contributions, the differential $J/\psi \pi$ invariant mass spectrum is highly sensitive to the underlying dynamics.

4. Results

• Total Cross-Section Fitting:
  • Both Scenario I (contact) and Scenario II (resonance) can reproduce the total cross-section data reasonably well when an additional intermediate state is included (Fit-2).
  • The inclusion of a state near 3.9 GeV significantly improves the fit to the data points around 3.9 GeV, but the total cross-section alone cannot confirm if this state is a pole or a kinematic effect.
• Invariant Mass Spectrum ($M_{J/\psi \pi}$):
  • Scenario I (Contact): The spectrum shows a "Cusp" structure at the $D\bar{D}$ threshold due to the TS mechanism. However, because the loop contribution is small compared to the tree amplitude, this effect is a small, narrow peak.
  • Scenario II (Resonance): If a genuine resonance exists in the $D\bar{D} \to J/\psi \pi$ transition, the TS mechanism strongly enhances the signal. This results in a pronounced peak near the $D\bar{D}$ threshold in the $J/\psi \pi$ invariant mass spectrum, significantly deviating from phase space expectations.
• Differentiation: The shape and magnitude of the peak in the $J/\psi \pi$ spectrum serve as a clear discriminator:
  • A small cusp suggests a kinematic TS effect without a genuine resonance.
  • A large, pronounced peak suggests the presence of a genuine exotic resonance (tetraquark or molecule) coupled to the $D\bar{D}$ channel.

5. Significance

• Theoretical Guidance: The paper offers a concrete theoretical prescription for future experiments (like BESIII or future $e^+e^-$ colliders) to search for exotic hidden-charm states. Instead of relying solely on total cross-section bumps, experiments should analyze the $J/\psi \pi$ invariant mass distribution.
• Nature of G(3900): It clarifies that if the G(3900) is a genuine P-wave hadronic molecule or tetraquark, it should manifest as a distinct, enhanced peak in the $J/\psi \pi$ spectrum driven by the triangle singularity. If it is purely a kinematic threshold effect, the signal will be much weaker.
• Exotic State Search: The study validates the $J/\psi \pi$ channel as an ideal "laboratory" for identifying states with exotic quantum numbers $(1, 1^-)$, providing a pathway to confirm the existence of charged charmonium-like states beyond the standard quark model.

In conclusion, Wang and Zhao demonstrate that while total cross-section data is ambiguous, the $J/\psi \pi$ invariant mass spectrum acts as a powerful filter to separate kinematic triangle singularities from genuine exotic resonances, offering a clear strategy for resolving the nature of the G(3900) and similar structures.