Decoding the structure near the $\pi^+\pi^-$ mass threshold in $\psi(3686) \rightarrow J/\psi \pi^+\pi^-$ decays

Using dispersion theory to account for strong pion-pion final-state interactions, this study demonstrates that the substructure near the $\pi^+\pi^-$ threshold in $\psi(3686) \rightarrow J/\psi \pi^+\pi^-$ decays can be reproduced without invoking an extra resonance, attributing the observed dip primarily to a helicity-flip amplitude rather than the virtual exchange of the $Z_c(3900)$ state.

The Gist

Imagine the subatomic world as a massive, chaotic dance floor. In this paper, physicists are trying to understand a very specific dance move: a heavy particle called the $\psi(3686)$ (let's call him "Big P") suddenly transforms into a slightly lighter particle called the $J/\psi$ ("Little P"), while spitting out two pions (tiny particles made of quarks, let's call them "Dancers A and B").

For a long time, scientists thought they knew the choreography. But recently, the BESIII experiment (a giant camera in China) took a super-high-definition video of this dance and noticed something weird. Right at the very beginning of the dance, when the two pions are just starting to move, there was a strange "dip" or a bump in the crowd's movement. It looked like a new, mysterious partner had joined the dance floor for a split second.

The big question was: Is this a new, exotic particle (a "ghost" dancer) appearing out of nowhere, or is it just a trick of the light caused by the existing dancers bumping into each other?

Here is how the authors of this paper solved the mystery, using simple analogies:

1. The Old Way vs. The New Way

• The Old Way (The "Guessing Game"): Previous studies tried to explain this bump by inventing a brand-new particle, like saying, "Oh, that bump must be a new ghost dancer named $Z_c$." They used a standard formula (like a Breit-Wigner function) that assumes a new object is there.
• The New Way (The "Physics of Echoes"): The authors in this paper decided to look closer at the physics of the existing dancers. They used a sophisticated mathematical tool called Dispersion Theory.
  • Analogy: Imagine two people clapping in a canyon. The sound you hear isn't just the clap; it's the clap plus the echo bouncing off the walls. The "bump" in the data might just be a complex echo of the two pions interacting with each other, not a new person entering the canyon.

2. The "Ghost" vs. The "Virtual"

The paper investigates a specific candidate for this "ghost" dancer: the $Z_c(3900)$. This is a strange, exotic particle that looks like a charmed quark and an anti-charmed quark holding hands with a pion.

• The Finding: The authors found that they do not need to invent a new, heavy resonance (a real, stable ghost) to explain the data.
• The Twist: However, they did find that a "virtual" exchange of this $Z_c$ helps the math fit the video slightly better.
  • Analogy: Think of it like a game of catch. You don't need a third person to throw the ball between you and your friend. But, if you pretend there was a third person briefly catching and throwing the ball back, the math describing the ball's path becomes a tiny bit more accurate. The $Z_c$ isn't a real, standing particle here; it's a "virtual" shadow that briefly influences the dance.

3. The "Helicity Flip" (The Spin-Off)

One of the most important discoveries in the paper is about something called the helicity-flip amplitude.

• Analogy: Imagine the dancers spinning. Usually, they spin in a predictable way. But in this specific dance, one of the dancers does a sudden, unexpected spin flip.
• Why it matters: The authors found that this "spin flip" is actually the main reason for the weird "dip" or "valley" in the data near the threshold. It's not a new particle causing the dip; it's the way the existing particles are spinning and interacting that creates the shape.

4. The Verdict

The authors ran three different simulations (Fits I, II, and III):

1. Fit I: Only the basic rules of the dance (chiral contact terms) and the echoes (final-state interactions). Result: It worked surprisingly well!
2. Fit II & III: Added the "virtual $Z_c$ ghost" to the mix. Result: It improved the fit slightly, but not enough to prove the ghost is real.

The Conclusion:
The strange structure near the threshold is not a new, heavy particle waiting to be discovered. Instead, it is a beautiful, complex interference pattern created by:

1. The basic rules of how pions interact (the "chiral contact").
2. The way the pions bounce off each other (the "final-state interactions").
3. A specific type of spin-flip in the dance.

The Takeaway for Everyone

This paper is like a detective story where the police thought a crime was committed by a new, unknown suspect. But after analyzing the fingerprints and the footprints with high-tech tools, they realized: "No, it was just the two victims bumping into each other in a very specific, complicated way."

While a "ghost" (the $Z_c$) might have briefly passed through the scene to make the story slightly more interesting, the mystery is solved without needing to add a new character to the universe. The laws of physics, as they currently stand, are enough to explain the dance.

Technical Summary

Here is a detailed technical summary of the paper "Decoding the structure near the $\pi^+\pi^-$ mass threshold in $\psi(3686) \to J/\psi\pi^+\pi^-$ decays".

1. Problem Statement

Recent high-precision data from the BESIII Collaboration on the decay $\psi(3686) \to J/\psi\pi^+\pi^-$ revealed a clear, resonance-like structure near the $\pi^+\pi^-$ mass threshold (around 282 MeV).

• The Conflict: The BESIII analysis described this structure using a Breit-Wigner parametrization with a mass of $\sim 282$ MeV. However, such a low-mass resonance is theoretically problematic. It is significantly lower than the known $f_0(500)$ (sigma) meson and contradicts the chiral structure of QCD, which suppresses two-pion interactions at very low energies.
• The Question: Is this structure a new, exotic resonance, or can it be explained by existing QCD dynamics, specifically final-state interactions (FSI) and virtual exchanges of known exotic states like the $Z_c(3900)$?

2. Methodology

The authors employ a model-independent dispersion theory approach to analyze the decay, avoiding the model-dependent assumptions of previous chiral unitary approaches.

• Theoretical Framework:

  • Effective Lagrangian: They utilize the QCD multipole expansion mapped onto a chiral effective Lagrangian. This includes contact terms for the $\psi' \to J/\psi \pi \pi$ interaction and interaction terms for the virtual exchange of the exotic $Z_c(3900)$ state.
  • Final-State Interactions (FSI): The strong $\pi\pi$ interactions are treated rigorously using dispersion relations (Omnès functions) constrained by experimental $\pi\pi$ scattering phase shifts. This ensures unitarity and consistency with low-energy QCD constraints.
  • Coulomb Enhancement: The Sommerfeld factor is included to account for the Coulomb enhancement of the $\pi^+\pi^-$ pair near the threshold.
  • Amplitude Decomposition: The decay amplitude is decomposed into partial waves (S-wave and D-wave). The analysis considers both helicity-conserving and helicity-flip amplitudes.

• Fitting Strategy:
  The authors perform a simultaneous fit to two datasets provided by BESIII:

  1. The $\pi\pi$ invariant-mass distribution.
  2. The helicity angular distribution ($\cos \theta$).

  They compare three scenarios:

  • Fit I: Only chiral contact terms (no $Z_c$ exchange).
  • Fit II: Contact terms + $Z_c$ exchange with fixed mass/width ($M_{Z_c} = 3.880$ GeV, $\Gamma_{Z_c} = 0.036$ GeV).
  • Fit III: Contact terms + $Z_c$ exchange with mass/width as free parameters within previous constraints.

3. Key Contributions

• Model-Independent FSI Treatment: Unlike previous studies that used the chiral unitary approach (which can introduce model dependence), this work uses dispersion theory, which strictly incorporates $\pi\pi$ scattering data and unitarity.
• Resolution of the "Low-Mass" Anomaly: The paper demonstrates that the observed threshold enhancement does not require the introduction of a new, unphysical low-mass resonance.
• Role of Helicity-Flip Amplitudes: The study highlights the critical importance of helicity-flip amplitudes in shaping the angular distribution and the dip in the invariant mass spectrum.
• Virtual $Z_c(3900)$ Contribution: The authors quantify the specific role of the virtual exchange of the $Z_c(3900)$ exotic state, showing it improves the fit quality, particularly for angular distributions, though it is not strictly necessary to reproduce the threshold peak.

4. Results

• Reproduction of Data: All three fits (I, II, and III) successfully reproduce the experimental data, including the substructure near the $\pi^+\pi^-$ threshold.
  • Fit I (No $Z_c$): Even without the $Z_c$ exchange, the chiral contact terms combined with rigorous FSI can describe the threshold enhancement. This directly challenges the need for a new resonance.
  • Fit II & III (With $Z_c$): Including the $Z_c$ exchange improves the $\chi^2$ values, particularly for the helicity angular distribution. However, the improvement is marginal regarding the mass spectrum itself.
• Amplitude Analysis:
  • S-wave vs. D-wave: The S-wave amplitudes dominate the mass spectrum, while D-wave amplitudes are roughly two orders of magnitude smaller.
  • Origin of the Dip: The "dip" or specific shape in the invariant mass distribution near the threshold is significantly influenced by the $j_1$ term (a specific contact interaction) and the helicity-flip amplitude.
  • Angular Distribution: The curved behavior of the observed angular distribution is primarily driven by the D-wave contribution, which is dominated by the helicity-flip term.
• Parameter Constraints:
  • In Fit III, the fitted $Z_c$ mass ($3.873$GeV) and width ($0.019$GeV) are consistent with previous determinations but with large uncertainties, indicating that the data does not strongly constrain the$Z_c$ parameters further than prior knowledge.
  • The coupling constants for the contact terms ($g_1, h_1, j_1, c_m$) are determined with high precision.

5. Significance

• No New Resonance Needed: The primary conclusion is that the "resonance-like" structure near the $\pi^+\pi^-$ threshold is a dynamical effect arising from strong $\pi\pi$ final-state interactions and chiral contact terms, rather than a new particle. This resolves the theoretical tension regarding a sub-300 MeV resonance.
• Validation of Dispersion Theory: The success of the dispersion-theory-based approach validates its utility in analyzing heavy quarkonium transitions, offering a more robust alternative to chiral unitary models for this specific process.
• Insight into Exotic Hadrons: While the $Z_c(3900)$ is not required to explain the threshold peak, its virtual exchange plays a non-negligible role in refining the description of the decay, particularly in angular correlations. This supports the hypothesis that $Z_c$ states may act as virtual intermediates in heavy quarkonium transitions.
• Methodological Benchmark: The paper provides a rigorous framework for disentangling contact terms, FSI, and virtual exchanges, setting a standard for future analyses of similar dipion transitions in heavy quarkonia (e.g., $\Upsilon$ systems).

In summary, the paper decodes the BESIII observation as a manifestation of known QCD dynamics (chiral symmetry breaking and FSI) rather than evidence for a new exotic low-mass meson, while simultaneously refining the understanding of the virtual role of the $Z_c(3900)$.