Determination of the $Z_c(3900)$ and the $Z_{cs}(3985)$ states from joint analysis of experimental and lattice data

This paper presents a unified analysis of experimental and lattice QCD data that confirms the $Z_c(3900)$ and $Z_{cs}(3985)$ as SU(3) flavor partner resonance states with specific pole masses and widths, while revealing that their internal structure requires significant components beyond simple meson-meson molecules.

The Gist

Imagine the subatomic world as a bustling, chaotic dance floor. For years, physicists have been watching a very specific, mysterious dancer: a particle called Zc(3900). This dancer is strange because it's made of four "quarks" (the building blocks of matter) instead of the usual two or three. It's like finding a four-legged cat in a world where everyone expects only two-legged dogs or four-legged cats.

For over a decade, scientists have argued about what this dancer actually is. Is it a tight-knit family of four quarks glued together? Is it two separate particles just bumping into each other (like a molecule)? Or is it just a trick of the light caused by the geometry of the dance floor itself?

This paper by Chen, Du, and Guo acts like a super-powered detective team that finally solves the mystery by combining two different types of evidence: real-world experiments and computer simulations.

The Two Types of Evidence

1. The Real-World Experiments (The "Live Concert"):
   The team looked at data from the BESIII experiment in China. Imagine this as recording a live concert. They watched particles collide and break apart, looking for the specific "sound" (energy peaks) that the Zc(3900) makes. They also looked at a new, similar dancer called Zcs(3985), which is like the Zc(3900) but with a "strange" flavor twist.

2. The Computer Simulations (The "Virtual Reality"):
   They also used Lattice QCD, which is like a super-accurate video game that simulates the laws of physics on a tiny grid. This allows scientists to see the "energy levels" of these particles in a controlled, virtual box.

The Detective Work: Putting the Puzzle Together

The authors didn't just look at one piece of data; they built a massive, unified model that tried to explain everything at once. They had to account for three tricky things:

• The "Triangle Trick" (Triangle Singularities): Sometimes, particles take a specific detour that looks like a peak in the data but isn't a real particle. It's like a mirage in the desert. The team had to make sure they weren't being fooled by these optical illusions.
• The "Crowded Dance Floor" (Coupled Channels): Particles don't just dance alone; they swap partners. The Zc(3900) can turn into different combinations of particles and back again. The team had to track all these potential partners.
• The "Echo Chamber" (Final State Interactions): When particles fly out, they sometimes bounce off each other, changing the sound of the final signal. The team had to correct for these echoes.

The Big Discovery

After running their complex calculations, the team found that you cannot explain the data without assuming these particles are real.

If they tried to explain the data using only the "Triangle Trick" (the mirage) or only random noise, the model failed. It was like trying to explain a song by only humming the background noise; it just didn't fit. The only way the math worked was if they included a real, physical "pole" (a resonance) in their equations.

The Verdict:

• Zc(3900) and Zcs(3985) are real resonances. They are not just fleeting tricks of geometry; they are actual, distinct states of matter.
• They are "Cousins": The team found that Zc(3900) and Zcs(3985) are SU(3) flavor partners. Think of them as two siblings in the same family. They have nearly identical personalities (masses and widths) but one has a "strange" ingredient while the other doesn't. This confirms they belong to the same "octet" (a group of eight) in the particle physics family tree.

What Are They Made Of?

The paper also asked: "What is the recipe for these particles?"

They calculated the "compositeness," which is like asking, "How much of this cake is flour, and how much is sugar?"

• They found that about 50% of the particle is made of a specific pair of open-charm mesons (like a $D\bar{D}^*$ molecule).
• However, the other 50% is something else. It's not just a simple molecule of two particles stuck together. There are other, more complex ingredients (short-distance components) needed to hold it together.

The Final Numbers

The team gave the most precise measurements to date for these mysterious dancers:

• Zc(3900): Weighs in at roughly 3880 MeV and has a "half-life" (width) of about 32 MeV.
• Zcs(3985): Weighs in at roughly 3977 MeV and has a "half-life" of about 29 MeV.

In Summary

This paper is the moment the detective closes the case file. By combining the "live concert" recordings with the "virtual reality" simulations, and by carefully filtering out the "mirages" and "echoes," the authors proved that the Zc(3900) and Zcs(3985) are real, exotic particles. They are a matched pair of cousins in the subatomic world, made of a mix of simple particle pairs and something more complex that we are still learning to understand.

Technical Summary

1. Problem Statement

The nature of the exotic hadrons $Z_c(3900)$ and its strange partner $Z_{cs}(3985)$ remains a subject of intense debate in high-energy physics. Key issues include:

• Nature of the State: Are they compact tetraquarks, hadronic molecules ($D\bar{D}^*$), kinematic effects (threshold cusps), or triangle singularities (TS)?
• Data Inconsistencies: Previous analyses often treated experimental data (from BESIII) and Lattice QCD data separately. Some lattice studies failed to find a clear signal for $Z_c(3900)$, while others suggested its existence only when specific coupled channels were included.
• Missing Physics: Previous theoretical models fitting experimental data often neglected crucial kinematic effects, specifically triangle singularities (TS) arising from open-charm meson loops, and final-state interactions (FSI) between $\pi\pi$ and $K\bar{K}$.
• SU(3) Flavor Symmetry: While $Z_c(3900)$ and $Z_{cs}(3985)$ are hypothesized to be SU(3) flavor partners, a unified framework confirming this relationship while simultaneously describing both experimental spectra and lattice energy levels was lacking.

2. Methodology

The authors performed a unified joint analysis combining experimental data from the BESIII collaboration and finite-volume energy levels from recent Lattice QCD simulations.

A. Data Sets

The analysis simultaneously fitted:

• Experimental Data (BESIII):
  • $e^+e^- \to J/\psi \pi^+ \pi^-$ and $J/\psi K^+ K^-$ invariant mass spectra across 17 center-of-mass energies (4.1271–4.3583 GeV).
  • $e^+e^- \to D^0 D^{*-} \pi^+$ mass spectra at 4.23 and 4.26 GeV.
  • $e^+e^- \to K^+(D^{*0}D_s^- + D^0 D_s^{*-})$ recoil-mass distribution.
  • $e^+e^- \to J/\psi K^+ K^-$ spectra at 4.68 GeV (search for $Z_{cs}$ in $J/\psi K$ channel).
• Lattice QCD Data:
  • Finite-volume energy levels from the Hadron Spectrum Collaboration, CLQCD Collaboration, and Sadl et al., focusing on the $J/\psi\pi$-$D\bar{D}^*$ coupled channels.

B. Theoretical Framework

The authors constructed a comprehensive amplitude model incorporating three essential physical ingredients:

1. Triangle Singularities (TS): Open-charm meson loop diagrams (e.g., $Y \to \bar{D}D_1 \to \bar{D}D^*\pi$) were calculated using non-relativistic effective field theory to account for kinematic enhancements that could mimic resonance peaks.
2. Coupled-Channel Interactions: The $J/\psi\pi$ ($J/\psi\bar{K}$) and $D\bar{D}^*$ ($D\bar{D}^*_s$) channels were treated via a Lippmann-Schwinger equation approach. The interaction kernel $V(s)$ included constant terms and energy-dependent terms constrained by Heavy Quark Spin Symmetry (HQSS) and SU(3) flavor symmetry.
3. Final-State Interactions (FSI): The $\pi\pi$-$K\bar{K}$ rescattering in the $J/\psi\pi\pi$ and $J/\psi K K$ channels was described using a model-independent dispersive approach (Muskhelishvili-Omnès representation).
4. Finite-Volume Formalism: To incorporate lattice data, the coupled-channel Lüscher method was employed. The infinite-volume loop function $G(t)$ was corrected by a finite-volume term $\Delta G(t)$ to match the discrete energy levels in the lattice box.

C. Fitting Strategy

Three distinct fits were performed to isolate the contributions of different mechanisms:

• Fit I (Full Model): Includes TS, coupled-channel poles, and FSI.
• Fit II (No TS): Removes triangle diagrams to test the necessity of TS.
• Fit III (No Pole): Replaces the dynamical $T$-matrix pole with a constant matrix to test if the data can be described without a resonance.

3. Key Contributions

• Unified Framework: First study to simultaneously describe BESIII experimental spectra and Lattice QCD finite-volume energy levels for both $Z_c(3900)$ and $Z_{cs}(3985)$ within a single theoretical framework.
• Inclusion of Kinematic Effects: Rigorously integrated triangle singularities and $\pi\pi$-$K\bar{K}$ FSI, which are often omitted in simpler resonance analyses.
• Model Selection: Demonstrated through comparative fits (I, II, III) that neither triangle singularities nor kinematic cusps alone are sufficient to explain the data; a genuine pole contribution is indispensable.
• Compositeness Analysis: Calculated the compositeness coefficients ($X_A$) to quantify the molecular ($D\bar{D}^*$) component of these states.

4. Results

A. Pole Parameters

The analysis confirmed the existence of poles on the unphysical Riemann sheet (RS-III) above the respective thresholds, identifying them as resonance states.

• $Z_c(3900)$:
  • Mass: $3879.6 \pm 4.8$ MeV
  • Half-width: $32.2 \pm 4.7$ MeV
• $Z_{cs}(3985)$:
  • Mass: $3976.9 \pm 5.1$ MeV
  • Half-width: $28.8 \pm 5.9$ MeV

B. Fit Quality and Necessity of Components

• Fit I (Full): Achieved $\chi^2/\text{d.o.f.} = 1.43$, successfully describing all experimental spectra and lattice energy levels.
• Fit II (No TS): Could reproduce most data but failed to describe the $K^+K^-$ and $J/\psi K^\pm$ spectra at 4.68 GeV and specific lattice energy levels.
• Fit III (No Pole): Failed to reproduce the peak structures around 3.9 GeV ($J/\psi\pi$) and 3.98 GeV ($K^+$ recoil).
• Conclusion: The pole contribution is indispensable for describing the peaks, while triangle diagrams significantly improve the overall fit quality, particularly for specific kinematic regions.

C. SU(3) Flavor Symmetry and Couplings

The extracted coupling ratios strongly support the interpretation of $Z_c(3900)$ and $Z_{cs}(3985)$ as SU(3) flavor partners within the same octet:

• $|g_{Z_c D\bar{D}^*} / g_{Z_c J/\psi\pi}| = 8.58 \pm 0.20$
• $|g_{Z_{cs} (D_s\bar{D}^* + D\bar{D}^*_s)} / g_{Z_{cs} J/\psi K}| = 8.59 \pm 0.19$
  The remarkable consistency between these ratios validates the SU(3) symmetry hypothesis.

D. Nature of the States (Compositeness)

The compositeness coefficient $X_A$ (probability of the two-body component) was calculated:

• $X_A(Z_c) \approx 0.50 \pm 0.04$
• $X_A(Z_{cs}) \approx 0.54 \pm 0.05$
  This indicates that while the $D\bar{D}^*$ (or $D_s\bar{D}^*$) molecular component is sizable, it accounts for only about 50% of the state. Additional components (likely compact tetraquark or short-distance contributions) are required to form these exotic states.

5. Significance

• Resolution of Controversy: The study provides robust evidence that $Z_c(3900)$ and $Z_{cs}(3985)$ are genuine resonance states, not merely kinematic artifacts like triangle singularities or threshold cusps.
• Bridging Experiment and Theory: By successfully reconciling BESIII data with Lattice QCD results, the work validates the theoretical framework used for extracting resonance properties from finite-volume data.
• Structural Insight: The finding that these states are roughly 50% molecular and 50% "other" suggests a complex internal structure, likely a mixture of molecular and compact tetraquark configurations, challenging simple "either/or" classifications.
• Future Directions: The precise determination of pole positions and couplings provides a solid foundation for further theoretical investigations into the short-distance dynamics of heavy-flavor exotic hadrons.