Multi-channel joint analysis of the exotic charmonium-like state $T_{c\bar{c}}(4020)$

Using a multi-channel joint analysis of BESIII data from $e^{+}e^{-}$ collisions, this study identifies the exotic charmonium-like state $T_{c\bar{c}}(4020)$ as a $J^{P}=1^{+}$ resonance with $11.7\sigma$ significance, while extracting its pole positions and relative branching fractions across three decay channels.

The Gist

Imagine the universe as a giant, cosmic LEGO set. For decades, physicists thought they understood all the basic bricks: some bricks stick together in pairs (like a proton and an electron), and others stick together in threes (like the protons and neutrons inside an atom). These are the "normal" particles.

But recently, scientists have started finding weird, "exotic" LEGO structures that don't fit the usual rules. They are made of four or more bricks stuck together in a way that shouldn't be possible according to the old rulebook. One of these mysterious structures is called $T_{c\bar{c}}(4020)$.

This paper is like a detective report from the BESIII experiment in China, where scientists finally solved a major mystery about this exotic particle: What is its personality (spin and parity), and what does it like to eat (decay)?

Here is the breakdown of their discovery in simple terms:

1. The Mystery: A Shape-Shifting Ghost

The $T_{c\bar{c}}(4020)$ is a short-lived particle that appears when they smash electrons and positrons (anti-electrons) together at very high speeds. It's like a ghost that appears for a split second and then vanishes, breaking apart into different pieces.

The problem was, nobody knew exactly what this ghost looked like.

• The "Spin" and "Parity" Question: Imagine the particle is a spinning top. Does it spin like a regular top (1+), or is it a weird, flat disc (2+)? Or maybe it's a mirror image of itself? Knowing this "spin" is crucial because it tells us if the particle is a tight-knit family of four quarks (a tetraquark) or just two separate particles holding hands loosely (a molecule).
• The "Eating Habits" Question: When the particle dies, what does it turn into? Does it mostly turn into heavy "open-charm" pieces (like $D^*$ mesons), or does it prefer "hidden-charm" pieces (like $J/\psi$ or $h_c$)? This tells us about its internal structure.

2. The Investigation: The Multi-Channel Detective Work

In the past, scientists looked at the particle's decay in just one way at a time. It's like trying to identify a suspect by only looking at their left shoe. You might get a clue, but you might miss the whole picture.

In this paper, the BESIII team did something new: They looked at all three "shoes" at once.
They analyzed three different ways the particle broke apart simultaneously:

1. Breaking into two heavy $D^*$ mesons.
2. Breaking into a $J/\psi$ (a heavy particle) and a pion.
3. Breaking into an $h_c$ (another heavy particle) and a pion.

They used a massive amount of data (over 1,500 "inverse picobarns" of collisions—think of this as a huge pile of digital evidence) collected at the BEPCII collider.

3. The Breakthrough: Solving the Puzzle

By using a sophisticated mathematical technique called Partial Wave Analysis (PWA)—which is like a super-advanced 3D scanner that reconstructs the particle's shape from all its decay angles—they found the answers:

• The Identity Revealed: The particle is definitely a $1^+$ state.

  • Analogy: Imagine you have a spinning top. The data proved it spins in a specific, upright way. This rules out it being a flat disc or a mirror image.
  • Significance: This is a huge deal. It's the first time this specific "spin" has been confirmed for this particle with extreme confidence (11.7 sigma, which is like being 99.9999999999% sure).

• The Structure Clue: The particle loves to break apart into $D^* \bar{D}^*$ (two heavy mesons) much more than it loves breaking into the hidden-charm particles ($J/\psi$ or $h_c$).

  • The Metaphor: Think of the particle as a house. If it were a "charmonium core" (a tight family), it would be like a house made of solid concrete; it would be hard to break apart, and when it did, it would likely stay together in a specific way.
  • However, since it breaks apart so easily into two heavy mesons, it suggests the particle is actually a "molecular" structure.
  • Analogy: It's like a house made of two magnets stuck together. They are holding hands, but not fused. It's easy to pull them apart. This supports the theory that $T_{c\bar{c}}(4020)$ is a molecule made of two $D^*$ mesons orbiting each other, rather than a single, tight-knit tetraquark.

4. The Result: A New Name

Because they finally figured out its spin and parity ($1^+$), the scientists are updating its name. Just like you might rename a pet once you learn its true personality, this particle is now officially called **$T_{c\bar{c}1}(4020)$**.

Why Does This Matter?

This isn't just about naming a particle. It helps us understand the glue that holds the universe together.

• QCD (Quantum Chromodynamics) is the theory of how quarks stick together.
• For a long time, we only knew about "pairs" and "triples."
• Finding these "molecules" proves that quarks can form complex, loose structures, just like atoms can form molecules. This helps physicists refine the rulebook of the universe.

In a nutshell: The BESIII team acted like cosmic detectives, using a multi-angle camera to finally identify a mysterious, four-quark particle. They proved it spins in a specific way and behaves like a loose molecular bond rather than a tight knot, giving us a clearer picture of how the building blocks of matter stick together.

Technical Summary

Here is a detailed technical summary of the paper "Multi-channel joint analysis of the exotic charmonium-like state $T_{c\bar{c}}(4020)$" by the BESIII Collaboration.

1. Problem Statement

The nature of exotic hadrons, specifically the charged charmonium-like states $T_{c\bar{c}}$ (also known as $Z_c$), remains a central puzzle in Quantum Chromodynamics (QCD). These states, composed of at least four quarks ($c\bar{c}q\bar{q}$), challenge the traditional quark model.

• The Specific Issue: The state $T_{c\bar{c}}(4020)$ (also referred to as $T_{c\bar{c}}(4025)$) has been observed in processes like $e^+e^- \to \pi\pi h_c$ and $\pi D^*\bar{D}^*$. However, previous analyses relied on one-dimensional fits of invariant mass spectra for individual decay channels.
• The Discrepancy: These single-channel fits yielded inconsistent values for the resonance width ($\Gamma$) between the hidden-charm ($\pi h_c$) and open-charm ($D^*\bar{D}^*$) channels, suggesting the line-shape models were insufficient.
• Missing Quantum Numbers: Unlike its lighter partner $T_{c\bar{c}}(3900)$, the spin-parity ($J^P$) of the $T_{c\bar{c}}(4020)$ had not been experimentally determined.
• Theoretical Ambiguity: Distinguishing between competing theoretical models (e.g., tetraquarks, hadronic molecules, or kinematic effects) requires precise knowledge of the relative branching fractions (BFs) between open-charm and hidden-charm decay modes.

2. Methodology

The authors performed the first multi-channel joint Partial Wave Analysis (PWA) to simultaneously extract the properties of the $T_{c\bar{c}}(4020)$.

• Data Source: Data collected by the BESIII detector at the BEPCII collider at center-of-mass energies $\sqrt{s} = 4.395$ GeV and $4.416$GeV, with a total integrated luminosity of **1598.9 pb$^{-1}$**.
• Channels Analyzed: The analysis simultaneously fitted three decay channels:
  1. $e^+e^- \to D^{*0}D^{*-}\pi^+$ (Open-charm)
  2. $e^+e^- \to \pi^+\pi^- J/\psi$ (Hidden-charm)
  3. $e^+e^- \to \pi^+\pi^- h_c$ (Hidden-charm)
• Amplitude Construction:
  • Used the helicity amplitude formalism in the $LS$-coupling scheme.
  • Modeled the $T_{c\bar{c}}(4020)$ resonance using a Breit-Wigner propagator with a running width that sums partial widths for $D^{*0}D^{*-}$, $\pi J/\psi$, $\pi h_c$, and an unknown component.
  • Included intermediate resonances such as $T_{c\bar{c}}(3900)$, $f_2(1270)$, and non-resonant (NR) contributions.
  • Parametrized the $\pi\pi$ S-wave using Flatté-like formulas for $f_0(980)$ and E791 propagators for $\sigma$.
• Statistical Approach:
  • Employed an unbinned maximum likelihood fit across all three channels and both energy points simultaneously.
  • Coupling parameters for production were constrained separately for each energy, while decay parameters were shared.
  • Spin-Parity Determination: Tested hypotheses $J^P = 1^+, 1^-, 2^+, 2^-$ by comparing negative log-likelihood (NLL) values and performing pseudo-experiment (toy MC) studies to assess statistical significance.
  • Pole Extraction: Extracted pole positions on the complex energy plane (Riemann sheets) by finding the zeros of the inverse Breit-Wigner function, accounting for three branch points ($D^*\bar{D}^*$, $\pi J/\psi$, $\pi h_c$).

3. Key Contributions

• First Joint Analysis: This is the first study to perform a simultaneous PWA of the $T_{c\bar{c}}(4020)$ across open-charm and hidden-charm channels, resolving previous inconsistencies in width measurements.
• Spin-Parity Determination: Definitively established the quantum numbers of the $T_{c\bar{c}}(4020)$ state.
• Pole Position Extraction: Provided the first extraction of pole positions on the multi-sheeted Riemann surface for this state, offering a more rigorous definition of the resonance parameters than mass/width fits.
• Relative Branching Fractions: Measured the ratios of branching fractions between hidden-charm and open-charm modes for the first time, providing critical data for model discrimination.

4. Key Results

A. Spin-Parity ($J^P$)

• The $J^P = 1^+$ hypothesis provides the best fit to the data.
• The significance of $1^+$over alternative hypotheses ($1^-, 2^+, 2^-$) is **$11.7\sigma$** (based on pseudo-experiments) and up to **$18.7\sigma$** (based on likelihood ratio tests).
• Conclusion: The state is identified as $T_{c\bar{c}1}(4020)^-$, consistent with the naming scheme of the $T_{c\bar{c}1}(3900)$.

B. Pole Positions

Two physically reasonable pole positions were identified on the Riemann sheets:

1. Pole 1: $m_{pole} = 4022.44 \pm 1.55$ MeV/$c^2$, $\Gamma_{pole} = 38.54 \pm 2.94$ MeV.
2. Pole 2: $m_{pole} = 4023.01 \pm 1.35$ MeV/$c^2$, $\Gamma_{pole} = 35.02 \pm 2.20$ MeV.
   (Note: Systematic uncertainties are included in the summary values in the paper's conclusion.)

C. Relative Branching Fractions

The analysis yielded the following ratios relative to the dominant $D^{*0}D^{*-}$ mode:

• $\mathcal{B}[T_{c\bar{c}}(4020)^- \to \pi^- J/\psi] / \mathcal{B}[T_{c\bar{c}}(4020)^- \to D^{*0}D^{*-}] = (3.6 \pm 0.6 \pm 1.6) \times 10^{-3}$
• $\mathcal{B}[T_{c\bar{c}}(4020)^- \to \pi^- h_c] / \mathcal{B}[T_{c\bar{c}}(4020)^- \to D^{*0}D^{*-}] = (8.9 \pm 1.3 \pm 2.3) \times 10^{-2}$
  (First uncertainty is statistical, second is systematic.)

D. Production Cross Sections

The product of the Born cross section and branching fraction for $e^+e^- \to T_{c\bar{c}}(4020)^-\pi^+$ was measured at both energy points (e.g., $\approx 335$ pb for the $D^{*0}D^{*-}$ channel at 4.395 GeV).

5. Significance and Implications

• Structural Insight: The results show a significantly stronger coupling to the open-charm $D^*\bar{D}^*$ channel compared to hidden-charm channels ($\pi J/\psi$ and $\pi h_c$).
  • If the state were a "charmonium-core" tetraquark, hidden-charm decays would be favored.
  • The dominance of open-charm decays strongly supports the hadronic molecular interpretation, specifically a $D^*\bar{D}^*$ molecule.
• Consistency with $T_{c\bar{c}}(3900)$: The alignment of $J^P=1^+$ with the $T_{c\bar{c}}(3900)$ and the similar molecular structure suggest these states form a family of $D^{(*)}\bar{D}^{(*)}$ molecules.
• Methodological Advance: The study demonstrates the necessity of multi-channel joint analyses to resolve discrepancies in resonance parameters that single-channel fits cannot address.
• Future Directions: The identification of multiple poles on different Riemann sheets suggests complex dynamics that require further theoretical quantification and additional experimental data to fully constrain.

In summary, this paper provides the most comprehensive characterization of the $T_{c\bar{c}}(4020)$ to date, confirming its $1^+$nature and providing strong evidence for its interpretation as a$D^\bar{D}^$ molecular state.