⚛️ phenomenology

Two-body strong decays of the pseudoscalar hidden-charm tetraquark states via the QCD sum rules

This paper investigates the two-body strong decays of pseudoscalar hidden-charm tetraquark states with specific diquark-antidiquark structures using QCD sum rules to calculate their hadronic coupling constants and total decay widths.

Original authors: Yu-Hang Xu, Zhi-Gang Wang

Published 2026-03-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yu-Hang Xu, Zhi-Gang Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, bustling construction site. For decades, physicists have been building a model of how the tiny bricks of matter (quarks) stick together to form larger structures like protons and neutrons. The standard rulebook, called the "Quark Model," says that most particles are built from either two bricks (a quark and an antiquark) or three bricks (like a proton).

But recently, construction workers (experimental physicists) have started finding strange, exotic structures that don't fit the standard blueprints. These are called tetraquarks—particles made of four bricks stuck together.

This paper is like a detailed engineering report on two specific, newly discovered (or predicted) tetraquarks, which the authors call $Z_c^+$ and $Z_c^-$ . Here is what they did, explained simply:

1. The Mystery of the "Hidden Charm" Tetraquarks

These specific particles are special because they contain a "hidden charm" pair. Imagine a four-person dance team where two partners are holding hands tightly in the middle (the charm quarks), while the other two partners are on the outside. The authors are trying to figure out exactly how these teams are arranged and, more importantly, how they break up.

In the world of particle physics, "breaking up" is called decay. These heavy particles are unstable; they don't last long. They quickly fall apart into lighter, more stable particles (like a J/ψ meson, a pion, or a D meson).

2. The Tool: QCD Sum Rules (The "Recipe Book")

To predict how these particles behave without being able to see them directly in a microscope, the authors used a powerful mathematical tool called QCD Sum Rules.

Think of this like trying to figure out the ingredients of a secret cake recipe just by tasting the crumbs.

The Crumbs (QCD Side): They started with the fundamental laws of physics (Quantum Chromodynamics) and the known properties of the "ingredients" (quarks and gluons). They calculated what the particle should look like based on pure math.
The Cake (Hadron Side): They then looked at the "real world" side, where these particles exist as actual objects with mass and decay rates.
The Match: They used a concept called Quark-Hadron Duality to match the mathematical crumbs to the physical cake. If the math matches the reality, the recipe is correct.

3. The Calculation: How They Break Up

The authors focused on two-body decays. Imagine a tetraquark as a fragile glass sculpture. When it shatters, it doesn't turn into dust; it breaks into two distinct, larger shards.

The paper calculates the "coupling constants." In everyday language, think of this as the strength of the glue holding the specific shards together before they break.

If the glue is weak, the particle breaks apart easily into that specific pair.
If the glue is strong, it's less likely to break into that pair.

They calculated this "glue strength" for every possible way the $Z_c^+$ and $Z_c^-$ could split. For example:

Could it split into a $J/\psi$ (a heavy charm particle) and a pion (a light particle)?
Could it split into a $D$ meson and a $\bar{D}$ meson?

4. The Results: The "Most Likely" Breakups

After doing the heavy math (which involved complex integrals and accounting for the vacuum of space acting like a foggy medium), they got some clear answers:

The $Z_c^-$ (The Negative One): This particle is a bit heavier. It is most likely to break apart into a $J/\psi$ and an $a_1$ meson. This is its "favorite" breakup path.
The $Z_c^+$ (The Positive One): This one prefers to break into a $D$ meson and a $\bar{D}^0$ meson.

They calculated the Total Decay Width, which is essentially a measure of how fast the particle lives and dies.

The $Z_c^-$ lives for a tiny fraction of a second and has a "width" of about 326 MeV.
The $Z_c^+$ is slightly more stable (in particle terms) with a width of about 92 MeV.

5. Why This Matters: The Treasure Map

Why do we care about these numbers? Because experimental physicists (the people with the giant particle colliders like the Large Hadron Collider) are currently hunting for these particles.

This paper acts like a treasure map.

Before this, scientists didn't know exactly how these particles would look when they decayed.
Now, the authors are saying: "If you want to find the $Z_c^-$ , look for a $J/\psi$ and an $a_1$ meson appearing together. If you want to find the $Z_c^+$ , look for a $D$ and a $\bar{D}^0$ ."

Summary Analogy

Imagine you are a detective trying to find a stolen, invisible diamond. You can't see the diamond, but you know that when it's stolen, it leaves behind specific footprints.

The Paper: Is the forensic report that says, "If the diamond was stolen by the $Z_c^-$ gang, you will find a red footprint and a blue footprint. If it was the $Z_c^+$ gang, you will find two green footprints."
The Result: The authors have calculated exactly how big those footprints are and how likely each gang is to leave them. This helps the police (experimentalists) know exactly where to look and what to look for to confirm the existence of these exotic particles.

In short, this paper uses advanced math to predict the "personality" and "breakup habits" of two mysterious four-quark particles, giving experimentalists a clear guide on how to catch them in the act.

1. Problem Statement

The paper addresses the theoretical characterization of pseudoscalar hidden-charm tetraquark states with quantum numbers $J^{PC} = 0^{-+}$ and $0^{--}$ . While the existence of exotic hadrons (like $X, Y, Z$ states) has been confirmed experimentally, their internal structure remains debated (molecular vs. compact multiquark).

Specific Gap: Previous studies (including the authors' own work) had established the mass spectrum of these states (approx. 4.56 GeV) using QCD sum rules. However, mass alone provides only a crude assessment. There was a lack of systematic theoretical predictions regarding their strong decay properties, specifically the hadronic coupling constants and partial/total decay widths, which are crucial for guiding experimental searches.
Target States: The study focuses on two specific diquark-antidiquark configurations:
- $Z_c^+$ ( $0^{-+}$ ): Structure $[uc]_A[\bar{d}\bar{c}]_V - [uc]_V[\bar{d}\bar{c}]_A$
- $Z_c^-$ ( $0^{--}$ ): Structure $[uc]_A[\bar{d}\bar{c}]_V + [uc]_V[\bar{d}\bar{c}]_A$
  (Where A and V denote axial-vector and vector diquarks, respectively).

2. Methodology

The authors employ the QCD Sum Rules (QCDSR) approach based on rigorous quark-hadron duality. The methodology involves the following steps:

Three-Point Correlation Functions: The authors construct three-point correlation functions involving the interpolating currents for the tetraquark states ( $J_{Z_c}$ $J_{Z_{c}}$ ) and the currents for the final-state mesons (e.g., $J/\psi, \pi, \rho, D, \bar{D}^*$ $J / ψ, π, ρ, D, \overset{ˉ}{D}^{*}$ , etc.).
- The analysis covers various decay channels, including hidden-charm modes ( $J/\psi \pi, \eta_c \rho, J/\psi a_1$ , etc.) and open-charm modes ( $D \bar{D}^*, D^* \bar{D}$ , etc.).
Operator Product Expansion (OPE): On the QCD side, the correlation functions are expanded using the OPE up to dimension-5 vacuum condensates. This includes:
- Perturbative terms.
- Quark condensates ( $\langle \bar{q}q \rangle$ ).
- Gluon condensates ( $\langle \frac{\alpha_s}{\pi} G^2 \rangle$ ).
- Quark-gluon mixed condensates ( $\langle \bar{q} g_s \sigma G q \rangle$ ).
Quark-Hadron Duality: The authors match the QCD side with the phenomenological (hadron) side.
- A critical technical step involves handling the mismatch between the triple dispersion relation on the hadron side and the double dispersion relation on the QCD side. They integrate over the third variable and apply continuum thresholds ( $s_0, u_0$ ) to establish duality rigorously.
- Borel Transformation: Double Borel transformations are applied to the variables $P^2$ and $Q^2$ to suppress higher resonance contributions and continuum states.
- Endpoint Divergences: To address divergences arising at thresholds ( $s=4m_c^2$ or $s=m_c^2$ ), a shift term ( $\Delta^2 = m_s^2$ ) is introduced in the denominators.
Parameter Extraction: The hadronic coupling constants ( $G$ ) are extracted by fitting the sum rules to obtain "flat Borel platforms" (regions where the results are stable against variations in the Borel parameter $T^2$ ).

3. Key Contributions

First Systematic Decay Analysis: This work provides the first comprehensive calculation of the two-body strong decay widths for these specific pseudoscalar tetraquark candidates using QCDSR.
Rigorous Treatment of Duality: The authors explicitly address the mathematical mismatch between hadronic and QCD dispersion relations, ensuring a more robust derivation of the sum rules compared to previous approximations.
Inclusion of Disconnected Diagrams: The calculation accounts for both connected and disconnected Feynman diagrams to ensure accuracy in the OPE.
Prediction of Dominant Channels: The study identifies specific decay modes that are most likely to be observed experimentally, distinguishing between the $0^{-+}$ and $0^{--}$ states.

4. Key Results

Using standard QCD parameters (vacuum condensates, $m_c$ , etc.) and fitted Borel platforms, the authors obtained the following results:

A. Hadronic Coupling Constants ( $G$ )
The coupling constants for various channels were extracted (e.g., $G_{J/\psi a_1 Z_c^-} \approx 6.40$ GeV, $G_{D \bar{D}^0 Z_c^+} \approx 5.67$ GeV).

B. Partial and Total Decay Widths

For $Z_c^-$ ( $0^{--}$ ):
- Total Width: $\Gamma_{Z_c^-} = 326.197^{+4.255}_{-3.106}$ MeV.
- Dominant Mode: $Z_c^- \to J/\psi a_1$ ( $\Gamma \approx 196$ MeV, ~60% of total).
- Other significant modes: $J/\psi \pi$ (~~39 MeV), $\chi_{c1} \rho$ (~~49 MeV), $D^* \bar{D}$ (~27 MeV).
For $Z_c^+$ ( $0^{-+}$ ):
- Total Width: $\Gamma_{Z_c^+} = 91.835^{+0.96}_{-0.76}$ MeV.
- Dominant Mode: $Z_c^+ \to D \bar{D}^0$ ( $\Gamma \approx 49.3$ MeV, ~54% of total).
- Other significant modes: $D^* \bar{D}_1$ (~35 MeV), $D \bar{D}^0$ (open charm).
- Note: The decay $Z_c^+ \to \eta_c a_0$ and $Z_c^+ \to J/\psi \rho$ are highly suppressed or zero.

C. Branching Ratios
The paper provides relative branching ratios, highlighting that:

$Z_c^-$ is best searched for in the $J/\psi a_1$ channel.
$Z_c^+$ is best searched for in the $D \bar{D}^0$ channel.

5. Significance

Experimental Guidance: The results provide concrete targets for experimental collaborations (such as BESIII, Belle II, LHCb) to search for these elusive pseudoscalar tetraquark states. By predicting the dominant decay channels and total widths, the paper suggests where to look in the invariant mass spectra.
Validation of Tetraquark Models: The calculated widths are consistent with the "fall-apart" mechanism expected for compact tetraquarks, where the decay occurs without quark-antiquark annihilation.
Theoretical Benchmark: The rigorous application of QCDSR with dimension-5 condensates and careful handling of dispersion relations sets a high standard for future theoretical studies of exotic hadron decays.

In conclusion, this paper successfully bridges the gap between mass predictions and decay phenomenology for pseudoscalar hidden-charm tetraquarks, offering a robust theoretical framework to interpret future experimental data.