Analysis of the strong decays of the $Y(4660)$ in tetraquark scenario via the QCD sum rules

Using three-point QCD sum rules with vacuum condensates up to dimension 5, this study investigates the strong decays of four vector tetraquark candidates for the $Y(4660)$ and finds that the predicted total width of $61.5\pm7.3\,\rm{MeV}$ for the $[sc][\bar{s}\bar{c}]$ configuration ($J^{PC}=1^{--}$) aligns excellently with experimental data, thereby supporting this specific tetraquark interpretation.

The Gist

Imagine the universe is a giant, bustling construction site. For decades, physicists have had a blueprint called the "Standard Model" that explains how the basic building blocks of matter (quarks) stick together to form particles. Usually, they stick together in pairs (like a proton and an antiproton) or triplets (like a proton).

But in recent years, construction workers (experimental physicists) have started finding strange, new structures that don't fit the old blueprints. These are called exotic particles. One of the most puzzling ones is a particle named Y(4660). It's like finding a house built with four bricks instead of the usual two or three, and nobody is sure exactly how those four bricks are arranged.

This paper is like a team of theoretical architects trying to figure out the blueprint of this Y(4660) house. They ask: "Is it a messy pile of four bricks, or is it a specific, organized structure?"

The Four Suspects (The Tetraquark Scenarios)

The authors propose that the Y(4660) is a tetraquark—a particle made of four quarks stuck together. Specifically, they look at four different ways these four quarks could be arranged, like four different architectural designs for a house:

1. Design A: A specific mix of "strange" and "charm" quarks arranged in a certain way.
2. Design B: The same ingredients, but arranged in a slightly different pattern.
3. Design C: A mix involving "up" and "down" quarks (the common ones) with charm quarks.
4. Design D: Another variation of the common quarks.

Think of these four designs as four different recipes for a cake. They all use similar ingredients (quarks), but the order in which you mix them and the shape of the pan (the quantum structure) changes the final result.

The Tool: QCD Sum Rules (The "Mathematical X-Ray")

How do you figure out which recipe makes the cake without baking it? You can't just look at it; you have to use math.

The authors use a powerful mathematical tool called QCD Sum Rules. Imagine this as a high-tech X-ray machine or a sound engineer's equalizer.

• On one side, they have the "theory" (the raw ingredients and the laws of physics).
• On the other side, they have the "experiment" (the actual particle observed in the lab).

They use this tool to calculate a specific property of the particle: how fast it falls apart. In physics, this is called the decay width.

• If a particle is very unstable, it falls apart quickly (a wide decay width).
• If it's more stable, it lasts a bit longer (a narrow decay width).

The Experiment: The "Falling Apart" Test

The authors simulate the four different designs (A, B, C, and D) and calculate how fast each one would break apart into smaller, known particles (like D-mesons and J/psi particles).

Here is the result of their "X-ray" analysis:

• Design A (The "AA" model): Predicts the particle would fall apart way too slowly. It's too stable. It doesn't match the real Y(4660).
• Design B (The "eAV" model): Predicts it would fall apart way too fast. It's too unstable.
• Design C (The "PA" model): Also predicts it would fall apart far too quickly.
• Design D (The "S eV" model): This is the winner! The math predicts this specific arrangement would fall apart at a speed of 61.5 MeV.

The Verdict: Matching the Real World

When the experimentalists (the people with the giant particle accelerators) measured the real Y(4660), they found it falls apart at a speed of roughly 48 to 60 MeV (depending on the experiment).

The authors' prediction for Design D (61.5 MeV) is a near-perfect match with the real-world data.

The Conclusion

The paper concludes that the mysterious Y(4660) is likely a tetraquark made of a specific combination of quarks: a "strange" quark and a "charm" quark paired up, and an anti-strange and anti-charm quark paired up, arranged in a very specific dance (the $[sc]S[\bar{s}\bar{c}]eV - [sc]eV[\bar{s}\bar{c}]S$ structure).

In simple terms:
The authors tried four different blueprints for a weird new particle. Three of the blueprints predicted a particle that would either last too long or break apart too fast. Only one blueprint predicted a particle that breaks apart at exactly the right speed to match what scientists see in the lab. Therefore, they are confident they have finally identified the correct "DNA" of the Y(4660).

This is a big deal because it helps us understand the "glue" (Quantum Chromodynamics) that holds the universe together, proving that nature can build complex structures out of quarks that we didn't expect to find.

Technical Summary

1. Problem Statement

The nature of the vector charmonium-like state $Y(4660)$ remains enigmatic. Observed initially by the Belle collaboration in $e^+e^- \to \pi^+\pi^-\psi(2S)$ and later confirmed by BaBar and BESIII, its properties (mass $\approx 4.66$ GeV) do not fit comfortably within the traditional quark model as a conventional $c\bar{c}$ state.

• Theoretical Ambiguity: Various interpretations exist, including $\psi'f_0(980)$ molecules, tetraquark states, hadrocharmonium, and higher radial excitations ($\psi(5S)$, $\psi(6S)$).
• Mass Degeneracy: Previous QCD sum rule studies have successfully reproduced the mass of $Y(4660)$ for several different tetraquark structures (e.g., $[sc]_S[\bar{s}\bar{c}]_V$, $[sc]_P[\bar{s}\bar{c}]_A$, etc.).
• The Core Issue: Since mass calculations alone cannot distinguish between these competing structural hypotheses, the authors argue that decay widths are the critical observable needed to determine the true nature of the $Y(4660)$. Specifically, they aim to test whether the $Y(4660)$ is a tetraquark state and, if so, which specific diquark-antidiquark configuration it represents.

2. Methodology

The authors employ the Three-Point QCD Sum Rules framework based on rigorous quark-hadron duality to investigate the strong decay behaviors of the $Y(4660)$.

• Interpolating Currents: Four specific vector tetraquark currents with quantum numbers $J^{PC} = 1^{--}$ are constructed using local four-quark operators:

  1. $J^{PA}_\mu$: $[uc]_P[\bar{u}\bar{c}]_A + [dc]_P[\bar{d}\bar{c}]_A - [uc]_A[\bar{u}\bar{c}]_P - [dc]_A[\bar{d}\bar{c}]_P$ (and $s$-quark analogues).
  2. $J^{AA}_{\mu\nu}$: $[uc]_A[\bar{u}\bar{c}]_A + [dc]_A[\bar{d}\bar{c}]_A$ (and $s$-quark analogues).
  3. $J^{\tilde{A}V}_\mu$: $[sc]_{\tilde{A}}[\bar{s}\bar{c}]_V + [sc]_V[\bar{s}\bar{c}]_{\tilde{A}}$ (Tensor diquark components).
  4. $J^{S\tilde{V}}_{\mu\nu}$: $[sc]_S[\bar{s}\bar{c}]_{\tilde{V}} - [sc]_{\tilde{V}}[\bar{s}\bar{c}]_S$.
     (Note: $S, P, A, V, \tilde{A}, \tilde{V}$ denote Scalar, Pseudoscalar, Axial-vector, Vector, and specific Tensor components).

• Correlation Functions: Three-point correlation functions are constructed involving the tetraquark current and two conventional meson currents (e.g., $D^{(*)}\bar{D}^{(*)}$, $J/\psi \omega$, $\eta_c \phi$).
  $$\Pi(p, q) = i^2 \int d^4x d^4y e^{ipx} e^{iqy} \langle 0 | T \{ J_{meson1}(x) J_{meson2}(y) J_{tetra}^\dagger(0) \} | 0 \rangle$$

• OPE and Duality:

  • QCD Side: The Operator Product Expansion (OPE) is performed up to dimension-5 vacuum condensates ($\langle \bar{q}q \rangle$, $\langle \bar{s}s \rangle$, $\langle \bar{q}g_s\sigma G q \rangle$, etc.). Gluon condensates are neglected due to small contributions.
  • Hadron Side: The ground state contributions are isolated using pole residues and hadronic coupling constants ($G$).
  • Continuum Subtraction: To handle the mismatch between the triple dispersion relation (hadron side) and double dispersion relation (QCD side), the authors introduce a free parameter $C$ to parameterize contributions from higher resonances in the $s'$ channel, ensuring a "flat platform" in the Borel window.

• Numerical Analysis:

  • Input parameters (masses, decay constants, condensates) are evolved to the energy scale $\mu = 1$ GeV using Renormalization Group Equations.
  • Borel windows ($T^2$) are selected to ensure stability (flat platforms) and pole dominance.
  • Hadronic coupling constants are extracted, and partial decay widths are calculated using the formula:
    $$\Gamma(Y \to FF') = \frac{|T|^2 p(m_Y, m_F, m_{F'})}{24\pi m_Y^2}$$

3. Key Contributions

• Systematic Decay Analysis: This is the first comprehensive study calculating the two-body strong decay widths for four distinct tetraquark configurations of the $Y(4660)$ within the same rigorous QCD sum rule framework.
• Model Discrimination: By comparing the predicted total widths against experimental data, the study provides a definitive method to rule out specific tetraquark configurations that were previously considered viable based solely on mass spectrum calculations.
• Handling Contaminations: The authors explicitly address the issue of axial-vector contaminations in channels where currents can couple to both $1^{--}$ and $1^{+-}$ states, introducing parameterized terms ($\bar{\lambda}$) to isolate the pure $1^{--}$ contributions.

4. Results

The study calculated the total decay widths for the four candidate tetraquark states:

1. $Y^{PA}$ (Pseudoscalar-Axial vector mix): $\Gamma \approx 391.1 \pm 21.4$ MeV.
2. $Y^{AA}$ (Axial-vector-Axial vector): $\Gamma \approx 27.8 \pm 2.3$ MeV.
3. $Y^{\tilde{A}V}$ (Tensor-Vector mix): $\Gamma \approx 266.7 \pm 23.9$ MeV.
4. $Y^{S\tilde{V}}$ (Scalar-Tensor mix): $\Gamma \approx 61.5 \pm 7.3$ MeV.

Comparison with Experiment:

• The experimental width of $Y(4660)$ is reported as $48 \pm 15 \pm 3$ MeV (Belle) and an average of $55 \pm 9$ MeV (PDG).
• The predictions for $Y^{PA}$, $Y^{AA}$, and $Y^{\tilde{A}V}$ are either too large or too small to match the experimental data.
• The prediction for $Y^{S\tilde{V}}$ ($61.5 \pm 7.3$ MeV) is in excellent agreement with the experimental values.

Conclusion on Structure:
The results strongly support the assignment of the $Y(4660)$ as a $[sc]_S[\bar{s}\bar{c}]_{\tilde{V}} - [sc]_{\tilde{V}}[\bar{s}\bar{c}]_S$ type tetraquark state (a mixture of scalar and tensor diquarks). The other configurations are disfavored.

5. Significance

• Resolution of the $Y(4660)$ Puzzle: The paper moves beyond mass spectroscopy to use decay dynamics as a discriminator, successfully identifying the most likely internal structure of the $Y(4660)$.
• Validation of Tetraquark Hypothesis: The agreement between the calculated width and experimental data reinforces the interpretation of $Y(4660)$ as a compact tetraquark state rather than a molecule or conventional charmonium.
• Future Guidance: The calculated partial widths for various channels (e.g., $D_s \bar{D}_s$, $D_s^* \bar{D}_s^*$, $J/\psi \phi$) provide specific predictions for future experimental searches at facilities like BESIII and Belle II to verify these decay modes.
• Methodological Robustness: The work demonstrates the efficacy of QCD sum rules in handling complex multi-body decays and distinguishing between competing exotic hadron models, even in the presence of model-dependent continuum parameters.