Study of $1^{--}$ P wave charmoniumlike and bottomoniumlike tetraquark spectroscopy

This paper calculates the masses and decay widths of $1^{--}$ P-wave charmonium-like and bottomonium-like tetraquark states using a constituent quark model, suggesting that the exotic Y states $\psi(4230)$, $\psi(4360)$, $\psi(4660)$, and $\Upsilon(10753)$ can be tentatively assigned as such tetraquark configurations.

The Gist

Imagine the universe is a giant, cosmic LEGO set. For decades, physicists believed they understood the basic rules: you could build stable structures using just two pieces (a quark and an antiquark, called mesons) or three pieces (three quarks, called baryons). These were the standard "bricks" of matter.

But recently, scientists have started finding strange, exotic structures that don't fit these simple rules. They are seeing "tetraquarks"—structures made of four pieces (two quarks and two antiquarks) stuck together. It's like finding a LEGO tower that somehow stays standing even though it's built with four pieces instead of the usual two or three.

This paper is a theoretical investigation into a specific family of these exotic four-piece towers. Here is the breakdown in simple terms:

1. The Mystery of the "Y" States

In the world of particle physics, there are some particles called "Y states" (like $Y(4230)$, $Y(4360)$, etc.). These are like mysterious guests at a party. We know they exist because we see them in experiments, but they act weirdly:

• They are too heavy to be normal two-piece particles.
• They decay (fall apart) in ways that standard physics can't easily explain.
• They seem to be made of "hidden charm" or "hidden bottom" (heavy quarks hidden inside a mix of lighter ones).

Scientists have been arguing: Are these just normal particles acting weird? Or are they actually these new four-piece tetraquarks?

2. The "Recipe Book" (The Model)

The authors of this paper decided to build a "recipe book" to predict what these tetraquarks should look like.

• The Ingredients: They used a "Constituent Quark Model." Think of this as a cooking recipe where they know the weight and flavor of the basic ingredients (quarks).
• The Cooking Method: They used a specific set of rules (mathematical potentials) to simulate how these four ingredients stick together. They didn't invent new rules; they used a recipe that had already been proven to work perfectly for normal particles (like the famous $J/\psi$ particle).
• The Goal: They wanted to calculate the mass (weight) and decay patterns (how they break apart) of these four-piece towers if they were indeed real.

3. The Prediction: "P-Wave" Towers

The paper focuses on a specific type of tetraquark called 1-- P-wave.

• Analogy: Imagine a spinning top. A "P-wave" means the pieces aren't just sitting still; they are orbiting each other in a specific, slightly higher-energy dance.
• The authors calculated that the lightest of these four-piece towers should weigh about 4.15 GeV (a unit of energy/mass).

4. The Match-Up: Theory vs. Reality

This is the exciting part. The authors took their calculated "recipe" and compared it to the real "guests" (the Y states) found in experiments.

• The 4.23 GeV Guest ($Y(4230)$): The paper suggests this mysterious particle is likely a four-piece tetraquark. The weight calculated by their recipe matches the weight seen in the lab almost perfectly.
• The 4.36 GeV Region: This area is crowded. The paper suggests there might be multiple different tetraquark towers hiding here, explaining why experiments see so many different peaks. They propose that what we call $Y(4360)$ might actually be a mix of a few different tetraquark states.
• The 4.66 GeV Guest ($Y(4660)$): This one also fits the tetraquark description well.
• The Bottomonium Twin ($Y(10753)$): Just as there are heavy "charm" particles, there are even heavier "bottom" particles. The paper predicts that the $Y(10753)$ is the heavy-bottom version of these tetraquarks.

5. How Do They Break? (Decay)

The paper also predicts how these towers fall apart.

• Analogy: If you drop a LEGO tower, it breaks into specific smaller piles.
• The authors calculated that if these are tetraquarks, they should break apart into specific pairs of particles (like an omega meson and a chi meson).
• They found that for some of these states, the "break-up" pattern matches what experiments are seeing. For example, the $Y(4230)$ seems to break into specific pairs exactly as the tetraquark model predicts.

The Big Conclusion

The paper argues that the "mystery guests" (the Y states) are likely not just weird versions of normal particles. Instead, they are strong evidence for the existence of compact tetraquarks—stable, four-piece structures made of quarks.

In a nutshell:
The universe has been hiding a secret: quarks can stick together in groups of four, not just two or three. This paper provides a mathematical "blueprint" showing that the strange, heavy particles we've been seeing in labs for the last 20 years are likely these four-piece structures, finally giving them a proper name and a place in the family tree of matter.

Technical Summary

1. Problem Statement

The nature of exotic hadrons, specifically the "Y states" (charmonium-like and bottomonium-like resonances with quantum numbers $J^{PC} = 1^{--}$), remains a significant challenge in Quantum Chromodynamics (QCD). While conventional mesons are described as quark-antiquark ($q\bar{q}$) pairs, experimental discoveries such as $Y(4230)$, $Y(4360)$, $Y(4660)$, and $\Upsilon(10753)$ exhibit properties (masses, decay patterns, and production mechanisms) that cannot be fully accommodated within the standard quarkonium picture, even when including $S-D$ wave mixing.

The specific problem addressed is whether these $1^{--}$ states can be interpreted as compact P-wave tetraquark states (bound states of two quarks and two antiquarks, $qQ\bar{q}\bar{Q}$). Previous studies struggled to fit these states into conventional heavy quarkonium models, necessitating a rigorous investigation of the tetraquark hypothesis, including mass spectrum predictions and strong decay characteristics.

2. Methodology

The authors employ a Constituent Quark Model (CQM) to calculate the mass spectra and decay properties of $1^{--}$ P-wave tetraquarks.

• Hamiltonian and Potential:

  • The system is treated non-relativistically using a Hamiltonian $H = H_0 + H_{so}$.
  • Central Potential ($V_{cen}$): A Cornell-like potential is used, comprising a linear confinement term ($A_{ij}r_{ij}$) and a Coulomb-like term ($-B_{ij}/r_{ij}$).
  • Spin-Dependent Interactions: The model includes the spin-spin interaction ($V_{ss}$) derived from one-gluon exchange and the spin-orbit interaction ($V_{so}$) treated as a perturbation.
  • Parameters: Model parameters (constituent quark masses $m_u, m_c, m_b$ and coupling constants) are predetermined and fixed by reproducing the mass spectra of low-lying conventional S- and P-wave mesons (light, charmed, and bottom mesons). The hyperfine coefficient $\sigma_0$ is tuned to 0.7 to match experimental data.

• Wave Function Construction:

  • Color Structure: Tetraquark states are constructed as color singlets. The model considers two primary color configurations: the color triplet-antitriplet ($\bar{3}_c \otimes 3_c$) and the color sextet-antisextet ($6_c \otimes \bar{6}_c$).
  • Spin and Spatial: The basis includes $S=0$ and $S=2$ spin configurations coupled with orbital angular momentum $L=1$ (P-wave). The spatial wave functions are constructed using harmonic oscillator (HO) basis functions in Jacobi coordinates.
  • Mixing: The Hamiltonian includes cross-terms from color and spin operators, leading to the mixing of different color-spin configurations (e.g., $\bar{3}_c \otimes 3_c$ mixing with $6_c \otimes \bar{6}_c$).

• Decay Mechanism:

  • Two-body strong decays are calculated using the quark rearrangement mechanism.
  • The decay amplitude is approximated by the overlap of the initial tetraquark wave function with the final two-meson wave functions (e.g., $\omega\chi_{cJ}$, $\eta J/\psi$, $\omega\chi_{bJ}$).
  • Branching ratios are estimated based on the squared modulus of the color-spin overlap factors ($|T_{cs}|^2$), assuming spatial contributions are similar across channels.

3. Key Contributions

• Systematic Spectroscopy: The paper provides a comprehensive mass spectrum for $1^{--}$ P-wave compact tetraquarks in both the charmonium-like ($Y_c$) and bottomonium-like ($Y_b$) sectors, covering a wide range of excitation levels ($E_1$ to $E_{16}$ for $Y_c$).
• Configuration Mixing: It explicitly calculates the mixing between different color configurations ($\bar{3}_c \otimes 3_c$ and $6_c \otimes \bar{6}_c$) and spin states ($S=0$ and $S=2$), demonstrating that $S=0$ and $S=2$ sectors do not mix, but color configurations within $S=0$ do.
• Decay Width Predictions: The study calculates theoretical branching ratios for specific decay channels, distinguishing between states that decay strongly into $\omega\chi_{cJ}$ versus those that might decay into $\eta J/\psi$ (which are found to be suppressed for $S=0$ states).
• Stability Analysis: The authors employ the real-scaling method to verify the stability of the calculated eigenstates against basis truncation, ensuring that the predicted states are genuine bound states rather than artifacts of the continuum or basis choice.

4. Results

The theoretical calculations yield the following key findings:

• Mass Spectrum:

  • The lowest $1^{--}$ tetraquark state is predicted at approximately 4.15 GeV.
  • A dense spectrum of states exists between 4.2 GeV and 4.6 GeV for charmonium-like tetraquarks, and between 10.5 GeV and 10.8 GeV for bottomonium-like tetraquarks.
  • $Y(4230)$: Assigned to the $E_2$ state ($S=0$) with a theoretical mass of 4220 MeV. This aligns with experimental observations in $\pi^+\pi^-J/\psi$, $K^+K^-J/\psi$, $\omega\chi_{c0}$, and $\pi^+\pi^-h_c$ channels.
  • $Y(4360)$ Region: The authors suggest this region contains multiple distinct states:
    • $Y(4300)$ and $Y(4340)$ are assigned to $S=2$ states (masses ~4285 MeV and 4351 MeV).
    • $Y(4390)$ (observed in $\pi^+\pi^-h_c$) is assigned to an $S=0$ state with a large branching ratio to $\omega\chi_{c0}$ (theoretical mass ~4390 MeV).
    • $Y'(4390)$ (observed in $\pi^+\pi^-J/\psi$) is assigned to a different state with a small branching ratio to $\omega\chi_{c0}$ (theoretical mass ~4409 MeV).
  • $Y(4660)$: Assigned to an $S=0$ state with a theoretical mass of 4618 MeV, consistent with its observation in $\pi^+\pi^-J/\psi$.
  • $\Upsilon(10753)$: Assigned to the $E_5$ state ($S=0$) with a theoretical mass of 10754 MeV, consistent with observations in $\pi^+\pi^-\Upsilon(nS)$ and $\omega\chi_{bJ}$ channels.
  • $Y(4484)$ and $Y(4544)$: Tentatively assigned to tetraquark states at 4499 MeV and 4567 MeV, respectively.

• Decay Patterns:

  • For $S=0$ charmonium-like tetraquarks, the decay into $\eta J/\psi$ is strongly suppressed (spin-color factor $\approx 0$). This implies that the experimentally observed $Y(4230)$ and $Y(4360)$ in the $\eta J/\psi$ channel likely correspond to different structures (e.g., molecular states or hybrids) rather than the compact $S=0$ tetraquarks predicted here.
  • The model predicts strong decays into $\omega\chi_{cJ}$ and $\omega\chi_{bJ}$ for the assigned states, supporting the experimental observation of $\Upsilon(10753) \to \omega\chi_{bJ}$.

5. Significance

• Resolution of Exotic States: The study provides a unified framework where several controversial $Y$ states ($Y(4230)$, $Y(4360)$, $Y(4660)$, $\Upsilon(10753)$) can be naturally interpreted as compact P-wave tetraquarks, resolving discrepancies found in conventional quarkonium models.
• Prediction of Multiplicity: The results suggest that the broad mass regions around 4.36 GeV and 4.66 GeV likely contain multiple overlapping states rather than single resonances, explaining the complex line shapes observed in different decay channels.
• Experimental Guidance: The paper offers specific predictions for future experimental searches:
  • It suggests searching for the $Y(4390)$ state in the $\omega\chi_{c0}$ channel.
  • It clarifies why certain states appear in $\omega\chi_{cJ}$ but not $\eta J/\psi$, guiding the interpretation of future BESIII and Belle II data.
• Model Validation: By successfully reproducing the mass spectra of conventional mesons and then applying the same parameters to predict exotic tetraquark properties, the study strengthens the validity of the constituent quark model with Cornell-like potentials for describing multi-quark systems.

In conclusion, the paper argues that the $1^{--}$ P-wave compact tetraquark picture is a viable and necessary explanation for the observed spectrum of exotic heavy quarkonium-like states, offering a coherent assignment for the $Y$ family and the $\Upsilon(10753)$.