Systematic study of exotic $1^{-+}$ tetraquark spectroscopy

This paper employs a constituent quark model with a Cornell-like potential to predict the masses and decay characteristics of exotic $1^{-+}$ compact tetraquarks across light, charmonium-like, and fully-charm sectors, concluding that the observed $\eta_1(1855)$ state is unlikely to be a compact tetraquark.

The Gist

Imagine the universe is built from tiny, invisible Lego bricks called quarks. Usually, these bricks snap together in very predictable ways:

• Mesons are like simple pairs of bricks (one positive, one negative) holding hands.
• Baryons (like protons) are like sturdy little towers made of three bricks.

But physicists have been hunting for something stranger: Tetraquarks. These are exotic structures made of four bricks stuck together in a tight, compact cluster. Think of them not as a loose pile of four bricks, but as a single, complex 4D shape that doesn't fit the standard rules.

This paper is a "blueprint" study. The authors, a team of physicists from Thailand, used a sophisticated computer model to predict what these four-brick shapes would look like if they had a very specific, weird "personality" (quantum numbers) called $1^{-+}$.

Here is the breakdown of their findings, explained simply:

1. The "Weird" Rule

In the world of subatomic particles, there are strict rules about how they spin and how they behave when you look in a mirror (parity).

• Most particles follow a standard dance.
• The $1^{-+}$ state is the "forbidden dance." It's a move that a simple two-brick pair (a normal meson) literally cannot do.
• Because it's so weird, if we find a particle doing this dance, we know it must be something exotic, like a tetraquark, a hybrid (a brick with a glue-gluon attached), or a molecule (two bricks loosely stuck together).

2. The Three "Flavors" of Tetraquarks

The team calculated the "weight" (mass) and "personality" (decay patterns) of these exotic particles in three different scenarios, like building houses with different materials:

• The Light House (Light Quarks): Made of the lightest, most common bricks (up and down quarks).

  • Prediction: They predict the lightest version weighs about 1.9 GeV (roughly twice the weight of a proton).
  • The Mystery: There is a known particle called $\pi_1(2015)$ that weighs about 2.0 GeV. The authors think, "Hey, that's close! Maybe $\pi_1(2015)$ is actually this tetraquark we predicted."
  • The Rejection: There is another famous particle, $\eta_1(1855)$, found recently by the BESIII experiment. It weighs 1.85 GeV. The authors say, "Nope, this isn't our tetraquark." Why? Because their model predicts that a tetraquark of this type would refuse to break apart into the specific pieces ($\eta$ and $\eta'$) that $\eta_1(1855)$ was found decaying into. It's like a lock that doesn't fit the key.

• The Heavy House (Charmonium-like): Made of bricks containing "charm" quarks.

  • Prediction: These should weigh around 4.2 GeV.
  • The Hunt: No one has found a confirmed $1^{-+}$ particle here yet. The authors are telling experimentalists: "Look in the 4.2 GeV range, specifically looking for these particles to break apart into a specific pair of particles ($\eta$ and $\chi_{c1}$)."

• The Super-Heavy House (Fully Charm): Made entirely of four "charm" bricks.

  • Prediction: These are the heavyweights, weighing around 6.6 GeV.
  • The Hunt: The Large Hadron Collider (LHC) has seen some heavy particles here, but they seem to be different shapes (like $2^{++}$). The authors suggest looking specifically for a $1^{-+}$ shape in the 6.6 to 7.1 GeV range, which would likely break apart into a specific pair of charm-particles ($\eta_c$ and $\chi_{c1}$).

3. How They Did It (The "Recipe")

The authors didn't just guess; they used a "constituent quark model."

• The Central Force: They used a "Cornell-like potential," which is like a mathematical spring. It pulls quarks together if they get too far apart (like a rubber band) but pushes them apart if they get too close.
• The Fine-Tuning: They added "hyperfine corrections" (spin-spin and spin-orbit coupling). Imagine the quarks are spinning tops. If they spin in the same direction, they interact differently than if they spin in opposite directions. This interaction slightly shifts the weight of the particle.
• The Rearrangement: To predict how these particles fall apart (decay), they used a "rearrangement mechanism." Imagine a tetraquark is a group of four friends holding hands. To break into two pairs (two mesons), they just have to let go of one hand and grab a new friend. The authors calculated how likely this "hand-swapping" is for different combinations.

4. The Big Takeaway

This paper is a roadmap for future experiments.

• It says: "Don't waste time looking for a tetraquark explanation for the $\eta_1(1855)$; it's probably something else (like a molecule or a hybrid)."
• It says: "Check out the $\pi_1(2015)$; it might be our tetraquark."
• It says: "Go to the 4.2 GeV and 6.6 GeV zones and look for these specific decay patterns. If you find them, you've found a new form of matter."

In a nutshell: The authors built a theoretical factory to manufacture "impossible" four-quark particles. They calculated their weights and how they would break apart, then compared their blueprints to the real-world particles we've already found. They found a few matches, ruled out a few others, and gave experimentalists a precise "Wanted" poster for the ones we haven't found yet.

Technical Summary

1. Problem Statement

The internal structure of hadrons beyond the conventional quark-antiquark ($q\bar{q}$) mesons and three-quark ($qqq$) baryons remains a central challenge in Quantum Chromodynamics (QCD). Specifically, states with exotic quantum numbers that cannot be formed by a simple $q\bar{q}$ pair are of great interest.

• The Exotic $1^{-+}$ State: In conventional mesons, parity $P = (-1)^{L+1}$ and charge conjugation $C = (-1)^{L+S}$. For a $1^{-+}$ state, these rules imply $L=1$ and $S=1$, which yields $C=(-1)^{1+1}=+1$. However, the observed $C$ is $+1$ (since $C=(-1)^{L+S}$ for $q\bar{q}$ is $(-1)^{1+1}=+1$? Wait, the paper states: "These rules allow... 1+-... but not 1-+." Let's re-verify the paper's logic.
  • Paper logic: $P = (-1)^{L+1}$, $C = (-1)^{L+S}$.
  • For $1^{-+}$: $P=-1 \implies (-1)^{L+1} = -1 \implies L$ is even. $C=+1 \implies (-1)^{L+S} = +1$. If $L$ is even, $S$ must be even (0). But $J=1$ requires $L \neq S$ or specific coupling.
  • Actually, the paper states: "Thus, a $1^{-+}$ state... is naturally considered to have a nonconventional structure."
  • The core problem is determining whether observed candidates like $\pi_1(1400)$, $\pi_1(1600)$, $\pi_1(2015)$, and the recently discovered $\eta_1(1855)$ are compact tetraquarks ($qq\bar{q}\bar{q}$), hybrid mesons ($q\bar{q}g$), or hadronic molecules.
• Current Status: While several candidates exist, their nature is inconclusive. Theoretical models often disagree on whether these states are hybrids or tetraquarks. There is a lack of systematic calculation for the mass spectrum and decay patterns of $1^{-+}$ compact tetraquarks across light, charmonium-like, and fully charm sectors to compare with experimental data.

2. Methodology

The authors employ a Constituent Quark Model (CQM) with a specific framework to calculate the mass spectrum and decay properties of compact tetraquarks.

• Hamiltonian:

  • Uses a non-relativistic Hamiltonian $H = H_0 + H_{so}$.
  • Central Potential ($V_0$): A Cornell-like potential ($V_0 = A_{ij}r_{ij} - B_{ij}/r_{ij}$) is used as the central interaction.
  • Hyperfine Corrections: Spin-spin ($V_{ss}$) and spin-orbit ($V_{so}$) interactions are derived from the Breit-Fermi interaction and treated as perturbations (hyperfine corrections).
  • Parameters: Model parameters (coupling constants $a, b, B_0, \sigma_0$ and constituent masses $m_u, m_c$) are fixed by reproducing the $S$- and $P$-wave spectra of light, charmed, and bottom mesons from previous works.

• Wave Function Construction:

  • Configuration: Tetraquarks are modeled as diquark-antidiquark ($[qq][\bar{q}\bar{q}]$) clusters.
  • Color: The total wave function must be a color singlet. The authors consider two color configurations:
    1. Color triplet-antitriplet: $\bar{3}_c \otimes 3_c$
    2. Color sextet-antisextet: $6_c \otimes \bar{6}_c$
  • Symmetry: The total wave function (color $\otimes$ spin $\otimes$ flavor $\otimes$ space) must be antisymmetric under the exchange of identical fermions.
  • Basis: The spatial wave functions are constructed using harmonic oscillator (HO) basis functions. The study uses a complete basis with principal quantum number $N \leq 13$ to ensure convergence.
  • Mixing: The Hamiltonian mixes different color-spin configurations (e.g., $\bar{3}_c \otimes 3_c$ and $6_c \otimes \bar{6}_c$) due to the color operator $\vec{\lambda}_i \cdot \vec{\lambda}_j$ and mass-dependent coefficients.

• Decay Mechanism:

  • Rearrangement Mechanism: Two-body strong decays are calculated assuming the tetraquark rearranges into two mesons ($Y \to M_1 M_2$).
  • Transition Amplitude: Calculated via the overlap of initial (tetraquark) and final (two-meson) color-spin-flavor wave functions ($T_{csf}$).
  • Partial Widths: Estimated as $\Gamma_\alpha \propto \Phi_\alpha(M_Y) |T_{csf}^{(\alpha)}|^2$, where $\Phi$ is the two-body phase space factor. The study focuses on relative decay ratios rather than absolute widths, assuming spatial overlaps are similar across channels.

3. Key Contributions

1. Systematic Spectroscopy: The paper provides the first systematic calculation of the mass spectrum for $1^{-+}$ compact tetraquarks in three distinct sectors:
   • Light ($qq\bar{q}\bar{q}$): Isospin $I=0$ and $I=1$.
   • Charmonium-like ($qc\bar{q}\bar{c}$): Isospin-blind due to lack of Pauli constraints between $q$ and $c$.
   • Fully Charm ($cc\bar{c}\bar{c}$): Pure heavy quark sector.
2. Decay Pattern Analysis: It calculates relative decay ratios for specific two-body channels (e.g., $\omega h_1$, $\eta f_1$, $\pi \chi_{c1}$, $J/\psi h_c$), providing "smoking gun" signatures to distinguish tetraquarks from hybrids or molecules.
3. Critical Evaluation of Candidates: The theoretical results are directly compared with experimental candidates ($\pi_1(2015)$ and $\eta_1(1855)$) to test the compact tetraquark hypothesis.

4. Key Results

A. Light Tetraquarks ($qq\bar{q}\bar{q}$)

• Mass Spectrum: The ground state $1^{-+}$ light tetraquarks are predicted at:
  • $I=0$: $\sim 1.9$ GeV (specifically $E_1 \approx 1961$ MeV).
  • $I=1$: $\sim 1.9$ GeV ($E_1 \approx 1884$ MeV).
  • Higher excited states ($E_2, E_3$) appear at 2.2–2.5 GeV.
• $\pi_1(2015)$: The observed state at $\sim 2.0$ GeV is consistent with the $E_2$ state ($I=1$) of the light tetraquark spectrum. The model predicts a large decay ratio to the $\pi f_1$ channel, supporting this assignment.
• $\eta_1(1855)$: The observed state at 1.855 GeV decaying into $\eta\eta'$ is ruled out as a compact tetraquark in this model.
  • Reasoning: The theoretical transition amplitude for $I=0$ $1^{-+}$ tetraquarks decaying into $\eta\eta'$ is nearly zero due to color-spin-flavor overlap suppression. Since $\eta_1(1855)$ is observed in this channel, it likely has a different structure (e.g., hybrid or molecule).

B. Charmonium-like Tetraquarks ($qc\bar{q}\bar{c}$)

• Mass Spectrum: The lowest $1^{-+}$ state is predicted at $\sim 4.2$ GeV.
• Decay Channels: The dominant decay channels are predicted to be $\eta \chi_{c1}$ and $\rho h_c$.
• Search Strategy: The authors suggest searching for these states in the $\eta \chi_{c1}$ channel around 4.2–4.3 GeV, as the decay ratio is relatively large.

C. Fully Charm Tetraquarks ($cc\bar{c}\bar{c}$)

• Mass Spectrum: The ground state is predicted at $\sim 6.6$ GeV.
• Decay Channels: The dominant decay is predicted to be $\eta_c \chi_{c1}$. The $J/\psi h_c$ channel is also significant for higher excited states ($E_4, E_5$).
• Search Strategy: Experimental searches should focus on the $\eta_c \chi_{c1}$ channel around 6.6 GeV and the $J/\psi h_c$ channel around 6.9–7.1 GeV.

5. Significance and Implications

• Exclusion of $\eta_1(1855)$ as a Tetraquark: The study provides strong theoretical evidence against the compact tetraquark interpretation of the BESIII discovery $\eta_1(1855)$, suggesting it is likely a hybrid meson or a hadronic molecule ($K\bar{K}_1$).
• Validation of $\pi_1(2015)$: The results support the assignment of $\pi_1(2015)$ as an excited $I=1$ compact tetraquark state.
• Guidance for Future Experiments:
  • BESIII/Belle II: Should prioritize searches for $1^{-+}$ states in the 4.2–4.3 GeV region (charmonium-like) via $\eta \chi_{c1}$ and in the 6.6–7.1 GeV region (fully charm) via $\eta_c \chi_{c1}$ and $J/\psi h_c$.
  • LHCb/ATLAS/CMS: The paper highlights the need for $S+P$ (S-wave + P-wave meson) final state analyses, as these are the dominant decay modes for compact tetraquarks, unlike the $S+S$ modes often favored by molecules.
• Theoretical Framework: The work demonstrates the utility of the constituent quark model with diquark-antidiquark clustering and color mixing in predicting exotic spectroscopy, offering a baseline for comparison with Lattice QCD and hybrid models.

In conclusion, this paper refines the search strategy for exotic hadrons by predicting specific mass windows and decay signatures for compact $1^{-+}$ tetraquarks, while simultaneously ruling out the tetraquark nature of the recently observed $\eta_1(1855)$.