Diquark-Antidiquark Model: Exotic Tetraquarks

• The diquark–antidiquark model is a framework that describes exotic tetraquark states as bound diquark and antidiquark clusters with distinct color configurations.
• It employs effective Hamiltonians, spin couplings, and numerical methods like the Lippmann–Schwinger equation to predict mass spectra and decay patterns.
• Its predictions, including mass splittings and branching ratios, align with experimental observations of exotic states like X(3872) and Y(10890).

The diquark–antidiquark model is a phenomenological approach for understanding hadrons that cannot be interpreted within the traditional quark–antiquark (meson) or three–quark (baryon) configurations. In this model, two constituent quarks form a color–antitriplet diquark, which serves as the effective building block, and binds to an antidiquark, giving rise to a compact tetraquark state. This framework is motivated by experimental observations of exotic hadrons that exhibit properties (quantum numbers, decay patterns, masses) inconsistent with conventional quark model assignments. The diquark–antidiquark model provides a systematic methodology for calculating the mass spectra, decay channels, and quantum numbers of exotic mesons, and has been extensively applied to interpret a range of experimentally observed candidates, such as X(3872), Y(10890), Z_c(3900), and their analogues in the bottomonium sector.

1. Theoretical Foundations and Quark Model Formalism

The diquark–antidiquark model is an extension of the constituent quark model in which tetraquark states are structured as color–correlated clusters: $[q_1q_2]_{\bar{3}_c} \, [\bar{q}_3\bar{q}_4]_{3_c}$ Here, each diquark or antidiquark forms a color antitriplet or triplet and may exhibit various flavor and spin configurations. For light quarks, the most attractive configuration is antisymmetric in color, flavor, and spin (color $\bar{3}$, flavor %%%%1%%%%, spin 0). For heavy–light diquarks (such as $[cq]$), both scalar ($S=0$) and axial–vector ($S=1$) diquark configurations can be relevant due to the suppressed heavy–quark spin interactions (Maiani, 2014).

The effective Hamiltonian for the diquark–antidiquark tetraquark is typically written as: $$H = 2 m_{\mathcal{Q}} + \sum_{i<j} 2\kappa_{ij} \,\mathbf{S}_i \cdot \mathbf{S}_j + H_{\text{SL}} + H_{\text{LL}}$$ where $m_{\mathcal{Q}}$ is the diquark mass (which can be fixed empirically, e.g., using the X(3872) mass for $[cq]$), $\kappa_{ij}$ are spin–spin couplings extracted from the meson and baryon spectra, and $H_{\text{SL}}$, $H_{\text{LL}}$ denote spin–orbit and orbital contributions, respectively. The values of $\kappa_{ij}$ depend on the flavor content and color configuration ($$\kappa_{qq}^{\bar{3}_c} = \frac12 \kappa_{q\bar{q}}^{1_c}$$), reflecting different short-range QCD dynamics (Rehman, 2011, Maiani, 2014).

The basis states are constructed as

$|J; S_{\mathcal{Q}}, S_{\bar{\mathcal{Q}}}\rangle$

and the eigenvalues are computed by diagonalizing the Hamiltonian in this basis, incorporating Clebsch–Gordan coefficients for spin couplings (Rehman, 2011).

2. Spectroscopy: Assignments and Mass Predictions

The model naturally predicts multiplets of tetraquark states grouped by their total spin and parity. For example, the X(3872) has been identified as an $S$–wave tetraquark with hidden charm content, assigned as: $$X(3872): \quad [cq]_{S=1} \left[\bar{c}\bar{q}\right]_{S=0},\quad J^{PC} = 1^{++}$$ with $q = u$ or $d$. The calculated mass,

$M(1^{++}) \simeq 3.872~\mathrm{GeV}$

is in excellent agreement with experiment (Rehman, 2011). In the hidden–bottom sector, Y(10890) is assigned as a $P$–wave tetraquark: $$Y(10890): \quad \left([bq]_{S=0} [\bar{b}\bar{q}]_{S=0}\right)_{\mathrm{P}\text{-wave}},\quad J^{PC}=1^{--}$$ which helps explain its anomalous decay patterns (Rehman, 2011).

The spectrum is populated by other calculated multiplets, e.g., $J^{PC} = 1^{+–}$ (Z_c(3900), Z_b(10610)), $0^{++}$, $2^{++}$, and their analogues in the $b$–sector (Maiani, 2014, Zhu, 2016). The model predicts mass splittings and multiplet structures due to explicit spin–spin and spin–orbit couplings, offering a physics-based explanation for the observed mass hierarchy and quantum number assignments.

The predictions are robust across different computational approaches—nonrelativistic constituent models (Rehman, 2011, Maiani, 2014), variational techniques (Bedolla et al., 2019), and lattice simulations (Padmanath et al., 2015).

3. Decay Mechanisms and Selection Rules

Diquark–antidiquark tetraquarks exhibit characteristic decay patterns driven by quark recombination and rearrangement. For instance, the decay of X(3872) proceeds dominantly via quark rearrangement: $$X(3872) \to J/\psi \,\pi^+\pi^- \,\,\text{(via intermediate } \rho^0),\quad X(3872) \to J/\psi \,\pi^+\pi^-\pi^0 \,\,\text{(via } \omega)$$ The decay amplitude is modeled by a contact interaction of strength $A$, with width formula (Rehman, 2011): $$\frac{d\Gamma(X \to J/\psi+f)}{ds} = \frac{2\,x_{l,V}\,|A|^2\,B(V\to f)}{8\pi M_X^2}\, \frac{M_V\Gamma_V}{\pi} \frac{p(s)}{(s - M_V^2)^2 + (M_V\Gamma_V)^2}$$ where $$p(s) = \frac{\sqrt{\lambda(M_X^2, M_{J/\psi}^2, s)}}{2M_X}$$ and $\lambda$ is the standard Källén function.

Radiative decays are related to hadronic decays through Vector Meson Dominance (VMD). For example, for the radiative decay: $X(3872) \to J/\psi\,\gamma$ the amplitude is given by (Rehman, 2011): $$\langle J/\psi\,\gamma\,|\,X\rangle = \frac{f_\rho\,A}{m_\rho^2}$$ where $f_\rho$ is the $\rho$ meson decay constant and $A$ is the hadronic amplitude from $X \to J/\psi\,\rho$. The predicted branching fraction ratios, e.g.,

$$\frac{\Gamma(X\to J/\psi\,\gamma)}{\Gamma(X\to J/\psi\,\pi^+\pi^-)}$$

are consistent with experimental measurements and support the tetraquark nature of these states (Rehman, 2011).

The model also predicts selection rules arising from spin–parity conservation and heavy–quark spin symmetry. Decays that preserve heavy quark spin (such as $S_{c\bar{c}}\to 1$) are favored, while those requiring spin flips are suppressed (Maiani, 2014).

4. Numerical Methods and Computational Strategies

The mass eigenvalues and binding energies of diquark–antidiquark systems can be computed via the Lippmann–Schwinger equation for an effective two–body system: $|\Psi\rangle = G_0\,V\,|\Psi\rangle$ Numerically, this is solved by discretizing the integral equation (for example, via Gauss–Legendre quadrature), casting it as an eigenvalue problem for the kernel $K(E_b)$, and finding the energy $E_b$ at which the largest eigenvalue equals 1 (Monemzadeh et al., 2015). The total mass is then

$M = m_1 + m_2 + E_b$

where $m_1$, $m_2$ are constituent diquark and antidiquark masses.

The confining and Coulomb-like potentials take the forms

$$V(r) = V_{\text{coul}}(r) + V_{\text{conf}}(r),\qquad V_{\text{coul}}(r) = -\alpha_s\,\frac{F_1(r)F_2(r)}{r},\qquad V_{\text{conf}}(r) = Ar + B$$

with form factors $F(r)=1-e^{-\xi r - \zeta r^2}$ accounting for the finite spatial size of the diquark (Monemzadeh et al., 2015).

This approach achieves tight agreement with experimental candidates: for the $1^{++}$ state (relevant for X(3872)), binding energies and predicted masses are within a few MeV of the observed value.

5. Lattice QCD and Operator Overlaps

Lattice QCD studies provide nonperturbative insights into the Fock space decomposition of tetraquark candidates. Interpolating operators considered can be constructed as either diquark–antidiquark–type or meson–meson–type, with formal equivalence up to Fierz transformations (Padmanath et al., 2015). The interpolator for the color–antisymmetric diquark–antidiquark is: $$O_{3_c}^{4q} = \left[\bar{c}C\gamma_5\bar{u}\right]_{\bar{3}_c}\left[c\gamma_i C u\right]_{3_c} + \left[\bar{c}C\gamma_i\bar{u}\right]_{\bar{3}_c}\left[c\gamma_5C u\right]_{3_c} + \{u\to d\}$$ However, the Fierz rearrangement demonstrates that this operator is a linear combination of two-meson operators (e.g., $D\bar{D}^*$, $J/\psi\,\omega$) and thus their spectra are entangled on the lattice. Lattice studies find that the X(3872) candidate emerges only when both conventional $\bar{c}c$ and meson–meson fields are included; diquark–antidiquark operators alone do not isolate the state (Padmanath et al., 2015). This suggests that the physical state is a mixture, with diquark–antidiquark configurations contributing as components in the full Fock space.

6. Phenomenological and Experimental Implications

The diquark–antidiquark picture provides a systematic account of experimental facts:

• Quantitative mass predictions and branching ratios for experimentally observed exotics, e.g., X(3872), Y(10890), Z_c(3900), Z_b(10610), etc.
• Natural explanations for charged exotics: states like Z_c(3900) have minimal $c\bar{c}$ content and require an interpretation beyond the $q\bar{q}$ model (Maiani, 2014).
• Large production rates and compactness at high-energy collisions support the tetraquark assignment over loosely bound molecular scenarios.
• The internal structure and compactness match observed decay widths and production cross sections.
• Predictions for radiative transitions consistent with measured ratios, utilizing VMD to relate electromagnetic and strong decays (Rehman, 2011).

Limitations arise due to parameter uncertainties (notably the extraction of $\kappa_{ij}$ from hadron data), possible mixing with conventional $q\bar{q}$ or meson–meson molecular components, and the dependence of multiplet completeness on unobserved partner states (Maiani, 2014). Nevertheless, the model remains predictive and consistent across multiple sectors, motivating ongoing and future collider searches.

7. Summary of Model Features and Outlook

The diquark–antidiquark model, based on well–established QCD color dynamics and constituent quark symmetries, provides a unified framework for describing exotic mesons that evade the $q\bar{q}$ structure. Its formalism yields concrete predictions for the mass spectrum, multiplet organization, quantum numbers, and decay patterns of observed and yet–unobserved tetraquark states through effective Hamiltonian methods, numerical solutions to the Lippmann–Schwinger or Schrödinger equations, and robust matching to experimental data. The consistency of the model’s predictions—both qualitatively and quantitatively—with the properties of established exotics underlines its central role in the current understanding of multiquark hadron spectroscopy. Experimental advances, higher statistics flavor factory data, and higher–precision lattice studies will further test and refine the parameters and reach of this framework (Rehman, 2011, Maiani, 2014, Monemzadeh et al., 2015, Padmanath et al., 2015).