Quark Delocalization Color Screening Model

• QDCSM is a constituent quark model that uses quark delocalization and a color-screened confinement potential to describe multiquark dynamics.
• The model integrates a detailed Hamiltonian, variational optimization of delocalized quark orbitals, and resonating-group methods to capture intermediate-range attractions.
• It predicts exotic states such as dibaryons, tetraquarks, and pentaquarks with experimental signatures in scattering experiments and heavy-ion collisions.

The Quark Delocalization Color Screening Model (QDCSM) is a constituent quark model framework formulated to describe the dynamics and interactions of multiquark systems, including baryon-baryon, baryon-meson, and multi-quark exotic states. Its defining features are the explicit use of quark delocalization to generate dynamical mixing between clusters, and a phenomenological color-screened confinement potential that simulates channel-coupling to hidden-color configurations. QDCSM has been systematically applied to systems ranging from deuteron and dibaryons to heavy tetraquarks and pentaquarks, providing a unified description of intermediate-range attractions and molecular formation in hadronic matter (Huang et al., 2015).

1. Hamiltonian Structure and Color-Screened Confinement

The Hamiltonian in QDCSM encompasses nonrelativistic kinetic energy, one-gluon-exchange (OGE) interactions, Goldstone-boson exchanges where relevant, and color-confinement with explicit screening between clusters. For an $n$-quark system ($n=4,5,6$ for tetraquarks, pentaquarks, or dibaryons), the generic form is: $$H = \sum_{i=1}^n \left( m_i + \frac{\mathbf{p}_i^2}{2m_i} \right) - T_{\rm CM} + \sum_{i<j} \left[ V^{\rm CON}_{ij} + V^{\rm OGE}_{ij} + V^{\chi}_{ij} \right].$$ Here, $T_{\rm CM}$ subtracts the center-of-mass motion; $V^{\rm OGE}$ encodes the color-Coulomb, color-magnetic, and tensor components of gluon exchange; $V^{\chi}$ is the Goldstone-boson (e.g., $\pi, K, \eta$) exchange between light quarks; and $V^{\rm CON}$ is the color-screened confining potential: $$V^{\rm CON}_{ij} = -a_c\,\lambda^c_i \cdot \lambda^c_j \begin{cases} r_{ij}^2 + V_0, & i,j \text{ in the same cluster}, \ \frac{1-e^{-\mu_{ij} r_{ij}^2}}{\mu_{ij}} + V_0, & i,j \text{ in different clusters}. \end{cases}$$ The parameter $\mu_{ij}$ is flavor-dependent and fitted to $NN$, $NY$, or deuteron data, typically $\mu_{uu}=0.45$ fm$^{-2}$, $\mu_{us}=0.19$ fm$^{-2}$, $\mu_{ss}=0.08$ fm$^{-2}$ for light quarks. For heavy-flavor pairs, much smaller values are used (e.g., $\mu_{cc}=0.01$ fm$^{-2}$) (Huang et al., 2015, Liu et al., 2023, Yan et al., 2023).

Color screening leads to a saturation of the inter-cluster confinement at large separations, simulating the physical effect of hidden-color channel-coupling without explicitly adding color-octet basis states. At short range, the screened form approximates a quadratic potential; at large range, it softens, facilitating quark exchange and dynamic channel coupling (Zhu et al., 2015).

2. Quark Delocalization Mechanism

Quark delocalization is implemented by constructing single-particle orbitals as linear combinations of Gaussians centered on the respective cluster centers: $$\psi_\alpha(\mathbf{r}; \mathbf{S}, \epsilon) = \frac{\phi_\alpha(\mathbf{r}; +\mathbf{S}) + \epsilon\, \phi_\alpha(\mathbf{r}; -\mathbf{S})}{N(\epsilon)},$$ where $N(\epsilon)$ is the normalization and the cluster separation is $\mathbf{S}$.

The delocalization parameter $\epsilon(\mathbf{S})$ is determined variationally at each cluster separation by minimizing the total system energy. This mechanism allows quark wave functions to “spread” between clusters, dynamically enabling overlap between color-singlet and hidden-color components. This is analogous to electron delocalization in covalent bonding, and it generates intermediate-range attraction in the baryon-baryon potential (Huang et al., 2015, Zhu et al., 2015).

In the refined QDCSM, full configuration mixing is implemented via symmetry-adapted bases—covering all clusterings, e.g., $\vert L^6\rangle$, $\vert L^5R\rangle$, $\dots$, $\vert R^6\rangle$—and solving the generalized eigenvalue problem for all allowed Young-tableau symmetries, further validating the physical efficacy of the $\epsilon(s)$ parametrization (Zhu et al., 2015).

3. Resonating-Group Method and Channel Coupling

The QDCSM adopts the Resonating-Group Method (RGM) to construct antisymmetrized multiquark wave functions as superpositions across all relevant cluster partitions and spin-flavor-color symmetries: $$\Psi = \sum_\alpha \int d\mathbf{R}\; \mathcal{A}\left[ \Phi_\alpha\, \chi_\alpha(\mathbf{R}) \right].$$ $\mathcal{A}$ is the antisymmetrizer over identical quarks. $\Phi_\alpha$ represents internal cluster quantum numbers, and $\chi_\alpha(\mathbf{R})$ is the inter-cluster wave function, expanded in a Gaussian basis up to a cutoff radius.

Coupled-channel dynamics are essential: physical ($q^3$–$q^3$ or baryon-meson) color-singlet channels are included, along with hidden-color configurations via color screening (or explicitly in some chiral-quark model analyses for benchmarking). The resulting multi-channel RGM integro-differential equations are reduced to generalized algebraic eigenvalue problems for bound-state and scattering calculations (Huang et al., 2015).

4. Physical Predictions and Systematics

QDCSM yields effective baryon-baryon or baryon-meson potentials that exhibit:

• Intermediate-range attraction only when color-screening (hidden-color coupling) is included; with pure quadratic confinement, the potentials are strongly repulsive with no intermediate pocket (Zhu et al., 2015).
• Dynamically generated weakly bound or resonance states in systems such as $N\Omega$, $ND$, $N\bar D$, $qqc\bar c c$ pentaquarks, $cc\bar q\bar s$ and $bb\bar b\bar b$ tetraquarks, and $^3_{\Lambda_c} \rm H$ (Huang et al., 2015, Zhao et al., 2016, Liu et al., 2023, Yan et al., 2023, Liu et al., 2023, Wu et al., 2023).
• A robust $N\Omega$ weakly bound state with binding $B=-5.2$ MeV, scattering length $a_0=2.80$ fm, effective range $r_0=0.58$ fm when full color-singlet channel coupling is included. In chiral quark models, explicit hidden-color channels are needed to achieve comparable binding, showing that screening in QDCSM encapsulates the hidden-color effect (Huang et al., 2015).
• For the ND system, a weakly bound $ND$ in the $I=0$, $J^P=1/2^-$ channel and a $ND^*$ molecular state matching $\Lambda_c(2940)^+$, with similar analogues in the $NB$ system (Zhao et al., 2016).
• Multiple narrow resonances and bound states in five- and four-quark systems, sensitive to channel coupling and delocalization; multi-channel treatments are critical to obtain physical binding (Liu et al., 2023, Liu et al., 2022, Liu et al., 2023).

5. Mechanisms of Binding: Delocalization and Screening

QDCSM identifies two dominant mechanisms for multiquark binding:

• Kinetic-energy reduction enabled by quark delocalization, which lowers the repulsion at intermediate distances.
• Effective intermediate-range attraction generated by color-screening, simulating mixing with hidden-color (color-octet) configurations, without explicitly enlarging the Hilbert space.

In contrast, chiral quark models attribute intermediate-range attraction primarily to scalar-meson exchanges and explicit hidden-color channel coupling; QDCSM shows (by quantitative comparison) that its combined delocalization plus screening reproduces the same effects, confirming the model's efficiency and validity (Huang et al., 2015, Zhu et al., 2015).

6. Model Parameters and Calibration

QDCSM employs parameter sets fitted to spectra of single-hadron and hadronic-cluster systems. Typical values for key parameters are:

Parameter	Light-Quark Value	Heavy-Quark Extensions
$m_u, m_d$ (MeV)	313
$m_s$ (MeV)	573/633
$m_c$ (MeV)	1675–1788	$m_b=5086$
$b$ (fm)	0.5–0.6	$b_{cc}=0.20$, $b_{bb}=0.126$
$a_c$	25–101 MeV fm$^{-2}$
$\mu_{qq}$	0.45 fm$^{-2}$	$\mu_{cc}=0.01$, $\mu_{bb}=0.001$ f$^{-2}$

All parameters are anchored by fits to $NN$, $N\Lambda$, or deuteron data and meson/baryon spectra. Screening parameters are constrained by light-sector fits and extended to heavy sectors via geometric means ($\mu_{qc}^2=\mu_{qq}\mu_{cc}$) (Zhu et al., 2015, Liu et al., 2023, Yan et al., 2023, Liu et al., 2023).

7. Experimental Signatures and Applications

QDCSM predictions have specific experimental consequences:

• Weakly bound or near-threshold resonances in $N\Omega$ manifest as enhancements in $N$–$\Omega$ correlation functions in heavy-ion collision experiments (RHIC, LHC).
• D-wave $\Lambda\Xi$ and $N\Omega$ correlation measurements provide further constraints (Huang et al., 2015).
• Predictions of exotic states (e.g., $ND$, $N\bar D$, $cc\bar q\bar s$, $qqc\bar c c$) inform targeted searches at hadron beam facilities and heavy-flavor experiments.
• The model reproduces known molecular states ($\Sigma_c(2800)$, $\Lambda_c(2940)^+$) and predicts analogues in the bottom sector (Zhao et al., 2016).
• QDCSM is extensible to hypernuclear and heavy multiquark systems, where it is used to compute effective potentials for three-body and higher systems (e.g., $^3_{\Lambda_c}$H) (Wu et al., 2023).

The approach provides a unified quantitative scheme for multiquark binding and resonance formation, supporting ongoing and future studies of exotic hadronic matter (Huang et al., 2015, Yan et al., 2023, Liu et al., 2023, Liu et al., 2023, Liu et al., 2022, Jin et al., 2020, Wu et al., 2023, Zhu et al., 2015, Zhao et al., 2016).