Nonrelativistic Quark Model

• The nonrelativistic quark model is a Hamiltonian framework that uses effective interquark potentials to describe hadronic spectra and wavefunctions under the heavy quark approximation.
• It employs both analytical and numerical methods, such as the Schrödinger equation and Matrix Numerov algorithm, to compute bound state energies and state mixings.
• Model calibration against experimental data allows precise predictions for mesons, baryons, and diquark partners, validating quantum chromodynamics-inspired potential forms.

The nonrelativistic quark model (NRQM) provides a Hamiltonian framework for describing the spectra, wavefunctions, and observables of hadronic systems under the assumption that constituent quarks are sufficiently heavy and their velocities are much less than the speed of light. The NRQM typically employs a Schrödinger equation with effective interquark potentials inspired by QCD, local gauge invariance, and symmetry considerations. NRQM methodologies have enabled quantitative predictions for mesons, baryons, and multiquark ensembles, facilitating direct comparison with experimental data and alternative QCD-based approaches.

1. Hamiltonian Formalism and Effective Potentials

The standard NRQM constructs the Hamiltonian for two-body systems as

$$H = m_Q + m_{\bar{q}} + \frac{\mathbf{p}^2}{2\mu} + V(r) + H_{SS} + H_{LS}$$

where $\mu = m_Q m_{\bar{q}}/(m_Q + m_{\bar{q}})$ is the reduced mass, and $V(r)$ is a symmetry-preserving, contact-interaction-inspired effective potential (Gutiérrez-Guerrero et al., 2021). For multiquark systems, e.g., baryons, the Hamiltonian generalizes to include all pairwise interactions, potentially augmented by spin-dependent corrections and color factors.

The effective potential $V(r)$ can take various forms:

• Contact-interaction-inspired (e.g., via dressed-gluon propagators):

$$V_0(r) = \frac{1}{2\pi^2 m_G^2 r^3}\left[\lambda r \cos(\lambda r) - \Lambda r \cos(\Lambda r) + \sin(\Lambda r) - \sin(\lambda r)\right]$$

• Cornell-type (Coulomb plus linear confinement):

$$V(r) = -\frac{4}{3}\frac{\alpha_s}{r} + b r + C + V_{SS}(r)$$

with additional smeared spin-spin hyperfine terms and spin-orbit/tensor corrections for heavy quark systems (Albertus et al., 2014, Li et al., 2019).

Spin-dependent terms are parametrized as:

• Contact hyperfine interaction:

$$H_{SS} = \frac{32\pi \alpha_s}{9 m_Q m_{\bar{q}}} \mathbf{S}_Q \cdot \mathbf{S}_{\bar{q}} |\psi(0)|^2$$

• Spin-orbit:

$$H_{LS} = -\frac{1}{2\mu^2 r} \frac{dV_0}{dr} \langle \mathbf{L} \cdot \mathbf{S} \rangle$$

These potentials are systematically calibrated against ground-state masses of heavy and heavy-light mesons and baryons.

2. Solution Methods: Schrödinger Equation and Numerov Algorithm

NRQM applies both analytic and numerical techniques to solve the Schrödinger equation for hadronic bound states. Heavy quark systems typically employ the following forms: $$\left[ -\frac{1}{2\mu} \frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{2\mu r^2} + V(r) + m_Q + m_{\bar{q}} \right] U_{n\ell}(r) = M_{n\ell} U_{n\ell}(r)$$ where $U_{n\ell}(r) = r \psi_{n\ell}(r)$, with boundary conditions $U_0=0$, $U_{N+1}=0$ on a finite grid.

The Matrix Numerov method discretizes $r$, replacing derivatives with finite differences and assembling the problem as a generalized eigenvalue system involving tri-diagonal matrices. The resultant spectrum $M_{n\ell}$ includes S, P, D, F states and can incorporate spin-mixing (Gutiérrez-Guerrero et al., 2021, Li et al., 2019).

For three-body systems (baryons, diquarks), the wavefunction is expanded in Jacobi coordinates and solved using Rayleigh–Ritz variational principles or the Gaussian Expansion Method, enabling extraction of spectra and radial wavefunctions (Song et al., 2023).

3. Model Calibration and Spectroscopic Predictions

NRQM parameter calibration adheres strictly to experimental data and theoretical constraints. Quark masses and potential strengths are fixed by minimizing

$$\chi^2 = \sum_i \left[ M_i^{\text{calc}}(m_Q, \lambda, ...) - M_i^{\text{exp}} \right]^2$$

with spin-spin splittings ($\alpha_s$) often fit state by state for heavy quarks (Gutiérrez-Guerrero et al., 2021).

After calibration, NRQM produces systematic predictions for:

• Radial excitations ($n=2,3,4$ S-waves) of $D$, $D_s$, $B$, $B_s$, $B_c$, charmonium, bottomonium.
• Scalar and axial-vector diquark partners, leveraging symmetry between meson and diquark quantum numbers with appropriate interaction scalings.

Typical mass predictions (in GeV) achieve agreement at the few MeV level for ground states, with radial excitation deviations at the $5$–$10\%$ level: $$\begin{aligned} M_{D^0}(1S;0^-) &= 1.86~(\text{exp}),~1.98~(V_0+H_{SS}) \ M_{J/\psi}(1S) &= 3.10~(\text{exp}),~3.57~(2S) \ M_{B_c}(1S,0^-) &= 6.27~(\text{exp}),~6.62~(2S) \end{aligned}$$ Diquark masses: $$M_{[cu]\,0^+}=1.88~\text{GeV},~ M_{\{cu\}\,1^+}=2.03~\text{GeV}$$ (Gutiérrez-Guerrero et al., 2021)

4. Comparison with Experimental Data and Other Theoretical Frameworks

Comprehensively, ground-state masses from NRQM match experimental values (PDG 2018) within a few MeV. Hyperfine splittings (e.g., $\Delta M_D \sim 150$ MeV, $\Delta M_B \sim 50$ MeV) are reproduced in close accord with observed data.

Radial and orbital excitations align with alternative quark-potential approaches (Cornell, Godfrey–Isgur, variational/lattice QCD) at the $5$–$10\%$ level. Diquark spectra conform to Schwinger–Dyson–Bethe–Salpeter calculations and other phenomenological extractions, maintaining correct level ordering and multiplet structure (Gutiérrez-Guerrero et al., 2021).

A plausible implication is that the chosen CI-inspired potential, combined with nonrelativistic techniques and phenomenological spin corrections, is robust for heavy and heavy-light hadron spectroscopy, and can reliably predict exotic states, e.g., diquarks and hybrids, for ongoing and future experimental searches.

5. Extensions: Diquarks, Thermodynamics, and Exotic States

NRQM accommodates the calculation of diquark partners by halving the interaction strength and adjusting parity transformation rules. Scalar diquarks ($0^+$) and axial-vector diquarks ($1^+$) mirror the properties of $0^-$ and $1^-$ mesons, respectively.

Thermodynamical extensions integrate temperature-dependent effective potentials (extended Cornell with Debye screening) and employ statistical mechanics formalisms to calculate partition functions, free energy, specific heat, and entropy for quarkonia (Aydın, 2022). At elevated temperatures, bound-state masses decrease due to potential screening and approach deconfined regimes.

Multi-quark and exotic states, including six-quark resonances and flavor-symmetric clusters, can be appraised within NRQM and TF extensions, with systematic calculations of masses and radii allowing rapid survey of binding energies and experimental detectability (Liu et al., 2012).

6. Limitations and Current Direction

The nonrelativistic approximation is most reliable for systems with heavy quarks ($m_Q \gg \Lambda_{\text{QCD}}$). For lighter systems, corrections become increasingly important, and relativistic or field-theoretic methods may be necessary for precision. Spin-dependent effects and mixing can be included perturbatively, but dynamical spin-orbit, tensor, and color correlations can demand more sophisticated treatments.

Ongoing research addresses NRQM's applicability to thermal spectroscopy, exotic multiquark systems, and nucleon structure under QCD-inspired models, with attention to chiral symmetry breaking, local gauge invariance, and asymptotic freedom (Musulmanbekov, 28 Jun 2025). Experimental searches for predicted states, especially narrow diquarks and thermally modified quarkonia, are informed by NRQM results.

The consensus from NRQM studies, as in (Gutiérrez-Guerrero et al., 2021), is that a contact-interaction-inspired potential combined with matrix-based solution algorithms and phenomenological spin terms achieves uniform, systematic description of heavy and heavy-light meson spectra and their diquark partners, maintaining fidelity with known experimental and lattice data within well-quantified uncertainties.