Non-Relativistic Potential Quark Model

• Non-Relativistic Potential Quark Model is a framework that uses effective potentials and the Schrödinger equation to describe hadronic structures and spectra.
• It incorporates central confining forces, short-range interactions, and spin-dependent corrections to reproduce observables like mass spectra and decay widths.
• The model is extended to address finite temperature effects, external fields, and multiquark dynamics, bridging traditional QCD with phenomenological approaches.

The non-relativistic potential quark model (NRPQM) provides a framework for describing the structure, spectrum, and transition dynamics of hadrons by modeling constituent quarks as bound by effective potentials. Despite being rooted in quantum chromodynamics (QCD), the model leverages the computational tractability of the Schrödinger equation, incorporating central confining and short-range interactions as well as spin-dependent corrections. NRPQM consistently reproduces a wide range of hadronic observables—including mass spectra, decay widths, and electromagnetic transitions—across the meson and baryon sectors, and supports systematic extensions to finite temperature, external fields, and multiquark configurations.

1. Model Hamiltonians and Interquark Potentials

The NRPQM formalism begins with the effective two-body Hamiltonian

$H = 2m_Q + \frac{\mathbf{p}^2}{2\mu} + V(r),$

where $m_Q$ is the constituent quark mass, $\mu$ the reduced mass, and $V(r)$ the effective potential. The canonical choice is a Cornell-type potential: $V(r) = -\frac{4}{3}\frac{\alpha_s}{r} + b r,$ where $\alpha_s$ denotes the strong coupling and $b$ the string tension, quantifying the color-Coulomb and linear confining forces respectively (Akbar et al., 2024, Ali et al., 2015, Monteiro et al., 2016). To capture the hyperfine, fine, and tensor structures, leading spin-dependent Breit–Fermi corrections are added: $$V_{\rm{SD}}(r) = V_{SS}(r)\,\mathbf{S}_Q\cdot\mathbf{S}_{\bar Q} + V_{LS}(r)\, \mathbf{L}\cdot\mathbf{S} + V_T(r)\, T,$$ with the explicit forms (in the contact/Gaussian-smeared or perturbative limit) given by

$$V_{SS}(r) = \frac{32\pi \alpha_s}{9 m_Q^2} \left(\frac{\sigma}{\sqrt{\pi}}\right)^3 e^{-\sigma^2 r^2},$$

$$V_{LS}(r) = \frac{1}{m_Q^2} \left( \frac{2\alpha_s}{r^3} - \frac{b}{2r} \right), \quad V_T(r) = \frac{4\alpha_s}{m_Q^2 r^3} T,$$

where $\sigma$ is the contact smearing parameter (Akbar et al., 2024, Debastiani et al., 2017).

High-precision quarkonium and mixed-flavor predictions also employ more general potentials—such as a screened Cornell potential at finite temperature (Hussain et al., 2022), trigonometric Rosen–Morse, or hypercentral power-law forms in baryonic systems (Abu-Shady et al., 2019, Salehi, 2016), as well as multi-body effective potentials encoding three-quark forces (Diakonos et al., 2010).

2. Spectra, Wave Functions, and Solution Techniques

NRPQM reduces the many-body bound-state problem to radial or hyperradial Schrödinger equations. For mesonic states: $$\left[ -\frac{1}{2\mu}\frac{d^2}{dr^2} + \frac{\ell(\ell+1)}{2\mu\,r^2} + V(r) \right] u_{n\ell}(r) = E_{n\ell} u_{n\ell}(r),$$ with imposed regularity at $r \rightarrow 0$ and vanishing at $r \rightarrow \infty$.

Eigenvalues $E_{n\ell}$ fix the meson mass,

$M_{n\ell} = 2m_Q + E_{n\ell},$

and the wave function $R_{n\ell}(r) = u_{n\ell}(r)/r$ provides access to spatial observables, e.g., the root-mean-square (rms) radius: $$\langle r^2 \rangle_{n\ell}^{1/2} = \sqrt{\int_0^\infty r^2 |R_{n\ell}(r)|^2 \, dr}.$$ Wavefunctions underpin calculations of decay constants, e.g., $f_{B_c} \sim \psi(0)$, directly linking potential model structure to weak processes (Monteiro et al., 2016).

Solving the eigenproblem employs various schemes:

• Shooting and matching algorithms (Akbar et al., 2024, Hussain et al., 2022)
• Matrix Numerov method for discretized systems (Gutiérrez-Guerrero et al., 2021)
• Tridiagonal (matrix) techniques for mesh-discretized Hamiltonians (Ali et al., 2015)

For baryonic and multiquark systems (baryons, tetraquarks), the hypercentral approach introduces hyperspherical coordinates and potentials $V(\rho)$, leading to a hyperradial wave equation with additional angular degrees of freedom (Diakonos et al., 2010, Salehi, 2016).

3. Radiative and Weak Decays

Electromagnetic transitions are parameterized by E1 (electric dipole) and M1 (magnetic dipole) operators. The partial widths are: $$\Gamma_{E1} = \frac{4}{3} C_{fi} \delta_{S S'} e_Q^2 \alpha \; |\langle \Psi_f | r | \Psi_i \rangle|^2 E_\gamma^3 \frac{E_f}{M_i},$$

$$\Gamma_{M1} = \frac{4}{3} \frac{2J'+1}{2\ell+1} \delta_{\ell\ell'} \delta_{S,S'\pm1} e_Q^2 \frac{\alpha}{m_Q^2} |\langle \Psi_f | \Psi_i \rangle|^2 E_\gamma^3 \frac{E_f}{M_i},$$

where $E_\gamma$ is the photon energy; explicit overlap integrals and angular coefficients are constructed from radial wavefunctions (Akbar et al., 2024, Monteiro et al., 2016).

Branching ratios and radiative hierarchies (E1 $\gg$ M1) emerge naturally, reflecting the $1/m_Q^2$ suppression of magnetic dipole rates for heavy quarks.

For strong and weak decays, the NRPQM supports both spectator-mode width calculations (as in $B_c$ mesons, employing constituent-level $b$ and $c$ decay widths, annihilation modes, and decay constants (Monteiro et al., 2016)) and PCAC-based approaches (as in $\Sigma_b \to \Lambda_b\,\pi$ (Hernández et al., 2011)). Non-relativistic wavefunctions and potential-derived parameters (e.g., $|\psi(0)|^2$) provide the input to decay amplitude evaluations.

4. Parameter Fitting and Phenomenological Successes

Model parameters—quark masses, $\alpha_s$, $b$, smearing widths, and, where relevant, screening or cutoff scales—are fitted to ground-state masses, hyperfine splittings, or charge radii (Akbar et al., 2024, Tan et al., 9 Mar 2025, Gutiérrez-Guerrero et al., 2021). The same parameter sets are then used to predict excited spectra and transition rates.

Notable fit results include:

• Toponium: $\alpha_s=0.1596$, $b=0.18\,\mathrm{GeV}^2$, $m_t=172.42\,\mathrm{GeV}$ (Akbar et al., 2024)
• Charmonium and Bottomonium: $\alpha_s \sim 0.48-0.53$, $b \sim 0.15\,\mathrm{GeV}^2$, with sub-percent accuracy on spectral lines (Ali et al., 2015, Debastiani et al., 2017)
• Hypercentral baryon spectra: constituent $m=293\,\mathrm{MeV}$, octic potential coefficients fit to N, $\Delta$, $\Lambda$, $\Sigma$ resonances (Salehi, 2016)
• Parameter values for color-magnetic extensions in classical analogies: $Z=2.98\times10^{-25}\,\mathrm{N\,m}^2$, $T=1.13\times10^{-43}\,\mathrm{N\,s}^2$ for $b\bar b$ fits (Tan et al., 9 Mar 2025)

The success in describing observables such as rms radii, binding energies, level ordering, and mass splittings justifies the NRQPM’s widespread use, even in precise determinations such as the bottom quark mass from non-relativistic sum rules (Beneke et al., 2016).

5. Extensions: Screening, Temperature, Multiquark, and Classical Analogy

Screening and Thresholds

NRPQM models are systematically extended to account for screening due to open-flavor meson-meson thresholds via piecewise “flattening” of the potential, generating plateau regions and level trapping above flavor thresholds (Gonzalez, 2014). This leads to enhanced state densities, especially in heavy-quark systems above open-flavor thresholds.

Finite Temperature and Fields

At finite temperature, color screening parameters ($\mu(T)$) attenuate the confining interaction, reducing $T_c$ for dissociation of quarkonia. Temperature-dependent $\alpha_s(T)$ and $\sigma(T)$ are tuned to lattice QCD data; the sequence of melting points follows binding energy and spatial extension, with ground-state bottomonia surviving to $T \sim 4\,T_c$ (Hussain et al., 2022). In external magnetic fields, Landau-level and Zeeman effects are explicitly incorporated, altering spectral patterns and thermodynamic observables (Kojo, 2021).

Multiquark and Baryon Extensions

NRPQM is generalized to diquark configurations, tetraquarks (e.g., $T_{4c}$ through the reduction to effective diquark–antidiquark dynamics) (Debastiani et al., 2017), and three-quark baryons using hypercentral power-law or octic potentials for SU(6)-invariant modeling and Gürsey–Radicati mass shifts for flavor–spin splittings (Salehi, 2016). Three-body and multiquark interactions, as in hypercentral or contact vector–vector potentials, improve agreement with observed baryon masses and parton distributions (Diakonos et al., 2010, Gutiérrez-Guerrero et al., 2021).

Classical Interpretations

A classical analogy—introducing explicit definitions for color charge, color flux, and magnetic self-energies—reproduces the phenomenology of the Cornell potential and yields a meson mass–radius relationship analogous to the Bohr atom (Tan et al., 9 Mar 2025). This approach quantitatively relates calculated structural radii to experimental r.m.s. or charge radii, highlighting the connection between classical and quantum viewpoints.

6. Scope, Limitations, and Theoretical Context

NRPQM achieves high precision (typical mass errors well below 1–2%) in the description of heavy quarkonia and baryons, validates hierarchies of radiative and weak transitions, and is adaptable across hadronic systems and external-field environments. However, several limitations persist:

• Non-relativistic approximation: Valid chiefly for heavy quarks; relativistic corrections are only partially included.
• Neglect of continuum and open channels: No direct coupling to hadronic decay channels or threshold mixing.
• Limited treatment of multi-body and chiral dynamics: While baryons and multiquarks are accessible, full coupled-channel and chiral effects are often omitted.
• Short lifetime issue: In systems like toponium, the lifetime is too short for well-defined narrow spectral lines (Akbar et al., 2024).

Despite these constraints, the NRQPM serves as a foundational tool for spectroscopy, transition rates, decay properties, and as a bridge to more fundamental QCD approaches (lattice QCD, NRQCD, potential non-relativistic QCD at high orders) (Chigodaev et al., 2013, Beneke et al., 2016). It offers an efficient, transparent, and physically motivated framework for heavy hadron phenomenology and exploratory studies of new states.