[Paper Review] A quadratic potential in a light cone QCD inspired model

Hans–Christian Pauli|ArXiv.org|Dec 20, 2003

Cold Atom Physics and Bose-Einstein Condensates5 references3 citations

TL;DR

This paper proposes a light-front QCD-inspired model with a quadratic potential $V(r) = -a + \frac{1}{2}fr^2$ to describe heavy meson spectra, incorporating spin-dependent interactions that generate a position-dependent effective mass $\widetilde{m}_r(r)$. Despite using a simplified oscillator approximation, the model achieves good agreement with experimental $\pi$, $\rho$, and heavy meson masses—within experimental errors and better than prior models—demonstrating the viability of the approach for hadron spectroscopy.

ABSTRACT

The general equation from previous work is specialized to a quadratic potential $V(r)=-a+\frac12 f r^2$ acting in the space of spherically symmetric S wave functions. The fine and hyperfine interaction creates then a position dependent mass $\widetilde m(r)$ in the effective kinetic energy of the associated Schrödinger equation. The results are compared with the available experimental and theoretical spectral data on the $π$ and $ρ$. Solving the eigenvalue problem within the usual oscillator approach induces a certain amount of arbitrariness. Despite of this, the agreement with experimental data is within the experimental error and better than other calculations, including Godfrey and Isgur \cite{GodIsg85} and Baldicchi and Prosperi \cite{BalPro02}. The short coming can be removed easily in more elaborate work.

Motivation & Objective

• To develop a tractable light-front QCD-inspired model for hadron spectroscopy using a quadratic potential.
• To incorporate spin-dependent interactions (hyperfine, kinetic, Darwin) that induce a position-dependent effective mass in the Hamiltonian.
• To test the model's predictive power against experimental S-wave meson masses, particularly for $\pi$, $\rho$, and heavy quarkonia.
• To assess the validity of the oscillator approximation with a mean-field replacement of the non-local position-dependent mass.
• To lay the groundwork for extending the model to baryons and nuclear systems using more sophisticated numerical treatments of the non-local Hamiltonian.

Proposed method

• The model Hamiltonian is derived from light-front QCD via unitary transformations, yielding a non-local Schrödinger-type equation with a position-dependent mass $\widetilde{m}_r(r)$.
• The central potential is taken as $V(r) = -a + \frac{1}{2}fr^2$, leading to a harmonic oscillator-like kinetic term with $\widetilde{m}_r(r)$.
• Spin-dependent interactions—hyperfine, kinetic, and Darwin—are included, contributing to the effective potential and spin splitting.
• The non-locality is approximated by replacing $\widetilde{m}_r(r)$ with a constant $\widetilde{m}_r$ determined via the mean square radius $\langle r^2 \rangle$, enabling analytical solution.
• The eigenvalues are computed using the harmonic oscillator basis, with energy levels given by $E_n = -\widetilde{a} + \omega\xi_0 + \omega\eta_n + \widetilde{c}\vec{\sigma}_1\vec{\sigma}_2$, where $\xi_0 = 3/2$, $\eta_n = 2n$, and $\omega = \sqrt{f/\widetilde{m}_r}$.
• Parameters are fixed by fitting to experimental masses of $\bar{D}^0$, $\bar{B}^+$, and $\bar{D}_s^-$, with $m_c$, $m_b$, $f$, and $a$ determined from three data points.

Experimental results

Research questions

• RQ1 Can a light-front QCD-inspired model with a quadratic potential accurately reproduce the S-wave meson spectrum, including spin splittings?
• RQ2 How does the inclusion of spin-dependent interactions (hyperfine, kinetic, Darwin) affect the effective mass and energy levels in a non-local Hamiltonian?
• RQ3 To what extent does the oscillator approximation with a constant effective mass $\widetilde{m}_r$ capture the physics of the full non-local model?
• RQ4 How does the model's agreement with experiment compare to established models like Godfrey and Isgur (1985) and Baldicchi and Prosperi (2002)?
• RQ5 What is the predictive power of the model for unobserved states, such as the $B_c^+$ and $B_c^{*+}$ mesons?

Key findings

• The model reproduces the $\bar{D}^0$ mass exactly by construction, with the $\bar{D}^{*0}$ mass predicted at 1.9594 GeV, overestimated by only 50 MeV compared to the experimental value of 1.9685 GeV.
• For the $\bar{B}^+$ and $\bar{B}^{*+}$ states, the model predicts 5.2790 GeV and 5.3085 GeV, respectively, matching experiment within 16 MeV for the triplet state.
• For the $D_s^-$ and $D_s^{*-}$ states, the model predicts 1.9961 GeV and 2.0718 GeV, differing from experiment by 27 MeV and 40 MeV, respectively.
• The model predicts the $B_c^+$ and $B_c^{*+}$ ground states at 6.5077 GeV and 6.5157 GeV, respectively, with no experimental confirmation yet.
• The model achieves better agreement with experiment than Godfrey and Isgur (1985) and Baldicchi and Prosperi (2002), particularly for the $B^*$ and $D^*$ states.
• The model's success suggests that the oscillator approximation with a mean-field effective mass is a robust first step, and the full non-local model with $\widetilde{m}_r(r)$ can be solved numerically to improve accuracy.