Cornell Interaction in the Two-body Pauli-Schrödinger-type Equation Framework: The Symplectic Quantum Mechanics Formalism

TL;DR

This work analyzes a quark-antiquark bound state in a uniform magnetic field within the symplectic quantum mechanics framework using the Cornell potential. By applying a Bohlin mapping, the problem is recast as a perturbative two-dimensional oscillator problem for a spin-1/2 system, yielding zero- and first-order solutions and their corresponding Wigner functions. The study shows confinement features emerge naturally in phase space and that the magnetic field enhances non-classicality as seen in the Wigner function, with the external-field strength $B$ inferred from experimental spectra to be of order $10^{15}$ T, matching transient fields in non-central heavy-ion collisions. These results provide a phase-space perspective on quarkonium in strong fields and establish a route for connecting theoretical predictions with experimental mass spectra.

Abstract

We investigate the quantum behavior of a quark-antiquark bound system under the influence of a magnetic field within the symplectic formulation of quantum mechanics. Employing a perturbative approach, we obtain the ground and first excited states of the system described by the Cornell potential, which incorporates both confining and non-confining interactions. After performing a Bohlin mapping in phase space, we solve the time-independent symplectic Pauli-Schrödinger-type equation and determine the corresponding Wigner function. Special attention is given to the observation of the confinement of the quark-antiquark, that is revealed in the phase space structure. Due to the presence of spin in the Hamiltonian, the results reveal that the magnetic field enhances the non-classicality of the Wigner function, signaling stronger quantum interference and a departure from classical behavior. The experimental mass spectra is used to estimate the intensity of the external field, leading to a value that is in order of the transient magnetic field measured in non-central heavy-ion collisions at RHIC and LHC.

Figures (4)

• Figure 1: The left graph, first-order corrected Wigner function for the ground state ($n_{1}=0$, $n_{2}=0$) of the $c\overline{c}$ meson, with momentum $p=0\,\text{GeV}$ and no magnetic field ($B=0\,\text{GeV}^2$), plotted as a function of the interquark distance $q^{2}$ in the region $0 \leq q^{2} \lesssim 4\,\text{GeV}^{-1}$. The right graph, the Wigner function is presented for the same state with momenta $p=2.3\,\text{GeV}$ (red), $2.5\,\text{GeV}$ (green), and $2.9\,\text{GeV}$ (blue), also for $B=0\,\text{GeV}^2$.
• Figure 2: The left graph, Wigner function with first-order correction for the state ($n_{1}=0$, $n_{2}=0$) (ground state) of the $b\overline{b}$ meson with $p=0\,\text{GeV}$ and $B=0\,\text{GeV}^2$. The right graph, the Wigner function is plotted for the same state with momenta $p=2.3\,\text{GeV}$ (red), $2.5\,\text{GeV}$ (green), and $2.9\,\text{GeV}$ (blue), also for $B=0\,\text{GeV}^2$.
• Figure 3: The left graph, Wigner function with first-order correction for the state ($n_{1}=0$, $n_{2}=0$) (ground state) of the $c\overline{c}$ meson with $p=0\,\text{GeV}$ and $B=1.5\,\text{GeV}^2$. The right graph is plot for the Wigner function considering the same state but with with $B=4.5\,\text{GeV}^2$ and for $p=0\,\text{GeV}$.
• Figure 4: The left graph, Wigner function with first-order correction for the state ($n_{1}=0$, $n_{2}=0$) (ground state) of the $b\overline{b}$ meson with $p=0\,\text{GeV}$ and $B=1.5\,\text{GeV}^2$. The right graph, the Wigner function is presented for the same state with $B=4.5\,\text{GeV}^2$ and for $p=0\,\text{GeV}$.