An innovative model for coupled fermion-antifermion pairs

Understanding the behavior of fermion-antifermion ( $f\overline{f}$ ) pairs is crucial in modern physics. These systems, governed by fundamental forces, exhibit complex interactions essential for particle physics, high-energy physics, nuclear physics, and solid-state physics. This study introduces a novel theoretical model using the many-body Dirac equation for $f\overline{f}$ pairs with an effective position-dependent mass (i.e., $m\to m+S\left(r\right)$ ) under the influence of an external magnetic field. To validate our model, we show that by modifying the mass with a Coulomb-like potential, $m\left(r\right)=m-\alpha /r$ , where $-\alpha /r$ is the Lorentz scalar potential $S\left(r\right)$ , our results match the well-established energy eigenvalues for $f\overline{f}$ pairs interacting through the Coulomb potential, without approximation. By applying adjustments based on the Cornell potential (i.e., $S\left(r\right)=kr-\alpha /r$ ), we derive a closed-form energy expression. We believe this unique model offers significant insights into the dynamics of $f\overline{f}$ pairs under various interaction potentials, with potential applications in particle physics. Additionally, it could be extended to various $f\overline{f}$ systems, such as positronium, relativistic Landau levels for neutral mesons, excitons in monolayer transition metal dichalcogenides, and Weyl pairs in monolayer graphene sheets.