An innovative model for coupled fermion-antifermion pairs

Abstract

Abstract Understanding the behavior of fermion-antifermion ( $$f\overline{f}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mover> <mml:mi>f</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> ) 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 $$f\overline{f}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mover> <mml:mi>f</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> pairs with an effective position-dependent mass (i.e., $$m \rightarrow m + \mathcal {S}(r)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>→</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mi>S</mml:mi> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> ) under the influence of an external magnetic field. To validate our model, we show that by modifying the mass with a Coulomb-like potential, $$m(r) = m - \alpha /r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>m</mml:mi> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> <mml:mo>/</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> </mml:math> , where $$-\alpha /r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> <mml:mo>/</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> </mml:math> is the Lorentz scalar potential $$\mathcal {S}(r)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , our results match the well-established energy eigenvalues for $$f\overline{f}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mover> <mml:mi>f</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> pairs interacting through the Coulomb potential, without approximation. By applying adjustments based on the Cornell potential (i.e., $$\mathcal {S}(r) = kr - \alpha /r$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>k</mml:mi> <mml:mi>r</mml:mi> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> <mml:mo>/</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> </mml:math> ), we derive a closed-form energy expression. We believe this unique model offers significant insights into the dynamics of $$f\overline{f}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mover> <mml:mi>f</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> pairs under various interaction potentials, with potential applications in particle physics. Additionally, it could be extended to various $$f\overline{f}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mover> <mml:mi>f</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:mrow> </mml:math> systems, such as positronium, relativistic Landau levels for neutral mesons, excitons in monolayer transition metal dichalcogenides, and Weyl pairs in monolayer graphene sheets.

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