Coupled fermion-antifermion pairs within a traversable wormhole

This study investigates the dynamics of fermion-antifermion ($f\bar{f}$) pairs within a traversable wormhole (TWH) spacetime by solving the two-body covariant Dirac equation with a position-dependent mass $m\to m\left(r\right)$. In the context of a static, radially symmetric (2+1)-dimensional TWH characterized by a constant redshift function and a given shape function, we explore two Lorentz scalar potentials: (i) a Coulomb-like potential and (ii) an exponentially decaying potential. The Coulomb potential leads to positronium-like binding energies, with the ground state ($n=0$) energy approximately $E_n^b\approx -m_e{c}^2{\alpha }^2/4\sim -6.803$ eV. On the other hand, the exponential potential establishes critical mass thresholds, $m_c=\frac{(n+\frac{1}{2})ħ}{2{\lambda }_{c}c}$, at which the energy approaches zero, causing the system to cease to exist over time. Stability is maintained when $n+\frac{1}{2}<2$, resulting in oscillatory behavior, while $n+\frac{1}{2}>2$ leads to decay. The energy spectrum reveals essential features of the system, and the wave function reflects the influence of the wormhole's throat, shaping spatial configurations and probability distributions. This work enhances our understanding of quantum phenomena in curved spacetimes and establishes connections to condensed matter physics.