Minimally coupled fermion-antifermion pairs via exponentially decaying potential

In this study, we explore how a fermion-antifermion ($f\bar{f}$) pair interacts via an exponentially decaying potential. Using a covariant one-time two-body Dirac equation, we examine their relative motion in a three-dimensional flat background. Our approach leads to coupled equations governing their behavior, resulting in a general second-order wave equation. Through this, we derive analytical solutions by establishing quantization conditions for pair formation, providing insights into their dynamics. Notably, we find that such interacting $f\bar{f}$ systems are unstable and decay over time, with the decay time depending on the Compton wavelength of the fermions.