Two-body Dirac equation in DSR: Results for fermion-antifermion pairs

This study investigates a modified two-body Dirac equation in $2+1$-dimensional spacetime, inspired by Amelino-Camelia's doubly special relativity (DSR). We begin by deriving a covariant two-body Dirac equation that, in the absence of DSR modifications, reduces to a Bessel-type wave equation. Incorporating corrections from the chosen DSR model modifies this wave equation, yielding solutions consistent with established results in the low-energy regime. We demonstrate that the effects of DSR modifications become particularly pronounced at large relative distances. For a coupled fermion-antifermion pair, we derive the modified binding energy solutions. By accounting for first-order Planck-scale corrections, we show that the fine-structure constant α behaves as an energy-dependent running parameter, given by ${\alpha }_{\text{eff}}\left(E\right)/\alpha \approx 1-\frac{E}{4E_p}$, where $E_p$ is the Planck energy. Binding energy levels are computed using a first-order approximation of the DSR modifications, and the results are applied to positronium-like systems. Our model reveals that DSR modifications induce shifts in the binding energy levels. To the best of our knowledge, DSR-modified two-body equations have not been previously studied. This model is the first of its kind, opening new avenues for further research in this area.