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

TL;DR

This work develops a covariant two-body Dirac equation in $(2+1)$-dimensional spacetime inspired by Amelino-Camelia's doubly special relativity (DSR) and analyzes both non-interacting and interacting fermion–antifermion pairs. The unmodified theory reduces to a Bessel-type radial equation for the relative motion, while the DSR corrections introduce a Planck-scale rescaling via $\lambda_k=[1+E/(4E_p)]$. For Coulomb-like interactions with a position-dependent mass, the energy spectrum acquires a DSR-modified form with a running fine-structure-like coupling $\alpha_{\mathrm{eff}}(E)=\alpha/[1+E/(4E_p)]$, yielding binding energies $E^{b^{DSR}}_n = E^b_n\left(1-\mathcal{E}_0/E_p\right)$ with $\mathcal{E}_0=mc^2$; the estimated shifts are tiny, of order $4.19\times10^{-23}$ for the ground state. Overall, the paper presents a first exploration of DSR-modified two-body dynamics in a simplified $(2+1)$-D setting, showing that DSR effects can induce small, energy-dependent modifications to binding energies and wavefunctions, and outlining avenues for extending this framework to more realistic systems such as positronium-like and quarkonium-like states.

Abstract

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 αbehaves as an energy-dependent running parameter, given by \(α_{eff(E)}/α\approx 1 - \frac{E}{4E_p}\), 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.

Figures (2)

• Figure 1: This figure illustrates the oscillatory behavior of the Bessel function of the first kind, $J_1(\lambda r)$, across a range of $r$ values (from 0 to 20) for different arbitrary values of $\lambda$ (0.5, 1, and 1.5). The curves demonstrate the characteristic oscillations and decay of the Bessel function, highlighting its dependence on the scaling factor $\lambda$. Each curve is labeled with its corresponding $\lambda$ value for clarity.
• Figure 2: This figure shows the behavior of the Bessel function-based solution $\varphi_1(r)$ across the range $r = [0, 200]$ for five different values of $\lambda_k$: 1, 1.005, 1.010, 1.015 and 1.020. Each curve represents the oscillatory solution $\varphi_1(r)$, with the amplitude adjusted by the factor $\frac{\lambda}{\lambda_k}$, illustrating how slight changes in $\lambda_k$ influence the oscillations in $\varphi_1(r)$. The plot provides insight into the sensitivity of the solution to variations in $\lambda_k$.