Double-pair Coulomb and Breit photon correction to the correlated relativistic energy

TL;DR

This work develops a framework to compute algebraic double-pair photon corrections to the correlated relativistic energy of two-fermion systems within an equal-time Bethe-Salpeter equation. It combines a no-pair Dirac-Coulomb(-Breit) reference with perturbative or with-pair treatments of instantaneous double-pair interactions, implemented in a sixteen-component spinor basis using explicitly correlated Gaussian functions and optimized via a targeted functional that includes an auxiliary basis for the $\mathcal{L}_{--}$ subspace. Numerical results for Ps, Mu, H, $\mu$H, and He-like ions demonstrate convergence and excellent agreement with nrQED at the $\alpha^3E_h$ level, while highlighting limitations of the instantaneous Breit approximation and the importance of auxiliary basis optimization. The approach provides a controllable, reference-based route to relativistic QED corrections beyond the no-pair approximation and lays groundwork for extending to non-instantaneous and radiative effects in future work.

Abstract

The simplest, algebraic quantum-electrodynamical corrections, due to the double-negative energy subspace and instantaneous interactions, are computed to the no-pair energy of two-spin-1/2-fermion systems. Numerical results are reported for two-electron atoms with a clamped nucleus and positronium-like genuine two-particle systems. The Bethe-Salpeter equation provides the theoretical framework, and numerical methods have been developed for its equal-time time-slice. In practice, it requires solving a sixteen-component eigenvalue equation with a two-particle Dirac Hamiltonian, including the appropriate interaction. The double-pair corrections can either be included in the interaction part of the eigenvalue equation or treated as a perturbation to the no-pair Hamiltonian. The numerical results have an $α$ fine-structure constant dependence that is in excellent agreement with the known $α^3E_\mathrm{h}\$-order double-pair correction of non-relativistic quantum electrodynamics.