Relativistic Three-Fermion Wave Equations in Reformulated QED and Relativistic Effects in Muonium Minus

TL;DR

This paper develops relativistic three-fermion wave equations within reformulated QED to calculate relativistic energy corrections for systems like Muonium negative ion, providing new insights into relativistic effects in such bound states.

Contribution

It introduces a variational method in reformulated QED to derive relativistic three-fermion wave equations and applies them to compute energy corrections for muonium and similar systems.

Findings

Relativistic correction for Mu$^-$ is -1.0773 x 10^{-4} eV.

Derived equations incorporate invariant $ ext{M}$ matrices at lowest order.

Results compare favorably with existing calculations.

Abstract

The variational method, within the Hamiltonian formalism of reformulated QED is used to determine relativistic wave equations for a system of three fermions of arbitrary mass interacting electromagnetically. The interaction kernels of the equations are, in essence, the invariant $\mathcal{M}$ matrices in lowest order. The equations are used to obtain relativistic $O(\alpha^2)$ corrections to the non-relativistic ground state energy levels of the Muonium negative ion ($\mu^+ e^- e^-$) as well as of $\mathrm{Ps}^-$ and $\mathrm{H}^-$, using approximate variational three-body wave functions. The results are compared with other calculations, where available. The relativistic correction for Mu$^-$ is found to be $-1.0773 \times 10^{-4}$ eV.