$\cal CPT$-invariant two-fermion Dirac equation with extended hyperfine operator

TL;DR

This paper derives and analyzes a $ ext{CPT}$-invariant two-fermion Dirac equation for S-states of muonium and positronium, calculating hyperfine shifts up to order $ ext{α}^8$ and introducing an extended operator to improve singularity behavior.

Contribution

It introduces an extended hyperfine operator that mitigates singularity issues, enabling nonperturbative calculations within a $ ext{CPT}$-invariant two-fermion framework.

Findings

Hyperfine shifts checked against nonrelativistic results up to order α^6.

Leading α^8 hyperfine shifts calculated.

An extended operator reduces singularity at r→0.

Abstract

For the S-states of muonium and positronium, the hyperfine shifts to the order $\alpha^6$ of a recently derived two-fermion equation with explicit $\cal CPT$-invariance are checked against the results of a nonrelativistic reduction, and the leading $\alpha^8$ shifts are calculated. An additional hyperfine operator is discovered which can milden the singularity for $r\to 0$ of the Dirac hyperfine operator, such that the resulting extended operator can be used nonperturbatively. The binding correction to magnetic moments is mentioned.