Two-body $P$-state energies at $\alpha^6$ order

Abstract

We present an analytical calculation of the complete $\alpha^6$ correction to energies of $nP$-levels of two-body systems consisting of the spin-$0$ or $1/2$ extended-size particles with arbitrary masses and magnetic moments. The obtained results apply to a wide class of two-body systems such as hydrogen, positronium, muonium, and pionic or aniprotonic helium ion. We found an additional $\alpha^6$ correction for $nP$-levels of positronium, which was previously overlooked. Our results are also relevant for light muonic atoms, whose accurate theoretical predictions are required for extracting the nuclear charge radii.