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

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.

Figure 1

Nuclear-recoil point-nucleus and fns corrections $E_{\mathrm{rec}}^{(6+)}$ for the $2p_{1/2}$ and $2p_{3/2}$ states of muonic atoms, as a function of the nuclear charge number $Z$. Units are $m_{\mu }^2/m_N{\left(Z\alpha \right)}^6$. The nonsmoothness of the fns plots is due to the irregular dependence of the nuclear charge radius on $Z$.