Quantum-electrodynamics corrections in pionic hydrogen

Abstract

We investigate all pure quantum-electrodynamics corrections to the $np\ensuremath{\rightarrow}1s$, $n=2$--4 transition energies of pionic hydrogen larger than $1\phantom{\rule{0.28em}{0ex}}\mathrm{meV}$, which requires an accurate evaluation of all relevant contributions up to order ${\ensuremath{\alpha}}^{5}$. These values are needed to extract an accurate strong interaction shift from experiment. Many small effects, such as second-order and double vacuum polarization contribution, proton and pion self-energies, finite size and recoil effects are included with exact mass dependence. Our final value differs from previous calculations by up to $\ensuremath{\approx}$$11\phantom{\rule{0.28em}{0ex}}\mathrm{ppm}$ for the $1s$ state, while a recent experiment aims at a $4\phantom{\rule{0.28em}{0ex}}\mathrm{ppm}$ accuracy.