Self-energy correction to the E1 transition amplitudes in hydrogen-like ions

We present calculations of the self-energy correction to the $E1$ transition amplitudes in hydrogen-like ions, performed to all orders in the nuclear binding strength parameter. Our results for the $1s$-$2p_{1\mathrm{/}2}$ transition for the hydrogen isoelectronic sequence show that the perturbed-orbital part of the self-energy correction provides the dominant contribution, accounting for approximately 99% of the total correction for this transition. Detailed calculations were performed for $ns$-$n^{\mathrm{\prime }}p$ and $np$-$n^{\mathrm{\prime }}d$ transitions in H-like caesium. We conclude that the perturbed-orbital part remains dominant also for other $ns$-$n^{\mathrm{\prime }}p$ transitions, whereas for the $np$-$n^{\mathrm{\prime }}d$ matrix elements this dominance no longer holds. Consequently, the self-energy corrections for the $np$-$n^{\mathrm{\prime }}d$ one-electron matrix elements cannot be well reproduced by means of effective QED operators constructed for energy levels.