We develop a numerical method to calculate the Bethe logarithm for resonant states. We use the complex coordinate rotation (CCR) formalism to describe resonances as time-independent Schrödinger solutions. To get a proper expression for the Bethe logarithm we apply the generalization of the second order perturbation theory to an isolated CCR eigenstate. Using the developed method we perform a systematic calculation of the Bethe logarithm for metastable states in the antiprotonic helium He${}^{+}\overline{p}$ atoms with a precision of 7–8 significant digits. We also recalculate the nonrelativistic energies with improved precision using CODATA10 recommended values of masses. Along with a complete set of corrections of $m{\alpha }^7$ order and the leading contributions of $m{\alpha }^8$ order, that has allowed us to get theoretical values for ro-vibrational transition frequencies for the He${}^{+}\overline{p}$ atoms with an uncertainty of 0.1–0.3 MHz.

• Received 16 December 2013

DOI:https://doi.org/10.1103/PhysRevA.89.014501