The Bethe Logarithm for resonant states. Antiprotonic helium

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 Schr\"odinger 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$^+\bar{p}\,$ atoms with 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 new theoretical values for ro-vibrational transition frequencies for the He$^+\bar{p}$ atoms with uncertainty of 0.1--0.3 MHz.