The Bethe Logarithm for resonant states. Antiprotonic helium

Abstract

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${}^{+}\bar{p}$ 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.

6 pages, no figures will be submitted to PRA

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