Eur. Phys. J. D 65, 357-366 (2011)
https://doi.org/10.1140/epjd/e2011-20338-2

Regular Article

Semi-analytic approach to higher-order corrections in simple muonic bound systems: Vacuum polarization, self-energy and radiative-recoil

¹ Department of Physics, Missouri University of Science and Technology, Rolla MO65409, USA
² Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

^a e-mail: ulj@mst.edu

Received: 12 July 2011
Published online: 22 November 2011

Abstract

The current discrepancy of theory and experiment observed recently in muonic hydrogen necessitates a reinvestigation of all corrections to contribute to the Lamb shift in muonic hydrogen (μH), muonic deuterium (μD), the muonic ${\mathrm{3}}^{}\mathrm{He}$ ion (denoted here as μ ³He⁺), as well as in the muonic ${\mathrm{4}}^{}\mathrm{He}$ ion (μ ⁴He⁺). Here, we choose a semi-analytic approach and evaluate a number of higher-order corrections to vacuum polarization (VP) semi-analytically, while remaining integrals over the spectral density of VP are performed numerically. We obtain semi-analytic results for the second-order correction, and for the relativistic correction to VP. The self-energy correction to VP is calculated, including the perturbations of the Bethe logarithms by vacuum polarization. Subleading logarithmic terms in the radiative-recoil correction to the 2S–2P Lamb shift of order α(Zα)⁵μ³ln(Zα) / (m_μm_N) are also obtained. All calculations are nonperturbative in the mass ratio of orbiting particle and nucleus.