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

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 hbox{³He{}^3{rm He}} 3 He ion (denoted here as μ ³He⁺), as well as in the muonic hbox{⁴He{}^4{rm He}} 4 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)⁵ μ ³ln(Zα) / (m _μ m _N ) are also obtained. All calculations are nonperturbative in the mass ratio of orbiting particle and nucleus.