Diagrams with One External Coulomb Line

The effects connected with the electron vacuum polarization contributions in muonic atoms were first quantitatively discussed in [4]. In electronic hydrogen polarization loops of other leptons and hadrons considered in Subsect. 3.2.5 played a relatively minor role, because they were additionally suppressed by the typical factors (mg/m). In the case of muonic hydrogen we have to deal with the polarization loops of the light electron, which are not suppressed at all. Moreover, characteristic exchange momenta mZa in muonic atoms are not small in comparison with the electron mass rUg, which determines the momentum scale of the polarization insertions m Za)jme 1.5). We see that even in the simplest case the polarization loops cannot be expanded in the exchange momenta, and the radiative corrections in muonic atoms induced by the electron loops should be calculated exactly in the parameter m Za)/me- 

Electron polarization insertion in the photon propagator in Fig. 2.2 induces a correction to the Coulomb potential, which may be easily written in the form 

The respective correction to the energy levels is given by the expectation value of this perturbation potential 

The radial wave functions depend on radius only via the combination p = rrUr-Za and it is convenient to write it explicitly as a function of this dimensionless variable 

Explicit dependence of the leading polarization correction on the parameters becomes more transparent after transition to the dimensionless integration variable p [4] 

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