Radiative corrections in hydrogen lowest order

In light atoms the parameter aZ is small aZ S. 1) and one can use an expansion in this parameter. Moreover, the radiative corrections can be evaluated with the nonrelativistic (Schrodinger) wave functions. 

Here A is the fictitious photon mass that is introduced to avoid the so called infrared divergency . This divergency occurs in the integral over the virtual photon frequency in Fig. 10a. Unlike the ultraviolet divergency it occurs in the small frequency region. To avoid it the photon propagator in the momentum representation, that is (where is the virtual 

Then the integral becomes finite. However, the infrared divergency occurs only for free electrons, and not for bound ones, since the bound electron graph Fig.la does not diverge when the virtual photon frequency tends to zero. Therefore the photon mass A should cancel at the end of the calculations. 

The term with the coefficient —1/5 in Eq(178) corresponds to the graph Fig.10b (vacuum polarization), the other terms correspond to the graph Fig.lOa (electron self-energy). 

The radiative correction to the energy arises after averaging the potential (178) with atomic wave functions. TVansformation to the nonrelativistic functions (see Bq(25)) yields  

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