Relativistic Binding Corrections to HFS

Relativistic and radiative corrections depend on the electron and muon masses only via the explicit mass factors in the electron and muon magnetic moments, and via the reduced mass factor in the Schrodinger wave function. All such corrections may be calculated in the framework of the external field approximation. 

In the external field approximation the heavy particle magnetic moment factorizes and the relativistic and radiative corrections have the form 

This factorization of the total muon magnetic moment occurs because the virtual momenta involved in calculation of the relativistic and radiative corrections are small in comparison with the muon mass, which sets the natural momentum for corrections to the muon magnetic moment. 

Purely relativistic corrections are by far the simplest corrections to h3rper-fine splitting. As in the case of the Lamb shift, they essentially correspond to the nonrelativistic expansion of the relativistic square root expression for the energy of the light particle in (1.3), and have the form of a series over Zo j /m . Calculation of these corrections should be carried out in the framework of the spinor Dirac equation, since clearly there would not be any hyperfine splitting for a scalar particle. 

The binding corrections to h q)erfine splitting as well as the main Fermi contribution are contained in the matrix element of the interaction Hamiltonian of the electron with the external vector potential created by the muon magnetic moment (A = V X /Lx/(47rr)). This matrix element should be calculated between the Dirac-Coulomb wave functions with the proper reduced mass dependence (these wave functions are discussed at the end of Sect. 1.3). Thus we see that the proper approach to calculation of these corrections is to start with the EDE (see discussion in Sect. 1.3), solve it with the convenient 

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