Estimation of Radiative Corrections to the Process of Muon-Electron Conversion

In detection of electrons from $\mu \to e$ conversion process the monochromatic electron spectrum is transformed due to a photon emission and fluctuations of energy loss in a target. The selection criterion of $\mu \to e (\gamma)$ conversion events is an electron momentum above the threshold momentum of 103.5 MeV/c, which corresponds to the maximum energy loss of 1.5 MeV. Radiative corrections including a virtual photon correction and soft photon emission below 1.5 MeV lead to a reduction by about 10% in the probability of $\mu \to e$ conversion process calculated without radiative corrections. The soft photons emission below 1.5 MeV contributes to a change of electron spectrum from monoenergetic one at 105 MeV to a spectrum with a low energy tail for the process of $\mu \to e (\gamma)$ conversion. However the effect of smearing of the initial momentum distribution due to the soft photon emission is small in comparison with a smearing due to energy loss fluctuations in a target. The average energy of soft photons emitted below 1.5 MeV is found to be 40 keV. The soft photon approximation is a good description for $\mu \to e (\gamma)$ conversion process with photons emitted below 1.5 MeV.