Radiative muon capture in a relativistic mean field theory: Fermi gas model.

We examine radiative muon capture in a nuclear medium using mean field theory and a relativistic Fermi gas model for the nucleus to obtain the single nucleon states. The aim is to explore, in a simple model, effects of the medium which are characterized primarily by a nucleon effective mass ${m}^{\mathrm{*}}$, as well as other relativistic effects. The relative rate, i.e., the photon spectrum divided by the nonradiative rate, and the photon asymmetry relative to the muon spin are calculated. The most important effect turns out to be the Fermi motion which reduces the relative rate by a factor of 2\char21{}3 in the experimentally accessible region, compared to the static case. The ${m}^{\mathrm{*}}$ effect further reduces the result by 10\char21{}50 %, depending on the photon energy. The relativistic nature of this calculation, unlike usual nonrelativistic calculations, allows these effects to be incorporated to all orders in 1/m. As a consequence some interesting effects can be studied in the small k region. We conclude that both relativistic kinematic and medium effects may be significant and thus that it is worthwhile investigating this reaction in more realistic relativistic nuclear models.