The g factor of the bound muon in medium-Z muonic atoms

We consider a theory of the g factor of a bound muon in a three-body atomic system, which consists of a spinless nucleus, a muon, and an electron in the medium range of $Z=10\text{–}30$. We show that the calculation at the one-ppm level of accuracy can be separated into a consideration of an internal subsystem $\mu -N$ (with a muon at the ground state) and an external subsystem with an electron and a compound $\mu -N$ nucleus. We discuss the most important contributions to the g factor of the bound muon in the $\mu -N$ system in the medium-Z approximation. In this range the list of relevant contributions contains kinematic pure Coulomb contributions, the finite-nuclear-size ones, and muonic-atom-specific contributions with closed electron loops. In the case of the medium Z one can apply a double limit $Z\alpha \ll 1$ and $Z\alpha m_{\mu }/m_e\gg 1$, at which a number of specific contributions is simplified. Special attention is paid to non-potential contributions, i.e., those which cannot be expressed in terms of an effective potential. We also consider the electron shielding of the magnetic moment of the compound nucleus, focusing on the corrections which are enhanced or suppressed comparing to the shielding of ordinary nuclei. In conclusion we discuss a generalization of our result for muonic atoms with a few electrons.