The hyperfine structure of the ground states in the helium muonic atoms

The hyperfine structures of the ground states in the ${}^3$He$^{2+} \mu^{-} e^{-}$and${}^{4}$He$^{2+} \mu^{-} e^{-}$ helium-muonic atoms are investigated with the use of highly accurate variational wave functions. The differences between corresponding levels of hyperfine structure (i.e. hyperfine splittings) are determined to very high numerical accuracy. In particular, we have found that the hyperfine structure splitting in the ground state of the ${}^4$He${}^{2+}{\mu }^{-}{e}^{-}$ atom is $\mathrm{\Delta }\nu \approx 4464.55454(3)$ $MHz$, while analogous splitting for the ${}^3$He${}^{2+}{\mu }^{-}{e}^{-}$ atom is $\mathrm{\Delta }\nu \approx 4166.43547(3)$.