The lowest-order correction in the hyperfine splitting of muonic ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ due to excitation of the effective ${(\mu -\mathrm{He})}^{+}$ nucleus is precisely evaluated numerically. The result is $\Delta {\nu }_1^e=-45.670\mathrm{}\left(8\right)\mathrm{}$ MHz, in agreement with previous approximate analytic calculations. The mass-polarization correction to the hyperfine splitting is found to be $\Delta {\nu }_1^m=0.0785\mathrm{}\left(2\right)\mathrm{}$ MHz. The total perturbation-theory value for the hyperfine splitting is $\Delta \nu =4464.3\mathrm{}\left(1.8\right)\mathrm{}$ MHz, which is consistent with experiments at the Swiss Institute for Nuclear Research and at the Clinton P. Anderson Meson Physics Facility at Los Alamos.

• Received 14 October 1983

DOI:https://doi.org/10.1103/PhysRevA.29.1047