Effect of the Spin-Orbit Force on Muon-Capture Matrix Elements for Closed-Shell Nuclei

Abstract

The vector, axial-vector, and pseudoscalar matrix elements for muon capture in closed-shell nuclei are studied in the case where a single-particle spin-orbit potential is included in the nuclear shell-model Hamiltonian. Because of sum rules arising from the rotational properties of the operators and wave functions involved, and in addition, because of the smallness of the ratio of the nuclear spin-orbit energy splitting to the emitted neutrino energy, it is found that the effect of the spin-orbit force is negligible. More precisely, if one considers a multipole expansion of the plane-wave neutrino wave function appearing in each of the matrix elements, it is found that the individual multipolar matrix elements are altered by less than 1% by the addition of the spin-orbit potential to the nuclear Hamiltonian. Therefore, to the same degree of accuracy, we conclude that the magnitude of each of the vector, axial-vector, and pseudoscalar matrix elements remains unchanged. Finally, based on calculations reported here and elsewhere, we conjecture that inaccuracies in the ${\mathrm{}\left(1p\right)\mathrm{}}^{-1\mathrm{}}2s$ and ${\mathrm{}\left(1p\right)\mathrm{}}^{-1\mathrm{}}1d$ wave functions of ${\mathrm{N}}^{16}$ are not the prime source of the disagreement between the observed muon-capture rate in ${\mathrm{O}}^{16}$ and that calculated using nuclear wave functions derived from a shell model plus residual interaction. We suggest other mechanisms which, if properly taken into account, may allow one to obtain better agreement with the experimental muon-capture rate in ${\mathrm{O}}^{16}$.