Angular Distribution of the Neutrino in Muon Capture and Nuclear Structure

Abstract

As a result of parity nonconservation, emitted neutrinos in muon capture by spin-zero nuclei have an asymmetric angular distribution with respect to the muon-polarization axis. The asymmetry coefficient $\alpha$ is defined by the distribution $1+\alpha Pcos\theta$, and it is given in cases in which the final states have definite spin and parity. In the unique forbidden transitions in which the final states have spin $J$ and parity ${\mathrm{}(-)\mathrm{}}^{J+1\mathrm{}}$, $\alpha$ is generally dependent on both nuclear structure and the pseudoscalar coupling constant $C_P$. However, in some cases, it is almost independent of nuclear structure, so that we can study the magnitude of $C_P$ from the asymmetry coefficient. In the nonunique forbidden transitions in which the final states have spin $J$ and parity ${\mathrm{}(-)\mathrm{}}^J$, $\alpha$ is independent of $C_P$, but is strongly dependent on nuclear structure. As extreme cases, $\alpha$ is + 1 for $A$-type and - 1 for $V$-type giant dipole excitations, with about 2% estimated error. This holds also for $A$- and $V$-type higher multipole excitations. If the final states are $0^{-}$ or $0^{+}$, then $\alpha$ is exactly - 1, and this case is independent of both nuclear structure and the magnitude of $C_P$, in contrast to the strong dependence of the capture rate on $C_P$ in the $0^{-}$ states. An averaged value of $\alpha$ over four kinds of the giant dipole states is equal to 0.4, in agreement with Primakoff's value in the closure approximation.