The theory of $\gamma$-neutrino angular correlation in nuclear muon capture is developed using the density matrix technique. A closed expression for the correlation coefficient, for unpolarized muon capture and including relativistic terms, is obtained in the particle-hole model for the nucleus. An interesting relation between the correlation coefficient and the longitudinal polarization of the recoil nucleus in muon capture is established. This relation is independent of nuclear models and the muon capture coupling constants, and its importance in connection with time-reversal invariance is pointed out. Utilizing the close analogy between muon capture and inelastic electron scattering, the numerical results for the process ${\mu }^{-}+{}_{}{}^{28}{}_{}{}^{}\mathrm{Si}\left(0^{+}\right)\to {}_{}{}^{28}{}_{}{}^{}\mathrm{Al}_{}^{*}{}_{}{}^{}\left(1^{+}\right)+{\nu }_{\mu }$, ${}_{}{}^{28}{}_{}{}^{}\mathrm{Al}_{}^{*}{}_{}{}^{}\left(1^{+}\right)\to {}_{}{}^{28}{}_{}{}^{}\mathrm{Al}\left(0^{+}\right)+\gamma$ are presented. It is found that the correlation coefficient is extremely sensitive to the nuclear model, contrary to common belief, and also sensitive to the induced pseudoscalar coupling constant in muon capture. The results are compared with the available experimental data and a range for $\frac{g_P}{g_A}$ is obtained as $3<\mathrm{}\frac{g_P}{g_A}<20\mathrm{}$, in agreement with other predictions, indicating a remote possibility of the quenching of $g_P$ due to virtual pion effects in the $A=28\mathrm{}$ system. With the "canonical" value for $g_P$, our results give $\frac{g_T}{g_A}=+4.5\mathrm{}\pm 7.5$ due to the large uncertainty in the experimental data.

NUCLEAR REACTIONS ${}_{}{}^{28}{}_{}{}^{}\mathrm{Si}({\mu }^{-}, {\gamma }_{\nu }){}_{}{}^{28}{}_{}{}^{}\mathrm{Al}$; $\gamma$-neutrino angular correlation coefficient; particle hole model; induced pseudoscalar and tensor form factors.

• Received 28 March 1978

DOI:https://doi.org/10.1103/PhysRevC.18.1796