Nuclear muon capture: Hyperfine effects in nuclear spin and isospin [1/$2^{\pm }$,1/2] → [1/$2^{\pm }$,1/2] and [$1^{+}$,0] → [$0^{+}$,1] transitions

Formulas for angular correlations and capture rates in muon-capture transitions between initial and final nuclear states characterized by spin and isospin${[1/2^{\pm },1/2]}_{\mathrm{ini}}$ and ${[1/2^{\pm },1/2]}_{\mathrm{fin}}$, and ${[1^{+},0]}_{\mathrm{ini}}$ and ${[0^{+},1]}_{\mathrm{fin}}$, are derived and applied to ${\mu }^{-}p\to {\nu }_{\mu }n$, ${\mu }^{-}{}_{}{}^3{}_{}{}^{}\mathrm{He}\to {\nu }_{\mu }{}_{}{}^3{}_{}{}^{}\mathrm{H}$, and ${\mu }^{-}{}_{}{}^6{}_{}{}^{}\mathrm{Li}\to {\nu }_{\mu }{}_{}{}^6{}_{}{}^{}\mathrm{He}$ in a model-independent (elementary-particle) approach. The results indicate that the angular correlations between the momenta of the recoiling final nuclei and the polarizations of the ${\mu }^{-}$ and/or the initial nuclei are sensitive to the value of the nuclear pseudoscalar form factors $F_P\left(q^2\right)$. The analysis also indicates that inconsistencies among the available data on ${\mu }^{-}{}_{}{}^6{}_{}{}^{}\mathrm{Li}\to {\nu }_{\mu }{}_{}{}^6{}_{}{}^{}\mathrm{He}$ are most likely of experimental origin.