Theory of Allowed and Forbidden Transitions in Muon Capture Reactions

Abstract

A general formula for the transition rate of the muon capture reaction, ${\ensuremath{\mu}}^{\ensuremath{-}}+(A, Z)\ensuremath{\rightarrow}\ensuremath{\nu}+(A, Z\ensuremath{-}1)$, where the final nuclear state has definite spin and parity, is given in terms of the total and orbital angular momenta of the emitted neutrino and of the spins of the initial and final nuclear states. The induced pseudoscalar interaction and the interaction due to the assumption of conserved vector current are taken into account, together with the vector and axial vector interactions. The forbiddenness of the muon capture reaction is defined in a manner analogous to the theory of the beta decay. The spin and parity changes can assume the values (0+, 1+), (0-, 1-, 2-), [$n{(\ensuremath{-})}^{n}, n+1{(\ensuremath{-})}^{n}$] for the allowed, first forbidden, and $n\mathrm{th}$ ($n\ensuremath{\ge}2$) forbidden transitions, respectively. (+ and - mean the parity change no and yes.) For these transitions, the number of reduced nuclear matrix elements involved is nine, sixteen, and fourteen, respectively. The transition rate of muon capture reaction is reduced by a factor of ${10}^{3}$, approximately, for a two-unit increase of the forbiddenness, if the atomic number and the energy of neutrino are constant. The contribution from the higher order transition to the lower one is less than 0.1% in the medium and light nuclei. Explicit formulas for the transition rate are given for the allowed, first forbidden and $n\mathrm{th}$ forbidden transitions. They are related to the corresponding formulas of beta decay. Our formalism was applied to the calculation of the partial muon capture rate by ${\mathrm{C}}^{12}$ ending in the ground state of ${\mathrm{B}}^{12}$. The numerical analysis indicates that measurement of this capture rate can determine whether the conserved vector current interaction term exists in nature only if the coupling constant of the induced pseudoscalar interaction and the nuclear wave functions are well known. The transition rates are given in Table V and Fig. 1, for the $j\ensuremath{-}j$ coupling shell model and harmonic oscillator wave functions. They are 9-13% smaller than those given by Fujii and Primakoff.