Coupling Constants in Muon Capture

The measured capture rate of muons in ${\mathrm{C}}^{12}$ leading to the ground state of ${\mathrm{B}}^{12}$, in combination with other data, is employed as a basis for a determination of the weak-interaction coupling constants associated with muon capture. The study of this transition has the advantages: (1) that the rate depends only weakly on the vector, induced pseudoscalar, and (possible) tensor coupling constants, but not at all on a possible scalar coupling constant; (2) that empirical information from inelastic electron scattering on ${\mathrm{C}}^{12}$ leading to the excitation of the 15.1-MeV level, the $M1\mathrm{}$ lifetime of this level, and the $f{\tau }_{\frac{1}{2}}$ value for the beta decay of ${\mathrm{B}}^{12}$ allows one to determine the required nuclear matrix elements of major importance (as well as some of minor importance) in practically a model-independent way. The capture rate can thus be expressed in terms of the axial vector coupling constant and the weak-magnetism coupling constants. Assuming the validity of the conserved-vector-current hypothesis (CVC), and $5<\mathrm{}{C}_p<28\mathrm{}$, one then finds that $\frac{{F_A}^{\mu }}{{F_A}^{\beta }}={{1.04}_{-0.10\mathrm{}}}^{+0.07\mathrm{}}$. On the other hand, if one assumes ${F_A}^{\mu }={F_A}^{\beta }$, one obtains for the isotopic vector magnetic moment of the nucleon (at the appropriate momentum transfer) $\mu \left({\nu }^2\right)={{5.7}_{-1.6\mathrm{}}}^{+1.1\mathrm{}}$ nuclear magnetons, which is consistent with the CVC prediction of 4.60 nm and can be considered as evidence for weak magnetism in muon capture.