Average polarization of the recoilB12from muon capture byC12

It is shown, in the "elementary-particle" treatment, that, if the source to produce the background ${}_{}{}^{12}{}_{}{}^{}\mathrm{B}$(g.s.) comes only from the cascade processes ${\mu }^{-}{}_{}{}^{12}{}_{}{}^{}\mathrm{C}(\mathrm{g}.\mathrm{s}.\mathrm{})\to {\nu }_{\mu }{}_{}{}^{12}{}_{}{}^{}\mathrm{B}_{}^{*}{}_{}{}^{}\mathrm{}\left(2.62\right)\mathrm{}\to {\nu }_{\mu }{}_{}{}^{12}{}_{}{}^{}\mathrm{B}(\mathrm{g}.\mathrm{s}.\mathrm{}) \gamma$, the average polarization of the recoil ${}_{}{}^{12}{}_{}{}^{}\mathrm{B}$(g.s.) produced by the direct polarized-muon capture ${\mu }^{-}{}_{}{}^{12}{}_{}{}^{}\mathrm{C}(\mathrm{g}.\mathrm{s}.\mathrm{})\to {\nu }_{\mu }{}_{}{}^{12}{}_{}{}^{}\mathrm{B}(\mathrm{g}.\mathrm{s}.\mathrm{})$ can be extracted reliably from the observed angular distribution of the electrons in the subsequent beta decays of ${}_{}{}^{12}{}_{}{}^{}\mathrm{B}$(g.s.). Using the recent data of Possoz et al, our result for the average polarization $P_{\mathrm{av}}$ of the recoil ${}_{}{}^{12}{}_{}{}^{}\mathrm{B}$(g.s.) from the direct polarized-muon capture is $P_{\mathrm{av}}=0.47\mathrm{}\pm 0.05$, which agrees very well with the prediction of the standard picture, namely, conservation of vector current, partial conservation of axial vector current, and absence of second-class axial currents.