Radiative Muon Capture in Hydrogen

It is shown that ${\Gamma }_{\mathrm{rad}}$, the rate (in ${\sec }^{-1\mathrm{}}$) of the radiative muon capture reaction ${\mu }^{-}+p\to \gamma +\nu +n$, is very different in the two eigenstates of total [${\mu }^{-}p$] spin (hyperfine effect in radiative muon capture). It is further shown that, in each of these two eigenstates, ${\Gamma }_{\mathrm{rad}}$ depends sensitively on the magnitudeo f the pseudoscalar form factor. In particular for ${g_P}^{\left(\mu \right)}=8\mathrm{}{g_A}^{\left(\mu \right)}$ (pseudoscalar form factor dominated by a single pion pole term) the ratio, $R=\frac{{\Gamma }_{\mathrm{rad}}}{{\Gamma }_{\mathrm{nonrad}}}$, of radiative to nonradiative muon capture rates is given by: $R_{\mathrm{sing}}=4.96\mathrm{}\times \frac{{10}^{-3\mathrm{}}}{634}=0.079\mathrm{}\times {10}^{-4\mathrm{}}$ (spin-singlet [$\mu p$] atom); $R_{\mathrm{trip}}=9.00\mathrm{}\times \frac{{10}^{-2\mathrm{}}}{13.3}=69\mathrm{}\times {10}^{-4\mathrm{}}$ (spin-triplet [$\mu p$] atom); $R_{\mathrm{mixt}}=6.87\mathrm{}\times \frac{{10}^{-2\mathrm{}}}{168}=4.1\mathrm{}\times {10}^{-4\mathrm{}}$ (statistical mixture of spin-singlet and spin-triplet [$\mu p$] atoms); $R_{\mathrm{molec}}=3.07\mathrm{}\times \frac{{10}^{-2\mathrm{}}}{560}=0.55\mathrm{}\times {10}^{-4\mathrm{}}$ (spin-doublet ortho [$p\mu p$] molecule). If, on the other hand, ${g_P}^{\left(\mu \right)}=16\mathrm{}{g_A}^{\left(\mu \right)}$ (with the additional contribution to the pseudoscalar form factor taken to be effectively independent of momentum transfer for $\left|q^2\right|\lesssim {m_{\pi }}^2$), the ratios $R$ become: