Comprehensive study of muon-catalyzed nuclear reaction processes in the $dt\mu$ molecule

Muon catalyzed fusion ($\mu$CF) has recently regained considerable research interest owing to several new developments and applications. In this regard, we have performed a comprehensive study of the most important fusion reaction, namely $(dt\mu)_{J=v=0}\to\alpha$+$n$+$\mu$+17.6 MeV or $(\alpha \mu)_{nl}$+$n$+17.6 MeV. The coupled-channels Schr\"{o}dinger equation for the reaction is thus solved, satisfying the boundary condition for the muonic molecule $(dt\mu)_{\rm J=v=0}$ as the initial state and the outgoing $\alpha n \mu$ channel with the $\alpha$-$n$ $D$ wave. We employ the $dt\mu$- and $\alpha n \mu$-channel coupled three-body model. All the $d$-$t$ and $\alpha$-$n$ potentials, and the $d t$-$\alpha n$ channel-coupling nonlocal tensor potential are chosen to reproduce the observed low-energy astrophysical $S$-factor of the reaction $d$+$t\to\alpha$+$n$+17.6 MeV, as well as the total cross section of the $\alpha$+$n$ reaction. The resultant $dt\mu$ fusion rate is 1.15x10$^{12}$ s$^{-1}$. Substituting the obtained total wave function into the $T$ matrix, we have calculated absolute values of the fusion rates going to the bound and continuum states of the outgoing $\alpha$-$\mu$ pair. We then derived the initial $\alpha$-$\mu$ sticking probability $\omega_S^0$=0.857%, which is $\sim$7% smaller than the literature values ($\simeq$0.91-0.93%), and can explain the recent observations (2001) at high D-T densities. We also calculate the absolute values for the momentum and energy spectra of the emitted muon. The most important result is that the peak energy is 1.1 keV although the mean energy is 9.5 keV. This is an essential result for the ongoing experimental project to realize the generation of an ultra-slow negative muon beam by utilizing the $\mu$CF for various applications e.g., a scanning negative muon microscope and an injection source for the muon collider.