Comprehensive study of nuclear reactions in muon catalyzed fusion: I. dt$\mu$ system

Muon catalyzed fusion ($\mu$CF) has recently regained considerable research interest owing to several new developments and applications. In this regard, we performed a comprehensive study on the most important fusion reaction, $(dt\mu)_{J=v=0} \to \alpha + n + \mu + 17.6$ MeV or $(\alpha \mu)_{nl} + n + 17.6$ MeV. For the first time, the coupled-channel Schrödinger equation has been solved for the reaction, satisfying the boundary condition for the muonic molecule $(dt\mu)_{J=v=0}$ as the initial state and the outgoing wave in the $\alpha n \mu$ channel. We employ the $dt\mu$- and $\alpha n \mu$-channel coupled three-body model. All the nuclear interactions, 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 $d+t\to \alpha+n + 17.6 \,{\rm MeV}$, as well as the total cross section of the $\alpha+n$ reaction. The resultant $dt\mu$ fusion rate is $1.03 \times 10^{12}\, {\rm s}^{-1}$. Substituting the obtained total wave function into the $T$-matrix based on the Lippmann-Schwinger equation, we have derived the reaction rates going to the individual bound and continuum states of the outgoing $\alpha$-$\mu$ pair. Using the rates, we obtain the initial $\alpha$-$\mu$ sticking probability $\omega_S^0=0.857 \%$, which is consistent with the most recent observations (2001) at high DT densities. We also calculate the momentum and energy spectra of muons emitted during the fusion process. The peak energy is 1.1 keV although the mean energy is 9.5 keV owing to the long higher-energy tail. This is a useful 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.