Investigating the muon catalyzed fusion in muonic lithium hydride

In this work, we consider the muonic LiH as a new stage of the $\mu$ catalyzed fusion. We calculate the bound states of the muonic LiH and LiH$^+$ and their wave functions based on the three and four body approximation. In order to solve the Schr$\ddot{\rm o}$dinger equation, we apply the Gaussian expansion method which gives us both the eigen energies and the wave functions. The existences of the bound states in muonic LiH and LiH$^+$ are confirmed. We find the dynamical behaviors between the $^7$Li and proton in muonic LiH and LiH$^+$ are similar as the ones in dt$\mu_2$ and dt$\mu$, respectively. The $\alpha$ sticking probability in the muonic LiH$^+$ after the nuclear fusion is studied. The calculated $\alpha$ sticking probability of being captured to the ground state of He$\mu$ is 5 times smaller than the one in dt$\mu$.