  In this paper, we simplify the proof of M. Hamano in \cite{Hamano2018},
scattering theory of the solution to \eqref{NLS system}, by using the method
from B. Dodson and J. Murphy in \cite{Dodson2018}. Firstly, we establish a
criterion to ensure the solution scatters in $ H^1(\mathbb{R}^5) \times
H^1(\mathbb{R}^5) $. In order to verify the correctness of the condition in
scattering criterion, we must exclude the concentration of mass near the
origin. The interaction Morawetz estimate and Galilean transform characterize a
decay estimate, which implies that the mass of the system cannot be
concentrated.
