  In this paper, using $I$-method, we establish $H^s_x\times H^s_x$ scattering
theories for the following Cauchy problem \begin{equation*} \left\{
\begin{array}{lll} iu_t+\Delta u=\lambda |v|^2u,\quad iv_t+\Delta
v=\mu|u|^2v,\quad x\in \mathbb{R}^3,\ t>0,\\ u(x,0)=u_0(x),\quad
v(x,0)=v_0(x),\quad x\in \mathbb{R}^3. \end{array}\right. \end{equation*} Here
$\lambda>0$, $\mu>0$, $(u_0,v_0)\in H^s_x(\mathbb{R}^3)\times
H^s_x(\mathbb{R}^3)$ and $\frac{1}{2}<s<1$.
