Scattering for the quadratic nonlinear Schr\"{o}dinger system in $\mathbb{R}^5$ without mass-resonance condition

We consider the quadratic nonlinear Schr\"{o}dinger system (NLS system) \begin{align*}\begin{cases} i\partial_t u + \Delta u = v \overline{u}, \\ i\partial_t v+\kappa \Delta v = u^2, \end{cases} \text{ on } I \times \mathbb{R}^5, \end{align*} where $\kappa>0$. The scattering below the standing wave solutions for NLS system was obtained by the first author when $\kappa = 1/2$. The condition of $\kappa=1/2$ is called mass-resonance. In this paper, we prove scattering below the standing wave solutions when $\kappa \neq 1/2$ under the radially symmetric assumption. Our proof is based on the concentration compactness and the rigidity by Kenig--Merle. Moreover, we discuss the concentration compactness and the rigidity for non-radial solutions.