  In this article, we consider the dynamics of the energy-critical quadratic
nonlinear Schr\"odinger system $\[ \left\{ \begin{aligned} & i u^1_t + \kappa_1
\Delta u^1 = -\overline{u^2}u^3, \\ & i u^2_t + \kappa_2 \Delta u^2 =
-\overline{u^1}u^3, \\ & i u^3_t + \kappa_3 \Delta u^3 = -u^1u^2, \\
\end{aligned} \right. \qquad (t, x) \in \R \times \R^6 \] in energy-space $
{\dot H}^1 \times {\dot H}^1\times{\dot H}^1 $, where the sign of potential
energy can not be determined. We prove the scattering theory with
mass-resonance (or with radial initial data) below ground state via
concentration compactness method. We discover a family of new physically
conserved quantities with mass-resonance which play an important role in the
proof of scattering.
