Sharp threshold of blow-up and scattering for the fractional Hartree equation

We consider the fractional Hartree equation in the $L^2$-supercritical case, and find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If $M{u_0]}^{\frac{s?s_c}{s_c}}E{u}_0]<M{Q]}^{\frac{s?s_c}{s_c}}EQ]$ and $M{u_0]}^{\frac{s?s_c}{s_c}}{6u_{0}6}_{{\dot{H}}^s}^2<M{Q]}^{\frac{s?s_c}{s_c}}{6Q6}_{{\dot{H}}^s}^2$, then the solution $u(t)$ is globally well-posed and scatters; if $M{u_0]}^{\frac{s?s_c}{s_c}}E{u}_0]<M{Q]}^{\frac{s?s_c}{s_c}}EQ]$ and $M{u_0]}^{\frac{s?s_c}{s_c}}{6u_{0}6}_{{\dot{H}}^s}^2>M{Q]}^{\frac{s?s_c}{s_c}}{6Q6}_{{\dot{H}}^s}^2$, the solution $u(t)$ blows up in finite time. This condition is sharp in the sense that the solitary wave solution $e^{it}Q(x)$ is global but not scattering, which satisfies the equality in the above conditions. Here, Q is the ground-state solution for the fractional Hartree equation.