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The present paper is concerned with the large data scattering problem for themass-energy double critical NLS \begin{align} i\partial_t u+\Delta u\pm|u|^{\frac{4}{d}}u\pm |u|^{\frac{4}{d-2}}u=0\tag{DCNLS} \end{align} in$H^1(\mathbb{R}^d)$ with $d\geq 3$. In the defocusing-defocusing regime, Tao,Visan and Zhang show that the unique solution of DCNLS is global and scatteringin time for arbitrary initial data in $H^1(\mathbb{R}^d)$. This does not holdwhen at least one of the nonlinearities is focusing, due to the possibleformation of blow-up and soliton solutions. However, precise thresholds for asolution of DCNLS being scattering were open in all the remaining regimes.Following the classical concentration compactness principle, we impose sharpscattering thresholds in terms of ground states for DCNLS in all the remainingregimes. The new challenge arises from the fact that the remainders of thestandard $L^2$- or $\dot{H}^1$-profile decomposition fail to haveasymptotically vanishing diagonal $L^2$- and $\dot{H}^1$-Strichartz normssimultaneously. To overcome this difficulty, we construct a double trackprofile decomposition which is capable to capture the low, medium and highfrequency bubbles within a single profile decomposition and possessesremainders that are asymptotically small in both of the diagonal $L^2$- and$\dot{H}^1$-Strichartz spaces.

On sharp scattering threshold for the mass-energy double critical NLS via double track profile decomposition