Copositive criteria for a two-component dark matter model

We consider a two-component scalar dark matter model in this work, where the scalars are stabilized by extra $Z_2 \times Z'_2$ symmetry. To guarantee the stability of the vacuum, we consider the copositive criteria and different choices of the signs of the couplings will contribute to different viable parameter spaces. Based on the copositive criteria, we systematically analyze the possible conditions and pick up 17 different cases with fixing some parameters to be negative. We randomly scan the parameter space under dark matter relic density constraint and direct detection constraint and focus on the quartic couplings with $\lambda_{13}$,$\lambda_{23}$,$\lambda_{14}$ and $\lambda_{24}$. The shapes of the viable parameter space for $|\lambda_{14}|-|\lambda_{24}|$ are almost similar among these cases, while for $|\lambda_{13}|$ and $|\lambda_{23}|$, the larger value are excluded as long as $\lambda_{13}\leqslant 0$ and $\lambda_{23}\leqslant 0$ due to the large interference effect of the $2 \to 2$ processes.