Hunting for direct CP violation in $\bar{B}_s^0 \to {\pi^+}{\pi^-}K^{*0}$

In perturbative QCD approach, based on the first order of isospin symmetry breaking, we study the direct $CP$ violation in the decay of $\bar{B}_s^0\to\rho(\omega )K^{*0}\to {\pi^+}{\pi^-}K^{*0}$. An interesting mechanism is applied to enlarge the $CP$ violating asymmetry involving the charge symmetry breaking between $\rho$ and $\omega$. We find that the $CP$ violation is large by the $\rho-\omega$ mixing mechanism when the invariant masses of the $\pi^+\pi^-$ pairs is in the vicinity of the $\omega$ resonance. For the decay process of $\bar{B}^0_{s}\to\rho(\omega )K^{*0}\to{\pi^+}{\pi^-}K^{*0}$, the maximum $CP$ violation can reach $-59.12\%$. Furthermore, taking $\rho-\omega$ mixing into account, we calculate the branching ratio for $\bar{B}_s^0 \rightarrow \rho(\omega) K^{*0}$. We also discuss the possibility of observing the predicted $CP$ violation asymmetry at the LHC.