The $CP$ violations and branching ratios for $B_c^{+}\to D_{(s)}^{+}{\pi }^{+}{\pi }^{-}(K^{+}{K}^{-})$ from interference of the vector mesons in Perturbative QCD

Within the framework of the perturbative QCD approach utilizing $K_T$ factorization, we have investigated the CP violations and branching ratios in the decay processes of $B_c^{+}\to D_{(s)}^{+}V(V\to {\pi }^{+}{\pi }^{-})$ and $B_c^{+}\to D_{(s)}^{+}V(V\to K^{+}{K}^{-})$, where V denotes three vector mesons ${\rho }^0$, $\omega$, and $\phi$. During the $V\to {\pi }^{+}{\pi }^{-}$ and $V\to K^{+}{K}^{-}$ decay processes, we incorporated the ${\rho }^0-\omega -\phi$ mixing mechanism to describe the amplitudes of these quasi-two-body decay processes. Within the interference regime of the three vector particles, we observed distinct changes in both CP violations and branching ratios. Furthermore, our study presents evidence for local CP violations and branching ratios that warrants further investigation through experiments.