Direct CP violation in semi-inclusive flavor-changing neutral current decays in the MSSM without $R-$parity

Semi-inclusive decays, $\text{hlx}$, are studied in the framework of the minimal supersymmetric standard model without $R-$parity, where $\text{qh}$ ($\text{ql}$) are the second or third (first or second) generation quarks with the same charge and $\xi$ is a vector meson formed by $q_j{\text{qb}}_j$. The study is focused on the contributions of sfermions with $m_{}<m_{\mathrm{t}\mathrm{o}\mathrm{p}}$. In this mass region, CP asymmetries in top decays can be induced by taking into account the decay-widths of the exchanged-bosons, while in light-quark decays it can be generated due to the long-distance effects. The contributions of sfermions also alter the branching-ratios destructively or constructively depend on the phases of complex couplings of the $R-$parity violation interactions.