Sum rules for $CP$ asymmetries of charmed baryon decays in the $SU(3)_F$ limit

Motivated by the recent LHCb observation of $CP$ violation in charm, we study $CP$ violation in the charmed baryon decays. A simple method to search for the $CP$ violation relations in the flavor $SU(3)$ limit, which is associated with a complete interchange of $d$ and $s$ quarks, is proposed. With this method, hundreds of $CP$ violation sum rules in the doubly and singly charmed baryon decays can be found. As examples, the $CP$ violation sum rules in two-body charmed baryon decays are presented. Some of the $CP$ violation sum rules could help the experiment to find better observables. As byproducts, the branching fraction of $\Xi^+_c\to pK^-\pi^+$ is predicted to be $(1.7\pm 0.5)\%$ in the $U$-spin limit and the fragmentation-fraction ratio is determined as $f_{\Xi_b}/f_{\Lambda_b}=0.065\pm 0.020$ using the LHCb data.