Probing Direct $CP$ Violation in $\Lambda_b^0 \to P_c^+ h^-$ $(h=\pi,K)$ with Final-State Rescattering

The LHCb collaboration has recently reported a measurement of the difference in direct CP asymmetries for the decays $\Lambda_b^0 \to J/\psi \, p \, h^-$ (with $h = K, \pi$), offering new experimental constraints on the decay dynamics of heavy baryons into charmonium final states. Inspired by these findings, we explore the branching ratios and direct CP violations for the decays $\Lambda_b^0 \to P_c^+(4312, 4440, 4457)\,h^-$ within the framework of final-state rescattering. Our analysis indicates that the branching fractions for $\Lambda_b^0 \to P_c^+ \pi^-$ lie around the $10^{-6}$ level, with the corresponding direct CP asymmetries approaching approximately $1\%$. In contrast, the direct CP violation for the decay $\Lambda_b^0 \to P_c^+ K^-$ is found to be very small, while its branching ratios show a strong dependence on the spin assignments of the $P_c$ states. These predictions may provide useful guidance for more precise CP measurements and amplitude analyses in the $P_c$ region in future experiments.