The study of weak decays induced by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msup><mml:mrow><mml:mfrac><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfrac></mml:mrow><mml:mrow><mml:mo linebreak="badbreak" linebreakstyle="aft

In this study, we analyzed the weak decays induced by $J^P = \frac{1}{2}^{+} \to \frac{3}{2}^{-} $ transitions within the light-cone sum rules. Specifically, semileptonic decays of the bottom baryons into the P-wave baryons $\Lambda_b \to \Lambda_c(2625) \ell \nu_l$ and $\Xi_b \to \Xi_c(2815) \ell \nu_l$ as well as nonleptonic $\Lambda_b \to \Lambda_c(2625) \pi (\rho)$ and $\Xi_b \to \Xi_c(2815) \pi (\rho)$ decays are investigated. The form factors for the considered transitions are obtained within the sum rules method. With the calculated form factors, the decay widths of the processes are determined. Up to now, only the decay width for $\Lambda_b^0 \to \Lambda_c^+ \mu^- \nu_\mu$ has been measured among the considered decays, and we observe that our finding is quite compatible with the measurement. We also compare our results with the predictions of other approaches.