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Predictive $CP$ Violating Relations for Charmless Two-body Decays of Beauty Baryons $Ξ^{-,\;0}_b$ and $Λ_b^0$ With Flavor $SU(3)$ Symmetry

Several baryons containing a heavy b-quark have been discovered. The decays of these states provide new platform for testing the standard model (SM). We study $CP$ violation in SM for charmless two-body decays of the flavor $SU(3)$ anti-triplet beauty baryon (b-baryon) ${\cal B} = (Ξ^-_b,\;Ξ^0_b,\;Λ_b^0)$ in a model independent way. We found, in the flavor $SU(3)$ symmetry limit, a set of new predictive relations among the branching ratio $Br$ and $CP$ asymmetry $A_{CP}$ for $\cal B$ decays, such as $A_{CP}(Ξ_b^-\to K^0 Ξ^-)/A_{CP}(Ξ_b^-\to \bar K^0 Σ^-) = - Br(Ξ_b^- \to \bar K^0 Σ^-)/Br(Ξ_b^- \to K^0 Σ^-)$, $A_{CP}(Λ_b^0 \to π^- p)/A_{CP} (Ξ^0_b \to K^- Σ^+) = -Br(Ξ^0_b \to K^- Σ^+)τ_{Λ_b^0}/Br(Λ_b^0 \to π^- p)τ_{Ξ^0_b}$,and $A_{CP}(Λ_b^0 \to K^- p)/ A_{CP} (Ξ^0_b \to π^- Σ^+) = -Br(Ξ^0_b \to π^- Σ^+)τ_{Λ_b^0}/ Br(Λ_b^0 \to K^- p)τ_{Ξ^0_b}$. Future data from LHCb can test these relations and also other relations found.