$U$-spin sum rules for $CP$ asymmetries of three-body charmed baryon decays

Triggered by a recent LHCb measurement and prospects for Belle II, we derive $U$-spin symmetry relations between integrated $CP$ asymmetries of three-body ${\mathrm{\Lambda }}_c^{+}$ and ${\mathrm{\Xi }}_c^{+}$ decays. The sum rules read $A_{CP}({\mathrm{\Lambda }}_c^{+}\to p{K}^{-}{K}^{+})+A_{CP}({\mathrm{\Xi }}_c^{+}\to {\mathrm{\Sigma }}^{+}{\pi }^{-}{\pi }^{+})=0$, $A_{CP}({\mathrm{\Lambda }}_c^{+}\to p{\pi }^{-}{\pi }^{+})+A_{CP}({\mathrm{\Xi }}_c^{+}\to {\mathrm{\Sigma }}^{+}{K}^{-}{K}^{+})=0$, and $A_{CP}({\mathrm{\Lambda }}_c^{+}\to {\mathrm{\Sigma }}^{+}{\pi }^{-}{K}^{+})+A_{CP}({\mathrm{\Xi }}_c^{+}\to p{K}^{-}{\pi }^{+})=0$. No such $U$-spin sum rule exists between $A_{CP}({\mathrm{\Lambda }}_c^{+}\to p{K}^{-}{K}^{+})$ and $A_{CP}({\mathrm{\Lambda }}_c^{+}\to p{\pi }^{-}{\pi }^{+})$. All of these sum rules are associated with a complete interchange of $d$ and $s$ quarks. Furthermore, there are no $U$-spin $CP$ asymmetry sum rules which hold to first order $U$-spin breaking.

Figure 1

Diagrams in $\mathcal{A}({\mathrm{\Lambda }}_c^{+}\to p{K}^{-}{K}^{+})=(\mathrm{\Sigma }+\mathrm{\Delta })(-T-C_1)+(-\mathrm{\Sigma }+\mathrm{\Delta })(-E_2)+\mathrm{\Delta }(-P_1)$. All diagrams have been drawn using JaxoDraw [69, 70].

Figure 2

Diagrams in $\mathcal{A}({\mathrm{\Xi }}_c^{+}\to {\mathrm{\Sigma }}^{+}{\pi }^{-}{\pi }^{+})=(-\mathrm{\Sigma }+\mathrm{\Delta })(-T-C_1)+(\mathrm{\Sigma }+\mathrm{\Delta })(-E_2)+\mathrm{\Delta }(-P_1)$.

Figure 3

Diagrams in $\mathcal{A}({\mathrm{\Lambda }}_c^{+}\to {\mathrm{\Sigma }}^{+}{\pi }^{-}{K}^{+})=(\mathrm{\Sigma }+\mathrm{\Delta })(-C_2)+(-\mathrm{\Sigma }+\mathrm{\Delta })(-E_1)+\mathrm{\Delta }(-P_2)$.

Figure 4

Diagrams in $\mathcal{A}({\mathrm{\Xi }}_c^{+}\to p{K}^{-}{\pi }^{+})=(-\mathrm{\Sigma }+\mathrm{\Delta })(-C_2)+(\mathrm{\Sigma }+\mathrm{\Delta })(-E_1)+\mathrm{\Delta }(-P_2)$.

Figure 5

Diagrams in $\mathcal{A}({\mathrm{\Lambda }}_c^{+}\to p{\pi }^{-}{\pi }^{+})=(-\mathrm{\Sigma }+\mathrm{\Delta })(-T-C_1-C_2-E_1-E_2)+\mathrm{\Delta }(-P_1-P_2)$.

Figure 6

Diagrams in $\mathcal{A}({\mathrm{\Xi }}_c^{+}\to {\mathrm{\Sigma }}^{+}{K}^{-}{K}^{+})=(\mathrm{\Sigma }+\mathrm{\Delta })(-T-C_1-C_2-E_1-E_2)+\mathrm{\Delta }(-P_1-P_2)$.