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

Abstract

Triggered by a recent LHCb measurement and prospects for Belle II, we derive $U$-spin symmetry relations between integrated $CP$ asymmetries of three-body ${\mathrm{\ensuremath{\Lambda}}}_{c}^{+}$ and ${\mathrm{\ensuremath{\Xi}}}_{c}^{+}$ decays. The sum rules read ${A}_{CP}({\mathrm{\ensuremath{\Lambda}}}_{c}^{+}\ensuremath{\rightarrow}p{K}^{\ensuremath{-}}{K}^{+})+{A}_{CP}({\mathrm{\ensuremath{\Xi}}}_{c}^{+}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Sigma}}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+})=0$, ${A}_{CP}({\mathrm{\ensuremath{\Lambda}}}_{c}^{+}\ensuremath{\rightarrow}p{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+})+{A}_{CP}({\mathrm{\ensuremath{\Xi}}}_{c}^{+}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Sigma}}}^{+}{K}^{\ensuremath{-}}{K}^{+})=0$, and ${A}_{CP}({\mathrm{\ensuremath{\Lambda}}}_{c}^{+}\ensuremath{\rightarrow}{\mathrm{\ensuremath{\Sigma}}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}{K}^{+})+{A}_{CP}({\mathrm{\ensuremath{\Xi}}}_{c}^{+}\ensuremath{\rightarrow}p{K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+})=0$. No such $U$-spin sum rule exists between ${A}_{CP}({\mathrm{\ensuremath{\Lambda}}}_{c}^{+}\ensuremath{\rightarrow}p{K}^{\ensuremath{-}}{K}^{+})$ and ${A}_{CP}({\mathrm{\ensuremath{\Lambda}}}_{c}^{+}\ensuremath{\rightarrow}p{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\pi}}^{+})$. 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.