Three-body charmed baryon Decays with SU(3) flavor symmetry

We study the three-body anti-triplet ${\mathbf{B}}_{\mathbf{c}}\to {\mathbf{B}}_{\mathbf{n}}M{M}^{\mathrm{\prime }}$ decays with the $SU(3)$ flavor ($SU(3{)}_f$) symmetry, where ${\mathbf{B}}_{\mathbf{c}}$ denotes the charmed baryon anti-triplet of $({\mathrm{\Xi }}_c^0,-{\mathrm{\Xi }}_c^{+},{\mathrm{\Lambda }}_c^{+})$, and ${\mathbf{B}}_{\mathbf{n}}$ and $M(M^{\mathrm{\prime }})$ represent baryon and meson octets, respectively. By considering only the S-wave $M{M}^{\mathrm{\prime }}$-pair contributions without resonance effects, the decays of ${\mathbf{B}}_{\mathbf{c}}\to {\mathbf{B}}_{\mathbf{n}}M{M}^{\mathrm{\prime }}$ can be decomposed into irreducible forms with 11 parameters under $SU(3{)}_f$, which are fitted by the 14 existing data, resulting in a reasonable value of ${\chi }^2\mathrm{/}d\mathrm{.}o\mathrm{.}f=2.8$ for the fit. Consequently, we find that the triangle sum rule of ${\cal A}(\Lambda_c^+\to n\bar K^0 \pi^+)-{\cal A}(\Lambda_c^+\to pK^- \pi^+)-\sqrt 2 {\cal A}(\Lambda_c^+\to p\bar K^0 \pi^0)=0$givenbytheisospinsymmetryholdsunder$SU(3)_f$,where${\cal A}$standsforthedecayamplitude\mathrm{.}Inaddition,wepredictthat${\cal B}(\Lambda_c^+\to n \pi^{+} \bar{K}^{0})=(0.9\pm 0.8)\times 10^{-2}$,whichis$3-4$ times smaller than the BESIII observation, indicating the existence of the resonant states. For the to-be-observed ${\mathbf{B}}_{\mathbf{c}}\to {\mathbf{B}}_{\mathbf{n}}M{M}^{\mathrm{\prime }}$ decays, we compute the branching fractions with the $SU(3{)}_f$ amplitudes to be compared to the BESIII and LHCb measurements in the future.