Equivalent $SU(3)_f$ approaches for two-body anti-triplet charmed baryon decays

For the two-body ${\bf B}_c\to{\bf B}M$ decays, where ${\bf B}_c$ denotes the anti-triplet charm baryon and ${\bf B}(M)$ the octet baryon (meson), there exist two theoretical studies based on the $SU(3)$ flavor [$SU(3)_f$] symmetry. One is the irreducible $SU(3)_f$ approach (IRA). In the irreducible $SU(3)_f$ representation, the effective Hamiltonian related to the initial and final states forms the amplitudes for ${\bf B}_c\to{\bf B}M$. The other is the topological-diagram approach (TDA), where the $W$-boson emission and $W$-boson exchange topologies are drawn and parameterized for the decays. As required by the group theoretical consideration, we present the same number of the IRA and TDA amplitudes. We can hence relate the two kinds of the amplitudes, and demonstrate the equivalence of the two $SU(3)_f$ approaches. We find a specific $W$-boson exchange topology only contributing to $\Xi_c^0\to{\bf B}M$. Denoted by $E_M$, it plays a key role in explaining ${\cal B}(\Xi_c^0\to \Lambda^0 K_S^0,\Sigma^0 K_S^0,\Sigma^+ K^-)$. We consider that $\Lambda_c^+ \to n \pi^+$ and $\Lambda_c^+ \to p\pi^0$ proceed through the constructive and destructive interfering effects, respectively, which leads to ${\cal B}(\Lambda_c^+ \to n \pi^+)\gg{\cal B}(\Lambda_c^+ \to p\pi^0)$ in agreement with the data. With the exact and broken $SU(3)_f$ symmetries, we predict the branching fractions of ${\bf B}_c\to{\bf B}M$ to be tested by future measurements.