The $\Lambda_c^+\to\Lambda\pi^+\pi^+\pi^-$ reaction, and a triangle singularity producing the $\Sigma^*(1430)$ state

We study the decay $\Lambda_c^+ \to \Lambda \pi^+ \pi^+ \pi^-$, focusing on the production of the $\Sigma^*(1430)$ resonance observed by the Belle Collaboration. Interpreted as a dynamically generated state from meson-baryon interactions in the chiral unitary approach, the $\Sigma^*(1430)$ signal is shown to be enhanced by a triangle singularity involving intermediate $K^{*-}$, $p$, and $\bar K^0$ states. This mechanism leads to a sharp peak near 1434 MeV in the $\pi^+ \Lambda$ invariant mass distribution, in agreement with the experimental observations, and predicts a secondary peak around 1875 MeV in the $\pi^- \Sigma^*(1430)$ spectrum tied to the triangle singularity. We also estimate the branching ratio of $\Lambda_c^+ \to \pi^+ \pi^- \Sigma^*(1430)$ to be about $3.5 \times 10^{-4}$. The results for the branching ratio and the $\pi^- \Sigma^*(1430)$ mass distributions are predictions of the theoretical approach, which could be tested with reanalysis of existing data.