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A $\Sigma^*$ resonance with spin-parity $J^P = 1/2^-$ and mass in thevicinity of the $\bar{K}N$ threshold has been predicted in the unitary chiralapproach and inferred from the analysis of CLAS data on the $\gamma p \to K^+\pi^0 \Sigma^0$ reaction. In this work, based on the dominant Cabibbo favoredweak decay mechanism, we perform a study of $\Lambda_c^+ \to \pi^+ \pi^0\Sigma^*$ with the possible $\Sigma^*$ state decaying into $\pi^- \Sigma^+$through a triangle diagram. This process is initiated by $ \Lambda_c^+ \to\pi^+ \bar{K}^*N $, then the $\bar{K}^*$ decays into $\bar{K} \pi$ and $\bar{K}N$ produce the $\Sigma^*$ through a triangle loop containing $\bar{K}^* N\bar{K}$ which develops a triangle singularity. We show that the $\pi^-\Sigma^+$ state is generated from final state interaction of $\bar{K}N$ in$S$-wave and isospin $I=1$, and the $\Lambda_c^+ \to \pi^+ \pi^0 \pi^-\Sigma^+$ decay can be used to study the possible $\Sigma^*$ state around the$\bar{K}N$ threshold. The proposed decay mechanism can provide valuableinformation on the nature of the $\Sigma^*$ resonance and can in principle betested by facilities such as LHCb, BelleII and BESIII.

Search for the $\Sigma^*$ state in $\Lambda^+_c \to \pi^+ \pi^0 \pi^-\Sigma^+$ decay by triangle singularity