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

A $\Sigma^*$ resonance with spin-parity $J^P = 1/2^-$ and mass in the vicinity of the $\bar{K}N$ threshold has been predicted in the unitary chiral approach 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 favored weak 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 valuable information on the nature of the $\Sigma^*$ resonance and can in principle be tested by facilities such as LHCb, BelleII and BESIII.