The role of the triangle singularity in $\mathrm{\Lambda }(1405)$ production in the ${\pi }^{-}p\to K^0\pi \mathrm{\Sigma }$ and $pp\to p{K}^{+}\pi \mathrm{\Sigma }$ processes

We have investigated the cross section for the $\pi^-p\rightarrow K^0\pi\Sigma$and$pp\rightarrow pK^+\pi\Sigma$ reactions paying attention to a mechanism that develops a triangle singularity. The triangle diagram is realized by the decay of a $N^{\ast }$ to $K^{\ast }\mathrm{\Sigma }$ and the $K^{\ast }$ decay into $\pi K$,andthe$\pi\Sigma$finallymergesinto$\Lambda(1405)$. The mechanism is expected to produce a peak around $2140$ MeV in the $K\mathrm{\Lambda }(1405)$ invariant mass. We found that a clear peak appears around $2100$ MeV in the $K\mathrm{\Lambda }(1405)$ invariant mass which is about $40$ MeV lower than the expectation, and that is due to the resonance peak of a $N^{\ast }$ resonance which plays a crucial role in the $K^{\ast }\mathrm{\Sigma }$ production. The mechanism studied produces the peak of the $\mathrm{\Lambda }(1405)$ around or below 1400 MeV, as is seen in the $pp\to p{K}^{+}\pi \mathrm{\Sigma }$ HADES experiment.