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

We have investigated the cross section for the ${\pi }^{-}p\to {K}^{0}\pi \Sigma $ and $pp\to p{K}^{+}\pi \Sigma $ reactions, paying attention to a mechanism that develops a triangle singularity. The triangle diagram is realized by the decay of a ${N}^{*}$ to ${K}^{*}\Sigma $ and the ${K}^{*}$ decay into $\pi K$, and the $\pi \Sigma $ finally merges into $\Lambda \left(1405\right)$. The mechanism is expected to produce a peak around $2140\phantom{\rule{4pt}{0ex}}\mathrm{MeV}$ in the $K\Lambda \left(1405\right)$ invariant mass. We found that a clear peak appears around $2100\phantom{\rule{4pt}{0ex}}\mathrm{MeV}$ in the $K\Lambda \left(1405\right)$ invariant mass, which is about $40\phantom{\rule{4pt}{0ex}}\mathrm{MeV}$ lower than the expectation, and that is due to the resonance peak of a ${N}^{*}$ resonance which plays a crucial role in the ${K}^{*}\Sigma $ production. The mechanism studied produces the peak of the $\Lambda \left(1405\right)$ around or below 1400 MeV, as is seen in the $pp\to p{K}^{+}\pi \Sigma $ HADES experiment.