$\Lambda(1405)$ mediated triangle singularity in the $K^-d\to p\Sigma^-$ reaction

We study for the first time the $p\Sigma^-\to K^-d$ and $K^-d\to p\Sigma^-$ reactions close to threshold and show that they are driven by a triangle mechanism, with the $\Lambda(1405)$, a proton and a neutron as intermediate states, which develops a triangle singularity close to the $\bar{K}d$ threshold. We find that a mechanism involving virtual pion exchange and the $K^-p\to\pi^+\Sigma^-$ amplitude dominates over another one involving kaon exchange and the $K^-p\to K^-p$ amplitude. Moreover, of the two $\Lambda(1405)$ states, the one with higher mass around $1420$ MeV, gives the largest contribution to the process. We show that the cross section, well within measurable range, is very sensitive to different models that, while reproducing $\bar{K}N$ observables above threshold, provide different extrapolations of the $\bar{K}N$ amplitudes below threshold. The observables of this reaction will provide new constraints on the theoretical models, leading to more reliable extrapolations of the $\bar{K}N$ amplitudes below threshold and to more accurate predictions of the $\Lambda(1405)$ state of lower mass.