Pole structure of \(P_\psi^N(4312)^+\) via machine learning and uniformized S-matrix

We probed the pole structure of the \(P_\psi^{N}(4312)^{+}\) using a trained deep neural network. The training dataset was generated using uniformized independent S-matrix poles to ensure that the obtained interpretation is as model-independent as possible. To prevent possible ambiguity in the interpretation of the pole structure, we included the contribution from the off-diagonal element of the S-matrix. Five out of the six neural networks we trained favor \(P_\psi^{N}(4312)^{+}\) as possibly having a three-pole structure, with one pole on each of the unphysical sheets - a first in its report. The two poles can be associated to a pole-shadow pair which is a characteristic of a true resonance. On the other hand, the last pole is most likely associated with the coupled-channel effect. The combined effect of these poles produced a peak below the \(\Sigma^{+}_C\bar{D}^0\) which mimic the line shape of a hadronic molecule.