Searching for low-lying multi-particle thresholds in lattice spectroscopy

We explore the Euclidean-time tails of odd-parity nucleon correlation functions in a search for the S$S$-wave pion–nucleon scattering-state threshold contribution. The analysis is performed using 2+1$2+1$ flavor 32³×64${32}^3\times 64$ PACS-CS gauge configurations available via the ILDG. Correlation matrices composed with various levels of fermion source/sink smearing are used to project low-lying states. The consideration of 25,600 fermion propagators reveals the presence of more than one state in what would normally be regarded as an eigenstate-projected correlation function. This observation is in accord with the scenario where the eigenstates contain a strong mixing of single and multi-particle states but only the single particle component has a strong coupling to the interpolating field. Employing a two-exponential fit to the eigenvector-projected correlation function, we are able to confirm the presence of two eigenstates. The lower-lying eigenstate is consistent with a Nπ$N\pi$ scattering threshold and has a relatively small coupling to the three-quark interpolating field. We discuss the impact of this small scattering-state contamination in the eigenvector projected correlation function on previous results presented in the literature.