The $\mathrm{\Lambda }(1405)$ from Lattice QCD: Determining the Finite-volume Spectra

This work presents technical details of determining the finite-volume energy spectra for the scattering amplitude of the coupled-channel $\pi \mathrm{\Sigma }-\bar{K}N$ from lattice QCD data. The importance of reliably extracting such spectra lies in the crucial dependence of the hadronic scattering amplitudes analysis on the energy spectrum when using Lüscher’s formalism. Results of the methods used are presented and the final finite-volume spectra are shown. The analysis of the scattering amplitude based on these results, exhibits a two-pole structure for the $\mathrm{\Lambda }(1405)$, a virtual bound state below the $\pi \mathrm{\Sigma }$ threshold and a resonance pole right below the $\bar{K}N$ threshold.