An approach for relating the nucleon excited states extracted from lattice QCD and the nucleon resonances of experimental data has been developed using the Hamiltonian effective field theory (HEFT) method. By formulating HEFT in the finite volume of
the lattice, the eigenstates of the Hamiltonian model can be related to the energy eigenstates observed in Lattice simulations. By taking the infinite-volume limit of HEFT, information from the lattice is linked to experiment. The approach opens a new window for the study of experimentally-observed resonances from the first principles of lattice QCD calculations. With the Hamiltonian approach, one not only describes the spectra of lattice-QCD eigenstates through the eigenvalues of the finite-volume Hamiltonian matrix, but one also learns the composition of the lattice-QCD eigenstates via the eigenvectors of the Hamiltonian matrix. One learns the composition of the states in terms of the meson-baryon basis states considered in formulating the effective field theory. One also learns the composition of the resonances observed in Nature. In this paper, we will focus on recent breakthroughs in our understanding of the structure of the $N^*(1535)$, $N^*(1440)$ and $Lambda^*(1405)$ resonances using this method.
Drawing on experimental data for baryon resonances, Hamiltonian effective field theory (HEFT) is used to predict the positions of the finite-volume energy levels to be observed in lattice QCD simulations. We have studied the low-lying baryons $N^*(15
35)$, $N^*(1440)$, and $Lambda(1405)$. In the initial analysis, the phenomenological parameters of the Hamiltonian model are constrained by experiment and the finite-volume eigenstate energies are a prediction of the model. The agreement between HEFT predictions and lattice QCD results obtained at finite volume is excellent. These lattice results also admit a more conventional analysis where the low-energy coefficients are constrained by lattice QCD results, enabling a determination of resonance properties from lattice QCD itself. The role and importance of various components of the Hamiltonian model are examined in the finite volume. The analysis of the lattice QCD data can help us to undertand the structure of these states better.
The pole structure of the $Lambda(1405)$ is examined by fitting the couplings of an underlying Hamiltonian effective field theory to cross sections of $K^- p$ scattering in the infinite-volume limit. Finite-volume spectra are then obtained from the t
heory, and compared to lattice QCD results for the mass of the $Lambda(1405)$. Momentum-dependent, non-separable potentials motivated by the well-known Weinberg-Tomozawa terms are used, with SU(3) flavour symmetry broken in the couplings and masses. In addition, we examine the effect on the behaviour of the spectra from the inclusion of a bare triquark-like isospin-zero basis state. It is found that the cross sections are consistent with the experimental data with two complex poles for the $Lambda(1405)$, regardless of whether a bare baryon basis state is introduced or not. However, it is apparent that the bare baryon is important for describing the results of lattice QCD at high pion masses.
