[Paper Review] Nucleon Excited States from Lattice QCD and Hamiltonian Effective Field Theory

Jia-Jun Wu, Jonathan M. M. Hall+7 more|arXiv (Cornell University)|May 14, 2018

Quantum Chromodynamics and Particle Interactions23 references3 citations

TL;DR

This paper develops a Hamiltonian Effective Field Theory (HEFT) framework that bridges lattice QCD calculations and experimental nucleon resonances by relating finite-volume energy eigenstates to infinite-volume scattering amplitudes. It reveals that the $N^*(1535)$ is predominantly a three-quark state, $\Lambda^*(1405)$ is a molecular $\bar{K}N$ bound state, and $N^*(1440)$ arises from strong meson-baryon rescattering, resolving long-standing puzzles in the low-lying baryon spectrum.

ABSTRACT

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 $Λ^*(1405)$ resonances using this method.

Motivation & Objective

• To resolve the long-standing ambiguity in the internal structure of low-lying nucleon and hyperon resonances, particularly $N^*(1535)$, $\Lambda^*(1405)$, and $N^*(1440)$.
• To establish a consistent theoretical framework linking finite-volume lattice QCD results with infinite-volume experimental observables.
• To determine the composition of lattice QCD eigenstates in terms of quark-model and meson-baryon basis states using eigenvectors of the finite-volume Hamiltonian.
• To test the validity of quark-model predictions against lattice QCD data and experimental resonance data.
• To clarify whether observed lattice states correspond to radial excitations or dynamically generated resonances.

Proposed method

• Formulate Hamiltonian Effective Field Theory (HEFT) in finite volume to model the spectrum of nucleon excited states.
• Construct a finite-volume Hamiltonian matrix using meson-baryon and three-quark basis states, with interactions derived from chiral dynamics.
• Solve the eigenvalue problem of the finite-volume Hamiltonian to extract energy levels matching lattice QCD simulations.
• Extract eigenvectors of the Hamiltonian matrix to determine the composition of lattice eigenstates in terms of basis states.
• Take the infinite-volume limit of the finite-volume HEFT framework to connect to experimental scattering amplitudes and $T$-matrix observables.
• Compare model predictions with lattice QCD data (e.g., from CSSM and Lang et al.) and experimental phase shifts to constrain resonance structure.

Experimental results

Research questions

• RQ1 What is the internal quark-baryon composition of the $N^*(1535)$ resonance as revealed by lattice QCD and HEFT?
• RQ2 Is the $\Lambda^*(1405)$ resonance primarily a molecular $\bar{K}N$ bound state or a three-quark excitation?
• RQ3 Does the $N^*(1440)$ resonance arise from a radial excitation of the nucleon quark core or from meson-baryon rescattering?
• RQ4 How do the eigenvectors of the finite-volume Hamiltonian matrix reflect the structure of lattice QCD eigenstates?
• RQ5 To what extent do lattice QCD results for the nucleon excited spectrum support or contradict the quark model and harmonic oscillator predictions?

Key findings

• The $N^*(1535)$ resonance is predominantly composed of a three-quark core, with a bare state component of approximately 50%, consistent with its dominance in local three-quark operator correlation functions.
• The $\Lambda^*(1405)$ resonance is best described as a molecular $\bar{K}N$ bound state, with its lowest-energy configuration arising from meson-baryon interaction rather than a three-quark core.
• The $N^*(1440)$ resonance is not a radial excitation of the three-quark core but is instead dynamically generated via strong $\pi N$ rescattering, with the first radial excitation predicted at around 1.9 GeV.
• Lattice QCD data show that the first excited state of the $\Lambda$ baryon is around 1.6 GeV, consistent with the $\Lambda^*(1405)$ resonance being a molecular state.
• The first radial excited nucleon state, predicted by the quark model and harmonic oscillator, appears at approximately 1.9 GeV, matching lattice QCD results and supporting the validity of the quark model for higher excitations.
• The finite-volume spectrum from HEFT, when compared with lattice data (e.g., CSSM and Lang et al.), confirms that states dominated by local three-quark operators are consistent with the quark model, while those dominated by non-local operators are molecular in nature.