Low-lying baryon resonances from lattice QCD

TL;DR

The paper develops a lattice QCD framework to extract baryon resonance properties from finite-volume spectra by combining a large basis of multi-hadron operators with a multichannel K-matrix quantization condition anchored by a box matrix. It demonstrates the method on the Δ resonance and on Λ(1405), the latter exhibiting a two-pole structure, and discusses the roadmap toward studying the Roper resonance with upcoming three-particle analyses. Technical innovations include variational diagonalization of correlation matrices, LapH smearing for all spatial sites, and stabilizing the quantization condition via a Cayley-transform reformulation. The results validate the approach and underscore the need for explicit multi-hadron and three-particle amplitudes to achieve a complete understanding of baryon resonances.

Abstract

Calculating the properties of baryon resonances from quantum chromodynamics requires evaluating the temporal correlations between hadronic operators using integrations over field configurations weighted by a phase associated with the action. By formulating quantum chromodynamics on a space-time lattice in imaginary time, such integrations can be carried out non-perturbatively using a Markov-chain Monte Carlo method with importance sampling. The energies of stationary states in the finite volume of the lattice can be extracted from the temporal correlations. A quantization condition involving the scattering $K$-matrix and a complicated ``box matrix'' also yields a finite-volume energy spectrum. By appropriately parametrizing the scattering $K$-matrix, the best-fit values of the $K$-matrix parameters are those that produce a finite-volume spectrum which most closely matches that obtained from the Monte Carlo computations. Results for the $Δ$ resonance are presented, and a study of scattering for energies near the $Λ(1405)$ resonance is outlined, showing a two pole structure. The prospects for applying this methodology to the Roper resonance are discussed.

Figures (7)

• Figure 1: The spatial arrangements of the quark-antiquark meson operators (top) and the three-quark baryon operators (bottom) that we use. Smeared quarks fields are shown as solid circles, each hollow circle indicates a smeared antiquark field, the solid line segments indicate covariant displacements, and each hollow box indicates the location of a Levi-Civita color coupling.
• Figure 2: (a) Effective energies associated with the diagonal elements of the original raw correlator matrix $C(t)$ of the toy model, whose energies are defined in Eq. (\ref{['eq:toy1']}) and the overlaps in Eq. (\ref{['eq:toy2']}). (b) Effective energies associated with the eigenvalues of the original correlator matrix $C(t)$ with eigenvector pinning used to label or order the eigenvalues. (c) Effective energies associated with the eigenvalues of $C(\tau_0)^{-1/2}C(t)C(\tau_0)^{-1/2}$ with eigenvector pinning, for $\tau_0=10$.
• Figure 3: The low-lying $I = 3/2$ (top) and $I = 1/2$ (bottom) nucleon-pion spectra in the center-of-momentum frame on the D200 ensemble as energies over the pion mass from Ref. Bulava:2022vpq. Each column corresponds to a particular irrep $\Lambda$ of the little group of total momentum $\bm{P}^2 = (2\pi/L_b)^2 \bm{d}^2$, denoted $\Lambda(\bm{d}^2)$. Dashed lines indicate the boundaries of the elastic region. Solid lines and shaded regions indicate non-interacting $N \pi$ levels and their associated statistical errors.
• Figure 4: $N\pi$ scattering phase shifts from Ref. Bulava:2022vpq for $I=\frac{3}{2}$$s$-wave (top left) and $p$-wave (top right) in their cotangent form multiplied with threshold momentum factors. The $p$-wave phase shift itself is shown in the bottom left. Similarly, $N\pi$ scattering phase shifts for $I=\frac{1}{2}$$s$-wave (bottom right). Lower panels indicate all of the energies used in the fits to obtain the phase shifts in the top panels.
• Figure 5: (Top) The $\pi N$ interacting two-hadron energy levels obtained in Ref. Alexandrou:2023elk. Box heights indicate estimated uncertainties. Horizontal dashed/dotted lines show various thresholds, as indicated by the legend. Noninteracting energies are shown by the green, thicker dashed lines. (Bottom) The $P$-wave scattering phase-shift as a function of the invariant mass $E_{\rm cm} = \sqrt{s}$. The error band is determined using jackknife resampling. The points with horizontal error bars show each fitted energy level included its jackknife error bar.