Two-pole nature of the $Λ(1405)$ from lattice QCD

Abstract

This letter presents the first lattice QCD computation of the coupled channel $πΣ-\bar{K}N$ scattering amplitudes at energies near $1405\,{\rm MeV}$. These amplitudes contain the resonance $Λ(1405)$ with strangeness $S=-1$ and isospin, spin, and parity quantum numbers $I(J^P)=0(1/2^-)$. However, whether there is a single resonance or two nearby resonance poles in this region is controversial theoretically and experimentally. Using single-baryon and meson-baryon operators to extract the finite-volume stationary-state energies to obtain the scattering amplitudes at slightly unphysical quark masses corresponding to $m_π\approx200$ MeV and $m_K\approx487$ MeV, this study finds the amplitudes exhibit a virtual bound state below the $πΣ$ threshold in addition to the established resonance pole just below the $\bar{K}N$ threshold. Several parametrizations of the two-channel $K$-matrix are employed to fit the lattice QCD results, all of which support the two-pole picture suggested by $SU(3)$ chiral symmetry and unitarity.

Figures (4)

• Figure 1: The $I=0$ and $S=-1$ coupled-channel $\pi\Sigma-\bar{K}N$ amplitude computed on a single lattice QCD gauge-field ensemble with $m_{\pi} \approx 200\,{\rm MeV}$ as a function of the energy difference to the $\pi\Sigma$ threshold in the center-of-mass frame. The upper panel shows the transition matrix elements, defined in Eq. (\ref{['eq:amplitude']}), using the $K$-matrix parametrization with the lowest AIC constrained by the finite-volume spectrum in the bottom panel. The second panel shows the model variation for the same quantities using several parametrizations. The third and fourth panels show the position of poles in the complex center-of-mass energy ($E_{\rm cm}$) plane on the sheets closest to the physical one: using the parametrization with lowest AIC (third panel), and for several parametrizations (fourth panel). In the second and fourth panel, the transparency of each line and corresponding pair of pole positions is proportional to $\exp{[- \left(\text{AIC} - \text{AIC}_\text{min}\right)/2]}$, where $\text{AIC}_\text{min}$ is the lowest AIC corresponding to the fit in Eq. (\ref{['eq:bestfit']}), which is also shown in the top panel. The subscripts $i,j$ index the two open scattering channels. In the lowest panel, the lattice QCD energy levels that serve as input to the amplitude analyses are displayed. For clarity, these energy levels are displaced vertically by the total spatial momentum $\boldsymbol d^2$ defined below Eq. (\ref{['e:det']}).
• Figure 2: Example determination of a finite-volume stationary-state energy, illustrating the sensitivity of the fitted energy to the lower end of the fit range ($t_{\rm min}$) for the lowest level of the $G_{\mathrm{1u}}(0)$ irrep. Each set of points corresponds to a different fit form. The two-exponential and geometric Bulava:2022vpq fits are performed to the diagonalized correlation function only. The single-exponential ratio fits are performed to the same correlator divided by either the product $\bar{K}(0)N(0)$ or $\pi(0)\Sigma(0)$ of correlators, and the lab frame energy $aE_{\rm lab}$ is reconstructed from the interaction shifts. The dark horizontal band and filled symbol denote the chosen fit.
• Figure 3: Finite-volume spectrum in the center-of-mass frame used as input data to constrain parametrizations of the coupled-channel $\pi\Sigma-\bar{K}N$ scattering amplitude. Each column corresponds to a particular irrep $\Lambda(\boldsymbol{d}^2)$ of the little group of total momentum $\boldsymbol{P}^2=(2\pi/L)^2\boldsymbol{d}^2$. Only irreps where the $\ell=0$ partial wave contributes are included. Dashed lines indicate various thresholds, as labeled. Model energies from the resultant scattering-amplitude fit are given by blue squares.
• Figure 4: The elastic $\pi\Sigma$ amplitude near threshold. The points are obtained from Eq. (\ref{['e:det']}) using a single channel and $\ell_{\rm max} = 0$. The shaded band denotes a fit of the four levels shown to a two-parameter effective range expansion. A pole on the real axis in the second Riemann sheet (a virtual bound state) occurs when $k_{\pi\Sigma}\cot \delta_{\pi\Sigma} - ik_{\pi\Sigma}=0$ below threshold. This is where the black dashed line intersects the fit.