Resonances in Coupled $\pi K\text{−}\eta K$ Scattering from Quantum Chromodynamics

Using a first-principles calculation within quantum chromodynamics, we are able to determine a pattern of $\text{strangeness}=1$ resonances that appear as complex singularities within coupled $\pi K\text{−}\eta K$ scattering amplitudes. We make use of numerical computation in the lattice discretized approach to the quantum field theory with light quark masses corresponding to $m_{\pi }\sim 400\mathrm{MeV}$ and at a single lattice spacing. The energy dependence of scattering amplitudes is extracted through their relationship to the discrete spectrum in a finite volume, which we map out in unprecedented detail.

Figure 1

Discrete spectrum of c.m.-frame energies obtained from variational analysis of correlation matrices featuring single-meson and meson-meson operators at three spatial volumes. Red bands are $\pi K$ noninteracting level positions, and green bands are $\eta K$ noninteracting level positions. (a) $\overrightarrow{P}=[000]$, $\mathrm{\Lambda }=A_1^{+}$ spectrum (dominated by $J^P=0^{+}$ with negligible contributions from $J\ge 4$). (b) $\overrightarrow{P}=[001]$, $\mathrm{\Lambda }=A_1$ spectrum ($J^P=0^{+},1^{-},2^{+}$ all contribute).

Figure 2

(a) $J^P=0^{+}$ amplitudes. Open circles on axis show $\pi K$ and $\eta K$ thresholds. Upper panel: $\pi K$ and $\eta K$ phase shifts in degrees. Lower panel: inelasticity. Points in center show the energy levels on three volumes used to constrain the $t$-matrix extraction; larger solid points show $\overrightarrow{P}=\overrightarrow{0}$, and smaller open circles show $\overrightarrow{P}\ne \overrightarrow{0}$. (b) $J^P=1^{-}$ around the $\pi K$ threshold. Points determined directly without parametrization of the vector amplitude from three volumes: $16^3$ (boxes), $20^3$ (circles), and $24^3$ (triangles). Curve shows the result of a relativistic Breit-Wigner parametrization $p^3\cot {\delta }_1=(m_R^2-s)[(6\pi \sqrt{s})/g_R^2]$. (c) $J^P=2^{+}$ amplitudes. Open circles on axis show $\pi K$, $\eta K$, and $\pi \pi K$ thresholds.

Figure 3

Pole singularities of partial-wave $t$ matrices in the complex plane for $J^P=0^{+}$ (red), $1^{-}$ (orange), and $2^{+}$ (green). Squares correspond to poles found on unphysical sheets, and the circle is a physical sheet bound state. Uncertainties include variation under changes in parametrization form.