$I=1/2$ $S$-wave and $P$-wave $K\pi$ scattering and the $\kappa$ and $K^{\ast }$ resonances from lattice QCD

Abstract

We present a lattice-QCD determination of the elastic isospin-$1/2$ $S$-wave and $P$-wave $K\pi$ scattering amplitudes as a function of the center-of-mass energy using Lüscher's method. We perform global fits of $K$-matrix parametrizations to the finite-volume energy spectra for all irreducible representations with total momenta up to $\sqrt{3}\frac{2\pi }{L}$; this includes irreps that mix the $S$- and $P$-waves. Several different parametrizations for the energy dependence of the $K$-matrix are considered. We also determine the positions of the nearest poles in the scattering amplitudes, which correspond to the broad $\kappa$ resonance in the $S$-wave and the narrow $K^{\ast }(892)$ resonance in the $P$-wave. Our calculations are performed with $2+1$ dynamical clover fermions for two different pion masses of $317.2(2.2)$ and $175.9(1.8)$ MeV. Our preferred $S$-wave parametrization is based on a conformal map and includes an Adler zero; for the $P$-wave we use a standard pole parametrization including Blatt-Weisskopf barrier factors. The $S$-wave $\kappa$-resonance pole positions are found to be $\left[0.86(12)-0.309(50)i\right]\mathrm{GeV}$ at the heavier pion mass and $\left[0.499(55)-0.379(66)i\right]\mathrm{GeV}$ at the lighter pion mass. The $P$-wave $K^{\ast }$-resonance pole positions are found to be $\left[0.8951(64)-0.00250(21)i\right]\mathrm{GeV}$ at the heavier pion mass and $\left[0.8718(82)-0.0130(11)i\right]\mathrm{GeV}$ at the lighter pion mass, which corresponds to couplings of $g_{K^{\ast }K\pi }=5.02(26)$ and $g_{K^{\ast }K\pi }=4.99(22)$, respectively.

Comments: 25 pages, 10 figures