S-wave pi-K scattering length from lattice QCD

The $S$-wave $\pi K$ scattering lengths are calculated for both the isospin 1/2 and 3/2 channels in the lattice QCD by using the finite size formula. We perform the calculation with $N_f=2+1$ gauge configurations generated on $32^3 \times 64$latticeusingtheIwasakigaugeactionandnonperturbatively$O(a)$-improvedWilsonactionat$1/a = 2.17$ GeV. The quark masses correspond to $m_{\pi }=0.30-0.70$ GeV. For $I=1\mathrm{/}2$, to separate the contamination from excited states, we construct a $2\times 2$ matrix of the time correlation function and diagonalize it. Here, we adopt the two kinds of operators, $\bar{s}u$ and $\pi K$. It is found that the signs of the scattering lengths are in agreement with experiment, namely attraction in $I=1\mathrm{/}2$ and repulsion in $I=3\mathrm{/}2$. We investigate the quark-mass dependence of the scattering lengths and also discuss the limitation of chiral perturbation theory.