S-wave $\pi-K$ scattering length from lattice QCD

We present the S-wave $\pi-K$ scattering lengths for both the isospin 1/2 and 3/2 channels evaluated by using the finite size formula. We utilize the $N_f=2+1$ gauge configurations generated on $32^3 \times 64$ lattice using the Iwasaki gauge action and the $O(a)$-improved Wilson action at $1/a = 2.17$ GeV. The quark masses correspond to $m_\pi$ = 0.30 - 0.70 GeV. For $I=1/2$, to separate the effects 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 $\pi-K$. Our preliminary results show signs of the scattering lengths in agreement with experiment, namely attraction in $I=1/2$ and repulsion in $I=3/2$. We investigate the quark-mass dependence of the scattering length and also discuss the limitation of chiral perturbation theory.