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The s-wave pion-kaon ($\pi K$) scattering lengths at zero momentum arecalculated in lattice QCD with sufficiently light $u/d$ quarks and strangequark at its physical value by the finite size formula. The light quark massescorrespond to $m_\pi = 0.330 - 0.466$ GeV. In the "Asqtad" improved staggeredfermion formulation, we measure the $\pi K$ four-point correlators for bothisospin $I=1/2$ and 3/2 channels, and analyze the lattice simulation data atthe next-to-leading order in the continuum three-flavor chiral perturbationtheory, which enables us a simultaneous extrapolation of $\pi K$ scatteringlengths at physical point. We adopt a technique with the moving wall sourceswithout gauge fixing to obtain the substantiable accuracy, moreover, for $I =1/2$ channel, we employ the variational method to isolate the contaminationfrom the excited states. Extrapolating to the physical point yields thescattering lengths as $m_\pi a_{3/2} = -0.0505(19)$ and $m_\pi a_{1/2} =0.1827(37)$ for $I=3/2$ and 1/2 channels, respectively. Our simulation resultsfor $\pi K$ scattering lengths are in agreement with the experimental reportsand theoretical predictions, and can be comparable with other latticesimulations. These simulations are carried out with MILC $N_f = 2+1$ flavorgauge configurations at lattice spacing $a \approx 0.15$ fm.

Lattice study on $\pi K $ scattering with moving wall source