$I=\frac{3}{2}$ $\pi K$ $s$-wave scattering length from lattice QCD

The $I=\frac{3}{2}$ $\pi K$ $s$-wave scattering phase shift is computed by lattice quantum chromodynamics with $N_f=3$ flavors of Asqtad-improved staggered fermions. The energy-eigenvalues of $\pi K$ systems at one center of mass frame and six moving frames using moving wall source technique are used to get phase shifts by L\"uscher's formula and its extensions. The calculations are good enough to acquire effective range expansion parameters: scattering length $a$, effective range $r$, and shape parameter $P$, which are in good agreement with our explicit analytical predictions in three-flavor chiral perturbation theory at next-to-leading order. All results are fairly consistent with experimental measurements, phenomenological studies, and lattice estimations. Numerical computations are implemented at a fine ($a\approx0.082$ fm, $L^3 T = 40^3 96$) lattice ensemble with physical quark masses.