Hadron-Hadron Interactions from $N_f=2+1+1$ Lattice QCD: $I=3/2$ $\pi K$ Scattering Length

In this paper we report on results for the s-wave scattering length of the $\pi$-$K$ system in the $I=3/2$ channel from $N_f=2+1+1$ Lattice QCD. The calculation is based on gauge configurations generated by the European Twisted Mass Collaboration with pion masses ranging from about $230$ to $450\,\text{MeV}$ at three values of the lattice spacing. Our main result reads $M_{\pi}\,a_0^{3/2,\text{phys}} = -0.059(2)$. Using chiral perturbation theory we are also able to estimate $M_{\pi}\,a_0^{1/2,\text{phys}} = 0.163(3)$. The error includes statistical and systematic uncertainties, and for the latter in particular errors from the chiral and continuum extrapolations.