Isospin breaking corrections to low-energy π-Kscattering

We evaluate the matrix elements for the processes ${\pi }^0{K}^0\to {\pi }^0{K}^0$ and ${\pi }^{-}{K}^{+}\to {\pi }^0{K}^0$ in the presence of isospin breaking terms at leading and next-to-leading order. As a direct application the relevant combination of the S-wave scattering lengths involved in the pion-kaon atom lifetime is determined. We discuss the sensitivity of the results with respect to the input parameters.