$K^{+}\to\pi^{+}\pi^{0}$ decays at next-to-leading order in the chiral expansion on finite volumes

We present the ingredients for determining $K^{+}\to\pi^{+}\pi^{0}$ matrix elements via the combination of lattice QCD and chiral perturbation theory ($\chi$PT). By simulating these matrix elements at unphysical kinematics, it is possible to determine all the low-energy constants (LECs) for constructing the physical $K^{+}\to\pi^{+}\pi^{0}$ amplitudes at next-to-leading order (NLO) in the chiral expansion. In this work, the one-loop chiral corrections are calculated for arbitrary meson four-momenta, in both $\chi$PT and quenched $\chi$PT (q$\chi$PT), and the finite-volume effects are studied.