QED-corrected Lellouch-L\"uscher formula for $K \rightarrow \pi\pi$ decay

A precise SM prediction for the direct CP violation in the $K \rightarrow \pi\pi$ decay process is of great importance in confronting experiments and constraining new physics. The state-of-art lattice QCD study of this process will soon achieve a precision that QED effects can no longer be neglected. The inclusion of QED in such calculations is planned, and the formalism to relate the finite-volume matrix element obtained from these calculations to the physical amplitude is underway. Here, we report on the progress towards an extension of the Lellouch-L\"uscher formalism in presence of QED, with the goal of enabling the extraction of physical amplitudes for the $K\rightarrow \pi\pi$ process with charged initial and/or final states.