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The rare decays of a kaon into a pion and a lepton/antilepton pair proceed via a flavour changing neutral current and therefore first arise in the Standard Model only as a second order electroweak interaction. This natural suppression makes these decays sensitive to the effects of potential New Physics. However the rare decay channels $K^+\to\pi^+\ell^+\ell^-$ are dominated by long-distance contributions, where the two electroweak processes are separated by distances over which non-perturbative QCD effects play a significant role. In this talk I provide an update on the progress of our exploratory calculations of the long-distance contributions to $K^+\to\pi^+\ell^+\ell^-$ amplitudes, which make use of the Domain Wall Fermion ensembles of the RBC and UKQCD collaborations, and outline the prospects for further progress of the calculation.

Progress in the exploratory calculation of the rare kaon decays $K\to\pi\ell^+\ell^-$