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

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.