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The rare kaon decays $K\to\pi\ell^+\ell^-$ and $K\to\pi\nu\bar{\nu}$ areflavor changing neutral current (FCNC) processes and hence promising channelswith which to probe the limits of the standard model and to look for signs ofnew physics. In this paper we demonstrate the feasibility of latticecalculations of $K\to\pi\ell^+\ell^-$ decay amplitudes for which long-distancecontributions are very significant. We show that the dominant finite-volumecorrections (those decreasing as powers of the volume) are negligibly small andthat, in the four-flavor theory, no new ultraviolet divergences appear as theelectromagnetic current $J$ and the effective weak Hamiltonian $H_W$ approacheach other. In addition, we demonstrate that one can remove the unphysicalterms which grow exponentially with the range of the integration over the timeseparation between $J$ and $H_W$. We will now proceed to exploratory numericalstudies with the aim of motivating further experimental measurements of thesedecays. Our work extends the earlier study by Isidori, Turchetti and Martinelliwhich focussed largely on the renormalization of ultraviolet divergences. In acompanion paper we discuss the evaluation of the long-distance contributions to$K\to\pi\nu\bar{\nu}$ decays; these contributions are expected to be at thelevel of a few percent for $K^+$ decays.

Prospects for a lattice computation of rare kaon decay amplitudes: $K\to\pi\ell^+\ell^-$ decays