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

TL;DR

This paper explores the feasibility of using lattice QCD to compute rare kaon decay amplitudes, addressing long-distance effects and finite-volume corrections, to aid experimental efforts and test the Standard Model.

Contribution

It demonstrates that lattice calculations of $K o ext{ extpi} ext{ extmu} ext{ extmu}$ decays are feasible, with negligible finite-volume effects and no new ultraviolet divergences, advancing theoretical methods for rare decay analysis.

Findings

Finite-volume corrections are negligible.

No new ultraviolet divergences appear in the four-flavor theory.

Unphysical exponential growth terms can be removed.

Abstract

The rare kaon decays $K\to\pi\ell^+\ell^-$ and $K\to\pi\nu\bar{\nu}$ are flavor changing neutral current (FCNC) processes and hence promising channels with which to probe the limits of the standard model and to look for signs of new physics. In this paper we demonstrate the feasibility of lattice calculations of $K\to\pi\ell^+\ell^-$ decay amplitudes for which long-distance contributions are very significant. We show that the dominant finite-volume corrections (those decreasing as powers of the volume) are negligibly small and that, in the four-flavor theory, no new ultraviolet divergences appear as the electromagnetic current $J$ and the effective weak Hamiltonian $H_W$ approach each other. In addition, we demonstrate that one can remove the unphysical terms which grow exponentially with the range of the integration over the time separation between $J$ and $H_W$. We will now proceed to exploratory numerical studies with the aim of motivating further experimental measurements of these decays. Our work extends the earlier study by Isidori, Turchetti and Martinelli which focussed largely on the renormalization of ultraviolet divergences. In a companion 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 the level of a few percent for $K^+$ decays.