A physical point simulation of the rare kaon decays $K \to \pi \ell^+ \ell^-$ with 4 active flavours is demanding for two reasons: first the simulation of a physical charm quark required for the GIM mechanism requires a fine lattice, and second a large volume is needed to suppress finite volume effects. A full lattice calculation of this decay is unfeasible with existing computational resources; a natural intermediate step therefore is to attempt this calculation with physical pion masses, but using a coarse lattice (and thus losing the ability to simulate with a physical charm quark). The charm quark is therefore integrated out in our lattice simulation, hence is treated using perturbation theory. This complicates our calculation somewhat, as the 4-flavour calculation has the advantage that unphysical divergences are cancelled automatically by the GIM mechanism. In this talk I will discuss the renormalisation procedure required for the calculation of the rare kaon decay $K \to \pi \ell^+ \ell^-$ with 3 flavours, in particular the cancellation of divergences between the lattice and perturbative sides of the calculation.

Weak Decays and Matrix Elements