Determination of the $A_2$ amplitude of $K \rightarrow ππ$ decays

We review the status of recent calculations by the RBC-UKQCD collaboration of the complex amplitude $A_2$, corresponding to the decay of a kaon to a two pion state with total isospin 2. In particular, we present preliminary results from two new ensembles: $48^3 \times 96$ with $a^{-1}=1.73$ GeV and $64^3 \times 128$ with $a^{-1}=2.3$ GeV, both at physical kinematics. Both ensembles were generated Iwasaki gauge action and domain wall fermion action with 2+1 flavours. These results, in comparison to our earlier ones on a $32^3$ DSDR lattice with $a^{-1}=1.36$ GeV, enable us to significantly reduce the discretization errors. The partial cancellation between the two dominant contractions contributing to Re($A_2$) has been confirmed and we believe that this cancellation is a major contribution to the $\Delta I=1/2$ rule.