Probing new physics with the kaon decays K→ππE

The latest search for the rare kaon decay ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}\ensuremath{\nu}\overline{\ensuremath{\nu}}$ by the NA62 experiment has produced evidence for it with a branching fraction consistent with the prediction of the standard model. The new result implies that in this decay, with the $\ensuremath{\nu}\overline{\ensuremath{\nu}}$ pair appearing as missing energy ($\overline{)E}$), the room for possible new physics is no longer sizable and that therefore its contributions to underlying four-particle $s\ensuremath{\rightarrow}d\overline{)E}$ operators with parity-even $ds$ quark bilinears have become significantly constrained. Nevertheless, we point out that appreciable manifestations from beyond the standard model induced by the corresponding operators with mainly parity-odd $ds$ quark bilinears could still occur in $K\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}\overline{)E}$ modes, on which there are only minimal empirical details at present. We find in particular that new physics of this kind may enhance the branching fraction of ${K}_{L}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}\overline{)E}$ to values reaching its current experimental upper limit and the branching fractions of ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{0}\overline{)E}$ and ${K}_{L}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}\overline{)E}$ to the levels of ${10}^{\ensuremath{-}7}$ and ${10}^{\ensuremath{-}6}$, respectively. Thus, quests for these decays in existing kaon facilities such as KOTO and NA62 or future ones could provide valuable information complementary to that gained from $K\ensuremath{\rightarrow}\ensuremath{\pi}\overline{)E}$.