Probing new physics with the kaon decays $K\to\pi\pi\!\not\!\!E$

The most recent search for the rare kaon decay $K^+\to\pi^+\nu\bar\nu$ 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 $\nu\bar\nu$ pair appearing as missing energy ($\not\!\!E$), the room for possible new physics is no longer sizable and that therefore its contributions to $s\to d\!\not\!\!E$ operators with parity-even $ds$ quark bilinears have become significantly constrained. Nevertheless, we point out that appreciable manifestations of new physics via operators with mainly parity-odd $ds$ quark bilinears may still occur in $K\to\pi\pi\!\not\!\!E$ modes, on which there are only minimal empirical details at present. We find in particular that new physics may enhance the branching fractions of $K^+\to\pi^+\pi^0\!\not\!\!E$ and $K_L\to\pi^0\pi^0\!\not\!\!E$ to their current experimental upper limits and of $K_L\to\pi^+\pi^-\!\not\!\!E$ to the level of $10^{-5}$. Thus, quests for these decays in existing kaon facilities, such as NA62 and KOTO, could provide valuable information complementary to that gained from $K\to\pi\!\not\!\!E$.