Recently, the KOTO experiment reported their new preliminary result of searching for the decay $K_L\to��^0��\bar��$. Three candidate events were observed in the signal region, which exceed significantly the expectation based on the standard model. On the other hand, the new NA62 and previous BNL-E949 experiments yielded a consistent result and confirmed the standard model prediction in the charged meson decay $K^+\to��^+��\bar��$. Furthermore, the two decays are bound by a well-motivated relation from an analysis of isospin symmetry that is hard to break by the new physics of heavy particles. In this work, we study the issue by a systematic effective field theory approach with three of the simplest scenarios, in which the $K_L$ may decay into a new light neutral particle $X$, i.e., $K_L\to��^0X$, $K_L\to ����X$, or $K_L\to��^0XX$. We assess the feasibility of the scenarios by simulations and by incorporating constraints coming from NA62 and other relevant experiments. Our main conclusion is that the scenario $K\to��XX$ for a long lived scalar $X$ seems more credible than the other two when combining distributions and other experimental constraints while the region below the KOTO's blind box provides a good detection environment to search for all three scenarios for a relatively heavy $X$.

10 pages, with 10 figures and 4 appendices; v2: added 4 refs [9,10,17,35], no changes in wording; v3: proofred version to appear in PRD, many improvements/explanations in response to referees' comments and suggestions, added a new figure 3, conclusions not altered, 11 pages, with 11 figures and 4 appendices