Imprint of a new light particle at KOTO?

Recently, the KOTO experiment reported their new preliminary result of searching for the decay $K_L\to\pi^0\nu\bar{\nu}$. 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 $K^+\to\pi^+\nu\bar{\nu}$. 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., $K_L\to\pi^0X$, $K_L\to \gamma\gamma X$, or $K_L\to\pi^0XX$. 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 $K\to\pi XX$ 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 $X$.