Prospects for $K_{L}\to\pi^{0}\nu\bar{\nu}$, $K_S\to\mu^+\mu^-$, $K_L\to\pi^0 \ell^+\ell^-$ and $\varepsilon^{\prime}/\varepsilon$ after the new $K^{+}\to\pi^{+}\nu\bar{\nu}$ result from NA62

Abstract

The recently announced NA62 measurement of the $K^{+}\to\pi^{+}\nu\bar{\nu}$ branching ratio, based on 2016--2024 data, is fully consistent with its very accurate Standard Model (SM) prediction. While it is not excluded that the final result based on 2016--2026 data will deviate from the SM prediction, the question arises whether a similar fate awaits the $K_{L}\to\pi^{0}\nu\bar{\nu}$ decay. This decay is currently being searched for by the KOTO experiment, with the present upper bound roughly two orders of magnitude above its very precise SM prediction. The proposed KOTO II experiment aims to provide the first discovery of the $K_{L}\to\pi^{0}\nu\bar{\nu}$ decay and measure its branching ratio. Building on the findings of several previous papers we demonstrate that in the presence of suitably chosen large new complex phases and of a small amount of new right-handed $\bar s d$ couplings, in addition to the left-handed ones, the $K_{L}\to\pi^{0}\nu\bar{\nu}$ branching ratio can still be enhanced by one order of magnitude with respect to the SM prediction while keeping both $K^{+}\to\pi^{+}\nu\bar{\nu}$ and $\varepsilon_K$ SM-like. Simultaneously the decays $K_S\to\mu^+\mu^-$, studied by LHCb, and $K_L\to\pi^0\ell^+\ell^-$, searched for by KOTO II, can be strongly enhanced and the anomaly in the ratio $\varepsilon^{\prime}/\varepsilon$, as claimed by Dual QCD, removed. We illustrate this with an example of a specific $Z^\prime$ scenario.