Systematic classification of one-loop models addressing the $b \to s \nu \bar{\nu}$ anomaly

The recent evidence for the decay $B^+ \to K^+ \nu \bar{\nu}$ reported by the Belle II collaboration, combined with the existing constraints on the neutral mode $B^0 \to K^{*0} \nu \bar{\nu}$, implies a deviation from the Standard Model prediction that necessitates New Physics contributions to both left- and right-handed vector currents. We perform a systematic topological classification of renormalizable one-loop completions capable of generating the required dimension-six operators while forbidding tree-level mediation. Based on this classification, we identify and construct two minimal benchmark scenarios -- a scalar-rich model and a fermion-rich model -- and perform a comprehensive phenomenological analysis. Our study demonstrates that while these one-loop models can yield enhancements in the $b \to s \nu \bar{\nu}$ branching fractions, the attainable magnitudes are significantly restricted by the combined effects of loop suppression and complementary flavor constraints, limiting their ability to fully accommodate the current anomaly.