Flavor Non-universal Vector Leptoquark Imprints in $K\to \pi \nu \bar{\nu }$ and $\mathrm{\Delta }F=2$ Transitions

We analyze $K\to \pi \nu \bar{\nu }$ rates in a model with a TeV-scale leptoquark addressing $B$-meson anomalies, based on the flavor non-universal 4321 gauge group featuring third-generation quark-lepton unification. We show that, together with the tight bounds imposed by $\mathrm{\Delta }F=2$ amplitudes, the present measurement of $\mathcal{B}(K^{+}\to {\pi }^{+}\nu \bar{\nu })$ already provides a non-trivial constraint on the model parameter space. In the minimal version of the model, the deviations from the Standard Model in $\mathcal{B}(K^{+}\to {\pi }^{+}\nu \bar{\nu })$ are predicted to be in close correlation with non-standard effects in the Lepton Flavor Universality ratios $R_D$ and $R_{D^{\ast }}$. With the help of future data, these correlations can provide a decisive test of the model.