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Leptoquarks (LQs) have been extensively studied in the context of $B$anomalies. When $U(1)_{L_\mu-L_\tau}$ is introduced to a scalar LQ model withthe LQ $S_1$ charged under the new symmetry, $S_1$ primarily couples to thethird-generation leptons while its couplings to first and second-generationleptons are naturally suppressed. Furthermore, only $S_1$ in the scalar LQmodels has the feature that down-type quarks merely couple to neutrinos but notthe charged leptons, avoiding strict restrictions from $b\to s \mu^+ \mu^-$.With this distinctive characteristic of $S_1$, we investigate its impact onrare processes involving the $d_i \to d_j \nu \bar\nu$ transitions. Under thedominant constraints from $\Delta F=2$ processes, we find that the $S_1$contributions to the branching ratios (BRs) of $B\to K(K^*) \nu \bar\nu$ and$K_L \to \pi^0 \nu \bar\nu$ can be factorized into the same multiplicativefactor multiplying the standard model predictions. Enhancement in the BRs canpossibly exceed a factor of 2. In particular, ${\cal B}(K^+\to \pi^+ \nu\bar\nu)$ can reach the upper $1\sigma$ error of the experimental value, i.e.,$\simeq 15.4 \times 10^{-11}$. We also show that the model can fit the newworld averages of $R(D)$ and $R(D^*)$.

Flavor anomalies in leptoquark model with gauged $U(1)_{L_\mu-L_\tau}$