Dark photon effect on the rare kaon decay $K_L\to {\pi }^0\nu \bar{\nu }$

We present an analysis of the effect of a dark photon on the rare kaon decay $K_L\to {\pi }^0\nu \bar{\nu }$. All relevant couplings of the dark photon to the Standard Model particles are derived explicitly in terms of the dark photon mass and the mixing parameter. We find that the dark photon yields no more than a few percent correction to the Standard Model branching ratio $\mathrm{B}\mathrm{r}(K_L\to {\pi }^0\nu \bar{\nu })$ in the region of interest.