Dark photon and dark $Z$ mediated $B$ meson decays

We study flavor changing neutral current decays of $B$ and $K$ mesons in the dark $U(1)_D$ model, with the dark photon/dark $Z$ mass between 10 MeV and 2 GeV. Although the model provides an improved fit (compared to the standard model) to the differential decay distributions of $B \to K^{(*)} \ell^+ \ell^-$, with $\ell= \mu, e$, and $B_s \to \phi \mu^+ \mu^-$, the allowed parameter space is ruled out by measurements of atomic parity violation, $K^+ \to \mu^+ + invisible$ decay, and $B_s - \overline{B}_s$ mixing, among others. To evade constraints from low energy data, we extend the model to allow for (1) additional invisible $Z_D$ decay, (2) a direct vector coupling of $Z_D$ to muons, and (3) a direct coupling of $Z_D$ to both muons and electrons, with the electron coupling fine-tuned to cancel the $Z_D$ coupling to electrons via mixing. We find that only the latter case survives all constraints.