Parity Violation and Rare Higgs Decays from a Dark Force

We outline the phenomenology of the "dark" $Z$, denoted by $Z_d$, which is a generalization of the "dark" photon hypothesis. Whereas the dark photon interacts with the Standard Model through kinetic mixing, $Z_d$ is assumed also to have mass-mixing with the $Z$ boson. In particular, we highlight the possibility of $Z_d$ contributions to low $Q^2$ parity violation measurements and rare Higgs decays $H\to Z Z_d \to 4 \ell$, where $\ell$ is a charged lepton. The parity violation effects of a $Z_d$ with an intermediate mass $\sim 10-35$ GeV can in principle relieve the mild $\sim 1.8\sigma$ tension among various measurements of the weak mixing angle $\theta_W$. We briefly comment on the prospects for future parity violation experiments at low $Q^2$ to probe this scenario, which could have correlated signals in rare Higgs decays at the LHC.