Oblique corrections in general dark $U(1)$ models

We investigate the impact of dark Abelian gauge bosons on the electroweak precision measurements at the one-loop level. The dark gauge boson couples to the standard model fermions generally via two kinds of mixing with the electroweak gauge bosons: the kinetic mixing and the mass mixing. We solve the Schwinger-Dyson equation for the gauge boson propagators and derive a renormalization scheme-independent representation of the scattering amplitudes for four-fermion processes, including the full oblique corrections. We define the running parameters at the one-loop level and show that the leading new physics effects, including the mixing, in the electroweak precision observables can be described by the oblique parameters $S$, $T$, and $U$ as in the standard electroweak gauge theory when the new physics scale is sufficiently high and the dark gauge boson mass lies away from the $Z$ pole. We consider the dark doublet scalar boson as an example and numerically show that a novel one-loop effect can drastically change the parameter region allowed by the electroweak precision tests.