$Z$ boson mixing and the mass of the $W$ boson

We explore the possibility of explaining the $W$ boson mass with an extra gauge boson mixing with the $Z$ boson at tree level. Extra boson mixing with the $Z$ boson will change the expression of the $Z$ boson mass, thus altering the $W$ boson mass. We explore two models in this work. We find that in the Derivative Portal Dark Matter model, there are parameters space which can give the observed $W$ boson mass, as well as the observed Dark Matter relic density. These parameters space can also fulfill the constraints from the electroweak oblique parameters and Dark Matter indirect detection. In the U(1) extension model, the kinetic mixing between extra boson and $B$ boson can also give the observed $W$ boson mass. However, to fulfill electroweak oblique parameters fit the kinetic mixing in the U(1) model can only contribute about $27~\mathrm{MeV}$ extra mass to the Standard Model $W$ boson mass. Both models indicate the extra vector boson with the best fit mass around $120~\mathrm{GeV}$.