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We enumerate various effective couplings that contribute to the mixingsbetween an exotic vector boson $Z^{\prime}$ and the neutral electroweak vectorbosons. The miscellaneous mixing patterns can be evaluated perturbatively. Theeffective oblique parameters $S^{\prime}$, $T^{\prime}$, and $U^{\prime}$ arecalculated to compare with the electroweak precision test results. With thecontributions to the non-negligible $U^{\prime}$ parameter from the$\epsilon_{B,W}$ parameters and the aid of some other parameters to cancel thenegative $T^{\prime}$, the recent CDF $W$-mass anomaly can therefore beexplained.

Corrections to electroweak precision observables from mixings of an exotic vector boson in light of the CDF $W$-mass anomaly