Electroweak precision test of axion-like particles

We study the contributions of an axion-like particle to the electroweak precision observables. The particle is assumed to couple with the standard model electroweak gauge bosons. We provide the formulae of the contributions valid for any mass of the axion-like particle. It is found that the effects arise not only via the oblique $S$ and $U$ parameters but also via radiative corrections to the gauge couplings. Besides, the decay of $Z\to a\gamma$ affects the total width of the $Z$ boson. All of those contributions are considered simultaneously in the global fit analysis of the electroweak precision observables. Also, we discuss the recent CDF result of the $W$-boson mass measurement. Since the model is tightly constrained by flavor and collider constraints, it is found that the discrepancy from the standard model prediction is solved only when the axion-like particle is heavier than 500 GeV and its coupling to di-photon is suppressed.