Polarization effects in the search for a dark vector boson at e^+ e^− colliders
We argue that the search for dark vector boson through e^+e^− → Z_d γ can determine the Lorentz structure of Z_d l^+ l^− couplings with the detection of leptonic decays Z_d → l^+l^-. We assume a general framework that the dark vector boson interacts with ordinary fermions through vector and axial-vector couplings. As a consequence of Ward-Takahashi identity, Z_d is transversely polarized in the limit m_Zd ≪ sqrt{s}. On the other hand, the fraction of longitudinal Z_d is non-negligible for m_Zd comparable to sqrt{s}. Such polarization effects can be analyzed through angular distributions of final-state particles in Z_d decays. Taking l ≡ μ, we study the correlation between Z_d angle relative to e^− beam direction in e^+e^- center of momentum frame and μ− angle relative to the boost direction of Z_d in Z_d rest frame. This correlation is shown to be useful for probing the Lorentz structure of Z_d l^+ l^− couplings. We discuss the measurement of such correlation in Belle II detector, taking into account the detector acceptance and energy resolution.
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