  We argue that the search for dark vector boson through $e^+e^-\to Z_d\gamma$
can determine the Lorentz structure of $Z_dl^+l^-$ couplings with the detection
of leptonic decays $Z_d\to l^+l^-$. 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 $m_{Z_d}\ll \sqrt{s}$. On the other hand, the fraction
of longitudinal $Z_d$ is non-negligible for $m_{Z_d}$ comparable to $\sqrt{s}$.
Such polarization effects can be analyzed through angular distributions of
final-state particles in $Z_d$ decays. Taking $l^{\pm}\equiv \mu^{\pm}$, we
study the correlation between $Z_d$ angle relative to $e^-$ beam direction in
$e^+e^-$ CM frame and $\mu^-$ 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 $Z_dl^+l^-$ couplings. We discuss the measurement of such
correlation in Belle II detector, taking into account the detector acceptance
and energy resolution.
