  The non-diagonal contributions to the trilinear gauge bosons couplings
$Z\gamma V^\ast$ ($V=Z$, $\gamma$) at the one-loop level are calculated using
complex flavor-changing neutral currents mediated by the $Z$ boson. We find
that it is not possible to generate non-diagonal contributions to the vertex
$Z\gamma\gamma^\ast$ using these couplings, whereas for the vertex $Z\gamma
Z^\ast$ only the $CP$-conserving form factor $h_3^Z$ is induced. Constraints on
the $Z\overline{t}c$ couplings are obtained from current LHC data. Our bounds
are $|g_{\text{V}}^{tu}|\text{,}|g_{\text{A}}^{tu}|\leqslant 0.007$ and
$|g_{\text{V}}^{tc}|\text{,}|g_{\text{A}}^{tc}|\leqslant 0.0095$ in terms of
the vector and axial couplings, while in terms of the coefficients of the
effective field theory framework these bounds are $|C^{(1+3)}_{\phi
\mu}|\text{,}|C^{-(1+3)}_{\phi q}|\leqslant 0.12$ and $|C^{(2+3)}_{\phi
\mu}|\text{,}|C^{-(2+3)}_{\phi q}|\leqslant 0.16$. The new contributions to
$h_3^Z$ are of order $10^{-6}-10^{-7}$. We also study the polarized partial
widths of the $V^\ast\rightarrow Z\gamma$ process, as they are useful for
expressing LHC cross-sections. Furthermore, two types of left-right asymmetries
are also discussed. The polarized $\Gamma(V^\ast\rightarrow Z\gamma)$ show
significant deviations from SM, whereas the asymmetries are only non-vanishing
in theories beyond the SM. Thus, the polarized observables are highly sensitive
to new physics effects, specially those related to $CP$-violation.
