  Flavour-changing-neutral currents (FCNCs) involving the top quark are highly
suppressed within the Standard Model (SM). Hence, any signal in current or
planned future collider experiments would constitute a clear manifestation of
physics beyond the SM. We propose a novel, interference-based strategy to
search for top-quark FCNCs involving the $Z$ boson that has the potential to
complement traditional search strategies due to a more favourable luminosity
scaling. The strategy leverages on-shell interference between the FCNC and SM
decay of the top quark into hadronic final states. We estimate the feasibility
of the most promising case of anomalous $tZc$ couplings using Monte Carlo
simulations and a simplified detector simulation. We consider the main
background processes and discriminate the signal from the background with a
deep neural network that is parametrised in the value of the anomalous $tZc$
coupling. We present sensitivity projections for the HL-LHC and the FCC-hh. We
find an expected $95\%$ CL upper limit of
$\mathcal{B}_{\mathrm{excl}}(t\rightarrow Zc) = 6.4 \times 10^{-5}$ for the
HL-LHC. In general, we conclude that the interference-based approach has the
potential to provide both competitive and complementary constraints to
traditional multi-lepton searches and other strategies that have been proposed
to search for $tZc$ FCNCs.
