  We present a study of the flavor changing decays $\phi\to tc$
($\phi=H_F,\,A_F$) of the $CP$-even and $CP$-odd scalar flavons at the large
hadron collider and its next stage, the high-luminosity large hadron collider.
The theoretical framework is an extension of the standard model that
incorporates an extra complex singlet and invokes the Froggatt-Nielsen
mechanism with an Abelian flavor symmetry. The projected exclusion and
discovery regions in terms of the model parameters are reported. We find that
$A_F$ could be detected at the LHC by considering a reasonable scenario of the
model parameter space. As far as $H_F$ is concerned, we also found promising
results that could be verified experimentally at the high-luminosity LHC.
