Tracing the anomalous $tqg$ and $tq\gamma$ flavor changing interactions at the FCC-he

We investigate the possible presence of \emph{ Flavor Changing Neutral Current (FCNC)} couplings of the top quark with gluon and photon through $e^-p\to e^-tj$ process. Focusing on disentangling the effects of different couplings that could be present, we exploit the presence of the scattered electron, the angular distribution of which is sensitive to the type of coupling involved. Further, we demonstrate the potential of electron beam polarisation in distinguishing the left-handed and right-handed couplings of both gluon and photon separately. Considering an $e^-p$ collider of beam energies of $E_{e(p)} = 60~(50000)$~GeV at 2~ab$^{-1}$ integrated luminosity, couplings can be probed at the level of $10^{-2}$ with the corresponding branching fractions of $BR(t\to u\gamma)\le 4 - 7 \times 10^{-6}$ and $BR(t\to c\gamma) \le 1-2 \times 10^{-5}$, depending on if the coupling is right-handed or left-handed. The corresponding limits on the gluon couplings lead to $BR(t\to ug)\le 1.7 \times 10^{-6}$ and $BR(t\to cg) \le 3-4 \times 10^{-5}$.