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The measurements of the top quark flavor changing neutral currentinteractions are one of the most important goals of the top quark physicsprogram in the present and the future collider experiments. These measurementsprovide direct information on non-standard interactions of the top quark.Within the framework of new physics beyond the Standard Model, theseinteractions can be defined by an effective Lagrangian. In this study, we haveinvestigated the potential of the future $\mu p$ colliders on the top quarkflavor changing neutral current interactions through the subprocesses $\gamma q\rightarrow t \rightarrow W b$ where $q=u,c$. These subprocesses have beenproduced through the main reaction $\mu p \rightarrow \mu \gamma p \rightarrow\mu W b X $ at the LHC$-\mu p$, the FCC$-\mu p$ and the SPPC-$\mu p$. For themain reaction, the total cross sections have been calculated as a function ofthe anomalous $ tq\gamma $ couplings. In addition, sensitivities on BR($t\rightarrow q \gamma$) at $95\%$ Confidence Level have been calculated. Weobtain that the best constraints on BR($t \rightarrow q \gamma$) are at theorder of 10$^{-7}$ which is four orders of magnitude better than the LHC'sexperimental results.

Probing the anomalous $tq\gamma$ couplings through single top production at the future lepton-hadron colliders