  As the heaviest known fundamental particle, the top quark ensures testing of
the Standard Model and occupies a significant role in a lot of theories of new
physics beyond the Standard Model. Up to now, the top quark has been only
generated at the Tevatron and the Large Hadron Collider. However, one of the
most important tasks of the physics program at the Large Hadron Collider is the
investigation of the anomalous top quark interactions. In addition, the
production of single top quarks, though rarer than production in pairs,
presents us a different way to study the top quark produced via the electroweak
interaction. This makes single top quark production an important signature for
studying the properties of the top quark. Hence, we examine the anomalous
$t\bar{t}\gamma$ interactions to investigate limits on the anomalous $a_A$ and
$a_V$ couplings through single top quark production of the process
$pp\rightarrow p \gamma^{*} p \rightarrow p t W X$ at the Large Hadron
Collider, the High Luminosity Large Hadron Collider and High Energy Large
Hadron Collider. The best limits obtained on the anomalous $a_A$ and $a_V$
couplings through the subprocess $\gamma^{*} b \rightarrow W t$ at the High
Energy Large Hadron Collider with $L_{int}=15$ ab$^{-1}$ at $68\%$ Confidence
Level are found to be $|a_{A}|=0.0590$ and $a_{V}=[-0.3202;0.0108]$. Therefore,
we understand that $\gamma^{*} p$ collisions at the High Energy Large Hadron
Collider improves the sensitivity limits of the anomalous coupling parameters
according to previous studies.
