Feasibility at the LHC, FCC-he and CLIC for sensitivity estimates on anomalous $\tau$-lepton couplings

In this paper, we present detailed studies on the feasibility at $pp$, $e^{-}p$ and $e^{+}{e}^{-}$ colliders for model-independent sensitivity estimates on the total cross-section and on the anomalous ${\tau }^{+}{\tau }^{-}\gamma$ interaction through the tau pair production channels $pp\to p\tau \bar{\tau }\gamma p$, $e^-p \to e^- \tau\bar \tau \gamma p$and$e^+e^- \to e^+ \tau\bar \tau \gamma e^-$atthe$\gamma^\gamma^ \to \tau^+\tau^-\gamma$ mode. Measurements of the anomalous couplings of the $\tau$-lepton ${\tilde{a}}_{\tau }$ and ${\tilde{d}}_{\tau }$ provide an excellent opportunity to probing extensions of the Standard Model. We estimate the sensitivity at the $95\mathrm{\%}$ Confidence Level, and we consider that the $\tau$-lepton decays leptonically or semi-leptonically. We found that of the three considered colliders, the future CLIC at high energy and high luminosity should provide the best sensitivity on the dipole moments of the $\tau$-lepton ${\tilde{a}}_{\tau }=[-0.00128,0.00105]$ and $ |\tilde{d}_\tau({\rm ecm})|= 6.4394\times 10^{-18}$, which show a potential advantage compared to those from LHC and FCC-he.