Search for the electromagnetic moments of the $\tau$ lepton in photon-photon collisions at the LHeC and the FCC-he

We examine the potential of the process $ep \rightarrow e\gamma^{} \gamma^{} p\rightarrow e\tau \bar{\tau}p$ at the Large Hadron Electron Collider (LHeC) and the Future Circular Hadron Electron Collider (FCC-he) to examine non-standard $\tau \bar{\tau }\gamma$ coupling in a model independent way by means of the effective Lagrangian approach. We perform pure leptonic and semileptonic decays for ${\tau }^{-}{\tau }^{+}$ production in the final state. Furthermore, we use $L=10,50,100,300,500,1000$ fb${}^{-1}$ with at $\sqrt{s}=1.30,1.98,7.07,10$ TeV and we consider systematic uncertainties of ${\delta }_{sys}=0,5,10\mathrm{\%}$. The best sensitivity bounds obtained from the process $ep\to e{\gamma }^{\ast }{\gamma }^{\ast }p\to e\tau \bar{\tau }p$ on the anomalous couplings are $-0.0025<{\tilde{a}}_{\tau }<0.0009$ and $\mathrm{\mid }\widetilde{d_{\tau }}\mathrm{\mid }<8.85\times {10}^{-18}ecm$, respectively. Therefore, our results show that the process $ep \rightarrow e\gamma^{} \gamma^{} p\rightarrow e\tau \bar{\tau}p$ at the LHeC and FCC-he are a very good prospect for probing the anomalous magnetic and electric dipole moments of the $\tau$ lepton at ${\gamma }^{\ast }{\gamma }^{\ast }$ mode of the future $ep$ collider.