Probing the electromagnetic properties of the neutrinos at future lepton colliders

In this study, we explore the non-standard $\nu\bar{\nu}\gamma \gamma$ couplings parametrized by dimension-seven operators via $e^{+}e^{-} \to \nu\bar{\nu}\gamma$ process at the FCC-ee/CEPC and $\mu^{+}\mu^{-}\to\nu\bar{\nu}\gamma$ process at the Muon Colliders. For the detailed Monte Carlo simulation, all signal and relevant background events are produced within the framework of Madgraph where non-standard $\nu\bar{\nu}\gamma \gamma$ couplings are implemented. After passing through Pythia for parton showering and hadronization, detector effects are included via tuned corresponding detector cards for each collider in Delphes. Projected sensitivities on $\nu\bar{\nu}\gamma \gamma$ couplings are obtained at a 5$\sigma$ confidence level without and with $5\%$ systematic uncertainties for the FCC-ee/CEPC and the Muon Colliders, showcasing the complementarity between lepton colliders. Our best limit on the anomalous $\nu\bar{\nu}\gamma \gamma$ couplings even with 5\% systematic uncertainties for muon collider with $\sqrt{s}=10$ TeV and $L_{int}=3$ ab$^{-1}$ are found to be thirteen orders of magnitude stronger than the upper bound obtained from rare decay $Z\to\gamma\gamma\nu\bar{\nu}$ analysis using LEP data.