Limits on the quartic couplings $Z\gamma\gamma\gamma$ and $ZZ\gamma\gamma$ from $e^+e^-$ colliders

We obtain limits on the quartic neutral gauge bosons couplings $Z\gamma\gamma\gamma$ and $ZZ\gamma\gamma$ using LEP 2 data published by the L3 Collaboration on the reactions $e^+e^-\to \gamma\gamma\gamma, Z\gamma\gamma$. We also obtain $95 \%$ C. L. limits on these couplings at the future linear colliders energies. The LEP 2 data induce limits of order $10^{-5}$ $GeV^{-4}$ for the $Z\gamma\gamma\gamma$ couplings and of order $10^{-3}$ $GeV^{-2}$ for the $ZZ\gamma\gamma$ couplings, which are still above the respective Standard Model predictions. Future $e^+e^-$ linear colliders may improve these limits by one or two orders of magnitude.