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Limits on the quartic couplings $Zγγγ$ and $ZZγγ$ from $e^+e^-$ colliders

We obtain limits on the quartic neutral gauge bosons couplings $Zγγγ$ and $ZZγγ$ using LEP 2 data published by the L3 Collaboration on the reactions $e^+e^-\to γγγ, Zγγ$. 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 $Zγγγ$ couplings and of order $10^{-3}$ $GeV^{-2}$ for the $ZZγγ$ 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.