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The anomalous trilinear gauge couplings of $Z$ and $\gamma$ are studied in$e^+e^-\to ZZ/Z\gamma$ with longitudinal beam polarizations using a completeset of polarization asymmetries for the $Z$ boson. We quantify the goodness ofthe beam polarization in terms of the likelihood and find the best-choice of$e^-$ and $e^+$ polarizations to be ($+0.16$, $-0.16$), ($+0.09$, $-0.10$) and($+0.12$, $-0.12$) for $ZZ$, $Z\gamma$ and combined processes, respectively.Simultaneous limits on anomalous couplings are obtained for these choices ofbeam polarizations using Markov-Chain--Monte-Carlo (MCMC) for an $e^+e^-$collider running at $\sqrt{s}=500$ GeV and $\mathcal{L}=100$ fb$^{-1}$. We findthe simultaneous limits for these beam polarizations to be comparable with eachother and also comparable with the unpolarized beam case.

On the choice of beam polarization in $e^+e^-\to ZZ/Z\gamma$ and anomalous triple gauge-boson couplings