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Probing the anomalous triple gauge boson couplings in $e^+e^-\to W^+W^-$ using $W$ polarizations with polarized beams

We study the anomalous $W^+W^-V$ ($V=γ,Z$) couplings in $e^+e^-\to W^+W^-$ using the complete set of polarization observables of $W$ boson with longitudinally polarized electron ($e^-$) and positron ($e^+$) beams. For the effective $W^+W^-V$ couplings, we use the most general Lorentz invariant form factor parametrization as well as $SU(2)\times U(1)$ invariant dimension-$6$ effective operators. We estimate simultaneous limits on the anomalous couplings using the Markov-Chain--Monte-Carlo (MCMC) method for an $e^+e^-$ collider running at center of mass energy of $\sqrt{s}=500$ GeV and the integrated luminosity of ${\cal L}=100$ fb$^{-1}$, $3.2$ ab$^{-1}$ and $4$ ab$^{-1}$. The best limits on the anomalous couplings are obtained for $e^-$ and $e^+$ polarization being $(\pm 0.8,\mp 0.6)$ for both $100$ fb$^{-1}$ and $3.2$ ab$^{-1}$ of luminosity.