Anomalous triple gauge boson couplings in $ZZ$ production at the LHC and the role of $Z$ boson polarizations

We study anomalous couplings among neutral gauge bosons in $ZZ$ production at the LHC for $\sqrt{s}=13$ TeV in $4$-lepton final state. We use the cross section and polarization asymmetries of the $Z$ boson to estimate simultaneous limits on anomalous coupling using Markov-Chain--Monte-Carlo (MCMC) method for luminosities $35.9$ fb$^{-1}$, $150$ fb$^{-1}$, $300$ fb$^{-1}$ and $1000$ fb$^{-1}$. The $CP$-even polarization asymmetry $A_{x^2-y^2}$ is sensitive mainly to the $CP$-odd couplings $f_4^{Z/\gamma}$ (quadratically) providing a probe to identify $CP$-odd nature of interaction at the LHC. We find that the polarization asymmetries significantly improve the estimation of anomalous couplings should a deviation from the Standard Model (SM) be observed.