Unravelling the anomalous gauge boson couplings in $ZW^\pm$ production at the LHC and the role of spin-$1$ polarizations

We study the anomalous triple gauge boson couplings (aTGC) in $ZW^\pm$ production in $3l$ + missing $E_T$ channel at the LHC for $\sqrt{s}=13$ TeV. We use cross sections, azimuthal asymmetry, forward-backward asymmetry, and polarization asymmetries of $Z$ and reconstructed $W$ to estimate simultaneous limits on the anomalous couplings for both the effective vertex formalism as well as the effective operator approach using the Markov-chain--Monte-Carlo (MCMC) method for luminosities $35.9$ fb$^{-1}$, $100$ fb$^{-1}$, $300$ fb$^{-1}$, $1000$ fb$^{-1}$, and $3000$ fb$^{-1}$. The trilepton invariant mass ($m_{3l}$)and the transverse momentum of $Z$ ($p_T(Z)$) are found to be sensitive to the aTGC for the cross sections as well as for the asymmetries. We observed that the asymmetries significantly improve the measurement of anomalous couplings at the high-luminosity large hadron collider (LHC) if a deviation from the Standard Model (SM) is observed.