Vector boson scattering at the LHC. A study of the WW -> WW channels with the Warsaw cut

We study W boson scattering in the same- and opposite-sign channels under the assumption that no resonances are present in the collider processes $pp\to l^{\pm }{\nu }_l{l}^{\pm }{\nu }_{l}jj$ and $pp\rightarrow l^{\pm}\nu_{l}l^{\mp}\nu_{l}jj$, respectively. Basic selection cuts together with a restriction on the combination of the final lepton and jet momenta (the Warsaw cut) makes it possible to argue that at the LHC a luminosity of 100 fb${}^{-1}$ and a center-of-mass energy of $\sqrt{s}=13$ TeV will allow to constrain the leading effective lagrangian coefficients at the permil level. We also discuss limits on the other coefficients of the effective lagrangian as well as stronger constraints provided by higher energy and luminosity. We show that the same-sign $WW\to WW$ channel suffices in providing the most stringent constraints.