Probing the Higgs self-coupling through double Higgs production in vector boson scattering at the LHC

In this work we explore the sensitivity to the Higgs self-coupling $\lambda$ in the production of two Higgs bosons via vector boson scattering at the LHC. Although these production channels, concretely $W^+W^- \to HH$ and $ZZ \to HH$, have lower rates than gluon-gluon fusion, they benefit from being tree level processes, being independent of top physics and having very distinctive kinematics that allow to obtain very clean experimental signatures. This makes them competitive channels concerning the sensitivity to the Higgs self-coupling. In order to give predictions for the sensitivity to this coupling, we first study the role of $\lambda$ at the subprocess level, both in and beyond the Standard Model, to move afterwards to the LHC scenario. We characterize the $pp\to HHjj$ case first and then provide quantitative results for the values of $\lambda$ that can be probed at the LHC in vector boson scattering processes after considering the Higgs boson decays. We focus mainly in $pp\to b\bar{b}b\bar{b}jj$, since it has the largest signal rates, and also comment on the potential of other channels, such as $pp\to b\bar{b}\gamma\gamma jj$, as they lead to cleaner, although smaller, signals. Our whole study is performed for a center of mass energy of $\sqrt{s}=14$ TeV and for various future expected LHC luminosities.