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In this work we explore the phenomenological implications at future $e^+e^-$colliders of assuming anomalous couplings of the Higgs boson to gauge bosons$HVV$ and $HHVV$ ($V=W,Z$) given by the $\kappa$-modifiers with respect to theStandard Model couplings, $\kappa_V$ and $\kappa_{2V}$, respectively. For thisstudy we use the Higgs Effective Field Theory (HEFT) where these two $\kappa$parameters are identified with the two most relevant effective couplings atleading order, concretely $a=\kappa_V$ and $b=\kappa_{2V}$. Our focus is put onthese two couplings and their potential correlations which we believe carryinteresting information on the underlying ultraviolet theory. The particularstudied process is $e^+e^- \to HH \nu \bar \nu$, where the vector bosonscattering subprocess $WW \to HH$ plays a central role, specially at thelargest planned energy colliders. Our detailed study of this process as afunction of the energy and the angular variables indicates that the producedHiggs bosons in the BSM scenarios will have in general a high transversality ascompared to the SM case if $\kappa_V^2 \neq \kappa_{2V}$. In order to enhancethe sensitivity to these HEFT parameters $\kappa_V$ and $\kappa_{2V}$ and theirpotential correlations we propose here some selected differential crosssections for the $e^+e^- \to HH \nu \bar\nu$ process where different kinematicproperties of the BSM case with respect to the SM are manifested. Finally, wewill focus on the dominant Higgs decays to $b \bar b$ pairs leading to finalevents with 4 $b$-jets and missing transverse energy from the undetectedneutrinos and will provide the expected accessibility to the$(\kappa_V,\kappa_{2V})$ effective couplings and their potential correlations.In our study we will consider the three projected energies for $e^+e^-$colliders of 500 GeV, 1000 GeV and 3000 GeV.

Exploring correlations between HEFT Higgs couplings $\kappa_V$ and $\kappa_{2V}$ via HH production at $e^+e^-$ colliders