In this work we explore the phenomenological implications at future $$e^+e^-$$
e
+
e
-
colliders of assuming anomalous couplings of the Higgs boson to gauge bosons HVV and HHVV$$(V=W,Z)$$
(
V
=
W
,
Z
)
given by the $$\kappa $$
κ
-modifiers with respect to the Standard Model couplings, $$\kappa _V$$
κ
V
and $$\kappa _{2V},$$
κ
2
V
,
respectively. For this study we use the Higgs Effective Field Theory (HEFT) where these two $$\kappa $$
κ
parameters are identified with the two most relevant effective couplings at leading order, concretely $$a=\kappa _V$$
a
=
κ
V
and $$b=\kappa _{2V}.$$
b
=
κ
2
V
.
Our focus is put on these two couplings and their potential correlations which we believe carry interesting information on the underlying ultraviolet theory. The particular studied process is $$e^+e^- \rightarrow HH \nu \bar{\nu },$$
e
+
e
-
→
H
H
ν
ν
¯
,
where the vector boson scattering subprocess $$WW \rightarrow HH$$
W
W
→
H
H
plays a central role, specially at the largest planned energy colliders. Our detailed study of this process as a function of the energy and the angular variables indicates that the produced Higgs bosons in the BSM scenarios will have in general a high transversality as compared to the SM case if $$\kappa _V^2 \ne \kappa _{2V}.$$
κ
V
2
≠
κ
2
V
.
In order to enhance the sensitivity to these HEFT parameters $$\kappa _V$$
κ
V
and $$\kappa _{2V}$$
κ
2
V
and their potential correlations we propose here some selected differential cross sections for the $$e^+e^- \rightarrow HH \nu \bar{\nu }$$
e
+
e
-
→
H
H
ν
ν
¯
process, where different kinematic properties of the BSM case with respect to the SM are manifested. Finally, we will focus on the dominant Higgs decays to $$b {\bar{b}}$$
b
b
¯
pairs leading to final events with 4 b-jets and missing transverse energy from the undetected neutrinos and will provide the expected accessibility to the $$(\kappa _V,\kappa _{2V})$$
(
κ
V
,
κ
2
V
)
effective couplings and their potential correlations. In our study we will consider the three projected energies for $$e^+e^-$$
e
+
e
-
colliders of 500 GeV, 1000 GeV and 3000 GeV.