HEFT's appraisal of triple (versus double) Higgs weak boson fusion

Multi-Higgs boson interactions with massive gauge bosons are known to be tell-tale probes of the vacuum manifold of electroweak symmetry breaking. Phenomenologically, a precise determination of these parameters is hampered through increasingly rare processes at the presently available energy frontier provided by the Large Hadron Collider. Contact interactions of three Higgs bosons with the $W$ and $Z$ bosons seem currently well out of experimental reach due to an irrelevant SM production cross section. From a theoretical perspective, in perturbative extensions of the SM such interactions are suppressed by weak loops and further diluted in a priori sensitive processes like weak boson fusion (WBF) when they admit a dimension-six Standard Model Effective Field Theory description. In this work, we identify scenarios that can indeed lead to large, and perhaps even observable modifications of WBF triple Higgs production most directly parametrised by Higgs Effective Field Theory. We critically analyse these enhancements at the LHC and future colliders from the perspective of unitarity and demonstrate the radiative stability of such analyses under QCD corrections at hadron colliders. Taking into account the restrictions from unitarity, we finally study the expected sensitivity to the electroweak triple Higgs production within HEFT, considering $HHVV$ and $HHHVV$ effective couplings, at both future hadron and lepton colliders. Particularly, we present numerical predictions for LHC, FCC, CLIC and muon colliders.