Bosonic multi-Higgs correlations beyond leading order

The production of multiple Higgs bosons at the LHC and beyond is a strong test of the mechanism of electroweak symmetry breaking. Taking inspiration from recent experimental efforts to move towards limits on triple Higgs production at the Large Hadron Collider, we consider generic bosonic deviations of $HH$ and $HHH$ production from the Standard Model in the guise of Higgs Effective Field Theory. Including one-loop radiative corrections within the HEFT and going up to ${\mathcal{O}}(p^4)$ in the momentum expansion, we provide a detailed motivation of the parameter range that the LHC (and future hadron colliders) can explore, through accessing non-standard coupling modifications and momentum dependencies that probe Higgs boson non-linearities. In particular, we find that radiative corrections can enhance the sensitivity to Higgs-self coupling modifiers and HEFT-specific momentum dependencies can vastly increase triple Higgs production thus providing further motivation to consider these processes during the LHC's high-luminosity phase.