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The electroweak symmetry-breaking sector is one of the most promising anduncharted parts of the Standard Model; but it seems likely that new electroweakphysics may be out of reach of the present accelerator effort and the hope isto observe small deviations from the SM. Given that, Effective Field Theorybecomes the logic method to use, and SMEFT has become the standard. However,the most general theory with the known particle content is HEFT, and whetherSMEFT suffices should be investigated in future experimental efforts. Buildingon investigations by other groups that established geometric criteria todistinguish SMEFT from HEFT (useful for theorists examining specific beyond-SMcompletions), we seek more phenomenological understanding and present ananalogous discussion aimed at a broader audience.  We discuss various aspects of (multi-) Higgs boson production fromlongitudinal electroweak gauge bosons $W_LW_L\to n\times h$ in the TeV regionas the necessary information to characterise the Flare function,$\mathcal{F}(h)$, that determines whether SMEFT or HEFT is needed. We alsopresent tree-level amplitudes including contact and exchange channels, as wellas a short discussion on accessing $\mathcal{F}(h)$ from the statistical limitof many bosons. We also discuss the status of the coefficients of its seriesexpansion, its validity, whether its complex-$h$ extension can be used topredict or not a tell-tale zero, and how they relate to the dimension-6 and -8SMEFT operators in the electroweak sector. We derive a set of new correlationsamong BSM corrections to the HEFT coefficients that help decide, fromexperimental data, whether we have a viable SMEFT. This analysis can be usefulfor machines beyond the LHC that could address the challenging final state withseveral Higgs bosons.

The flair of Higgsflare: Distinguishing electroweak EFTs with $W_LW_L \to n\times h$