  We derive the contact terms contributing to the four-point amplitudes of the
standard-model particles, keeping terms with up to quartic energy growth.
Imposing just the unbroken low-energy symmetry, and treating the electroweak
gauge bosons and the Higgs as independent degrees of freedom, we obtain the
most general four-point contact-term amplitudes, corresponding to the HEFT
framework. The contact terms are spanned by a basis of Stripped Contact Terms
(SCTs), which carry the polarization information, multiplied by polynomials in
the Mandelstam invariants. For terms with quadratic energy growth, we also
derive the low-energy SMEFT predictions, via on-shell Higgsing of the massless
SMEFT contact terms. We discuss several aspects of bottom-up versus top-down
on-shell derivations of the HEFT and SMEFT amplitudes, highlighting in
particular the simple counting of HEFT dimensions in the on-shell approach and
the transparent relation between perturbative unitarity and gauge-invariance in
the little-group covariant massive spinor formalism. Our results provide a
formulation of EFT analyses directly in terms of observable quantities. For
terms with quadratic energy growth, we also provide the mapping to the Warsaw
basis.
