  As there seems to be a large mass gap between the SM and new physics
particles, the EFT framework emerges as the natural approach for the analysis
and interpretation of collider data. However, this large gap and the fact that
(so far) all the measured interactions look pretty much SM-like does not imply
that the linear Higgs representation as a complex doublet $\phi$ in the SMEFT
is always appropriate. Although there is a wide class of SM extensions that
accept this linear description, this realization does not always provide a good
perturbative expansion. The HEFT and its organization according to a chiral
expansion cures these issues. Path integral functional methods allow one to
compute the corrections to the NLO effective action: at tree level, the heavy
resonances only contribute to the $O(p^4)$ low-energy couplings (or higher)
according to a pattern that depends on their quantum numbers and is suppressed
by their masses; at the one-loop level, all the new-physics corrections go to
the $O(p^4)$ effective action (or higher) and are suppressed by an intrinsic
scale of the LO HEFT Lagrangian, which has been related in recent times with
the curvature of the manifold of scalar fields (being the SM its flat limit and
having the HEFT effective action an expansion in the curvature). This two
suppression scales (resonance masses and non-linearity/curvature) compete at
NLO and may, in principle, be very different/similar depending on the
particular BSM scenario and the observable at hand. Likewise, experiments do
not tell us yet whether $O(p^4)$ loops are essentially negligible or not, as it
is shown with a couple of examples. In summary, the HEFT extends the range of
applicability of the SMEFT, which, although justified for a pretty wide class
of BSM scenarios, introduces a bias in the data analyses and does not provide
the most general EFT framework.
