Within the current situation where there seems to be a large mass gap between the Standard Model (SM) and new physics (NP) particles, the effective field theory (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 Standard Model Effective Field Theory (SMEFT) is always appropriate. Although there is a wide class of SM extensions (e.g., supersymmetry) that accept this linear description, this realization does not always provide a good perturbative expansion. The non-linear Higgs Effetive Field Theory (HEFT) (also denoted as Electroweak Chiral Lagrangian, EW$\chi$L) considers an EFT with a singlet scalar Higgs $h$ and the triplet of electroweak Goldstones $\omega^a$, and its organization according to a chiral expansion cures these issues (even if the analysis sometimes becomes slightly more cumbersome). Path integral functional methods allow one to compute the corrections to the next-to-leading order (NLO), $O(p^4)$, 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 leading order (LO), $O(p^2)$, HEFT Lagrangian, which has been related in recent times with the curvature of the manifold of scalar fields (being the SM its flat limit). Thus, the contribution of the leading HEFT Lagrangian to the effective action might be understood as an expansion in its curvature. From the theoretical point of view, 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 beyond-SM (BSM) scenario and the observable at hand. This is what happens in Quantum Chromodynamics, where there are multiple scales and the relevant meson mass and the loop suppression scale factor depend on the channel at hand -and even have a different scaling with the number of colours-. 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.