  We classify effective field theory (EFT) deformations of the Standard Model
(SM) according to the analyticity property of the Lagrangian as a function of
the Higgs doublet H. Our distinction in analytic and non-analytic corresponds
to the more familiar one between linearly and non-linearly realized electroweak
symmetry, but offers deeper physical insight. From the UV perspective,
non-analyticity occurs when the new states acquire mass from electroweak
symmetry breaking, and thus cannot be decoupled to arbitrarily high scales.
This is reflected in the IR by the anomalous growth of the interaction strength
for processes involving many Higgs bosons and longitudinally polarized massive
vectors, with a breakdown of the EFT description below a scale $O(4 \pi v)$.
Conversely, analyticity occurs when new physics can be pushed parametrically
above the electroweak scale.
  We illustrate the physical distinction between these two EFT families by
discussing Higgs boson self-interactions. In the analytic case, at the price of
some unnaturalness in the Higgs potential, there exists space for $O(1)$
deviations of the cubic coupling, compatible with single Higgs and electroweak
precision measurements, and with new particles out of the direct LHC reach.
Larger deviations are possible, but subject to less robust assumptions about
higher-dimensional operators in the Higgs potential. On the other hand, when
the cubic coupling is produced by a non-analytic deformation of the SM, we show
by an explicit calculation that the theory reaches strong coupling at $O(4 \pi
v)$, quite independently of the magnitude of the cubic enhancement.
