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Which EFT

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 $O(4 πv)$. 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,