Abstract
There are two canonical approaches to treating the Standard Model as an Effective Field Theory (EFT): Standard Model EFT (SMEFT), expressed in the electroweak symmetric phase utilizing the Higgs doublet, and Higgs EFT (HEFT), expressed in the broken phase utilizing the physical Higgs boson and an independent set of Goldstone bosons. HEFT encompasses SMEFT, so understanding whether SMEFT is sufficient motivates identifying UV theories that require HEFT as their low energy limit. This distinction is complicated by field redefinitions that obscure the naive differences between the two EFTs. By reformulating the question in a geometric language, we derive concrete criteria that can be used to distinguish SMEFT from HEFT independent of the chosen field basis. We highlight two cases where perturbative new physics must be matched onto HEFT: (i) the new particles derive all of their mass from electroweak symmetry breaking, and (ii) there are additional sources of electroweak symmetry breaking. Additionally, HEFT has a broader practical application: it can provide a more convergent parametrization when new physics lies near the weak scale. The ubiquity of models requiring HEFT suggests that SMEFT is not enough.
Highlights
Treating the Standard Model (SM) as an Effective Field Theory (EFT) is a principled way to organize the observable impact of new physics [1].1 Defining an EFT entails choosing a set of low energy degrees of freedom, and specifying a UV cutoff and a set of symmetries
A complementary approach was introduced by Falkowski and Rattazzi (FR) in [33], where it was argued that Higgs EFT (HEFT) is required when the scalar potential as expressed in terms of the electroweak doublet H is non-analytic at H = 0.4 As FR [33] emphasizes, such non-analyticity is a hallmark of integrating out new states that acquire all of their mass from electroweak symmetry breaking, thereby violating decoupling
10As we will emphasize in appendix B, our criteria only hold for O(N ) groups with N > 2, i.e., when the Higgs transforms as a non-trivial representation of a non-Abelian group. This will set us up for an exploration of concrete UV scenarios in section 6, where these leading order criteria are applied to expose that HEFT is required at the fixed point when one is integrating out a state that receives all of its mass from electroweak symmetry breaking
Summary
Treating the Standard Model (SM) as an Effective Field Theory (EFT) is a principled way to organize the observable impact of new physics [1].1 Defining an EFT entails choosing a set of low energy degrees of freedom, and specifying a UV cutoff and a set of symmetries. A complementary approach was introduced by Falkowski and Rattazzi (FR) in [33], where it was argued that HEFT is required when the scalar potential as expressed in terms of the electroweak doublet H is non-analytic at H = 0.4 As FR [33] emphasizes, such non-analyticity is a hallmark of integrating out new states that acquire all of their mass from electroweak symmetry breaking, thereby violating decoupling Both approaches highlight the irrelevance of the linear versus non-linear parametrization of the Higgs field itself in distinguishing HEFT from SMEFT, insofar as the two are related by field redefinitions; the distinction depends on the properties of the Lagrangian in a given parametrization.
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