An EFT hunter's guide to two-to-two scattering: HEFT and SMEFT on-shell amplitudes

Abstract

We derive the contact terms contributing to the four-point amplitudes of the standard-model particles, keeping terms with up to quartic energy growth. Imposing just the unbroken low-energy symmetry, and treating the electroweak gauge bosons and the Higgs as independent degrees of freedom, we obtain the most general four-point contact-term amplitudes, corresponding to the HEFT framework. The contact terms are spanned by a basis of Stripped Contact Terms (SCTs), which carry the polarization information, multiplied by polynomials in the Mandelstam invariants. For terms with quadratic energy growth, we also derive the low-energy SMEFT predictions, via on-shell Higgsing of the massless SMEFT contact terms. We discuss several aspects of bottom-up versus top-down on-shell derivations of the HEFT and SMEFT amplitudes, highlighting in particular the simple counting of HEFT dimensions in the on-shell approach and the transparent relation between perturbative unitarity and gauge-invariance in the little-group covariant massive spinor formalism. Our results provide a formulation of EFT analyses directly in terms of observable quantities. For terms with quadratic energy growth, we also provide the mapping to the Warsaw basis. 4.3.9 γγγγ 4.3.10 gggZ, gggγ and γγγZ 4.3.11 ggZZ and γγZZ 4.3.12 ggW W and γγW W 4.3.13 γZW W 4.3.14 γZZZ 4.3.15 ggγγ 4.3.16 ggγZ 4.3.17gggg 4.4 Fermionic Amplitudes with Massless Vectors 4.4.1 f c f γh and f c f gh 4.4.2 ggf c f and γγf c f 4.4.3 γgf c f 4.4.4 γZf c f , γW f c f ′ , gZf c f and gW f c f ′ 5 Conclusions 25 A W W hh: On-shell construction of the HEFT and SMEFT amplitudes and on-shell Higgsing 26 A.1 The structure of the full amplitude A.2 Four-point contact terms from on-shell Higgsing A.3 The W W h coupling from on-shell Higgsing B Energy-growth of factorizable amplitudes 33