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 Higgs Effective Field Theory (HEFT) framework. The contact terms are spanned by a basis of Stripped Contact Terms, which carry the polarization information, multiplied by polynomials in the Mandelstam invariants. For terms with quadratic energy growth, we also derive the low-energy Standard Model Effective Field Theory (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 Effective Field Theory analyses directly in terms of observable quantities. For terms with quadratic energy growth, we also provide the mapping to the Warsaw basis.
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Acknowledgments
We thank Csaba Csaki, Gauthier Durieux, Christophe Grojean and Markus Luty for discussions. YS and MW thank the Mainz Institute for Theoretical Physics (MITP) of the Cluster of Excellence PRISMA+ (Project ID 39083149), for its hospitality and support during the workshop Amplitudes meet BSM. YS thanks the Aspen Center for physics, which is supported by the National Science Foundation (grant PHY-1607611), where parts of this work were completed. Research supported in part by the Israel Science Foundation (Grant No. 751/19), and by the NSF-BSF (Grant No. 2020-785). The research of TM is also supported by “Study in Israel” Fellowship for Outstanding Post-Doctoral Researchers from China and India by PBC of CHE. The research of MW is also supported by a Zuckerman Fellowship. HL is supported by the ISF, BSF and the Azrieli Foundation
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Liu, H., Ma, T., Shadmi, Y. et al. An EFT hunter’s guide to two-to-two scattering: HEFT and SMEFT on-shell amplitudes. J. High Energ. Phys. 2023, 241 (2023). https://doi.org/10.1007/JHEP05(2023)241
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DOI: https://doi.org/10.1007/JHEP05(2023)241