Abstract
We apply on-shell methods to the bottom-up construction of electroweak amplitudes, allowing for both renormalizable and non-renormalizable interactions. We use the little-group covariant massive-spinor formalism, and flesh out some of its details along the way. Thanks to the compact form of the resulting amplitudes, many of their properties, and in particular the constraints of perturbative unitarity, are easily seen in this formalism. Our approach is purely bottom-up, assuming just the standard-model electroweak spectrum as well as the conservation of electric charge and fermion number. The most general massive three-point amplitudes consistent with these symmetries are derived and studied in detail, as the primary building blocks for the construction of scattering amplitudes. We employ a simple argument, based on tree-level unitarity of four-point amplitudes, to identify the three-point amplitudes that are non-renormalizable at tree level. This bottom-up analysis remarkably reproduces many low-energy relations implied by electroweak symmetry through the standard-model Higgs mechanism and beyond it. We then discuss four-point amplitudes. The gluing of three-point amplitudes into four-point amplitudes in the massive spinor helicity formalism is clarified. As an example, we work out the ψcψ Zh amplitude, including also the non-factorizable part. The latter is an all-order expression in the effective-field-theory expansion. Further constraints on the couplings are obtained by requiring perturbative unitarity. In the ψcψ Zh example, one for instance obtains the renormalizable-level relations between vector and fermion masses and gauge and Yukawa couplings. We supplement our bottom-up derivations with a matching of three- and fourpoint amplitude coefficients onto the standard-model effective field theory (SMEFT) in the broken electroweak phase. This establishes the correspondence with the usual Lagrangian approach and paves the way for SMEFT computations in the on-shell formalism.
Highlights
The bottom-up construction of effective-field-theory (EFT) Lagrangians proceeds from a field content and imposed — spacetime, local, and global — symmetries [1]
We adopted massive on-shell amplitude methods to examine a theory of the electroweak spectrum from the bottom up, including both renormalizable and non-renormalizable interactions
Our main objectives were first to explicitly demonstrate the emergence of the patterns due to electroweak symmetry breaking and, second, to provide the necessary basis for on-shell amplitude computations equivalent to those obtained from the Lagrangian of the standard-model effective field theory (SMEFT)
Summary
The bottom-up construction of effective-field-theory (EFT) Lagrangians proceeds from a field content and imposed — spacetime, local, and global — symmetries [1]. Given the electroweak particle content, the SU(2)L × U(1)Y local symmetry should follow from perturbative tree-level unitarity imposed on a sufficient number of four- and higher-point amplitudes [35,36,37,38]. At the non-renormalizable level, perturbative unitarity would yield relations between various couplings, and in particular, between amplitudes differing only by the numbers of external Higgs legs The resulting theory is characterized by the coefficients of a finite number of three-point amplitudes and of an infinite set of higher-point contact terms These correspond to the coefficients of independent non-renormalizable operators. We turn to the calculation of four-point tree-level amplitudes, and derive the ψcψZh amplitude as an example These amplitudes have factorizable parts, featuring single-particle poles, and non-factorizable parts, with no poles. Subsequent appendices contain a matching of amplitude coefficients to the SMEFT in the broken phase (section B), a compendium of massless amplitudes (section C), and a derivation of the non-factorizable ψcψZh amplitudes for massless fermions (section D)
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