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Register a new userScalar and fermion on-shell amplitudes in generalized Higgs effective field theory
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Beyond standard model (BSM) particles should be included in effective field theory in order to compute the scattering amplitudes involving these extra particles. We formulate an extension of Higgs effective field theory which contains arbitrary number of scalar and fermion fields with arbitrary electric and chromoelectric charges. The BSM Higgs sector is described by using the non-linear sigma model in a manner consistent with the spontaneous electroweak symmetry breaking. The chiral order counting rule is arranged consistently with the loop expansion. The leading order Lagrangian is organized in accord with the chiral order counting rule. We use a geometrical language to describe the particle interactions. The parametrization redundancy in the effective Lagrangian is resolved by describing the on-shell scattering amplitudes only with the covariant quantities in the scalar/fermion field space. We introduce a useful coordinate (normal coordinate), which simplifies the computations of the on-shell amplitudes significantly. We show the high energy behaviors of the scattering amplitudes determine the curvature tensors in the scalar/fermion field space. The massive spinor-wavefunction formalism is shown to be useful in the computations of on-shell helicity amplitudes.
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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 $psi^c psi Zh$ 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 $psi^c psi Zh$ 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 four-point amplitude coefficients onto the standard-model effective field theory (SMEFT) in the broken electroweak phase.
We formulate a generalization of Higgs effective field theory (HEFT) including arbitrary number of extra neutral and charged Higgs bosons (generalized HEFT, GHEFT) to describe non-minimal electroweak symmetry breaking models. Using the geometrical form of the GHEFT Lagrangian, which can be regarded as a nonlinear sigma model on a scalar manifold, it is shown that the scalar boson scattering amplitudes are described in terms of the Riemann curvature tensor (geometry) of the scalar manifold and the covariant derivatives of the potential. The coefficients of the one-loop divergent terms in the oblique correction parameters S and U can also be written in terms of the Killing vectors (symmetry) and the Riemann curvature tensor (geometry). It is found that perturbative unitarity of the scattering amplitudes involving the Higgs bosons and the longitudinal gauge bosons demands the flatness of the scalar manifold. The relationship between the finiteness of the electroweak oblique corrections and perturbative unitarity of the scattering amplitudes is also clarified in this language: we verify that once the tree-level unitarity is ensured, then the one-loop finiteness of the oblique correction parameters S and U is automatically guaranteed.
We consider the phenomenological implications of charged scalar extensions of the SM Higgs sector in addition to EFT couplings of this new state to SM matter. We perform a detailed investigation of modifications of loop-induced decays of the 125 GeV Higgs boson, which receives corrections from the propagating charged scalars alongside one-loop EFT operator insertions and demonstrate that the interplay of $Hto gammagamma$ and $Hto Zgamma$ decays can be used to clarify the additional states phenomenology in case a discovery is made in the future. In parallel, EFT interactions of the charged Higgs can lead to a decreased sensitivity to the virtual presence of charged Higgs states, which can significantly weaken the constraints that are naively expected from the precisely measured $Hto gammagamma$ branching ratio. Again $Hto Zgamma$ measurements provide complementary sensitivity that can be exploited in the future.
The off-shell one-loop renormalization of a Higgs effective field theory possessing a scalar potential $simleft(Phi^daggerPhi-frac{v^2}2right)^N$ with $N$ arbitrary is presented. This is achieved by renormalizing the theory once reformulated in terms of two auxiliary fields $X_{1,2}$, which, due to the invariance under an extended Becchi-Rouet-Stora-Tyutin symmetry, are tightly constrained by functional identities. The latter allow in turn the explicit derivation of the mapping onto the original theory, through which the (divergent) multi-Higgs amplitude are generated in a purely algebraic fashion. We show that, contrary to naive expectations based on the loss of power counting renormalizability, the Higgs field undergoes a linear Standard Model like redefinition, and evaluate the renormalization of the complete set of Higgs self-coupling in the $Ntoinfty$ case.
We review some of the modern approaches to scattering amplitude computations in QCD and their application to precision LHC phenomenology. We emphasise the usefulness of momentum twistor variables in parameterising general amplitudes.
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