Symmetry and geometry in a generalized Higgs effective field theory: Finiteness of oblique corrections versus perturbative unitarity

We formulate a generalization of Higgs effective field theory (HEFT) including an arbitrary number of extra neutral and charged Higgs bosons - a generalized HEFT (GHEFT) - to describe nonminimal 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 the perturbative unitarity of the scattering amplitudes involving the Higgs bosons and the longitudinal gauge bosons demands that the scalar manifold be flat. The relationship between the finiteness of the electroweak oblique corrections and the perturbative unitarity of the scattering amplitudes is also clarified in this language: we verify that once the tree-level unitarity is ensured, the one-loop finiteness of the oblique correction parameters S and U is automatically guaranteed.