Higgs Scalars' Self-Couplings in Gildener-Weinberg Models

In Gildener-S. Weinberg (GW) models of electroweak symmetry breaking, the mass and alignment of the $125\,{\rm GeV}$ Higgs boson, $H$, are protected by approximate scale symmetry. We show that, due to spontaneous scale-symmetry breaking at the classical level, certain cubic and quartic scalar couplings in the scalar potential must vanish. These scalar couplings can acquire values somewhat larger than in the Standard Model at one-loop level of the Coleman-E. Weinberg expansion. In a two-Higgs doublet model of the GW mechanism, the triple-Higgs coupling $\lambda_{HHH}$ typically is 1.5--3.0 times its standard model value, $(\lambda_{HHH})_{\rm SM} = M_H^2/2v \cong 32\,{\rm GeV}$. This corresponds to $\sigma(pp \to HH)$ near its minimum value of 15--$20\,{\rm fb}$ for $\sqrt{s} = 13$--$14\,{\rm TeV}$ at the LHC. The coupling $\lambda_{HH'H'} = {\cal{O}}(M_{H'}^2/v)$, where $H'$ is the heavier CP-even (or odd) Higgs boson. We also calculate the quartic coupling $\lambda_{HHHH}$ and find it to be several times larger than its standard model value, $(\lambda_{HHHH})_{\rm SM} = M_H^2/8v^2 \cong 0.0323$. We conclude that searches at the LHC for new charged and neutral Higgs bosons, expected below $500\,{\rm GeV}$ in this and similar models, are the more promising path to testing them