Unavoidable Higgs coupling deviations in the $Z_2$-symmetric Georgi-Machacek model

The $Z_{2}$-symmetric version of the Georgi-Machacek model does not possess a decoupling limit in which all the new particles can be made arbitrarily heavy, opening the possibility that the model can be entirely excluded if experiments reveal no deviations from the Standard Model. We explore this model, focusing on the part of parameter space in which the vacuum expectation value of the triplets, $\nu_\chi$, is small. In the small-$\nu_\chi$ limit, the second custodial-singlet scalar field $S$ necessarily becomes very light and can contribute to the total width of the 125 GeV Higgs boson $h$ via $h \to SS$. We show that this process, together with LHC measurements of the $h \to \gamma\gamma$ rate, entirely excludes masses $m_S < m_h/2$ and thereby severely constrains the parameter space, setting an experimental lower bound $\nu_\chi \gtrsim 12.5$ GeV on the vacuum expectation value of the triplets. This lower bound makes it impossible to avoid deviations from the Standard Model in the couplings of $h$ to fermion and vector boson pairs. We study the remaining parameter space after imposing constraints from direct searches for the additional Higgs bosons, and show that it is on the edge of being fully excluded at $95\%$ confidence level by LHC measurements of the 125 GeV Higgs boson's couplings. Measurements of these couplings at the future high-luminosity run of the LHC will have sufficient precision to entirely exclude the model at $5\sigma$ if no deviations from the Standard Model are observed.