Searches of exotic Higgs bosons in general mass spectra of the Georgi-Machacek model at the LHC

We derive the most general sets of viable mass spectra of the exotic Higgs bosons in the Georgi-Machacek model that are consistent with the theoretical constraints of vacuum stability and perturbative unitarity and the experimental constraints of electroweak precision observables, $Zb \bar b$ coupling and Higgs boson signal strengths. Branching ratios of various cascade decay channels of the doubly-charged Higgs boson in the ${\bf 5}$ representation, the singly-charged Higgs boson in ${\bf 3}$, and the singlet Higgs boson are further computed. As one of the most promising channels for discovering the model, we study the prospects for detecting the doubly-charged Higgs boson that is produced via the vector boson fusion process and decays into final states containing a pair of same-sign leptons at the 14-TeV LHC and a 100-TeV future $pp$ collider. For this purpose, we evaluate acceptance times efficiency for signals of the doubly-charged Higgs boson with general viable mass spectra and compare it with the standard model background estimates.