We perform in the type II seesaw setting, a detailed study of the dynamical features of the corresponding general renormalizable doublet/triplet Higgs potential that depends on five dimensionless couplings and two mass parameters after spontaneous symmetry breaking, and highlight the implications for the Higgs phenomenology. In particular, we determine i) the complete set of tree-level unitarity constraints on the couplings of the potential and ii) the exact tree-level {sl all directions} boundedness from below constraints on these couplings. When combined, these constraints delineate precisely the theoretically allowed parameter space domain within our perturbative approximation. Among the seven physical Higgs states of this model, the mass of the lighter (heavier) CP-even state h0 (H0) will always satisfy a theoretical upper (lower) bound that is reached for a critical value mu_c of mu (the mass parameter controlling triple couplings among the doublet/triplet Higgses). Saturating the unitarity bounds we find m_h0 < {cal O}(0.7 - 1 TeV), while the upper bound for the remaining Higgses lies in the several tens of TeV. However, the actual masses can be much lighter. We identify two regimes corresponding to mu > mu_c and mu < mu_c. In the first regime the Higgs sector is typically very heavy and only h0 that becomes SM-like could be accessible to the LHC. In contrast, in the second regime, somewhat overlooked in the literature, most of the Higgs sector is light. In particular the heaviest state H0 becomes SM-like, the lighter states being the CP-odd Higgs, the (doubly) charged Higgses and a decoupled h0, possibly leading to a distinctive phenomenology at the colliders.
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