Constraints on Parameter Space from Perturbative Unitarity in Models with Three Scalar Doublets

We calculate an $s$-wave amplitude matrix for all the possible 2--to--2 body scalar boson elastic scatterings in models with three scalar doublets, including contributions from the longitudinal component of weak gauge bosons via the Equivalence Theorem Approximation. Specifically, we concentrate on the two cases with two[one] active plus one[two] inert doublet fields, referred to as I(1+2)HDM[I(2+1)HDM], under CP conservation. We obtain three analytically irreducible sub-matrices with the $3\times 3$ form and eighteen eigenvalues for the amplitude matrix as an independent set, where the same formula can be applied to both models. By requiring a perturbative unitarity condition, we can constrain the magnitude of quartic coupling constants in the Higgs potential. This constraint, in particular in the I(1+2)HDM, can be translated into a bound on masses of extra active scalar bosons. Furthermore, when Standard Model-like Higgs boson couplings with weak gauge bosons are deviated from the Standard Model predictions, the unitarity condition provides an upper limit on the masses. We find that stronger upper bounds on the masses of the active CP-even and CP-odd Higgs bosons are obtained under the constraints from the unitarity and vacuum stability conditions, as well as the electroweak $S$, $T$ and $U$ parameters, as compared to those in 2-Higgs Doublet Models with a softly-broken $Z_2$ symmetry.