condition of partial wave unitarity is given for the S-wave amplitude matrix by [@Gunion:1992hs; @Lee:1977yc] | \_3 \_4 | a\^0 |\_1 \_2 | &lt; . \[PU\] We employ this condition in the high energy limit to constrain the model parameters in the next section. Thanks to the equivalence theorem [@Cornwall:1974km], the S-matrix elements in which longitudinally-polarized weak bosons are in initial and final states are equivalent to those in which these weak bosons are replaced by the corresponding Nambu-Goldstone bosons in the high energy limit [@Lee:1977yc]. In addition, in this limit, only quartic couplings (scalar contact interactions) of the Higgs-Goldstone couplings are relevant to the unitarity conditions, which can be translated into the bounds on the related Higgs-boson masses after Eq.(\[PU\]) is imposed. Therefore, we here evaluate the matrix $\langle \varphi_3 \varphi_4 | a^0 |\varphi_1 \varphi_2 \rangle$ taking into account all possible two-body scalar channels in the high energy limit, and obtain all the eigenstates and the eigenvalues.

Under $O(4) (\simeq SU(2)_L \otimes SU(2)_R)$, the field components of $\phi$, $\chi$ and $\xi$ are expressed by a and a representations as \_D &=&( \_1, \_2, \_r\^0, \_i\^0 ),\
\_T &=&( \_1, \_2, \_3, \_4, \_1, \_2, \_r\^0, \_i\^0, \_r\^0 ), where 