\_U g\^[  j]{}\_[U i]{} + ’\_U g\^[  j]{}\_[U k]{} \[(g\_U\^)\^[k]{}\_[ l]{} (g\_U)\^[l]{}\_[ i]{}\] + ,\
Y\^[  j]{}\_[D i]{} &=& \_D g\^[  j]{}\_[D i]{} + ’\_D g\^[  j]{}\_[D k]{} \[(g\_U\^)\^[k]{}\_[ l]{} (g\_U)\^[l]{}\_[ i]{}\] + . We decompose the second scalar doublet as S = (

[c]{} [S\^+]{}\
[S\^0]{}

), where $S^{0} = (S^{0} _R + i S^{0}_I)/\sqrt{2}$.

The scalar potential is \[pot\] V &=& (H\^[ i]{} H\_i - )\^2 + m\_1\^2 (S\^[i]{} S\_i),\
&+& (m\_2\^2 H\^[ i]{} S\_i + [h.c.]{}) + \_1 (H\^[ i]{} H\_i) (S\^[ j]{} S\_j),\
&+& \_2 (H\^[i]{} H\_j) (S\^[j]{} S\_i) + ,\
&+& ,\
&+& \_6 (S\^[i]{} S\_i)\^2.where $i,j$ are $\rm SU(2)$ indices. Here $v \simeq 246{\rm GeV}$ is the vacuum expectation value (vev) of the Higgs. Since we adopted the convention that the doublet $S$ does not get a vev the parameters $m_2^2$ and $\lambda_5$ are related by, $$m_2^2+\lambda_5^\star{v^2 \over 2}=0.$$ The spectrum of neutral real scalars consists of the Higgs scalar $h$ and $S_R^0$ and $S_I^0$. However, these are not mass eigenstates. In the $(h, S_R^0,S_I^0)$ basis, the neutral mass squared matrix ${\cal M}^2 $, where $\lambda_3$ is chosen real and positive, is

$${\cal M}^2 =\begin{pmatrix}
        m_h^2  &  \lambda_5^Rv^2 & \lambda_5^Iv^2   \\
          \lambda_5^Rv^2 &m_S^2+\lambda_3 v^2 