& 0  \\
        \lambda_5^Iv^2  & 0 & m_S^2-\lambda_3^2 \\
    \end{pmatrix}.$$

Where $m_S^2 = m_1^2 + (\lambda_1 + \lambda_2) v^2/2$. Within the convention that $\lambda_3$ is real, the couplings $\eta_U$, $\eta'_U$, $\eta_D$ and $\eta'_D$ and $\lambda_5=\lambda_R^R + i \lambda_5^I$ are in general complex. The mass eigenstate scalars $N_j^0$ with mass $m_j$ are related to $h, S_R^0, S^0_I$ by the orthogonal transformations h &=& \_j O\_[hj]{} N\_j,\
S\_R &=& \_j O\_[Rj]{} N\_j, S\_I = \_j O\_[Ij]{} N\_j. We find the CP violating NP contribution to $B_q-{\bar B}_q$ mixing from neutral scalar exchange is \[o2coeff\] C\^[NP]{}(m\_t)= ( ’\_D m\_b/v )\^2 ( F ( m\_t/v))\^2 , where $ F(x)=x^2+ ~...~,$ and = \_j . For the rest of this paper we truncate the expansions in ${\sqrt 2} m_t/v$ at the leading non trivial term. So, for example, in Eq. (\[o2coefff\]) we use $F(x)=x^2$.

For simplicity we now focus on the case where $\lambda_5=0$. Then $h$, $S_R^0$ and $S_I^0$ are mass eigenstates and the term in the potential proportional to $\lambda_3$ is of interest as it leads to the mass splitting between the real neutral fields given by $m^2_R - m^2_I = 2\lambda_3 \, v^2$. When this term is non vanishing[^7], the tree level exchange 