of $S_{R/I}$ generates a CP violating NP contribution to $B_q-{\bar B}_q$ mixing and \[o2coefff\] C\^[NP]{}(m\_t) =(’\_D)\^2 ()\^4 (). In the above equation, the bottom quark mass $m_b\simeq 2.93~{\rm GeV}$ is evaluated at the top quark mass scale and $m^2_{S_{R/I}}=m_S^2 \pm \lambda_3 v^2$.

Using Eq. (\[key\]) the mass scale of the new scalars is given by m\_S\^4 + \_3\^2 v\^4. Using the best fit value $h_q =0.255$ [@Lenz:2010gu] we find that \[upperlimitmass\] m\_S\^4 (154 [GeV]{})\^4 |’\_D|\^2 \_3+ (246 [GeV]{})\^4 \_3\^2. Then for example with a value $|\eta'_D| = 5$ and $\lambda_3=1$ the scalar mass scale is $m_S \simeq 360 \,  {\rm GeV}$. As the mass splitting is significant we show in Fig. 1 the masses of the neutral scalars $S_R,S_I$ as a function of $\eta_D'$. Moderate enhancements of $\eta_D'$ avoid a light neutral state.

We have checked that the mass scale $m_S$ required for $B_q-{\rm B}_q$ mixing is compatible with the constraints from $K - \bar{K}$ mixing. This compatibility is due to MFV, which causes the ratio of the relevant Wilson coefficients to scale as $m_s^2/m_b^2 \, |V_{td}/V_{tb}|^2$.

Next we derive constraints on $\eta_U \eta_D$ that come from one loop Feynman diagrams with charged $S$ scalar exchange. We will show that 