that obey MFV. [^4] This scenario is argued to be a better fit to the current data then the SM in [@Lenz:2010gu], which is disfavoured with a p-value of $3.1 \sigma$. In this case, the best fit values are $h_q =0.255$ and $2\sigma_q=180^{o}+63.4^o$. The best fit magnitude of the correction $h_q$ is small but its phase is large.

For simplicity in this paper we treat perturbative QCD in the leading logarithmic approximation and evaluate the needed matrix elements of four quark operators using the vacuum insertion approximation at the bottom mass scale. At the $t$-quark mass scale, in the SM, the effective Hamiltonian for $B_q-\bar B_q$ mixing is, $${\cal H}^{\rm SM}_q=  (V_{tq}^\star \, V_{tb})^2 C^{\rm SM}(m_t) {\bar b}^{\alpha}_L \gamma^{\mu} q^{\alpha}_L{\bar b}^{\beta}_L \gamma_{\mu} q^{\beta}_L,$$ where $\alpha$ and $\beta$ are color indices and $$C^{\rm SM}(m_t) =  \frac{G_F^2}{4 \, \pi^2} \, M_W^2 \,  S(m_t^2/M_W^2).$$ Here $S(m_t^2/M_W^2) \simeq 2.35$ is a function of $m_t^2/M_W^2$ that results from integrating out the top quark and $W$-bosons. Using $$\left(M_{12}^q\right)^{\rm SM}={\langle  B_q|{\cal H}^{\rm SM}_q|{\bar B}_q\rangle \over 2 m_{B_q}},$$ we have after running down to the $b$-quark mass scale that, $$\left(M_{12}^q\right)^{\rm SM}= (V_{tq}^\star \, V_{tb})^2C^{\rm SM} (m_t) \left({1 \over 3}\right)\eta f_{B_q}^2 m_{B_q}.$$ Here $\eta \simeq 0.84$ is a QCD 