ConstraintDeltaMd < UTfit

Constraint from $\Delta m_d$

In the Standard Model, $B^0-\bar{B}^0$ oscillations occur through a second order process (a box diagram) with a loop containing W and up-type quarks. The box diagram with the top quark gives the dominant contribution. The $B^0-\bar{B}^0$ oscillation frequency, which is related to the mass difference between the light and the heavy mass eigenstates of the system is expressed, in the Standard Model, as function of $(\bar{\rho},~\bar{\eta})$ and the other elements of CKM matrix:
$\Delta m_d & = & \frac {G_F^2} {6 \pi^2} m_W^2 \eta_b S(x_t) m_{B_d} f_{B_d}^2 \hat{B}_{B_d} \left | V_{tb} \right | ^2 \left | V_{td} \right | ^2 =\\ & = & \frac {G_F^2} {6 \pi^2} m_W^2 \eta_b S(x_t) m_{B_d} f_{B_d}^2 \hat{B}_{B_d} \left | V_{cb} \right | ^2 \lambda^2 [(1-\bar{\rho})^2 + \bar{\eta}^2]$
where is the Inami-Lim function and , is the $\overline{\mathrm{MS}}$ top mass, is the W mass, and $\eta_c$ is the perturbative QCD short distance NLO correction. The remaining factor $f_{B_d}^2 \hat{B}_{B_d}$ encodes the information of non-perturbative QCD. The experimental values we use are summarized in the Table of Inputs. The representation of this constraint in the $(\bar{\rho},~\bar{\eta})$ plane is given below.

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