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 S(x_t) is the Inami-Lim function and x_t=m_t^2/M_W^2, m_t is the \overline{\mathrm{MS}} top mass, m_W 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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