ConstraintDeltaMs < UTfit

Constraint from $\Delta m_d /\Delta m_s$

The $B^0_s-\bar{B}^0_s$ oscillation frequency, which is related to the mass difference between the light and the heavy mass eigenstates of the system, is proportional to $|V_{ts}|^2$ in the Standard Model. Neglecting terms giving small contributions, $|V_{ts}|^2$ is independent of $\bar{\rho}$ and $\bar{\eta}$ . Instead, the ratio $\Delta m_d / \Delta m_s$ is proportional to $\bar{\rho}$ and $\bar{\eta}$ and the advantage of using this ratio instead of only $\Delta m_s$ is that the ratio $\xi=f_{B_d}\sqrt{B_{B_d}} / f_{B_s}\sqrt{B_{B_s}}$ is expected to be better determined from the theory than the individual quantities entering its expression. The measurement of $\Delta m_d / \Delta m_s$ gives a similar constraint as $\Delta m_d / \Delta m_d$ :
$\frac{\Delta m_d}{\Delta m_s} & = & \frac{m_{B_d}f^2_{B_d}\hat{B}_{B_d}}{m_{B_s}f^2_{B_s}\hat{B}_{B_s}} \frac{|V_{td}|^2}{|V_{ts}|^2} =\\ & = & \frac{m_{B_d}f^2_{B_d}\hat{B}_{B_d}}{m_{B_s}f^2_{B_s}\hat{B}_{B_s}} \left ( \frac {\lambda} {1-\lambda^2/2} \right ) ^2 \frac {(1-\bar{\rho})^2+\bar{\eta}^2}{\left ( 1+ \frac{\lambda^2}{1-\lambda^2 /2} \bar{\rho} \right ) ^2 +\lambda^4\bar{\eta}^2}$
In practice a problem arises in the (possible) use of the information from $\Delta m_d$ twice in the fit: in the $\Delta m_d$ constraint itself and in the ratio $\Delta m_d / \Delta m_s$ . The correlation has to be taken into account. One should recall that $\Delta m_s$ alone would give a constraint on $|V_{ts}^2|$ and so $f_{B_s}\sqrt{B_{B_s}}$ . It should also be considered that due to the error coming from chiral extrapolations the quantities which are best known and which are effectively calculated are $f_{B_s}\sqrt{B_{B_s}}$ and $\xi$ while $f_{B_d}\sqrt{B_{B_d}}$ is derived from the other two. For this reason we write the constraints in the following way :
$\Delta m_d \simeq [(1-\bar{\rho})^2+\bar{\eta}^2] \frac{f_{B_s}^2 B_{B_s}}{\xi^2}$
$\Delta m_s \simeq f_{B_s}^2 B_{B_s}$
Here the quantities entering in the expression of $\Delta m_d$ are the calculated ones: $f_{B_s}^2 B_{B_s}$ and $\xi$ . To make the constraint on $\Delta m_d$ more effective both quantities have to be improved. The constraint on $\Delta m_s$ helps in improving the knowledge of one of those: $f_{B_s}^2 B_{B_s}$ . We implemented this bound using directly the information from the experimental likelihood provided by CDF. The representation of this constraint in the $(\bar{\rho},~\bar{\eta})$ plane is given below.

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