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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