Including the experimental information about $\Delta m_s$

The treatment of the experimental information about $\Delta m_s$ cannot be done by the usual uncertainty propagation, just because the simplifying hypotheses (linearization and Gaussian model for $e/\Delta m_s$) do not hold. In fact, the likelihood, which has the role of reweighting the probability, is open, in the sense described in Section 7 of Ref. [7], i.e. it does not go to zero at both ends of the kinematical region ($\Delta m_s=0$ and $\Delta m_s=\infty$ in this case), as shown in the top plot of Fig. 7. The reason is simple: in this kind of measurement $\Delta m_s$ is not yet incompatible with $\infty$ (no oscillation). Though the open likelihood does not allow to renormalize the p.d.f. (unless strong priors forbidding high values are used), the likelihood can still be used to reweigh the points in the $\bar {\rho}$-$\bar{\eta}$ plane (see Refs. [5] and [8] for other examples and discussions about the $R$ function).

It is instructive to see how the reweighting of $\Delta m_s$ is turned into the reweighting of the square radius of circle centered in $\bar {\rho}=1$ and $\bar{\eta}=0$ given by the constraint $C_4$, i.e. $r^2=e/\Delta m_s$. This is illustrated in the central plot of Fig. 7. We see that the reflection of $R\rightarrow 0$ for $\Delta m_s< 14$ ps$^{-1}$ essentially kills values above $r = 1$, resulting in a strong sensitivity bound (in the sense of Ref. [7]) on the angle $\gamma$ of the CKM matrix, forced to be below 90$^\circ$. The bottom plot of Fig. 7 shows the reweighting function in the $\bar {\rho}$-$\bar{\eta}$ plane.

For small values of $r^2$ the reweighting function is divergent, since the whole region of high $\Delta m_s$ is squeezed into a small region of $r^2$. This is no serious problem, since these points are already ruled out by the other constraints. The cutoff shown in the central and bottom plot of Fig. 7 is due to a cutoff at $\Delta m_s=100$ps$^{-1}$. Note, moreover, that the values of $\Delta m_s$ preferred by the data (around 15-20 ps$^{-1}$) overlap well with the $\bar {\rho}$-$\bar{\eta}$ region indicated by the other constraints. Note also that even if one goes through the academic exercise of removing by hand the peak around 15-20 ps$^{-1}$, chopping $R$ to 1, the effect on the values of $r$ above 1 (and hence of $\gamma$ above 90$^\circ$) does not change.

Figure: Top plot: likelihood of $\Delta m_s$ rescaled to the region of insensitivity. Central plot: reweighting factor of $r^2=e/\Delta m_s$. The peak just below 0.2 is an artifact of the cutoff in $\Delta m_s$ (see text). Bottom plot: same reweighting factor in the $\bar {\rho}$-$\bar{\eta}$ plane (note also here the low $r$ cutoff).