Global combination and results

Reweighting $f(\bar {\rho}, \bar{\eta}\,\vert\,C1,C2,C3)$ with $R$ function of $\Delta m_s$ we get the final result shown in Fig. 8.

Expected values and standard deviations are obtained by numerical integration. The result is

$$f(\bar {\rho}, \bar{\eta}\,\vert\,C1,C2,C3,C4)\hspace{0.5cm}... ...4 \\ \sigma(\bar{\eta})=0.039 \end{array}\right. \end{array}$$ (17)

practically identical to $\bar {\rho}=0.224\pm 0.038$ and $\bar{\eta}=0.317\pm 0.040$ given in Ref. [1] using the full Monte Carlo integration (obviously, those who pay attention the tenths of standard deviations should use the more accurate result of Ref. [1]).

Figure: Probability density function and contour plot $f(\bar {\rho}, \bar{\eta}\,\vert\,C1,C2,C3,C4)$ obtained by the constraint given by $\left \vert \frac{V_{ub}}{V_{cb}} \right \vert$, $\bar{\eta}$, $\Delta m_d$ and $\Delta m_s$ (see remarks in the text and in caption of Fig. 1 about the interpretation of the contour plot).

Figure: Top plot: $\Delta (\ln L)=1/2$ contours from the constraints $C_1$, $C_2$, $C_3$ and their combination. The bottom plot shows the effect of the experimental information on $\Delta m_s$, limited to the region of interest. The probability that $\bar {\rho}$ and $\bar{\eta}$ are both in the ``ellipses'' is about 37%.

It is interesting to show partial and global results as contour lines at $e^{-1/2}=0.61$ of the maximum of the reweighting functions, equivalent to the $\Delta (-\ln{L})=1/2$ or $\Delta\chi^2=1/2$ rules²(I refer to Ref. [1] for the relation between ``standard'' methods based on $\chi^2$ minimization and the more detailed inferential scheme illustrated there). The top plot of Fig. 9 shows the contour ``roads'' given by the first three constraints, together with the (almost) ellipse of their combination. The probability that the values of $\bar {\rho}$ and $\bar{\eta}$ are both in the ellipse is about 37%.³ Instead, the projections of the ellipse on each axis gives an interval of about $68\%$ probability in each variable.

The bottom plot of Fig. 9 shows the effect of the constraint $C_4$. First we notice the perfect agreement between the $\Delta m_d$ and $\Delta m_s$ roads, indicating that the values of $\Delta m_s$ suggested by the data are absolutely consistent with the other constraints within the Standard Model. Furthermore, the effect of the $\Delta m_s$ on the ``ellipse'' of the final inference is to reshape the left side, increasing the value of $\bar {\rho}$ and decreasing its uncertainty, with almost no effect on $\bar{\eta}$, as also shown by the results (16) and (17).