Improved determination of the CKM angle $\alpha$ from $B\to \pi \pi$ decays

Motivated by a recent paper that compares the results of the analysis of the CKM angle $\alpha$ in the frequentist and in the Bayesian approaches, we have reconsidered the information on the hadronic amplitudes, which helps in constraining the value of $\alpha$ in the standard model. We find that the Bayesian method gives consistent results irrespective of the parametrization of the hadronic amplitudes and that the results of the frequentist and Bayesian approaches are equivalent when comparing meaningful probability ranges or confidence levels. We also find that from $B\to \pi \pi$ decays alone the 95% probability region for $\alpha$ is the interval [80°, 170°], well consistent with recent analyses of the unitarity triangle where, by using all the available experimental and theoretical information, one gets $\alpha =(93\pm 4)°$. Last but not least, by using simple arguments on the hadronic matrix elements, we show that the unphysical region $\alpha \sim 0$, present in several experimental analyses, can be eliminated.

Figure 1

Comparison of Bayesian (left) and frequentist (right) results in the minimal assumptions case. The dark band in the left figure corresponds to the 95% probability region, to be compared with the 95% C.L. interval obtained in the right figure. The same interval is essentially selected in the two cases. The same conclusion holds when comparing the 68% probability region to the 68% C.L. interval.

Figure 2

Same as Fig. 1 but with the experimental errors reduced by a factor 10. Obviously, also in the Bayesian case, an eightfold ambiguity appears.

Figure 3

C.L. frequentist plot using the updated experimental value from [8].

Figure 4

Probability distribution for $\alpha$ obtained removing ${\mathcal{B}}_{\pi \pi }^{00}$. No significant information can be obtained in this case.

Figure 5

Probability distribution for $\alpha$ obtained with three different prior assumptions: $ES$ (left), $MA$ ( center), and $RI$ (right). In all the cases the hadronic parameters are constrained by $|T^{ij}|\le 10$, $|P|\le 2.5$. Explicitly, in the RI case we generate Re $P$, $T^{ij}$ and Im $P$, $T^{ij}$ with flat distribution and assign zero weight if $|P|>2.5$ or $|T^{ij}|>10$. In all the cases the region with “low” $\alpha$ is excluded. This is not due to the statistical approach, but to the physical assumptions on the hadronic matrix elements.

Figure 6

We show the range of allowed values of $\alpha$ corresponding to the 95% probability regions for the three parametrizations ($MA$—red, left line; $RI$—green, middle line; $ES$—blue, right line) as a function of the maximum allowed value of $|P|$, $P_{\max }$, scanning the range $1\le P_{\max }\le 5.5$. This figure shows that the residual dependence on $|P|$ is very mild.

Figure 7

We show the p.d.f.s for $|P|$, $|T^{+-}|$, and $|T^{00}|$, and the relative phases, obtained in the $MA$ parametrization, using only $B\to \pi \pi$ decays (first two rows) or the full UT analysis (last two rows).