Constraints on the CKM angle $\alpha$ from the measured CP asymmetries and branching ratios of $B\to {\pi }^{+}{\pi }^{-},{\pi }^{+}{\pi }^0$ and $K^0{\pi }^{+}$ decays

Abstract

In this paper, we draw constraints on the Cabibbo-Kobayashi-Maskawa (CKM) angle $\alpha$ and the strong phase $\delta$ from the experimental measurements of $\text{ssb}$, the CP-violating asymmetries and branching ratios of $B\to {\pi }^{+}{\pi }^{-},{\pi }^{+}{\pi }^0$ and $K^0{\pi }^{+}$ decays. In the SM, the measured $\text{ssb}$ leads to an upper limit on $\alpha$: $\alpha \le 180^{\circ }-\beta$. Taking the weighted-average of the newest BaBar and Belle measurements of $\text{spp}$ and $\text{app}$ in $B^0\to {\pi }^{+}{\pi }^{-}$ decay as the experimental input, we find the allowed range of angle $\alpha$, $76^{\circ }\le \alpha \le 135^{\circ }$ for $r=|P/T|=0.3\pm 0.1$. From the measured CP-averaged branching ratios of $B\to {\pi }^{+}{\pi }^{-},{\pi }^{+}{\pi }^0$ and $K^0{\pi }^{+}$ decays, we found an inequality, $\cos \alpha \cos \delta \ge 0.45$. The region of $70^{\circ }\le \alpha \le 110^{\circ }$ is thus excluded by applying this inequality. The combined constraints on $\alpha$ and $\delta$ are $117^{\circ }\le \alpha \le 135^{\circ }$ and $-160^{\circ }\le \delta \le -132^{\circ }$ if we take $\text{spp}=0.49\pm 0.27$ and $\text{app}=0.51\pm 0.19$ as the experimental input. The new lower limit on the angle $\alpha$ is dominated by the measured branching ratios considered in this paper, and is much stronger than those obtained before.

Comments: 31 pages, Revtex, 10 PS/EPs figures, some corrections have been made, several new references added. Repot-no: BIHEP-TH-2003-7