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

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\mathrm{.}IntheSM,themeasured$\ssb$leadstoanupperlimiton$\alpha$:$\alpha \leq 180^\circ-\beta$. 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 $r=\mathrm{\mid }P\mathrm{/}T\mathrm{\mid }=0.3\pm 0.1$. From the measured CP-averaged branching ratios of $B \to \pi^+ \pi^-, \pi^+ \pi^0$and$K^0 \pi^+$decays,wefoundaninequality,$\cos{\alpha} \cos{\delta} \geq 0.45$\mathrm{.}Theregionof$70^\circ \leq \alpha \leq 110^\circ$ 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.