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

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