Constraint on the CKM angle $α$ from the experimental measurements of CP violation in $B_d^0 \to π^+ π^-$ decay

In this paper, we study and try to find the constraint on the CKM angle $\alpha$ from the experimental measurements of CP violation in $B_d^0 \to \pi^+ \pi^-$ decay, as reported very recently by BaBar and Belle Collaborations. After considering uncertainties of the data and the ratio $r$ of penguin over tree amplitude, we found that strong constraint on both the CKM angle $\alpha$ and the strong phase $\delta$ can be obtained from the measured CP asymmetries $\spp$ and $\app$: (a) the range of $\delta < 0^\circ $ is excluded by the positiveness of observed $\app$, and the range of $86^\circ \leq \alpha \leq 137^\circ \bigvee 139^\circ \leq \alpha \leq 155^\circ$ is allowed by $1\sigma$ of the averaged data for $r =0.31$ and $\delta =[-\pi, \pi]$; (b) for Belle's result alone, the limit on $\alpha$ is $95^\circ \leq \alpha \leq 139^\circ$ for $ 0.32 \leq r \leq 0.41$; and (c) the angle $\alpha$ larger than $90^\circ$ is preferred.