A Simplified Approach to Determine $\alpha$ and the Penguin Amplitude in $B_{d}\to \pi\pi$

Abstract

The effect of inelastic final-state interactions (IFSI's) on the determination of the weak phase $\alpha$ from the isospin triangles of $B\rightarrow \pi\pi$ is qualitatively illustrated. Neglecting the electroweak penguins and IFSI's and assuming the dominance of the top-quark loop in strong penguin diagrams, we propose an experimentally accessible way to approximately determine $\alpha$ and the penguin amplitude in $B_{d}\rightarrow \pi\pi$. This approach relies on a simplified isospin consideration and the factorization approximation, and its feasibility is irrelevant to the time-dependent measurements of $B_d\rightarrow \pi\pi$.