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

Abstract

The effect of inelastic final-state interactions (IFSI's) on the determination of the weak phase $α$ from the isospin triangles of $B\rightarrow ππ$ 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 $α$ and the penguin amplitude in $B_{d}\rightarrow ππ$. This approach relies on a simplified isospin consideration and the factorization approximation, and its feasibility is irrelevant to the time-dependent measurements of $B_d\rightarrow ππ$.