Model-independent Constraints on the Weak Phase $\alpha$ (or $\phi_2$) and QCD Penguin Pollution in $B \to \pi\pi$ Decays

We present an {\it algebraic} isospin approach towards a more straightforward and model-independent determination of the weak phase $\alpha$ (or $\phi_2$) and QCD penguin pollution in $B\to \pi\pi$ decays. The world averages of current experimental data allow us to impose some useful constraints on the isospin parameters of $B\to \pi\pi$ transitions. We find that the magnitude of $\alpha$ (or $\phi_2$) extracted from the indirect CP violation in $\pi^+\pi^-$ mode is in agreement with the standard-model expectation from other indirect measurements, but its four-fold discrete ambiguity has to be resolved in the near future.