Strong and Weak Phases from Time-Dependent Measurements of $B \to \pi \pi$

Abstract

Time-dependence in $B^0(t) \to \pi^+ \pi^-$ and $\ob(t) \to \pi^+ \pi^-$ is utilized to obtain a maximal set of information on strong and weak phases. One can thereby check theoretical predictions of a small strong phase $\delta$ between penguin and tree amplitudes. A discrete ambiguity between $\delta \simeq 0$ and $\delta \simeq \pi$ may be resolved by comparing the observed charge-averaged branching ratio predicted for the tree amplitude alone, using measurements of $B \to \pi l \nu$ and factorization, or by direct comparison of parameters of the Cabibbo-Kobayashi-Maskawa (CKM) matrix with those determined by other means. It is found that with 150 fb$^{-1}$ from BaBar and Belle, this ambiguity will be resolvable if no direct CP violation is found. In the presence of direct CP violation, the discrete ambiguity between $\delta$ and $\pi - \delta$ becomes less important, vanishing altogether as $|\delta| \to \pi/2$. The role of measurements involving the lifetime difference between neutral $B$ eigenstates is mentioned briefly.