Convention-independent study of CP-violating asymmetries in B to π π

CP-violating asymmetries in the decay $B^0(t)\to \pi^+ \pi^-$ are a potentially rich source of information about both strong and weak phases. In a previous treatment by the present authors use was made of an assumption about the relative magnitude of tree and penguin amplitudes contributing to this process. This assumption involved an ambiguity in relating the tree amplitude to the amplitude for $B \to \pi \ell \nu$. It is shown here that one can avoid this assumption, which adopted a particular convention for tree and penguin amplitudes, and that the results are convention-independent.