Measuring<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>CP</mml:mi></mml:math>asymmetry in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>B</mml:mi><mml:mo>→</mml:mo><mml:mi>ρ</mml:mi><mml:mi>π</mml:mi></mml:math>decays without ambiguities

We reexamine the question of isospin analysis applied to the $\ensuremath{\rho}\ensuremath{\pi}$ decay modes of the $B$ as a tool to extract the CKM angle between ${V}_{\mathrm{ub}}^{*}{V}_{\mathrm{ud}}$ and ${V}_{\mathrm{tb}}^{*}{V}_{\mathrm{td}}$, without ambiguities or uncertainties due to penguin amplitude contributions. We find that a maximum-likelihood fit of the parameters to the full Dalitz plot distribution can successfully extract the parameters---perhaps with as few as 1000 events. This result is from a study of Monte Carlo-generated events without inclusion of detector efficiency or background effects. The interference between channels is a large effect and provides additional constraints on the parameters compared to channel-by-channel studies. These can usually lift the degeneracy between the multiple solutions that plague such studies. The inclusion of charged $B$ decays in the analysis would provide a direct check of the standard model prediction that the penguin amplitude weak phase for this channel is canceled by the weak mixing phase.