Rescattering and chiral dynamics in B\to \rho\pi decay

We examine the role of B^0(\bar B^0) \to \sigma \pi^0 \to \pi^+\pi^- \pi^0 decay in the Dalitz plot analysis of B^0 (\bar B^0) \to \rho\pi \to \pi^+\pi^-\pi^0 decays, employed to extract the CKM parameter \alpha. The \sigma \pi channel is significant because it can break the relationship between the penguin contributions in B\to\rho^0\pi^0, B\to\rho^+\pi^-, and B\to\rho^-\pi^+ decays consequent to an assumption of isospin symmetry. Its presence thus mimics the effect of isospin violation. The \sigma\pi^0 state is of definite CP, however; we demonstrate that the B\to\rho\pi analysis can be generalized to include this channel without difficulty. The \sigma or f_0(400-1200) ``meson'' is a broad I=J=0 enhancement driven by strong \pi\pi rescattering; a suitable scalar form factor is constrained by the chiral dynamics of low-energy hadron-hadron interactions - it is rather different from the relativistic Breit-Wigner form adopted in earlier B\to\sigma\pi and D\to\sigma\pi analyses. We show that the use of this scalar form factor leads to an improved theoretical understanding of the measured ratio Br(\bar B^0 \to \rho^\mp \pi^\pm) / Br(B^-\to \rho^0 \pi^-).