Rescattering and chiral dynamics inB→ρπdecay

We examine the role of $B^0\left(B^0\right)\to \sigma {\pi }^0\to {\pi }^{+}{\pi }^{-}{\pi }^0$ decay in the Dalitz plot analysis of $B^0\left(B^0\right)\to \rho \overrightarrow{\pi }{\pi }^{+}{\pi }^{-}{\pi }^0$ 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 $\overrightarrow{B}{\rho }^0{\pi }^0,$ $\overrightarrow{B}{\rho }^{+}{\pi }^{-},$ and $\overrightarrow{B}{\rho }^{-}{\pi }^{+}$ 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 $\overrightarrow{B}\rho \pi$ analysis can be generalized to include this channel without difficulty. The $\sigma$ or $f_0\left(400\mathrm{}-1200\right)$ “meson” is a broad $I=J=0\mathrm{}$ 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 $\overrightarrow{B}\sigma \pi$ and $\overrightarrow{D}\sigma \pi$ analyses. We show that the use of this scalar form factor leads to an improved theoretical understanding of the measured ratio $\mathrm{Br}(B^0\to {\rho }^{\mp }{\pi }^{\pm })/\mathrm{Br}{(B}^{-}\to {\rho }^0{\pi }^{-}).\mathrm{}$