CP Violation in $B^\pm\to \rho^0\pi^\pm$ and $B^\pm\to \sigma\pi^\pm$ Decays

The decay amplitude of $B^+\to \pi^+\pi^-\pi^+$ in the Dalitz plot has been analyzed by the LHCb using three different approaches for the $S$-wave component. It was found that the mode with $\sigma$ (or $f_0(500)$) exhibited a CP asymmetry of 15\% in the isobar model, whereas the $f_2(1270)$ mode had a 40\% asymmetry. On the contrary, CP asymmetry for the dominant quasi-two-body decay $B^-\to\rho^0\pi^-$ was found to be consistent with zero in all three approaches, while all the existing theoretical predictions lead to a negative CP asymmetry ranging from $-7\%$ to $-45\%$. We show that the nearly vanishing CP violation in $B^-\to\rho^0\pi^-$ is understandable in the framework of QCD factorization (QCDF). It arises from the $1/m_b$ power corrections to the penguin amplitudes due to penguin annihilations and to the color-suppressed tree amplitude due to hard spectator interactions. Penguin annihilation and hard spectator interactions contribute destructively to $A_{CP}(B^-\to\rho^0\pi^-)$ to render it consistent with zero. The branching fraction and CP asymmetry in $B^-\to\sigma/f_0(500)\pi^-$ are investigated in QCDF with results in agreement with experiment.