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CP asymmetry from hadronic charm rescattering in $B^\pm \to π^-π^+π^\pm$ decays at the high mass region

A model for the $B^\pm \to π^-π^+π^\pm$ decay amplitude is proposed to study the large CP violation observed at the high mass region of the Dalitz plane. A short distance $ b \to u $ amplitude with the weak phase $γ$ is considered together with the contribution of a hadronic charm loop and a s-wave $D\bar{D}\to ππ$ rescattering. In the model, the $χ_c^0$ appears as a narrow resonant state of the $D\bar D$ system below threshold. It is introduced in an unitary two channel S-matrix model of the coupled $D\bar D$ and $ππ$ channels, where the $χ_c^0$ complex pole in $D\bar D$ channel shows its signature in the off-diagonal matrix element and in the associated $D\bar{D}\to ππ$ transition amplitude. The strong phase of the resulting decay amplitude has a sharp sign change at the $D\bar D$ threshold, changing the sign of the CP asymmetry, as it is observed in the data. We conclude that the hadronic charm loop and rescattering mechanism are relevant to the broadening of the CP asymmetry around the $χ_c^0$ resonance in the $ππ$ channel. This novel mechanism provides a possible interpretation of the CP asymmetry defier experimental result presented by the LHCb collaboration for the $B^\pm \to π^-π^+π^\pm$ decay in the high mass region.