We show that, besides the usual short distance contribution for $CP$ violation, final state interactions together with $CPT$ invariance can play an important role in the recent observation of $CP$ violation in three-body charmless $B^{\pm }$ decays. A significant part of the observed $CP$ asymmetry distribution in the Dalitz plot is located in a region where hadronic channels are strongly coupled. We illustrate our discussion comparing the recent observation of $CP$ violation in the $B^{\pm }\to K^{\pm }{K}^{+}{K}^{-}$ and $B^{\pm }\to K^{\pm }{\pi }^{+}{\pi }^{-}$ phase space, with a calculation based on $\pi \pi \to KK$ scattering.

• Received 5 August 2013

DOI:https://doi.org/10.1103/PhysRevD.89.094013