Resonances and final state interactions (FSI) play a role in the formation of charge parity ($CP$) violation (CPV) constrained by $CPT$ invariance. We provide a general formulation of CPV including resonances and FSI starting from the $CPT$ constraint. Our discussion is elaborated within a simple $B$ decay model with the $\rho$ and $f_0(980)$ resonances plus a nonresonant background including the $\pi \pi \to KK$ coupled amplitude. We consider a few illustrative examples to show the interference patterns appearing in the $CP$ asymmetry, namely, that from the $\rho$ resonance plus a nonresonant amplitude, and that from the interference of the $\rho$ and $f_0(980)$ resonances. We perform the fit of the $CP$ asymmetry for the charmless three-body $B^{\pm }$ decay channel $B^{\pm }\to {\pi }^{\pm }{\pi }^{+}{\pi }^{-}$ and obtain as outcome the $B^{\pm }\to {\pi }^{\pm }{K}^{+}{K}^{-}$ for $\pi \pi$ channel asymmetry in the mass region below 1.6 GeV in fair agreement with the new LHCb data. Analogously, we also describe the $CP$ asymmetry of the $B^{\pm }\to K^{\pm }{\pi }^{+}{\pi }^{-}$ decay, with that from the $B^{\pm }\to K^{\pm }{K}^{+}{K}^{-}$ channel obtained as output. As in the previous case, we also found agreement with LHCb experimental data.

• Received 1 July 2015

DOI:https://doi.org/10.1103/PhysRevD.92.054010