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CP violation: Dalitz interference, CPT and FSI

Resonances and final state interactions (FSI) play a role in the formation of 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 $ρ$ and $f_0(980)$ resonances plus a non resonant background including the $ππ\to KK$ coupled amplitude. We consider few illustrative examples to show the interference patterns appearing in the CP asymmetry, namely, that from the $ρ$ resonance plus a non-resonant amplitude, and that from the interference of the $ρ$ and $f_0(980)$ resonances. We perform the fit of the CP asymmetry for the charmless three-body $B^\pm$ decay channel $B^{\pm}\to π^{\pm} π^+π^-$ and obtain as outcome the $B^{\pm}\to π^{\pm} K^+K^-$ for $ππ$ channel asymmetry in the mass region below $1.6$ GeV in fair agreement with the new data LHCb data. Analogously, we also describe the CP asymmetry of the $B^\pm\to K^\pmπ^+π^-$ 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.