Direct $CP$ violation in charmless three-body hadronic decays of $B$ mesons is studied within the framework of a simple model based on the factorization approach. Three-body decays of heavy mesons receive both resonant and nonresonant contributions. Dominant nonresonant contributions to tree-dominated and penguin-dominated three-body decays arise from the $b\to u$ tree transition and $b\to s$ penguin transition, respectively. The former can be evaluated in the framework of heavy meson chiral perturbation theory with some modification, while the latter is governed by the matrix element of the scalar density $⟨M_1{M}_2|{\overline{q}}_1{q}_2|0⟩$. Resonant contributions to three-body decays are treated using the isobar model. Strong phases in this work reside in effective Wilson coefficients, propagators of resonances, and the matrix element of scalar density. In order to accommodate the branching fraction and $CP$ asymmetries observed in $B^{-}\to K^{-}{\pi }^{+}{\pi }^{-}$, the matrix element $⟨K\pi |\overline{s}q|0⟩$ should have an additional strong phase, which might arise from some sort of power corrections such as final-state interactions. We calculate inclusive and regional $CP$ asymmetries and find that nonresonant $CP$ violation is usually much larger than the resonant one and that the interference effect between resonant and nonresonant components is generally quite significant. If nonresonant contributions are turned off in the $K^{+}{K}^{-}{K}^{-}$ mode, the predicted $CP$ asymmetries due to resonances will be wrong in sign when confronted with experiment. In our study of $B^{-}\to {\pi }^{-}{\pi }^{+}{\pi }^{-}$, we find that ${\mathcal{A}}_{CP}({\rho }^0{\pi }^{-})$ should be positive in order to account for $CP$ asymmetries observed in this decay. Indeed, both BABAR and LHCb measurements of $B^{-}\to {\pi }^{+}{\pi }^{-}{\pi }^{-}$ indicate positive $CP$ asymmetry in the $m({\pi }^{+}{\pi }^{-})$ region peaked at $m_{\rho }$. On the other hand, all theories predict a large and negative $CP$ violation in $B^{-}\to {\rho }^0{\pi }^{-}$. Therefore, the issue with $CP$ violation in $B^{-}\to {\rho }^0{\pi }^{-}$ needs to be resolved. Measurements of $CP$-asymmetry Dalitz distributions put very stringent constraints on the theoretical models. We check the magnitude and the sign of $CP$ violation in some (large) invariant mass regions to test our model.

• Received 29 July 2016

DOI:https://doi.org/10.1103/PhysRevD.94.094015