Direct $CP$ asymmetries of three-body $B$ decays in perturbative QCD

We propose a theoretical framework for analyzing three-body hadronic $B$ meson decays based on the perturbative QCD approach. The crucial nonperturbative input is a two-hadron distribution amplitude for final states, whose timelike form factor and rescattering phase are fit to relevant experimental data. Together with the short-distance strong phase from the $b$-quark decay kernel, we are able to make predictions for direct $CP$ asymmetries in, for example, the $B^{\pm }\to {\pi }^{+}{\pi }^{-}{\pi }^{\pm }$ and ${\pi }^{+}{\pi }^{-}{K}^{\pm }$ modes, which are consistent with the LHCb data in various localized regions of phase space. Applications of our formalism to other three-body hadronic and radiative $B$ meson decays are mentioned.

Figure 1

Single-pion emission diagrams for the $B^{+}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}$ decay, where $Ms$ stands for the pion pair.

Figure 2

Two-pion emission diagrams, where $q$ denotes a $u$ or $d$ quark.

Figure 3

Annihilation diagrams.

Figure 4

More annihilation diagrams.