Abstract
We study the three-body radiative decays $B\to \phi(\rho) K\gamma$ induced by a flavor-changing neutral current in the perturbative QCD approach. Pseudoscalar-vector ($PV$) distribution amplitudes (DAs) are introduced for the final-state $\phi K$ ($\rho K$) pair to capture important infrared dynamics in the region with a small $PV$-pair invariant mass. The dependence of these $PV$ DAs on the parton momentum fraction is parametrized in terms of the Gegenbauer polynomials, and the dependence on the meson momentum fraction is derived through their normalizations to time-like $PV$ form factors. In addition to the dominant electromagnetic penguin, the subleading chromomagnetic penguin, quark-loop and annihilation diagrams are also calculated. After determining the $PV$ DAs from relevant branching-ratio data, the direct $CP$ asymmetries and decay spectra in the $PV$-pair invariant mass are predicted for each $B\to \phi(\rho) K\gamma$ mode.
Highlights
A large number of experimental investigations on three-body hadronic B meson decays have been carried out by the BABAR [1,2,3,4,5,6], Belle [7,8,9,10] and LHCb [11,12,13] Collaborations in recent decades
Pseudoscalar-vector (PV) distribution amplitudes (DAs) are introduced for the final-state φK pair to capture important infrared dynamics in the region with a small PV-pair invariant mass. The dependence of these PV DAs on the parton momentum fraction is parametrized in terms of the Gegenbauer polynomials, and the dependence on the meson momentum fraction is derived through their normalizations to timelike PV form factors
Most of perturbative QCD (PQCD) studies of the above decays focus on the kinematic configuration corresponding to edges of a Dalitz plot, whose formalism can be simplified by the introduction of two-hadron distribution amplitudes (DAs) [15]
Summary
A large number of experimental investigations on three-body hadronic B meson decays have been carried out by the BABAR [1,2,3,4,5,6], Belle [7,8,9,10] and LHCb [11,12,13] Collaborations in recent decades. Most of PQCD studies of the above decays focus on the kinematic configuration corresponding to edges of a Dalitz plot, whose formalism can be simplified by the introduction of two-hadron distribution amplitudes (DAs) [15]. In these regions two of the three final state hadrons collimate with each other in the rest frame of the B meson. The two-hadron DAs ΦPV can be introduced to collect the dominant contribution from the region with a small PV-pair invariant mass mPV. The summary is given in the last section, and the factorization formulas for the B → φðρÞKγ decay amplitudes are collected in the Appendix
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