Branching ratio and polarization ofB→ρ(ω)ρ(ω)decays in perturbative QCD approach

Abstract

In this work, we calculate the branching ratios, polarization fractions and $CP$ asymmetry parameters of decay modes $B\ensuremath{\rightarrow}\ensuremath{\rho}(\ensuremath{\omega})\ensuremath{\rho}(\ensuremath{\omega})$ in the perturbative QCD approach, which is based on ${\mathbf{k}}_{T}$ factorization. After calculation, we find that the branching ratios of ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{+}{\ensuremath{\rho}}^{\ensuremath{-}}$, ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{+}{\ensuremath{\rho}}^{0}$, and ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{+}\ensuremath{\omega}$ are at the order of ${10}^{\ensuremath{-}5}$, and their longitudinal polarization fractions are more than 90%. The above results agree with BaBar's measurements. We also calculate the branching ratios and polarization fractions of ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}{\ensuremath{\rho}}^{0}$, ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{0}\ensuremath{\omega}$, and ${B}^{0}\ensuremath{\rightarrow}\ensuremath{\omega}\ensuremath{\omega}$ decays. We find that their longitudinal polarization fractions are suppressed to 60--80% due to a small color suppressed tree contribution. The dominant penguin and nonfactorization tree contributions equally contribute to the longitudinal and transverse polarization, which will be tested in the future experiments. We predict the $CP$ asymmetry of ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{+}{\ensuremath{\rho}}^{\ensuremath{-}}$ and ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\rho}}^{+}{\ensuremath{\rho}}^{0}$, which will be measured in $B$ factories.