Quasi-two-body decays B→ηc(1S,2S) [ρ(770),ρ(1450),ρ(1700)→] ππB \to \eta_c {(1S ,2S)}\;[\rho(770),\rho(1450),\rho(1700) \to ]\; \pi\pi in the perturbative QCD approach

In this paper, we calculated the branching ratios of the quasi-two-body decays $B \to \eta_c (1S ,2S)$ $[\rho(770), \rho(1450),\rho(1700)\to ] \pi\pi$ by employing the perturbative QCD (PQCD) approach. The contributions from the $P$-wave resonances $\rho(770)$, $\rho(1450)$ and $\rho(1700)$ were taken into account. The two-pion distribution amplitude $\Phi_{\pi\pi}^{\rm P}$ is parameterized by the vector current time-like form factor $F_{\pi}$ to study the considered decay modes. We found that (a) the PQCD predictions for the branching ratios of the considered quasi-two-body decays are in the order of $10^{-7} \sim 10^{-6}$, while the two-body decay rates ${\cal B}(B \to \eta_c{(1S,2S)} (\rho(1450),\rho(1700)))$ are extracted from those for the corresponding quasi-two-body decays; (b) the whole pattern of the pion form factor-squared $|F_\pi|^2$ measured by the BABAR Collaboration could be understood based on our theoretical results; (c) the general expectation based on the similarity between $B \to \eta_c \pi\pi$ and $B \to J/\psi \pi\pi$ decays are confirmed: $R_2(\eta_c)\approx 0.45$ is consistent with the measured $R_2(J/\psi)\approx 0.56\pm 0.09$ within errors; and (d) new ratios $R_3(\eta_c(1S))$ and $R_4(\eta_c(2S))$ among the branching ratios of the considered decay modes are defined and could be tested by future experiments.Comment: 10 pages, 3 figure

Branching ratios, CPCP asymmetries and polarizations of B→ψ(2S)VB\rightarrow \psi(2S) V decays

We analyzed the nonleptonic decays $B/B_s\to \psi(2S) V$ with $V=(\rho, \omega, K^{*}, \phi)$ by employing the perturbative QCD (PQCD) factorization approach. Here the branching ratios, the $CP$ asymmetries and the complete set of polarization observables are investigated systematically. Besides the traditional contributions from the factorizable and nonfactorizable diagrams at the leading order, the next-to-leading order (NLO) vertex corrections could also provide considerable contributions. The PQCD predictions for the branching ratios of the $B_{(s)}\to \psi(2S)K^{*}, \psi(2S) \phi$ decays are consistent with the measured values within errors. As for $B\to \psi(2S) \rho, \psi(2S) \omega$ decays, the branching ratios can reach the order of $10^{-5}$ and could be measured in the LHCb and Belle-II experiments. The numerical results show that the direct $CP$ asymmetries of the considered decays are very small. Thus the observation of any large direct $CP$ asymmetry for these decays will be a signal for new physics. The mixing induced $CP$ asymmetries in the neutral modes are very close to $\sin 2\beta_{(s)}$, which suggests that these channels can give a cross-check on the measurement of the Cabbibo-Kobayashi-Maskawa (CKM) angle $\beta$ and $\beta_s$. We found that the longitudinal polarization fractions $f_0$ are suppressed to $\sim 50\%$ due to the large nonfactorizable contributions. The magnitudes and phases of the two transverse amplitudes $\mathcal {A}_{\parallel}$ and $\mathcal {A}_{\perp}$ are roughly equal, which is an indication for the approximate light quark helicity conservation in these decays. The overall polarization observables of $B\to \psi(2S) K^{*0}$ and $B_s\to \psi(2S) \phi$ channels are also in good agreement with the experimental measurements as reported by LHCb and BaBar. Other results can also be tested by the LHCb and Belle-II experiments.Comment: 14 pages, 1 figure, 6 table