Quasi-two-body decays $B \to \eta_c {(1S ,2S)}\;[\rho(770),\rho(1450),\rho(1700) \to ]\; \pi\pi$ 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.


I. INTRODUCTION
In recent years, due to the great progress in the theoretical studies and experimental measurements, the three-body hadronic B meson decays become much more attractive than ever before, and begin to play an important role in testing the standard model (SM) and in searching for the signal of the possible new physics beyond the SM.
In the theory side, the three-body hadronic decays of the heavy B meson are clearly much more complicated to be described theoretically than those two-body decays. We firstly can not separate the nonresonant contributions from the resonant ones clearly, and secondly do not know how to calculate or estimate the nonresonant and FSI contributions reliably [26]. As a first step, however, we can restrict ourselves to specific kinematical configurations, in which two energetic final state mesons almost collimating to each other, the three-body interactions for such topologies are expected to be suppressed strongly. Then it seems reasonable to assume the validity of factorization for these quasi-two-body B decays. In the "quasi-two-body" mechanism, the two-body scattering and all possible interactions between the two involved particles are included but the interactions between the third particle and the pair of mesons are neglected.
As discussed in Ref. [42], we here assume that the hard b-quark decay kernels containing two virtual gluons at leading order is not important due to the power-suppression. The contributions from the dynamical region, where there is at least one pair of the final state light mesons having an invariant mass below O(Λm B ) [42],Λ = m B − m b being the B meson and b quark mass difference, is dominant. It's reasonable that the dynamics associated with the pair of mesons can be factorized into a two-meson distribution amplitude Φ h1h2 . In the PQCD approach, one can write down the decay amplitude for a B → h 1 h 2 h 3 decay symbolically in the following form [42] where the hard kernel H describes the dynamics of the strong and electroweak interactions in three-body hadronic decays in a similar way as the one for the two-body B → h 1 h 2 decays, the function Φ B and Φ h3 are the wave functions for the B meson and the final-state h 3 meson. Up to now, the decays of B mesons to the charmonium state plus a pion pair, such as the decay 16], have been measured by BABAR and LHCb Collaboration. ForB 0 → J/ψπ + π − decay [62], six interfering π + π − states, ρ(770), f 0 (500), f 2 (1270), ρ ′ (1450), ω(782) and ρ ′′ (1700), are required to give a good description of invariant mass spectra and decay angular distributions. Along with the rapid progress of the LHCb experiment, more information of the B meson three-body decays involving various charmonium states (η c (1S, 2S) etc.) will become available. To improve the description of the invariant mass spectra, more resonant structures should be taken into account. Very recently, based on the PQCD factorization approach, we studied the S-wave resonance contributions to the decays B 0 (s) → η c (1S, 2S)π + π − [47, 51] and B 0 s → ψ(2s)π + π − [50], as well as the P -wave contributions (i.e. ρ(770), ρ(1450) and ρ(1700) 1 ) to the decays B → P ρ → P ππ [49,52].
In this paper, we will extend our previous analysis to the cases for the P -wave resonance (ρ, ρ ′ and ρ ′′ ) contributions to the three-body decays B → η c (1S, 2S)ππ. For the quasi-two-body decays B → η c (1S, 2S)(ρ, ρ ′ , ρ ′′ ) → η c (1S, 2S)ππ, the relevant Feynman diagrams are illustrated in Fig. 1. The vector current time-like form factor F π [67] will be adopted to describe the strong interactions between the P -wave resonant state (ρ, ρ ′ , ρ ′′ ) and the final-state pion pair in our work. In Sec. II, we give a brief introduction for the theoretical framework. The numerical values, some discussions and the conclusions will be given in last two sections. The explicit PQCD factorization formulas for all the decay amplitudes are collected in the Appendix.

II. FRAMEWORK
For the quasi-two-body B → η c (1S, 2S)(ρ, ρ ′ , ρ ′′ ) → η c (1S, 2S)ππ decays, the B meson momentum p B , the total momentum of the pion pair p = p 1 + p 2 and the final-state η c momentum p 3 , can be expressed in the light-cone coordinates as the following form: where m B is the mass of B meson, η = ω 2 (1−r 2 )m 2 B with r = m ηc /m B and the invariant mass squared ω 2 = p 2 . In the same way, we also define the momentum k B of the spectator quark in the B meson, the momentum k = zp + and k 3 = x 3 p 3 for the quark in the resonant state (ρ, ρ ′ , ρ ′′ ) and in the final state η c in the following form: where the parameter x B , z, x 3 denotes the momentum fraction of the quark in each meson and runs from zero to unity. If we define ζ = p + 1 /p + as one of the pion pair's momentum fraction, other kinematic variables of the two pions can be chosen as We assume that the B → η c (ρ, ρ ′ , ρ ′′ →)ππ decays can proceed mainly via quasi-two-body channels, which contain a Pwave resonant state by introducing the two-pion DAs Φ P ππ . As done in Ref. [46], we should introduce the time-like form factor F π (s), which involves the strong interactions between the P -wave resonance and two pions, as well as elastic rescattering of pion pair to parameterize the P -wave two-pion distribution amplitudes Φ P ππ . We adopt the same F π (s) in this work as the one in Ref. [46], the approximate relations will also be used in the following section. By taking the ρ − ω interference and the excited states into account, the form factor F π (s) can be written in the form of where s = m 2 (ππ) is the two-pion invariant mass squared, i = (ρ(1450), ρ(1700), ρ(2254)), Γ i is the decay width for the relevant resonance, m ρ,ω,i are the masses of the corresponding mesons, respectively. The function GS ρ (s, m ρ , Γ ρ ) has been parameterized in the Gounaris-Sakurai (GS) model [67] based on the Breit-Wigner (BW) model [21], The explicit expressions of the resonant state function GS ρ , GS i and BW ω and the values of the involved parameters can be found for example in Ref. [68]. We here adopt the same two-pion distribution amplitude as the one being used in Ref. [46], with where the Legendre polynomial P 1 (2ζ − 1) = 2ζ − 1. In the numerical calculations, we will use the same set of Gegenbauer moments a 0,s,t 2ρ in the two-pion distribution amplitude Φ P ππ as those used in Refs. [49,52],

III. NUMERICAL RESULTS AND DISCUSSIONS
The following input parameters (in units of GeV) will be adopted [69] for numerical calculations, The values of the Wolfenstein parameters are the same as given in Ref. [69]: A = 0.811 ± 0.026, λ = 0.22506 ± 0.00050, ρ = 0.124 +0.019 −0.018 ,η = 0.356 ± 0.011. For the decay B → η c (ρ → ππ), the differential decay rate is written as with the kinematic variables | − → p 1 | and | − → p 3 | where τ B ± = 1.638 ps, τ B 0 = 1.520 ps is the mean lifetime of B ± and B 0 meson. By using the differential decay rate as defined in Eq. (13) and the relevant decay amplitudes as given in the Appendix, we make the PQCD predictions for the branching rations B(B → η c (1S, 2S)(ρ, ρ ′ , ρ ′′ →)ππ) and find the following numerical results (in units of 10 −6 ) For the decays B → η c (1S)(ρ →)ππ, the first error of the PQCD predictions comes from the uncertainty of ω B = (0.40 ± 0.04) GeV, the following three errors are due to a t 2ρ = −0.40 ± 0.10, a s 2ρ = 0.70 ± 0.20 and a 0 2ρ = 0.30 ± 0.05 respectively. For the decay modes involving ρ ′ and ρ ′′ resonant states, the fifth error results from the uncertainty of the form factor F π (s) as given in Eq. For the considered decay modes B → η c (1S, 2S)ππ decay, the dynamical limit on the value of invariant mass ω is 2m π ≤ ω ≤ (m B − m ηc(1S,2S) ). For B → η c (2S)ππ decays, since m(ρ ′′ ) > ω max = (m B − m ηc(2S) ), the resonant ρ ′′ can not contribute to this decay. We therefore have the following PQCD predictions for the branching ratios ( in units of 10 −6 ): The errors in above equations have the same meaning as those in Eqs. (15,16).
From the curves as illustrated in Fig. 2 and Fig. 3 and the PQCD predictions for the decay rates as given in Eqs. (15)(16)(17)(18)(19)(20)(21), we have the following observations: (1) According to the full Dalitz-plot analysis to the B → J/ψπ + π − decay by the LHCb experiment [62], the dominant contributions come from the P -wave resonance ρ(770) and S-wave resonance f 0 (500). The relative rate between the two contributions was measured to be in the range here only the fraction of the helicity λ = 0 component of the P -wave resonance has been taken into account. Because of the analogous properties of the η c and J/ψ meson, it is reasonable for us to expect a similar invariant mass distribution for B → η c π + π − decay when compared with that of the B → J/ψπ + π − decay.
In a previous work [47], we calculated the S-wave resonance contributions to B 0 → η c (1S)π + π − decay, and confirmed that the largest contribution is from the f 0 (500). The PQCD predictions for the branching ratios are By using the PQCD prediction as given in Eq. (16), one can define the relative ratio of the P -wave and S-wave contribution as the following form: 6 , in BW model [21] , 1.7 , in Bugg model [22] .
The ratio R 1 agrees well with the ratio R J/ψ for the case of B → J/ψπ + π − decay and will be tested by the future LHCb and Belle-II experiment.
(2) From Fig. 2(a), one can see one prominent ρ peak, a shoulder around the ρ(1450) and a deep dip near ω ≈ 1.6 GeV, followed by an enhancement (the second lower but wider peak ) in the ρ(1700) region. Because the differential decay rate dB/dω depends on the values of |F π | 2 , the position of the first peak and deep dip, as well as the pattern of the whole curve do agree well with the curve in Fig. 45 of the Ref. [68], where the pion form factor-squared |F π | 2 measured by BABAR are illustrated as a function of √ s ′ (i.e.m(ππ)) in the region from 0.3 to 3 GeV.
The first dip around ω ≈ 1.6 GeV is in fact caused by the strong destructive interference between the resonant state ρ(1450) and ρ(1700). Taking B + → η c (1S)ρ(1450) → η c (1S)π + π 0 and B + → η c (1S)ρ(1700) → η c (1S)π + π 0 decay as an example, we calculated the interference terms between ρ(1450) and ρ(1700) amplitudes and found the large negative contribution to the total branching ratio. Numerically, the PQCD predictions for the individual decay rate and the interference term are: By comparing with other two individual contributions, we find that the interference term is indeed large and negative, which leads to the first deep dip in the region around ω ≈ 1.6 GeV, as illustrated in Fig. 2(a).
From the numerical results as given in Eqs. (15)(16)(17)(18), we obtain the relative ratio R 2 between the branching ratios of B meson decays involving η c (2S) and η c (1S) respectively, Owing to the same quark structures between η c and J/ψ mesons, one generally expect that the B → η c ππ decays should be similar in nature with the decays B → J/ψππ: i.e. R 2 (η c ) ≈ R 2 (J/ψ). This general expectation, in fact, agrees well with the LHCb measurement [66]: Here the main contribution also come from B 0 → J/ψρ(770) → J/ψπ + π − .
Based on the similarity between (η c (1S), η c (2S)) and (J/ψ, ψ(2S)) mesons, furthermore, it also be reasonable for us to expect similar R 3 and R 4 ratios for the cases of B → J/ψππ and B → ψ(2S)ππ decays. Fortunately, the ratio R 3 (J/ψ) analogous to R 3 (η c (1S)) has been measured by LHCb Collaboration recently [62]. If we take only the contributions from the longitudinal component ρ(1450) 0 and ρ(1700) 0 into account, we can obtain the value of the ratio R 3 (J/ψ) from the "Fit fractions of contributing components" as listed in Table VI of Ref. [62]: which indeed agrees very well with R 3 (η c (1S)) ≈ 0. 26. Other predictions will be tested by the forthcoming LHCb and Belle-II experimental measurements.

IV. CONCLUSION
In this work, we studied the contributions from the P -wave resonance ρ(770), ρ(1450) and ρ(1700) to the B → η c (1S, 2S)ππ decays in the PQCD framework. We calculated the branching ratios of the quasi-two-body decays B → η c (1S, 2S)(ρ(770), ρ(1450), ρ(1700) →)ππ by utilizing the vector current time-like form factor F π (s) with the inclusion of the final state interactions between the pion pair in the resonant regions.
From the analytical analysis and the numerical results, we found the following points: (1) The PQCD predictions for the branching ratios of the considered quasi-two-body decays are generally in the order of 10 −7 to 10 −6 . We obtained the theoretical predictions for the branching ratios of the two-body decays B(B → η c (1S, 2S)(ρ, ρ(1450), ρ(1700)) out of the PQCD predictions for the corresponding quasi-two-body decay modes, which will be tested by future LHCb and Belle II experiments.
(2) The whole pattern of the ω-dependence of the pion form factor-squared |F π | 2 measured by the BABAR Collaboration could be understood based on our studies, as illustrated in Fig. 2(a). The dominant contribution comes from the ρ(770) resonance, while the deep dip around ω ≈ 1.6 GeV is induced by the strong destructive interference between the contribution from ρ(1450) and ρ(1700).
(3) The general expectation based on the similarity between B → η c ππ and B → J/ψππ decays are confirmed: the value of newly defined ratio R 2 (η c ) ≈ 0.45 agrees well with the measured value R 2 (J/ψ) = 0.56 ± 0.09 as reported by LHCb experiments.
(4) The new ratios R 3 (η c (1S)) and R 4 (η c (2S)) among the branching ratios of the considered decay modes are defined, and the PQCD predictions for their values will be tested by future experiments.