Quasi-two-body decays $B_c \to D_{(s)} [\rho(770),\rho(1450),\rho(1700) \to ] \pi \pi$ in the perturbative QCD factorization approach

In this paper, we studied the quasi-two-body $B_c \to D_{(s)} [ \rho(770),\rho(1450),\rho(1700) \to ] \pi \pi$ decays by employing the perturbative QCD (PQCD) factorization approach. The two-pion distribution amplitudes $\Phi_{\pi\pi}$ are applied to include the final-state interactions between the pion pair, while the time-like form factors $F_{\pi}(w^2)$ associated with the $P$-wave resonant states $\rho(770)$, $\rho(1450)$ and $\rho(1700)$ are extracted from the experimental data of the $e^+e^-$ annihilation. We found that: (a) the PQCD predictions for the branching ratios of the quasi-two-body $B_c \to D_{(s)} [\rho(770), \rho(1450), \rho(1700) \to ] \pi \pi$ decays are in the order of $10^{-9}$ to $ 10^{-5}$ and the direct $CP$ violations around $(10-40)\%$ in magnitude; (b) the two sets of the large hierarchy $R_{1a,1b,1c}$ and $R_{2a,2b,2c}$ for the ratios of the branching ratios of the considered decays are defined and can be understood in the PQCD factorization approach, while the self-consistency between the quasi-two-body and two-body framework for $B_{c} \to D_{(s)} [\rho(770) \to ] \pi \pi$ and $B_{c} \to D_{(s)} \rho(770)$ decays are confirmed by our numerical results; (c) taking currently known ${\cal B}(\rho(1450)\to\pi\pi)$ and ${\cal B}(\rho(1700) \to \pi\pi)$ as input, we extracted the theoretical predictions for ${\cal B}(B_{c} \to D \rho(1450)) $ and ${\cal B}(B_{c} \to D \rho(1700))$ from the PQCD predictions for the decay rates of the quasi-two-body decays $B_{c} \to D [\rho(1450), \rho(1700) \to ] \pi \pi$. All the PQCD predictions will be tested in the future experiments.

As known, in three-body decays, one can measure the distribution of CP asymmetry in the Dalitz plot [40] experimentally. However, from a theoretical point of view, it is too difficult to calculate CP violation in the whole Dalitz plot but practical to analyze a process of quasi-two-body decay. Experimentally, three-body B meson decays are known to be usually dominated by the low energy resonances on ππ, KK and Kπ channels and most of the quasi-two-body decays are extracted from the Dalitzplot analysis of three-body ones. In a quasi-two-body decay, the final-state interactions between the pair of mesons are considered while the rescattering between the third particle and the meson pair is usually ignored. In the views of PQCD [17,18], a direct evaluation of the hard kernels which contain two virtual gluons at lowest order is not important, the dominant contributions come from the region where the two energetic light mesons are almost collimating to each other with an invariant mass below O(Λm B )(Λ = m B − m b ), and the two-meson distribution amplitudes [17,18,[41][42][43][44] have been introduced to include both resonant and nonresonant contributions for the meson pair. In the previous work, the parameters in the P -wave two-pion distribution amplitudes were determined in PQCD approach [21]. Following Ref. [21], we have studied the quasi-two body decays B (s) → P/D[ρ(770), ρ(1450), ρ(1700) →]ππ [25][26][27][28] where P = π, K, η, η ′ , and D represents the charmed D meson.
In the past several years, a series of semileptonic B c decays [45] and nonleptonic two-body B c decays [46][47][48][49][50][51][52][53][54][55][56] have been studied in the PQCD framework. End-point singularity is avoid by keeping the transverse momentum k T of the quarks, and the Sudakov formalism makes this approach more reliable. From those literatures, we know the following points which can be also helpful for us to study the three-body B c decays: (1) The size of annihilation contributions is a meaningful issue in B c physics since the two-body nonleptonic charmless decays B c → h 1 h 2 (h 1 , h 2 represent the light pseudoscalar mesons, vector mesons, axial-vector mesons, scalar mesons and so on) occur through the weak annihilation diagrams only. As a feature of PQCD, the diagrams including factorizable, nonfactorizable and annihilation type are all calculable. From numerical calculation, the contribution from nonfactorizable and annihilation-type diagrams is also found to be of great importance in charmed decays B c → Dh (D stands for charmed D meson); (2) Since only tree operators are involved, the direct CP -violating asymmetries for those charmless B c decays are absent naturally, while there are both penguin and tree diagrams involved in B c → Dh decays and the possibly large direct CP violations in some channels were predicted [53,54].
The paper is organized as follows. In Sec. II, we give a brief introduction for the theoretical framework and perturbative calculations for the considered decays. Then, the numerical values and phenomenological analysis are given in Sec. III. Finally, the last section contains a short summary.

II. THE THEORETICAL FRAMEWORK
In the PQCD approach based on k T factorization, one separates the hard and soft dynamics in a QCD process [17]. The hard part is calculable in the perturbation theory while the soft part is not calculable perturbatively but have to be treated as an universal input determined from experiments. The amplitude of the process, consequently, could be expressed as a convolution of a hard kernel H with the hadron wave functions Φ(x, k T ) (x means a longitudinal momentum fraction and k T represents a transverse momentum). For a quasi-two-body B c -meson decay, its decay amplitude A in PQCD approach can then be written conceptually as the following convolution [17,18] The symbols ⊗ means the convolution integrations over the parton kinematic variables and the specific calculation formula will be shown in the following subsections.

A. Coordinates and wave functions
In the light-cone coordinates, the B c meson momentum p B , the momenta p 1 , p 2 for each pion and the total momentum of the pion pair p = p 1 + p 2 , and the D meson momentum p 3 in the rest frame of B c meson are chosen as where η = w 2 /[(1 − r 2 )m 2 Bc ] with the mass ratio r = m D /m Bc and the invariant mass squared of the pion pair w 2 = p 2 = m 2 (ππ), ζ is the momentum fraction for one of the pion pair. The momenta of the light quarks in the B c meson and the final state mesons are defined as k B , k and k 3 respectively where the momentum fraction x B , z and x 3 run between zero and unity. For B c meson, we use the same wave function as in Refs. [46][47][48][49][50][51][52][53][54][55][56]: where the exponent term describes the k T -dependence of φ Bc (x, b); while the parameter ω B = (0.60 ± 0.05) GeV, m c is the charm quark mass, m Bc is the B c meson mass, and f Bc is the decay constant of B c meson.

B. Decay amplitudes
For the considered B c → D (s) [ρ, ρ ′ , ρ ′′ →]ππ decays, the effective Hamiltonian H ef f [73] can be written as: where G F = 1.16639 × 10 −5 GeV −2 is the Fermi coupling constant, C i (µ) are the Wilson coefficients at the renormalization scale µ, O i (µ) are the effective four quark operators and V ij are the CKM matrix elements. At the leading order, there are eight diagrams contributing to the considered decays as shown in the Fig. 1. The four diagrams in first line are the emission type diagrams while the diagrams in the second line are the four annihilation type diagrams. By making analytical evaluations for those Feynman diagrams in Fig. 1, we can obtain the total decay amplitudes of these considered decays.
The F LL eD 1 and other individual amplitudes relevant with the eight sub-diagrams in Fig. 1 can be written in the following forms: (1) From the factorizable emission diagrams Fig. 1(a) and 1(b): (2) From the nonfactorizable emission diagrams Fig. 1(c) and 1(d): (3) From the factorizable annihilation diagrams Fig. 1(e) and 1(f): (4) From the nonfactorizable annihilation diagrams Fig. 1(g) and 1(h): whereη = 1 − η, C F = 4/3 is the color factor. The explicit expressions of the hard functions (h a , · · · , h h ), the hard scales (t a , · · · , t h ), the evolution factors (E a , · · · , E n ) and the threshold resummation factor S t (x i ) will be given in Appendix A.
For the decays involving ρ ′ and ρ ′′ mesons, one can get the relevant expressions for the corresponding decay amplitudes by simple replacements of φ 0,s,t for ρ meson to the ones for ρ ′ or ρ ′′ , respectively.

III. NUMERICAL RESULTS
Besides those specified in previous sections, the following input parameters will also be used in our numerical calculations (the masses, decay constants and QCD scale are in units of GeV) [2]: For the considered B c → D (s) [ρ, ρ ′ , ρ ′′ →]ππ decays, the differential decay rate can be written in the following form where τ Bc is the mean lifetime of B c meson, | p π | and | p D | denote the magnitudes of the π and D momenta in the center-of-mass frame of the pion pair, Based on the decay amplitudes as given in Eqs. (12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24) and the differential decay rate in Eq. (26), we obtain the PQCD predictions for the CP -averaged branching ratios (B) and the direct CP -violating asymmetries (A CP ) of the B c → D (s) [ρ, ρ ′ , ρ ′′ → ]ππ decays, and list the numerical results in Table I. The first error of these PQCD predictions comes from ω B = (0.60 ± 0.05) GeV for B c meson, the following three errors are from the Gegenbauer coefficients in the two-pion distribution amplitudes: a 0 2 = 0.30 ± 0.05, a s 2 = 0.70 ± 0.20, a t 2 = −0.40 ± 0.10 and the last error is from C D = 0.5 ± 0.1 ( C Ds = 0.4 ± 0.1 ) in D (D s ) meson wave function. The total theoretical error is about 10% to 30% of the central values.
In Fig. 2, we show the PQCD predictions for the differential decay rate dB/dw (Fig. 2(a) ) and for the CP -violating asymmetry A CP (Fig. 2(b) ) for the considered B + c → D 0 [ρ + →]π + π 0 decay and its charged-conjugation B − c →D 0 [ρ − →]π − π 0 decay. In Fig. 3, we show the same kinds of PQCD predictions as in Fig. 2 but for Figs. (2,3) and the numerical results as listed in Table I, we have the following observations: (1) For the considered quasi-two-body decays, the PQCD predictions are in the order of 10 −9 to 10 −5 for the CP -averaged branching ratios, and around (10 − 40)% in size for the direct CP violations. The B + c → D 0 [ρ + →]π + π 0 decay has the largest branching ratio, ∼ 1.64 × 10 −5 , and to be measured in LHCb experiment. (2) Among the three decays involving ρ(770) meson, there is a large hierarchy between their decay rates: For the special B + c → D 0 [ρ + →]π + π 0 decay, the factorizable emission diagram ( i.e. the term proportional to a 1 F LL eD of the decay amplitude in Eq. (12)) provide the dominant contribution. For B + c → D + [ρ 0 →]π + π − decay, however, the dominant contribution comes from the term proportional to a 2 F LL eD of the decay amplitude in Eq. (13). The small ratio R 1a ≈ 0.04 can be understood basically by the strong suppression due to the ratio |a 2 /a 1 | 2 ∼ 0.04. For B + c → D + s [ρ 0 → ]π + π − decay, besides the strong suppression due to |a 2 /a 1 | 2 , a new suppression factor |V us /V ud | 2 ∼ λ 2 may also be responsible for the ratio R 1c .  4. (a) The CP averaged differential decay rates for the B + c → D + s [ρ, ρ ′ , ρ ′′ →]π + π − decays; and (b) The total differential decay rate when the interference effects between ρ, ρ ′ and ρ ′′ also be included.
decays as a example, we can define the new ratios R 2a,2b,2c : Here the main reason for the hierarchy as shown by above ratios R 2a,2b and R 2c is the difference between the pion pair form factor F π for different intermediate resonance ρ, ρ ′ and ρ ′′ . For other two sets of decay modes, we find the similar hierarchy. From the three curves as shown in Fig. 4(a), one can see directly the large difference between the differential decay rates dB/dw for B + c → D + s [ρ, ρ ′ , ρ ′′ →]π + π − decays.
(4) By using the following well-known relation of the decay rates between the quasi-two-body and the corresponding twobody decay modes, one can extract the theoretical predictions for B(B c → D (s) ρ(ρ ′ , ρ ′′ )) from those for relevant quasi-two-body decays, if the decay rates B(ρ(ρ ′ , ρ ′′ ) → ππ) can be determined by employing other theoretical methods or measured directly by experiments.
For the decays involving ρ meson, for example, since B(ρ → ππ) ≈ 100%, we therefore do expect a similar branching ratio for the two-body B c → D (s) ρ decay and the corresponding quasi-two-body one. In order to examine this general expectation, we do the calculations for B(B c → D (s) ρ) directly by employing the PQCD approach in the same way as Ref. [53]. We used the same formulae as in Ref. [53], but with the updated input parameters and the new wave functions.
In the second and third column of Table II, we list our PQCD predictions obtained in the framework of the "Quasi-twobody" and two-body decay. In the forth, fifth and sixth columns of the Table II, as a comparison, we also show the relevant PQCD predictions as given in Ref. [53], and the theoretical predictions obtained by using the relativistic constituent quark model (RCQM) [57], or by using the light-front quark model (LFQM) [58].
From Table II, one can see that the PQCD predictions as listed in the column two and three agree very well with each other. This is a new confirmation for the self-consistency between the quasi-two-body and two-body framework of the PQCD approach for the considered B c meson decays. Although we used the same two-body framework and the decay amplitudes as in Ref. [53], but one can see that the PQCD predictions obtained in this work (column three) are much larger than those (column four) as given in Ref. [53], since we here used different distribution amplitude φ Bc (x, b), different wave function φ D (s) (x, b) for D (s) meson and the updated Gegenbauer moments, masses and decay constants as well. In Ref. [53], we set φ Bc (x, b) = δ(x− mc mB c ). In this paper, however, we take φ Bc (x, b) = δ(x− mc mB c )·exp[−ω 2 Bc b 2 /2] as given in Eq. (7).
used in Ref. [53] also be very different from the TABLE II. In the framework of the quasi-two-body or two-body decays, we list the PQCD predictions for the CP averaged branching ratios B(Bc → D (s) [ρ →]ππ) decays. As a comparison, we also list the theoretical predictions as given in Refs. [53,57,58].

Decays
Quasi-two-body Two-body PQCD [ one as given in Eq. [7] of this paper. By direct examination, we find that the dominant changes of the PQCD predictions are induced by the difference between the wave function φ D (s) (x, b) used here and the one used in Ref. [53]. More studies for the structure of the heavy mesons, such as B c , D and D s are clearly required. Precise experimental measurements for more B c meson decays can also test our predictions and help us to improve the theoretical framework itself.
(5) Due to the lack of the distribution amplitudes for ρ ′ and ρ ′′ , we can not calculate the branching ratios of the two-body decays B c → Dρ ′ and B c → Dρ ′′ by using the traditional way in the PQCD approach. In the framework of the quasi-twobody decays, fortunately, we can extract the PQCD predictions for the branching ratios of the two-body decays B c → Dρ ′ and B c → Dρ ′′ from the PQCD predictions for the branching ratios of the quasi-  Table I, we can then extract the PQCD predictions for the following two-body B c meson decays: where the individual errors have been added in quadrature. These PQCD predictions will be tested at the future LHCb experiments.
(6) In Fig. 4(b), we show the total differential decay rate after the inclusion of the contributions from all three resonant states ρ, ρ ′ and ρ ′′ . From the magnitude and the shape of the curve as illustrated in 4(b), one can see clearly the strong destructive interference near 1.6 GeV: a clear dip at w ≈ 1.6 GeV, similar with the one as shown in Fig. 45 of the Ref. [65], where the pion form factor-squared |F π | 2 measured by BABAR are illustrated as a function of the invariant mass of the pion pair in the range from 0.3 to 3 GeV. In our work, the same dip is induced by the strong destructive interference between ρ ′ and ρ ′′ , as shown in Fig. 4(b). Numerically, the PQCD predictions for the individual decay rate of ρ ′ and ρ ′′ and the interference term between them are the following: It is easy to see that the interference term is indeed large and negative when compared with other two individual contributions.

IV. SUMMARY
In this paper, we studied the quasi-two-body B c → D (s) [ρ, ρ ′ , ρ ′′ →]ππ decays in PQCD factorization approach. The twopion distribution amplitudes have been applied to include the final-state interactions between the pion pair. The contributions from the ρ, ρ ′ and ρ ′′ intermediate resonant states were estimated by introducing the time-like form factor F π involved in the P -wave two-pion distribution amplitudes. The PQCD predictions for the CP -averaged branching ratios and direct CPviolating asymmetries of the considered quasi-two-body decays are obtained and listed in Table I and II. Based on the relation as given in Eq. (34), we extract the theoretical predictions for the branching ratios of the two-body decays B c → D (s) X with X = (ρ, ρ ′ , ρ ′′ ) from those PQCD predictions for B(B c → D (s) [ρ, ρ ′ , ρ ′′ →]ππ) and those previously determined decay rates B(ρ ′ → ππ) and B(ρ ′′ → ππ).
From the analytical analysis and numerical calculations, we found the following points: (1) The PQCD predictions for the branching ratios of the quasi-two-body B c → D (s) [ρ, ρ ′ , ρ ′′ →]ππ decays are in the order of 10 −9 to 10 −5 , the direct CP violations are around (10 − 40)% in magnitude. The decay mode B + c → D 0 [ρ + →]π + π 0 has a large branching ratio, ∼ 1.64 × 10 −5 , and could be measured in the future LHCb experiment.
(2) The two sets of the large hierarchy R 1a,1b,1c for the ratios between the branching ratios B(B c → D (s) [ρ →]ππ) and R 2a,2b,2c among the branching ratios B(B + c → D 0 [ρ, ρ ′ , ρ ′′ →]π + π 0 ) are defined and can be understood in the PQCD factorization approach. The self-consistency between the quasi-two-body and two-body framework for B c → D (s)  where