$P$-wave contributions to $B\to\psi\pi\pi$ decays in perturbative QCD approach

We present the differential branching fractions for the $B\rightarrow \psi \pi\pi$ decays with the charmonia $\psi=J/\psi,\psi(2S)$ in the invariant mass of the $P$-wave pion pairs in the perturbative QCD approach. The two-pion distribution amplitudes (DAs) corresponding to both longitudinal and transverse polarizations are constructed to capture important final state interactions in the processes. The time-like form factors, normalizing the two-pion DAs, contains contributions from the $\rho$ resonance and radial excitations fitted to the BaBar data. Given the hadronic parameters for the two-pion DAs associated with the longitudinal polarization which were determined in our previous study, and tuning those associated with the transverse polarization, we accommodate well the observed branching ratios and polarization fractions of the $B\rightarrow J/\psi \pi\pi$ decays. Our predictions for the $B\rightarrow \psi(2S) \pi\pi$ modes from the same set of parameters can be tested in future LHCb and BelleII experiments. We also investigate the sources of theoretical uncertainties in our calculation.

described by a Breit-Wigner (BW) propagator. More recently, final state interactions in the B (s) → J/ψππ decays were extracted from data in a framework based on dispersion theory [11]. These works mainly focus on the ρ(770) contribution to the P -wave di-pion system, with the two radial excitations ρ(1450) and ρ(1700) and the ρ-ω interference being neglected. As stressed in Ref. [12], the contributions from the two excited ρ states to the time-like pion form factor are indispensable, if one intends to accommodate the measured space-like pion form factor from the time-like one through analytic continuation. Several collaborations [13,14] have also successfully fitted the e + e − → π + π − cross section in the vicinity of the ρ(770) resonance, with a small but clearly visible ω-meson admixture.
It has been argued [15] that the dominant kinematic region for three-body B meson decays is restricted to the edges of a Dalitz plot, where two of the three final state mesons form a collimated pair in the rest frame of the B meson. In this region, the proof of the corresponding factorization theorem is basically similar to that for two-body cases [16][17][18]. Hence, the perturbative QCD (PQCD) approach [19,20] is applicable to three-body B meson decays, albeit the underlying k T factorization has not been proven rigorously [21,22]. With the introduction of two-hadron distribution amplitudes (DAs) [23][24][25][26] to absorb the final state interaction involved in the meson pair, the factorization formalism can be greatly simplified. As a result, a typical amplitude for the B → ψππ decays, ψ = J/ψ, ψ(2S), is written as [15] in which Φ B and Φ ψ are the B meson and charmonium DAs, respectively. The two-pion DA Φ ππ collects the nonperturbative dynamics in the ππ hadronization process. The hard kernel H, similar to that in two-body decays, can be evaluated in perturbation theory. The symbol ⊗ denotes the convolution in parton momenta of all the perturbative and nonperturbative objects.
In this paper we will analyze the decays B → ψ(ππ) P with the P -wave di-pion system. We do not consider the corresponding decays of a B s meson, in which the isovector resonant contributions are forbidden as explained before. The decays B (s) → J/ψ(ππ) S and B (s) → J/ψ(Kπ) S as well as the ψ(2S) counterparts, with the S-wave ππ and Kπ pairs, have been studied under the quasi-two-body approximation in the PQCD approach [27][28][29]. The charmless B meson decays into P -wave pion pairs in the longitudinal polarization were investigated in Refs. [30][31][32]. The three possible polarizations of the spin-1 ψ meson generate the longitudinal (0), parallel ( ), and perpendicular (⊥) amplitudes, such that the two-pion DAs corresponding to both the longitudinal and transverse polarizations are necessary nonperturbative inputs in our analysis. We will include the two-pion P -wave DAs corresponding to the transverse polarization into the PQCD formalism for the B → ψ(ππ) P decays. It will be explained that the total momentum (angular momentum) of the pion pair mimics the longitudinal (transverse) polarization of the P -wave di-pion system.
The decomposition of the longitudinal two-pion DAs up to the twist-3 accuracy has been presented in Ref. [30], but that of the transverse DAs is not yet available. Following the derivation in Refs. [33,34], the two-pion DAs can be parametrized in terms of the Gegenbauer polynomials that depend on parton momentum fractions, and the Legendre polynomials that depend on meson momentum fractions. Moreover, the two-pion DAs are normalized to the time-like form factors, which contain both resonant and nonresonant contributions to the di-pion system. To be specific, we adopt the vector-dominance-model parameterization for these form factors, which has been used to fit the pion form factor measured via the e + e − annihilation process [14]. Apart from the dominant ρ(770) component, the two radial excitations ρ(1450) and ρ(1700) as well as the ρ-ω interference effect were also taken into account. Besides, the B → J/ψππ modes are relevant to the determination of the CP violation phases in the B system, which is, however, not the theme of the present work. For recent progresses on this subject, refer to [35][36][37][38][39].
The paper is organized as follows. In Sec. II we define the involved kinematic variables and construct the two-pion DAs for the longitudinal and transverse polarizations. The numerical results are presented and discussed in Sec. III. The last section contains the conclusion. The factorization formulas for the considered decay amplitudes are collected in the Appendix.

II. FRAMEWORK
We begin with the parametrization of the kinematic variables involved in the decay B(P B ) → ψ(P 3 )(ππ)(P ). The momenta in the light-cone coordinates are chosen as in the B meson rest frame, with the mass ratio r = m/M , m(M ) being the charmonium (B meson) mass, and the variable η = ω 2 /(M 2 − m 2 ), ω 2 = P 2 being the invariant mass squared of the pion pair. The momenta p 1 and p 2 of the two pions, obeying p 1 + p 2 = P , are defined as with the pion momentum fraction ζ. We focus on the kinematic configuration, where p 1 and p 2 are almost collimated to each other with small amount of relative transverse momenta. The valence quark momenta labelled by k B , k 3 , and k in Fig. 1 (a) are parametrized as in which x B , x 3 , z denote the longitudinal momentum fractions, and k iT represent the transverse momenta. The hadronic matrix element for the B meson is written as [40] with the impact parameter b conjugate to the transverse momentum k BT , and the number of colors N c . The B meson DA φ B (x, b) is the same as in Refs. [40,41], where the shape parameter ω b = 0.40 ± 0.04 GeV has been fixed in the study of the B meson transition form factors [42,43], and the coefficient N is determined by the normalization 1 0 dxφ B (x, b = 0) = 1. The hadronic matrix elements for the longitudinally and transversely polarized vector charmonia are decomposed into respectively, with the longitudinal and transverse polarization vectors The explicit expressions of ψ i are referred to our previous works [44,45]. The two-pion DAs can be related to the pion DAs through a perturbative evaluation of the matrix elements [33,34], as a time-like di-pion production process, where Γ denotes the possible spin projectors I, γ 5 , γ µ , γ µ γ 5 , σ µν , and σ µν γ 5 . The complete set of pion meson DAs up to twist 3 is given by with the chiral scale m 0 . The above decompositions, in which the B meson four-velocity v B = (1, 0, 0, 0) is invariant under the frame rotation, hold for the pion momenta p 1 and p 2 in arbitrary directions. It is easy to see that the third structure in Eq. (10) approaches to the conventional one in [41], as p 1 is aligned with the plus direction n + = (1, 0, 0 T ), where the dimensionless vector n − = (0, 1, 0 T ) is along the direction of the displacement between the quarks q and q ′ in Eq. (9).
The key to construct the transverse polarization vector ǫ T µ for the di-pion system in terms of the kinematic variables in Eq. (10) is to relate it to the orbital angular momentum with the Levi-Civita tensor ǫ µνρσ under the convention ǫ 0123 = −1. The transverse polarization vector is then normalized into To arrive at the above expression, we have added p ρ 1 to p ρ 2 in Eq. (12) to get the total momentum P ρ of the pion pair without changing the result, and replaced v B by n − , because P is dominated by the plus component.
Employing the pion DAs in Eq. (10), adopting the definition in Eq. (13), and following the prescription in [33,34], we obtain the nonlocal matrix elements in Eq. (9) for various spin projectors Γ up to twist 3: with the two-pion DAs φ 0,T and φ s,t,v,a being of twist 2 and twist 3, respectively.
Some detailed derivation of Eqs. (14)- (19) are outlined here. For Eq. (14), we have applied the parametrizations for the longitudinal and transverse components of p 1 − p 2 , where the ζ-dependent factors will be absorbed into the corresponding two-pion DAs below. The matrix element in Eq. (16) for the choice µ, ν = +, y is proportional to in which Eq. (13) has been inserted. It is pointed out that the structure (p 1µ p 2ν − p 1ν p 2µ ) in Eq. (18) corresponds to / ǫ L / P for the twist-3 DAs in the longitudinally polarized pseudoscalar-vector meson pair [33,34]. We summarize the hadronic matrix elements Φ L ππ (Φ T ππ ) for the pion pair associated with the longitudinal (transverse) polarization from Eqs. (14)-(18) as where the projectors γ 5 / ǫ T / P , γ 5 / ǫ T , and ǫ µνρσ γ µ ǫ T ν P ρ n −σ come from Eq. (16), Eq. (17), and the second line of Eq. (14), respectively. Our result for the longitudinal piece Φ L ππ has the same form as in [30], while the transverse one Φ T ππ is new. The two-pion DAs for various twists are expanded in terms of the Gegenbauer polynomials, such as C in which we have introduced one Gegenbauer moment a 2 for each DA. The decomposition of the above DAs is similar to that of the ρ meson DAs, but with the vector (tensor) decay constant f ρ (f T ρ ) being replaced by the time-like pion form factors F (F ⊥ ).
For the form factor F (ω 2 ), we adopt the parametrization in Ref. [14], with i = ρ ′ (1450) and ρ ′′ (1700). The values of the masses m i , the widths Γ i , the complex coefficients c i , and the BW functions of various resonances are referred to [14]. For the form factor F ⊥ (ω 2 ), we employ the approximate [30]. Because the tensor decay constants f T for ρ(1450) and ρ(1700) are not known yet, we treat the corresponding modules |c i | in F ⊥ as free parameters, but keep their phases the same as in [14]. The global fit to the existing data for the B → J/ψππ branching ratios and polarization fractions [4] determines the central values of the dimensionless parameters appearing in the two-pion DAs, The differential branching fraction for the B → ψππ decays into P -wave pion pairs is expressed as where the pion and charmonium three-momenta in the ππ center-of-mass frame are given by respectively, with the pion mass m π and the Källén function λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + ac + bc). The terms A 0 , A , and A ⊥ represent the longitudinal, parallel, and perpendicular polarization amplitudes in the transversity basis, respectively. The polarization fractions f λ with λ = 0, , and ⊥ are then defined by The resultant branching ratios B and the polarization fractions f λ together with the available experimental measurements from the LHCb Collaboration for the J/ψ involved modes are summarized in Table I, and the corresponding ones for ψ(2S) are listed in Table II. Since the charged and neutral B meson decays differ only in the lifetimes and the isospin factor in our formalism, one can derive the branching ratios for the B + meson by multiplying those for the B 0 meson by the ratio 2τ B + /τ B 0 . The theoretical errors in Tables I and II are from some typical sources, namely, the two Gegenbauer moments in the twist-2 two-pion DAs, a 0 2 = 0.2±0.2 and a T 2 = 0.5±0.5, and the variation of the hard scales t from 0.75t to 1.25t, which characterize the energy release in decay processes (see the factorization formulas in the Appendix). It is worthwhile to mention that the hard kernels are evaluated only up to leading order plus the vertex corrections in this work, so the theoretical accuracy still needs to be improved. This is the case especially for B meson decays into charmonia, whose energy release may not be high enough for justifying the leading-order calculation. It is then expected that the hadronic parameters extracted from the data in the present framework should suffer larger theoretical uncertainty. Therefore, we have considered a wide range for the variation of the Gegenbauer moment a 0 2 = 0.2 ± 0.2, which covers the central value a 0 2 = 0.3 extracted from the data for charmless B meson decays in Ref. [31]. Eventually, we will improve the accuracy of our analysis and perform a global fit to all relevant data, when determining the involved hadronic parameters.
One can see that the errors from the two Gegenbauer moments are comparable and contribute to the major uncertainties as shown in Tables I and II , while the last one from the hard scales is only of a few percents due to the inclusion of the vertex corrections. We have also examined the sensitivity of our results to the choice of other Gegenbauer moments in the twist-3 two-pion DAs, a s 2 , a t 2 , a a 2 , and a v 2 , in Eq. (25). The first two give a comparable effect on the longitudinal branching ratio as a 0 2 does. With the increase (decrease) of a s 2 (a t 2 ), the total branching ratios and the longitudinal polarization factions become larger. On the contrary, the last two have a little impact on the total branching ratios, but can modify the relative importance of the parallel and perpendicular polarization amplitudes. As we set a a 2 = a v 2 = 0, the polarization fractions f and f ⊥ are roughly equal. When a a 2 and a v 2 are changed in the opposite direction, as indicated in Eq. (25), the difference between f and f ⊥ is enhanced and matches the data. It can be understood from the the factorization formulas presented in the Appendix: the contribution from φ a to the parallel polarization amplitudes plays a role similar to that from φ v to the perpendicular polarization I: PQCD results for the branching ratios and the polarization fractions of the P -wave resonance channels in the B 0 → J/ψπ + π − decay. The theoretical errors are attributed to the variation of the Gegenbauer moments a 0 2 and a T 2 , and the hard scales t, respectively. The data are taken from [3,4,38], where the first uncertainty is statistical and the second is systematic. The uncertainties from [38] are statistical only.
amplitudes, so the inputs of a a 2 and a v 2 opposite in sign increase the difference between the two amplitudes. It is also found that the coefficients |c i | in F ⊥ cause a significant effect on the branching ratios for the ρ(1450) and ρ(1700) channels. The variation of |c i | by 20% results in the change of the branching ratios by 40% ∼ 50%. The uncertainties from other parameters in our formalism, such as the decay constants and the CKM matrix elements, are not discussed here. The polarization fractions are not sensitive to these parameters, because they mainly yield an overall effect, which cancels in the ratios defined by Eq. (28).
Summing over all the contributing P -wave resonances in the ππ invariant mass spectra [2m π , M − m], we have the total branching ratios where the sources of the errors have been interpreted before. The former amounts up to 78% of the total three-body branching ratio B(B 0 → J/ψπ + π − ) = (3.96 ± 0.17) × 10 −5 [46]. As noticed in [3], the S-wave f 0 (500) and D-wave f 2 (1270) resonances, besides the P -wave ones, were also produced significantly in the the J/ψπ + π − final states. The best fit model in [3] implies that one full ρ(770) meson width contains 11.9% S-wave component and 0.72% D-wave component. Therefore, it is reasonable to leave the remaining 22% to the S-wave and D-wave contributions, as well as the nonresonant one and their interference in the entire invariant mass range. We estimate from Eq. (31) the ratio of the branching fractions, in which all the uncertainties have been added in quadrature. The value is slightly lower than the LHCb measurement where the third uncertainty corresponds to the one from the dilepton branching fractions of the J/ψ and ψ(2S) charmonium decays. The minor discrepancy may be resoled by including other partial wave contributions. The resonant decay rate obeys a simple factorization relation under the narrow width approximation, from which we extract the two-body B → ψR branching ratios, given the input of B(R → π + π − ). Combining the experimental fact B(ρ → ππ) ∼ 100% [46] and the estimates of B(ρ ′ → ππ) = 10.04 +5.23 −2.61 % and B(ρ ′′ → ππ) = 8.11 +2.22 −1.47 % in Ref. [32], we obtain the central values It is seen that both B(B 0 → J/ψρ) and B(B 0 → ψ(2S)ρ) are consistent with those derived in the PQCD framework for two-body decays [47]. We plot in Fig. 2 the total differential branching fractions in the P -wave π + π − invariant mass for the considered decays. The curve for the B 0 → J/ψπ + π − mode is similar to those for the charmless B → P ππ decays [32], since the same time-like form factors for the two-pion DAs, fitted by the BaBar Collaboration via the e + e − annihilation process [14], have been adopted. One finds a dip appearing at the invariant mass around 1.5 ∼ 1.6 GeV in Fig.  2(a), that is usually interpreted as the destructive interference between the ρ(1450) and ρ(1700) channels [14,48]. In fact, the best fit model also shows that the destructive interference between ρ(1450) and ρ(1700) is comparable with their individual fit fractions (see Tables VII and IX in Ref. [4]). However, the dip is not observed in Fig. 2(b), because the ρ(1700) state is beyond the di-pion invariant mass spectra for the B 0 → ψ(2S)π + π − mode. Both cases exhibit a clear ρ-ω interference pattern in the ρ peak region. The individual resonance contributions are displayed in   Tables I and II with Eq. (31), the ρ(770) resonance accounts for 83% of the total P -wave branching fractions in both the B 0 → J/ψπ + π − and ψ(2S)π + π − decays, while the higher ρ(1450) and ρ(1700) resonances contribute less than 10%. The obtained distributions in the P -wave ππ mass as well as the individual resonance contributions agree fairly well with the LHCb data shown in Fig. 13 of Ref. [4] and in Fig. 4 of Ref. [8].

IV. CONCLUSION
In this paper we have performed the analysis of the B → ψππ decays under the quasi-two-body approximation in the PQCD framework by introducing the two-pion DAs. Since both the charmonium and the P -wave pion pair in the final state carry the spin degrees of freedom, the two-pion DAs corresponding to both the longitudinal and transverse polarizations are the necessary nonperturbative inputs, and were constructed through a perturbative evaluation of the associated hadronic matrix elements as a time-like process. It was observed that the total momentum and the orbital angular momentum of the P -wave di-pion system mimics its longitudinal and transverse polarizations, respectively. The two-pion DAs for various spin projectors were then decomposed in terms of the Gegenbauer polynomials that depend on parton momentum fractions, and the Legendre polynomials that depend on meson momentum fractions up to twist 3. The time-like form factors, normalizing the two-pion DAs, were parametrized to consist of a linear combination of the ρ, ρ ′ , and ρ ′′ resonant contributions together with the ρ-ω interference.
We have determined the hadronic parameters involved in the two-pion DAs from a global fit to the data of the B 0 → J/ψρ(→ π + π − ) branching ratios and polarization fractions with good consistency. In particular, the resultant differential branching fractions in the P -wave di-pion invariant mass and individual resonance contributions match the LHCb data. We have also predicted the branching ratios and the polarization fractions of the B 0 → ψ(2S)ρ(→ π + π − ) decays, which can be confronted with future measurements. As a by-product, we extracted the two-body B 0 → ψρ branching ratios from the results for the corresponding quasi-two-body modes by employing the narrow width approximation. The predictions for the ρ(770) channels are in accordance with our previous PQCD calculations performed for two-body decays. The consistency between the three-body and two-body analyses supports the PQCD approach to exclusive charmonium B meson decays. The predictions for the higher excited intermediate states still need to be tested at the ongoing and forthcoming experiments. In this Appendix we present the explicit factorization formula for each B 0 → ψππ decay amplitude. The contributions from the longitudinal polarization, the normal polarization, and the transverse polarization are labelled by the subscripts L, N and T , respectively. The contributions from the (V − A) ⊗ (V − A), (V − A) ⊗ (V + A), and (S − P ) ⊗ (S + P ) operators are labelled by the superscripts LL, LR, and SP , respectively. The total decay amplitude is decomposed into where the three individual polarization amplitudes are written as with the CKM matrix elements V ij and the Fermi coupling constant G F . The Wilson coefficients C i encode the hard dynamics of weak decays. The above amplitudes are related to those in Eq. (26) via The explicit amplitudes F (M) from the factorizable (non-factorizable) diagrams in Fig. 1 read as {[−φ 0 (r 2 (−2ηz + 2z + 1) + (η − 1)(z + 1)) − η(1 − r 2 )(φ s (η + r 2 (2(η − 1)z + 1) − 2ηz + 2z − 1) +φ t (η + r 2 (2(η − 1)z − 1) − 2ηz + 2z − 1))]E e (t a )h a (x B , z, b B , b) +[2φ s ( η(1 − r 2 )(−η + r 2 x B − r 2 + 1)) − φ 0 (−η 2 + η + η 2 r 2 − 2ηr 2 + r 2 with r c = m c /M , m c being the charm quark mass, the color factor C F = 4/3, and the decay constant f ψ of the charmonium. The expressions for the evolution functions E, the hard kernels h, and the hard scales t a,b,c,d can be found in the Appendix of Ref. [27]. In addition, the vertex corrections to the factorizable diagrams in Fig. 1 are included through the modification of the Wilson coefficients as usual [49][50][51]. We point out that the amplitudes F correspond to the B → ππ transition form factors, which have been computed in QCD light-cone sum rules [52,53].