Quasi-two-body decays $B_{(s)}\to P f_2(1270)\to P\pi\pi$ in the perturbative QCD approach

In this work, we calculate the $CP$-averaged branching ratios and direct $CP$-violating asymmetries of the quasi-two-body decays $B_{(s)}\to P f_2(1270)\to P\pi\pi$ with the two-pion distribution amplitude $\Phi_{\pi\pi}^{\rm D}$ by using the perturbative QCD factorization approach, where $P$ represents a light pseudoscalar meson $K, \pi, \eta$ and $\eta^{\prime}$. The relativistic Breit-Wigner formula for the $D$-wave resonance $f_2(1270)$ is adopted to parameterize the timelike form factor $F_{\pi}$, which contains the final state interactions between the pions in the resonant regions. The consistency of theoretical results with data can be achieved by determining the Gegenbauer moments of the $D$-wave two-pion distribution amplitudes. The decay rates for the considered decay modes are generally in the order of $10^{-9}$ to $ 10^{-6}$. The integrated direct $CP$ asymmetries for the charged modes agree with the {\it BABAR} and Belle measurements. As a by-product, we extract the branching ratios of $B_{(s)}\to Pf_2(1270)$ from the corresponding quasi-two-body decay modes, which still need experimental tests at the ongoing and forthcoming experiments.


I. INTRODUCTION
Three-body hadronic B meson decays are a rich field for studying the direct CP violation and testing the standard model and QCD. Recent measurements by BABAR [1][2][3][4][5][6] and Belle [7][8][9] Collaborations of a number of B → πππ, B → Kππ, or B → J/ψππ decays have triggered considerable theoretical interests in understanding three-body hadronic B decays. These three-body decays are known experimentally to be dominated by the low energy resonances on ππ, KK and Kπ channels. As the LHCb Collaboration reported, the ππ final states are found to comprise the decay products of the ρ(770), f 0 (500), f 0 (980), f 2 (1270) 1 , and f 0 (1370) (etc.) mesons in case of B 0 or B 0 s decays [10][11][12][13][14][15]. It is known that such three-body B decay modes are more intractable than those two-body decays due to the entangled resonant and nonresonant contributions, as well as the possible final-state interactions (FSIs) [16][17][18], whereas the relative strength of these contributions vary significantly for different decay modes. The analysis of these three-body decays utilizing the Dalitz plots [19,20] enables one to investigate the properties of various tensor, vector, and scalar resonances with the isobar model [21,22] in terms of the usual Breit-Wigner (BW) model [23]. Unfortunately, no proof of factorization has been given for the decays of the B into three mesons. However, we can restrict ourselves to specific kinematical configurations, in which two energetic final state mesons almost collimate to each other. Then the three-body interactions are expected to be suppressed strongly in such conditions. It seems reasonable to assume the validity of factorization for these quasi-two-body B decays. In the "quasi-twobody" 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.
There are several theoretical approaches for describing hadronic three-body decays of B mesons based on the symmetry principles and factorization theorems. U -spin and flavor SU (3) symmetries were adopted in Refs. [24][25][26][27][28][29]. The QCD-improved factorization (QCDF) [30] has been widely applied to the studies of three-body hadronic B meson decays in Refs. [31][32][33][34][35][36][37][38][39][40]. The authors investigated the detailed factorization properties of the B + → π + π + π − mode in different regions of phase space [31]. In Ref. [35], the authors studied the decays of B ± → π ± π ∓ π ± within a quasi-two-body QCD factorization approach and introduced the scalar and vector form factors for the S and P waves, as well as a relativistic BW formula for the D wave to describe the meson-meson final state interactions. In Ref. [38], for instance, the authors studied the nonresonant contributions using heavy meson chiral perturbation theory [41][42][43] with some modifications and analyzed the resonant contributions with the isobar model by using the usual BW formalism.
In the PQCD factorization approach, the contribution 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 ) [44],Λ = m B −m b being the B meson and b quark mass difference, is dominant. Because the hard b-quark decay kernels containing two hard gluons at leading order is not important due to the power-suppression, the configuration involving two energetic mesons almost parallel to each other may provide the dominant contribution. Then it's reasonable that the dynamics associated with the pair of mesons can be factorized into a two-meson distribution amplitude Φ h1h2 . The typical PQCD factorization formula for the B → h 1 h 2 h 3 decay amplitude can be described as the form of [44] where the hard kernel H contains only one hard gluon and describes the dynamics of the strong and electroweak interactions in the three-body hadronic decays as in the formalism for the two-body B meson decays. The Φ B and Φ h3 are the wave functions for the B meson and the final state h 3 , which absorb the non-perturbative dynamics in the relevant processes. In the PQCD approach based on the k T factorization theorem, we adopt the widely used wave function for B meson [68], which includes the intrinsic b dependence with b being a variable conjugate to k T . In Ref. [69], the authors pointed out that the operator-level definition of the transverse-momentum-dependent (TMD) hadronic wave functions is highly nontrivial in order to avoid the potential light-cone divergence and the rapidity singularity. A well-defined TMD can be found in Ref. [70]. Meanwhile, the Sudakov factors from the k T resummation have been included to suppress the long-distance contributions from the large b region in this work. The more precise joint resummation derived in [71] can be included in the future. For the QCD resummation, one can include its effect as going beyond the tree level in PQCD analysis, which will be done in the future by taking into account the results as given in Refs. [70,71].
In this work, we will extend our previous work on S and P -wave resonances to the D-wave ones in the PQCD framework. Taking the decays B (s) → P f 2 (1270) → P ππ, P = (π, K, η or η ′ ) as examples, the relevant Feynman diagrams are illustrated in Fig. 1 and 2. Since the tensor resonance cannot be created from the (V ± A), (S ± P ) or tensor current, the diagrams with a D-wave ππ pair emitted in Fig. 2 are prohibited in naive factorization. Phenomenologically there are growing appeals for the two-body charmless hadronic B decays involving a light tensor meson in the final states in the past few years [72][73][74][75][76][77][78][79]. More recently, one of us has investigated the two-body decays of the B c meson into the tensor charmonium using the PQCD approach [80]. Experimentally the CP -averaged branching ratios and direct CP -violating asymmetries of quasi-twobody decays B → πf 2 (1270) → πππ [2] and B → Kf 2 (1270) → Kππ [4,5,8,9] have been measured. One can see that the measured CP violation is just a number in two-body B decays, while the CP asymmetry depends on the invariant mass displayed in the Dalitz plot in the three-body modes [81]. It is meaningful to study the decays B → P f 2 (1270) → P ππ in the three-body framework, which provide useful information for understanding the CP -violation mechanisms. For the D-wave resonant state f 2 (1270), the relativistic BW formula is adopted to parametrize the timelike form factors F π , which contains the final state interactions between the pions in the resonant regions. The agreement of theoretical results with data can be achieved by determining the appropriate Gegenbauer moments of the D-wave two-pion distribution amplitudes. Just like the η-η ′ mixing in the pseudoscalar case, the isoscalar tensor states f 2 (1270) and f ′ 2 (1525) also have a similar mixing. The mixing angle between the f 2 (1270) and f ′ 2 (1525) is really small due to a fact that the f 2 (1270)(f ′ 2 (1525)) predominantly decays into ππ (KK). In our paper, we focus on the resonances on ππ channel and leave the detailed discussion about mixtures of f 2 (1270)-f ′ 2 (1525) for future studies associated with precise experimental measurements.
The present paper is organized as follows. 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
In the light-cone coordinates, the B meson momentum p B , the total momentum of the pion pair, p = p 1 + p 2 , the momentum p 3 of the final state meson P , and the momentum k B of the spectator quark in the B meson, the momentum k for the resonant state f 2 , k 3 for the final state P are in the form of where m B is the mass of B meson, the variable η is defined as η = ω 2 /m 2 B with the invariant mass squared ω 2 = p 2 = (p 1 + p 2 ) 2 . The parameters x B , z, x 3 are chosen as the momentum fraction of the positive quark in each meson and run from zero to unity. k BT , k T , and k 3T denote the transverse momentum of the positive quark, respectively. We define ζ = p + 1 /p + as one of the pion pair's momentum fraction, the two pions momenta p 1,2 can be described as where We here adopt the D-wave two-pion distribution amplitude similar as the one being used in Ref. [61], For I = 0, the distribution amplitude Φ I=0 vν=− contributes at twist-2, Φ I=0 s and Φ I=0 tν=+ contribute at twist-3. It is worthwhile to stress that this π-π system has similar asymptotic distribution amplitudes as the ones for a tensor meson [75][76][77], but replacing the tensor decay constants with the timelike form factor: π π π π π π π π FIG. 1: Typical Feynman diagrams for the quasi-two-body decays B → P (f2(1270) →)ππ, where B stands for the B ± , B 0 or Bs meson and P denotes K, π, η or η ′ . With α = a-d and β = e-h, the diagrams (α1) for the B → f2(1270) → ππ transition and the diagrams (β1) for annihilation contributions.   [74], while other parameters are adopted in PDG [91] in our numerical calculations.
The Wolfenstein parameters A = 0.811 λ = 0.22506ρ = 0.124η = 0.356 Lifetimes (ps) τB s = 1.51 where the Legendre polynomial P 2 (2ζ −1) = 1−6ζ(1−ζ). The twist-3 distribution amplitudes should be fixed by the equations of motion [82,83] related to twist-2 ones. The moment a 0 1 is regarded as a free parameter and determined in this work. Now, we focus on the dipion electromagnetic form factor. Taking the resonance contribution to the pion form factor into account, the pion electromagnetic form factor is defined in the standard way π . The form factor F π (s) can be analytically continued to the spacelike region s < 0, corresponding to the hadronic matrix element π + (p 1 )|j em µ |π + (−p 2 ) by crossingsymmetry. Even so, the continuation from the timelike to spacelike region does not work well for the resonance J/ψ as shown in Ref. [84]. As is well known, the electromagnetic form factor of pion at large (spacelike) momentum transfer on the basis of one-pion distribution amplitude has been computed in Ref. [85] with the PQCD approach at NLO. Applying the analytical continuation of the obtained result in the kinetic variable of momentum-transfer squared, one should be able to compute the timelike pion electromagnetic form factor directly without resorting to fitting the experimental measurements [86]. However, it's difficult to make the analytical continuation from the spacelike to timelike region for the dipion form factor. When we start from the spacelike region, it is not easy to identify the decay width piece, which can be interpreted as contribution of multihadron states to the imaginary part of the form factor in the resonant contribution [84,87]. In other word, it is impossible to generate the pole from the spacelike region by the analytical continuation.
Certainly, an alternative way to account for the hadronic resonance effect is that the electromagnetic form factor of pion at large (timelike) momentum transfer can be computed from perturbative QCD factorization approach at large momentum transfer with the parton-hadron duality ansatz. In order to account for the hadronic resonance effect, one can apply the hadronic dispersion relation for the electromagnetic form factor of the pion in the entire kinematic region and then implement the constraints from QCD calculation at large Q 2 for the determination of the unknown hadronic parameters entering the nonperturbative parametrization of the dispersion form. We will make efforts to calculate the electromagnetic form factor of pion at large (timelike) momentum transfer in the future. In this work, the relativistic BW formula for the D-wave resonance f 2 (1270) is adopted to parametrize the timelike form factor F π (s) and the explicit simplified expressions are in the following form, with the two-pion invariant mass squared s = ω 2 = m 2 (ππ) and the m f2 = 1.276GeV and Γ f2 = 0.187GeV are the pole mass and width of resonance state f 2 (1270), respectively. We find that the variations of radius parameter r BW = 4 [2] do not significantly change the values in our calculations. Hence, it is reasonable to set r BW = 0 in our latter numerical calculations.
will also be used in the following section.

III. NUMERICAL RESULTS
For the numerical calculations, those parameters such as meson masses, the Wolfenstein parameters, decay constants, and the lifetimes of B (s) mesons are given in Table I, while the B meson and kaon (pion) distribution amplitudes are the same as widely adopted in the PQCD approach [62,[88][89][90].
For the decays B → P f 2 (1270) → P ππ, the differential branching ratio is written as [91], The kinematic variables | − → p π | and | − → p 3 | denote one of the pion pair's and P 's momentum in the center-of-mass frame of the pion pair, By using the differential branching fraction in Eq. (11), and the decay amplitudes in the Appendix, we calculate the CP averaged branching ratios (B) and direct CP -violating asymmetries (A CP ) for the decays B (s) → P (f 2 →)ππ. In this work, we consider two scenarios for the tensor meson f 2 (1270). In scenario I, it is assumed that f 2 (1270) is a pure nonstrange isospin singlet state (uū + dd)/ √ 2, while in scenario II, the strange state ss enters the contributions with a nonvanishing mixing angle just like the η − η ′ mixing in the pseudoscalar sector. Thus, the isoscalar tensor states can be written as The detailed discussions about the mixing angle could be found in Refs. [92,93]. Here we employ the most recent updated value (9 ± 1) • from PDG2016 [91]. The predictions on the CP averaged branching ratios and direct CP -violating asymmetries from scenario I and II are enumerated distinctly in Tables II and III, as well as the current available data, respectively. The fit to the data [91] determines the Gegenbauer moment a 0 1 = 0.40, which differs from that in the distribution amplitudes for a longitudinally polarized f 2 (1270) meson [75][76][77].
In our numerical calculations, the first theoretical uncertainty results from the variations of the shape parameter ω B (s) of the B (s) meson distribution amplitude. We adopt the value ω B = 0.40 ± 0.04 GeV or ω Bs = 0.50 ± 0.05 GeV and vary its value with a 10% range, which is supported by intensive PQCD studies [68,[94][95][96]. It is shown that the shape parameter ω B can reach about 20% in magnitude for the main uncertainties. We note that another value ω B (1GeV) = 0.354 +0.038 −0.030 GeV implied by the light-cone sum rules calculations of the semileptonic B → π form factors with B-meson DAs on the light-cone [97] has been taken in the Refs. [98,99]. This number is very close to our error range, which result in the branching ratios changing 20 percents as mentioned above. Model-independent determinations of the inverse moment of the B-meson light-cone distribution amplitude ω B in HQET have been discussed extensively from the radiative leptonic B-meson decays with the QCD factorization approach and the dispersion relations in Refs. [98,99]. The opportunity of constraining the inverse moment ω B should be explored TABLE II: CP averaged branching ratios of B (s) → P (f2(1270) →)π + π − decays calculated in PQCD approach together with experimental data [91]. The theoretical errors correspond to the uncertainties due to the shape parameters ωB (s) in the wave function of B (s) meson, the Gegenbauer moment a 0 1 and the next-to-leading-order effects (the hard scale t and the QCD scale ΛQCD), respectively.   III: Direct CP -violating asymmetries of B (s) → P (f2(1270) →)π + π − decays calculated in PQCD approach together with experimental data [91]. The theoretical errors correspond to the uncertainties due to the shape parameters ωB (s) in the wave function of B (s) meson and the next-to-leading-order effects (the hard scale t and the QCD scale ΛQCD), respectively. For the CP averaged branching ratios, the second error contributes the main uncertainties in our approach, while the other two errors are comparable and less than 30%. For the direct CP -violating asymmetries, the error from the Gegenbauer moment is largely cancelled between the numerator and denominator, and the main uncertainty refer to the hard scale. The uncertainties from τ B ± , τ B 0 , τ Bs and the Wolfenstein parameters in [91] are small and have been neglected. The significance of the radius parameter r BW has been verified in our calculations. Taking the decay channel B + → K + (f 2 →)π + π − in scenario I as an example, the branching ratio B = 11.09 × 10 −7 and direct CP asymmetry A CP = −48.2% for r BW = 0, while for r BW = 4.0, the corresponding values are B = 10.77 × 10 −7 and A CP = −48.4%. One can find that the difference between the results with two choices is really small.

Modes Scenario I Scenario II Experiment
It is observed that the branching ratios from the two scenarios are comparable for the same B meson decay modes, while in the case of B s decays, the two scenarios are rather different. For example, the value of B(B s → η ′ (f 2 →)π + π − ) in scenario II are nearly twice as that in scenario I. The main reason is that the factorizable contributions from the P emission diagrams [see Fig. 1(a1) and Fig. 1(b1)] in ss component enhance the corresponding decay amplitudes. However, for the B meson decays, such factorizable contributions come from the nn (n = u, d) component, while the ss will contribute to the annihilation diagrams or nonfactorizable emission diagrams, which are power suppressed when compared with the factorizable emission diagrams according to the power counting rules in the factorization assumption.
It is shown that f 2 (1270) is primarily an (uū + dd)/ √ 2 state for the case of B meson decay modes, whereas for the B s decays, the ss component makes a significant contributions albeit with suffering a large suppression from the mixing angle, especially for the process B s → η ′ (f 2 →)π + π − . Therefore, we recommend the LHCb and/or Belle-II experiments to measure this mode to probe the precise structures of f 2 (1270). In addition, from Table II Since both η q and η s will contribute in these modes, but the relative sign of the η q state with respect to the η s state is negative for η and positive for η ′ , which leads to destructive interference between η q and η s for the former, but constructive interference for the latter.
Combined with the Clebsch-Gordan Coefficients, we can describe the relation Isospin conservation is assumed for the strong decays of an I = 0 resonance f 2 to ππ when we compute the branching fraction of the quasi-two-body process B → P f 2 , namely, According to the relation of the decay rates between the quasi-two-body and the corresponding two-body decay modes  . 3: (a) Differential branching ratios for the B ± → K ± f2(1270) → K ± π + π − decays. (b) Differential distribution of Acp in ω for the decay modes with B(f 2 → ππ) = (84.2 +2.9 −0.9 )% [91], we further obtain the PQCD predictions for B(B/B s → P f 2 ) from the values as listed in the second column of Table II: where the individual errors have been added in quadrature.
One can see that some channels have both large branching ratios and direct CP asymmetries, letting the corresponding measurement appear feasible. In fact, some physical observables (such as the CP averaged branching ratios and direct CP violations) of the two charged decay modes like B + → K + (f 2 →)π + π − and B + → π + (f 2 →)π + π − have been searched by BABAR and Belle Collaborations [2,4,8,9]. For example, BABAR Collaboration [2] reported a measurement, B(B + → π + (f 2 →)π + π − ) = (0.90 +0. 37 −0.24 ) × 10 −6 , which agrees with our calculations in both scenarios I and II. Furthermore, for the K mode, the measurements from BABAR [4] and Belle [8] are the following Their weighted average, enter the numbers given in Table II, are typically smaller than our prediction. None the less, taking the errors into consideration, the theoretical prediction and experimental data can still agree with each other. For the direct CP asymmetries, although the error bars from the data are still large, we are happy to see that all these measured entries have the same sign as our theoretical calculations [see Table III].
From Table II, one can see that the branching ratio of B + → K + (f 2 →)π + π − decay is a little larger than that of B 0 → K 0 (f 2 →)π + π − decay due to the extra contribution from the tree diagrams [see Eq. (A1) and (A2)] and the larger lifetime of the B + meson for the former decay mode. The similar situations also appear in the previous calculations from QCDF [75] and PQCD [77]. However, the data [91] shows that the latter decay mode has a relative large decay rate. It is worth of noting that this mode also has much larger relative errors because of limited statistics and the Dalitz-plot signal model dependence [5]. Such a difference should be clarified in the forthcoming experiments based on much larger data samples.
According to the full Dalitz-plot analysis to the B ± → π ± π ± π ∓ decay by the BABAR experiment [2], the dominant contributions come from the P -wave resonance ρ(770) and nonresonant contributions. The relative rate between the contributions from the P -wave resonance ρ(770) and D-wave resonance f 2 (1270) was measured to be For a more direct comparison with this available experimental data, we use our previous PQCD calculation of the P -wave resonance contribution B(B + → π + (ρ 0 (770) →)π + π − ) = (8.84 +1.91 −1.69 ) × 10 −6 , where all errors are combined in quadrature, as an input. Combined with the prediction on B(B + → π + (f 2 (1270) →)π + π − ) in Table II, we obtain the ratio which is consistent with above BABAR data quite well. For the similar ratio for the K counterpart, the calculated value is 3.64 +2.91 −0.93 , which is compatible with BABAR data R exp = 7.12 +2.30 −1.08 [4] within about two standard deviations and Belle data R exp = 5.19 +1. 35 −0.47 [8] within about one standard deviation. These results suggest that the PQCD factorization approach is suitable for describing the quasi-two-body B meson decays through analyzing various resonances by reconstructing ππ final states and reproducing the invariant mass spectra of Dalitz plots.
Different from the fixed kinematics of the two-body decays, the decay amplitudes of the quasi-two-body decays depends on the ππ invariant mass, which resulting in the differential distribution of branching ratios and direct CP asymmetries. In Fig. 3(a), we plot the differential branching ratios of the B ± → K ± f 2 → K ± π + π − decays. One can see that the differential branching ratios of the B ± → K ± f 2 → K ± π + π − decays exhibit peaks at the f 2 meson mass. Thus, the main portion of the branching ratios lies in the region around the pole mass of the f 2 (1270) resonance as expected. The central values of the branching ratio B are 6.03 × 10 −7 and 8.51 × 10 −7 when the integration over ω is limited in the range of ω = [m f2 − 0.5Γ f2 , m f2 + 0.5Γ f2 ] or ω = [m f2 − Γ f2 , m f2 + Γ f2 ] respectively, which amount to 54% and 77% of the total branching ratio B = 11.09 × 10 −7 as listed in Table II. In Fig. 3(b), we display the differential distributions of A CP for the four decay modes It implies that the direct CP asymmetries in the above three quasi-two-body decays, if calculated as the two-body decays with the f 2 resonance mass being fixed to m f2 , may be overestimated. The ascent of the differential distribution of A CP with ω for B + → π + [f 2 →]π + π − suggests that its direct CP asymmetry, if calculated in the two-body formalism, may be underestimated. In two-body B decays, the measured CP violation is just a number. But in three-body decays, one can measure the distribution of CP asymmetry in the Dalitz plot. Hence, the Dalitz-plot analysis of A CP distributions can reveal very rich information about CP violation. In the future, we will make more efforts to describe the distributions of CP asymmetries for various resonances in the Dalitz plot.

IV. CONCLUSION
In this work, we calculated the quasi-two-body decays B (s) → P f 2 (1270) → P ππ with P = (π, K, η, η ′ ) by utilizing the timelike form factor F π (s) within the PQCD factorization framework. The relativistic Breit-Wigner formula for the D-wave resonance f 2 (1270) was adopted to parametrize the timelike form factors F π , which contains the final state interactions between the pions in the resonant regions. Using the determined Gegenbauer moments of the D-wave two-pion distribution amplitudes, we have predicted the branching ratios and the direct CP asymmetries of the B (s) → P f 2 → P ππ channels, and compared their differential branching ratios with currently available data. General agreements between the PQCD predictions and the data could be achieved, although there is no enough data at present. We have taken two scenarios of constituents of f 2 (1270) into consideration and found that the interference between (uū + dd)/ √ 2 and ss part can result in remarkable effects on some decay modes. The branching ratios of the corresponding two-body decays have been extracted from the quasi-two-body decay modes. More precise data from the LHCb and the future Belle II will test our predictions.

Appendix A: Decay amplitudes
When the meson f 2 (1270) is treated as an pure 1 √ 2 (uū + dd) state, the total decay amplitude for each considered decay mode in this work are given as follows: On the other hand, the meson f 2 (1270) is more like an pure ss state, the total decay amplitude for each considered decay mode can be written as: 3 + C 10 ))(F LL ef2 + F LL af2 ) + ( C 5 3 + C 6 − 1 2 ( C 7 3 + C 8 ))(F SP ef2 + F SP af2 ) + (C 3 − where G F is the Fermi coupling constant. V ij 's are the Cabibbo-Kobayashi-Maskawa matrix elements. The functions (F LL ef2 , F LL af2 , M LL ef2 , M LL af2 , · · · ) appeared in above equations are the individual decay amplitudes corresponding to different currents, and their explicit expressions can be found in the Appendix of Ref. [61].