S-wave contributions to in the perturbative QCD framework

is induced by the / transitions, which can interfere if a CP-eigenstate is formed. The interference contribution is sensitive to the CKM angle . In this work, we study the S-wave contributions to the process in the perturbative QCD factorization. In the factorization framework, we adopt two-meson light-cone distribution amplitudes, whose normalization is parametrized by the S-wave time-like two-pion form factor with resonance contributions from , . We find that the branching ratio of is of the order of , and that significant interference exists in . Future measurement could not only provide useful constraints on the CKM angle , but would also be helpful for exploring the multi-body decay mechanism of heavy mesons.

In recent years, three-body hadronic B/ meson decays have attracted considerable attention of the experiments [1][2][3]. These processes provide new ways of studying the phenomenology of the Standard Model and of probing new physics effects. For instance, the LHCb collaboration has measured sizable direct CP asymmetries in the phase space of the three-body B decays [4,5]. In addition, these processes are also valuable for understanding the mechanism of multi-body heavy meson decays. On the theoretical side, the perturbative QCD (PQCD) framework, based on the factorization, has been applied to analyze the B/ semi-leptonic and twobody decays processes . The PQCD framework has also been used to study three-body decays [31][32][33][34][35][36][37][38][39][40][41]. Generally, the multi-scale decay amplitude may be written as a convolution, including the nonperturbative wave functions, hard kernel at the intermediate scale and short-distance Wilson coefficients. The factorization is greatly simplified if two of the final hadrons move collinearly. In (D 0 )π + π − b → cūs b → ucs γ this case, the three-body decays are reduced to quasi-twobody processes. Therefore, nonperturbative wave functions include two-meson light-cone distributions, which contain both resonant and nonresoant contributions. For instance, the measurement of by LHCb [5] indicates that the resonances , of the S-wave -pair are dominant, which is confirmed by the theoretical calculation in the framework of PQCD [42][43][44][45][46][47]. In this work, we focus on and include the contributions. More explicitly, a Breit-Wigner (BW) model is used for the resonances [48] , and the Flatté model is adopted for the resonance [49]. , with the CP eigenstate containing the interference amplitude from ( ), is sensitive to the angle of the CKM Unitarity Triangle, whose precise measurement is one of the primary objectives in flavour physics. PQCD framework. In Sec. 4, we give the numerical results, and a conclusion is presented in the last section.

Wave functions
Φ αβ α, β I αβ , γ µ αβ , (γ µ γ 5 ) αβ , γ 5 αβ , σ µν αβ B s In general, the wave function with Dirac indices can be decomposed into 16 independent components, . For the pseudoscalar meson, the light-cone matrix element is defined as where the light-cone vectors are and . The two independent parts of the meson light-cone distribution amplitude obey the following normalization conditions: where is the decay constant of meson. Since the contribution of is numerically small [28], we neglect it and keep only the part in the above equation. In the momentum space the light-cone matrix of meson can be expressed as follows: Usually, the hard part is independent of or/and , thus one can integrate one of them out from . With b as the conjugate space coordinate of , we can express where x is the momentum fraction of the light quark in meson. In this paper, we adopt the following expression for where is the normalization factor, which is determined by the above equation with b = 0. In our calculation, we adopt [9] and [10], from which we determine . The wave function of the charmed D meson, treated as the heavy-light system, is defined by the light-cone matrix element as follows [11]: which satisfies the normalization Here, is the decay constant, and the chiral D meson mass is taken as . For the numerical calculation, we adopt the parametrization [50], where the free shape parameter is [14], and , read as [10] and [14].
The S-wave two-pion distribution amplitude is then given as [46] where is the momentum fraction carried by the spectator positive quark, , and are twist-2 and twist-3 distribution amplitudes.
is the invariant mass of the pion pair. We consider that the two-pion system moves in the n direction. as the momentum fraction of in the pion pair. The asymptotic forms are parametrized as [51][52][53] Here, and are the timelike scalar form factor and the Gegenbauer coefficient, respectively. As a first approximation, the S-wave resonances used to parametrize include both the resonant and nonresonant parts of the S-wave two-pion distribution amplitude. Therefore, we take into account and in the density operator, and in the density operator: .
Chinese Physics C Vol. 43 , and , , are tunable parameters. is the pole mass of the resonance, and is the energy dependent width of the S-wave resonance which decays into two pions. For the contribution of , an anomalous structure was found around 980 MeV in the scattering [54,55]. This was accompanied by the observation of a narrow anomaly (less than 100 MeV wide) in the S-wave phase shift associated with an enhancement in the system at threshold. It was shown that the anomaly could be understood as a narrow two-channel resonance, which combines the and channels [56]. Generally, the Breit-Wigner (BW) model can be applied to describe an unstable particle as an isolated resonance. Since the resonance is near the threshold of of about 992 MeV, the model should be modified to include the coupled channels and [56]. Therefore, the Breit-Wigner form proposed by Flatté and adopted widely in many studies of the and system is also used in this work. In the Flatté model, the phase space factors and are given as [48] 3 Perturbative calculations According to the factorization theorem, the amplitude of a process can be calculated as an expansion in and , where Q denotes a large momentum transfer, and is a small hadronic scale. Usually, the factorization formula for the nonleptonic b-meson decays can be expressed as where the Wilson coefficients and the typical scale t. The hard kernel , representing b-quark decay subamplitude, and the nonperturbative meson wave function , describe the evolution from scale t to the lower hadronic scale . For a review of this approach, see Ref. [7].B 0 s → D 0 The effective Hamiltonian for is given as , for the process, and , for the process . In particular, the penguin operators do not contribute to the processes. Using the above effective Hamiltonian, we obtain the typical Feynman diagrams for the process shown in Fig. 1, in which the first row represents the color-suppressed emission process, and the second row indicates the W-exchange process. In the factorization framework, the factorizable diagrams in Fig. 1  (a,b,e,f) are relevant for , and the non-factorizable diagrams in Fig. 1 (c,d,g,h) are proportional to [57], where We will work in the light-cone coordinates. The momenta of the mesons are defined as follows: Accordingly, the momentum transfer and the light-cone components can be obtained as , , and . In the heavy quark limit, the mass difference between b-quark (c-quark) and meson is negligible, ( is of the order of the QCD scale). Since , we expand the amplitudes in terms of , , and for high order . For the leading order of the expansion, . The momenta of the light quarks in the mesons ( represent the momenta of the light quarks in and D mesons, is the momentum of the positive quark in the pion-pair system) are given as In the -factorization, the color-suppressed emission Feynman diagrams can be calculated out, with the formulas labeled as (x = 1,2,3,4) in the subscript. Thus, the factorization formulas for the color-suppressed -emission diagrams are given as , is the color factor. represents the two-pion distribution amplitude defined by the operator. The hard kernels and are given in the following.
The factorization formulas for the W-exchange diagrams and are given as where , represents the distribution amplitude of the operator. Due to the helicity suppression, the contribution of the factorizable diagrams is suppressed significantly. Therefore, the dominant contribution comes from the non-factorizable diagrams In the -emission process, the two factorizable diagrams have the same factorization . Accordingly, we give the factorization formulas for the non-factorizable emission diagrams , the factorizable W-exchange diagrams and the non-factorizable W-ex- β e x β w x In the following, we give the forms for the offshellness of the intermediate gluon / and quarks The hard kernel functions ( ) and ( ) are written as i, k = 1, 2 j, l = 3, 4 where and , and , and are the Bessel functions. The threshold re-summation factor is parametrized as with the parameter in this work. The evolution factors in the factorization formulas are given by with the quark anomalous dimension . The explicit expression for can be found, for example, in Appendix A of Ref. [9]. The hard scales are chosen as Therefore, we obtain the total decay amplitudes, The differential branching ratio for the decays follows the formula given in [58,59] dB dm with the meson mean lifetime . The kinematic variables and denote the magnitudes of the and D momenta in the center-of-mass frame of the pion pair,

Numerical results
We adopt the following inputs (in units of GeV) [58,59]  , and the CKM matrix elements are taken as: The parameters of the scalar form factor are extracted from the LHCb data for the process , given in [48,60] (mass and widths are given in units of GeV): We calculate the branching ratios for the different resonances in the S-wave pion-pair wave function, which are given in Table 1. In this table, the first uncertainties are  from in the wave function, the second arise from in the pion-pair wave function, and the third are from the QCD scale . The errors from the parameter in the D meson wave function, the variations of the CKM matrix elements and the mean lifetime of are small and have been omitted. However, the above results are sensitive to and , namely the and S-wave two-pion wave functions. Future measurements of decay branching ratios will be valuable for understanding physics and the S-wave two-pion resonances.
Including all S-wave resonances , , and in the scalar form factor, we obtain the total branching ratio We found the contributions of , , and to be respectively 16.4%, 59.3%, 14.6% and 4.5% of the total decay rate. For the process, the corresponding rates are respectively 24.6%, 35.2%, 8.3% and 2.4% . This indicates that the and contributions are dominant, and that the contribution from is larger than in the ( ) final state. LHCb collaboration measured the upper limit of the branching ratio of [61], which roughly agrees with our value.B In order to compare the two channels and , we determine the rate of their branching ratios which significantly deviates from the ratio of the CKM factors: In these two decays, there are competition effects from the CKM factors and dynamical decay amplitudes. In these processes, the dominant contributions come from the emission diagrams and non-factorizable W-exchange diagrams. Although the emission diagrams result in similar factorization formulas and numerical results for the two channels, the formulas for the non-factorizable W-exchange diagrams are different. We found that the non-factorizable W-exchange process for is numerically larger than for , with the CKM factor inversed. As a result, their final branching ratios are sim- The CKM element for is ( ), where is sensitive to . Therefore, we can get the dependence of our results on by providing a parameter defined as [62] √ Accordingly, the dependence of the branching ratio on is shown in Fig. 2(a,b). The corresponding physical observable measured by the experiments is defined as . The dependence of on is shown in Fig. 2(c,d). The current bound for is [63]. The predicted dependence of the differential branching ratio on the pion-pair invariant mass is presented in Fig. 3(a) and Fig. 3(b) for the resonances , , and in the decays and . The figures show that the main contribution to the two decays lies in the region around the pole mass , while gives a contribution primarily in the region below . The other resonances, and , still give considerable contributions to the processes. Therefore, we hope that more precise data from LHCb and the future KEKB may test our theoretical calculations. In the past decades, two-body B decays have provided an ideal platform for extracting the Standard Model parameters, and for probing new physics beyond SM [64,65]. In this work, we studied the three-body decays within the PQCD framework, and in particular the S-wave contribution which was explicitly calculated. The S-wave two-pion light-cone distribution amplitudes can have both resonant , and nonresonant contributions. Furthermore, the processes proceed via tree level operators, and the branching ratios were found to be in the range from 10 −7 to 10 −6 . It was found that the branching ratios are sensitive to the parameters and in the and twopion distribution amplitudes. Therefore, we expect that future measurement could help to better understand the multi-body processes and the S-wave two-pion resonance and distribution amplitudes.