Study of the quasi-two-body decays B^{0}_{s} \rightarrow \psi(3770)\pi^+\pi^- with perturbative QCD approach

In this note, we study the contributions from the S-wave resonances, f_{0}(980) and f_{0}(1500), to the B^{0}_{s}\rightarrow \psi(3770)\pi^ {+}\pi^{-} decay by introducing the S-wave \pi\pi distribution amplitudes within the framework of the perturbative QCD approach. Both resonant and nonresonant contributions are contained in the scalar form factor in the S-wave distribution amplitude \Phi^S_{\pi\pi}. Since the vector charmonium meson \psi(3770) is a S-D wave mixed state, we calculated the branching ratios of S-wave and D-wave respectively, and the results indicate that f_{0}(980) is the main contribution of the considered decay, and the branching ratio of the \psi(2S) mode is in good agreement with the experimental data. Our calculations show that the branching ratio of B^{0}_{s}\rightarrow \psi(3770)\pi^ {+}\pi^{-} can be at the order of 10^{-5}, which can be tested by the running LHC-b experiments.


I. INTRODUCTION
In recent decades, the B mesons' three-body hadronic decays have drew lots of attention in both experimental and theoretical aspects, since it can test SM(standard model) and help us to have a better understanding of the scheme of QCD dynamics through researching in-depth. For the three-body decay mode, it's more complicate than the two-body case because it includes both resonant and nonresonant contributions and contains two virtual gluons exchange at lowest order, which will lead to the hard kernel involves 3-body matrix elements, and maybe introduce the possible final-state interactions [1][2][3], so just as stated in Ref. [4], because of the interference between the resonance and nonresonance, it is difficult to make a direct calculations of the involved contributions respectively.
In the pQCD approach based on the k T factorization theorem, the three-body decay can be simplified into two-body case by bringing in two-hadron distribution amplitudes [42,43], which contain the messages of both resonance and nonresonance. The dominant contributions come from the region, where the two light pair mesons moves parallelly with an invariant mass below O(ΛM B ), whereΛ = M B − m b is the difference between the B meson and b-quark. Therefore, one can express the typical pQCD factorization formula of the B meson's three-body decay amplitude as [34,35] where the hard decay kernel H can be calculated in the perturbative theory, describes the contribution from the region with only one gluon exchange diagram at leading order. The φ B , φ h1h2 and φ h3 , which can be extracted from experiment or calculated by several nonperturbative approach and can be regarded as nonperturbative input, is the distribution amplitude of B meson, h 1 h 2 pair and h 3 , respectively.
The B 0 s → ψ(2S)π + π − decay was first observed by the LHCb collaboration [6], the data was displayed based on an integrated luminosity of 1 f b −1 in the pp collisions at a centre-of-mass energy of √ s = 7 TeV. It is found that f 0 (980) is the main source of the decay rate by a way called sPlot technique. The B 0 s → ψ(3770)π + π − decay have not been observed yet, so it is desirable to make a theoretical prediction for the branching ratios of this decay mode. In this work, we will calculate the branching ratio of the quasi-two-body decay mode B 0 s → ψ(3770)π + π − , since the vector charmonium meson ψ(3770), the lowest-lying charmonium state just above the DD threshold, is mainly regarded as a S-D mixture, we will take the S-wave and D-wave contribution into account respectively. Here, ψ(2S) is the first radially excited charmonium meson, and the pure 1D state indicates the principle quantum number n = 1 and the orbital quantum number l = 2. The S-D mixing angle θ obtains from the ratio of the leptonic decay widths of ψ ′ and ψ ′′ 1 [44]. In Ref. [45], the authors make tentative calculations for different mixing solutions of the B meson exclusive decay B → ψ(3770)K with the QCD factorization, and they draw a conclusion that when taking account of higher-twist effects and adopting the S-D mixing angle θ = −(12 ± 2) • , the widely accepted value, the branching ratio of the decay B → ψ(3770)K can fit the experimental data well. Also, in Refs. [46][47][48], the authors provide two sets of mixing scheme within the nonrelativistic potential model: θ = −(12 ± 2) • or θ = (27 ± 2) • . Here, ψ(3770) may be almost described as [46][47][48][49] ψ(3770) = cos θ|cc(1D) − sin θ|cc(2S) . ( The content syllabus of this paper is as follows. After the introduction, we describe the theoretical framework and the wave function of the excited charmonium mesons ψ(2S) and ψ(1D) in section II. And in section III, we list the decay amplitude of the considered decay modes. The numerical results and analysis about the results we have got will be shown in section IV. Finally we will finish this paper with a brief summary.
FIG. 1. The lowest order Feynman diagrams for the B 0 s → ψ(f0 →)π + π − decays other component's kinematic variables of pion-pair can be expressed as In our calculations, the hadron B 0 s usually treated as a heavy-light system, the wave function of which can be found in Refs. [51][52][53] where the distribution amplitude(DA) φ Bs (x B , b B ) of B 0 s meson is written as mostly used form, which is the normalization factor N B can be calculated by the normalization relation is the color number. Here, we choose shape parameter ω Bs = 0.50 ± 0.05 GeV [54].
For the vector charmonium meson ψ(3770), as mentioned above, it's commonly regarded as a S-wave and D-wave mixing state. We adopt the wave function form of this vector charmonium meson with the basis of harmonic-oscillator potential, and which have been applied to the charmonium state successfully, such as J/ψ, ψ(2S), ψ(3S) and so on [49,[55][56][57], and the theoretical results agree well with the measured experimental data, which indicate the reasonability to adopt this form of the function. For the wave function of the pure 2S state, ψ(2S), and the pure 1D state, ψ(1D), whose longitudinal polarized component is defined as [56,57] where p 3 is the momentum of the charmonium mesons ψ(2S), ψ(1D) with the longitudinal polarization vector ǫ L = and M ψ is the corresponding mass. Here the ψ L and ψ t attribute to twist-2 and twist-3 distribution amplitudes(DAs). The explicit form is: [49,56] with for ψ(1D). The shape parameter ω 1D in the DAs of the ψ(1D), we choose ω 1D = 0.5 ± 0.05 GeV, for the reason we have discussed in Ref. [49], and ω 2S = 0.2 ± 0.1 GeV [56]. The N i (i = L, t) are the normalization constant, which satisfy the normalization conditions: and the decay constant of the orbitally excited ψ(2S) state and the angular excitation state ψ(1D) have been given in Table I. Both the wave functions Eq. (10) and Eq. (11) are symmetric under x ↔ x.
The differential branching ratios for the B 0 s → ψ(2S, 1D)π + π − decay in the B 0 s meson rest frame can be written as [58] dB with in the pion-pair center-of-mass system and the B 0 s meson lifetime τ B 0 s .

III. THE DECAY AMPLITUDES
In the pQCD factorization apporach, the B 0 s → ψ(2S)π + π − decay amplitude A express as in form of where the explicit form of with r c = mc MB s and the Wilson coefficients a 1 = C 1 + C 2 /N c , a 2 = C 3 + C 4 /N c + C 9 + C 10 /N c , a 3 = C 5 + C 6 /N c + C 7 + C 8 /N c , a 4 = C 4 + C 10 , and a 5 = C 6 + C 8 . C F = 4 3 is the group factor of the SU (3) c gauge group. The S B 0 s (t), S M (t), S ψ (t) used in the decay amplitudes, the hard functions h i (i = a, b, c, d), and the hard scales t i are collected in the Appendix.
As for the decay amplitude of the B 0 s → ψ(3770)π + π − decay, we give the expression based on the idea of S-D mixing scheme:

IV. NUMERICAL RESULTS AND DISCUSSIONS
In our numerical calculation, the input parameters are listed in Table I, where the mass of the involved mesons, the lifetime of meson and Wolfenstein parameters are got from 2018 PDG [58], while the decay constant of the ψ(1D) is calculated in the Ref. [59]. By using the differential branching ratio formula Eq. (14), first we make predictions of branching ratios of decay mode B 0 s → ψ(2S)π + π − for different intermediate state, which including f 0 (980) and f 0 (1500) two resonances, and the numerical results are listed as follows: where the three main errors come from the shape parameter ω Bs of the wave function of B 0 s meson, the hard scale t, which varies from 0.9t ∼ 1.1t (not changing 1/b i , i = 1, 2, 3), and the Gegenbauer moment a 2 = 0.2 ± 0.2 [10] in the ππ distribution  amplitude, respectively. The other errors from the uncertainty of the input parameters, for example, the decay constants of the B 0 s and charmonium mesons and the Wolfenstein parameters, are tiny and can be neglected safely. We can see that the input parameter ω Bs of the B 0 s meson is the primary source of the uncertainties, which take up approximately 6% ∼ 23%, and then the hard scale t, which characterizes the size of the next-leading-order contribution. The uncertainty from the Gegenbauer moment is relatively small. When we consider the total S-wave contributions of the f 0 (980) and f 0 (1500), we can get: which is agree with the new experiment data (7.1 ± 1.3) × 10 −5 in allowed errors [58]. From the numerical results, we can see that f 0 (980) is the principal contribution, which take the percentage of 91.2%, just as the experiment observed, and the f 0 (1500) is 2.4%, while the constructive interference between this two resonance can contribute nearly 6.4% to the total branching ratio.
In experiment, the calculated ratio of the branching fraction have been given in Ref. [6], which is ± 0.03(syst) ± 0.01(B).
By using the previous prediction about the branching ratio of the decay mode B 0 s → J/ψ(π + π − ) S [33], we obtain the ratio B(B 0 s → ψ(2S)π + π − )/B(B 0 s → J/ψπ + π − ) = 0.42 +0.10 −0.12 , which is consistent with the experiment measurement, and indicate that the harmonic-oscillator wave function for excited charmonium is applicable and reasonable. Besides the decay mode B 0 s → ψ(2S)π + π − , we make calculation for the part of 1D, and we also consider the similar contributions from the containing S-wave resonance state, f 0 (980) and f 0 (1500), the reason is that these two resonances mass is also within the scope of the ππ invariant mass spectra, which is 2m π < ω < M B 0 s − M ψ , after making integral over ω, the results are: Also the total S-wave contribution of the B 0 s → ψ(1D)π + π − decay is: In Fig. 2(a) and Fig. 3, we plot the differential branching ratio of the B 0 s → ψ(2S, 1D)π + π − decay as a function of the ππ invariant mass ω, in which we can clearly see that the peak arises from f 0 (980), while f 0 (1500) is unsharp that also make contribution for the decay. For contrast and comparison, at the same time, we present the experiment data from LHCb [6] in Fig. 2(b), which shows a basic agreement with our predictive results. Comparing the results between ψ(2S) and ψ(1D), it is easy to find that the results of ψ(1D) is more sensitive to the Gegenbauer moment a 2 = 0.2 ± 0.2, and this means that although the value is in good agreement with many decay modes, there is still a necessity to explore more accurate data to facilitate a FIG. 3. The S-wave differential branching ratio of the B 0 s → ψ(1D)π + π − decay better understanding of the non-perturbative hadron dynamics. In the ψ(2S) and ψ(1D) mode, since f 0 (1500) mass is near the maximum of ππ invariant mass, the corresponding contributions is very small compared to the total contributions of the S-wave. We can note that the branching ratio of the ψ(1D) is smaller than the ψ(2S), which should be attributed to the dependence of the corresponding wave function and the decay constant.
Furthermore, we calculate the branching fraction of the mode B 0 s → ψ(3770)π + π − based on S-D mixing scheme, whose two sets of mixing angle has been introduced in Section I, and we list the computational results in Table II. Comparing with the pure D-wave state, we can notice that the branching ratio of the S-D mixing state will be raised approximately two times when the mixing angle is −12 • , whose reason is mainly owing to the small decay constant of ψ(1D), which is compatible with what is summarized in Refs. [45,48,[59][60][61]. Considering the size of the data set in LHC-b, we can expect the measurement of this decay mode coming in the near future, that will help us to understand the structure of ψ(3770) and the three body decay mechanism. II. Branching fractions of the quasi-two-decay B 0 s → ψ(3770)(f0 →)π + π − in the pQCD approach based on two sets of S-D mixing angle.
In this work, we have calculated the contributions from the S-wave resonances, f 0 (980) and f 0 (1500), to the B 0 s → ψ(3770)π + π − decay by introducing the S-wave ππ distribution amplitudes within the framework of the perturbative QCD approach. Due to the character of 2S-1D mixing scheme of ψ(3770), we calculate the branching ratios of S-wave and D-wave respectively, and the results indicate that the f 0 (980) is the main contribution of the considered decay, and the differential result of the ψ(2S) mode is in good agreement with the experimental data. We also analyzed the theoretical uncertainties in this paper, and find that the result of ψ(1D) is sensitive to the Gegenbauer coefficient, which we need more accurate data to understand the non-perturbative hadron dynamics. In the end, by introducing the mixing angle θ = −12 • and θ = 27 • , we make further calculation of B 0 s → ψ(3770)π + π − , and our calculations show that the branching ratio can be at the order of 10 −5 based on the small mixing angle θ = −12 • , which will be tested by the running LHC-b experiments.
Fourier transform of virtual quark and gluon propagators and are written as follows: where J 0 is the Bessel function and K 0 , I 0 are modified Bessel function with H (1) 0 (x) = J 0 (x) + iY 0 (x). The threshold resummation factor S t (x) have been parameterized in [63], which is: with the parameter c = 0.04Q 2 − 0.51Q + 1.87 and Q 2 = M 2 B (1 − r 2 ) [64]. For killing the large logarithmic radiative corrections, the hard scale t i in the amplitudes are chosen as