Study of B 0 s → T T ( a 2 (1320) , K ∗ 2 (1430) , f 2 (1270) , f ′ 2 (1525)) in the perturbative QCD approach

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I. INTRODUCTION
The model elucidating the two-body decay of B mesons has been the subject of extensive scrutiny over the past two decades, both from a theoretical [1][2][3][4]and experimental [5][6][7] perspective.The emergence of charmless hadronic B decays featuring a light tensor meson in the final state has invigorated interest in tensor mesons.Based on the principles of flavor SU(3) symmetry, we investigate nine mesons [8,9], comprising isovector mesons a 2 (1320), isodoulet states K * 2 (1430), and two isosinglet mesons f 2 (1270), f ′ 2 (1525), which forms the first 1 3 P 2 nonet [10].According to the latest experimental data of 2022 PDG [8], the B 0 s → K * 2 (1430)K * 2 (1430) decay has already appeared, signifying the emergence of numerous decay modes involving the final state of two tensor mesons may appear.In Ref. [2], some researchers have calculated the decay channel about B 0 s → V T , at the same time, they also adopted some other methods to compare.Further, these calculations indicated that predictions based on the pQCD that can accommodate experimental data well.By comparing their predictions with the experimental data, we find that BR(B 0 s → Φ(K * 2 (1430), K * 2 (1430))) is similar to BR(B 0 s → Φ(K * (1430), K * (1430))), but the magnitude is smaller.For B 0 s → V V, T V , they are similar but a little different.
In the exploration of two-body decay of B mesons, the perturbative QCD factorization approach assumes paramount significance.This method based on factorization [11][12][13][14][15][16][17] to calculate the decay process of B 0 s → M 1 M 2 , in which M i are composed of light noncharmed mesons.In the perturbative QCD framework, the factorization scale about 1/b is employed to demarcate the boundary between the perturbative and nonperturbative regimes.The nonleptonic decay of the B meson is postulated to be primarily governed by the exchange of hard gluons, permitting the isolation and direct computation of the hard portion of the decay process through perturbative methodologies.Simultaneously, the nonperturbative component is absorbed into the universal hadron wave function.On this foundation, the two-body decay amplitude of B 0 s meson is generically expressed as Here the hard decay kernel H represents the contributions emanating from Feynman diagrams, amenable to computation through perturbative theory.The nonperturbative inputs φ B 0 s , φ h1 and φ h2 denote the wave functions of B 0 s meson, tensor mesons, respectively.These wave functions may be deduced via the extraction of pertinent empirical data or calculated through various nonperturbative methodologies.
Theoretical investigations into the calculations pertaining to tensor mesons have garnered the attention of several researchers.In comparison to vector mesons, tensor mesons are more special and complex.For B → P T , V T [1,18], some scholars have investigated the tensor meson in the final state.However, for B 0 s → T T , the case where the final states are all tensor mesons has not yet been studied in the literature, and this article is the first in this perspective.For the decays of B 0 s → T T , the amplitude can be defined as three invariant helicity components: A 0 , for which the polarizations of the tensor meson are longitudinal with the respect to their momenta, and A , A ⊥ are for transversely polarized tensor meson [19,20].
The branching ratios of ) are contingent upon the mixing angle θ of f 2 (1270) and f ′ 2 (1525), analogous to η and η ′ mixing in the pseudoscalar sector [4,[21][22][23].From the experiment that ππ is the dominant decay mode of f 2 (1270), and f ′ 2 (1525) decays dominantly into KK, we can know that the physical f 2 (1270)-f ′ 2 (1525) mixing angle is smaller than the decoupling value: θ T,ph − θ dec.= 29.5 [25].The corresponding helicity amplitudes are characterized as follows [26,27] where Moreover, it is also found that the mixing angle θ f2 = 7.8 • [25] and This paper is structured as follows: In Sec.II, we expound upon the theoretical underpinnings of the perturbative QCD (pQCD) framework and elaborate upon the wave functions integral to our calculation for the B 0 s → T T decays.Section III assembles the helicity amplitudes.Subsequently, in Sec.IV, we present the numerical results and engage in discussions.The key content of Sec.V comprises a summarization of the principal contributions of this study.Finally, the explicit formulations of all the helicity amplitudes are provided in the Appendix for reference.

A. Hamiltonian and kinematics
The pertinent weak effective Hamiltonian governing the decays B 0 s → T T is defined by the following expression [29,30] where V * ub V us and V * tb V ts are Cabibbo-Kobayashi-Maskawa factors, the Fermi coupling constant G F =1.66378 × 10 −5 GeV −2 , and C i is the Wilson coefficient corresponding to the quark operator, O i represents the local four-quark operators, which can be expressed as Within the framework of the pQCD approach, the decay amplitude can be meticulously into three constituent components: the hard scattering kernel, the wave functions characterizing the mesons, and the convolution of the Wilson coefficients.For B 0 s → M 1 M 2 decay, the decay amplitude is presented as follows [20,[31][32][33]] where x i are the proportions of the momenta for the spectator quark inside the mesons B 0 s , T 2 and T 3 , respectively, with the values ranging from 0 to 1, b i are the conjugate space coordinates of the transverse momenta k i for the light quarks.T r denotes the trace over all Dirac structure and color indices.C(t) is the short distance Wilson coefficients at the hard scale t. t denotes the largest energy scale of the hard part H.The threshold resummation S t (x i ) stems from the large double logarithms [34], which can remove the end point singularities.The last term e −S(t) is the Sudakov factor, which can suppress soft dynamics [35].
In the context of the light cone coordinate system, the associated physical quantities are represented as follows.Assuming that the initial state of the meson B 0 s is stationary, the tensor mesons T 2 and the T 3 move in the direction of the lightlike vector v = (0, 1, 0 ⊤ ) and n = (1, 0, 0 ⊤ ), respectively.Here we use p 1 , p 2 and p 3 to represent the momenta of the mesons B 0 s , T 2 and The wave function of the meson is expressed as a decomposition of Lorentz structures where φ B (x 1 , b 1 )and φ B (x 1 , b 1 ) are the twist distribution amplitudes, the contribution of φ B (x 1 , b 1 ) is relatively small, so we neglect it.Therefore, the meson B 0 s is deemed to be a heavy-light model, with the wave function defined as [36][37][38][39]] where N c = 3 is the number of colors, because we calculate the relevant parameters in the b space, and the distribution amplitude φ B can be expressed as [40,41] This distribution amplitude adheres to the normalization condition where N B = 91.784GeV is the normalization constant, f B is the decay constant.For B 0 s meson, we use the shape parameter ω Bs = 0.50 ± 0.05 GeV [4].

Tensor meson
For the spin-2 polarization tensor ǫ uv (λ) with helicity λ, satisfies ǫ uv p v 2 = 0 [42,43], which can be constructed based on the polarization vectors of vector mesons ǫ, they can be written as With the tensor meson moving in the plus direction of the z axis, the polarizations ǫ are defined as where E T represents the energy of the tensor meson.In the subsequent calculations, the introduction of a new polarization vector ǫ T for the tensor meson under consideration is deemed necessary for the sake of convenience [9] which satisfies The contraction is evaluated as ǫ(0) It is obvious that the new vector ǫ T is similar to the ordinary polarization vector ǫ, regardless of the dimensionless constants mT .The decay constants of the tensor mesons are defined as [9] Where the currents are expressed as with , respectively.Here we adopted these decay constants from Ref. [42] that have been calculated in the QCD sum rules [44][45][46], which can be seen from Table I.We can find that the transverse decay constants are approximately equal to the longitudinal one for a 2 (1320), f 2 (1270), but it is different from K * 2 (1430) and f ′ 2 (1525), their ratio relationship: f T T /f T ∼ (50% − 65%).
From earlier studies [42], we obtain insights into the light cone distribution amplitudes of the tensor mesons.The light cone distribution amplitudes up to twist-3 for generic tensor mesons are defined as follows The convention ǫ 0123 =1 has been adopted.Equation ( 18) is for the longitudinal polarization(λ = 0), and Eq. ( 19) is for the transverse polarizations(λ = ±1), respectively.Here n is the moving direction of the tensor meson and v is the opposite direction.The new vector ǫ • which plays the same role with the polarization vector ǫ, which is defined by With the momenta and polarizations, which can be reexpressed as In earlier studies [9,42,43], the amplitudes are expressed as The twist-2 distribution amplitudes can be expanded in terms of Gegenbauer polynomials , with the asymptotic form given by Adhering to normalization conditions The twist-3 distribution amplitudes also assume an asymptotic form, as delineated in [42] h

III. DECAY AMPLITUDES
In this section, we provide the perturbative QCD formulas for all the Feynman diagrams, as illustrated in Fig. 1.The first row showcases the annihilation-type diagrams, with the first two being factorizable and the last two being nonfactorizable.The second row consists of nonfactorizable emission diagrams.For the B 0 s → T T decays, both the longitudinal polarization and the transverse polarization contribute.The symbol F and M represent the factorizable and nonfactorizable contributions, respectively.The superscripts LL denotes the amplitude of the (V − A)(V − A) operators, and LR describe the amplitude of the (V − A)(V + A) operators.The symbol SP is Fierz transformation of LR.Notably, the decay amplitudes for longitudinal and transverse polarizations exhibit the same form after simplification, as follows in Eq. ( 26)- (49).
The longitudinal polarization amplitudes of the factorizable annihilation diagrams are FIG. 1.The Feynman diagrams for the B 0 s → T T decays The longitudinal polarization amplitudes of the nonfactorizable annihilation diagrams are given below The longitudinal polarization amplitudes of the nonfactorizable emission diagrams are as follows The transverse polarization amplitudes of the factorizable annihilation diagrams are The transverse polarization amplitudes of the nonfactorizable annihilation diagrams are For B 0 s → T T decays, the transverse polarization amplitudes of the nonfactorizable emission diagrams are as follows

IV. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we initiate our calculations by enumerating the input parameters.These encompass the decay constant f B , the Wolfenstein parameters, the masses of B meson and tensor mesons, and the corresponding lifetime, as detailed in Table II [47].Our numerical calculations within the pQCD framework are focused on branching ratios, direct CP violations, and polarization fractions, as summarized in Tables III-V.It is crucial to acknowledge that there exist uncertainties in our calculation results.In Table III, the errors stem from induced by the uncertainties in the the shape parameter ω B = (0.50 ± 0.05) GeV pertaining to the B 0 s meson distribution amplitude [4].The second source of uncertainty pertains to the B 0 s meson and the final state tensor mesons, as documented in Table I.The third error arises from Λ QCD = (0.25 ± 0.05) GeV, and varies 20% from hard scale t max = (1.0 ± 0.2)t detailed in the Appendix.Other uncertainties such as the Cabibbo-Kobayashi-Maskawa matrix elements V from the η and ρ, angles of the unitary triangle that can be neglected.

Mass of mesons
With the amplitudes calculated in Sec.III, the decay width is determined as The branching ratio is got through BR = Γ • τ B 0 s .In Ref. [48,49], we can learn about direct CP violations, A dir CP is defined by Here the two amplitudes are defined as follows where the B meson has a b quark and f is the CP conjugate state of f .The results of the polarization fractions f i , which are defined as where A i (i = 0, , ⊥) is the amplitude of the longitudinal or transverse polarization contributions.Based on the helicity amplitudes A i (i = 0, , ⊥) for longitudinal, parallel, and perpendicular polarizations, the three part amplitudes are given as   Table III displays the pertinent data.Several observations can be made: (1) For B 0 s → a 0 2 a 0 2 , a + 2 a − 2 , f 2 f 2 , they are only the pure annihilation diagrams, whose branching ratios are at the order of they have the annihilation and emission diagrams, whose branching ratios are at the order of 10 −6 .Under the SU (3) limit, since the Bose statistics are satisfied, the meson wave function will be antisymmetric when the momenta fractions of the quark and antiquark of tensor mesons are exchanged [42,43].Due to the commutative antisymmetry of the tensor meson wave function, the nonfactorizable emission diagrams will be more pronounced and provide a greater contribution [42,43].Take for example, if without nonfactorizable emission contributions, the branching ratio will decrease 90%, and its longitudinal polarization fraction will also reduce largely, which is different from the B 0 s → f 1 (1420)f 1 (1420) in Ref. [26].For B 0 s → f 1 (1420)f 1 (1420), the annihilation diagrams could contribute a large imaginary part and play an important role in calculating the branching ratios.The reason for the difference may be that, relative to the commutative antisymmetry of the tensor meson wave function, nonfactorizable emission contributions do not get offset but enhanced.
(3) For B 0 s → V T (a 2 , f 2 ), when a vector meson is emitted, the factorizable emission contribution of the penguin diagrams will offset the contribution of the tree annihilation diagrams, and the branching ratio turn to be very small.However, the contribution of the annihilation diagrams does not get offset due to the absence of factorizable emission diagrams in B 0 s → T T .Therefore, the branching ratio of B 0 s → T T (a 2 , f 2 ) is one or two orders of magnitude larger than that of B 0 s → V T (a 2 , f 2 ), making it more beneficial to experimental observation. ( 1525) mixing, just as the η − η ′ mixing.To see the variation clearly with the mixing angle, we show the branching ratios )) varying with θ ∈ [0, π] in Fig. 2. In Ref. [26], the authors plotted the related figures about the branching ratios of B 0 s → f 1 f 1 decays dependent on the free parameter θ.By comparing figures about , we can find that when the θ is large enough, and its influence on the branching ratio is more obvious, which can be seen from the line shapes.When θ reaches a certain angle, the branching ratios of ) and B 0 s → f 1 f 1 will vary an order of magnitude.But for f 2 (1270) − f ′ 2 (1525) mixing, in the contrast to f 1 (1285) − f 1 (1420), the mixing angle is very small.From the Refs.[24,25,50], we have known that the mixing θ is about 5.8 • ∼ 10 • , and the related branching ratios are close to that of 0 • .In addition, the branching ratio of the 1525) decay is larger than that of the B 0 s → f 2 (1270)f 2 (1270) decay by one order of magnitude, which is caused due to the reason that the former has more Feynman diagrams.B 0 s mesons can be produced in the Υ(5S) [51].Since of the kinematic smearing from excited B * s production and contamination from B +/0 decays, measurements are not as easy as at the Υ(4S).Although full B 0 s reconstruction would mitigate B +/0 background, such a technique will only be possible with Belle II [52].The number of B 0 s mesons in a dataset can be calculated as: The parameter f s is a key component to calculate the total B 0 s yield in the sample.L is the integrated luminosity of the data and σ is the cross section of the process e + e − → bb [51,53].By using the values from the Refs.[54,55], the number of B 0 s mesons is estimated to be ∼ 5.9 × 10 8 in the dataset of L = 5ab −1 taken at Υ(5S) in Belle II [51], which indicates that a 5ab −1 Υ(5S) sample contains approximately 300 million B 0 s B 0 s pairs.The branching ratios of our calculation for B 0 s → T T are at the order of 10 −6 and 10 −7 .Therefore, the decays of B 0 s → T T will hopefully be observed by the Belle II experiments in the near future.Moving on to Table IV, we delve into predictions pertaining to the polarization fraction of mesons.The pQCD approach has effectively elucidated the theoretical underpinnings of pure annihilation diagrams about B 0 s → π + π − and B 0 → D − S K + theoretically, and corresponding numerical results has been confirmed by experiments.Based on this success, it is plausible to assert that the pQCD approach holds substantial predictive power for processes primarily governed by annihilation diagrams [4,[56][57][58].In line with prior research [3,59] .it has been ascertained that the contributions to these processes are chiefly orchestrated by longitudinal polarization in the case of pure annihilation of two-body decays.The fractions pertaining to these decays have been observed to approach nearly 100%.This trend corroborates our predictions of Notably, the longitudinal polarizations of these three decays approximate 90%, underscoring the indispensability of accounting for transverse polarization, which can yield noteworthy contributions in pure annihilation decays.To illustrate this point, we take B 0 s → a 0 2 a 0 2 and B 0 s → f 2 f 2 as an example, from Table VI reveals that the contribution of longitudinal polarization surpasses that of transverse polarization, rendering the former the predominant factor.
Across all the decays calculated in this study, longitudinal polarization predominantly steers the main decay modes.Through a factorial power estimation, it is evident that longitudinal polarization will play a leading role in the B meson decay [60,61].For B 0 s → T T , in the longitudinal polarization part, the contribution of nonfactorizable emission diagrams is substantial, and the extent of cancellation with annihilation diagrams is notably feeble.Conversely, in the transverse polarization segment, nonfactorizable emission diagrams contribute minimally, and this contribution effectively counterbalances the annihilation contribution.Consequently, longitudinal polarization prevails, and the transverse polarization fraction is about 10%-30%.Direct CP asymmetries of B 0 s → T T decays are listed in Table V.The magnitude of the direct CP violation is proportional to the ratio of the penguin and tree contributions [50].For the B 0 s → V V and B 0 s → V T , when penguin contributions and tree contributions stay at the same level, the direct CP violation appears.Since the decays presented in this paper are dominated by the penguin contributions, so the direct CP violation is very small.However, we can find that the direct CP violations of some special channels in the transverse polarization are sizable, and the tree contributions become comparable, which brings relatively large direct CP violation.
For pure annihilation-type decay, the CP -violating asymmetry is small, which has been pointed out in previous predictions of two-body decays [3,59].From Table V, the CP -violating asymmetries about B 0 s → a 0 2 a 0 2 , f 2 f 2 and a + 2 a − 2 , which also suggest that the CP -violating asymmetry of pure annihilation decay is very small.For B 1525) (without considering the mixing angle), in the standard model, there is no contribution of the tree diagrams operator, so the direct CP violation is zero.

V. SUMMARY
In this study, we predicted the relevant parameters of decays B 0 s → T T in the pQCD approach, where the tensor mesons are a 2 (1320), f 2 (1270), K * 2 (1430), and f ′ 2 (1525).We calculate the branching ratios, the polarization fractions, and the direct CP violations of these decays.Our calculation results suggest that (1) the production of tensor mesons via either vector or tensor currents is prohibited, highlighting the significance of nonfactorizable emission and annihilation contributions.Notably, the nonfactorizable emission diagram exhibits an augmented contribution owing to the antisymmetry inherent in the tensor meson wave function.(2) For decays characterized exclusively by annihilation processes, the branching ratio is situated at an order of 10 Based on the mixing scheme, the helicity amplitudes of B 0 s → f n f n (f s f s ) and B 0 s → f n f s decays are given by [26] A h (B 0 s → f 2 (1270)f (A.9) In this part, we summarize the functions that appear in the previous sections.For the factorizable annihilation diagrams that the first two diagrams in the Fig. 1, whose hard scales t i can be written by where α and β are color indices, X ′ = u, d, s, c, or b quarks, and they are the active quarks at the scale m b .O 1 and O 2 are current-current operators, O i (i = 3, ..., 10) are penguin operators, in which O i (i = 7, ..., 10) are the electroweak penguin operators.The operators O 7γ and O 8g are not listed, because their contribution is neglected.

TABLE IV .
Presents the polarization fraction of the decay.The accompanying errors arise from considerations encompassing the shape parameter, decay constants, hard scale and QCD scale.

TABLE V .
The CP -violating asymmetries of the B 0 s → T1T2 decay, the errors come from the shape parameter, decay constants, hard scale and QCD scale.

TABLE VI .
Decay amplitudes (in unit of 10 −3 GeV 3 ) of the B 0 s → T1T2 modes with three polarizations in the pQCD approach, where only the central values are quoted for clarification.
−7.And for f