Perturbative QCD Analysis of Exclusive Processes $e^+e^-\rightarrow VP$ and $e^+e^-\rightarrow TP$

We study the $e^+e^-\to VP$ and $e^+e^-\to TP$ processes in the perturbative QCD approach based on $k_T$ factorization, where the $P,V$ and $T$ denotes a light pseudo-scalar, vector and tensor meson, respectively. We point out in the case of $e^+e^-\to TP$ transition due to charge conjugation invariance, only three channels are allowed: $e^+e^-\to a_2^{\pm} \pi^\mp$, $e^+e^-\to K_2^{*\pm} K^\mp$ and the V-spin suppressed $e^+e^-\to K_2^{*0} \bar K^0+\overline K_2^{*0} K^0 $. Cross sections of $e^+e^-\to VP$ and $e^+e^-\to TP$ at $\sqrt{s}=3.67$ GeV and $\sqrt{s}=10.58$ GeV are calculated and the invariant mass dependence is found to favor the $1/s^4$ power law. Most of our theoretical results are consistent with the available experimental data and other predictions can be tested at the ongoing BESIII and forthcoming Belle-II experiments.

tensor meson, respectively, have the topology with annihilation diagrams in B decays, and thus they can provide an ideal laboratory to isolate power correction effects.
It is anticipated that hard exclusive processes with hadrons involve both perturbative and nonperturbative strong interactions. Factorization, if it exists, allows one to handle the perturbative and non-perturbative contributions separately. The short-distance hard kernels can be calculated perturbatively. With the nonperturbative inputs determined from other sources, hard exclusive processes provide an effective way to explore the factorization scheme. The factorization theorem ensures that a physical amplitude can be expressed as a convolution of hard scattering kernels and hadron distribution amplitudes. However if one directly applies the collinear factorization to the e + e − → V P, T P , the amplitude diverges in the end point region x → 0. Here x is the momentum fraction of the involved quark.
Then the physical amplitude is written as a convolution of the universal non-perturbative hadronic wave functions and hard kernels in both longitudinal and transverse directions. Double logarithms, arising from the overlap of the soft and collinear divergence, can be resumed into Sudakov factor, while single logarithms from ultraviolet divergences can be handled by renormalization group equation (RGE). With Sudakov factor taken into account, the applicability of perturbative QCD can be brought down to a few GeV. In this work, we will study the e + e − → V P and e + e − → T P in the perturbative QCD (PQCD) approach [1][2][3][4][5][6] based on k T factorization.
The rest of this paper is organized as follows. In section II, we first collect the input parameters including decay constants and light-cone wave functions. Then we present the PQCD framework and give factorization formulas for the time-like form factors. Numerical results and detailed discussions are presented in section III. The last section contains the conclusion.

A. Notations
We consider the e + e − → V (T )P , in which V (T ) is a vector (tensor) meson with momentum P 1 and polarization vector ǫ µ (polarization tensor ǫ µν ), and P denotes a pseudoscalar meson with momentum P 2 in the center of mass frame. The collision energy is denoted as Q = √ s. In the standard model, such processes proceed through a virtual photon or a Z 0 boson. At low energy with √ s ∼ a few GeV, the amplitude is dominated by a photon. In this case the hadron amplitude is parameterized in terms of a form factor: Notice that in Eq.(1) the vector meson is transversely polarized. We have adopted the convention ǫ 0123 = 1 for the Levi-Civita tensor.
For a tensor meson, its polarization tensor ǫ µν can be constructed via the polarization vector Using the Clebsch-Gordan coefficients [26], one has In the calculation it is convenient to introduce a new polarization vector ξ: where q = P 1 + P 2 is the four momentum of the virtual photon and q 2 = s. Then Eq.(3) becomes where η = 1−m 2 T /Q 2 , with m T as the mass of the tensor meson. Here the mass of the pseudoscalar meson has been neglected. The new vector ξ plays a similar role with the ordinary polarization vector ǫ, regardless of some dimensionless constants.
Then like Eq. (1), one can define the T P form factor as in which the final state tensor meson is also transversely polarized.
Using the form factors in Eqs. (1,6), one can derive the cross sections for e + e − → V P, T P σ(e + e − → T P ) = πα 2 with the fine structure constant α em = 1/137, and

B. Decay constants and Light cone wave functions
Decay constants for a pseudoscalar meson and a vector meson are defined by: Tensor meson decay constants are defined as [27] T (P, λ)|j The interpolating currents are chosen as with the covariant derivative . The pseudoscalar and vector decay constants can be determined from various reactions, π − → e −ν , τ − → (π − , K − ρ − , K * − )ν τ and V 0 → e + e − [26]. For tensor mesons, their decay constants can be calculated in QCD sum rules [28,29] and we quote the recently updated results from Ref. [27].
Results for decay constants are collected in Table I.
We use the following form for leading twist LCDAs derived from the conformal symmetry: where N C = 3 and t = 2x − 1. C

3/2
i (i = 1, 2) are Gegenbauer polynomials, with the definition The Gegenbauer moments at µ = 1GeV are used as [31,32]: In this paper, we will study the collision at √ s = 3.67GeV and 10.58GeV, and then it is plausible to adopt the asymptotic forms for twist-3 DAs for simplicity: As for the η − η ′ mixing, we use the quark flavor basis with the mixing scheme [33,34]: The mixing angle is φ = 39.3 • ± 1.0 • [33,34] and Their decay constants are defined as: In the following calculation, we will assume the same wave functions for the nn and ss as the pion's wave function, except for the different decay constants [33,34] and the chiral scale parameters [35]: Similar with pseudoscalar mesons, the two-particle LCDAs for transversely polarized vector mesons up to twist-3 are parameterized as [36,37]: 6 The twist-2 LCDA can be expanded as: with Gegenbauer moments at µ = 1GeV [38,39]: As for the twist-3 LCDAs, we will also use the asymptotic forms: For a generic tensor meson, the LCDAs up to twist-3 can be defined as [20]: These LCDAs are related to the ones given in [27]: The asymptotic forms will be used in the calculation: In the above, we have only discussed the longitudinal momentum distributions. It is reasonable that the transverse momentum also plays an important role. Thus we will include the transverse momentum dependent parton distributions (TMDs) of the final-state light mesons. Following Ref.
[7], we assume no interference between the longitudinal and transverse distributions, and thus one can use the following Gaussian forms to factorize the wave functions [40,41]: In the above equation φ(x) is the longitudinal momentum distribution amplitude, and the exponential factor describes the transverse momentum distribution. The parameters β and a characterize the shape of the transverse momentum distributions. The parameter β is expected at the order of Λ QCD and related with the root of the averaged transverse momentum square k 2 T 1/2 . If we According to Ref. [41], the size parameter a follows where f M is the decay constant of the related hadron.

C. PQCD Calculation
In the PQCD scheme, a form factor can be written as the convolution of a hard scattering kernel with universal hadron wave functions. In small-x region, the parton transverse momentum k T is at the same order with the longitudinal momentum. Once k T is introduced in the hard kernel, a transverse momentum dependent (TMD) wave function is requested. Then the form factor is factorized as: and are both the hadron wave functions, relying on k T and b respectively.
Double logarithms arising from the overlap of soft and collinear divergences, can be resumed into Sudakov factor [42,43]: The Sudakov factor s(ξ, b i , Q), ξ = x i or 1 − x i , is given as [44,45]: where the notations have been used: The running coupling constant is given as and the coefficients A (i) and β i are Here n f is the number of the quark flavors and γ E is the Euler constant.
Apart from the double logarithms, single logarithms from ultraviolet divergence emerge in the radiative corrections to both the hadronic wave functions and hard kernels. These are summed by the renormalization group (RG) method: Here the quark anomalous dimension is γ q = −α s /π. In terms of the above equations, we can get the RG evolution of the hadronic wave functions and hard scattering amplitude as where t is the largest energy scale in the hard scattering. Then from equations (35) and (42), the large-b behavior of P can be summarized as with Furthermore, QCD loop corrections for the electromagnetic vertex can induce another type of double logarithms α s ln 2 x i . They are usually factorized from the hard amplitude and resummed into the jet function S t (x i ) to further suppress the end-point contribution. It should be pointed out that Sudakov factor from threshold resummation is universal and independent on the flavors of internal quarks, twist and topologies of hard scattering amplitudes and the specific process [46][47][48][49][50]. The following approximate parametrization is proposed in [51] for the convenience of phenomenological applications in which the c is a parameter depending on Q. Ref. [52] proposed a parabolic parametrization of the Q 2 dependence: The threshold resummation modifies the end point behavior of the hadron wave functions, rendering them vanish faster in this region. Taking into account all the above ingredients, one can obtain the analytic results of the first four diagrams in Fig. 1 in k T factorization: where E(t i ) and h are given as where J 0 and H 0 are both Bessel functions. We takex = 1 − x for short and define r i = m i /Q, with the index i = 1, 2 for the cases of final state meson is vector(tensor) or pseudoscalar meson.
The factorization scale t is chosen as the largest mass scale involved in the hard scattering: If the final state meson is not a strange meson, the distribution amplitudes are completely symmetric or antisymmetric under the interchange of the quark and antiquark's momentum fraction x and 1 − x. Then one can obtain The contributions from a photon radiated from the interaction point into a vector meson, shown as the last two panels in Fig. 1, might be sizable. Although these diagrams are suppressed by α em , they are enhanced by the almost on-shell photon propagator (1/m 2 V ) compared with the gluon propagator in the first four diagrams (∼ 1/s) [53][54][55][56]. These two amplitudes can be calculated in collinear factorization due to the absence of endpoint singularities in these two diagrams. In particular, they are equal after integrating out the momentum fractions: Finally, the form factors for the explicit channels of e + e − → V P process are combinations of the six amplitudes F a−f : The form factors for e + e − → V (T )η (′ ) are mixtures of the η q and η s components: where V = ρ 0 , ω, φ and Similarly, based on Eq.(56), form factors of the e + e − → T P channels can be written as: The abbreviations a 2 , K * 2 correspond to the tensor meson a 2 (1320) and K * 2 (1430), respectively.

III. NUMERICAL RESULTS AND DISCUSSIONS
Using Eqs.(48)- (51), and other input parameters, we can calculate cross sections for the processes e + e − → V P and e + e − → T P . In Tab. II, we have collected the results for cross sections at √ s = 3.67GeV, together with the experimental data from CLEO-c collaboration [58,59] (see Ref. [57] for BES measurements), and the results at √ s = 10.58GeV, together with the data measured by Belle [60] and Babar [61] collaborations. As we have discussed before, three different types of transverse momentum distribution functions were used, denoted as S1, S2 and S3 respectively.    Channel ρ ± π ∓ 6.80 ± 1. 1.15 ± 0.10 0.94 ± 0.08 1.02 ± 0.08 0.18 +0.14 −0.12 ± 0.02 A few remarks are in order.
• Results at different center of mass energy √ s can be used to study the 1/s n dependence of cross sections. From our results at √ s = 3.67GeV and 10.58GeV, the averaged value is about n = 4.1 for e + e − → V P and n = 3.9 for e + e − → T P 1 . This favors the 1/s 4 scaling, which is consistent with the constituent scaling rule [62,63]. The fitted result from experimental data is n = 3.83 ± 0.07 and 3.75 ± 0.12 for e + e − → K * (892) 0K 0 and ωπ 0 , respectively [60].
• From Table II, we can see that, cross sections for many processes are large enough to be measured, such as the e + e − → ρπ, ρη, ωπ and a ± 2 π ∓ at √ s = 10.58GeV, and e + e − → a ± 2 π ∓ , K * ± 2 K ∓ at √ s = 3.67GeV. We suggest the experimentalists to measure these channels especially at BESIII [64] and Belle-II in future.
• For the channels e + e − → K * ± K ∓ , there are very poor measurements from CLEO collaboration [59], since the charged K * meson is reconstructed by three-body decays: K * ± → K 0 π ± → 3π, with large systematic uncertainties. Our results are larger than the central of experimental data. We hope the future experimental measurements can clarify this difference more clearly.
• In the SU(3) limit, we expect σ(ωπ 0 )/σ(K * 0K 0 +K * 0 K 0 ) = 9/8 > 1, however our calculation has indicated that the cross section σ(ωπ 0 ) is smaller than that for e + e − → K * 0K 0 + K * 0 K 0 by a factor of 2 to 3. One reason arises from the fact that the decay constants f π f ω is about 30% smaller than f K f K * . The chiral scale parameter m K 0 will further enhance the cross sections.
• On the experiment side, the ratios R V P and R T P are introduced to explore the SU(3) symmetry breaking effect in the e + e − → K * K and e + e − → K * 2 K processes, with the definition In the PQCD framework, this ratio can be written as In SU(3) symmetry limit, the wave functions of K, K * and K * 2 is symmetric or antisymmetric under the exchange of the momentum fractions of quark and antiquark, and thus the relations in Eq.(55) are obtained. Then one can drive R V P = 4. One source of the SU(3) symmetry breaking is that the s quark is heavier than q(= u, d) quark and carries more momentum in the final state meson, therefore the gluon which generatesss is harder than theqq one. In this case, the coupling constant in thess process is smaller. Consequently, the amplitude |F a + F b | will be smaller than |F c + F d |, and thus R V P is expected larger than 4.
From Table II, one can obtain theoretical results for R V P : • At √ s = 3.67 GeV, the CLEO-c collaboration [59] has measured the ratio: Note that in the region near 10.58GeV, Belle result is significantly larger than our expectation, which might come from the Υ(4S) resonance contribution. Off the Υ(4S) resonance, the experimental results are consistent with our theoretical calculations.
• Due to the charge conjugation invariance, we have the relations for the e + e − → T P transition amplitude given in Eq. (56). Thus only three channels are allowed: e + e − → a ± 2 π ∓ , e + e − → K * ± 2 K ∓ and e + e − → K * 0 2K 0 + K * 0 If one further assume V-spin symmetry, the process e + e − → K * 0 2K 0 + K * 0 2 K 0 is highly suppressed since F a + F b ∼ −(F c + F d ). From Table II, one can obtain theoretical results for R T P : This is consistent with the Belle data [60]: R Exp T P < 1.1, 0.4, 0.6. (83) • The theoretical uncertainties in our calculation are mainly from the uncertainties of the meson wave functions. The longitudinal distribution amplitudes in exclusive B decays will give about 10% − 20% uncertainties [41]. When the transverse momentum distribution functions are introduced in Eqs. (32) and (33), the contribution from the large-b region will be suppressed. This suppression makes the PQCD approach more self-consistent. Comparing the different results in Table II, one can observe severe suppressions especially at √ s = 3.67GeV: the suppression is about 50% for S2 and about 40% for S3. Since the results depend on the explicit form of transverse momentum distribution, more accurate transverse momentum dependent wave functions and more experimental results would be valuable.
• In this calculation, we have limited ourselves to the leading-order accuracy. The next-toleading order (NLO) calculation is complicated [65][66][67] that will be presented in a future publication. As an estimation of the size of the NLO contribution, we vary Λ QCD and the factorization scale t in Eq.(54): Λ QCD = (0.25 ± 0.05)GeV, and changing the hard scale t from 0.75t to 1.25t (without changing 1/b i ). We find that our results are not sensitive to these variations. It implies that the NLO contributions are presumably not very large.

IV. CONCLUSION
Hard exclusive processes e + e − → V P and e + e − → T P at center of mass energy √ s = 3.67GeV and 10.58GeV are investigated in the perturbative QCD framework in this work. For the wave functions of the light mesons involved in the factorization amplitudes, we have employed various models of transverse momentum dependence of wave functions. At the center of mass energy √ s = 3.67GeV, two different transverse momentum distribution functions can give about 50% and 40% suppressions, respectively. The value R V P and R T P obtained from our results are consistent with the experimental data. We found that our theoretical results favor the 1/s 4 scaling law for the cross sections. Most of our results are consistent with the experimental data and the others can be tested at the ongoing BESIII and forthcoming Belle-ILL experiments.