Abstract. We perform a perturbative QCD analysis of the quark transverse momentum

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1 BIHEP-TH The erturbative ion-hoton transition form factors with transverse momentum corrections Fu-Guang Cao, Tao Huang, and Bo-Qiang Ma CCAST (World Laboratory), P.O.Box 8730, Beijing , China and Institute of High Energy Physics, Academia Sinica, P.O.Box 918(4), Beijing , China 1 (Received 31 January 1996) Abstract We erform a erturbative QCD analysis of the quark transverse momentum eect on the ion-hoton transition form factors F and F in the standard light-cone formalism, with two henomenological models of wavefunction as the inut of the non-erturbative asect of the ion. We oint out that the transverse momentum deendence in both the numerator and the denominator of the hard scattering amlitude is of the same imortance and should be considered consistently. Itisshown that after taking into account the quark transverse momentum corrections, the results obtained from dierent model wavefunctions are consistent with the available exerimental data at nite Q 2. PACS number(s): Bx, Ki, G, Aq 1 Mailing address. address: caofg@bec3.ihe.ac.cn. 1

2 I. Introduction The ion-hoton transition form factor F (Q 2 ) is a simle examle for the erturbative analysis to exclusive rocesses and was rst analysed by Leage and Brodsky [1]. They redicted F (Q 2 )by neglecting k? relative toq?, F (Q 2 )= 2 Z [dx] (x) 3Q 2 x 1 x 2! 1+O s ; m2 ; (1) Q 2 and Q 2 F (Q 2 )would be essentially constant asq 2!1. This aroximation would be valid if the wavefunction is eaked at low k? ( k? is the transverse momentum of quark) so that x 1 x 2 Q 2 in the hard scattering amlitude dominates the denominator. However, at the end-oint region x i! 0; 1 and Q 2 a few GeV 2 the wavefunction does not guarantee the k? negligible. One should takeinto account k? corrections from both the hard scattering amlitude and the wavefunction. Recently, Refs. [2, 3] calculated the - transition form factor within the covariant hard scattering aroach including transverse momentum eects and Sudakov corrections [4] by neglecting the quark masses, the mass of the ion meson and the k? - deendence in the numerator of T H. Their results show that Sudakov suression in the form factor F (Q 2 ) is less imortant than in other exclusive channels and the Chernyak-Zhitnitsky(CZ) wavefunction should be discarded by tting the exerimental data. However, as we know, the k? -deendence of the wavefunction in Ref. [2] is the same in the dierent models and it may be dicult to draw a conclusion which excludes the CZ wavefunction. We will re-examine this roblem in the resent aer. The light-cone formalism rovides a convenient framework for the relativistic descrition of hadrons in terms of quark and gluon degrees of freedom, and for the alication of erturbative QCD (QCD) to exclusive rocesses [5, 6]. In this formalism, the hadronic wavefunction which describes the hadronic comosite state at a articular 2

3 is exressed in terms of a series of light-cone wavefunctions in Fock-state basis, ji = X jqqi q q + X jqqgi qqg + ; (2) and the temoral evolution of the state is generated by the light-cone Hamiltonian H LC = P = P 0 P 3. Furthermore the vacuum state in the light-cone Fock basis is an exact eigen-state of the full Hamiltonian H LC. Thus all bare quanta in a hadronic Fock state are art of the hadron (This oint isvery dierent from that in the equal-t erturbative theory in which the quantization is erformed at a given time t). Light-cone QCD is very convenient for light-cone dominated rocesses. For the detail quantization rules we refer to literatures [1, 5, 6, 7]. The more imortant oint for ractical calculation is that the contributions coming from higher Fock states are suressed by 1=Q n, therefore we can emloy only the valence state to the leading order for large Q 2. In this aer, we analyze the quark transverse momentum eects on the ion-hoton transition form factors F and F at nite Q 2 in the standard light-cone formalism, with two henomenological models of wavefunction as the inut of the non-erturbative asect of the ion. We demonstrate that the QCD redictions with the dierent models of wavefunction are consistent with the available exerimental data by taking into account the quark transverse momentum. II. The ion-hoton transition form factors F and F The - transition form factor F is dened from the 0 vertex in the amlitude of e! e, = ie 2 F q ; (3) where and q are the momenta of the incident ion and the virtual hoton resectively, and is the olarization vector of the nal (on-shell) hoton. We adot the 3

4 standard momentum assignment at the \innite-momentum frame [1] = ( + ; ;? )=(1;0;0? ); q = (0;q 2? ;q?); (4) where + is arbitrary. For simlicity wechoose + = 1, and we haveq 2 = q 2?= Q 2. Then the F is given by F (Q 2 )= + ie(? q? ) ; (5) where =(0;0;? ),? q? = 0 is chosen and? q? =?1 q?2 +?2 q?1. Since the contributions coming from higher Fock states are suressed, we takeinto account only the conventional lowest Fock state of ion meson, = a b 1 u u 2 u u d d d d nc 2 2 (x i ;k? ) x1 x 2 : (6) The leading-order contribution to F is calculated from Fig. 1 in light-cone QCD [1], nc (e 2 F (Q 2 u e 2 )= d) i(? q? ) v (x 2 ; k? ) Z 1 0 [dx] Z 1 0 d 2 k? 16 3 (x i ;k? ) x2 / u (x 1 ;k? +q? ) x1 u (x 1 ;k? +q? ) x1 +u (x 1 ;k? ) x1 1 D +(1$2) ; (7) where [dx]=dx 1 dx 2 (1 x 1 x 2 );e u;d are the quark charges in unites of e, and D is the \energy-denominator, D = q 2? (k? + q? ) 2 + m 2 x 1 2 k? + m 2 : (8) x 2 The quark masses relative to Q 2 can be neglected since they are the current quark masses in QCD calculation. Thus Eq. (7) becomes F (Q 2 )=2 n c (e 2 u e 2 d) Z 1 0 [dx] Z d 2 k? 16 3 (x i ;k? )T H (x 1 ;x 2 ;k? ); (9) 4

5 where T H (x 1 ;x 2 ;k? )= q?(x 2 q? +k? ) q? 2 (x +(1$2): (10) 2 2q? +k? ) The leading behavior of T H (at large Q 2 ) is obtained by neglecting k? relative tox i q? [6], T LO H (x 1 ;x 2 ;k? )= 1 x 1 x 2 Q2: (11) Thus Eq. (9) become Eq. (1) to the leading order. The higher twist corrections to T LO H take the forms of k? x i Q n. Eq. (10) tells us that there are two factors to contribute for the k? -deendence. One is from the QCD hard scattering amlitude T H (x i ;Q;k? ), and another one is from the non-erturbative wavefunction (x i ;k? ). Although one hoes that the end-oint behavior of the wavefunction can guarantee the reliability of neglecting these higher twist corrections and can suress the end-oint singularity, these corrections may substantially modify the redictions for F at the momentum transfer Q of a few GeV, esecially for the wavefunction with a milder suression factor in the end-oint region. It should be emhasized that the k? -deendence in the numerator and the denominator of T H is of the same imortance. Thus one can not simly ignore the k? term in the numerator of T H and Eq. (10) gives the comlete exression in the leading order. The - transition form factor F is extracted from the 0 vertex in the two-hoton hysics. Once again, we emloy the standard momentum assignment at the \innite-momentum frame = ( + ; ;? )=(1;0;0? ); q = (0;q 2? Q 02 ;q? ); q 0 = (1;q 2? Q 02 ;q? ); (12) 5

6 where q and q 0 are the momenta of the two hotons resectively, and q 2 = q 2? = Q 2, q 02 = Q 02. F may be calculated from Fig. 1 by substituting a virtual hoton for the on-shell hoton, which gives F (Q 2 ;Q 02 ) = 2 n c (e 2 u e 2 d) Z 1 Z d 2 k? [dx] (x i ;k? ) q? (x 2 q? +k? ) q? 2 [(x 2q? +k? ) 2 +x 1 x 2 q? 02 ] +(1$2) : (13) The leading order behavior of F can be obtained from Eq. (13) by neglecting k? relative tox i q? [1], F (Q 2 ;Q 02 )=2 n c (e 2 u e 2 d) Z 1 0 [dx] (x) 1 x 2 Q 2 + x 1 Q +(1$2) 02 : (14) Similar to the F, Eq. (13) may substantially modies the redictions obtained from Eq. (14). III. Numerical calculations In order to see the transverse momentum corrections, we emloy two models of wavefunction: (a) the Brodsky-Huang-Leage (BHL) wavefunction [5] BHL (x; k? )=Aex k 2? +m2 ; (15) 8 2 x(1 x) where A = 32GeV 1, =0:385 GeV and m = 289 MeV [8]; (b) the CZ-like wavefunction [9] CZ (x; k? )=A(1 2x) 2 ex k 2? +m2 ; (16) 8 2 x(1 x) where A = 136GeV 1, =0:455 GeV and m = 342 MeV [8]. These models exress that the Fock state wavefunction (x i ;k? ) in the innite momentum frame deends P on the o-shell energy variable = n k 2 +m2?i i x i, which was ointed out in Ref. [5]. i 6

7 Substituting the models (15) and (16) into Eqs. (9), (10) and (13), one can get the transverse momentum corrections to the ion-hoton transition form factor. The results of F calculated with BHL and CZ are lotted in Fig. 2. The dashed curves are calculated from the hard scattering amlitude T LO H in the leading order without the transverse momentum corrections (see Eq. (1)), and the constant redictions with the dierent wavefunctions can not describe the exerimental data at momentum transfer of a few GeV 2 exlicitly. The solid curves are obtained from the comlete exression of T H (see Eq. (10)) with the transverse momentum corrections. As exected, the higher twist correction are suressed by 1=Q 2 and the rediction aroaches to a constant which deends on the wavefunction at large Q region. The erturbative redictions are smaller than the exerimental data, esecially for Q 2 of 1 3 GeV 2, which suorts the suggestion that the higher order eects should rovide some contributions at exerimental accessible momentum transfer and become more imortant with Q 2 decreasing. Although the asymtotic behaviors of F redicted from the BHL model and CZ-like model of wavefunction are quit dierent, their redictions at nite Q 2 obtained with transverse momentum corrections are consistent with the exerimental data. The reason is as following: There are two factors to aect the rediction with the CZ-like wavefunction. First, the CZ-like model emhasizes the end-oint region in a strong way, which enhance its rediction of F. Second, the transverse momentum corrections become more imortant in the end-oint region, which make its rediction decrease. Combining these two factors, the CZ-like model gives a very similar rediction as the BHL model in the nite momentum transfer region. Thus, neither of the two models of wavefunction can be excluded by the available data of this exclusive rocess. 7

8 The results of F calculated with BHL and CZ are lotted in Fig. 3. Once again, the higher twist corrections are suressed by 1=Q 2 and rovide more contributions as Q 2 decreasing. The redictions of the two models are not dierent dramatically, no matter the transverse momentum corrections are taken into account or not, since the energy scale Q 0 coming from the other virtual hoton makes the hard scattering amlitude is not as singular as that in the case of F. It is also dicult to exclude one of the two models of wavefunction basing on F.Atresent, the lack of exeriment data make the examination of higher twist eects in F more comlex than that in F. But the future high-luminosity e + e colliders in the \-charm factory or \B factory will make this examination feasible. III. Summary In summary, we emhasize again that the light-cone erturbative QCD is a natural framework to calculate the large-momentum-transfer exclusive rocesses. It is reasonable to get the higher twist corrections by taking into account the quark transverse momentum deendence. As Q 2!1, these corrections become negligible. After taking into account the transverse momentum deendence, QCD may give correct rediction for the ion-hoton transition form factor which is consistent with the exerimental data. The transverse-momentum-deendence in both the numerator and the denominator of the hard scattering amlitude T H is of the same imortance and should be considered consistently. Neither the BHL model nor the CZ-like model, the two tyical models of wavefunctions, can be excluded by the available data of the ionhoton transition form factors. The future \-charm factory as well as \B factory will rovide the oortunity to examine the higher twist eects in the erturbative 8

9 calculation of F and to test the validity of the erturbative analysis. Acknowledgments We would like to thank S.J. Brodsky and H.N. Li for helful discussions. References [1] G.P. Leage and S.J. Brodsky, Phys. Rev. D 22, 2157 (1980), ibid. 24, 1808 (1981). [2] R. Jakob, P. Kroll, and M. Raulfs, rerint he-h (1994); P. Kroll, rerint he-h (1995). [3] S. Ong, Phys. Rev. D52, 3111 (1995). [4] J. Botts and G. Sterman, Nucl. Phys. B325, 62 (1989); H.N. Li and G. Sterman, Nucl. Phys. B381, 129 (1992); F.G. Cao, T. Huang, and C.W. Luo, Phys. Rev. D52, 5358 (1995). [5] S.J. Brodsky, T.Huang, and G.P. Leage, in Particles and Fields-2, Proceedings of the Ban Summer Institute, Ban, Alberta, 1981, edited by A.Z. Cari and A.N. Kamal (Plenum, New York, 1983), T. Huang, Proceedings of XX-th International Conference on High Energy Physics, Madison, Wisconsin, [6] S.J. Brodsky and G.P. Leage, In Perturbative Quantum Chromodynamics, edited by A.H. Mueller (World Scientic, Singaore 1989),

10 [7] See, e.g., J.B. Kogut and D.E. Soer, Phys. Rev. D1, 2901 (1970); J.B. Bjorken, J.B. Kogut, and D.E. Soer, ibid. 3, 1328 (1971); S.J. Brodsky, R. Roskies, and R. Suaya, ibid. 8, 4574 (1973). [8] T. Huang, B.-Q. Ma, and Q.-X. Shen, Phys. Rew. D49, 1490 (1994). [9] T. Huang and Q.-X. Shen, Z. Phys. C50, 139 (1991). [10] CELLO Collab., H.-J. Behrend et al., Z. Phys. C49, 401 (1991), there is a factor of 4 for the denition of form factors between our analysis and this reference. [11] CLEO Collab., V. Savinov et al., he-ex/ (1995). 10

11 Figure Cations Fig. 1 The lowest order diagrams contributing to F in light-cone QCD. Fig. 2 The - transition form factor. The solid curves are obtained by taking into account the k? -deendence, while the dashed curves are results without k? -deendence. In both of the cases, the thick curves are calculated from the BHL wave function and the thin curves are for the CZ-like wavefunction. The data are taken from Refs. [10, 11]. Fig. 3 The - form factor at Q 02 = 2 GeV 2. The exlanation of the curves is similar to Fig