Study of exclusive processes e^+ e^- \to VP
aa r X i v : . [ h e p - ph ] O c t Study of ex lusive pro esses e + e − → V P .V.V. Braguta,1, ∗ A.K. Likhoded,1, † and A.V. Lu hinsky1, ‡ e + e − → V P , where V = ρ, φ ; P = η, η ′ . Experimental measurement of the ross se tion of the pro ess e + e − → φη at BaBar ollaboration at large enter mass energy √ s = 10 . GeV and some low energy experimental data √ s ∼ − GeV give us the possibility to study the ross se tion in the broad energy region.As the result, we have determined the asymptoti behavior of the ross se tion of e + e − → φη inthe limit s → ∞ , whi h is in agreement with perturbative QCD predi tion. Assuming that thesame asymptoti behavior is valid for the other pro esses under onsideration and using low energyexperimental data we have predi ted the ross se tions of these pro esses at energies √ s = 3 . , . GeV. In addition, we have al ulated the ross se tions of these pro esses at the same energies withinperturbative QCD. Our results are in agreement with available experimental data.PACS numbers: 12.38.-t, 12.38.Bx, 13.66.B ,I. INTRODUCTIONEx lusive hadron produ tion in high energy ele tron-positron annihilation is a very interesting task for theoreti aland experimental investigations. The presen e of high energy s ale √ s that is mu h greater than typi al hadroni s ale allows one to separate the amplitude of su h pro esses into hard part ( reation of quarks at very small distan es)and soft part (subsequent hadronization of these quarks into experimentally observed mesons at larger distan es).The (cid:28)rst part of the amplitude an be al ulated within perturbative QCD. The se ond part of the amplitude isdes ribed by distribution amplitudes (DA), whi h ontain nonperturbative properties of (cid:28)nal hadrons.The des ription of hard ex lusive hadron produ tion within this pattern gives some very interesting predi tions ofthe properties of hard ex lusive pro esses [1, 2℄. One of su h predi tion is the asymptoti behavior of the amplitudesand ross se tions of hard ex lusive pro esses in the limit s → ∞ [3, 4, 5℄. It turns out that this behavior is determinedby the perturbative part of amplitude and quantum numbers of (cid:28)nal hadrons and does not depend on DAs of (cid:28)nalhadrons.To give quantitative predi tion for the ross se tion of hard ex lusive pro ess one needs to know DAs. It is interestingto note that if DAs are known the theory an equivalently well predi t the ross se tions for the produ tion of hadrons omposed of light ( u, d, s quarks) or heavy quarks ( b, c quarks). There is well known example of ex lusive pro esswith heavy quarkonia produ tion e + e − → J/ Ψ η c measured at Belle [6℄ and BaBar [7℄ ollaborations. This pro esswas extensively studied in many papers [8, 9, 10, 11, 12, 13, 14, 15, 16, 17℄ within di(cid:27)erent approa hes, what led toa better understanding of harmonia properties and produ tion pro esses. It is important to note that the approa hto hard ex lusive pro esses des ribed above leads to a reasonable agreement with the experiments.In this paper we study the pro esses e + e − → V P , where V = ρ, φ ; P = η, η ′ . Experimental measurement of the ross se tion of the pro ess e + e − → φη at BaBar ollaboration at large enter mass energy √ s = 10 . GeV andsome low energy experimental data √ s ∼ − GeV allow us to study the ross se tion of this pro ess in the broadenergy region. Our (cid:28)rst purpose is to use these data in order to determine the asymptoti behavior of the rossse tion of the pro ess e + e − → φη . Assuming that the same asymptoti behavior is valid for the other pro essesunder onsideration and using low energy experimental data one an predi t the ross se tions of the e + e − → V P atenergies √ s = 3 . , . GeV. In addition, we apply perturbative QCD approa h to estimate the values of the rossse tions for the pro esses under onsideration.This paper is organized as follows. In the next se tion we analyze the experimental data for the pro ess e + e − → φη and determine the asymptoti behavior of this ross se tion. Then we apply the result of this study to predi t the rossse tions of the other pro esses under onsideration at the enter mass energies √ s = 3 . , . GeV. In se tion III wegive theoreti al predi tions for the ross se tions σ ( e + e − → ρ η ) , σ ( e + e − → ρ η ′ ) , σ ( e + e − → φη ) and σ ( e + e − → φη ′ ) ∗ Ele troni address: bragutamail.ru † Ele troni address: Likhodedihep.ru ‡ Ele troni address: Alexey.Lu hinskyihep.ru P ( p ) V ( p , λ ) γ q ¯ q γ P ( p ) V ( p , λ ) γ ¯ qg gqe − ( q ) e + ( q ) e − ( q ) e + ( q ) P ( p ) V ( p , λ ) γ g ¯ qq q ¯ qe − ( q ) e + ( q )( a ) ( b ) ( c ) Figure 1: Typi al diagrams for the e + e − → V P pro ess.at √ s = 3 . GeV and 10.6 GeV and ompare them with available experimental data. The (cid:28)nal se tion is devoted tothe dis ussion of the results of this paper.II. THE ASYMPTOTIC BEHAVIOR.The amplitude of the pro ess involved an be written in the following form: M ( e + e − → V P ) = 4 πα ¯ v ( q ) γ µ u ( q ) s (cid:10) V ( p , λ ) P ( p ) (cid:12)(cid:12) J emµ (cid:12)(cid:12) (cid:11) , where α is the ele tromagneti oupling onstant, u ( q ) and ¯ v ( q ) are ele tron and positron bispinors, s = ( q + q ) is the invariant mass of e + e − system squared and J emµ is ele tromagneti urrent. The matrix element (cid:10) V P (cid:12)(cid:12) J emµ (cid:12)(cid:12) (cid:11) an be parameterized by the only formfa tor F ( s ) : (cid:10) V ( p , λ ) P ( p ) (cid:12)(cid:12) J emµ (cid:12)(cid:12) (cid:11) = ie µναβ ǫ νλ p α p β F ( s ) , (1)where ǫ νλ is the polarization ve tor of meson V . The ross se tion of the pro ess under onsideration equals σ ( e + e − → V P ) = πα (cid:18) | p |√ s (cid:19) | F ( s ) | . (2)In the last formula p is the momentum of the ve tor meson V in the enter mass frame of (cid:28)nal mesons.In this se tion we will be interested in the asymptoti behavior of the formfa tor F ( s ) in the high energy region.Typi al diagrams of the pro ess under onsideration are shown in Fig. 1. The asymptoti behavior of the diagramshown in Fig. 1a is σ ∼ /s . At extremely large energies this diagram gives the dominant ontribution. However,the amplitude of this diagram is suppressed by the smallness of the ele tromagneti oupling onstant and our studyshows that in the energy region analyzed in this paper the ontribution of this diagram is negligible. Further let us onsider the diagrams shown in Fig. 1b, . A ording to perturbative QCD [2℄ amplitudes (1) from su h diagramshave the following asymptoti behavior (cid:10) H ( p , λ ) H ( p , λ ) (cid:12)(cid:12) J emµ (cid:12)(cid:12) (cid:11) ∼ (cid:18) √ s (cid:19) | λ + λ | +1 , where H and H are mesons with momenta p , p and heli ities λ and λ . For the pro ess under onsideration H and H are the ve tor and pseudos alar mesons respe tively. The heli ity of the pseudos alar meson is, obviously, λ = 0 . Be ause of antisymmetri tensor in (1), longitudinal polarization of the ve tor meson is forbidden and it istransversely polarized ( λ = ± . So, we have F ( s ) ∼ /s , σ ( e + e − → V P ) ∼ /s . On the other hand, in papers[18, 19℄, it was stated that experimental data an be des ribed only by the dependen e σ ∼ /s . To larify thissituation we will parameterize the formfa tor F ( s ) by the expression F ( s ) = a n ( s )( √ s ) n , (3)where a n ( s ) slightly depends on s due to power and logarithmi orre tions to the leading asymptoti behavior. Laterin this se tion we will negle t su h dependen e. This approximation allows us to (cid:28)x the onstants a n from the lowenergy data and predi t the ross se tions at √ s = 3 . GeV and 10.6 GeV. We he k three di(cid:27)erent hypothesis( n = 3 , and 5) and ompare the results with existing experimental data.First we are going to onsider the pro ess e + e − → φη . To (cid:28)x the onstants a n one an use low energy data [20℄.When the onstants a n for di(cid:27)erent hypothesis are (cid:28)xed, one an use them to predi t the ross se tion in the highenergy region and then ompare this predi tion with available high energy data [21℄ measured by BaBar ollaboration σ BaBar ( e + e − → φη ) = 2 . ± . . Our results are shown in Fig. 2 and Table I. From these (cid:28)gure and table it an be learly seen, that only n = 4 des ribessatisfa tory low and high energy experimental BaBar data [20, 21℄. This result is in agreement with the predi tionsof perturbative QCD. Form Fig. 2 one an see that low energy CLEO- data [22℄ are in disagreement with hypothesis n = 3 , . Only hypotheses n = 5 does not ontradi t to the experimental results. However, if we assume that thishypotheses is orre t we will fa ed with dramati ontradi tion with the high energy BaBar data (see Fig. 2 and TableI). It should be also noted that in papers [18, 19℄ it was stated that the energy dependen e σ ( e + e − → φη ) ∼ /s des ribes experimental data more a urately. We believe that the disagreement of this statement with our on lusionarises from the fa t that the authors of papers [18, 19℄ did not take into a ount low energy experimental result [20℄.In view of this the question arises: is it possible to use asymptoti behavior of the ross se tion (3) in the region √ s ∈ (2 , . GeV. First, one an estimate the ross un ertainty due to the power orre tions as ∼ M /s ∼ . Even, forthe heaviest meson φ and the smallest √ s from the region √ s ∈ (2 , . the error is ∼ . . We an also determinethe size of power orre tions from the (cid:28)tting data [20℄ by the asymptoti form (3) plus next-to-leading-order power orre tion. Our analysis shows that the un ertainty is not greater than ∼ . We believe that these arguments on(cid:28)rm the appli ability of the asymptoti expression for the ross se tion.Now let us onsider the pro esses e + e − → ρη , e + e − → ρη ′ . Unfortunately at the moment only the low energyCLEO- data [22℄ are available for these pro esses. As it is seen from Tab. I the experimental error of these data arerather large. More pre ise experimental data an be obtained from the de ays J/ψ → ρη, J/ψ → ρη ′ . Correspondingbran hing fra tion are equal to [23℄ Br (cid:0) e + e − → ρη (cid:1) = (1 . ± . × − , Br (cid:0) e + e − → ρη ′ (cid:1) = (1 . ± . × − . Generally speaking, these de ays an pro eed both via strong and ele tromagneti intera tion (see Fig. 3 for thetypi al diagrams). Be ause of the isospin violation, however, the gluon indu ed diagrams are strongly suppressed, andpurely ele tromagneti diagram gives the dominant ontribution. The bran hing fra tion of these de ays are equal toBr ( J/ψ → V P ) = (cid:18) | p | M J/ψ (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a n ( M J/ψ ) M nJ/ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M J/ψ Br ( J/ψ → e + e − ) . (4)From this relation we an determine the value of the fun tion a n ( M J/ψ ) and, negle ting energy dependen e in a n ( s ) (this assumptions leads to additional errors, whi h an be estimated as ∼ M /s ∼ ), predi t the ross se tionsof the pro esses e + e − → ρη and e + e − → ρη ′ over the large energy region. The results are shown in Figs. 4, 5 andTable I. From these results it is seen that the low energy data do not allow to understand what hypotheses gives thebest agreement with the experiments. So, to determine the asymptoti behavior unambiguously the high energy dataare needed. However, if we assume that the asymptoti behavior for the ross se tion of the pro ess e + e − → φη isthe same as that for the pro esses e + e − → ρη , e + e − → ρη ′ , one an predi t the ross se tions of these pro esses atenergies √ s = 3 . , . GeV σ √ s =3 . GeV ( e + e − → ρη ) = 8 ± pb , σ √ s =3 . GeV ( e + e − → ρη ′ ) = 5 ± pb ,σ √ s =10 . GeV ( e + e − → ρη ) = 2 . ± . fb , σ √ s =10 . GeV ( e + e − → ρη ′ ) = 1 . ± . fb . (5)The ross se tions at the energy √ s = 3 . GeV are in agreement with the CLEO- data.Let us now pro eed with the rea tion e + e − → φη ′ . In this ase experimental information is even more poor. In the ase one annot use the bran hing fra tion of the de ay Br ( J/ Ψ → φη ′ ) , sin e in this ase isospin is onserved and thediagram shown in Fig. 3a gives the main ontribution to J/ψ → φη ′ de ay. In addition, CLEO- gives us only upperbound on the ross se tion σ ( e + e − → φη ′ ) , and no experimental information in high energy region is available. Using η − η ′ mixing it is possible, however, to estimate the ross se tion σ ( e + e − → φη ′ ) from the value of the e + e − → φη n = = = s , GeV110100100010 Σ H ee ®ΦΗ L , fb Figure 2: Di(cid:27)erent hypothesis on the energy dependen e for the σ ( e + e − → φη ) . The onstants a n are (cid:28)xed from the experi-mental values of this ross se tion in the low energy region ( GeV ≤ √ s ≤ . GeV).
J/ψ c ¯ c ggg VPqq ¯ q ¯ q J/ψ VPc ¯ c γ Figure 3: Typi al diagrams for
J/ψ → V P de ay n = = = s @ GeV D Σ H ee ®ΡΗ L @ fb D Figure 4: Di(cid:27)erent hypothesis on the energy dependen e for the σ ( e + e − → ρη ) . The onstants a n are (cid:28)xed from the J/ψ → ρη bran hing fra tion (the rightmost point represents the ross se tion at √ s = M J/ψ al ulated using relations (2), (3) and (4).) n = = = s @ GeV D Σ H ee ®ΡΗ ’ L @ fb D Figure 5: Di(cid:27)erent hypothesis on the energy dependen e for the σ ( e + e − → ρη ′ ) . The onstants a n are (cid:28)xed from the J/ψ → ρη ′ bran hing fra tion (the rightmost point represents the ross se tion at √ s = M J/ψ al ulated using relations (2), (3) and (4). n = = = s @ GeV D Σ H ee ->ΦΗ ’ L @ fb D Figure 6: Di(cid:27)erent hypothesis on the energy dependen e for the σ ( e + e − → φη ′ ) . The onstants a n are (cid:28)xed from theexperimental values of this ross se tion at √ s = 10 . GeV. ross se tion. The mixing of pseudos alar mesons an be des ribed by di(cid:27)erent parameterizations [24℄. In our paperwe will use the parametrization of η − η ′ mixing in quark (cid:29)avor basis with one mixing angle [25℄: (cid:18) ηη ′ (cid:19) = (cid:18) cos Φ − sin Φsin Φ cos Φ (cid:19) (cid:18) η n η s (cid:19) , (6)where η n = ( u ¯ u + d ¯ d ) / √ and η s = s ¯ s represents the basis of the quark mixing s heme. Using this mixing s heme itis easy to obtain the following relation between e + e − → φη and e + e − → φη ′ ross se tions: σ ( e + e − → φη ′ ) σ ( e + e − → φη ) = cot Φ . There are plenty of theoreti al and experimental works dedi ated to the determination of the mixing angle. The wellknown estimation, based on Gell-Mann-Okubo mass formulas give the value of Φ about o for linear mass formulaand about o for a quadrati ase. The analysis of the axial anomaly generated de ays η, η ′ → γγ was performed inpapers [26, 27℄ and the estimate Φ = 30 o ÷ o was obtained. In papers [28, 29℄ another anomaly based investigationof a large set of de ay pro esses was performed and the value Φ = 38 o ± was presented. The authors of the re entwork [30℄ used dispersive approa h to η − η ′ mixing and obtained the value Φ = 39 . o ± o . It is interesting to note that this value is lose to phenomenologi al value
Φ = 39 . o ± o , presented in pioneeringwork [25℄. We will use this value of the mixing angle in our arti le. The error of this angle is small in omparisonwith the error in e + e − → φη ross se tion.Knowing the value of the ross se tion σ ( e + e − → φη ) one an determine the value of the ross se tion σ ( e + e − → φη ′ ) at the energy √ s = 10 . GeV. σ √ s =10 . GeV ( e + e − → φη ′ ) = 4 . ± . fb . From this we an al ulate the value of the onstants a n for di(cid:27)erent hypothesis and predi t the ross se tion of thepro ess e + e − → φη ′ in the low energy region. We show the energy dependen e of this ross se tion in Fig. 6. Fromthis (cid:28)gure one sees that the value n = 5 ontradi ts experimental results at √ s = 3 . GeV, while n = 3 and n = 4 do not. From theoreti al arguments and presented above analysis of e + e − → φη rea tion we think, that the value n = 4 is more preferable. For this hypothesis we an predi t the value of the ross se tion at energy √ s = 3 . GeV σ √ s =3 . GeV ( e + e − → φη ′ ) = 12 . ± . pb , whi h does not ontradi t to the CLEO- data.In table I we present the results of this se tion. Se ond olumn ontains onstants a , , for di(cid:27)erent (cid:28)nal states,obtained from low energy (cid:28)ts. In the third and (cid:28)fth olumns experimental results for the ross se tions σ ( e + e − → V P ) at the enter mass energies √ s = 3 . GeV and 10.6 GeV are presented. The results of the al ulation are shown inthe forth and sixth olumns. From this table it is lear, that only relation σ ∼ /s agrees with experiment.III. CALCULATION OF THE CROSS SECTIONS.A. Numeri al parameters and distribution amplitudes.Now let us try to estimate the ross se tions of the pro esses studied in the last se tion theoreti ally. To al ulatethe ross se tions of the pro esses e + e − → V P for V = φ, ρ, P = η, η ′ one needs to know distribution amplitudes(DA) of (cid:28)nal mesons. For the ve tor mesons the DAs needed in the al ulation an be written as [2, 13℄ h V λ ( p ) | ¯ q β ( z ) q α ( − z ) | i µ = f V M V Z o dx e i ( pz )(2 x − (cid:26)b p ( e λ z )( pz ) V L ( x ) + (cid:18)b e λ − b p ( e λ z )( pz ) (cid:19) V ⊥ ( x ) + f T ( µ ) f V M V ( σ µν e µλ p ν ) V T ( x ) + 12 ( ǫ µναβ γ µ γ e νλ p α z β ) V A ( x ) (cid:27) αβ , (7)where x is the fra tion of momentum arried by quark, f V , f T ( µ ) , M V are the leptoni , tensor onstants and the massof ve tor meson. In the al ulation φ meson is assumed to be omposed of s quarks, so in (7) q = s . For ρ meson σ ( √ s = 3 . GeV ) pb σ ( √ s = 10 . GeV ) fb V P parrams exp (cid:28)t results exp (cid:28)t results a = 1 . ± . GeV ± ± ρη a = 5 . ± . GeV ± . ± (cid:21) . ± . a = 17 . ± . GeV ± . ± . a = 1 . ± . GeV ± ± ρη ′ a = 4 . ± . GeV . ± . ± (cid:21) . ± . a = 14 . ± . GeV . ± . . ± . a = 2 . ± . GeV ± ± φη a = 6 . ± . GeV . ± . . ± . . ± . . ± . a = 14 . ± . GeV . ± . . ± . a = 0 . ± . GeV . ± . . ± . φη ′ a = 8 . ± . GeV < . . ± . (cid:21) . ± . a = 85 . ± . GeV ±
18 4 . ± . Table I: The onstants a n and the ross se tions at √ s = 3 . and 10.6 GeV in omparison with the experimental data. These ond olumn ontains the onstants a , , for the di(cid:27)erent (cid:28)nal states, obtained from the low energy data. In the third and(cid:28)fth olumns the experimental results for the ross se tions σ ( e + e − → V P ) at the enter mass energy √ s = 3 . GeV and √ s = 10 . GeV are presented. The results of the al ulation are shown in the forth and sixth olumns.isospin 1 ombination ¯ qq = (¯ uu − ¯ dd ) / √ is assumed. The models for DAs and the de ay onstants f V , f T ( µ ) for φ and ρ mesons will be taken from paper [31℄.In the framework of the quark mixing s heme (6) the de ay onstants h P ( p ) | ¯ nγ µ γ n | i = if nP p µ , h P ( p ) | ¯ sγ µ γ s | i = if sP p µ , needed in the al ulation an be expressed through the onstants h η n ( p ) | ¯ nγ µ γ n | i = if n p µ , h η s ( p ) | ¯ sγ µ γ s | i = if s p µ , as follows f nη f sη f nη ′ f sη ′ ! = cos Φ − sin Φsin Φ cos Φ ! f n f s ! . In turn, the onstants f n , f s and the mixing angle Φ an be determined from experiment [25℄ f n = (1 . ± . f π , f s = (1 . ± . f π . Within this mixing pattern the DAs needed in the al ulation an be written in the following form [2, 13℄ h P ( p ) | ¯ n β ( z ) n α ( − z ) | i µ = i f nP M P Z dye i ( pz )(2 y − (cid:26) ˆ p γ M P P nA ( y ) − f np ( µ ) γ P nP ( y ) (cid:27) , h P ( p ) | ¯ s β ( z ) s α ( − z ) | i µ = i f sP M P Z dye i ( pz )(2 y − (cid:26) ˆ p γ M P P sA ( y ) − f sp ( µ ) γ P sP ( y ) (cid:27) , where f np ( µ ) = 12 m n ( µ ) (cid:20) m η cos Φ + m η ′ sin Φ − √ f s f n ( m η ′ − m η ) cos Φ sin Φ (cid:21) ,f sp ( µ ) = 12 m s ( µ ) (cid:20) m η ′ cos Φ + m η sin Φ − f n √ f s ( m η ′ − m η ) cos Φ sin Φ (cid:21) . The al ulation will be done with the masses m s (1 GeV ) = 150
MeV, m n (1 GeV ) = ( m u (1 GeV ) + m d (1 GeV )) / MeV and it will be assumed that P sA ( y ) = P nA ( y ) = P A ( y ) and P sP ( y ) = P nP ( y ) = P P ( y ) . The models of the leadingtwist DAs P A for the η and η ′ mesons will be taken from paper [32℄. For the fun tion P P ( y ) the asymptoti form willbe used.It should be noted that the wave fun tions and the onstants introdu ed above depend on renormalization s ale µ .Our al ulation shows that the s ale dependen e of the DAs is not very important and below it will be ignored. Atthe same time the s ale dependen e of the onstants is important and it will be taken into a ount. The al ulationwill be done at the s ale µ = √ s/ . B. Numeri al results and dis ussion.Having introdu ed the designations of the DAs one an pro eed with the al ulation of the ross se tions. First weare going to onsider the diagrams similar to that shown in Fig.1a. It is not di(cid:30) ult to al ulate the ontribution ofthese diagrams to the formfa tor F ( s ) | F ( s ) | = 4 παs f V M V (cid:18) e u + e d √ f nP + e s f sP (cid:19) Z dyy y P A ( y ) , where e u , e d , e s are the harges of u, d, s quarks orrespondingly. For the pro ess e + e − → φη at the energy √ s = 10 . GeV we have σ = 0 . fb, what is by two orders of magnitude less than the experimental result. Form this it is learthat the ontribution of the diagrams shown in Fig.1a is negligible and below it will be ignored. It should be notedthat in the limit s → ∞ the ontribution of Fig.1a diagrams to the ross se tion has the following behavior ∼ /s .So, at extremely large energy this diagrams give the dominant ontribution. Our al ulation shows that this happensat energies √ s ≥ GeV.Now we are going to onsider the other diagrams shown in Fig. 1. The leading asymptoti behavior of thesediagrams is σ ∼ /s . First, it should be noted that the ontribution of the diagrams shown in Fig. 1b is very small[18℄ and it will be ignore below. This fa t results from rather small admixture of | GG i fo k state in the pseudos alarmesons η, η ′ . Now we pro eed to the al ulation of the diagrams shown in Fig. 1 . The ontribution of these diagramsto the F ( s ) for the pro esses under study an be written as follows [13℄ | F φη ( s ) | = 32 π f V f s M V sin Φ s e s I , | F φη ′ ( s ) | = 32 π f V f s M V cos Φ s e s I , | F ρη ( s ) | = 32 π f V f n M V cos Φ2 s ( e u − e d ) I , | F ρη ′ ( s ) | = 32 π f V f n M V sin Φ2 s ( e u − e d ) I . (8)In the above expressions I = Z dx Z dy α s ( µ ) (cid:26) f t ( µ ) M V f p ( µ ) V T ( x ) P P ( y ) x y + (9) + 12 V L ( x ) P A ( y ) x y + (1 − y ) V ⊥ ( x ) P A ( y ) x y + 18 (1 + y ) V A ( x ) P A ( y ) x y (cid:27) , where f t ( µ ) = f T ( µ ) /f V , f p ( µ ) = f sp ( µ ) for the pro esses with φ meson in the (cid:28)nal state and f p ( µ ) = f np ( µ ) for thepro esses with ρ meson in the (cid:28)nal state.The following point deserves onsideration. The models for the DAs that we use in our al ulation are trun atedseries in Gegebauer polynomials. This means that the end point behavior ( x → , ) of these DAs oin ides with theend point behavior of the orresponding asymptoti DAs. From this it is not di(cid:30) ult to see that the integral I islogarithmi ally divergent. One way to regularize this divergen e is to introdu e ut o(cid:27) parameter x . In our al ulationwe take the following value of ut o(cid:27) parameter x = Λ / √ s . Λ is of order of typi al hadroni s ale, whi h is of orderof several hundreds MeV. Physi al meaning of this ut o(cid:27) an be understood as follows: if x ∼ x quark momentum isof order of ∼ Λ and in this region one must take into a ount transverse motion in hadron what regularizes the wholeintegral I . The parameter Λ an be determined from available experimental results. We determined this parameterfrom the ross se tion of the pro ess e + e − → φη at √ s = 10 . GeV. Thus we get
Λ = 130 +25 − MeV. The variation ofparameter Λ orresponds σ deviation from the entral value measured at Belle ollaboration.The results of the al ulation are shown in Table II. The se ond and (cid:28)fth olumns ontain the experimental resultsfor the ross se tions at the energies √ s = 3 . GeV and √ s = 10 . GeV orrespondingly. The result obtained in σ ( √ s = 3 . GeV ) pb σ ( √ s = 10 . GeV ) fb V P exp [22℄ [18℄ this work exp [21℄ [18℄ this wok ρη ± . . ÷ . . ÷ . (cid:21) . ÷ . . ÷ . ρη ′ . ± . . ÷ . . ÷ . (cid:21) . ÷ . . ÷ . φη . ± . . ÷ . . ÷ . . ± . . ÷ . . ÷ . φη ′ < . . ÷ . . ÷ . (cid:21) . ÷ . . ÷ . Table II: The results of the al ulation. The se ond and (cid:28)fth olumns ontain experimental results for the ross se tions atenergies √ s = 3 . GeV and √ s = 10 . GeV orrespondingly. The result obtained in paper [18℄ are shown in the third andsixth olumns. The results obtained in this paper are shown in the fourth and seventh olumns.paper [18℄ are shown in the third and sixth olumns. The results obtained in this paper are shown in the fourth andseventh olumns. The variation of the results are due to the variation in parameter Λ . It is seen from this table thatthe results of the al ulation are in satisfa tory agreement with the experiment.It should be noted here that in addition to the diagrams shown in Fig. 1, there is additional ontribution tothe formfa tor F ( s ) whi h was not onsidered in this paper. This ontribution appears if one takes into a ounthigher fo k state of the ve tor and pseudos alar mesons | q ¯ qG i and it's asymptoti behavior is also σ ∼ /s . This ontribution was onsidered in papers [2, 33℄. In paper [2℄ the authors asserted that the fo k state | q ¯ qG i gives veryimportant ontribution to the ross se tion and must be taken into a ount. Contrary to this on lusion, the author ofpaper [33℄ asserted that the main ontribution arises from the diagrams shown in Fig. 1 . So, the question about therole of higher fo k state | q ¯ qG i in the total ross se tion deserves separate onsideration and it will not be onsideredhere. IV. DISCUSSION AND CONCLUSION.In this work we have studied the produ tion of light mesons ρη , ρη ′ , φη and φη ′ in the high energy ele tron-positronannihilation.The (cid:28)rst question studied in this paper is the asymptoti behavior of the ross se tions of the pro esses under onsideration. Perturbative QCD predi ts, that the ross se tion of the rea tion e + e − → V P has the followingasymptoti behavior σ ∼ /s in the limit s → ∞ . Experimental measurement of the ross se tion of the pro ess e + e − → φη at the large enter mass energy √ s = 10 . GeV [21℄ and the low energy experimental data √ s ∼ − GeV [20℄ give us the possibility to study the ross se tion in the broad energy region. As the result, we havedetermined the asymptoti behavior of the ross se tion of e + e − → φη in the limit s → ∞ , whi h is in agreementwith perturbative QCD predi tion. As to the other pro esses under study, there are no high energy experimentaldata, whi h allow us to on(cid:28)rm perturbative QCD predi tion for these pro esses. We would like to stress here thatthe high energy experimental data turned out to be ru ial in the determination of the asymptoti behaviour of the ross se tions. Assuming that the asymptoti behavior predi ted by perturbative QCD is valid for the other pro essesunder onsideration, we have al ulated the ross se tions of the pro esses e + e − → ρη, ρη ′ , φη, φη ′ at the energies √ s = 3 . , . GeV.In addition, we have applied perturbative QCD approa h to al ulate the ross se tions of the pro esses understudy at the energies √ s = 3 . GeV and √ s = 10 .6