Eta-photon transition form factor

The eta-photon transition form factor S. Nogueraa), S. Scopettab)∗ arXiv:1110.6402v3 [hep-ph] 8 Feb 2012 a) Departamento de Fisica Teorica and Instituto de Fı́sica Corpuscular, Universidad de Valencia-CSIC, E-46100 Burjassot (Valencia), Spain. and b) Dipartimento di Fisica, Università di Perugia, and INFN, Sezione di Perugia, via A. Pascoli, I-06100 Perugia, Italy. (Dated: October 22, 2018) The eta-photon transition form factor is evaluated in a formalism based on a phenomenological description at low values of the photon virtuality, and a QCD-based description at high photon virtualities, matching at a scale Q20 . The high photon virtuality description makes use of a Distribution Amplitude calculated in the Nambu-Jona-Lasinio model with Pauli-Villars regularization at the matching scale Q20 , and QCD evolution from Q20 to higher values of Q2 . A good description of the available data is obtained. The analysis indicates that the recent data from the BaBar collaboration on pion and eta transition form factor can be well reproduced, if a small contribution of higher twist is added to the dominant twist two contribution at the matching scale Q20 . PACS numbers: 12.38.-t, 12.39.St, 13.40.Gp, 13.60.Le I. INTRODUCTION Meson Distribution Amplitudes (DA) are fundamental theoretical ingredients in the description of exclusive high energy processes. The pseudoscalar transition form factors (T F F ), Fγγ ∗ P , describing the process P → γγ ∗ , where P is a pseudoscalar meson, are directly connected with the DAs. Recently, the BaBar Collaboration has provided new data at high virtuality for the pion and eta T F F (πT F F and ηT F F ) [1, 2]. The implications of these results on πT F F in our understanding of the pion structure have been widely discussed [3–13] In particular, these results have cast doubts on the behavior, as a function of the light-cone momentum fraction x, of the pion distribution amplitude (πDA) φπ (x) [14, 15], a quantity for which some investigations have predicted a flat behavior, i.e., a constant value for any x [3, 4], in good agreement with the data of the form factor. These scenarios are compatible with QCD sum rules [15] and lattice QCD [16, 17] calculations which provide values for the second moment of the πDA which are large compared to the asymptotic value 6x(1 − x). Several model calculations, such as the ones performed in the Nambu-Jona-Lasinio (NJL) [18–20] or in the ”spectral” quark model [21] frameworks, give a constant πDA, i.e. φπ (x) = 1. With the availability of data about the eta, it is important to analyze all the proposed theoretical schemes. Some work in this direction has been already done [11, 22–27]. In particular, in Refs. [11, 22–25] the importance of the transverse momentum of the quarks is emphasized, making use of some parametrization of the eta wave function. In Refs. [26, 27], the TFFs of pseudoscalar mesons are considered by using a dispersive representation of the axial anomaly, considering also the violation of factorization and possible higher twist corrections. The parton distributions, generalized parton distributions and distribution amplitudes have been used as a test of hadron models. The procedure consists of three ingredients: i) the hadron model provides a low energy description of the studied distribution; ii) a high energy description is obtained by QCD evolution, which needs an input at some low scale Q20 ; iii) a matching condition between the two descriptions at a scale Q20 characterizing the separation between the two regimes. This procedure has been useful in the study of nucleon parton distributions [28–30] as well as in that of pion distributions [31–35]. In Ref. [8], a version of the previous program, but in a rather model-independent formalism, has been used to calculate the πT F F . An excellent description of experimental data has been obtained in the whole range of virtuality. Summarizing, the evaluation of the πT F F at high Q2 values in Ref. [8] is based on the following arguments: i) chiral symmetry and soft pion theorems, which explain that, at some point Q20 , the πDA has a flat behavior, φπ (x, Q20 ) = 1; ii) applying QCD evolution to the πDA, one can obtain the πT F F at any Q2 ≫ Q20 , iii) for Q2 < Q20 , the experimental parametrization of Fγγ ∗ π (Q2 ) given in ref. [36] is assumed; iv ) for Q2 > Q20 the πT F F is given by its standard expression in terms of the πDA modified in two directions, the quark propagator is corrected , as suggested by Radyushkin [3], and a term originated by other higher twist contributions is included. ∗ Electronic address: [email protected], [email protected] 2 This scheme, successful for the pion, requires further tests. The most natural one consists in the evaluation of the same quantity for the other pseudoscalar mesons. In this paper, the program is developed for the η meson, for which a few sets of data are available [2, 36]. The importance of the η − η ′ system for our understanding of the QCD symmetries, and for their treatment in effective, low energy descriptions, is well known (see, i.e., Ref. [37] and references therein). To implement this program, the approach of Ref. [8] has to be complicated, and some hints have to be obtained within a specific model. In particular, a generalized SU(3) Nambu-Jona-Lasinio model with Pauli-Villars regularization, along the lines of Ref. [38], will be used. The NJL model is the most realistic model for the pseudoscalar mesons based on a local quantum field theory built with quarks. It respects the realization of chiral symmetry and gives a good description of the low energy physics of pseudoscalar mesons. It allows to describe mesons in a field theoretical framework treating them as bound states in a fully covariant manner using the Bethe-Salpeter amplitude. In this way, the Lorentz covariance of the problem is preserved. The NJL model is a non-renormalizable field theory and therefore a cut-off procedure has to be implemented. The Pauli-Villars regularization procedure has been chosen because it respects the gauge symmetry of the problem. The NJL model together with its regularization procedure is regarded as an effective theory of QCD. In the chiral limit, it predicts for the pion φπ (x) = 1, 0 ≤ x ≤ 1, in agreement with the model independent study of the pion in Ref. [8]. At this point, due to the lack of fits in the SU (3) NJL model with the Pauli-Villars regularization , it is performed a new analysis of the parameters of the model. It is interesting to notice that an early attempt to use the NJL model, but in a U (3) invariant scheme, has been presented in Ref. [39], where the η parton distribution has been evaluated. The paper is organized as follows. In section II, the theoretical description of the γγ ∗ → P is reviewed, extracting the soft (non-perturbative) part, to be described by the model. In section III, the η calculation in the NJL model is presented. In the following section, numerical results are presented and discussed. The conclusions are drawn in the last section. In the Appendix, a summary of the NJL model is given, including the description of the fit which has been used. II. THE γγ ∗ → P PROCESS: THEORETICAL DESCRIPTION. The subject of this study is the transition form factor, Fγγ ∗ P , i.e., the form factor for the coupling of a real photon and a virtual photon to a pseudoscalar meson, P . The TFF is a very important quantity in the QCD description of exclusive processes. In particular, it can be used to obtain information on the shape of the meson DA [14, 40, 41]. Experimentally, it has been measured for the π, η and η ′ mesons by the CELLO [42], by the CLEO [36] and, recently, by the BaBar [1, 2] collaborations. The latter results, for the pion, have been found in disagreement with theoretical expectations. In order to establish the proper formalism, in this section the theoretical description of the γ (q1 ) γ ∗ (q2 ) → P (k) process is reviewed. From general arguments it is well known that the transition amplitude of this process can be written as (see, i.e., [43]): hP (k) out|γ (q1 , ε1 ) , γ ∗ (q2 , ε2 ) ini = i (2π)4 δ 4 (q1 + q2 − k) T , (1) T = 4 π α ε1µ ε2ν q1ρ q2σ εµνρσ Fγγ ∗ P , (2) with where α is the fine structure constant. On the other hand, applying the reduction formalism of Lehmann, Symanzik and Zimmermann [43], this process is described by hP (k) out|γ (q1 , ε1 ) , γ ∗ (q2 , ε2 ) ini Z  z   z  4 = − (2π) δ 4 (q1 + q2 − k) d4 z e−i(q1 −q2 )z/2 hP (k) out| T ε1 · j ε2 · j − |0i , (3) 2 2 P where jµ (z) = j ej ψ̄j (z) γµ ψj (z) is the electromagnetic current for the quarks. Evaluating the time ordered product at leading order and after some algebra, one has Z Z d4 t −itz T = i d4 z e−i(q1 −q2 )z/2 4e (2π)  z z X
1 γβ γ5 ψj − |0i . (4) hP (k) out| ψ̄j 2 i e2j ε1µ ε2ν −iεµανβ tα 2 2 t − mj + iǫ 2 2 j 3 Defining hP (k) out| ψ̄ja Z  z d4 ℓ −i( k2 −ℓ)z j ψjb − |0i = i χab (ℓ) , e 2 2 (2π)4 z  (5) where a, b are quadrispinor indices, one can write the amplitude T as, T = ε1µ ε2ν εµανβ Iαβ , (6) with Iαβ = −i Z d4 ℓ 42 (2π) (q1 − ℓ)α X j 1 2 (q1 − ℓ) − m2j + iǫ   e2j (γβ γ5 )ba i χjab (ℓ) . (7) Since the symmetric part of Iαβ doesn’t give contribution to T , the attention is focused in the antisymmetric part. With the momenta q1 and q2 , one can build two antisymmetric tensors, q1α q2β − q1β q2α and εαβρσ q1ρ q2σ . Nevertheless, there is not enough structure in the integrand of Eq. (7) to generate a tensor like εαβρσ q1ρ q2σ , at least at the leading order. Therefore, the tensor structure of Iαβ is Iαβ = 1 (q1α q2β − q1β q2α ) I + symmetric part . 2 (8) Turning back to the Eqs. (2) and (6), and using Eq. (8), one gets Fγγ ∗ P = −I. Contracting (8) with q1α q2β − q1β q2α and using the explicit expression of Iαβ , Eq. (7), yields i Z d4 ℓ X 1 e2 2 + iǫ j (q − ℓ) − m 1 j j   [q1 . (q1 − ℓ) (/q2 γ5 )ba − q2 . (q1 − ℓ) (/q1 γ5 )ba ] i χjab (ℓ) . Fγγ ∗ P = − (q1 .q2 ) 2 42 (2π) 2 (9) This expression for the transition form factor is quite general. At this stage the assumptions made are i) the free quark propagator has been used in going from Eq. (3) to Eq. (4) and ii) the corrections to the elctromagnetic vertex −1 has not been considered. A more general expression could be obtained by changing the free propagator, (/t − m + iǫ) , by the general one associated to a dressed quark, (A (t) /t − B (t) + iǫ)−1 , studied in actual lattice QCD calculations [44] and including a term with the neglected structure, εαβρσ q1ρ q2σ . Looking at the kinematics of the process, one can choose the reference frame in such a way that the pion and photons four momenta are k = (Ek , kx , 0, kz ) , q1 = (E1 , 0, 0, −E1 ) , q2 = (Ek − E1 , kx , 0, E1 +√kz ) , respectively. It is interesting to express the quantities in terms of the light-front variables, k ± = (Ek ± kz ) / 2, ~k⊥ = (kx , 0) ,

2 q1− = Q2 + m2P /2k + , q1+ = 0, ~q1⊥ = 0, q2+ = k + , q2− = − Q2 − k⊥ /2k + , ~q2⊥ = ~k⊥ , where Q2 = −q22 . In the limit 2 of large Q , some of the quantities in Eq. (9) can be approximated by /q1 ≃ − /q2 ≃ Q2 + γ , 2k + 1 2 Q , 2 ℓ+ (q1 − ℓ)2 − m2j + iǫ ≃ − + Q2 , k q1 .q2 ≃ giving for the transition form factor the expression    Z  1 d4 ℓ X 1 2 1 + j Fγγ ∗ P ≃ − 2 i 2 e γ γ i χ (ℓ) . 5 + j ab 4 ℓ Q k+ (2π) ba j k+ Finally, defining x = ℓ+ k+ , one arrives to the usual expression Z
dx 1 2 ΦP x, Q2 , Fγγ ∗ P (Q ) ≃ 2 Q x (10) (11) (12) (13) (14) 4 with
ΦP x, Q2 = −i Z = −2i dℓ− d2 ℓ⊥ 4 (2π) Z 2 X j e2j γ + γ5
ba   i χjab (ℓ) z  z X dz − iz− k+ (x− 12 ) e2j ψ̄j hP (k) out| γ + γ5 ψj − |0i e 2π 2 2 j . (15) z + =0,~ z⊥ =0 As will be discussed later, in
Eq. (14), besides the explicit Q2 dependence, also an implicit one appears, through the QCD evolution of ΦP x, Q2 . In the SU(3) formalism, the quark operator has the form z  z z   z X e2j ψ̄j γ + γ5 ψj − = ψ̄ γ + γ5 Oψ − , (16) 2 2 2 2 j √ √ with O = 2 λ0 /(3 6)+λ3 /6+λ8 /(6 3) where λa are the SU(3) generators. In the present case it is more interesting to √ use the flavor basis in describing the η particle (see the Appendix). In this basis, one has O = 5 λq /18+λs /(9 2)+λ3 /6. As usual, the DA of P in the flavor basis is defined as Z  z z  dz − iz− k+ (x− 12 ) λj j j 2 hP (k)| ψ̄j ifP φP (x, Q ) = − γ + γ5 √ ψj − |0i , (17) e 2π 2 2 2 z + =0,~ z⊥ =0 with j = 3, q, s. This yields √ 5√ q q 2 3 3 2 fP φP (x, Q2 ) + (18) ΦP (x, Q ) = 2fP φP (x, Q2 ) + fPs φsP (x, Q2 ) . 3 9 9 √ In the pion case, this equation corresponds to Φπ (x) = 2 fπ φπ (x) /3, where φπ (x) is the πDA and fπ = 0.131 GeV is the pion decay constant. One should notice that, in going from Eq. (1) to the final result Eq. (14), a few approximations have been done: the free expression has been used for the quark propagator, with the additional simplification given by Eq. (12); besides, the approximations Eqs. (10) and (11) have been applied in the numerator of Eq. (9) and a new tensor structure in Iαβ has been neglected. Some of these corrections have a kinematic character, while others are certainly dynamical. Both type of corrections imply the presence of higher twist distribution amplitudes. In Ref. [8] it has been argued that the approximations, leading to the simple expression Eq. (14) for the transition form factor, are too crude to explain the BaBar experimental data, and corrections at the next order in the Q−2 expansion have been added. The simplest way to implement these corrections is to start from the following expression: Z 1 dx C3 Q2 Fγγ ∗ P (Q2 ) = Φ (x, Q2 ) + 2 . (19) M2 P Q x + 0 Q2 2 The mass M in Eq. (19) was introduced by Radyushkin [3], to cure the divergence of the integrand in Eq. (14),
occurring when a DA ΦP x, Q20 , not vanishing at x = 0, 1, is used. This was justified as a consequence of the existence of some transverse component in the quark momentum. As it has been shown in the previous section, M contains not only effects associated to the mean transverse momentum, but also the ones associated to the constituent quark masses, among others. In Ref. [8] it has been shown that it is necessary to introduce the C3 -dependent term in Eq. (19), otherwise the data cannot be well described around the region of Q2 = 10 − 20 GeV2 . The inclusion of this term has been thoroughly motivated in this section, where it has been shown that the perturbative approach leading to Eq. (14) is correct only for high enough values of the virtuality. We call the C3 term as “the higher twist term”, although it is clear that also the mass term, M, is of the same order. III. THE η TRANSITION FORM FACTOR In this section, we evaluate the ηT F F . To this aim, according to Eq. (19), in calculating Fγγ ∗ η (Q2 ) the ηDA is needed. From Eq.(18), the ηDA is expressed by
5√ q q

2 2fη φη x, Q2 + fηs φsη x, Q2 . Φη x, Q2 = 9 9 (20) 5 In the calculation, we use the following values of the η weak decay constants fηq = (0.828 ± 0.019)fπ , fηs = (−0.848 ± 0.042)fπ , (21) with fπ = 131 MeV, obtained in the phenomenological study of Ref. [37]. Now,it is necessary to calculate the ηDA at some initial scale Q20 within a model. To this end, we have obtained the DAs corresponding to the q and s flavors within the Nambu-Jona Lasinio (NJL) model, which has a long tradition of successful predictions of meson parton structure [31–35]. In particular, we use in the present calculation the three quark flavor version of the model with the Pauli-Villars regularization [38, 45]. A brief summary of the model and of the regularization procedure is given in the Appendix. In the NJL model, mesons are described through Bethe-Salpeter amplitudes. For the η meson one has: Z d4 ℓ iℓ(x1 −x2 ) ik(x1 +x2 )/2 hη (k)| ψβ (x2 ) ψ̄α (x1 ) |0i = e 4e (2π)      k k iSF ℓ − igηqq (cos φ λq − sin φ λs ) iγ5 iSF ℓ + (22) 2 2 βα where the index α, β stand for color, flavour and quadrispinor index. Inserting this expression in Eq. (17) one obtains √ ifηj φjη (x) = 2gηqq (cos φ δj,q − sin φ δj,s ) Nc          Z k k d4 ℓ 1 + + + . (23) tr SF ℓ − γ5 S F ℓ + γ γ5 x− 4δ ℓ −k 2 2 2 (2π) Evaluating the trace and using Eq. (A18) of the Appendix, one has φηq(s) (x, Q20 ) = 1 I˜η,q(s) (x, mq(s) , m2η ) , I2 (mq(s) , mq(s) , m2η ) (24)
where I˜2 x, mi , m2P is given in Eq. (A23) and I2 (mq(s) , mq(s) , m2η ) is the two-propagator integral defined in Eq. (A2). The flavor DA defined in Eq. (24) satisfies the normalization condition: Z dxφηq(s) (x, Q20 ) = 1 . (25) For Q2 > Q20 , the ηT F F is obtained through QCD evolution [14, 46, 47]. The ηDA can be expressed in terms of the Gegenbauer polynomials, 2 φq,s η (x, Q ) = 6 x (1 − x) ∞ X n(even)=0 where γn is the anomalous dimension CF γn = β 1+4 n+1 X k=2  3/2  aq,s n Cn (2x − 1) 1 2 − k (n + 1)(n + 2) 2 log Λ2Q log ! QCD Q20 Λ2QCD −γn  , , (26) (27) N 2 −1 2N C C β = 11N − 3 f is the beta function to lowest order and CF = 2N . If the aq,s are known, using Eq. n coefficients 3 C
2 2 q,s 2 (26) in Eq. (20) and (19), the ηT F F is obtained for any Q ≫ ΛQCD . Once φη x, Q0 are known at a given scale Q20 , the aq,s n coefficients are obtained using the orthogonality property of the Gegenbauer polynomials aq,s n = 2 2n + 3 3 (n + 1) (n + 2) Z 0 1 dx Cn3/2 (2x − 1) φq,s x, Q20 η
. (28) In this scheme, the η meson cannot couple to two gluons. This should not be a serious drawback of the approach, having the η essentially an octet character under SU (3) transformations. 6 2.0 2 q (x) Q q (x) Q =50 GeV =1. GeV 0 2 Asymptotic 6 2 2 x (1-x) 1.0 q (x) 1.5 0.5 0.0 0.0 0.5 1.0 x s 2.0 s (x) Q 2 0 =1 GeV 2 2 Q =50 GeV Asymptotic 1.5 6 2 x (1-x) 1.0 s (x ) (x) 0.5 0.0 0.0 0.5 1.0 x FIG. 1: The DA for the q (upper panel) and s (lower panel) flavor in the η meson, at the initial scale Q20 = 1 GeV2 (dotted line) and after evolution to the scale Q2 = 50 GeV2 (full line). The asymptotic behavior is also shown for comparison (dashed line). We fix now the values of Q0 , C3 and M. The Q0 scale is closely related to the choice of the ΛQCD value. We fix a scale of Q0 = 1 GeV, together with ΛQCD = 0.226 GeV, in analogy with the previous analysis [8]. A natural condition to be satisfied is continuity between the low virtuality description of the ηT F F and the high virtuality description, provided by Eq (19). To minimize the model dependence, we use the parameterization of the CLEO collaboration [36] for the description of the TFF in the LV region:  
1 F (0) 64πΓ(η → γγ) 2 LV = Fγγ ∗ η Q = (29) 2 , (4πα)2 m3η 1 − Λt2 1+ Q Λ2 η η with F (0) = 0.272 ± 0.007 GeV−1 , obtained using Γ(η → γγ) = 0.510 ± 0.026 10−3 MeV as given by the Particle Data Group [48] together with mη = 547.85 MeV, and Λη = 774 ± 29 MeV [36]. The value of the mass M can be obtained equating the ηT F F given by Eq. (19) at Q2 = Q20 , using as Φη (x, Q20 ) the one provided by the NJL model, to the value given, at the same scale, by the monopole parametrization Eq. (19), Z 1
Q20 F (0) C3 dx 2 LV (30) Φ (x, Q20 ) + 2 . = = Q20 Fγγ ∗ η Q0 M2 η Q20 Q x + 0 0 1 + Λ2 Q20 η Finally, the only unknown is C3 , for which several reasonable values have been used, as discussed in the following section. IV. RESULTS AND DISCUSSION We present now our results of the calculation of the ηT F F . The starting point is the ηDA, evaluated at the low energy scale of the model. According to Eq. (24), all we need is the value of the η mass, the quark masses and 0.2 ( 2 Q) (GeV) 7 Q F 0.1 CLEO direct 0 2 CLEO 3 CLEO + - 0 BABAR 0.0 0 10 20 Q 2 30 40 50 2 (GeV ) 0.2 0.1 CLEO direct 2 Q F ( 2 Q) (GeV) FIG. 2: Calculation of the transition form factor via the ηDA with M = 0.557 GeV, C3 = 2.04 10−2 GeV3 and using Q20 = 1 GeV2 (full-line) compared with the available experimental data [2, 36]. The gray region describes the indeterminacy on C3 . The dashed line represents the result obtained taking C3 = 0. The dotted line corresponds to the asymptotic value for the form factor. CLEO 3 CLEO + 0 - 0 BABAR 0.0 0 2 4 Q 2 6 8 2 (GeV ) FIG. 3: The same as in Fig 2, but in the low virtuality region 10 8 Set I Set II Exp. µs ms hs̄si1/3 171.2 430 −186 184.2 435 −184 −194 mK 497 515 495 mη 541 554 548 mη′ fK /fπ 1157 1.07 1148 1.07 958 1.18 fηq /fπ 0.603 0.832 0.828 fηs /fπ −0.664 −0.840 −0.848 TABLE I: We show results for two different parametrizations of the NJL model for several physical quantities, together with their experimental or phenomenological values. Explicit expressions for these are reported in the Appendix. The masses and the quark condensate are given in MeV. the regularization parameter Λ. We use for the η mass the experimental value, mη = 548 MeV . The quark masses mq(s) and the regularization parameter Λ have to be fixed within the NJL model. It is important to work in the Pauli-Villars regularization scheme, in order to preserve gauge invariance. Unfortunately, to our knowledge, all the available fits for the NJL model in SU (3) are done within the cutoff regularization scheme. The only exception is the paper by Bernard and Vautherin [45], where anyway an approximate expression for the I2 (mi , mj, q 2 ) integral is used. Therefore, we have performed a new fit of the model parameters. The SU(3) NJL model gives a very good description for the meson properties [49], but one has to be careful, since it does not include confinement. To avoid problems, we impose a value of mu > mη /2. The details of the model (whose Lagrangian is given by Eq. (A1)) are given in the Appendix. Here it is worth to recall only that the model has five parameters, which can be chosen as the current quark masses, µu and µs , the dressed quark masses, mu and ms , and the cutoff parameter, Λ. Our strategy for the fits has been: i) the mu mass has been fixed to 275 MeV; ii) µu and Λ are determined by fitting mπ and fπ ; iii) µs and ms are chosen looking for an overall good description of the strange sector. In table I two different sets of the relevant quantities, obtained by the above described fitting procedure, are reported. It is seen that their experimental values are reproduced very well. In the Set I, we have imposed the additional condition for the resulting eta mass: mη ≤ 2mu . In this set, a very good description of masses in the strange sector is obtained, but paying the price of a worse description of fηq,s . In Set II, the description of the masses is slightly worse, but the fηq,s are very well reproduced. One should notice that, in going from SU (2) to SU (3) within the NJL model, the number of parameters moves from 3 to 5. One has therefore at hand two more parameters for explaining three new masses, three new decay constants and a new quark condensate. Actually, for calculating the ηDA only mu and Λ, which have the same value in both sets, are needed, together with ms , which changes from 430 MeV in Set I to 435 MeV in Set II. With respect to these quantities, the predictions of the model are therefore reasonably stable. The results for the ηDA are presented for the following values of the parameters: Λ = 740 MeV, mu = 275 MeV and ms = 435 MeV . It may be useful to reiterate that the obtained values of fηq,s are not relevant for the present calculation, because we used the experimental ones.
In Fig. 1, the ηDA are shown. We observe that φqη x, Q20 is peaked around the central point x = 0.5 while φsη (x) is relatively flat. This is a consequence of the masses of quarks u and d, which are close to half the mass of the eta, while it is not the case for the mass of the strange quark. What is clearly seen is the following reasonable feature: the less bound is a system, the more narrow is its DA around the point x = 0.5. Our DAs have an infinite expansion in terms of the Gegenbauer polynomials. The firsts coefficients aq,s η , defined in Eq. (28), at Q20 = 1 GeV2 are: aq2 = 0.134 aq4 = 0.352 as2 = 0.377 as4 = 0.245 (31) The asη coefficients are close to the values predicted by a flat distribution. On the other hand side, we observe that aq2 < aq4 , at variance with what is commonly used in the field. This feature is due to the narrow structure of φqη (x) . We can compare our results with the values used by other authors. The aq2 and as2 coefficients are to be compared to the parameter B = 0.3 used in Ref. [24]. In Ref. [11] the values a12 = −0.06 ± 0.06, a82 = −0.07 ± 0.04 and aπ2 = 0.22 ± 0.06 are given, but at Q2 = 4 GeV2 . From Eq. (31) and using φ1η = (5 φqη + φsη )/6 and φ8η = (5 φqη − 2 φsη )/3 we obtain

(32) a82 1 GeV2 = −0.028 . a12 1 GeV2 = 0.17 Evolving these results we have
a12 4 GeV2 = 0.14
a82 4 GeV2 = −0.022 . (33) The value for a82 is therefore consistent with that used in Ref. [11], while some difference
is found for the value of a12 . In the pion case, if we use a flat distribution at Q20 = 1 GeV2 , a value aπ2 4 GeV2 = 0.31 is obtained. In Ref. 9 [25] it has been noted that the values for these parameters found in [11] suggest a very large SU(3) breaking between the DA of the π 0 and the one of η 8 , and a very little U(1) symmetry breaking between η 8 and η 1 . Our results show the same structure of those of Ref [11], at least for a82 and for the big difference between aπ2 and a82 . The origin of this difference is in the small value of aq2 due to the narrow structure of φqη (x), originated by the fact that mu is close to mη /2. At the same time, a small value of aq2 explains a small value for a12 . One should remember that the present scheme reproduces the SU(3)F and U(3)F symmetry breaking in the pseudoscalar meson sector. The results of Eq. (32) are also in good agreement with those from [23, 50]. In the latter references, the authors give their results for Q20
= 1 GeV2 in terms of the quantities
B2q and B2g ,defined in [50] and related to our expressions as follows : a12 1 GeV2 = −B2q + B2g /102 and a82 1 GeV2 = −B2q . Using the numerical results of Ref [23], we

have a12 1 GeV2 = 0.149 ± 0.048 and a82 1 GeV2 = −0.0425 ± 0.0175, to be compared to our results, Eq. (32). In these papers, the coupling of a two gluon state to the singlet q̄q component of the η mesons is introduced explicitely, providing a contribution, B2g , which is an important part of the final result. In absence of gluons, the symmetry U (1) is not broken. In our case, the U (1) symmetry is broken through the ’t Hooft interaction term [51] introduced in the Lagrangian (A1), making our results consistent with those of refs. [23, 50]. As stated at the end of the previous section, once the ηDA has been obtained at the scale Q20 and evolved to Q2 according to Eqs. (26)-(28), the only remaining unknown for the evaluation of the ηT F F according to Eq. (19) is the constant C3 of the higher twist term. To this aim, three different scenarios have been considered, corresponding to a contribution from this term to the form factor at Q20 = 1 GeV2 of 10% (C3 = 1.02 · 10−2 GeV3 ), 20% (C3 = 2.04 · 10−2 GeV3 ) and 30% (C3 = 3.06 · 10−2 GeV3 ). The cutoff parameter M varies between 487 MeV, for C3 = 1.02 · 10−2 GeV3 , 557 Mev, for C3 = 2.04 · 10−2 GeV3 , and 638 MeV, C3 = 3.06 · 10−2 GeV3 . We show in Fig 2 the obtained result for ηT F F and in Fig. 3 a detail of the region between Q2 = 0 and Q2 = 10 GeV2 . The results for the ηT F F , shown in Figs. 2 and 3, exhibit a very good description of the experimental data in the whole kinematic region. For
completeness we have included in the figures the C3 = 0 case and the asymptotic √ vale for the ηT F F, Q2 Fγγ ∗ η Q2 = (5 2fηq + 2fηs ) = 0.181 GeV . It may be interesting to notice that the value √
of the asymptotic ηT F F is very close to the value of the asymptotic πT F F, Q2 Fγγ ∗ π Q2 = 2fπ = 0.185 GeV . It is also clear that, at variance with what happens in the π case, to explain the eta data some C3 contribution may be needed only in the region around Q2 = 20 − 30 GeV2 . Anyway, a complete discussion is obtained only by comparing the present results for the πT F F with those of Ref. [8]. In the π case, the C3 contribution was crucial to reproduce the data in the region Q2 = 10 − 20 GeV2 and the calculated πT F F crossed the asymptotic curve quite early (around Q2 = 10 GeV2 ) and with a significative slope. In the η case, the situation is less dramatic: the higher twist term improves the TFF description only slowly, and the theoretical result crosses softly the asymptotic value around Q2 = 40 GeV2 . Another interesting point is the stability of the parameters. In calculating the ηT F F , we have adopted a procedure independent from that used in Ref. [8], namely, C3 and M have been fitted using the η data only. The parameters used in both calculations have been ΛQCD = 0.226 GeV and Q0 = 1 GeV . Otherwise, in Ref [8], a fully model independent calculation was performed, choosing φπ (x) = 1 on the basis of chiral symmetry. Here one is forced to choose a model for the description of the ηDA at Q20 , and C3 and M have been fixed within this model, independently from the π case. Despite of this, the result obtained in the two calculations are quite consistent. Varying the weight of the higher twist term from 10% to 30% produces a change in C3 from 1.02 · 10−2 GeV3 to 3.06 · 10−2 GeV3 in the η case, to be compared to a variation of C3 from 0.99 · 10−2 GeV3 to 2.98 · 10−2 GeV3 in the π case. The agreement is impressive. On the other hand side, we found for the mass parameter a wider variation. In the η case one gets M = 560 ± 70 MeV,taking into account the uncertainty in C3 , to be compared to M = 620 ± 70 MeV for the π case. Despite of these differences, the results can be considered perfectly consistent with each other. The difference in the central value of M could imply that, for the pion, a larger contribution from the transverse momentum is expected with respect to that for the eta particle. It is indeed what has been obtained in Refs. [11, 24]. The values of M could be compared with the value of hk⊥ i given by P. Kroll [11], hk⊥ i ≃ 710 MeV for the pion and hk⊥ i ≃ 390 − 440 MeV for the eta. It can be also compared with the βπ parameter used by [24], which is related with the width of the gaussian distribution of transverse momentum used by these authors, with the values βπ = 668 MeV for the u-quark and βπ ≃ 530 MeV for the s-quark. A comparison of our parameters, based on a quark-flavor decomposition of the relevant quantities (DAs, decay constants), with those used in Ref. [22], obtained within a singlet-octet decomposition, is instead rather involved. The spirit of the present calculation and those of Refs. [22, 24] are rather different. In our calculation, the known QCD evolution of the DA governs the Q2 dependence of the form factor. The same Q2 dependence is obtained, in Refs. [22, 24], through the kT dependence assumed for the light-cone wave function of the mesons. It is therefore significant that the two approaches provide similar results, describing probably, using different tools, a similar mechanism. In the present discussion, the result on γ ∗ (q) → P γ, reported by BaBar for a time like q 2 = 112 GeV2 ,
2 2 q Fγγ ∗ η q = 0.229 ± 0.030 ± 0.008 GeV, has not been included. The reason is that the kinematics and dynam- 10 ics of this process is different from the ones studied here. There is no symmetry relating Fγγ ∗ P (q 2 ) at one point q 2 to Fγγ ∗ P (Q2 ) in the point Q2 =
−q 2√ . The coincidence
is in the asymptotic value, which has been predicted for
this process to be[40] −q 2 Fγγ ∗ P q 2 = 2fP (1 − 5 αs q 2 /3π), when the contribution coming from the αs q 2 term could be disregarded. In the present scheme we obtain Q2 Fγγ ∗ P (Q2 ) = 0.19 GeV at Q2 = 112 GeV2 , which implies a very slowly growing behavior of the T F F even for these high values of the virtuality. In closing this section, it is useful to list items that prevent from using the same formalism for the description of the η ′ T F F . First of all, as it has been previously noted, the NJL does not include confinement. Therefore, if one uses the same expression, Eq. (24), in the η ′ case, an imaginary part will appear in the DA at some value of x. Secondly, the η ′ is basically a singlet state and it can mix strongly with the two gluons state or, later, with some cc̄ component. These ingredients are not included in the present formalism. V. CONCLUSIONS In this paper, the ηT F F has been discussed in a formalism which connects a low energy description of the hadron involved with a high energy description based on a QCD perturbative formulation. The two descriptions are matched at some scale Q20 . The scheme has been applied to describe the parton and generalized parton distributions with notable success[28–30, 32, 35] and, in particular, to the πT F F in [8]. The formalism selects therefore two regions of virtuality, separated at Q20 . For Q2 < Q20 , use has been made of the experimental parametrization of the ηT F F data. This has been done to avoid model dependence in this region. At Q2 > Q20 , use has been made of a high virtuality description, which incorporates the following important physical ingredients: i) a ηDA obtained in the NJL model ; ii) a mass cut-off in the definition of the ηT F F from the ηDA, M, [3] which, interpreted from the point of view of constituent models, takes into account the constituent mass, transverse momentum effects and also higher twist effects; iii) an additional higher twist term into the definition of the ηT F F in the high virtuality description, parameterized by a unique constant, C3 ; iv) the two descriptions have to match at a virtuality Q20 , a scale which is universal and should be the same for all observables. In section II it has been shown that the dominant, twist two, expression for the pseudoscalar T F F, given in Eq. (14) has to be corrected, for including higher twist effects. The minimal correction would be the one given in Eq. (19). The ηDA at Q20 has been obtained in the NJL model. For that, the parameters have been adjusted for a good reproduction of the η sector with the Pauli-Villars regularization. The obtained fits represent an overall good description of the strange sector, not only for the masses, but also for the meson decay constants. It is worth to strees that, in going from SU(2) to SU(3), the number of parameters is increased by two, while the number of new physical quantities, included in Table I, are seven. The obtained DAs in this model show consystency with other analyses,where the DAs are parametrized [11, 23, 24]. Using Q0 = 1 GeV as matching point, the higher virtuality results of the ηT F F are well reproduced. The C3 term turns out to be relatively small. Its effect is to reduce the value of the contribution to the twist two ηT F F only for Q2 < 5 GeV2 . Its value, C3 = 2.04 · 10−2 GeV3 for a 20% of higher twist contamination at Q20 , is in perfect agreement with the one obtained for the πT F F in Ref. [8], C3 = 1.98 · 10−2 GeV3 . Moreover, the results are very stable with respect to variations of this parameter. The value obtained for M = 560 MeV is comparable with that of the pion case in Ref [8], M = 620 MeV . The relative high value of M in both cases can be understood thinking that it includes the constituent quark mass, the mean value of the transverse quark momentum and other higher twist contributions. In turn, the higher value of M for the π than for the η can be related to the fact that the contribution of the quark transverse momentum in the π case is expected to be more important with respect to the η case [11, 24]. The calculation proves that all the BaBar results can be accommodated in the present scheme, which only uses standard QCD ingredients and low virtuality data. It must be emphasized that, in order to have a good description for both π and η, higher twist effects are important, as the modification from Eq. (14) to Eq. (19) signals. It must be also noted that the matching scale is as high as 1 GeV, a feature already found in the description of parton distributions when precision was to be attained. With these ingredients, the calculation shows an excellent agreement with the data. Let us conclude by stressing that we have justified the formalism developed in Ref. [8] to describe the πT F F and we have extended it to the ηT F F. The idea of the approach is that one can use models or effective theories to describe the non perturbative sector, and QCD to describe the perturbative one. In here, we have preferred to use data for the low virtuality sector to avoid model dependence, but in building the ηDA at Q20 we have used the NJL model. Higher twist effects (parametrized in our case by M and C3 ) are small but crucial in order to attain an excellent description of the π and η experimental results. 11 Appendix A: The SU (3) NJL model for pseudoscalars mesons. In calculating the ηDA, the minimal extension of the NJL model for describing pseudoscalar mesons in SU(3), proposed in Ref [52], has been used: L = q̄ (x) (i/ ∂ − µ) q (x) + G 8 h i X (q̄λa q)2 + (q̄iγ5 λa q)2 − a=0 − K [det (q̄ (1 + γ5 ) q) + det (q̄ (1 − γ5 ) q)] , (A1) where µ = diag[µu , µd , µs ] is the matrix of the current quark masses and λa , a = 0, ..., 8, are the SU(3) generators. SU(2) will be considered a good symmetry, and, therefore, µu = µd . As it is well known, the first consequence of the scalar interaction term is to provide the constituent quark masses, mu = md , ms , different from the current ones. The main results are summarized here, while the reader is referred to the section IV-B of Ref. [38] for details. By defining the integrals: Z d4 p 1 I1 (m) = i 4 (p2 − m2 + iǫ) , (2π) Z
d4 p 1 i , h I2 mi , mj , q 2 = i (A2) 4 2 2 (2π) (p − mi + iǫ) (p − q)2 − m2j + iǫ the constituent masses are given by mu = µu + 16 G Nc mu I1 (mu ) + 32 K Nc2 mu I1 (mu ) ms I1 (ms ) , 2 ms = µs + 16 G Nc ms I1 (ms ) + 32 K Nc2 [mu I1 (mu )] , (A3) where Nc is the number of colors. The vacuum expectation values for the condesates of the quarks of flavor qi are hq̄i qi i = 4 mi I1 (mi ) − 4 µi I1 (µi ) , (A4) where the expectation value of the q̄q in the perturbative vacuum has been substracted from the expectation value in the true vacuum. The last term in Eq. (A4) is negligible in the u − d quark sector, while it becomes important in the strange sector. The next step is the description of the pseudoscalar states. For the pion and kaon case, by defining the quantities K3 = G + 2 Nc K ms I1 (ms ) , K6 = G + 2 Nc K mu I1 (mu ) , (A5) and  

2 D mi , mj , q 2 = I1 (mi ) + I1 (mj ) + (mi − mj ) − q 2 I2 mi , mj , q 2 , one has that the pion and kaon masses are obtained solving the equations:
1 − 8 Nc K3 D mu , mu , m2π = 0 ,
1 − 8 Nc K6 D mu , ms , m2K = 0 . (A6) (A7) The couplings of the pion and the kaon to the quarks are given by:  
−1 d 2 2 gπqq = 4 Nc 2 D mu , mu , q , dq q2 =m2 π 2 gKqq  
−1 d 2 = 4 N c 2 D mu , ms , q , dq q2 =m2 (A8) K and the decay constants are: Fπ = −4 Nc gπqq mu I2 mu , mu , m2π
, (A9) 12 FK =
2 Nc gKqq  (ms + mu ) D mu , ms , m2K − 2 mu I1 (mu ) − 2 ms I1 (ms ) 2 mK , (A10) √ where Fπ,K = fπ,K / 2. The η particle deserves a more careful discussion, due to its mixing with the η ′ particle. Working in the flavor basis, one can define Kq = G − 2 Nc K ms I1 (ms ) , √ Kqs = −2 2 Nc K mu I1 (mu ) , K = Kq G − 2 Kqs (A11) . The interaction in the η − η ′ sector can be described by the expression

iγ5 λi [Mij ] iγ5 λj √ with i, j = q, s, λq = diag[1, 1, 0], λs = diag[0, 0, 2] and the interaction matrix is given by ! 1 aqq aqs , [Mij ] = Dη aqs ass (A12) (A13) with

aqq = 2 Kq − 8 K Nc D ms , ms , q 2 ,

ass = 2 G − 8 K Nc D mu , mu , q 2 , Dη aqs = 2 Kqs ,
q 2 = (aqq ass − a2qs )/(4K) . (A14) The η mass is obtained solving the equation
Dη m2η = 0 . (A15) In a neighborhood of q 2 = m2η the interaction can be written as (− sin φ λs + cos φ λq ) = −gηqq (− sin φ λs + cos φ λq ) − m2η q2 aqq (ǫη λs + λq ) (ǫη λs + λq ) , Dη (A16) with ǫη = aqs /aqq . In obtaining the right hand side of this equation, use has been made of Eq. (A15), which implies ass = a2qs /aqq . From (A16) one has 1 , cos φ = q 1 + ǫ2η 2 gηqq −ǫη sin φ = q , 1 + ǫ2η
1 + ǫ2η aqq =− dDη /dq 2 . (A17) q2 =m2η For the flavor decay constants, one has
Fηq = −12 gηqq cos φ mu I2 mu , mu , m2η ,
Fηs = 12 gηqq sin φ ms I2 ms , ms , m2η , (A18) √ where Fηq,s = fηq,s / 2. We need to evaluate the integrals defined in Eq. (A2). Due to the point-like character of the interaction, the lagrangian Eq. (A1) is not renormalizable and a regularization procedure for these integrals has to be defined. We 13 use the Pauli-Villars regularization in order to render the occurring integrals finite. This means that, for integrals like the ones defined in Eq. (A2), we make the following replacement, I1 (mi ) −→ 2 X 2 X

cℓ I2 Mℓ,i , Mℓ,j , q 2 I2 mi , mj , q 2 −→ cℓ I1 (Mℓ,i ) (A19) ℓ=0 ℓ=0 2 with Mℓ,j = m2j + ℓ Λ2 , c0 = c2 = 1, c1 = −2. Here, for simplicity, we choose the same Λ value for the strange and the nonstrange sector. According to these prescriptions one finds # " 2 2 M1,i M2,i 1 2 2 , (A20) −2 M1,i ln 2 + M2,i ln 2 I1 (mi ) = 16 π 2 mi mi I2 mi , mj , q with Φℓ = 2 q2 2
2 1 X = cℓ 32π 2 ℓ=0 " 2 2 M1,j M1,i ln 2 + ln mi m2j ! 2 2 2 Mℓ,j M1,j − M1,i + ln 2 + Φℓ q2 Mℓ,i # ,   2 2 2 2 2 2  q + M1,j − M1,i q − M1,j + M1,i  2 , M 2 , q 2 arctan r  −λ Mℓ,i ℓ,j    + arctan r   2 , M 2 , q2 2 , M 2 , q2 −λ Mℓ,i −λ Mℓ,i ℓ,j ℓ,j r  (A21) (A22)   2 2 where λ Mℓ,i , Mℓ,j , q 2 is the Källén lambda. Now, we fix the parameters of the model. Looking at the lagrangian, we have a five parameters model, µu , µs , G, K and Λ. Nevertheless, it is more intuitive to organize the fit of the parameters in terms of µu , µs , mu , ms and Λ, using equations (A3) to determine G and K. We impose mu = 275 MeV, in order to have mexp < 2mu . Then, µu and η Λ are obtained in recovering the values of Fπ and mπ . At this step one has µu = 6.69 MeV, mu = 275 MeV, Λ = 740 MeV −→ 3 mπ = 138 MeV, Fπ = 92.2 MeV, hūui = (−227 MeV) determining the SU(2) sector. Then, µs and ms have been fixed by requiring a good overall fit of masses (mK , mη , mη′ ) and decay constants (FK , Fηq , Fηs ). In table I two different sets of parameters are given, together with the obtained results. Using set I, by imposing mη ≤ 2mu , a good agreement for the masses and a slightly less good agreement for the Fηq,s is obtained. On the other hand, in set II a very good agreement for Fηq,s is obtained, with a slightly worse result for the masses. In the light-front calculation one needs the integral   + Z 4 δ x − 1 + Pk +
d k I˜2 x, mi , m2P = i (2π)4 [(k − P )2 − m2i + iǫ] (k 2 − m2i + iǫ) = −θ(x)θ(1 − x) 2 1 X m2i − (1 − x) x m2P c ln ℓ 2 − (1 − x) x m2 (4π)2 Mℓ,i P . (A23) ℓ=1 Clearly, all one needs to calculate the ηDA are the η and quark masses and the value of the cutoff parameter. For the DA calculations, the values mu = 275 MeV, ms = 435 MeV, Λ = 740 MeV and mη = 548 MeV have been chosen. Acknowledgements This work was supported in part by HadronPhysics3, a FP7-Infrastructures-2011-1 Program of the European Commission under Grant 283288, by the MICINN (Spain) grant FPA2010-21750-C02-01, by Generalitat Valenciana, grant Prometeo2009/129, and by “Partonic structure of nucleons, mesons and light nuclei”, an INFN (Italy, Perugia) - MICINN (Spain, Valencia) exchange agreement. [1] B. Aubert et al. [The BABAR Collaboration], Phys. Rev. D 80 (2009) 052002 [arXiv:0905.4778 [hep-ex]]. 14 [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] P. del Amo Sanchez et al. [ BABAR Collaboration ], Phys. Rev. D84 (2011) 052001. [arXiv:1101.1142 [hep-ex]]. A. V. Radyushkin, Phys. Rev. D 80 (2009) 094009 [arXiv:0906.0323 [hep-ph]]. M. V. Polyakov, JETP Lett. 90 (2009) 228 [arXiv:0906.0538 [hep-ph]]. S. V. Mikhailov and N. G. Stefanis, Mod. Phys. Lett. A 24 (2009) 2858 [arXiv:0910.3498 [hep-ph]]. A. E. Dorokhov, Nucl. Phys. Proc. Suppl. 198 (2010) 190-193. [arXiv:0909.5111 [hep-ph]]. N. I. Kochelev, V. Vento, Phys. Rev. D81, 034009 (2010). [arXiv:0912.2172 [hep-ph]]. S. Noguera and V. Vento, Eur. Phys. J. A 46, 197 (2010) [arXiv:1001.3075 [hep-ph]]. E. R. Arriola, W. Broniowski, Phys. Rev. D81, 094021 (2010). [arXiv:1004.0837 [hep-ph]]. O. Leitner, J.-F. Mathiot, N.A. Tsirova, Eur. Phys. J. A47, 17 (2011). [arXiv:1009.5484 [hep-ph]]. P. Kroll, Eur. Phys. J. C71, 1623 (2011). [arXiv:1012.3542 [hep-ph]]. X-G. Wu, T. Huang, Phys. Rev. D82, 034024 (2010). [arXiv:1005.3359 [hep-ph]]. S. S. Agaev, V. M. Braun, N. Offen, F. A. Porkert, Phys. Rev. D83, 054020 (2011). [arXiv:1012.4671 [hep-ph]]. A. V. Efremov and A. V. Radyushkin, Phys. Lett. B 94 (1980) 245. V. L. Chernyak and A. R. Zhitnitsky, Nucl. Phys. B 201 (1982) 492 [Erratum-ibid. B 214 (1983) 547]. L. Del Debbio, Few Body Syst. 36 (2005) 77. V. M. Braun et al., Phys. Rev. D 74 (2006) 074501 [arXiv:hep-lat/0606012]. E. Ruiz Arriola and W. Broniowski, Phys. Rev. D 66 (2002) 094016 [arXiv:hep-ph/0207266]. A. Courtoy and S. Noguera, Phys. Rev. D 76 (2007) 094026 [arXiv:0707.3366 [hep-ph]]. A. Courtoy, Ph. D. Thesis, Valencia University, 2009. http://arxiv.org/abs/arXiv:1010.2974 E. Ruiz Arriola and W. Broniowski, Phys. Rev. D 67 (2003) 074021 [arXiv:hep-ph/0301202]. B. -W. Xiao and B. -Q. Ma, Phys. Rev. D 71 (2005) 014034 [hep-ph/0501160]. S. S. Agaev, Eur. Phys. J. C70, 125-137 (2010). X.-G. Wu, T. Huang, Phys. Rev. D84, 074011 (2011). [arXiv:1106.4365 [hep-ph]]. S. J. Brodsky, F. -G. Cao, G. F. de Teramond, Phys. Rev. D84, 033001 (2011). [arXiv:1104.3364 [hep-ph]]; [arXiv:1105.3999 [hep-ph]]. Y. N. Klopot, A. G. Oganesian and O. V. Teryaev, Phys. Rev. D 84 (2011) 051901 [arXiv:1106.3855 [hep-ph]]. Y. N. Klopot, A. G. Oganesian and O. V. Teryaev, Phys. Lett. B 695 (2011) 130 [arXiv:1009.1120 [hep-ph]]. M. Traini, A. Mair, A. Zambarda and V. Vento, Nucl. Phys. A 614 (1997) 472. S. Scopetta, V. Vento and M. Traini, Phys. Lett. B 421 (1998) 64 [arXiv:hep-ph/9708262]. S. Scopetta, V. Vento, M. Traini, Phys. Lett. B442, 28-37 (1998). [arXiv:hep-ph/9804302 [hep-ph]]. R. M. Davidson and E. Ruiz Arriola, Phys. Lett. B 348 (1995) 163. L. Theussl, S. Noguera and V. Vento, Eur. Phys. J. A 20 (2004) 483 [arXiv:nucl-th/0211036]. S. Noguera and V. Vento, Eur. Phys. J. A 28 (2006) 227 [arXiv:hep-ph/0505102]. W. Broniowski, E. R. Arriola and K. Golec-Biernat, Phys. Rev. D 77 (2008) 034023 [arXiv:0712.1012 [hep-ph]]. A. Courtoy and S. Noguera, Phys. Lett. B 675 (2009) 38 [arXiv:0811.0550 [hep-ph]]. J. Gronberg et al. [CLEO Collaboration], Phys. Rev. D 57 (1998) 33 [arXiv:hep-ex/9707031]. T. Feldmann, Int. J. Mod. Phys. A 15 (2000) 159 [arXiv:hep-ph/9907491]. S. P. Klevansky, Rev. Mod. Phys. 64 (1992) 649. R. M. Davidson, E. Ruiz Arriola, Acta Phys. Polon. B33, 1791-1808 (2002). [hep-ph/0110291]. G. P. Lepage and S. J. Brodsky, Phys. Rev. D 22 (1980) 2157. V. L. Chernyak and A. R. Zhitnitsky, Phys. Rept. 112 (1984) 173. H. J. Behrend et al. [CELLO Collaboration], Z. Phys. C 49 (1991) 401. C. Itzykson and J. B. Zuber, New York, Usa: Mcgraw-hill (1980) 705 P.(International Series In Pure and Applied Physics) S. Furui, H. Nakajima, Phys. Rev. D73, 074503 (2006). V. Bernard, D. Vautherin, Phys. Rev. D40 (1989) 1615. G. P. Lepage and S. J. Brodsky, Phys. Lett. B 87 (1979) 359. D. Mueller, Phys. Rev. D 51 (1995) 3855 [arXiv:hep-ph/9411338]. K. Nakamura et al. [ Particle Data Group Collaboration ], J. Phys. G G37 (2010) 075021. S. Klimt, M. F. M. Lutz, U. Vogl, W. Weise, Nucl. Phys. A516 (1990) 429-468. S. S. Agaev, N. G. Stefanis, Phys. Rev. D70, 054020 (2004). [hep-ph/0307087]. G. ’t Hooft, Phys. Rept. 142, 357-387 (1986). V. Bernard, R. L. Jaffe, U. G. Meissner, Nucl. Phys. B308 (1988) 753.