The ${\gamma^* \gamma^* \to \eta_c (1S,2S)}$ transition form factors for spacelike photons

We derive the light-front wave function (LFWF) representation of the $\gamma^* \gamma^* \to \eta_c(1S)\,,\eta_c(2S)$ transition form factor $F(Q_1^2, Q_2^2)$ for two virtual photons in the initial state. For the LFWF, we use different models obtained from the solution of the Schr\"odinger equation for a variety of $c \bar c$ potentials. We compare our results to the BaBar experimental data for the $\eta_c(1S)$ transition form factor, for one real and one virtual photon. We observe that the onset of the asymptotic behaviour is strongly delayed and discuss applicability of the collinear and/or massless limit. We present some examples of two-dimensional distributions for $F (Q_1^2,Q_2^2)$. A factorization breaking measure is proposed and factorization breaking effects are quantified and shown to be almost model independent. Factorization is shown to be strongly broken, and a scaling of the form factor as a function of $\bar Q^2 = (Q_1^2 + Q_2^2)/2$ is obtained.


I. INTRODUCTION
The description of the hadronic structure in terms of the quark and gluon degrees of freedom is one of the main goals of the Quantum Chromodynamics (QCD). During the last years, our understanding about the partonic distributions has been substantially improved by the experimental data obtained in ep and pp colliders. Complementary information about the internal structure of mesons can be accessed by the study of the electromagnetic form factors and the meson -photon transition form factors. There has been a lot of interest recently in the exclusive production of mesons via photon fusion processes studied mainly at the e + e − colliders [1]. Such studies are strongly motivated by the expectation that at large photon virtualities the measurements of the cross sections will provide strong constrains in the probability amplitude for finding partons in the mesons [2][3][4]. The meson -photon transition form factors are also of interest because of the role they play in the hadronic light-by-light contribution to the muon anomalous magnetic moment [5].
During the last years, a lot of attention has been paid to the case of pseudoscalar light meson -photon transition form factors [6,7], mainly motivated by the experimental data from the CLEO, BaBar, Belle and L3 Collaborations for the π 0 , η and η ′ production in e + e − collisions. These collaborations have extracted the transition form factor from single -tag events where only one of the leptons in the final state is measured. In this case, one of the photons is far off the mass shell, while the other is almost real. Such data have allowed to test the collinear factorization approach and the onset of the asymptotic regime, as well motivated the improvement of the theoretical approaches. Similar results have been obtained for the η c production. In this case, the η c mass provides a hard scale that justifies to use a perturbative approach even for zero virtualities. In the past this transition form factor has been studied in different approaches, (although often only for one virtual photon), such as: perturbative QCD [8,10], lattice QCD [11,12], non-relativistic QCD [13,14], QCD sum rules [15], as well as from Dyson-Schwinger and Bethe-Salpeter equations [16]. In the light-front quark model (LFQM) the case of one virtual and one real photon has been studied in [17,18].
In the present paper we will treat the heavy meson -photon transition form factor, focusing our analysis on the pseudoscalar charmonium state η c (1S) and its radial excitation η c (2S). Here we will focus on calculating transition form factor for both virtual photons, which was not studied so far in the light-front approach. These double virtual transition form factors can be measured in e + e − collisions in the double -tag mode, where both electron and positron are detected in the final state. Recent results for the η ′ production by the BaBar Collaboration [19] have demonstrated that this study is feasible. Our study is motivated by the possibility of an accurate measurement of the double virtual transition form factors considering the high luminosity expected at Belle2. This may open new possibilities, the issue of factorization breaking of the transition form factors, which will be addressed in this paper. Regarding the wave functions of the quarkonia, we wish to use also cc wave functions obtained from realistic potential models. Here we will make use of the solutions obtained in [20]. We shall investigate how well they can describe the recent BaBar data [21] for γγ * → η c (1S). We shall also calculate transition form factors for γγ * → η c (2S), not yet measured, but could be considered for Belle 2 program.
The paper is organized as follows. In Section II, we present the light-front formalism that is used for computations of the γ * γ * → η c form factor. Here, we provide the details of the light front η c (1S, 2S) wave function calculations in the framework of the Schrödinger equation with a chosen set of cc interaction potentials, as well as derive the basic relations for the corresponding amplitude and the transition form factor. In Section III, we discuss the most relevant numerical results on the η c wave functions and the transition form factors for different cc potentials, which are also compared to the existing BaBar data for the η c (1S) state. Section IV summarizes the most important results of our analysis.
Our goal in this Section is to describe the γ * γ * → η c form factor, which can be measured in e + e − collisions using two -photon events in which both photons are far off the mass shell. The typical diagram is represented in Fig. 1. We will consider the light -front formalism, which allows to describe the meson in terms of the quark degrees of freedom. Let us first start from general kinematical considerations.
The amplitude for the photon fusion γ * γ * → η c has the general form dictated by the J P C = 0 −+ quantum numbers of the η c : Here Q 2 i = −q 2 i ≥ 0, i = 1, 2 are the virtualities of photons, which we take both to be space-like. The form factor F (Q 2 1 , Q 2 2 ) above is the object of interest in this paper. It is normalized such that the two-photon decay-width of the meson is obtained from For further calculation it is useful to choose a frame in which incoming photon four-momenta have the form Here and we will denote the transverse vectors by boldface vectors, e.g.
As the polarization vectors of off-shell photons we will choose n + and n − for the first and second photon, respectively. The four-momenta of photons satisfy n + · q 1 = n − · q 2 = 0 1 We finally note that the transverse momentum of the meson is P = q 1 + q 2 , and the photon light-front momenta fulfill Projected onto the photon polarizations, the amplitude takes the simple form where [q 1 , FIG. 1. The η c production by the interaction of two virtual photons in e + e − collisions.

A. Light front wave function of η c
We treat the η c as a bound state of a charm quark and antiquark, thus assuming that the dominant contribution comes from the cc component in the Fock-state expansion: Here the c-quark andc-antiquark carry a fraction z and 1 − z respectively of the η c 's plus-momentum. The light-front helicites of quark and antiquark are denoted by λ,λ, and take values ±1. The transverse momenta of quark and antiquark are p c = k + zP , pc = −k + (1 − z)P . (2.9) The contribution of higher Fock states is expected to be suppressed at large Q 2 . Moreover, the fact that non -relativistic potential models are able to describe the quarkonia properties implies that the valence cc Fock state probability of η c is almost 1 (see e.g. [8], [9]). In Ref. [17], the authors have assumed the η c wave function is a combination of the uū, dd, ss and cc Fock states. The resulting predictions for the η c form factor become dependent on the assumptions for the mixing angles, but are similar to those obtained assuming that the η c is a pure cc state. Therefore, our assumption for the η c wave function as a pure cc state is a very good approximation. The light-front wave function encodes all the necessary information on the bound state. Recently there has been a lot of interest in calculating light-front wave functions of heavy quarkonia, see for example [23][24][25]. Here we follow a different approach which relies on a prescription due to Terentev [26], valid for weakly bound non-relativistic systems, which expresses the light-front wave function in terms of the rest-frame Schrödinger wave function.
In the 2S+1 L J = 1 S 0 state the wave function of the two-body system for canonical spinprojections σσ of quark and antiquark has the form (2.10) Here χ † , χ are the Pauli spinors, and p = | p|. The normalization condition reads In order to apply Terentev's transformation, we first introduce the relative momentum p of quark and antiquark in the center -of -mass frame at fixed invariant mass M cc , so that We then decompose the helicity dependent light-front wave function (LFWF) into a spin/momentum and radial part as (2.14) To obtain the helicity dependent part of the LFWF one needs to transform the rest-frame spinors to the light-front spinors, which is effected by means of a Melosh transform [27]. This has been done e.g. in Ref. [28], and the the helicity-dependent vertex reads Then, if we take the meson state to obey the canonical relativistic normalization the radial light-front wave function φ(z, k) will be normalized as To relate the radial LFWF to the rest-frame wave function, we should still take into account a jacobian from changing the integration measure so that we obtain the identification To lighten up the notation of the amplitude, we also use which we also refer to as the radial wave function.
Let us now present the details of computation of the radial wave function by means of the Schödinger equation with a set of chosen cc interaction potentials. For reviews on these topics, see e.g. Refs. [29,30].

B. Schrödinger equation and cc interaction potentials
The charmonium wave function is found in the quark-antiquark rest frame by solving the Schrödinger equation which for the radial wave function ψ(r) can be written as (for more details see Appendix in Ref. [20]) in terms of the interaction cc potential, V (r). Here, we briefly describe several models for V (r) chosen for our analysis.
• Harmonic oscillator: where for charmonia we adopt For such a simple choice of the interaction potential one finds an analytic solution of the Schrödinger equation (2.22) yielding a Gaussian shape of the wave function.
• Buchmüller-Tye (BT) potential [37]: where γ E = 0.5772 is the Euler constant, and the function v(x) is known numerically from Ref. [37], and Here, the charm quark mass is taken to be m c = 1.48 GeV. One notices that the BT potential at small r has a Coulomb-like behaviour, while at large r -a string-like behaviour such that its difference from the Cornell potential is mainly at small r.

C. Light-front representation of the amplitude and transition form factor
Our calculation follows the standard procedure of perturbative QCD for exclusive processes. We regard the γ * γ * → cc transition as a hard, perturbatively calculable process and convolute the amplitude with the bound state wave-function. We will assume later, that the charm quark mass m c by itself is large enough to justify perturbation theory and apply our results even in the limit of vanishing photon virtualities. Following [38] (see Eq. (A7) therein), the photon fusion amplitude can be expressed as (2.32) Here withp ≡ p µ γ µ , is the standard Feynman amplitude including the t-channel and u-channel exchange of a quark. We can express the denominators of the propagators though LFvariables: Here we introduced the notation and Contracting the Feynman amplitude with n + µ n − ν allows us to reduce the amplitude to a form where only simple spinor products of the formūn + v,ūn − u,vn − v need to be performed. Using the spinors from Ref. [38], we obtain then 2 (2.37) Here, subscripts ± of the wave functions stand for helicities ± 1 2 of (anti-)quarks. Inserting the explicit expressions for the helicity dependent wave functions given in Eq. (2.15), we observe that there is a large cancellation between the parallel and antiparallel helicity configurations, and only the last term ∝ [q 1 , q 2 ] survives. It is then straightforward to read off our result for the γ * γ * → η c form factor by comparing to Eq. (2.7): and then to perform the integration over the azimuthal angle of k. Using 2π 0 dφ 2π we can obtain .

(2.40)
This form puts into evidence, that the invariant form factor F (Q 2 1 , Q 2 2 ) is a function of Q 2 1 = q 2 1 and Q 2 2 = q 2 2 only. Notice that by the Bose-symmetry, the form factor must be a symmetric function of Q 2 1 , Q 2 2 . This is evidently not obvious from the representations Eq. (2.38) or Eq. (2.40), as the integrand is manifestly asymmetric in q 1 , q 2 . However, as will be demonstrated below by the numerical results, our representation has the required symmetry. In particular, in the limit of one on-shell photon, one must have, that F (Q 2 ) ≡ F (Q 2 , 0) = F (0, Q 2 ), and the two different integral representations for F (Q 2 ) which follow from Eq. (2.38) coincide with the ones found in Ref. [18], where their equivalence was also demonstrated numerically.
A number of limits of Eq. (2.38) are interesting. Firstly, the value F (0, 0) for two on-shell photons is related to the two-photon decay width by Eq. (2.2). First note, that (2.41) Let us write this result as an integral over the three-momentum p and the radial wave function u(p).
Here we used, that in polar coordinates k 2 = p 2 sin 2 θ, hence k 2 + m 2 the velocity v/c of the quark in the cc cms-frame. Similar results for the relativistic corrections to the decay width exist in the literature, see e.g. Ref. [39] and references therein. Notice that in these approaches typically the running of the mass M cc with p is neglected.
In the non-relativistic (NR) limit, where p 2 /m 2 c ≪ 1, β ≪ 1, the invariant mass M cc approaches M cc = 2m c . If we neglect the binding energy, and identify 2m c = M ηc , we obtain Here R(0) is the value of the radial wave function R(r) = u(r)/r at the origin. This yields the well known result for the γγ-width (see e.g. Table 2.2 in Ref. [40]) which serves as a check on our normalization. In the same limit, which amounts to an expansion around k = 0 and z = 1/2 in Eq. (2.38), we can derive the transition form factor in the NRQCD-limit, Still another interesting limit exists, namely at very large Q 2 i , the k-smearing becomes unimportant, and one can neglect k in the hard matrix element of Eq. (2.38). Then only the LFWF appears under the k integral, and the hard scattering factorization in terms of the distribution amplitude emerges. We introduce the distribution amplitude (DA) at a scale µ 2 0 as The DA is conveniently normalized as The transition form factor simplifies to This representation is valid in the limit of large photon virtualities Q 2 1 , Q 2 2 .

III. NUMERICAL RESULTS
In this Section we will present our results for the doubly virtual transition form factor, which will be calculated for the different models for the meson wave function. A comparison with the current experimental data will be presented and the onset of the asymptotic regime will be discussed. Moreover, in order to estimate the factorization breaking in the transition form factor we will also estimate the normalized form factor, defined by: which nicely quantifies the deviation from point-like coupling. A popular model for the transition form factor is based on the vector meson dominance approach (see e.g. Ref. [21]), and readsF It features a factorized dependence on the photon virtualities, which we expect to be broken. In our analysis, we will quantify the factorization breaking of the transition form factor by estimating the quantity defined by: .

(3.3)
A. cc wave functions of η c (1S) and η c (2S) Our wave functions u(p) were obtained by Fourier transform from the r-dependent cc wave functions obtained as a solution of the Schrödinger equation with different, realistic, potentials from the literature as described in the previous Section. In Fig. 2 we show the wave function u(p) for different potentials for the 1S state (left panel) and for the 2S radial excitation (right panel). We have that the different models predict similar shapes for the wave functions, but differ in its predictions for the position of the peaks and the approach to the large momentum limit.
Using the Terentev prescription (see Eq. (2.12)) we obtain wave functions in the lightfront variables z and k T = |k|. In Fig. 3 we show as an example the light-front wave function of Eq. (2.21) in the (z, k T )-space for the Buchmüller-Tye potential. As expected, the wave function is strongly peaked at z ≈ 1/2 and is strongly suppressed in the endpoints. These properties are shared by all wave functions for the potentials used by us. The Cornell potential wave function somewhat stands out as it has the hardest tail at large momenta.
B. γ * γ * transition form factor We start the presentation of our results from the value of F (0, 0) for η c (1S) and η c (2S) obtained from the different potential model wave functions. The obtained results for F (0, 0), together with the resulting decay width into photons, Γ γγ are collected in Table 1 for the η c (1S) and Table 2 for the η c (2S), respectively. Rather different results are obtained for different potentials, and they are not always consistent with the one obtained from radiative decay width Γ γγ , rather well known from recent experiments [41]. It is interesting to compare these -fully relativistic-results to the ones obtained from the wave function at the origin collected in Table 3 and Table 4 respectively for η c (1S) and η c (2S). We observe that the relativistic corrections are fairly strong, especially for the Cornell-potential.
Also shown in Tables 1 and 2 are the values for the so-called decay constant f ηc . In the case of the 1S state, the agreement with a value extracted by the CLEO collaboration [42] is generally quite good. We wish to point out that the decay constant f ηc is not directly related to the two-photon decay width Γ γγ . Such a relation only exists in the non -relativistic limit, where both quantities are expressed in terms of the wave function at the origin.
In the upper panel of Fig. 4 we show our results for the normalized γ * γ → η c transition form factorF (Q 2 , 0) as a function of photon virtuality (Q 2 ) for different potential models. Below we will also use the notation F (Q 2 ) = F (Q 2 , 0) = F (0, Q 2 ), when one of the photons is on-shell. For reference we also show the BaBar experimental data [21]. The oscillator and power-law potentials give the best description of the BaBar data. This appears to be related to the lower value of m c used with these potentials. A modification of the quark mass to m c = 1.3 GeV in the hard matrix element in fact leads to a much better agreement with the BaBar data. The γ * γ transition form factor for the η c (2S) is shown in the lower panel of Fig. 4 . In this case theF (Q 2 , 0) =F (0, Q 2 ) has a somewhat harder tail as a function of Q 2 than for the η c (1S).    In Fig.5 we show the rate of approaching of Q 2 F (Q 2 ) to its asymptotic value predicted by Brodsky and Lepage [38]. For the asymptotic distribution amplitude ϕ(z, µ 2 ) = 6z(1 − z), one would obtain Q 2 F (Q 2 ) = 8 3 f ηc . Therefore the horizontal lines 8 3 f ηc are shown for reference (upper panel -η c (1S), lower panel -η c (2S)), considering the values for f ηc presented in Tables 1 and 2. We do not observe approaching towards BL asymptotic value for Q 2 ≤ 50 GeV 2 . While our results flatten out at large Q 2 , their asymptotic value is much smaller then the one predicted within the hard scattering formalism.
In order to understand the results for Q 2 F (Q 2 ) shown above, let us discuss the applicability of collinear approach, commonly used for light pseudoscalar mesons, for the η c case. From Eq. (2.49), one has that the collinear formula with the correct asymptotics reads: The evolution with the factorization scale is easily implemented in the formalism of distribution amplitudes. This is routinely done for light pseudoscalar mesons (π 0 , η , η ′ ) see e.g. a recent NLO analysis [45]. For cc (or bb) quarkonia the situation is much more complicated and quark mass effects and/or higher twists must be included. In Fig. 6 we show the distribution amplitude at a factorization scale µ = 3 GeV calculated from our wave functions TABLE 5. Extracted coefficients a n (µ 0 ), for the Buchmüller-Tye potential n a n (µ 0 ) η c (1S) a n (µ 0 ) η c (2S) for both η c (1S) and η c (2S). To perform the evolution with the hard scale, the distribution amplitude is expanded with the help of the Gegenbauer C 3/2 n polynomials: We extract the Gegenbauer coefficients by means of a n (µ 0 ) = 2(2n + 3) 3(n + 1)(n + 2) They evolve according to a n (µ) = a n (µ 0 ) · αs(µ) with the the anomalous dimensions γ n , which can be found for example in Ref. [38]. For η c (1S), the a 2 coefficient dominates and is typically -0.3 at the initial evolution scale. For the η c (2S) the n = 4, 6 coefficients dominate. We show the Gegenbauer coefficients at the scale µ 0 = 3 GeV for both η c (1S) and η c (2S) in Table 5.
In Fig. 7 we illustrate the effect of evolution on Q 2 F (Q 2 ). We compare the result obtained with our original formula, given by Eq. (2.38). In addition, we show results obtained with the collinear formula (3.4) using distribution amplitudes obtained from our light-front wave functions (see Fig.6). There is only some difference at low Q 2 . Finally we show also result obtained within collinear approach with QCD evolution of distribution amplitudes built in, starting from µ 0 = 3 GeV. The effect of evolution is very weak. The reader is asked to notice much broader range of Q 2 in the figure compared to that in previous figures. Summarizing the effect of evolution can be safely neglected for Q 2 < 100 GeV 2 , i.e. in the range of our interest, i.e. where F (Q 2 ) can be measured. Now we wish to present also two-dimensional distributions for the γ * γ * transition form factor as a function of the photon virtualities Q 2 1 and Q 2 2 . As an example in Fig.8 we again show our results for the Buchmüller-Tye potential. To investigate the scaling properties, we show the transition form factor as a function of the variables One can see that F (and F ) is almost independent of the asymmetry parameter ω. For comparison γ * γ * → π 0 transition the dependence on ω is somewhat stronger [44]. Note that for the VDM model (3.2) some dependence on ω would be obtained. Future investigation of the slow ω dependence would be in our opinion an interesting task for Belle 2.

IV. CONCLUSIONS
The description of transition form factors is directly related to our understanding of the structure of bound states in QCD. In the present paper we have studied the transition form factors for γ * γ * → η c (1S,2S) for two space-like virtual photons, which can be accessed experimentally in future measurements of the cross section for the e + e − → e + e − η c process in the double -tag mode. The light-front wave function representation of these observables has been derived and discussed.
We have calculated the transition form factor for different wave functions obtained as a solution of the Schrödinger equation for the cc system for different phenomenological cc potentials from the literature. The rest-frame momentum space wave functions have been transformed to the light-front representation using the Terentev prescription. Firstly we have presented the transition form factor for only one off-shell photon as a function of its virtuality and compared to the BaBar data for the η c (1S) case. We have also presented the delayed convergence of the form factor to its asymptotic value 8 3 f ηc as predicted by the standard hard scattering formalism. Our results for Q 2 F (Q 2 ) approach to a lower asymptotic value. The Brodsky-Lepage limit can only be obtained after including QCD evolution of the distribution amplitudes for massless quarks, but appears irrelevant in the accessible kinematic domain. We conclude that it is not necessary to include the QCD evolution for Q 2 < 100 GeV 2 . This justifies, a posteriori, our results obtained within the approach using cc wave functions. Furthermore, we have presented two-dimensional distributions in the virtualities of photons of the γ * γ * η c transition form factor for η c (1S) and η c (2S). We have predicted a very slow dependence on the asymmetry parameter ω, which could be verified experimentally at Belle 2. We have also defined a measure of factorization breaking and have calculated it for different potentials as a function of (Q 2 1 , Q 2 2 ). The results on the Q 2 1 , Q 2 2 dependence are almost model independent. Finally, it is important to emphasize that the η c production in hadronic collisions is dominated, at lowest order, by the g * g * → η c subprocess, which is identical (up to global color factors) to the γ * γ * → η c amplitude derived in this paper. Consequently, our results can also be useful to estimate the η c production at the LHC, which is currently a theme of intense debate (see e.g. Refs. [46][47][48]).