Electroproduction of tensor mesons in QCD

Due to multiple possible polarizations hard exclusive production of tensor mesons by virtual photons or in heavy meson decays offers interesting possibilities to study the helicity structure of the underlying short-distance process. Motivated by the first measurement of the transition form factor $\gamma^*\gamma \to f_2(1270)$ at large momentum transfers by the BELLE collaboration we present an improved QCD analysis of this reaction in the framework of collinear factorization including contributions of twist-three quark-antiquark-gluon operators and an estimate of soft end-point corrections using light-cone sum rules. The results appear to be in a very good agreement with the data, in particular the predicted scaling behavior is reproduced in all cases.


Introduction
In recent years there has been increasing interest to hard exclusive production of tensor mesons a 2 (1320), K * 2 (1430), f 2 (1270) and f 2 (1520) by virtual photons or in heavy meson decays. In particular the possibility of three different polarizations of tensor mesons in weak B meson decays can shed light on the helicity structure of the underlying electroweak interactions. A different symmetry of the wave function and hence a different hierarchy of the leading contributions for the tensor mesons as compared to the vector mesons can lead to the situations that the color-allowed amplitude is suppressed and becomes comparable to the color-suppressed one. This feature can give an additional handle on penguin contributions. The early work was devoted mainly on the identification of the interesting decay modes and their basic theoretical description using various factorization techniques at the leading-order and the leading-twist level, see e.g. [1][2][3][4][5][6]. These studies are to a large extent exploratory. The physics potential of tensor meson production will depend on the accuracy of the theoretical description of such processes that can be achieved in QCD.
The recent study [7] of hard exclusive production of tensor mesons in single-tag twophoton processes is an important step forward in this context. This is a "gold-plated" reaction where the theoretical formalism can be tested and the relevant nonperturbative functions -tensor meson distribution amplitudes (DAs) -determined, or at least constrained. Our work aims to match this experimental progress with a development of the robust QCD framework for the study of the transition form factor γ * γ → f 2 (1270) in collinear factorization.
This reaction has already attracted some attention. Useful kinematic relations and estimates of the transition form factors for the mesons built of light and heavy quarks can be found in [8]. In Ref. [9] it was pointed out that hard exclusive production of f 2 (1270) with helicity λ = ±2 is dominated by the gluon component in the meson wave function and can be used to determine gluon admixture in tensor mesons in a theoretically clean manner. In Ref. [10] the helicity difference sum rule for the weighted integral of the γ * γ fusion cross section was derived and shown to provide constraints on the transition form factor in question. A phenomenological model for the tensor meson form factor can also be found in [11]. A related reaction γ * γ → ππ near the threshold has been discussed in [12][13][14].
Theory of the transition form factors goes back to the classical work on hard exclusive reactions in QCD [15][16][17]. The case of tensor mesons does not bring in complications of principle as compared to the pseudoscalar meson transition form factors that have been studied in great detail, but the tensor meson case is much less developed on a technical level. Our paper can be viewed as a major update of a earlier work [9] where the leading contributions to this process have been identified and calculated at the leading order. The new elements are: • We introduce twist-three and twist-four DAs and calculate the corresponding contributions to the form factors; • We calculate meson mass corrections terms in the higher-twist DAs and estimate the leading "genuine" three-particle contributions; • We include the next-to-leading (NLO) corrections and calculate the charm-loop contribution for the helicity amplitude with λ = ±2 taking into account for the c-quark mass; • We estimate quark-gluon coupling constants entering on the higher-twist level using QCD sum rules and the leading-twist gluon couplings using QCD sum rules and, alternatively, from the quarkonium decay Υ(1S) → γ f 2 ; • We estimate the soft (end-point) correction for the leading, helicity-conserving amplitude.
The main conclusion from our study is that the experimental results on the γ * γ → f 2 (1270) transition form factors reported in Ref. [7] appear to be in a very good agreement with the QCD scaling predictions starting already at moderate Q 2 5 GeV 2 . This is in contrast to the transition form factors for pseudoscalar π, η, η mesons where large scaling violations have been observed [18][19][20]. The absolute normalization for all helicity form factors can be reproduced assuming a 10-15% lower value of the tensor meson coupling to the quark energy-momentum tensor as compared to the estimates existing in the literature, which is well within the uncertainty. The presentation is organized as follows. Section 2 is introductory. It contains the definition of helicity amplitudes for the γ * γ → f 2 (1270) transition and the necessary kinematic relations. For the reader's convenience, the relation of our conventions to other definitions existing in the literature is explained in Appendix A. Section 3 contains a detailed discussion of the leading-twist and higher-twist DAs of the tensor meson, which are the main nonperturbative input in the calculations. This section contains several new results. The relevant nonperturbative parameters are calculated in Appendix C using QCD sum rules. In Appendix D we estimate one of the leading-twist gluon couplings from the decay Υ(1S) → γ f 2 . In Section 4 we calculate the three existing helicity amplitudes in collinear factorization, including higher-twist and, partially, radiative corrections. In Section 5 we discuss the power suppressed corrections ∼ 1/Q 2 arising from the end-point regions. We explain how such corrections can be estimated using dispersion relations and duality and construct the light-cone sum rule for the largest, helicity conserving amplitude. In Section 6 we compare our results to the experimental data [7] and summarize.

f (1270) production in two-photon reactions
We consider the reaction with one real and one virtual photon, P = q 1 + q 2 . Here and below m = 1270 MeV is the meson mass.
The transition amplitude can be related to the matrix element of the time-ordered product of two electromagnetic currents . . . The correlation function T µν can be decomposed in contributions of three Lorentz structures defined as The polarization tensor e (λ) αβ is symmetric and traceless, and satisfies the condition e (λ) αβ P β = 0. Polarization sums can be calculated using where M µν = g µν − P µ P ν /m 2 and the normalization is such that e (2.7) All three amplitudes (form factors) have mass dimension equal to one and scale as T k ∼ Q 0 (up to logarithms) in the Q 2 → ∞ limit. The two-photon decay width of f 2 (1270) is given by [21] Γ[f 2 → γγ] = πα 2 5m where α 1/137 is the electromagnetic coupling constant. Assuming that |T 2 (0)| |T 0 (0)| we obtain The relation of our definition of helicity form factors to the other existing in the literature definitions is given in Appendix A.

Distribution amplitudes
In the standard classification the tensor J P C = 2 ++ SU (3) f nonet is composed of f 2 (1270), f 2 (1525), a 2 (1320) and K * 2 (1430). Isoscalar tensor states f 2 (1270) and f 2 (1525) have a dominant decay mode in two pions (or two kaons). The isovector a 2 (1320) decays only in three pions and is more difficult to observe in hard reactions. In the quark model these mesons are constructed from a constituent quark-antiquark pair in the P-wave and with the total spin equal to one. In QCD they can be represented by a set of Fock states in terms of quarks and gluons, that further reduce to DAs in the limit of small transverse separations.
In the exact SU (3)-flavor symmetry limit the f 2 (1270) meson is part of a flavor-octet, f 2 = T 8 , and f 2 (1525) is a flavor-singlet, f 2 = T 1 . However, it is known empirically that the SU (3)-breaking corrections are large. Since f 2 (1270) and f 2 (1525) decay predominantly in ππ and KK, it follows that they are close to the nonstrange and strange flavor eigenstates, respectively, with a small mixing angle, see [21,22]. In this paper we assume ideal mixing at a low scale which we take to be µ 0 = 1 GeV, for definiteness. In other words, we assume that f 2 (1270) at this scale is a pure nonstrange isospin singlet. This assumption can easily be relaxed when more precise data on the form factors become available. In what follows the notationq . . . q refers to the SU (2)-flavor-singlet combination where u ans d are the usual "up" and "down" quark flavors. Let n µ be an arbitrary light-like vector, n 2 = 0, and We define the f 2 -meson quark-antiquark light-cone DAs as matrix elements of nonlocal light-ray operators [9,23] f 2 (P, λ)|q(z 2 n)γ µ q(z 1 n)|0 = f q m 2 e (λ) * nn (pn) 2 p µ 1 0 du e iz u 12 (pn) φ 2 (u, µ) and we use a shorthand notation Note that In all expressions light-like Wilson lines between the quark fields are implied. The DAs defined in (3.3) satisfy the following symmetry relations: and are normalized as The integral of the DA g a (u) vanishes and the first nonzero (second) moment, , involves contributions of three-particle operators, see below.
The coupling f q is defined as the matrix element of the local operator Dµ is the covariant derivative. This coupling is scale dependent and gets mixed with the gluon coupling and the similar coupling for strange quarks. In Appendix B we summarize the scale dependence of all DA parameters introduced in this Section.
The numerical value of f q has been estimated in the past [23][24][25] (see also Appendix C) using the QCD sum rule approach. Another possibility is to use the experimental result on the decay width Γ(f 2 → ππ) and estimate f q assuming that the matrix element of the energy-momentum tensor π + π − |Θ µν |0 is saturated by the tensor meson [23][24][25][26][27]. These two estimates agree with each other surprisingly well, although this agreement should not be overrated as in both cases the non-resonant two-pion background is not taken into account. We use (cf. [23] and Appendix C) (at the scale 1 GeV) as the default value for the present study. Note that the positive sign for this coupling is a phase convention, whereas the relative signs of the other matrix elements with respect to f q are physical and can be determined by considering suitable correlation functions as explained in Appendix C.
The operator product expansion (OPE) of quark bilinears close to the light cone x 2 → 0 takes the form where φ 4 (u) is another twist-four two-particle DA that can be expressed in terms of the other functions using QCD equations of motion (EOM), see below.
In addition we define three-particle twist-three DAs as The conformal expansion of the three-particle DAs reads [28,29] 14) The two-particle DAs g a (u) and g v (u) have collinear twist three and contain contributions of geometric twist-two and twist-three operators. The contributions of lower geometric twist are traditionally referred to as Wandzura-Wilczek (WW) contributions. They can be calculated in the terms of the leading-twist DA φ 2 (u) as [9,23] Assuming for simplicity the asymptotic expression for the leading-twist quark DA where C 1/2 n (x) are Legendre polynomials. The Legendre expansion can be motivated by the properties of these DAs under conformal transformations [28,29]. The "genuine" geometric twist-three contributions can be related to the three-particle DAs using EOM: The twist-three matrix elements can be estimated using QCD sum rules, see Appendix C.
We obtain (at the scale 1 GeV) The DAs φ 4 (u) and g 4 (u) have collinear twist four and receive contributions of the geometric twist-two, -three and -four operators. The Wandzura-Wilczek-type twist-two contributions assuming the asymptotic expression for φ 2 (u) (3.16) have the form We expect that these contributions are the dominant source of the power-suppressed corrections ∼ 1/Q 2 because of the large mass of the f 2 (1270) and will neglect "genuine" geometric twist-three and twist-four contributions. The derivation of these expressions proceeds similar to the case of the DAs of vector mesons considered in [28,30,31] so that we omit the details. Finally, the leading-twist gluon DAs of f 2 (1270) can be defined as [9] g µµ The distribution amplitudes φ T g (u) and φ S g (u) are both symmetric to the interchange of u ↔ū and describe the momentum fraction distribution of the two gluons in the f 2 -meson with the same and the opposite helicity, respectively. The asymptotic distributions at large scales are equal to φ T,as The normalization constants f T g and f S g are defined through the matrix element of the local two-gluon operator: The coupling f S g can be estimated from the radiative decay Υ(1S) → γf 2 , see Appendix D. The result is consistent with the assumption that f S g is very small at hadronic scales and is generated mainly by the evolution. In the numerical analysis we use the value The coupling to a helicity-aligned gluon pair, f T g , is difficult to quantify. The calculation of the leading contributions to the relevant correlation functions suggests that the two couplings, f S g and f T g , have the same sign, see Appendix C. In what follows we use as a ballpark estimate.
As already mentioned, all couplings considered here are scale dependent. The relevant expressions are collected in Appendix B.

QCD factorization
QCD description of the transition form factors in two-photon reactions is based on the analysis of singularities in the product of two electromagnetic currents in (2.2) in the limit (x − y) 2 → 0. Typical Feynman diagrams contributing to the leading-order accuracy are shown in Fig. 1.
The leading contributions in the Q 2 → ∞ limit have been calculated already in [9]. The form factor T 0 (Q 2 ) is of the leading twist and is dominated by the quark DA. In this case we include, in addition, NLO perturbative corrections to the leading twist contribution, which can be extracted from the corresponding expressions for the two-pion production in [13]. We also include the twist-four meson-mass correction m 2 /Q 2 which is a new result.
The form factor T 1 (Q 2 ) is of twist-three. It receives the Wandzura-Wilczek-type contributions calculated in [9] and the "genuine" twist-three contributions of three-particle quark-antiquark gluon DAs (new result).
As already noticed in [9], the T 2 (Q 2 ) form factor is rather peculiar: The leading contribution at Q 2 → ∞ comes in this case from the two-gluon DA with aligned helicity that we refer to as gluon transversity DA. However, this contribution is suppressed by the factor α s /π ∼ 0.1 which is the standard perturbation theory factor for an extra loop, and also the two-gluon coupling to a "conventional" quark-antiquark meson is expected to be small as compared to the quark-antiquark coupling. By this reason the true QCD asymptotics for this form factor may be postponed to very large momentum transfers that are out of reach on the existing experimental facilities. The result for T 2 (Q 2 ) given below includes the leading term and the Wandzura-Wilczek-type higher-twist power correction that does not involve such small factors. We also calculate and add the leading-twist c-quark contribution.
With these new additions, the expressions for the form factors are where the notation f q stands for the sum of the light quark couplings weighted with the electromagnetic charges The coefficient function of the three-particle DAs to T 1 is given by 5) and the NLO quark and gluon coefficient functions for T 0 read [13] C q (u) = C F ln 2ū + 3 ln u − 9 , The c-quark contribution to the transversity gluon distribution (this is a new result) is given by Here m c 1.4 GeV is the c-quark mass.
We have checked the electromagnetic gauge invariance of our results by explicit calculation. Note that electromagnetic Ward identities relate the contributions of three-particle DAs (in the last diagram in Fig. 1) to the contributions corresponding to gluon emission from the external quark legs in the hard scattering amplitude that are encoded in the "genuine" twist-three contributions to the two-particle DAs. Thus it is not surprising that the twist-three form factor T 1 (Q 2 ) can be written in two equivalent representations as in (4.2): either contributions of the three-particle DAs can be eliminated in favor the two particle ones, or, vice verse, the "genuine" twist-three contributions to the two-particle DAs can be rewritten in terms of the three-particle DAs.
Evaluating (4.1), (4.2), (4.3) on the asymptotic expressions for all DAs that are collected in the previous Section we obtain where all nonperturbative parameters and the QCD coupling have to be taken at the hard scale µ ∝ Q. The function λ(m 2 c /Q 2 ) takes into account suppression of the charm quark contribution in comparison to the light flavors; it is given by where β = √ 1 + 4x. The normalization is such that λ(0) = 1. Note that λ(0.1) 0.091 so that the c-quark contribution at Q 2 ∼ 20 GeV 2 is still suppressed by an order of magnitude as compared to the contributions of u, d, s quarks.
The expressions for the helicity form factors collected in this Section present our main result.

Soft (end-point) contributions
Transition form factors with one real photon receive power corrections ∼ 1/Q 2 coming from the region of large separation (x − y) 2 ∼ 1/Λ 2 QCD between the electromagnetic currents in (2.2). Such contributions are missing in the OPE and involve overlap integrals of the nonperturbative light-front wave functions at large transverse separations between the constituents and cannot be calculated in terms of DAs. They are revealed, nevertheless, as end-point divergences in the momentum fraction integrals in the framework of QCD factorization if one tries to extend it beyond the leading power accuracy. Such divergences are a clear indication that some extra contributions have to be added.
The technique that we adopt in what follows has been suggested originally [32] for the γ * γ → π 0 , see [33,34] for two recent state-of-the-art analysis. In this section we demonstrate how the same approach can be applied to the production of tensor mesons (cf. [35]). To this end we consider the simplest example: the form factor T 0 to the leadingorder accuracy, leaving the NLO corrections to T 0 and the other two form factors, T 1 and T 2 , for a future study. Our presentation follows closely the work [33] where further details and generalizations can be found.
The idea is to calculate the transition form factor for two large virtualities using collinear factorization or, equivalently, OPE, and perform analytic continuation to the real photon limit q 2 = 0 using dispersion relations. In this way explicit evaluation of contributions of the end-point regions is avoided (since they do not contribute for sufficiently large q 2 ) and effectively replaced by certain assumptions on the physical spectral density in the q 2 -channel. For our purposes it is sufficient to assume that the second photon is transversely polarized. Then there are no new Lorentz structures and the only difference is that the form factors depend on two virtualities. The starting point is that the form factor satisfies an unsubtracted dispersion relation in the variable q 2 for fixed Q 2 . Separating the contribution of the lowest-lying vector mesons ρ, ω one can write where s 0 is a certain effective threshold. Here we assumed that the ρ and ω contributions are combined in one resonance term and the zero-width approximation is adopted; f ρ ∼ 200 MeV is the usual vector meson decay constant. Since there are no massless states, the real photon limit is recovered by the simple substitution q 2 → 0 in this equation. If both virtualities are large, Q 2 , q 2 Λ 2 QCD , the same form factor can be calculated using OPE. Assume this calculation is done to some accuracy. The result is an analytic function that satisfies a similar dispersion relation The basic assumption of the method is that the physical spectral density above the threshold s 0 coincides (if integrated with a smooth test function) with the spectral density calculated in OPE, in the very similar way as the total cross section of e + e − -annihilation above the resonance region coincides with the QCD prediction, We expect that OPE becomes exact as q 2 → ∞ so that in this limit the calculation has to reproduce the "true" form factor. Equating the two representations in Eqs. (5.2), (5.3) at q 2 → ∞ and subtracting the contributions of s > s 0 from the both sides one obtains This relation explains why s 0 is usually referred to as the interval of duality (in the vector channel). The (perturbative) QCD spectral density Im T γ * γ * →f 2 0,OPE (Q 2 , −s) is a smooth function, very different from the physical spectral density Im T γ * γ * →f 2 0 (Q 2 , −s) ∼ δ(s − m 2 ρ ). Nevertheless, their integrals over a certain region of energies coincide. In this sense QCD description of correlation functions in the terms of quark and gluons is dual to the description in the terms of hadronic states.
In practical applications one uses a certain trick [36] which allows to reduce the sensitivity on the duality assumption in (5.4) and simultaneously suppress contributions of higher twists in the OPE. This is done going over to the Borel representation 1/(s + q 2 ) → exp[−s/M 2 ] the net effect being the appearance of an additional weight factor under the integral that suppresses the large invariant mass region: Varying the Borel parameter M 2 within a certain window, usually 1-2 GeV 2 one can test sensitivity of the results to the particular model of the spectral density. With this refinement, substituting Eq. (5.6) in (5.2) and using Eq. (5.4) one obtains for q 2 → 0 (cf. [32]) The abbreviation LCSR stands for the Light-Cone Sum Rules [37], as this approach is usually referred to. Adding and subtracting the contribution of s < s 0 in the second term 1 , one can rewrite the result as where the first term is the original OPE expression which is possible but not necessary to write in the dispersion representation, and the second term is the correction of interest: An attractive feature of this technique is that one does not need to evaluate the nonperturbative wave function overlap contributions explicitly: They are taken into account effectively via the modification of the spectral density.
As an illustration, consider the leading-twist QCD result at the leading order in strong coupling: This expression can easily be brought to the form of a dispersion relation changing variables u → s =ū u Q 2 +ūm 2 and integrating by parts. In this way one obtains after some rewriting, where and, for comparison, to the same accuracy (5.14) Note that u 0 → 1 as Q 2 → ∞ so that the integration region shrinks to the end-point u → 1 and the correction is power suppressed ∼ 1/Q 2 in this limit, as expected. Numerical results are presented in the next Section.

Results and discussion
The effective form factor averaged over polarizations is calculated using default values of the nonperturbative parameters and compared with the experimental data [7] in Fig. 2. We observe a perfect scaling behavior for Q 2 4−5 GeV 2 as predicted by QCD, whereas the normalization is slightly off -about 1 − 1.5σ if systematic errors in the data are taken into account. This difference can easily be compensated by a 10 − 15% decrease of the value of the quark coupling f q which serves as an overall normalization factor in the calculation, or, alternatively, by a moderate deviation of the leading twist DA φ 2 (u) from its asymptotic form. For illustration we show in the same Such a 10 − 15% smaller value of f q is certainly possible and does not contradict the existing estimates which are not very reliable. A more precise value can eventually be obtained from lattice QCD, however, this calculation is rather complicated and will take time. It would be very interesting to measure the time-like transition form factor e + e − → f 2 (1270)+γ at large virtualities q 2 ∼ 100 GeV 2 (cf. [38]) where the nonperturbative uncertainties are considerably reduced. This would give a direct measurement of the f qcoupling.
Our results for the helicity-separated form factors T 0 (Q 2 ), T 1 (Q 2 ), T 2 (Q 2 ) are compared with the experimental data [7] in Fig. 3. All three form factors are described rather well, the QCD calculation being slightly above the data as we have already seen for the helicity-averaged form factor in Fig. 2. Note that our result for T 1 (Q 2 ) only includes the leading-power contribution at large Q 2 in contrast to T 0 (Q 2 ) and T 2 (Q 2 ) where we also calculated the 1/Q 2 correction. Terms ∼ 1/Q 2 in T 1 (Q 2 ) correspond to collinear-twist-five and soft contributions and are more difficult to estimate. They should be expected, however, to be negative and of the same order of magnitude as for T 2 (Q 2 ) so that the increase of the QCD curve for T 1 (Q 2 ) in Fig. 3 at smaller Q 2 will almost certainly be compensated by power corrections and is not a reason for concern. As expected, T 1 (Q 2 ) is also more sensitive to the twist-three quark-antiquark-gluon contributions as compared to the other two form factors, and the uncertainties in the corresponding parameters are not negligible, they are shown by the shaded area.
As discussed in [9], the form factor T 2 (Q 2 ) at asymptotically large Q 2 is dominated by the two-gluon contribution with aligned helicity that we refer to as gluon transversity DA. This contribution is suppressed, however, by the factor α s /π ∼ 0.1 which is the standard penalty for an extra loop. Also the two-gluon coupling to a "conventional" quark-antiquark  Figure 3. The form factors T 0 (Q 2 ), T 1 (Q 2 ), T 2 (Q 2 ) (from top to bottom) normalized to T 2 (0) = 339 MeV. The result for T 0 (Q 2 ) shown by the solid line includes the estimate of soft end-point contributions using light-cone sum rules. The result of a pure pQCD calculation is shown by dashes. The error band for T 1 (Q 2 ) (shaded area) corresponds to variation of the twist-three parameters in the range specified in (3.19), whereas for T 2 (Q 2 ) we also include variation of the tensor gluon coupling f T g in the range ±50 MeV. The experimental data are taken from Ref. [7]. Only statistical errors are shown.
meson is unlikely to be large as compared to the quark-antiquark coupling. By this reason, T 2 (Q 2 ) at realistic Q 2 is still dominated by the Wandzura-Wilczek-type higher-twist power correction that does not involve such small factors: The shaded area in the plot for T 2 (Q 2 ) includes variation of the tensor gluon coupling f T g in a rather broad range, ±50 MeV, but the effect is barely visible. Our result does not mean that measurements of the T 2 form factor at large Q 2 are not interesting. On the contrary, a broad resonance structure in the two-pion channel with a scaling behavior T 0 ∼ Q 0 would be a clear signature of a tensor gluonium state.
To summarize, the main conclusion from our study is that the experimental results on the γ * γ → f 2 (1270) transition form factors reported in Ref. [7] appear to be in a very good agreement with QCD scaling predictions starting already at moderate Q 2 5 GeV 2 . The absolute normalization for all helicity form factors can be reproduced assuming a 10-15% lower value of the tensor meson coupling to the quark energy-momentum tensor as compared to the estimates existing in the literature, which is well within the uncertainty. These findings are in contrast to the transition form factors to pseudoscalar π, η, η mesons where large scaling violations have been observed [18][19][20]. If confirmed by future higherstatistics measurements that can come from BELLE II, perfect scaling behavior can be an indication that higher-twist and soft corrections are less of an issue for tensor as compared to pseudoscalar mesons. This can be interesting in context of the studies of heavy meson decays [1][2][3][4][5][6] where the effective hard scale is not very large and estimates of preasymptotic corrections are difficult. In turn, the QCD description implemented in our analysis can still be improved in many ways, e.g., taking into account deviation from ideal SU (3)-flavor mixing at hadronic scales, two-loop scale dependence of the couplings, higher-twist and endpoint corrections to T 1 (Q 2 ), more elaborate models for the DAs, etc. The corresponding studies will become necessary if the accuracy of the experimental data is increased.

A Other conventions
The experimental results in Ref. [7] are presented for a different set of transition form factors F i (Q 2 ) suggested in [8]. The form factors T i (Q 2 ) defined in (2.4) are more convenient for the QCD study but in order to compare our results with the data we need to establish the precise correspondence between these two descriptions.
In Ref. [8], the cross section σ λ 1 λ 2 for the production of f 2 (1270) by photons with helicities λ 1 and λ 2 is written as where s = (q 1 + q 2 ) 2 and Γ γγ denotes the two-photon decay width (2.8). The form factors are defined in terms of the helicity cross sections as [8] .
(A.2) Calculation of the helicity cross sections (A.1) in terms of the Lorentz covariant amplitudes similar to T i was done in Ref. [10], see Appendix C3. Using the expressions presented there we obtain where Γ Λ γγ stands for the two-photon decay width of f 2 (1270) with the polarization Λ: Using these expressions and the definitions in (A.2) one finds Experimentally the ratio of the decay widths with Λ = 0 and Λ = 2 is small [39]: Hence the expressions in (A.7-A.9) can be simplified neglecting the contribution of Γ Λ=0 γγ in the full decay width: , (A.12) Ideal mixing at the reference scale implies The relevant renormalization group equation reads [40][41][42] µ ∂ ∂µ where from one finds The last expression is based on the calculation of the relevant anomalous dimension by Hoodbhoy and Ji [43]. Note that the following combination of the quark and gluon couplings is scale-independent: √ n f f (1) (µ) + 2f s g (µ) = √ n f f (1) (µ 0 ) + 2f s g (µ 0 ) , (B.7) as it corresponds to the matrix element of a conserved current: the traceless part of the QCD energy-momentum tensor. The scale dependence of the flavor-nonsinglet twist-three couplings ζ 3 , ω 3 and ω 3 can be found, e.g., in [28,44]. Since the twist-three gluon DAs are completely unknown, using flavor-singlet evolution equations is not justified, and also the numerical difference between flavor-singlet and flavor-nonsinglet evolution is negligible as compared with the errors on the parameters. Staying with the flavor-nonsinglet evolution one obtains ζ 3 (µ) = L 3(C A −C F )/β 0 ζ 3 (µ 0 ) . (B.8) The remaining couplings ω 3 and ω 3 mix with each other: with the anomalous dimension matrix Γ = Here we tacitly assumed that this coupling is positive (with respect to f q ), as suggested by the QCD sum rule analysis in Appendix C. The given error bar reflects experimental uncertainties only. The theoretical uncertainties are much larger so that we estimate the overall accuracy of the value in (D.6) as 30-50%. This result appears to support an intuitive picture that the gluon coupling of "ordinary" quark-antiquark mesons is very small at hadronic scales and is generated entirely by the evolution. Indeed, assuming f q (1 GeV) = 101(10) MeV and f s g (1 GeV)=0 and using the expressions collected in Appendix B one finds This number is in a reasonable agreement with the above extraction from the bottomonium radiative decay having in mind the theoretical uncertainties.