The γ∗\documentclass[12pt]{minimal}
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\begin{document}$$\gamma ^*$$\end{document}–f0(980)\documentclass[12pt]{minimal}
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\begin{document}$$f_{0}(980)$$\end{document} transition form factors are calculated within the QCD factorization framework. The f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document}-meson is assumed to be mainly generated through its ss¯\documentclass[12pt]{minimal}
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\begin{document}$$s\bar{s}$$\end{document} Fock component. The corresponding spin wave function of the f0(980)\documentclass[12pt]{minimal}
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\begin{document}$$f_{0}(980)$$\end{document} meson is constructed and, combined with a model light-cone wave function for this Fock component, used in the calculation of the form factors. In the real-photon limit the results for the transverse form factor are compared to the large momentum-transfer data measured by the BELLE collaboration recently. It turns out that, for the momentum-transfer range explored by BELLE, the collinear approximation does not suffice, power corrections to it, modeled as quark transverse moment effects, seem to be needed. Mixing of the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document} with the σ(500)\documentclass[12pt]{minimal}
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\begin{document}$$\sigma (500)$$\end{document} is also briefly discussed.
volume-issue-count12issue-article-count78issue-toc-levels0issue-typeRegularissue-online-date-year2017issue-online-date-month3issue-online-date-day20issue-pricelist-year2017issue-copyright-holderSIF and Springer-Verlag Berlin Heidelbergissue-copyright-year2017article-contains-esmNoarticle-numbering-styleContentOnlyarticle-toc-levels0article-registration-date-year2017article-registration-date-month2article-registration-date-day1article-grants-typeOpenChoicemetadata-grantOpenAccessabstract-grantOpenAccessbodypdf-grantOpenAccessbodyhtml-grantOpenAccessbibliography-grantOpenAccessesm-grantOpenAccessIntroduction
Recently the BELLE collaboration [1] has measured the cross section for γ∗γ→π0π0\documentclass[12pt]{minimal}
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\begin{document}$$\gamma ^*\gamma \rightarrow \pi ^0\pi ^0$$\end{document} for large photon virtuality, Q12\documentclass[12pt]{minimal}
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\begin{document}$$Q_1^2$$\end{document}, and small energy in the γ∗γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma ^*\gamma $$\end{document} center-of-mass system. From these data the photon–meson transition form factors have been extracted for the scalar, f0(980)\documentclass[12pt]{minimal}
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\begin{document}$$f_0(980)$$\end{document}, and tensor, f2(1270)\documentclass[12pt]{minimal}
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\begin{document}$$f_2(1270)$$\end{document}, mesons for . These transition form factors are similar to those for the pseudoscalar mesons which have been extensively studied by both experimentalists and theoreticians. In Ref. [2] the γ-f0\documentclass[12pt]{minimal}
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\begin{document}$$\gamma -f_0$$\end{document} and the γ-f2\documentclass[12pt]{minimal}
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\begin{document}$$\gamma -f_2$$\end{document} form factors have been investigated within the NRQCD factorization framework [3, 4], in which relativistic corrections and higher Fock state contributions are suppressed by powers of the relativistic velocity of the quarks in the meson, i.e. up to some minor modifications, the light mesons are treated like heavy Quarkonia. Super-convergence relations have been derived in [5] and shown to provide constraints on the γ-f2\documentclass[12pt]{minimal}
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\begin{document}$$\gamma - f_2$$\end{document} transition form factor. The latter form factor has also been studied within the framework of collinear factorization [6]. A phenomenological model for this form factor is discussed in [7, 8]. The process γ∗γ→ππ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma ^*\gamma \rightarrow \pi \pi $$\end{document} has been discussed in the framework of generalized distribution amplitudes, time-like versions of generalized parton distributions [9]. In this paper the interest is focused on the γ-f0\documentclass[12pt]{minimal}
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\begin{document}$$\gamma - f_0$$\end{document} transition form factor.
The f0(980)\documentclass[12pt]{minimal}
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\begin{document}$$f_0(980)$$\end{document} meson is a complicated system whose nature is not yet fully understood. Its peculiar properties have led to many speculations about its quark content. A comparison of the partial widths for the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document} decays into pairs of pions and Kaons [10] under regard of the respective phase spaces reveals that the matrix element for f0→K+K-\documentclass[12pt]{minimal}
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\begin{document}$$f_0\rightarrow K^+K^-$$\end{document} is much larger than that for f0→π+π-\documentclass[12pt]{minimal}
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\begin{document}$$f_0\rightarrow \pi ^+\pi ^-$$\end{document}. Thus, if the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document} is viewed as a quark–antiquark state, it is dominantly an ss¯\documentclass[12pt]{minimal}
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\begin{document}$$s\bar{s}$$\end{document} state. The comparison of the branching ratios for the radiative decays of the ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document}-meson into the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document} and π0\documentclass[12pt]{minimal}
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\begin{document}$$\pi ^0$$\end{document} leads to the same conclusion. However, the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document}-meson is not a pure ss¯\documentclass[12pt]{minimal}
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\begin{document}$$s\bar{s}$$\end{document} state as is, for instance, obvious from the decay widths for J/Ψ→f0ω\documentclass[12pt]{minimal}
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\begin{document}$$J/\varPsi \rightarrow f_0\omega $$\end{document} and J/Ψ→f0ϕ\documentclass[12pt]{minimal}
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\begin{document}$$J/\varPsi \rightarrow f_0\phi $$\end{document}. This fact is interpreted as f0\documentclass[12pt]{minimal}
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\begin{document}$$ \sigma (500)$$\end{document} mixing. Detailed phenomenological analyses of f0\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} mixing in various decay processes [11–14] revealed two ranges for the mixing angle, φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document},(25-40)∘(140-165)∘\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (25{-}40)^\circ \quad (140{-}165)^\circ \end{aligned}$$\end{document}A light scalar glueball may affect this result [11].
As an alternative to the quark–antiquark interpretation other authors [15, 16] have suggested a tetraquark configuration for the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document}-meson. This appears as a natural explanation for the fact that the a0(980)\documentclass[12pt]{minimal}
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\begin{document}$$a_0(980)$$\end{document} and the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document} mesons are degenerate in mass and are the heaviest particles of the lightest scalar-meson nonet. For the tetraquark interpretation there seems to be no f0\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document} mixing [13]. The drawback of this picture is that the two-pion decay of the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document} is too small as compared to experiment whereas the a0→ηπ\documentclass[12pt]{minimal}
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\begin{document}$$a_0\rightarrow \eta \pi $$\end{document} is too large. In [17] it has been suggested that the lightest scalar-meson nonet, considered as tetraquarks states, mixes with the scalar-meson nonet with masses around 1200 MeV under the effect of the instanton force. The latter nonet is believed to have a predominant qq¯\documentclass[12pt]{minimal}
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\begin{document}$$q\bar{q}$$\end{document} structure. This mixing leads to a better description of the light scalar-meson decays. The f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document} may also have a substantial KK¯\documentclass[12pt]{minimal}
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\begin{document}$$K\bar{K}$$\end{document} molecule component [18]. It goes without saying that the real f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document}-meson is a superposition of all these configurations.
The goal of he present paper is the calculation of the γ∗-f0\documentclass[12pt]{minimal}
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\begin{document}$$\gamma ^* - f_0$$\end{document} transition form factors at large photon virtualities. For this calculation the pQCD framework developed by Brodsky and Lepage [19] is utilized in which the process is factorized in a perturbatively calculable hard subprocess (here γ∗γ∗→qq¯\documentclass[12pt]{minimal}
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\begin{document}$$\gamma ^*\gamma ^*\rightarrow q\bar{q}$$\end{document}) and a soft hadronic matrix element, parametrized as a light-cone wave function, which is under control of soft, long-distance QCD. As any hadron the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document}-meson possesses a Fock decomposition [20] starting with the simple quark–antiquark components|f0;p⟩=∑β∫[dτ]2[d2k⊥]2Ψ2,β(τ,k⊥)|qq¯,β;k1,k2⟩+higher Fock states\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} |f_0;p\rangle= & {} \sum _\beta \int [\mathrm{d}\tau ]_2[\mathrm{d}^2\mathbf{k}_\perp ]_2 \varPsi _{2,\beta }(\tau ,\mathbf{k}_\perp ) |q\bar{q},\beta ;k_1,k_2\rangle \nonumber \\&+\,\text {higher Fock states} \end{aligned}$$\end{document}where Ψ2,β\documentclass[12pt]{minimal}
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\begin{document}$$\varPsi _{2,\beta }$$\end{document} is the light-cone wave function of the qq¯\documentclass[12pt]{minimal}
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\begin{document}$$q\bar{q}$$\end{document} Fock state; the index β\documentclass[12pt]{minimal}
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\begin{document}$$\beta $$\end{document} labels its decomposition in flavor, color and helicity. The integration measures are defined by[dτ]2=dτ1dτ2δ(1-τ1-τ2),[d2k⊥]2=d2k⊥1d2k⊥216π3δ(2)(k⊥1+k⊥2-p⊥).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&[\mathrm{d}\tau ]_2 = \mathrm{d}\tau _1\mathrm{d}\tau _2\,\delta (1-\tau _1-\tau _2), \nonumber \\&{[\mathrm{d}^2 \mathbf{k}_{\perp }]}_2= \frac{\mathrm{d}^2\mathbf{k}_{\perp 1}\mathrm{d}^2\mathbf{k}_{\perp 2}}{16\pi ^3}\, \delta ^{(2)}(\mathbf{k}_{\perp 1}+\mathbf{k}_{\perp 2}-\mathbf{p}_\perp ). \end{aligned}$$\end{document}In the photon—photon interactions at large photon virtualities the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document}-meson is generated through its lowest Fock components, mainly the ss¯\documentclass[12pt]{minimal}
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\begin{document}$$s\bar{s}$$\end{document} one. As can be shown [19] the hard generation of the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document} through higher Fock components is suppressed by inverse powers of the photon virtuality and is therefore neglected. Once the meson is produced it gets dressed by fluctuations into higher Fock components under the effect of long-distance QCD. The calculation of the γ∗\documentclass[12pt]{minimal}
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\begin{document}$$\gamma ^*$$\end{document}–f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document} transition form factors is similar to the one of the photon–pseudoscalar-meson form factors [19]. The latter calculation is to be generalized in such a way that also hadrons with non-zero orbital angular between their constituents can be treated.
The paper is organized as follows: In the next section the spin part of the light-cone wave function, termed the spin wave function, of the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document} is constructed assuming that this mesons is an ss¯\documentclass[12pt]{minimal}
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\begin{document}$$s\bar{s}$$\end{document} state. In Sect. 2.1 the collinear reduction of the spin wave function is discussed and, in Sect. 2.2, an example of a light-cone wave function of the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document} is introduced and compared to the twist-2 and 3 distribution amplitudes. The γ∗-f0\documentclass[12pt]{minimal}
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\begin{document}$$\gamma ^*-f_0$$\end{document} transition form factors are defined in Sect. 3.1, followed by a LO perturbative calculation within the modified perturbative approach in which quark transverse degrees of freedom are retained (Sect. 3.2). Numerical results for the form factors in the real-photon limit are given in Sect. 4.1 and compared to the BELLE data. Some comments on the behavior of the γ∗-f0\documentclass[12pt]{minimal}
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\begin{document}$$\gamma ^* - f_0$$\end{document} form factors are presented in Sect. 4.2. Finally, the summary will be given in Sect. 5.
The spin wave function of the <inline-formula id="IEq68"><alternatives><mml:math><mml:msub><mml:mi>f</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><tex-math id="IEq68_TeX">\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document}</tex-math><inline-graphic xlink:href="10052_2017_4661_Article_IEq68.gif"/></alternatives></inline-formula>-meson
For the description of the hadron the light-cone approach is used which enables one to completely separate the dynamical and kinematical features of the Poincaré invariance [21, 22]. The overall motion of the hadron is decoupled from the internal motion of the constituents, i.e. the light-cone wave function of the hadron, Ψ\documentclass[12pt]{minimal}
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\begin{document}$$\varPsi $$\end{document}, is independent of the hadron’s momentum and is invariant under the kinematical Poincaré transformations (boosts along and rotations around the 3-directions as well as transverse boosts). Hence, Ψ\documentclass[12pt]{minimal}
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\begin{document}$$\varPsi $$\end{document} is determined if it is known at rest. The ss¯\documentclass[12pt]{minimal}
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\begin{document}$$s\bar{s}$$\end{document} Fock component given in (2), is split in a spin part (hereafter denoted spin wave function) and a reduced light-cone wave function, Ψ0\documentclass[12pt]{minimal}
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\begin{document}$$\varPsi _0$$\end{document}, which represents the full, soft wave function, Ψ\documentclass[12pt]{minimal}
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\begin{document}$$\varPsi $$\end{document}, with a factor Kμ\documentclass[12pt]{minimal}
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\begin{document}$$K^\mu $$\end{document} removed from it. As discussed in detail in Ref. [23] the covariant spin wave function can be constructed starting from the observation [24] that, in zero binding energy approximation, an equal-time hadron state (in the spin basis) in the constituent center-of-mass frame equals the (helicity) light-cone state at rest. Consequently, one can use the standard ls coupling scheme in order to couple quark and antiquark to a state of given spin and parity. On boosting the results to a frame with arbitrary hadron momentum one easily reads off the covariant spin wave function.1
Since the f0(980)\documentclass[12pt]{minimal}
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\begin{document}$$f_0(980)$$\end{document}-meson is a JPC=0++\documentclass[12pt]{minimal}
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\begin{document}$$J^{PC}= 0^{++}$$\end{document} state the quark and antiquark have to couple in a spin-1 state and one unit of orbital angular momenta is required.2 The ls coupling scheme leads to the following ansatz for the spin wave function of a final state meson in its rest frame [23, 25] (S¯0=γ0S†γ0\documentclass[12pt]{minimal}
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\begin{document}$$\bar{S}_0=\gamma _0S^\dagger \gamma _0$$\end{document}):Note that μ1,μ2\documentclass[12pt]{minimal}
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\begin{document}$$\mu _1, \mu _2$$\end{document} denote spin components and v,u¯\documentclass[12pt]{minimal}
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\begin{document}$$v, \bar{u}$$\end{document} are equal-t spinors here. In the meson’s rest frame the meson and the constituent momenta readp^μ=(M0,0),p^1μ=(m1,k),p^2μ=(m2,-k),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \hat{p}^{\mu } =(M_0,\mathbf{0}), \quad \hat{p}_1^{\mu } =(m_1,\mathbf{k}), \quad \hat{p}_2^{\mu }=(m_2,-\mathbf{k}), \end{aligned}$$\end{document}where k\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{k}$$\end{document} is the three-momentum part of the relative momentum of quark and antiquarkk=12(p^1-p^2).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathbf{k} =\frac{1}{2}(\hat{\mathbf{p}}_1 - \hat{\mathbf{p}}_2). \end{aligned}$$\end{document}In order to retain a covariant formulation, the four-vector K=(0,k)\documentclass[12pt]{minimal}
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\begin{document}$$K=(0,\mathbf{k})$$\end{document} is introduced.3 As is customary in the parton model, the binding energy is neglected and the constituents are considered as quasi-on-shell particles. That possibly crude approximation can be achieved by putting the minus components of the constituents to zero. Hence, k3=0\documentclass[12pt]{minimal}
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\begin{document}$$k_3=0$$\end{document} and our relative vector reduces toK=[00k⊥].\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} K = [0\, 0\, \mathbf{k}_{\perp }]. \end{aligned}$$\end{document}In this case the spin wave function (4) readsS¯0=12[k⊥+v(p^2,+)u¯(p^1,+)-k⊥-v(p^2,-)u¯(p^1,-)]\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \bar{S}_0= \frac{1}{\sqrt{2}} [k_{\perp +}v(\hat{p}_2,+)\bar{u}(\hat{p}_1,+) -k_{\perp -}v(\hat{p}_2,-)\bar{u}(\hat{p}_1,-)]\nonumber \\ \end{aligned}$$\end{document}where k⊥±=k⊥1±ik⊥2\documentclass[12pt]{minimal}
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\begin{document}$$k_{\perp \pm }=k_{\perp 1} \pm i k_{\perp 2}$$\end{document}. This spin wave function is of the same type as is discussed in [26] for the l=1\documentclass[12pt]{minimal}
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\begin{document}$$l=1$$\end{document} Fock components of ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} and π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document}-mesons.
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\begin{document}$$p\cdot K=0$$\end{document} holds and the quark and antiquark momenta are parametrized as4p1=τp+K,p2=τ¯p-K\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} p_1=\tau p + K, \quad p_2=\bar{\tau } p - K \end{aligned}$$\end{document}where τ¯=1-τ\documentclass[12pt]{minimal}
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\begin{document}$$\bar{\tau }=1-\tau $$\end{document} andp12=m12=τ2M02+O(k⊥2),p22=m22=τ¯2M02+O(k⊥2).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} p_1^2= & {} m_1^2=\tau ^2M_0^2 + \mathcal{O}(k^2_\perp ), \nonumber \\ p_2^2= & {} m_2^2=\bar{\tau }^2M_0^2 + \mathcal{O}(k^2_\perp ). \end{aligned}$$\end{document}The boost to the IMF leads toFor convenience the variable ξ=1-2τ\documentclass[12pt]{minimal}
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\begin{document}$$\xi =1-2\tau $$\end{document} is introduced. For ξ=0\documentclass[12pt]{minimal}
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\begin{document}$$\xi =0$$\end{document} this covariant spin wave function coincides with the one employed for the χc0\documentclass[12pt]{minimal}
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\begin{document}$$\chi _{c0}$$\end{document} in [28]. The normalization of the spin wave function is chosen such thatTr(S0†S0)=4E2k⊥2+O(k⊥4)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm{Tr}(S_0^\dagger S_0)= 4E^2k^2_\perp + \mathcal{O}(k^4_\perp ) \end{aligned}$$\end{document}where E is the meson’s energy. The meson’s ss¯\documentclass[12pt]{minimal}
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\begin{document}$$s\bar{s}$$\end{document} Fock state (2) explicitly reads⟨f0;p|=δcc¯2Nc∫dξd2k⊥16π3Ψ0(ξ,k⊥2)×S¯0⟨sc;p1,λ1|⟨s¯c¯;p2,λ1|.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle f_0;p|= & {} \frac{\delta _{c\bar{c}}}{2\sqrt{N_c}}\int \frac{\mathrm{d}\xi \mathrm{d}^2k_\perp }{16\pi ^3} \varPsi _0(\xi ,k^2_\perp )\nonumber \\&\times \,\bar{S}_0 \langle s_c;p_1,\lambda _1| \langle \bar{s}_{\bar{c}};p_2,\lambda _1|. \end{aligned}$$\end{document}The number of colors is denoted by Nc\documentclass[12pt]{minimal}
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\begin{document}$$N_c$$\end{document} and c,c¯\documentclass[12pt]{minimal}
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\begin{document}$$c, \bar{c}$$\end{document} are color labels. Proper state normalization requires the condition12∫dξd2k⊥16π3k⊥2|Ψ0(τ,k⊥2)|2=Pf0≤1\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{1}{2}\int \frac{\mathrm{d}\xi \mathrm{d}^2k_\perp }{16\pi ^3} k^2_\perp |\varPsi _0(\tau ,k^2_\perp )|^2=P_{f_0}\le 1 \end{aligned}$$\end{document}where Pf0\documentclass[12pt]{minimal}
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\begin{document}$$P_{f_0}$$\end{document} is the probability of the ss¯\documentclass[12pt]{minimal}
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\begin{document}$$s\bar{s}$$\end{document} Fock component.
Collinear reduction
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\begin{document}$$k_\perp \rightarrow 0$$\end{document} in the hard subprocess is to be taken in general. However, terms ∝Kα\documentclass[12pt]{minimal}
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\begin{document}$$\propto K^\alpha $$\end{document} in it combine with terms linear in K in the spin wave function and therefore survive the k⊥\documentclass[12pt]{minimal}
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\begin{document}$$k_\perp $$\end{document}-integration of the wave function. These terms are in general of the same order as the other terms in the spin wave function and it is therefore unjustified to neglect these terms.5 Consider the expansion of the subprocess amplitude with respect to K:M=A0(ξ)+KαA1α(ξ)+O(KαKβ)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathcal{M} = A_0(\xi ) + K^\alpha A_{1\alpha }(\xi ) + \mathcal{O}(K^\alpha K^\beta ) \end{aligned}$$\end{document}where A0\documentclass[12pt]{minimal}
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\begin{document}$$1/p^+$$\end{document} for dimensional reason. The k⊥\documentclass[12pt]{minimal}
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\begin{document}$$k_\perp $$\end{document}-integration yieldsFormally this is equivalent to the replacementThe transverse metric tensor is defined by g⊥11=g⊥22=-1\documentclass[12pt]{minimal}
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\begin{document}$$g_\perp ^{11}=g_\perp ^{22}=-1$$\end{document}, while all other components are zero in a frame where the meson moves along the 3-direction. In the collinear limit the spin wave function becomesMultiplying the spin wave function with the reduced wave function and integrating over k⊥\documentclass[12pt]{minimal}
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\begin{document}$$k_\perp $$\end{document}, one arrives at the associated distribution amplitudes. The first term in (18) generates the twist-2 distribution amplitudef¯022NcΦ0(ξ)=2ξ1-ξ2∫dk⊥216π2k⊥2M0Ψ0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \frac{\bar{f}_0 }{2\sqrt{2N_c}}\varPhi _0(\xi )= \frac{2\xi }{1-\xi ^2} \int \frac{\mathrm{d}k_\perp ^2}{16\pi ^2}\,\frac{k^2_\perp }{M_0}\,\varPsi _0. \end{aligned}$$\end{document}Because of charge conjugation invariance the twist-2 distribution amplitude is antisymmetric in ξ\documentclass[12pt]{minimal}
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\begin{document}$$\xi $$\end{document}. It possesses the Gegenbauer expansion and depends on the factorization scale, μF\documentclass[12pt]{minimal}
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\begin{document}$$\mu _F$$\end{document} [12, 29, 30],Φ0(ξ,μF)=Nc2(1-ξ2)∑m=1,3,⋯Bm(μF)Cm3/2(ξ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varPhi _0(\xi ,\mu _F)=\frac{N_c}{2} (1-\xi ^2) \sum _{m=1,3,\dots }B_m(\mu _F)C_m^{3/2}(\xi ). \end{aligned}$$\end{document}Evidently, the reduced wave function must be symmetric in ξ\documentclass[12pt]{minimal}
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\begin{document}$$\xi $$\end{document}. The Gegenbauer coefficients in (20), which encode the soft, non-perturbative QCD, evolve with the factorization scale asBm(μF)=Bm(μ0)αs(μ0)αs(μF)-γm/β0\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} B_m(\mu _F)=B_m(\mu _0) \left( \frac{\alpha _\mathrm{s}(\mu _0)}{\alpha _\mathrm{s}(\mu _F)}\right) ^{-\gamma _m/\beta _0} \end{aligned}$$\end{document}whereγm=CF1-2(m+1)(m+2)+4∑j=2m+11j.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \gamma _m=C_F\left( 1 - \frac{2}{(m+1)(m+2)}+4\sum _{j=2}^{m+1}\frac{1}{j}\right) . \end{aligned}$$\end{document}Here, β0=(11Nc-2nf)/3\documentclass[12pt]{minimal}
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\begin{document}$$n_f$$\end{document} denotes the number of active flavors. For the initial scale, μ0\documentclass[12pt]{minimal}
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\begin{document}$$\mu _0$$\end{document}, the value 1.41GeV\documentclass[12pt]{minimal}
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\begin{document}$$1.41~\mathrm{GeV}$$\end{document} is chosen in this article. The decay constant f¯0\documentclass[12pt]{minimal}
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\begin{document}$$\bar{f}_0$$\end{document} depends on the scale too [12]f¯0(μF)=f¯0(μ0)αs(μ0)αs(μF)4/β0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \bar{f}_0(\mu _F)=\bar{f}_0(\mu _0) \left( \frac{\alpha _\mathrm{s}(\mu _0)}{\alpha _\mathrm{s}(\mu _F)}\right) ^{4/\beta _0}. \end{aligned}$$\end{document}The other terms in (18) are of twist-3 nature although they do not correspond to the full twist-3 contributions since, in general, they also receive contributions from a second reduced wave function. This is, however, of no relevance for the purpose of the present paper, namely the calculation of the γ∗-f0\documentclass[12pt]{minimal}
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\begin{document}$$\gamma ^* -f_0$$\end{document} transition form factors. As we shall see in the following there is no twist-3 contribution to it. Anyway the k⊥\documentclass[12pt]{minimal}
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\begin{document}$$k_\perp $$\end{document}-integration of the other terms leads to two further distribution amplitudes which are related to Φ0\documentclass[12pt]{minimal}
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\begin{document}$$\varPhi _0$$\end{document} in the case at hand:Φ0s(ξ,μF)=1ξΦ0(ξ,μF),Φ0σ(ξ,μF)=1-ξ24ξΦ0(ξ,μF).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varPhi _{0s}(\xi ,\mu _F)= & {} \frac{1}{\xi } \varPhi _0(\xi ,\mu _F), \nonumber \\ \varPhi _{0\sigma }(\xi ,\mu _F)= & {} \frac{1-\xi ^2}{4\xi } \varPhi _0(\xi ,\mu _F). \end{aligned}$$\end{document}Both these distribution amplitudes are symmetric in ξ\documentclass[12pt]{minimal}
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\begin{document}$$\xi $$\end{document} and only the even terms appear in their Gegenbauer expansions.
With the help of these distribution amplitudes one can transform the product of wave function and collinear spin wave function (18), integrated over k⊥\documentclass[12pt]{minimal}
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\begin{document}$$k_\perp $$\end{document}, into the formThis expression resembles the corresponding pion spin wave function to twist-3 accuracy [31, 32].
A wave function for the <inline-formula id="IEq134"><alternatives><mml:math><mml:msub><mml:mi>f</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><tex-math id="IEq134_TeX">\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document}</tex-math><inline-graphic xlink:href="10052_2017_4661_Article_IEq134.gif"/></alternatives></inline-formula> meson
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\begin{document}$$k^2_\perp /(1-\xi ^2)$$\end{document} times the most general ξ\documentclass[12pt]{minimal}
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\begin{document}$$\xi $$\end{document} dependenceΨ0=c∑n=0,2…B~nCn3/2(ξ)exp-4a02k⊥21-ξ2\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varPsi _{0}=c\sum _{n=0,2\ldots } \tilde{B}_n C_n^{3/2}(\xi )\exp {\left[ -4\frac{a_0^2 k_\perp ^2}{1-\xi ^2}\right] } \end{aligned}$$\end{document}withc=16π22Ncf¯0M0a04.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} c=16\pi ^2 \sqrt{2N_c} \bar{f}_0 M_0 a_0^4. \end{aligned}$$\end{document}This wave function is similar to the one for the pion advocated for in [20]. It has been used for instance in the calculation of the photon–pseudoscalar transition form factors [33] or in the analysis of pion electroproduction [34]. Insertion of the wave function into Eq. (19) leads to the associated distribution amplitude (20) with the Gegenbauer coefficients (m is an odd integer)Bm=m2m+1B~m-1+m+32m+5B~m+1.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} B_m=\frac{m}{2m+1} \tilde{B}_{m-1} + \frac{m+3}{2m+5} \tilde{B}_{m+1}. \end{aligned}$$\end{document}As a consequence of charge conjugation invariance which forces Ψ0\documentclass[12pt]{minimal}
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\begin{document}$$\varPsi _0$$\end{document} to be symmetric in ξ\documentclass[12pt]{minimal}
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\begin{document}$$\xi $$\end{document}, the matrix element⟨f0;p|s¯(0)γμs(0)|0⟩=Nc2∫dξdk⊥216π2Ψ0Tr[S¯0γμ]\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle f_0;p|\bar{s}(0)\gamma _\mu s(0)|0\rangle = \frac{\sqrt{N_c}}{2} \int \mathrm{d}\xi \frac{\mathrm{d}k^2_\perp }{16\pi ^2} \varPsi _0 \mathrm{Tr}[\bar{S}_0\gamma _\mu ] \end{aligned}$$\end{document}vanishes in accord with the result quoted in [12]. On the other hand, the scalar density provides⟨f0;p|s¯(0)s(0)|0⟩=M0f¯0=Nc2∫dξdk⊥216π2Ψ0Tr[S¯0].\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle f_0;p|\bar{s}(0)s(0)|0\rangle = M_0\bar{f}_0 = \frac{\sqrt{N_c}}{2} \int \mathrm{d}\xi \frac{\mathrm{d}k^2_\perp }{16\pi ^2} \varPsi _0 \mathrm{Tr}[\bar{S}_0].\nonumber \\ \end{aligned}$$\end{document}Evaluation of the integral leads to B~0≃-1\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{B}_0\simeq -1$$\end{document}. This estimate is to be taken with caution since Φ0s\documentclass[12pt]{minimal}
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\begin{document}$$\varPhi _{0s}$$\end{document} in (24) is likely not the full twist-3 distribution amplitude, but it provides orientation. As is obvious from the vacuum-particle matrix element of quark field operators given in (30), the decay constant is a short-distance quantity; it represents the wave function at the origin of the configuration space. It is also clear that only the ss¯\documentclass[12pt]{minimal}
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\begin{document}$$s\bar{s}$$\end{document} Fock component of the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document}-meson contributes to this matrix element.
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\begin{document}$$\gamma ^*-f_0$$\end{document} transition form factors the wave function will be restricted to the first Gegenbauer term, all others are neglected. We haveΨ01=3cB1exp-4a02k⊥21-ξ2\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varPsi _{01}= 3cB_1\exp {\left[ -4\frac{a_0^2 k_\perp ^2}{1-\xi ^2}\right] } \end{aligned}$$\end{document}withB1≃B~0/3≃-1/3.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} B_1\simeq \tilde{B}_0/3 \simeq -1/3. \end{aligned}$$\end{document}In this case the twist-2 distribution amplitude readsΦ01=Nc2(1-ξ2)B1C13/2(ξ).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varPhi _{01}=\frac{N_c}{2} (1-\xi ^2) B_1 C_1^{3/2}(\xi ). \end{aligned}$$\end{document}For the transverse-size parameter, a0\documentclass[12pt]{minimal}
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\begin{document}$$0.8~\mathrm{GeV}^{-1}$$\end{document} is taken in the following. This value is very close to the corresponding value for the pion; see [33]. The r.m.s. k⊥\documentclass[12pt]{minimal}
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\begin{document}$$k_\perp $$\end{document} is related to the transverse-size parameter by⟨k⊥2⟩=3141a0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sqrt{\langle k^2_\perp \rangle }=\sqrt{\frac{3}{14}}\,\frac{1}{a_0}. \end{aligned}$$\end{document}For the value a0=0.8GeV-1\documentclass[12pt]{minimal}
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\begin{document}$$0.58~\mathrm{GeV}$$\end{document} which is similar to the corresponding results for the valence Fock components of other hadrons. For the decay constant the valuef¯0(μ0)=(180±15)MeV\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \bar{f}_0(\mu _0)=(180\pm 15)\,~\mathrm{MeV}\end{aligned}$$\end{document}is adopted, which has been derived by De Fazio and Pennington [35] from radiative ϕ→f0γ\documentclass[12pt]{minimal}
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\begin{document}$$\phi \rightarrow f_0\gamma $$\end{document} decays with the help of QCD sum rules (see also [36]). In [35] the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document}-meson is considered as a (dominantly) ss¯\documentclass[12pt]{minimal}
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\begin{document}$$s\bar{s}$$\end{document} state. The value (35) is extracted from the stability window for the Borel parameter between 1.2 and 2GeV2\documentclass[12pt]{minimal}
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\begin{document}$$2~\mathrm{GeV}^2$$\end{document}. This is consistent with the initial scale chosen in this article.
The <inline-formula id="IEq154"><alternatives><mml:math><mml:msup><mml:mi mathvariant="italic">γ</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math><tex-math id="IEq154_TeX">\documentclass[12pt]{minimal}
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\begin{document}$$\gamma ^* $$\end{document}</tex-math><inline-graphic xlink:href="10052_2017_4661_Article_IEq154.gif"/></alternatives></inline-formula>–<inline-formula id="IEq155"><alternatives><mml:math><mml:msub><mml:mi>f</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math><tex-math id="IEq155_TeX">\documentclass[12pt]{minimal}
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\begin{document}$$ f_0$$\end{document}</tex-math><inline-graphic xlink:href="10052_2017_4661_Article_IEq155.gif"/></alternatives></inline-formula> transition form factorsThe definition of the form factors
Let us consider the general case of two virtual photonsγ∗(q1,λ1)+γ∗(q2,λ2)→f0(p)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \gamma ^*(q_1,\lambda _1) + \gamma ^*(q_2,\lambda _2)\rightarrow f_0(p) \end{aligned}$$\end{document}where qi\documentclass[12pt]{minimal}
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\begin{document}$$q_i$$\end{document} and p denote the momenta of the photons and the mesons while λi\documentclass[12pt]{minimal}
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\begin{document}$$\lambda _i$$\end{document} are the helicities of the photons. One hasq12=-Q12,q22=-Q22,p2=M02.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} q_1^2=-Q_1^2, \quad q_2^2=-Q_2^2, \quad p^2=M_0^2. \end{aligned}$$\end{document}It is convenient to introduce the following variables [37]:Q¯2=12(Q12+Q22),ω=Q12-Q22Q12+Q22\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \overline{Q}^2=\frac{1}{2}(Q_1^2+Q^2_2),\quad \omega =\frac{Q^2_1 - Q^2_2}{Q^2_1 + Q^2_2} \end{aligned}$$\end{document}where, obviously, -1≤ω≤1\documentclass[12pt]{minimal}
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\begin{document}$$-1\le \omega \le 1$$\end{document}.
The transition vertex is defined by the matrix element of the time-ordered product of two electromagnetic currents,Γμν=-ie02∫d4xe-iq1x⟨f0;p∣T{jemμ(x)jemν(0)}∣0⟩\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varGamma ^{\mu \nu }=-ie_0^2 \int \mathrm{d}^4x e^{-iq_1x}\langle f_0;p\mid T\{j^\mu _\mathrm{em}(x)j^\nu _\mathrm{em}(0)\}\mid 0\rangle \nonumber \\ \end{aligned}$$\end{document}wherejemμ=euu¯(x)γμu(x)+edd¯(x)γμd(x)+ess¯(x)γμs(x)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} j^\mu _\mathrm{em}= e_u \bar{u}(x)\gamma ^\mu u(x) + e_d \bar{d}(x)\gamma ^\mu d(x) + e_s \bar{s}(x)\gamma ^\mu s(x)\nonumber \\ \end{aligned}$$\end{document}and ei\documentclass[12pt]{minimal}
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\begin{document}$$e_i$$\end{document} are the quark charges in units of the positron charge, e0\documentclass[12pt]{minimal}
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\begin{document}$$e_0$$\end{document}. Following [5] the vertex is covariantly decomposed asΓμν=ie02q1·q2M0-gμν+1Q¯4κ[q1·q2(q1μq2ν+q2μq1ν)+Q12q2μq2ν+Q22q1μq1ν)]FT(Q¯2,ω)-q1·q2Q¯4κq1μ+Q12q1·q2q2μq2ν+Q22q1·q2q1ν×FL(Q¯2,ω)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varGamma ^{\mu \nu }= & {} ie_0^2 \frac{q_1\cdot q_2}{M_0} \left\{ \left[ -g^{\mu \nu } + \frac{1}{\overline{Q}^4\kappa } [q_1\cdot q_2(q_1^\mu q_2^\nu +q_2^\mu q_1^\nu ) \right. \right. \nonumber \\&\left. +\, Q_1^2 q_2^\mu q_2^\nu + Q_2^2 q_1^\mu q_1^\nu )]\right] \,F_T(\overline{Q}^2,\omega )\nonumber \\&-\,\frac{q_1\cdot q_2}{\overline{Q}^4\kappa } \left[ q_1^\mu + \frac{Q_1^2}{q_1\cdot q_2}q_2^\mu \right] \left[ q_2^\nu + \frac{Q_2^2}{q_1\cdot q_2}q_1^\nu \right] \nonumber \\&\left. \times \, F_L(\overline{Q}^2,\omega ) \phantom {\frac{1}{\overline{Q}^4\kappa }}\right\} \end{aligned}$$\end{document}whereκ=(q1·q2)2Q¯4-1+ω2=ω2+M02Q¯2+M044Q¯4\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \kappa =\frac{(q_1\cdot q_2)^2}{\overline{Q}^4} - 1 + \omega ^2= \omega ^2 + \frac{M_0^2}{\overline{Q}^2}+\frac{M_0^4}{4\overline{Q}^4}\, \end{aligned}$$\end{document}and e0\documentclass[12pt]{minimal}
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\begin{document}$$e_0$$\end{document} denotes the positron charge. Current conversation is manifest:q1μΓμν=0,q2νΓμν=0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} q_{1\mu }\varGamma ^{\mu \nu }=0,\quad q_{2\nu }\varGamma ^{\mu \nu }=0. \end{aligned}$$\end{document}As one sees from (41) there are two form factors, one for transverse photon polarization, FT\documentclass[12pt]{minimal}
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\begin{document}$$F_T$$\end{document}, and another one for longitudinal polarization, FL\documentclass[12pt]{minimal}
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\begin{document}$$F_L$$\end{document}. By definition the form factors are dimensionless.
Contracting the vertex function with the polarization vectors of the photons and using transversality (ϵiqi=0\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon _iq_i=0$$\end{document}), one arrives atϵ1μϵ2νΓμν=ie02q1·q2M0-ϵ1·ϵ2+q1·q2κQ¯4ϵ1·q2ϵ2·q1FT-1-ω2κq1·q2ϵ1·q2ϵ2·q1FL.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \epsilon _1^\mu \epsilon _2^\nu \varGamma _{\mu \nu }= & {} ie_0^2 \frac{q_1\cdot q_2}{M_0} \left\{ \left[ -\epsilon _1\cdot \epsilon _2 + \frac{q_1\cdot q_2}{\kappa \overline{Q}^4}\epsilon _1\cdot q_2 \epsilon _2\cdot q_1\right] \,F_T \right. \nonumber \\&\left. -\,\frac{1-\omega ^2}{\kappa q_1\cdot q_2} \epsilon _1\cdot q_2 \epsilon _2\cdot q_1\,F_L\right\} . \end{aligned}$$\end{document}One can show, most easily in the equal-energy brick wall frame (see Fig. 1), defined byq1=(ν00a1),q2=(ν00a2),p=(2ν00a1+a2),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} q_1= (\nu \,0\,0\, a_1),\quad q_2= (\nu \,0\,0\, a_2), \quad p=(2\nu \,0\,0\, a_1+a_2),\nonumber \\ \end{aligned}$$\end{document}that the contraction with transverse photon polarization vectors with the same helicity projects out the form factor FT\documentclass[12pt]{minimal}
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\begin{document}$$F_T$$\end{document} and with longitudinal ones FL\documentclass[12pt]{minimal}
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\begin{document}$$F_L$$\end{document}:ϵ1μ(λ1)ϵ2ν(λ2)Γμν=-ie02q1·q2M0FT(Q¯2,ω)δλ1λ2,ϵ1μ(0)ϵ2ν(0)Γμν=ie021-ω2Q¯2M0FL(Q¯2,ω).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\epsilon _1^\mu (\lambda _1) \epsilon _2^\nu (\lambda _2)\varGamma _{\mu \nu } = -ie_0^2\frac{q_1\cdot q_2}{M_0} F_T(\overline{Q}^2,\omega ) \delta _{\lambda _1\lambda _2}, \nonumber \\&\epsilon _1^\mu (0) \epsilon _2^\nu (0)\varGamma _{\mu \nu } = \phantom {-}ie_0^2\sqrt{1-\omega ^2} \frac{\overline{Q}^2}{M_0}F_L(\overline{Q}^2,\omega ). \end{aligned}$$\end{document}If the photons have different helicities the vertex function is zero.
The equal-energy brick wall frame
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The LO perturbative calculation
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\begin{document}$$f_0$$\end{document}-meson is neglected whenever this is possible. From the Feynman graphs shown in Fig. 2 one finds for the vertex function (39)where the parton virtualities read (see also Fig. 2)g12=-Q¯2(1+ξω)-k⊥2,g22=-Q¯2(1-ξω)-k⊥2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} g_1^2= -\overline{Q}^2(1+\xi \omega )-k_\perp ^2,\quad g_2^2= -\overline{Q}^2(1-\xi \omega )-k_\perp ^2.\nonumber \\ \end{aligned}$$\end{document}Taking into consideration that the traces are only non-zero for even numbers of γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} matrices, one notices that only the first term of the spin wave function (11), i.e. the leading-twist piece, contributes to the traces. The twist-3 terms lead to an odd number of γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} matrices in the traces, the fourth term is neglected. With the help of (46) one finally arrives at the following expressions for the form factors:FT(Q¯2,ω)=-42Nces2Q¯2ω2+12M02Q¯21+12M02Q¯2×∫dξdk⊥216π2k⊥2Ψ0(ξ,k⊥)×ξ21-ξ211-ξ2ω2+2k⊥2/Q¯2,FL(Q¯2,ω)=-12M02Q¯21+12M02Q¯2ω2+12M02Q¯2FT(Q¯2,ω).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} F_T(\overline{Q}^2,\omega )= & {} - 4\sqrt{2N_c}\frac{e_s^2}{\overline{Q}^2} \frac{\omega ^2 + \frac{1}{2}\frac{M_0^2}{\overline{Q}^2}}{1+\frac{1}{2}\frac{M_0^2}{\overline{Q}^2}}\nonumber \\&\times \int \frac{\mathrm{d}\xi \mathrm{d}k^2_\perp }{16\pi ^2}k^2_\perp \varPsi _0(\xi ,k_\perp )\nonumber \\&\times \frac{\xi ^2}{1-\xi ^2}\, \frac{1}{1-\xi ^2\omega ^2 + 2 k^2_\perp /\overline{Q}^2}, \nonumber \\ F_L(\overline{Q}^2,\omega )= & {} -\frac{1}{2} \frac{M_0^2}{\overline{Q}^2} \frac{1 + \frac{1}{2}\frac{M_0^2}{\overline{Q}^2}}{\omega ^2+\frac{1}{2}\frac{M_0^2}{\overline{Q}^2}} F_T(\overline{Q}^2,\omega ). \end{aligned}$$\end{document}Because of the variation of ω2\documentclass[12pt]{minimal}
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\begin{document}$$\omega ^2$$\end{document} between 0 and 1 the mass dependent terms in front of the integral are kept. For ω→±1\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}. One also notices from (49) thatFT,L(Q¯2,-ω)=FT,L(Q¯2,ω).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} F_{T,L}(\overline{Q}^2,-\omega )= F_{T,L}(\overline{Q}^2,\omega ). \end{aligned}$$\end{document}For ω≫M02/(2Q¯2)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} F_T \propto 1/\overline{Q}^2,\quad F_L \propto 1/\overline{Q}^4. \end{aligned}$$\end{document}For ω→0\documentclass[12pt]{minimal}
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\begin{document}$$\sim M_0^2/(2\overline{Q}^2)$$\end{document} remains andFT,L∝1/Q¯4.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} F_{T,L} \propto 1/\overline{Q}^4. \end{aligned}$$\end{document}Explicitly, for ω→1\documentclass[12pt]{minimal}
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\begin{document}$$Q_2^2=0$$\end{document})FT(Q12,1)=-82Nces2Q12∫dξdk⊥216π2k⊥2Ψ0(ξ,k⊥)×ξ21-ξ211-ξ2+4k⊥2/Q12.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} F_T(Q_1^2,1)= & {} -8\sqrt{2N_c}\frac{e_s^2}{Q_1^2} \int \frac{\mathrm{d}\xi \mathrm{d}k^2_\perp }{16\pi ^2} k^2_\perp \varPsi _0(\xi ,k_\perp )\nonumber \\&\times \frac{\xi ^2}{1-\xi ^2}\, \frac{1}{1-\xi ^2+4k^2_\perp /Q_1^2}. \end{aligned}$$\end{document}For a wave function of the type (26) one can write Eq. (53) asFT(Q12,1)=ϱ(a02Q12)FTcoll(Q12,1)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} F_T(Q_1^2,1)= \varrho (a_0^2Q_1^2)\, F_T^\mathrm{coll}(Q_1^2,1) \end{aligned}$$\end{document}withFTcoll=-2es2Q12f¯0M0∫dξξΦ0(ξ)1-ξ2=-2Nces2Q12f¯0M0∑m=1,3…Bm\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} F_T^\mathrm{coll}= & {} -2\frac{e_s^2}{Q_1^2} \bar{f}_0M_0\int \mathrm{d}\xi \frac{\xi \varPhi _0(\xi )}{1-\xi ^2} \nonumber \\= & {} -2N_c\frac{e_s^2}{Q_1^2} \bar{f}_0M_0\sum _{m=1,3\ldots } B_m \end{aligned}$$\end{document}andϱ(x)=∫dKKe-K1+K/x.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varrho (x)=\int \mathrm{d}K\frac{K e^{-K}}{1+K/x}. \end{aligned}$$\end{document}For this type of wave functions the transition form factor is given by the collinear result multiplied by a universal reduction factor ϱ\documentclass[12pt]{minimal}
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\begin{document}$$\varrho $$\end{document}. The latter function is shown in Fig. 3. It is interesting that, in the collinear approximation, the LO perturbative result for the form factor is related to the sum over all Gegenbauer coefficients. The γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma -f_0$$\end{document} transition form factor data. This coefficient is to be regarded as an effective one. NLO corrections may allow one to fix a second coefficient [37]. The situation improves for |ω|<1\documentclass[12pt]{minimal}
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\begin{document}$$|\omega | < 1$$\end{document} as will be discussed in Sect. 4.2.
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\begin{document}$$a_0^2Q_1^2$$\end{document}
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\begin{document}$$M_0=(990\pm 20)\,~\mathrm{MeV}$$\end{document} [10])Γ(f0→γγ)=π4αem2M0|FT(0)|2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varGamma (f_0\rightarrow \gamma \gamma )=\frac{\pi }{4}\alpha _\mathrm{em}^2 M_0 |F_T(0)|^2. \end{aligned}$$\end{document}From the average decay width quoted in [10], one obtains|FT(0)|=0.0865±0.0141.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} |F_T(0)|= 0.0865 \pm 0.0141. \end{aligned}$$\end{document}The BELLE collaboration uses the slightly different value |FT(0)|BELLE=0.0832±0.0136\documentclass[12pt]{minimal}
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\begin{document}$$|F_T(0)|_\mathrm{BELLE}=0.0832 \pm 0.0136$$\end{document}.
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\begin{document}$$\begin{aligned} P_{f_0}=\frac{12}{5}N_c\big [\pi \bar{f}_0M_0a_0^2 B_1\big ]^2=0.18. \end{aligned}$$\end{document}The results of the fit are shown in Fig. 4. Reasonable agreement with experiment is to be seen within rather large errors although the shape of the fit is opposite to that of the data: the collinear result for the scaled form factor, Q12FTcoll\documentclass[12pt]{minimal}
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\begin{document}$$|F_T(0)|$$\end{document} the value (58) is taken). Data are taken from [1]; only the statistical errors are shown. The dashed and solid lines are the results of the collinear approximation and the modified perturbative approach evaluated from wave function (31), respectively. The shaded band represents the normalization uncertainty of the second result
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\begin{document}$$\gamma $$\end{document} transition form factor [33] and will be used here as well. In the impact-parameter plane the transition form factor (53) readsFT(Q12,1)=-es22NC2π∫-11dξξ21-ξ2∫01/ΛQCDdbb×[k⊥2Ψ0]e-SK0(bQ1/21-ξ2).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} F_T(Q_1^2,1)= & {} -\frac{e_s^2\sqrt{2N_C}}{2\pi }\,\int _{-1}^1 \mathrm{d}\xi \frac{\xi ^2}{1-\xi ^2} \int _0^{1/\varLambda _\mathrm{QCD}} \mathrm{d}b b \nonumber \\&\times [k_\perp ^2\varPsi _0]\,e^{-S} K_0\Big (bQ_1/2\sqrt{1-\xi ^2}\Big ). \end{aligned}$$\end{document}The integrand is completed by the Sudakov factor, exp(-S)\documentclass[12pt]{minimal}
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\begin{document}$$\exp {(-S)}$$\end{document}, its explicit form can be found for instance in [33]. The Sudakov factor provides the sharp cut-off of the b-integral at 1/ΛQCD\documentclass[12pt]{minimal}
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\begin{document}$$1/\varLambda _\mathrm{QCD}$$\end{document}. Since 1 / b in the Sudakov factor marks the interface between the non-perturbative soft momenta which are implicitly accounted for in the meson wave function, and the contributions from soft gluons, incorporated in a perturbative way in the Sudakov factor [33, 38], it naturally acts as the factorization scale. The Bessel function K0\documentclass[12pt]{minimal}
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\begin{document}$$K_0$$\end{document} is the Fourier transform of the hard scattering kernel and [k⊥2Ψ0]\documentclass[12pt]{minimal}
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\begin{document}$$[k^2_\perp \varPsi _0]$$\end{document} is the Fourier transform of the wave function (31) multiplied by k⊥2\documentclass[12pt]{minimal}
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\begin{document}$$k^2_\perp $$\end{document}. It reads[k⊥2Ψ0]=3π42NCf¯0M0B1(1-ξ2)2×1-1-ξ216a02b2e-1-ξ216a02b2.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}{}[k^2_\perp \varPsi _0]= & {} \frac{3\pi }{4} \sqrt{2N_C}\bar{f}_0 M_0 B_1(1-\xi ^2)^2\nonumber \\&\times \left( 1-\frac{1-\xi ^2}{16a_0^2}b^2\right) e^{-\frac{1-\xi ^2}{16a_0^2}b^2}. \end{aligned}$$\end{document}Evaluating the form factor within the modified perturbative approach and fitting B1\documentclass[12pt]{minimal}
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\begin{document}$$B_1$$\end{document} to the BELLE data [1] one arrives at the results shown in Fig. 4. The fit provides the following value for the Gegenbauer coefficient6:B1mpa(μ0)=-0.57±0.05\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} B_1^\mathrm{mpa}(\mu _0)=-0.57\pm 0.05 \end{aligned}$$\end{document}and χ2=5.9\documentclass[12pt]{minimal}
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\begin{document}$$\chi ^2=5.9$$\end{document} for 9 data points. The normalization uncertainty of the theoretical result follows from the errors of B1\documentclass[12pt]{minimal}
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\begin{document}$$B_1$$\end{document} and FT(0)\documentclass[12pt]{minimal}
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\begin{document}$$F_T(0)$$\end{document}; see (58). The agreement of the result obtained within the modified perturbative approach, with experiment is somewhat better than for the collinear approximation—the scaled form factor increases with Q12\documentclass[12pt]{minimal}
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\begin{document}$$Q^2_1$$\end{document} as the data do. This increase is the effect of the k⊥\documentclass[12pt]{minimal}
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\begin{document}$$k_\perp $$\end{document} corrections shown in Fig. 3, the Sudakov factor plays a minor role in this context.7 In passing it is noted that the predictions presented in [2] lie markedly below experiment for .
The value (62) of the Gegenbauer coefficient B1\documentclass[12pt]{minimal}
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\begin{document}$$B_1$$\end{document} not far from the QCD sum result [12]:f¯0(μ0)=(410±22)MeV,B1(μ0)=-0.65±0.07.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \bar{f}_0(\mu _0)=(410\pm 22)\,~\mathrm{MeV}, \quad B_1(\mu _0)=-0.65\pm 0.07.\nonumber \\ \end{aligned}$$\end{document}The coefficient B3\documentclass[12pt]{minimal}
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\begin{document}$$B_3$$\end{document} is found to be zero within errors. However, the value of f¯0\documentclass[12pt]{minimal}
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\begin{document}$$\bar{f}_0$$\end{document} is substantially larger than the value (35) used in the form factor calculation. More precisely, the fit to the BELLE data fixes the product of f¯0\documentclass[12pt]{minimal}
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\begin{document}$$\bar{f}_0$$\end{document} and B1\documentclass[12pt]{minimal}
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\begin{document}$$B_1$$\end{document} for which the following results exist:f¯0(μ0)B1(μ0)=(-0.079±0.007)GeVcollinear,=(-0.103±0.990)GeVmpa=(-0.267±0.029)GeV[12].\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \bar{f}_0(\mu _0)B_1(\mu _0)= & {} (-0.079 \pm 0.007)~\mathrm{GeV}\quad \quad \mathrm{collinear},\nonumber \\= & {} (-0.103 \pm 0.990)~\mathrm{GeV}\quad \quad \mathrm{mpa}\nonumber \\= & {} (-0.267 \pm 0.029)~\mathrm{GeV}\quad \quad [12]. \end{aligned}$$\end{document}The product of f¯0\documentclass[12pt]{minimal}
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\begin{document}$$\bar{f}_0$$\end{document} and B1\documentclass[12pt]{minimal}
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\begin{document}$$B_1$$\end{document} derived in [12] is substantially larger than the BELLE data [1] on the γ-f0\documentclass[12pt]{minimal}
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\begin{document}$$\gamma - f_0$$\end{document} transition form factors allow. This product of f¯0\documentclass[12pt]{minimal}
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\begin{document}$$B_1$$\end{document} is also in conflict with a light-cone wave function interpretation since it leads to a probability larger than 1. Of course a smaller value of the transverse-size parameter would cure this problem for the prize of an implausible compact valence Fock component. For instance, if one halves a0\documentclass[12pt]{minimal}
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\begin{document}$$a_0$$\end{document} the probability is about 0.12 but ⟨k⊥2⟩≃1.2GeV\documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{\langle k^2_\perp \rangle }\simeq 1.2~\mathrm{GeV}$$\end{document}.
The last issue to be discussed is the contribution from the non-strange qq¯\documentclass[12pt]{minimal}
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\begin{document}$$ f_0$$\end{document} transition form factor. This is usually considered as f0\documentclass[12pt]{minimal}
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\begin{document}$$ \sigma $$\end{document} mixing [11–17]. As for the η\documentclass[12pt]{minimal}
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\begin{document}$$ \eta '$$\end{document} system [39] this mixing is treated in the quark-flavor basis. As a consequence of the smallness of OZI-rule violations η\documentclass[12pt]{minimal}
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\begin{document}$$ \eta '$$\end{document} mixing is particularly simple in that basis—there is a common mixing angle for the states and the decay constants. It is assumed that this mixing scheme also holds for the case of interest here. Let σn\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _s$$\end{document} be states with the lowest Fock components nn¯=(uu¯+dd¯)/2\documentclass[12pt]{minimal}
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\begin{document}$$n\bar{n}=(u\bar{u}+d\bar{d})/\sqrt{2}$$\end{document} and ss¯\documentclass[12pt]{minimal}
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\begin{document}$$s\bar{s}$$\end{document}, respectively. In analogy to (30) the corresponding decay constants are defined by the σi\documentclass[12pt]{minimal}
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\begin{document}$$\sigma _i$$\end{document}-vacuum matrix elements of the quark field operators:⟨σn|n¯(0)n(0)|0⟩=Mσnf¯n,⟨σs|s¯(0)s(0)|0⟩=Mσsf¯s.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle \sigma _n|\bar{n}(0)n(0)|0\rangle= & {} M_{\sigma _n}\bar{f}_n, \nonumber \\ \langle \sigma _s|\bar{s}(0)s(0)|0\rangle= & {} M_{\sigma _s}\bar{f}_s. \end{aligned}$$\end{document}Since in hard processes only small spatial quark–antiquark separations are of relevance it seems plausible to embed the particle dependence and the mixing behavior of the qq¯\documentclass[12pt]{minimal}
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\begin{document}$$q\bar{q}$$\end{document} Fock components solely into the decay constants8 (for a detailed discussion of this procedure in the η\documentclass[12pt]{minimal}
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\begin{document}$$ \eta '$$\end{document} case see [40]). In generalization of (30) one may also define the decay constants f¯iq\documentclass[12pt]{minimal}
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\begin{document}$$q=n,s$$\end{document})⟨i|q¯(0)q(0)|0⟩=Mif¯iq.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \langle i | \bar{q}(0)q(0)|0\rangle = M_i \bar{f}_i^q. \end{aligned}$$\end{document}These decay constants mix according tof¯σn=f¯ncosφ,f¯σs=-f¯ssinφ,f¯0n=f¯nsinφ,f¯0s=f¯scosφ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \bar{f}_\sigma ^n= & {} \bar{f}_n\cos {\varphi },\quad \bar{f}_\sigma ^s=-\bar{f}_s\sin {\varphi },\nonumber \\ \bar{f}_0^n= & {} \bar{f}_n\sin {\varphi },\quad \bar{f}_0^s=\phantom {-}\bar{f}_s\cos {\varphi }. \end{aligned}$$\end{document}Hence, the γ∗\documentclass[12pt]{minimal}
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\begin{document}$$ f_0$$\end{document} transition form factors are made of two contributions,FT,L=FT,Ln+FT,Ls\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} F_{T,L} = F_{T,L}^n + F_{T,L}^s \end{aligned}$$\end{document}where the n and s contributions differ from (49) only by the decay constants, f¯n\documentclass[12pt]{minimal}
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\begin{document}$$n\bar{n}$$\end{document} Fock state is taken into account if in (49), and in other expressions derived for the form factors, the decay constant, f¯0\documentclass[12pt]{minimal}
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\begin{document}$$\bar{f}_0$$\end{document}, is to be replaced by an effective one defined byf¯0eff=f¯nsinφ12eu2+ed2es2+f¯scosφ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \bar{f}^\mathrm{eff}_0=\bar{f}_n\sin {\varphi }\frac{1}{\sqrt{2}}\frac{e_u^2+e_d^2}{e_s^2} + \bar{f}_s\cos {\varphi }. \end{aligned}$$\end{document}According to [12, 41] f¯n≃f¯s\documentclass[12pt]{minimal}
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\begin{document}$$\bar{f}_n\simeq \bar{f}_s$$\end{document}. Since the decay constant quoted in (35) is to be identified with f¯0s\documentclass[12pt]{minimal}
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\begin{document}$$40)^\circ $$\end{document} of the mixing angle quoted in (1) one findsf¯0eff/f¯0=2.4-3.0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \bar{f}^\mathrm{eff}_0/\bar{f}_0 = 2.4{-}3.0. \end{aligned}$$\end{document}Clearly, this leads to a transition form factor which is in conflict with the BELLE data [1]. Using the second range of mixing angles in (1) one obtains reasonable agreement with experiment. Particularly favored is the range φ=(145\documentclass[12pt]{minimal}
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\begin{document}$$ 151)^\circ $$\end{document} for which the form factor stays within the uncertainty band displayed in Fig. 4. An exact determination of the mixing angle is not possible at present given the poor information available for the basic decay constants, f¯n,f¯s\documentclass[12pt]{minimal}
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\begin{document}$$\bar{f}_n, \bar{f}_s$$\end{document}, and the assumption on the explicit form of the light-cone wave function.
The case of two virtual photons
The ratio of the transition form factors evaluated from (49) and from the collinear result (71) versus ω\documentclass[12pt]{minimal}
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\begin{document}$$\overline{Q}^2$$\end{document} values. The form factors are evaluated from the wave function (31) and the associated distribution amplitude (33), respectively
Here, in this subsection, we will comment on the γ∗\documentclass[12pt]{minimal}
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\begin{document}$$ f_0$$\end{document} transition form factor. As is the case for ω=1\documentclass[12pt]{minimal}
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\begin{document}$$\omega =1$$\end{document}, the Sudakov factor plays a minor role. In order to estimate the importance of the power corrections taken into account in the modified perturbative approach the ratio of the form factors evaluated from (49) (transformed to the impact parameter plane and with the Sudakov factor included) and from the collinear approximationFTcoll(Q¯2,ω)=-es2Q¯2f¯0M0ω2+12M02Q¯21+12M02Q¯2∫dξξΦ0(ξ)1-ξ2ω2\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} F_T^\mathrm{coll}(\overline{Q}^2,\omega )= -\frac{e_s^2}{\overline{Q}^2} \bar{f}_0M_0 \frac{\omega ^2 + \frac{1}{2}\frac{M_0^2}{\overline{Q}^2}}{1+\frac{1}{2}\frac{M_0^2}{\overline{Q}^2}} \int \mathrm{d}\xi \frac{\xi \varPhi _0(\xi )}{1-\xi ^2\omega ^2}\nonumber \\ \end{aligned}$$\end{document}is displayed in Fig. 5. As expected the power corrections become smaller with increasing Q¯2\documentclass[12pt]{minimal}
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\begin{document}$$\overline{Q}^2$$\end{document} and their importance decreases if ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} deviated from 1. The same observation has been made in [37] in the case of the γ∗\documentclass[12pt]{minimal}
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\begin{document}$$ \pi $$\end{document} transition form factor. As noticed in [37] the reason for this effect is the term 1-ξ2ω2\documentclass[12pt]{minimal}
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\begin{document}$$1-\xi ^2\omega ^2$$\end{document} in the hard scattering kernel which controls to which extent the form factor is sensitive to contributions from the end-point regions ξ→±1\documentclass[12pt]{minimal}
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\begin{document}$$\xi \rightarrow \pm 1$$\end{document} where soft effects can be important.
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\begin{document}$$\omega $$\end{document}, it is of interest to look at the transition form factor (71) in this region. Using the Gegenbauer expansion of the distribution amplitude the integral can be carried out term by term. The full result is a power series in ω2\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}-dependence of the prefactor. The first terms of this series readFTcoll(Q¯2,ω)=-25Nces2f¯0M0Q¯2ω2+12M02Q¯21+12M02Q¯2×B1+ω237B1+2027B3+ω4521B1+4033B3+56143B5+⋯.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} F_T^\mathrm{coll}(\overline{Q}^2,\omega )= & {} -\frac{2}{5}N_c e_s^2 \frac{\bar{f}_0 M_0}{\overline{Q}^2} \frac{\omega ^2 + \frac{1}{2}\frac{M_0^2}{\overline{Q}^2}}{1+\frac{1}{2}\frac{M_0^2}{\overline{Q}^2}}\nonumber \\&\times \left[ B_1 + \omega ^2\frac{3}{7}\left( B_1+\frac{20}{27} B_3\right) \right. \nonumber \\&+\left. \, \omega ^4\frac{5}{21} \left( B_1 + \frac{40}{33} B_3 + \frac{56}{143} B_5\right) +\cdots \right] .\nonumber \\ \end{aligned}$$\end{document}As one notices the mth Gegenbauer coefficient comes with the power ωm-1\documentclass[12pt]{minimal}
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\begin{document}$$\omega ^{m-1}$$\end{document} first. For Q¯2\documentclass[12pt]{minimal}
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\begin{document}$$\overline{Q}^2$$\end{document} larger than 4GeV2\documentclass[12pt]{minimal}
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\begin{document}$$4~\mathrm{GeV}^2$$\end{document} the difference between the modified perturbative approach and the collinear result is smaller than 10%\documentclass[12pt]{minimal}
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\begin{document}$$10\%$$\end{document}. Hence, the result in the modified perturbative approach, evaluated from the wave function (26), is not far from the collinear result (72). Thus, as is the case for the γ∗\documentclass[12pt]{minimal}
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\begin{document}$$ \pi $$\end{document} transition form factor [37], a measurement of the γ∗\documentclass[12pt]{minimal}
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\begin{document}$$ f_0$$\end{document} transition form factors for a range of small ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} would therefore provide valuable constraints on the f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document} distribution amplitude.
The γ∗\documentclass[12pt]{minimal}
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\begin{document}$$ f_0$$\end{document} transition form factor, scaled by |FT(0)|/Q¯2\documentclass[12pt]{minimal}
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\begin{document}$$|F_T(0)|/\overline{Q}^2$$\end{document}, evaluated from the wave function (31) (with B1=-0.57\documentclass[12pt]{minimal}
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\begin{document}$$B_1=-0.57$$\end{document}) within the modified perturbative approach versus Q¯2\documentclass[12pt]{minimal}
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\begin{document}$$\overline{Q}^2$$\end{document} for a set of ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} values
In Fig. 6 the γ∗\documentclass[12pt]{minimal}
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\begin{document}$$ f_0$$\end{document} transition form factor, evaluated from the wave function (31) within the modified perturbative approach, is shown for several small values of ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}. It is clearly seen that the form factor drops with Q¯2\documentclass[12pt]{minimal}
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\begin{document}$$1/\overline{Q}^4$$\end{document} (aside from evolution logarithms).
Summary
In this article the spin wave function of the f0(980)\documentclass[12pt]{minimal}
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\begin{document}$$f_0(980)$$\end{document} meson is constructed under the assumption that the meson is dominantly a strange–antistrange quark state. The collinear limit of the spin wave function is also discussed and the connection to the twist-2 and twist-3 distribution amplitudes is made. The spin wave function is applied in a calculation of the γ∗\documentclass[12pt]{minimal}
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\begin{document}$$ f_0$$\end{document} transition form factors. In the real-photon limit the results for the transverse form factor are compared to the large momentum-transfer data measured by the BELLE collaboration recently. It turns out that, for the momentum-transfer range explored by BELLE, the collinear approximation does not suffice, power corrections to it, modeled as quark transverse moment effects, seem to be needed. The parameters required in this calculation in order to achieve agreement with BELLE form factor data, the transverse-size parameter, a0\documentclass[12pt]{minimal}
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\begin{document}$$a_0$$\end{document}, the decay constant, f¯0\documentclass[12pt]{minimal}
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\begin{document}$$\bar{f}_0$$\end{document}, and the lowest (effective) Gegenbauer coefficient, B1\documentclass[12pt]{minimal}
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\begin{document}$$B_1$$\end{document}, have plausible values. However, Cheng et al. [12] in their analysis of charmless B-meson decays, adopt a much larger value for f¯0\documentclass[12pt]{minimal}
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\begin{document}$$\bar{f}_0$$\end{document} than (35). It remains to be seen whether the B-meson decays can be reconciled with the decay constant (35). The implications of σ\documentclass[12pt]{minimal}
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\begin{document}$$ f_0$$\end{document} mixing for the transition form factors are also briefly discussed. A mixing angle of about 150∘\documentclass[12pt]{minimal}
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\begin{document}$$150^\circ $$\end{document} seems to be favored. The paper is completed by presenting results on the γ∗\documentclass[12pt]{minimal}
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\begin{document}$$ f_0$$\end{document} form factors and on their collinear limits. It turns out that, in many aspects, the photon f0\documentclass[12pt]{minimal}
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\begin{document}$$f_0$$\end{document} form factors have properties similar to the form factors for the transition from a photon to the π0\documentclass[12pt]{minimal}
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\begin{document}$$Q_1^2\rightarrow \infty $$\end{document} are different. Whereas for the pseudoscalar mesons the limits of the scaled form factors are finite (e.g. Q12Fγπ0→2fπ\documentclass[12pt]{minimal}
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\begin{document}$$Q_1^2F_{\gamma \pi ^0}\rightarrow \sqrt{2}f_\pi $$\end{document}) the γ\documentclass[12pt]{minimal}
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\begin{document}$$ f_0$$\end{document} form factor FT\documentclass[12pt]{minimal}
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\begin{document}$$F_T$$\end{document} tends to zero ∼f0(μ0)B1(μ0)(αs(μ0)/αs(Q12))-4/25\documentclass[12pt]{minimal}
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\begin{document}$$\sim f_0(\mu _0)B_1(\mu _0)(\alpha _s(\mu _0)/\alpha _s(Q_1^2))^{-4/25}$$\end{document}. The γ∗\documentclass[12pt]{minimal}
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\begin{document}$$ f_0$$\end{document} transition form factors also play a role in the calculation of the hadronic light-by-light contribution to the muon anomalous magnetic moment [42–45]. In particular, the results presented in this article clarify the asymptotic behavior of the γ∗\documentclass[12pt]{minimal}
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\begin{document}$$ f_0$$\end{document} form factors.
Acknowledgements
Thanks to Volodya Braun and Andreas Schäfer for suggesting this study and for discussions. The work is supported in part by the BMBF, contract number 05P12WRFTE.
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\begin{document}$$\pi - a_1(1260)$$\end{document} form factors.
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\begin{document}$$K^\mu $$\end{document} introduced in the line after (6), is strictly speaking K⊥μ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} p_1=\tau p + K + \frac{k^2_\perp }{2\tau p\cdot \bar{p}} \bar{p}, \quad p_2=\bar{\tau } p - K + \frac{k^2_\perp }{2\bar{\tau }p\cdot \bar{p}} \bar{p} \end{aligned}$$\end{document}where p¯\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf{p}$$\end{document}. For this parametrization momentum conservation only holds up to corrections of order k⊥2/p\documentclass[12pt]{minimal}
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\begin{document}$$k^2_\perp /p$$\end{document}. It, however, also leads to the spin wave function (11) up to corrections of order k⊥3/M0\documentclass[12pt]{minimal}
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\begin{document}$$k_\perp ^2$$\end{document}; the leading term is k⊥\documentclass[12pt]{minimal}
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\begin{document}$$ \pi $$\end{document} form factor in [33] the contributions from the higher Gegenbauer terms are suppressed as compared to the lowest one. This property of the modified perturbative approach comes into effect here, too.
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\begin{document}$$\gamma - f_2$$\end{document} form factor performed in [6] the collinear factorization framework does also not suffice. In order to achieve fair agreement with experiment [1] soft end-point corrections have to be included in the analysis.
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\begin{document}$$\sigma _s$$\end{document}, are assumed to be the same.