1. Leptoproduction of pseudoscalar mesons
In this paper, we analyze pseudoscalar mesons electroproduction (π0, η) on the basis of the handbag approach. Its essential ingredients are the generalized parton distributions (GPDs) that were proposed in [1–3] and provide an extensive information on the hadron structure. GPDs are complicated nonperturbative objects which depend on x -the momentum fraction of proton carried by parton, ξ- skewness and t- momentum transfer. GPDs are connected in the forward limit with parton distribution functions (PDFs), they contain information about hadron form factors and the parton angular momentum [4]. They give information on the 3D structure of the hadrons, see e.g. [5]. More details on GPDs can be found, e.g. in [6–9].
GPDs were proposed to investigate exclusive reactions such as deeply virtual Compton scattering (DVCS) [4, 10, 11], time-like Compton scattering (TCS) [12–14] and deeply virtual meson production (DVMP) [6, 7]. Such processes at large photon virtuality Q2 can be factorized into the hard subprocess that can be calculated perturbatively and GPDs [4, 10, 11]. Generally, this factorization was proved in the leading-twist amplitudes with longitudinally polarized photons. This factorization formula is valid up to power corrections of the order 1/Q to the leading twist results which are unknown.
The study of exclusive meson electroproduction is one of the effective ways to access GPDs. An experimental study of π0 production was performed by CLAS [15] and COMPASS [16]. For η production, CLAS results are available at [17]. These experimental data can be adopted to constrain the models of GPDs.
Theoretical investigation of DVMP in terms of GPDs is based on the handbag approach where, as previously mentioned, the amplitudes are factorized into the hard subprocess and GPDs [2–4, 10] see figure 1. This amplitude has an ingredient, the non-perturbative meson distribution amplitudes, which probe the two-quark component of the meson wave functions. GPDs can be constructed using double distribution (DD) representation [22]. The DD generates ξ-dependence of GPDs by integration of the DD function together with PDFs, modified by t- dependent term. The handbag approach with the DD form of GPDs was successfully applied to the light vector mesons (VM) leptoproduction at high photon virtuality Q2 [23] and the pseudoscalar mesons (PM) leptoproduction [24].
Figure 1. The handbag diagram for the meson electroproduction off a proton. Parton helicities for transversity GPDs contribution are shown. |
In this work, we continue our previous study of π0 production [25] at the kinematics for EIC in China (EicC) based on the handbag approach. As shown in [24], the leading twist longitudinal cross section σL is rather small with respect to the predominant contribution determined by transversely polarized photons σT. This result was proved experimentally by the JLab Hall A collaboration [26]. The transversity dominance σT ≫ σL is confirmed in [25] at all EicC energy ranges for π0 production.
This paper is organized as follows. In section 2 , we discuss the contributions to the meson production amplitudes from the transversity GPDs HT, ${\bar{E}}_{T}$. More information can be found in [27] and in our previous paper [25]. Using the handbag approach, the transversity GPDs together with the twist-3 meson wave functions [27] contribute to the amplitudes with transversely polarized photons which produce transverse cross sections σT. They give essential contributions to the cross sections that are consistent with the experiment [15, 16].
In section 3 , we consider two models for transversity GPDs that give results for the cross sections of the π0 and η leptoproduction that are consistent with experiments at CLAS and COMPASS energies [15–17]. Predictions for η cross section at EicC energies are done. In addition to these results that are associated with [25], we analyse what information on the transversity GPDs can be extracted from future EicC experiments on PM leptoproduction. We discuss the possibility to perform u, d flavor separation for transversity GPDs HT and ${\bar{E}}_{T}$ using π0 and η cross sections [28, 29]. Finally, we give some discussion and conclusions in section 4 .
2. Handbag approach properties of meson production amplitudes
The process amplitude in the handbag approach is depicted in figure 1. In the handbag approach, the meson photoproduction amplitude is factorized into a hard subprocess amplitude ${ \mathcal H }$ which is shown in the upper part of figure 1 and GPDs F which includes information on the hadron structure at sufficiently high photon virtuality Q2. For the leading twist amplitudes, with longitudinally polarized photons, its factorization has been proved [2, 3].
In what follows, we consider the twist-3 contributions from transversity GPDs HT and ${\bar{E}}_{T}$ as well. Factorization for these twist-3 amplitudes is an assumption now. However, factorization models give results that are consistent with the experiment [27].
In the handbag method, the subprocess amplitude is calculated employing the modified perturbative approach (MPA) [30], where the quark transverse momenta k⊥ is taken into account. The power ${k}_{\perp }^{2}/{Q}^{2}$ correction is considered in the propagators of the hard subprocess ${ \mathcal H }$ together with the nonperturbative k⊥-dependent meson wave functions [31]. The gluonic corrections are regarded as the form of the Sudakov factor. Resummation of the Sudakov factor can be done in the impact parameter space [30].
The unpolarized ep → e(π0, η)p cross sections can be decomposed into a number of partial cross sections which are expressed in terms of the γ*p → (π0, η)p helicity amplitudes. The amplitude M0−,−+ is close to zero and can be omitted in the calculation. When we consider relation between amplitudes M0+,−+ = M0+,++, the partial cross sections can be written as follows:
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}{\sigma }_{L}}{{\rm{d}}t} & = & \displaystyle \frac{1}{\kappa }\left(| {M}_{0+,0+}{| }^{2}+| {M}_{0-,0+}{| }^{2}\right),\\ \displaystyle \frac{{\rm{d}}{\sigma }_{T}}{{\rm{d}}t} & = & \displaystyle \frac{1}{2\kappa }(| {M}_{0-,++}{| }^{2}+2| {M}_{0+,++}{| }^{2}),\\ \displaystyle \frac{{\rm{d}}{\sigma }_{{LT}}}{{\rm{d}}t} & = & -\displaystyle \frac{1}{\sqrt{2}\kappa }\mathrm{Re}\left[{{M}^{* }}_{0-,++}{M}_{0-,0+}\right],\\ \displaystyle \frac{{\rm{d}}{\sigma }_{{TT}}}{{\rm{d}}t} & = & -\displaystyle \frac{1}{\kappa }| {M}_{0+,++}{| }^{2}.\end{array}\end{eqnarray}$ 
Here, κ is the phase space factor, it reads $\begin{eqnarray}\kappa =16\pi ({W}^{2}-{m}^{2})\sqrt{\lambda ({W}^{2},-{Q}^{2},{m}^{2})}.\end{eqnarray}$ 
λ(x, y, z) is expressed as λ(x, y, z) = (x2 + y2 + z2) – 2xy − 2xz − 2yz. σLT is the interference contributions of the longitudinal and transverse amplitudes while σTT contains transverse amplitudes only.In (2 ), W2 = (q + p)2 is a squared energy in the photon–proton channel, that is connected with the EicC lepton–proton collision energy squared s = (l + p)2 as
$\begin{eqnarray}{W}^{2}={sy}-{Q}^{2}\quad \mathrm{with}\quad y=\displaystyle \frac{(q.p)}{(l.p)},\end{eqnarray}$ 
where y is an inelasticity variable that is usually used in lepton–proton reactions, (see e.g. [18, 19]). The typical W-energy interval for EicC is 6 GeV < W < 16 GeV.The leading twist amplitudes M0−,0+ and M0+,0+ are listed in our previous paper [25]. The transversity amplitudes that are essential in our study can be written in terms of convolutions as
$\begin{eqnarray}\begin{array}{rcl}{M}_{0-,++} & = & \displaystyle \frac{{e}_{0}}{Q}\sqrt{1-{\xi }^{2}}\langle {H}_{T}\rangle ,\\ {M}_{0+,++} & = & -\displaystyle \frac{{e}_{0}}{Q}\displaystyle \frac{\sqrt{-t^{\prime} }}{4m}\langle {\bar{E}}_{T}\rangle ,\end{array}\end{eqnarray}$ 
where e0 is a positron charge. The other variables are defined as $\begin{eqnarray}\begin{array}{l}\xi =\displaystyle \frac{{x}_{B}}{2-{x}_{B}}\left(1+\displaystyle \frac{{m}_{P}^{2}}{{Q}^{2}}\right),\,\,t^{\prime} =t-{t}_{0},\\ {t}_{0}=-\displaystyle \frac{4{m}^{2}{\xi }^{2}}{1-{\xi }^{2}}.\end{array}\end{eqnarray}$ 
xB is the Bjorken variable which is given as xB = Q2/(W2 + Q2 − m2). m is the proton mass and mP is the pseudoscalar meson mass.The GPDs F(x, ξ, t) are calculated as the integration of the double distributions function [22]
$\begin{gathered}F(x,\xi,t) =\int_{-1}^{1}d\rho\int_{-1+|\rho|}^{1-|\rho|}d\gamma\delta(\rho+\xi\gamma-x) \\\times f(\rho,t)\upsilon(\rho,\gamma,t). \end{gathered}$ 
For the valence quark double distributions read as $\begin{eqnarray}\upsilon (\rho ,\gamma ,t)=\displaystyle \frac{3}{4}\,\displaystyle \frac{\left[{\left(1-| \rho | \right)}^{2}-{\gamma }^{2}\right]}{{\left(1-| \rho | \right)}^{3}}.\end{eqnarray}$ 
The t- dependence in PDFs f is expressed as the Regge form $\begin{eqnarray}f(\rho ,t)=N\,{e}^{\left(b-\alpha ^{\prime} \mathrm{ln}\rho \right)t}{\rho }^{-\alpha (0)}{\left(1-\rho \right)}^{\beta },\end{eqnarray}$ 
and $\alpha (t)=\alpha (0)+\alpha ^{\prime} t$ is the corresponding Regge trajectory factor.It was found that for PM leptoproduction the contributions of the transversity GPDs HT and ${\bar{E}}_{T}=2{\tilde{H}}_{T}+{E}_{T}$ are essential [27]. We use the following form for hT based on the model [32]6 ). In equation (9 ), f and Δ f are valence and polarized quark distributions, respectively. The parameters in equation (8 ) for these PDFs are fitted from the known information about CTEQ6 PDF [33] e.g, or from the nucleon form factor analysis [34]. Some knowledge on GPDs ${\bar{E}}_{T}$ can only now be obtained from the lattice QCD [35]. We parameterized ${\bar{E}}_{T}$ as equation (8 ). The evolution of transversity GPDs was analyzed in [36]. We show that if we consider the evolution of GPDs via Q2 evolution of hT PDF in equation (9 ), the evolution of HT GPDs, found in [36] is reproduced quite well. The evolution of ${\bar{E}}_{T}$ GPDs is rather weak [36] and we do not consider Q2 evolution in GPDs. Generally, in this work, the explicit form of GPDs evolution is not so important because we work at very limited Q2 intervals.
$\begin{eqnarray}{h}_{T}(\rho ,0)=N\,\sqrt{\rho }(1-\rho )[f(\rho ,0)+{\rm{\Delta }}\,f(\rho ,0)].\end{eqnarray}$ 
It generates HT with the help of equation (GPDs HT and ${\bar{E}}_{T}$ determine the amplitudes M0−,++ and M0+,++ respectively, see equation (4 ). With the handbag approach the transversity GPDs are accompanied by a twist-3 PM wave functions in the hard amplitude ${ \mathcal H }$ [27] which is the same for both the M0±,++ amplitudes in equation (4 ). This property is demonstrated in figure 1, where the parton helicities of the subprocess amplitude ${ \mathcal H }$ are presented. For corresponding transversity convolutions we have forms:
$\begin{eqnarray}\begin{array}{rcl}\langle {H}_{T}\rangle & = & {\int }_{-1}^{1}{\rm{d}}x{{ \mathcal H }}_{0-,++}(x,\ldots ){H}_{T};\\ \langle {\bar{E}}_{T}\rangle & = & {\int }_{-1}^{1}{\rm{d}}x{{ \mathcal H }}_{0-,++}(x,\ldots ){\bar{E}}_{T}.\end{array}\end{eqnarray}$ 
There is a parameter μP in twist-3 meson wave function that is large and enhanced by the chiral condensate. In our calculation, we use μP = 2 GeV at scale of 2 GeV.More details of leading twist polarized GPDs $\tilde{H}$ and $\tilde{E}$ which contribute to the leading twist amplitudes with longitudinally polarized virtual photons can be found in papers [24, 27]. These amplitudes contribute to longitudinal cross section σL which is rather small with respect to transversity contribution σT for π0 and η production.
3. Model results for π0 and η leptoproduction and convolution extraction from the data
We consider the transversity effects described in equation (10 ) and take into account the leading twist contribution in equation (1 ). The amplitudes are transferred from the program produced by PARTONS collaboration codes [37] which was changed into Fortran employing results of GK model for GPDs [27].
In our previous paper [25], two models for transversity GPDs were analyzed. Model-1 was applied in [27] and described fine low energy CLAS data [15], but gave results about two times larger with respect to COMPASS data [16]. This was the reason to change GPDs parameters, especially for ${\bar{E}}_{T}$ contribution that is important in σT and σTT cross sections. Some changes were done for HT as well. The parameters for new models labeled as Model-2 are exhibited in table 1 [38].
Results of this model are shown at COMPASS energies in figure 2 by dashed lines. It can be seen that there is some discrepancy between Model-2 results and COMPASS data [16] at large −t > 0.3 GeV2. That was the reason to test in addition to the new Model-3 results for π0 and η leptoproduction. The parameters for the new Model-3 are listed in table 2 [38]. Note that in this model parameters are close to model I in [25], only parameters of ${\bar{E}}_{T}$ were changed. It can be seen from the N parameters that Model-2 has larger ${\bar{E}}_{T}$ and smaller HT values with respect to Model-3. In Model-3, we have smaller ${\bar{E}}_{T}$ and larger HT. Both models describe well π0 production at COMPASS. Model-3 gives better results for large −t > 0.3 GeV2, see figure 2.
Figure 2. Models results at COMPASS kinematics. Experimental data are taken from [16], dashed line represents the results of Model-2 and solid curve indicates the prediction of Model-3. |
Model-2 and 3 results for π0 production at CLAS energy are exhibited in figure 3. It can be seen that both models are in accordance with unseparated cross sections σ = σT + εσL, where σT predominated. At the same time, Model-3 gives closer results for σTT that is smaller with respect to Model-2. This confirms the previously mentioned smaller values of ${\bar{E}}_{T}$ in the Model-3. σLT cross sections are shown as well.
Figure 3. Cross sections of π0 production in the CLAS energy range together with the data [15]. Left graph is for σ and σTT while right graph is for σLT. |
Calculation of the amplitudes of η production is based on the singlet-octet decomposition of the η-state [39] where the amplitude is presented in the form
$\begin{eqnarray}{M}_{\eta }=\cos {\theta }_{8}{M}^{(8)}-\sin {\theta }_{1}{M}^{(1)}.\end{eqnarray}$ 
In the case if we omit the strange sea contribution which is small and can be neglected, the GPDs contribution to these amplitudes has a form $\begin{eqnarray}\begin{array}{c}{F}^{\left(8\right)}=\displaystyle \frac{1}{\sqrt{6}}({e}^{u}{F}^{u}+{e}^{d}{F}^{d});\\ {F}^{\left(1\right)}=\sqrt{2}\,{F}^{\left(8\right)}.\end{array}\end{eqnarray}$ 
We use the values of mixing angles and decay coupling constant from [39] $\begin{eqnarray}\begin{array}{l}{\theta }_{8}=-{21.2}^{0},\,\,{\theta }_{1}=-{9.2}^{0};\\ {f}_{8}=1.26{f}_{\pi },\,\,{f}_{1}=1.17{f}_{\pi }.\end{array}\end{eqnarray}$ 
This gives us the possibility to calculate the η production amplitude in a similar way as was used for the π0 case [27].The flavor factors for π0 and η production appear in combinations
$\begin{eqnarray}\begin{array}{rcl}{F}^{{\pi }^{0}} & = & \displaystyle \frac{1}{3\sqrt{2}}(2{F}^{u}+{F}^{d});\\ {F}^{\eta } & = & \displaystyle \frac{1}{3\sqrt{6}}(2{F}^{u}-{F}^{d}).\end{array}\end{eqnarray}$ 
Here, the explicit values eu = 2/3, ed = −1/3 of quark charges are used.From table 1 and 2, it can be seen that ${\bar{E}}_{T}$ has the same signs for u and d quarks but HT has the different signs, respectively. This means that for π0 case ${\bar{E}}_{T}$ contributions for u, d quarks are added but HT are subtracted. For η production, we have opposite cases: HT contributions are added but ${\bar{E}}_{T}$ compensated.
Table 1. Regge parameters and normalizations of the GPDs at a scale of 2 GeV for Model-2. |
| GPD | α(0) | βu | βd | ${\alpha }^{{\prime} }[\,{\mathrm{GeV}}^{-2}]$ | b[ GeV−2] | Nu | Nd |
|---|---|---|---|---|---|---|---|
| ${\bar{E}}_{T}$ | −0.1 | 4 | 5 | 0.45 | 0.67 | 29.23 | 21.61 |
| HT | — | — | — | 0.45 | 0.04 | 0.68 | −0.186 |
Table 2. Regge parameters and normalizations of the GPDs at a scale of 2 GeV. Model-3. |
| GPD | α(0) | βu | βd | ${\alpha }^{{\prime} }[\,{\mathrm{GeV}}^{-2}]$ | b[ GeV−2] | Nu | Nd |
|---|---|---|---|---|---|---|---|
| ${\bar{E}}_{T}$ | −0.1 | 4 | 5 | 0.45 | 0.77 | 20.91 | 15.46 |
| HT | — | — | — | 0.45 | 0.3 | 1.1 | −0.3 |
Thus we have ${\bar{E}}_{T}$ enhancement for the π0 case. For η production, HT is increased. Therefore, π0 process is more sensitive to ${\bar{E}}_{T}$ effects but for η production HT influences are more visible.
Model results for η production at CLAS energy [17] are depicted in figure 4. It can be seen that Model-3 with a larger HT contribution describes experimental data better at small momentum transfers. Model-2 with smaller HT produces an essential dip in the cross section that is not observed experimentally. Cross sections σTT and σLT are described properly for both models.
Figure 4. Cross section of η production in the CLAS energy ranges together with the data [17]. Left graph is for σ and σTT, while right graph is for σLT. |
Model-2 predictions at EicC energies for π0 production are presented at [25]. For Model-3 at these energies, we have results similar to those shown in figure 2. The π0 cross sections for Model-3 don’t have deep near $| {t}^{{\prime} }| =$ 0 GeV2 as we have for Model-2. Model-2 and 3 results are similar for $| {t}^{{\prime} }| \,\sim $ 0.2 GeV2 and cross section is a bit smaller for Model-3 with respect to Model-2 at $| {t}^{{\prime} }| \,\gt $ 0.3 GeV2.
Our results for EicC energies W = 7–16 GeV for η production are exhibited in figures 5 and 6. It can be concluded that Model-3 results are higher for the cross section σ with respect to Model-2 and for σTT result is opposite- Model-2 gives higher results. This is caused by larger HT contribution in Model-3 and larger ${\bar{E}}_{T}$ effects in Model-2 that is important in σTT. These model results can be checked experimentally by EicC and determine what Model-2 or 3 is more adequate to experiment.
Now we shall discuss how we can get information about transversity convolutions ${\bar{E}}_{T}$ and HT from experimental data. From equation (1 ), we can obtain4 ) we can determine HT and ${\bar{E}}_{T}$ convolutions. This procedure was adopted to extract transversity convolutions from CLAS experimental data in [28, 29].
$\begin{eqnarray}\begin{array}{rcl}| {M}_{0+++}| & = & \sqrt{-\kappa \displaystyle \frac{{\rm{d}}{\sigma }_{{TT}}}{{\rm{d}}t}},\\ | {M}_{0-++}| & = & \sqrt{2\kappa \left(\displaystyle \frac{{\rm{d}}{\sigma }_{T}}{{\rm{d}}t}+\displaystyle \frac{{\rm{d}}{\sigma }_{{TT}}}{{\rm{d}}t}\right)},\end{array}\end{eqnarray}$ 
we can determine the absolute values of the amplitudes. Employing the normalization factor from equation (At present, we do not have experimental data from China EicC. To analyse what information on the transversity GPDs can be extracted from future EicC experiments on PM leptoproduction, we shall instead use realistic experimental data, our model calculations for the cross sections $\tfrac{{\rm{d}}{\sigma }_{T}}{{\rm{d}}t}$ and $\tfrac{{\rm{d}}{\sigma }_{{TT}}}{{\rm{d}}t}$. Our results for HT and ${\bar{E}}_{T}$ convolutions for π0 and η production are depicted in figure 7. They are close to results found in [28, 29] at CLAS energies. As expected we find that HT convolution is larger for Model-3, and for Model-2 we get a larger ${\bar{E}}_{T}$. Using these results, we can extract convolutions for u and d flavors with the help of14 ). Here, F is the corresponding transversity of HT or ${\bar{E}}_{T}$ convolution functions. Such analyses were performed at CLAS energies in [29].
$\begin{eqnarray}\begin{array}{l}{F}^{u}=\displaystyle \frac{3}{4}(\sqrt{2}{F}^{{\pi }^{0}}+\sqrt{6}{F}^{\eta }),\\ {F}^{d}=\displaystyle \frac{3}{2}(\sqrt{2}{F}^{{\pi }^{0}}-\sqrt{6}{F}^{\eta }),\end{array}\end{eqnarray}$ 
which is a consequence of equation (
Figure 7. Extracted from the cross section transversity convolutions ∣⟨HT⟩∣ and $| \langle {\bar{E}}_{T}\rangle | $ for π0 (upper part) and η production (lower part) at CLAS energy range. |
We will not do this here, because we have model results for flavor convolutions, but the extraction of transversity convolution functions from future experimental data can be important in results for later experiments.
Our predictions for HT or ${\bar{E}}_{T}$ convolution functions that were extracted from the cross sections at the energies W = 8, 12 GeV which are typical at EicC energy ranges are exhibited in figures 8 and 9. Experimental analyses of these quantities can give information on the preferable models for transversity GPDs.
Figure 8. Extracted from the cross section transversity convolutions for π0 (upper part) and η (lower part) production at EicC (W = 8 GeV). |
Figure 9. Extracted from the cross section transversity convolution functions for π0 (upper part) and η (lower part) production at EicC (W = 12 GeV). |
In figure 10, we present our model predictions for energy dependencies of transversity convolution functions at fixed Q2 and momentum transfer. Such analyses will be important to give constraints on the W- dependence of HT or ${\bar{E}}_{T}$ GPDs from future experimental data.
Figure 10. Energy dependencies of extracted transversity convolutions for π0 (upper part) and η (lower part) production at EicC (Q2 = 2 GeV2, $| t^{\prime} | $ = 0.1 GeV2). |
Note that the transversity dominance σT ≫ σL that was tested for π0 production at high energies is valid for η production at the energies W = 2 ∼ 15 GeV. This means that in experimental analyses of transversity convolutions, unseparated cross section σ = σT + εσL can be applied instead σT for both processes of π0 and η production.
4. Conclusion
In this paper, we investigate the exclusive electroproduction of pseudoscalar π0 and η meson at China EicC energies. The process amplitudes are calculated in the model where amplitudes are factorized into subprocess amplitudes and GPDs. For the transversity twist-3 effects, the subprocess amplitude ${{ \mathcal H }}_{0-,++}$ is the same in equation (10 ) for both contributions that contain HT and ${\bar{E}}_{T}$ GPDs.
We consider two GPDs parameterization Model-2 and Model-3. Both models describe properly π0 and η production at CLAS energies. It seems that Model-3 gives better results for π0 production at COMPASS for large momentum transfer and gives better descriptions of η production at CLAS at momentum transfer ∣t∣ < 0.5 GeV2.
We perform predictions for unseparated σ and σTT cross sections for EicC kinematics for η production with Model-2 and Model-3. We observe that transversity dominance σT ≫ σL, found at low CLAS energies [27] and confirmed at EicC energies in [25] for the π0 process is valid at all these energies for η production too.
Adopting a combination of the cross sections, we extract GPDs HT and ${\bar{E}}_{T}$ convolutions determined in equation (4 ) for the cases of π0 and η mesons. These results for EicC energies are quite different that give possibilities to determine a preferable model at future experiments. In addition, we analyze energy dependencies of transversity convolutions at fixed $t^{\prime} $ and Q2 that can give information about energy parameters of GPDs HT and ${\bar{E}}_{T}$ from the data.
Note that the reactions π0 and η production considered have different flavor contributions to the amplitudes. This provides a possibility to perform u and d flavor separation for transversity GPDs [29].
Our results can be useful in future experiments at China EicC on the pseudoscalar mesons production and give more important knowledge on transversity influences at these energy ranges.