Abstract
Hadron tomography can be investigated by three-dimensional structure functions such as generalized parton distributions (GPDs), transverse-momentum-dependent parton distributions, and generalized distribution amplitudes (GDAs). Here, we extract the GDAs, which are $s$-$t$ crossed quantities of the GPDs, from cross-section measurements of hadron-pair production process $\gamma^* \gamma \rightarrow \pi^0 \pi^0$ at KEKB. This work is the first attempt to obtain the GDAs from the actual experimental data. The GDAs are expressed by a number of parameters and they are determined from the data of $\gamma^* \gamma \rightarrow \pi^0 \pi^0$ by including intermediate scalar- and tensor-meson contributions to the cross section. Our results indicate that the dependence of parton-momentum fraction $z$ in the GDAs is close to the asymptotic one. The timelike gravitational form factors $\Theta_1$ and $\Theta_2$ are obtained from our GDAs, and they are converted to the spacelike ones by the dispersion relation.From the spacelike $\Theta_1$ and $\Theta_2$, gravitational densities of the pion are calculated. Then, we obtained the mass (energy) radius and the mechanical (pressure and shear force) radius from $\Theta_2$ and $\Theta_1$, respectively. They are calculated as $\sqrt {\langle r^2 \rangle _{\text{mass}}} =0.32 \sim 0.39$ fm, whereas the mechanical radius is larger $\sqrt {\langle r^2 \rangle _{\text{mech}}} =0.82 \sim 0.88$ fm. This is the first report on the gravitational radius of a hadron from actual experimental measurements. It is interesting to find the possibility that the gravitational mass and mechanical radii could be different from the experimental charge radius $\sqrt {\langle r^2 \rangle _{\text{charge}}} =0.672 \pm 0.008$ fm for the charged pion.
Highlights
Internal structure of hadrons has been investigated in terms of form factors and parton distribution functions (PDFs)
generalized distribution amplitudes (GDAs) studies will provide another information on the ERBL generalized parton distributions (GPDs) it is in the unphysical region of t > 0
The GDAs are one of three-dimensional structure functions, and they are related to the GPDs by the s − t crossing relation
Summary
Internal structure of hadrons has been investigated in terms of form factors and parton distribution functions (PDFs). Form factors contained in the GDAs should provide information whether exotic hadron candidates are diffuse molecular states or compact multiquark ones [4]. Another advantage is that the GDAs and GPDs contain information on form factors of the energy-momentum tensor so that the gravitational-interaction radius can be investigated. The Belle collaboration recently reported the cross sections for the pion-pair production in two-photon process γÃγ → π0π0 at KEKB with various kinematical conditions [6,7] It is our research purpose of this paper to extract the pion GDAs from the Belle measurements.
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