Hadron tomography by generalized distribution amplitudes in pion-pair production process $\gamma^* \gamma \rightarrow \pi^0 \pi^0 $ and gravitational form factors for pion

TL;DR

This paper extracts generalized distribution amplitudes from experimental data of pion-pair production, enabling the first measurement of the pion's gravitational form factors and radii, revealing insights into hadron structure.

Contribution

It is the first to determine GDAs and gravitational form factors of the pion directly from experimental data, linking hadron tomography with gravitational properties.

Findings

Gravitational radii of the pion are estimated as 0.32-0.39 fm (mass) and 0.82-0.88 fm (mechanical).

GDAs show dependence close to the asymptotic form.

First experimental determination of hadron gravitational radii.

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.