  Generalized distribution amplitudes (GDAs) are one type of three-dimensional
structure functions, and they are related to the generalized distribution
functions (GPDs) by the $s$-$t$ crossing of the Mandelstam variables. The GDA
studies provide information on three-dimensional tomography of hadrons. The
GDAs can be investigated by the two-photon process $\gamma^* \gamma \to h\bar
h$, and the GPDs are studied by the deeply virtual Compton scattering $\gamma^*
h \to \gamma h$. The GDA studies had been pure theoretical topics, although the
GPDs have been experimentally investigated, because there was no available
experimental measurement. Recently, the Belle collaboration reported their
measurements on the $\gamma^* \gamma \to \pi^0 \pi^0$ differential cross
section, so that it became possible to find the GDAs from their measurements.
Here, we report our analysis of the Belle data for determining the pion GDAs.
From the GDAs, the timelike gravitational form factors $\Theta_1 (s)$ and
$\Theta_2 (s)$ can be calculated, which are mechanical (pressure, shear force)
and mass (energy) form factors, respectively. They are converted to the
spacelike form factors by using the dispersion relation, and then gravitational
radii are evaluated for the pion. The mass and mechanical radii are obtained
from $\Theta_2$ and $\Theta_1$ as $\sqrt {\langle r^2 \rangle_{\text{mass}}}
=0.56 \sim 0.69$ fm and $\sqrt {\langle r^2 \rangle_{\text{mech}}} =1.45 \sim
1.56$ fm, whereas the experimental charge radius is $\sqrt {\langle r^2
\rangle_{\text{charge}}} =0.672 \pm 0.008$ fm for the charged pion. Future
developments are expected in this new field to explore gravitational physics in
the quark and gluon level.
