Hadron tomography for pion and its gravitational form factors

Abstract: Generalized parton distributions (GPDs) are three-dimensional structure functions of hadrons, and they can reveal the orbital-angular-momentum contributions to the nucleon spin. Therefore, GPDs are important for solving the proton spin puzzle. The generalized distribution amplitudes (GDAs) are the $s$-$t$ crossed quantities of the GPDs, and the GDAs can be investigated in two-photon process ($\gamma^* \gamma \to h\bar h$) which is accessible at KEKB. The pion GDAs were obtained by analyzing the Belle measurements for $\pi^0 \pi^0$ production in the $e^+ e^-$ collision. From the obtained GDAs, the form factors of energy-momentum tenor were calculated for pion in the timelike region. In order to study the gravitational radii for the pion, the form factors of energy-momentum tenor were obtained in the spacelike region by using the dispersion relation. Then, the mass radius was calculated as 0.32 $\sim$ 0.39 fm and the mechanical radius was also estimated for the pion as 0.82 $\sim$ 0.88 fm by using the spacelike form factors. This is the first finding on gravitational form factors and radii of hadrons from actual experimental measurements. In the near future we can expect more precise measurements of $\gamma^* \gamma \to h\bar h$ as the Belle II started data taking by the higher luminosity Super KEKB, so that the GDAs of other hadrons could be studied as well.