Hadron tomography for pion and its gravitational form factors

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.

GDA is an important quantity of hadron, it is defined as In the process !! * →h bar{h}, an hard part describing the process !! * →q bar{q} with produced collinear and on-shell quark, and a soft part describing the production of the hadron h pair from a q bar{q}.This soft part is called Generalized Distribution Amplitude (GDA). The process ! * ! h bar{h} γ(q 2 λ 2 )

Generalized distribution amplitude for pion
GDA is closely related to generalized parton distribution (GPD) by the s-t crossing, so GDA could provide another way to obtain GPD information.
GPD can be used to study the proton spin puzzle!
The cross section of process ! * ! π 0 π 0 A λ1λ2 is the helicity amplitude, and there are 3 independent helicity amplitudes, they are A ++ , A 0+ and A +-. The leading-twist amplitude A ++ has a close relation with the generalized distribution amplitude (GDA) Φ q (z, ξ, W 2 ).
Higher-twist contribution A 0+ requires a helicity flip along the fermion line, and it decreases as 1/Q. Higher-order contribution A +-contributes with the GDA of gluon, since A +-indicates the angular momentum L z =2 .Therefore A +-is suppressed by running coupling constant α s.

Higher twist and higher order contributions
The GDAs are related to the energy-momentum form factor in the timelike region.
where the energy-momentum form factor for quarks is defined as By using this sum rule we can obtain where R π is the momentum fraction carried by quarks in the pion. At very high energy Q 2 , we can have the asymptotic form of the GDA

GDA expression
In 2016, the Belle Collaboration released the measurements of differential cross section for ! * !→π 0 π 0 . The GDAs can be obtained by analyzing the Belle data.
Differential cross section for ! * ! π 0 π 0 In these figures, the resonance f 2 (1270) is clearly seen around W = 1.25 GeV, however, other resonances are not clearly seen due to the large errors.

Scale violation of GDA based on Belle data
The scale dependence of the Belle data. We have red color for W = 0. 525 GeV, blue color for W = 0. 975 GeV, and green color for W = 1. 55 GeV.
The scaling violation of the GDAs is not so obvious in the Belle data on account of the large errors, so that the Q 2 -independent GDAs could be used in analyzing the Belle data.

Q 2 -independent (asymptotic form) GDAs
In the above equation δ 0 and δ 2 and are the ππ elastic scattering phase shifts in the isospin=0 channel (see the figure). Above the KK threshold, the additional phase is introduced for S-wave

Resonance effects
In the process ! * !→π 0 π 0 , the π 0 π 0 can be produced through intermediate meson state h. The q bar{q}→h amplitude should be proportional to the decay constant f h or the distribution amplitude (DA), and the h→π 0 π 0 amplitude can be expressed by the coupling constant g hππ . These resonance contributions read The resonance effects play an important role in the resonance regions.
We adopt a simple expression of GDA to analyze Belle data, here resonance effects of f 0 (500) and f 2 (1270) are introduced.
The function F h (W 2 ) is the form factor of the quark part of the energymomentum tensor, and the parameter Λ is the momentum cutoff in the form factor. The parameter n is predicted as n = 2 at very high energy, because we have dσ/d|cosθ|/ 1/W 6 by the counting rule. In the asymptotic limit, " = 1. By analyzing the Belle data, the values of parameters are obtained

Set
Set 1 is the analysis without the resonance effect f 0 (500), in Set 2 the resonance effect f 0 (500) is included.