Non-diagonal DVCS and transition GPDs: A unified framework for spinless hadron case $\gamma^* \pi \to \gamma \pi \pi$

Hadron-to-two-hadron transition generalized parton distributions (GPDs) extend the concept of hadron-to-resonance transition GPDs and provide a unified description of non-diagonal hard exclusive reactions in the generalized Bjorken limit. We present the formalism for the case of spinless hadrons addressing the non-diagonal deeply virtual Compton scattering $\gamma^*\pi\to\gamma\pi\pi$ in terms of $\pi\to\pi\pi$ transition GPDs, which generalize GPDs for $\pi \to f_0, \, \rho, \, f_2, \, \cdots$ transitions. We work out the basic properties of $\pi\to\pi\pi$ transition GPDs and establish the soft pion theorems at the $2\pi$ production threshold. We construct the partial wave expansion of $\pi\to\pi\pi$ transition GPDs in the two-pion decay angles and employ the dispersive approach to constrain $\pi\to\pi\pi$ transition GPDs in terms of $\pi\pi$-scattering phases with help of the Omn\`{e}s representation. We estimate the $e^-\pi^+ \to e^- \gamma \pi^+ \pi^0$ cross section in the kinematics of the JLab@12 GeV incorporating the isolated $\rho(770)$ resonance state and work out the angular distributions of the cross section, specifying the observables sensitive to the polarization states of the produced $\rho(770)$ resonance.