Non-diagonal DVCS with spinless hadron-to-two-hadron transition $\gamma^* \pi \to \gamma \pi \pi$

We present the formalism of hadron-to-two-hadron transition generalized parton distributions (GPDs) for spinless hadron case. Definitions of the twist-2 unpolarized and polarized $\pi\to\pi\pi$ transition GPDs are introduced with a particular choice of kinematic variables that characterize the produced two-pion system. In the vicinity of $\rho(770)$, we work out the two-pion decay angular distributions of the $e^-\pi\to e^-\gamma\pi\pi$ cross section, incorporating both the Bethe-Heitler and deeply virtual Compton scattering processes. Each cross section exhibits a distinctive angular distribution, which is sensitive to the polarization states of the produced $\rho(770)$ resonance. In addition, we construct the double partial-wave expansion of $\pi\to\pi\pi$ transition GPDs in the two-pion decay angles as a generalization of GPDs for transition from a pion to a spin-$\ell$ resonance state. With the help of the Omn\`es representation, we constrain the $\pi\to\pi\pi$ transition GPDs in terms of the low energy $\pi\pi$ scattering phase shift and build a phenomenological model for these GPDs.