Diffractive rho + lepton pair production at an electron-ion collider

In high energy electron-ion colliders, a new way to probe nucleon structure becomes available through diffractive reactions, where the incident particle produces a very energetic almost forward particle. QCD describes these reactions as due to the exchange of a Pomeron which may be perturbatively described as a dressed two-gluon state, provided a hard scale allows the factorization of the amplitude in terms of two impact factors convoluted with a Pomeron propagator. We consider here a process where such a description allows to access hadronic structure in terms of the generalized parton distributions, namely the electroproduction of a forward $\rho$ meson and a timelike deeply virtual photon, separated by a large rapidity gap. We explore the dependence of the cross section on the kinematic variables and study the dependence on the non-perturbative inputs (generalized parton distributions, distribution amplitude). Our leading order studies show the cross section is mainly sensitive to the GPD model input, but the small size of the cross sections could prohibit straightforward analysis of this process at planned facilities.