Elucidating the rho-meson's role as intermediate resonance in the time-like electromagnetic pion form factor

Motivated by the planned measurements of the time-like electromagnetic proton form factors an exploratory study of the time-like electromagnetic pion form factor is presented in a formalism which describes mesons as Poincar\'e-invariant bound states. In the respective quark interaction kernel, beyond the gluon-intermediated interactions for valence-type quarks, non-valence effects are included by allowing the pions to couple back in a self-consistent manner to the quarks. Consequently, the opening of the dominant $\rho$ decay channel, $\rho \to \pi\pi$, and the presence of a multi-particle branch cut, setting in when the two-pion threshold is crossed, are included consistently, first in the correspondingly calculated quark-photon vertex, and then consequently in the time-like electromagnetic pion form factor. The obtained results are in agreement with the available experimental data and provide further evidence for the efficacy of a vector-meson dominance model. Effects of going beyond the here employed isospin-symmetric limit are discussed. Last but not least, an outlook is provided on how to include in Poincar\'e-covariant Faddeev approaches the effects of intermediate resonances in the calculation of baryon properties.