Annihilation of $bar p+pto e^++e^-+ pi^0$ and $bar p+pto gamma + pi^0$ through $omega$-meson intermediate state

The s-channel annihilation of proton and antiproton into a neutral pion and a real or virtual photon followed by lepton pair emission is studied. Such mechanism is expected to play a role at moderate values of the total energy $sqrt{s}$, when the pion is emitted around $90^{circ}$ in the center of mass. A fair comparison with the existing data is obtained taking into account scattering and annihilation channels. The cross section is calculated and numerical results are given in the kinematical range accessible in the PANDA experiment at FAIR.