Purely electro-magnetically before a proton (+) and an anti-proton (-) annihilate each other /assume they do/, they form a dipole. What happens after that? In other words, what does an external observer detect - a sudden disturbance of the dipole field or some continuous change from a dipole-field to a monotone vacuum background?
The proton and the antiproton are in a quantum mechanical framework,not in a classical one with dipoles and antennas. In the quantum framework the interactions of particles are governed by the solution of quantum mechanical equations which will give the probabilities for the annihilation taking place, and for the appearance of new particles. The necessary integrals are shown in the Feyman diagrams where strict rules apply transferring line to functions and integrals.
Here is one possibility of what happens when proton annihilates on an antiproton at high energies. A top antitop pair can be produced through the strong interaction.
An equivalent diagram exists where the exchange is a photon. Due to the much smaller coupling constant of the electromagnetic interaction that you require the probability of this happening is very small with respect to the strong annihilation, and this holds true from low energies to higher.
There exist experimental limits for the reaction antiproton proton annihilating to e+e-.