Consider the lowest order interaction e.g. e- e+ -> virtual photon-> muon anti muon. I appreciate that the electron-positron pair cannot annihilate into a real photon due to conservation of 4-momentum, but why is the pair permitted to produce a virtual photon? I know that virtual particles are "off the mass shell" insofar as energy,momentum and charge are conserved at the vertices and thus virtual photons are allowed to have non-zero mass, but how can I proceed from here to convince myself? I could argue that since this virtual photon has non-zero mass, it can stay at rest in the COM frame unlike the massless real photons, but why do physicists represent this intermediate stage with a photon in the first place? I could also argue that since it only exist for a short duration of time Δt and can only be localized with uncertainty Δx=cΔt the HUP tells me there would be a minimum uncertainty in momentum Δp such that the virtual photon can appear to have zero momentum, but it doesn't sound very convincing either and that I might have misused HUP here.