The only known "charged massive photons" are the two W-bosons, carrying charged weak interactions. Exchange of charged quanta cannot bind two electrons into a Cooper pair, because the emission/absorption of one such quantum would change the charge of the emitting/absorbing electron -- after which it is no longer an electron.

I am puzzled why you ask these questions.

The only known "charged massive photons" are the two W-bosons, carrying charged weak interactions. Exchange of charged quanta cannot bind two electrons into a Cooper pair, because the emission/absorption of one such quantum would change the charge of the emitting/absorbing electron -- after which it is no longer an electron.

I am puzzled why you ask these questions.

Dear Kare,

Thanks you for  your reply. Why should you be puzzled? Shouldn't I ask such questions? Why not? Can you tell me? It is possible to have this possibility, and see below the Feynman diagram for such an interaction.

There are no known "massive photons" with charge -2e; hence the process you depict does not exist as a realistic possibility in condensed matter physics.

It is not necessarily to call it a "massive photon", but it is some kind of a boson field that mass and charge that could be present inside the superconductor to mediate the electrons interactions. It is a new suggestion: let the future decide!

But there are no such thing in any known material! It is unclear why/if the presence of something like that in your future, cleverly designed, material would trigger the formation of Cooper pairs. Or, more precisely, breakdown of the Fermi surface. Keep in mind that superconductivity occur in materials with a finite density of electrons. Therefore, the best description is in terms of particle-hole excitations in the vicinity of the Fermi surface, which is not described by vacuum Feynman diagrams like the one you have drawn.

What is great in diagram technique is the fact that there is no need for the agent in the middle to exist. Absolutely. Put there a dragon or a vampire.

But

1. You have to sum over all possible diagrams including vampires and dragons. 

2. The points where lines meet stand for interaction. There is an electron-phonon interaction in the Hamiltonian of the solid. That's why it contributes.

Existence of charged boson is can rise some question about it's mass, or other properties. it's absorption by a charged particle can be even more problematic. If we ignore some problem, you diagram seems more like to creation-annihilation process for two electron than interaction between cooper pairs. 

The photon doesn't carry electric charge. The electrons, in a medium, are subject to two interactions: the repulsive interaction, mediated by the photon (that carries spin 1) and the attractive interaction, mediated by the phonon (that carries spin 0, but doesn't carry electric charge, either: it's the excitation of the medium, the electrons are in and its interaction with the electrons describes mathematically the statement that the electrons are in a definite medium.). When the attraction dominates, bound states can form, the Cooper pairs. They carry net electric charge and so do continue to interact with photons. 

But  the electromagnetic interaction is screened-which is one way of saying that the photon acquires a mass. 

So the only charged bosons in this context, are the Cooper pairs, in fact, whose condensation leads to superconductivity. 

All this is described in standard texts on superconductivity. 

(For completeness, since it was brought up: the Z bosons are electrically neutral but do carry weak charge; the W bosons carry both electric and weak charge. Neither is relevant to the discussion, however. While, in superconductivity, the photon does acquire a mass, the mechanism is different in a subtle way, noticed by W. Gilbert,  from the Brout-Englert-Higgs mechanism, that leads to  the W and Z bosons of the weak interactions becoming massive.)

I think @Arbab question is about hypothetical process and not current known physical processes.

One can say that the two electrons condense in a massive boson and this massive boson then disintegrates over a distance L=h/mc, where here m is the boson mass which is 10^-36 kg.

No, one can't say this. What one can say is known as superconductivity. Cooper pairs aren't hypothetical and the mechanism that binds them is known-the competition between photon and phonon exchange. 

Phonons don't carry any charge. 

Arbab> One can say that the two electrons condense in a massive boson

You seem to be thinking of the Landau-Ginzburg phenomenological description of superconductivity, where the Cooper-pairs are described by an effective, non-relativistic, charge 2e, scalar field. This is a useful description, given that the pairs have already formed (it is more a pairing in reciprocal space than a binding into two-electron bound states in configuration space), not for explaining the physical mechanism through which they were formed (through phonon exchange in ordinary BCS superconductivity).