Feynman diagram of electron- electron interaction and cool down of an electron gas Reading this answer I wondered if during an electron-electron interaction real photons have to be emitted. Since the interaction of two electrons is accompanied by the change of direction and velocity it takes place an acceleration and every acceleration of electrons is accompanied by the emission of photons (and slowdown of the electrons). On the other side I've never seen a Feynman diagram with e-e interaction and real photon emission.
Will an electron gas cool down, not taking in account the interaction with the boundary walls?
 A: Here is the Feynman diagram, it is the Brehmstrahlung one for an electron in the field of another electron. One should add the symmetric one to find the total amplitude at this order.

Yes, electrons scattering off electrons will radiate as in the diagram, with smaller probability by 1/137 at least, due to the extra electromagnetic vertex, and the gas will slowly cool.
A: 
Reading this answer I wondered if during an electron-electron interaction real photons have to be emitted. 

They don't. The virtual photons of the electromagnetic interaction are virtual. They aren't short-lived real particles that pop in and out of existence like magic. See anna's answer here. Virtual particle only exist in the mathematics of he model. 

Since the interaction of two electrons is accompanied by the change of direction and velocity it takes place an acceleration and every acceleration of electrons is accompanied by the emission of photons

Not every acceleration. If you drop an electron it doesn't emit a photon. If you put it in a uniform electric field it doesn't emit a photon. 

On the other side I've never seen a Feynman diagram with e-e interaction and real photon emission.

You have in the guise of electron-positron annihilation. There is also cyclotron radiation. I imagine you could contrive that with just electrons.  

Will an electron gas cool down, not taking in account the interaction with the boundary walls?

Maybe, see above. It depends. But if you simplify it to one electron in vacuum, it won't slow down unless it interacts with something else. 
