Temperature reduction in 4D QED I would like to find references for the following topic.
Consider QED with non-zero temperatures, which is naively constructed by Wick rotation. Then, consider the case of high temperatures, $\beta\rightarrow 0$. In this limit (I am not sure!) reduction occurs. It means that 4-potential should be considered as two separate quantities: massless vector field ${\bf A}$ (which corresponds to goldstone mode for U$(1)$) and massive scalar field $\phi$.
So, it seems that in thermal 4D QED after temperature reduction U$(1)$ symmetry is broken. It is very strange for me.
However, my lecturer describes this phenomenon briefly and gives only one reference: "The Hagedorn transition and the number of degrees of freedom of string theory", Atick & Witten, 1988.
Unfortunately, I cannot find any relevant info about 4D thermal QED in this paper.
 A: My question is relatively well-known. Indeed, in hot thermal loops one can see that 4-vector decouples into times and spatial components. The theory now has only 3D gauge invariance. Time-component of 4-vector field obtains mass by loop corrections.
In one loop approximation, the photon mass is given by the series of fermion bubbles with external photon legs. It is retively easy to show the first correction, vacuum polarization, gives the following photon mass
$$m_{\gamma}^2=\frac{e^2T^2}{3}.$$
My naive understanding of the problem was wrong. I forget about large temperatures. Indeed, the answer for 4-legs correction is very similar with the expression for gap in BCS theory,
$$-\frac{7\xi(3)e^2m_e^4}{16\pi^4T^2},$$
but the point is that in the limit $T\rightarrow \infty$ this contribution vanishes. It will be interesting to consider case of very heavy fermions in theory, $m_e/T\sim 1$. For heavy fermions it seems that  this contribution doest not vanish. But of course, may be higher order terms will contain higher powers of $m_e$ and 1-loop action will be divergent.
