In the preceding chapter we have learned that the leading behaviour in temperature of the gauge particle and fermion self-energy is proportional to T2, and that this behaviour is obtained without too much effort in the HTL approximation. In the present chapter we generalize these results to N-point functions, computed at the one-loop approximation. We shall show that some (but not all!) N-point functions also behave as T2, and that these N-point functions have rather simple expressions in the HTL approximation. This situation should be constrasted with that of the ϕ4-theory, where only the two-point function behaves as T2: once more, gauge theories are much richer than scalar theories.

We shall also discover that these N-point functions obey remarkable Ward identities, and that there are again striking similarities between QED and QCD. All our results can be expressed in a compact way by writing an effective Lagrangian. Most importantly, we shall show how to correct naïve perturbation theory, which breaks down for soft external momenta, by using a resummed (or effective) perturbative expansion. Some applications to physical processes will be given in section 7.3, and more will be given in the following chapters. We conclude with a kinetic derivation of hard thermal loops, which generalizes the results of section 6.4.