Isolating vacuum amplitudes in quantum field calculations at finite temperature

In calculating Feynman diagrams at finite temperature, it is sometimes convenient to isolate subdiagrams which do not depend explicitly on the temperature. We show that, in the imaginary time formalism, such a separation can be achieved easily by exploiting a simple method, due to Gaudin, to perform the sum over the Matsubara frequencies. In order to manipulate freely contributions which may be individually singular, a regularization has to be introduced. We show that, in some cases, it is possible to choose this regularization in such a way that the isolated subdiagrams can be identified with analytical continuations of vacuum n-point functions. However, at least with the regularization used in this paper, this simple analytical structure does not hold for arbitrary diagrams, as revealed by counter-examples. As an aside illustration of Gaudin's method, we use it to prove the main part of a recent conjecture for a relation, in the imaginary time formalism, between the expressions of a Feynman diagram at zero and finite temperature.