section \[sec:conc\], the first three of these assumptions can be easily relaxed, but at the cost of making the notation somewhat more convoluted. Note, however, that we have made absolutely no assumptions about the masses of the propagators, so that they can and will be considered independent.

Under the above assumptions, let us consider an arbitrary 1PI $N$-loop $n$-point Feynman graph $F(\{P_k\},\mu)$, where the $P_k$, $k=1,2,...,n$ stand for the external momenta. According to the cutting rules, we may write this function in the form $$\begin{aligned}
F(\{P_k\},\mu)&=&F_\text{0-cut}(\{P_k\})+F_\text{1-cut}(\{P_k\},\mu)+\cdots+F_\text{$N$-cut}(\{P_k\},\mu), \end{aligned}$$ where $F_\text{0-cut}(\{P_i\})$ is simply the original graph evaluated at vanishing chemical potential, $\mu=0$, while the remaining pieces result from the cutting procedure. In particular, $F_\text{$j$-cut}(\{P_k\})$ denotes the sum of all so-called “$j$-cut” diagrams, in which exactly $j$ of the internal fermionic propagators have been cut off. This cutting procedure involves the following steps:

1.  Removing the cut propagators from the original graph.

2.  Evaluating the resulting $N-j$ -loop $n+2j$ -point amplitude at $T=\mu=0$, assuming all external momenta to be real-valued.

3.  Setting the cut momenta $Q_i$ on shell, i.e. writing $q_0^i=iE_i$ for each of them.

4.  Integrating the resulting expression over the cut three-dimensional momenta with the weights $-\theta(\mu-E_i)/(2E_i)$.

An important additional rule 