E_{q}) }{2 E_{q}}
        \begin{gathered}
            \begin{fmfgraph*}(80,40)
                \fmfleft{ind4,inu4}
                \fmfright{outd4,outu4}
                \fmf{dashes,tension=0.3}{ind4,vin4,inu4}
                \fmf{dashes,tension=0.3}{outd4,vout4,outu4}
                \fmf{dots,width=2,tension=0.2}{vin4,vout4}
            \end{fmfgraph*}
        \end{gathered}      
            \Bigg{|}_{p_{0}\rightarrow iE_{p},q_{0}\rightarrow iE_{q}}, \label{sunset2}\end{aligned}$$ which is nothing but the last term of eq. (\[I2int\]). Thus, the cutting rules do indeed reproduce the full result for the graph considered. It is worth pointing out here that this result is by no means new and only serves as a pedagogical introduction to our discussion; in a finite-$T$ context, a similar result has been obtained, e.g., in Appendix A of [@Laine:2006cp].

Proof of the rules \[sec:proof\]
================================

Organization of the proof
-------------------------

In this section, we provide a proof of the cutting rules in the case of a generic 1PI Feynman graph at zero temperature but finite chemical potential. For reasons of clarity, we do this in three sequential steps, of which the first two can be considered useful lemmas while the connection to the cutting rules is made in the third part of the proof:

1.  *Vanishing-chemical-potential case*: Consider a generic Euclidean Feynman integral at zero temperature and vanishing chemical potential, corresponding to a 1PI $N$-loop $n$-point diagram. Such a graph consists of some (potentially large) number $M$ of internal lines, which we enumerate by the index $\alpha=1,2,...,M$. The corresponding 