that do not belong to the set $S_r$ and the explicit forms of the $R_\alpha$ in terms of the $Q_i$ and $P_k$ are dictated by $S_r$. Note that each set $S_r$ is to be counted only once here, and the labeling of the momenta within $S_r$ (including changing the direction of some loop momenta, $Q_i\to-Q_i$) plays no role: it is only the choice of propagators that counts.

2.  *Generalization to finite density*: Assume next that some of the internal propagators of the graph are fermionic in the sense that they carry a chemical potential $\mu$ in the way stated in the previous section. The above result for $I$ is then modified only by the factors $1/(2E_i)$ corresponding to the internal fermion lines changing according to $$\begin{aligned}
    \frac{1}{2E_i}&\to& \frac{\theta(E_i-\mu)}{2E_i} \, .\end{aligned}$$ Note that for different $S_r\in S$, the numbers of fermionic momenta and thus $\theta$-function factors are typically different.

3.  *Connection to the original cutting rules*: Writing the $\theta$-functions in the form $\theta(E_i-\mu) = 1-\theta(\mu-E_i)$ and rearranging terms, the above results reduce to the cutting rules stated in section \[sec:cut\].

We now proceed to prove these three claims, thereby deriving the cutting rules.

Vanishing-chemical-potential case
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Given a random choice of 