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Notation and statement of the rules \[sec:state\]
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We work with Euclidean signature Feynman graphs at zero temperature and finite chemical potentials. This means that we can think of our diagrams as consisting of two types of propagators, “fermionic” $1/((q_0+i\mu)^2+(E_q^i)^2)$ and “bosonic” $1/(q_0^2+(E_q^i)^2)$, where $E_q^i\equiv \sqrt{q^2+m_i^2}$ and $\mathbf{q}$ represents a spatial momentum vector. Consistently with the fermionic nature of the chemical potential, we assume $\mu$ to be larger than the mass of the corresponding field. Divergences are finally regulated via dimensional regularization by working in $d=3-2\epsilon$ spatial dimensions, defining $$\begin{aligned}
\int_Q\,\equiv\, \int_{-\infty}^\infty \frac{dq_0}{2\pi}\int_q \,\equiv\,\int_{-\infty}^\infty \frac{dq_0}{2\pi} \int\frac{d^{d}q}{(2\pi)^{d}}\, ,\end{aligned}$$ where $Q$ denotes a Euclidean four-vector, such that $Q^2\equiv q_0^2+q^2$. Before stating the cutting rules, we make a few simplifying assumptions, the purpose of which is to keep our presentation as concise and readable as possible:

-   There is no structure in the numerator of the original Feynman integral, i.e. we consider scalar propagators and trivial vertex functions.

-   No individual propagator is raised to a power higher than one.

-   There is only one chemical potential appearing in the graph.

-   In the external momenta $P_k=(p_0^k,\mathbf{p}_k)$, the $p_0^k$ are always real-valued, corresponding to imaginary frequencies $\omega_k$.

As will be discussed in 