is that those cuts that divide the original graph into two or more disconnected pieces are to be thrown out.

The usefulness of the cutting rules originates from the fact that they isolate the chemical-potential dependence of the original graph in the $\theta$-function factors in the three-dimensional “phase space” integrations. Owing to the abundance of analytic results for vacuum amplitudes in the literature, one typically only needs to perform (some of) these phase space integrations numerically, which is an enormous simplification.

Example calculation: standard technique
---------------------------------------

![\[fig:mot1\] A two-loop diagram contributing to the partition functions of QED and QCD, with the wiggly line corresponding to a gauge boson (photon or gluon) and the solid lines to fermions (electrons or quarks).](twoloop_gluons.pdf){width=".8\linewidth"}

To illustrate the use of the cutting rules as well as motivate their introduction, we consider next an elementary practical example in the form of a two-loop integral encountered when considering the graph of fig. \[fig:mot1\], appearing in the ${\mathcal O}(\alpha_s)$ contribution to the EoS of QED or QCD matter. At zero temperature, the most nontrivial master integral resulting from this diagram is the two-loop “sunset”

[2loop]{} $$\begin{aligned}
\label{eq:abc2}
\!\!I_2(\mu)&\equiv&\!\!\!\!\!\!
\begin{gathered}
            \begin{fmfgraph*}(65,60)
                \fmfleft{in1}
                \fmfright{out1}
                \fmf{phantom,tension=5}{in1,vin1}
                \fmf{phantom,tension=5}{vout1,out1}
                \fmf{plain,left,tension=0.4}{vin1,vout1,vin1}
                \fmf{dots,width=2}{vin1,vout1}
            \end{fmfgraph*}
        \end{gathered}
 