proceed is by first evaluating the corresponding graph with the propagator raised to power 1, and then (repeatedly) differentiating the result with respect to the mass squared of the propagator in question, relying on the formula $$\begin{aligned}
      \frac{1}{(Q^2+m_i^2)^n}&=&\frac{(-1)^{n-1}}{(n-1)!}\frac{d}{dm_i^2}  \frac{1}{Q^2+m_i^2}\,.
     \end{aligned}$$ A possible caveat here has to do with massless propagators raised to higher powers and the associated physical IR divergences. If we introduce a mass term for such a line and then differentiate with respect to it, this will in general produce a $1/m_i^k$ term in the $m_i\to 0$ limit. Some extra effort will then be required to convert this divergence into a $1/\epsilon$ pole, as expected in dimensional regularization.

-   Having several closed fermion loops in the graph, each with an independent chemical potential, clearly produces a mere notational complication, and the form of the result stays exactly the same as above. We only need to keep track of the correspondence of the chemical potentials with the cut fermion lines.

Together, these three generalizations allow us to tackle all Feynman integrals encountered in gauge field theories coupled to Dirac fermions, such as QED or QCD.

As discussed already in section \[sec:intro\], the cutting rules become increasingly important when one 