but not explicitly derived in ref. [@Kurkela:2009gj]. They concern Feynman integrals at zero temperature and finite fermionic chemical potentials, and reduce the evaluation of the original One-Particle-Irreducible (1PI) Feynman graph to the computation of three-dimensional phase space integrals over on-shell vacuum ($T=\mu=0$) amplitudes. This represents a remarkable simplification for practical calculations, as there is a vast amount of literature on vacuum amplitudes that can be directly taken over. This significantly streamlines the evaluation of multi-loop Feynman graphs.

Although our derivation of the cutting rules utilizes the imaginary-time formalism of thermal field theory, it is interesting to note that the result appears to have an intimate connection to the real-time formalism as well. In particular, it can be shown that (modulo some simple additional assumptions) our cutting rules would emerge from a naive replacement of Euclidean propagators by the time-ordered propagators of the real-time formalism, closely reminiscent of eq. (4) of ref. [@Andersen:2000zn]. This is sometimes referred to as the “naive real-time formalism”. This result dates back to the much earlier work of Dashen et al. [@Dashen:1969ep], where a connection between certain statistical-physics quantities and scattering amplitudes was proposed, and it has since then been developed, e.g., in [@Frenkel:1992az; @Bugrii:1995vn]. It 