tries to extend perturbative studies of the thermodynamics of cold and dense systems to higher loop orders. The rules were an integral part of the determination of the three-loop EoS of cold quark matter in ref. [@Kurkela:2009gj], and it is because of them that an extension of this result to the full four-loop order is feasible. In this context, there are in fact two separate challenges: in addition to the evaluation of all four-loop bubble diagrams one needs to determine (a specific component of) the gluon polarization tensor to two-loop order. The latter of these two computations is alone sufficient for determining the logarithmic contributions $\alpha_s^3\ln^2\,\alpha_s$ and $\alpha_s^3\ln\,\alpha_s$ to the EoS. This work is near completion, and the results will be presented in a separate publication later [@nextpaper].

To conclude, let us briefly return to the connection between our work and the naive real-time formalism discussed in sec. \[sec:intro\]. In proving the validity of the zero-temperature cutting rules, we have, in effect, also shown that the naive real-time formalism is applicable not only for vacuum diagrams, contributing to the free energy, but also for Euclidean $n$-point functions in the $T=0$ limit. At the same time, we know that the multitude 