generalized to nonzero quark masses [@Kurkela:2009gj] (see also [@Fraga:2004gz]) and small but nonvanishing temperatures [@Kurkela:2016was]. Extending these zero-temperature results to higher orders, however, presents a considerable technical challenge. Similarly to the case of high temperatures [@Kajantie:2002wa; @DiRenzo:2006nh], part of the problem in extending these results lies in understanding how to handle the contributions of the soft momentum scales to the quantity. These difficulties will be addressed in a forthcoming publication, containing the logarithmic $\alpha_s^3\ln\,\alpha_s$ and $\alpha_s^3\ln^2\,\alpha_s$ contributions to the perturbative EoS [@nextpaper]. A more challenging part of the full ${\mathcal O}(\alpha_s^3)$ result is, however, the contribution of the hard energy scale $\mu_B$, i.e. the baryon chemical potential, which is obtained from the sum of all four-loop bubble diagrams of QCD. The high-temperature counterpart of this computation has turned out to be extremely challenging, and has only been worked out in $\phi^4$ theory [@Gynther:2007bw] as well as for the large-$N_f$ limit of QCD [@Gynther:2009qf].

In the paper at hand, we present a new technical tool for perturbative calculations at zero temperature but finite chemical potentials that we argue enables a high-order determination of many important thermodynamic quantities. This tool is referred to as a set of “cutting rules”, which were proposed 