is, however, important to note that the formalism has been proposed only for vacuum diagrams, and even there no proof to all orders exists; rather, the validity of the replacement has only been checked on a case-by-case basis up to partial three-loop order. In contrast, our proof of the zero-temperature cutting rules covers all Euclidean $n$-point functions up to an arbitrary order in perturbation theory, thereby validating the use of the naive real-time formalism for these quantities.

Our paper is organized as follows. In section \[sec:cut\], we introduce our notation and state the cutting rules. In addition, as an illustration, we present a simple two-loop computation in two ways: both without and with the help of the cutting rules. Section \[sec:proof\] then contains a detailed proof of the rules, as well as two intermediate lemmas that are each interesting in their own right. After this, we discuss the regularization of the most common divergences occurring in the cut graphs in section \[sec:reg\], while section \[sec:conc\] presents an outlook on the applications of the cutting rules, in particular in dense QCD. Lastly, many details of the more subtle parts of our proof have been relegated to the appendixes.

Cutting rules \[sec:cut\]
