Wilson line correlators beyond the large-$N_c$

We study hard $1\to 2$ final-state parton splittings in the medium, and put special emphasis on calculating the Wilson line correlators that appear in these calculations. As partons go through the medium their color continuously rotates, an effect that is encapsulated in a Wilson line along their trajectory. When calculating observables, one typically has to calculate traces of two or more medium-averaged Wilson lines. These are usually dealt with in the literature by invoking the large-$N_c$ limit, but exact calculations have been lacking in many cases. In our work, we show how correlators of multiple Wilson lines appear, and develop a method to calculate them numerically to all orders in $N_c$. Initially, we focus on the trace of four Wilson lines, which we develop a differential equation for. We will then generalize this calculation to a product of an arbitrary number of Wilson lines, and show how to do the exact calculation numerically, and even analytically in the large-$N_c$ limit. Color sub-leading corrections, that are suppressed with a factor $N_c^{-2}$ relative to the leading scaling, are calculated explicitly for the four-point correlator and we discuss how to extend this method to the general case. These results are relevant for high-$p_T$ jet processes and initial stage physics at the LHC.