Multi-particle potentials from light-like Wilson lines in quark-gluon plasmas: a generalized relation of in-medium splitting rates to jet-quenching parameters $\hat q$

A powerful historical insight about the theory of in-medium showering in QCD backgrounds was that splitting rates can be related to a parameter $\hat q$ that characterizes the rate of transverse-momentum kicks to a high-energy particle from the medium. Another powerful insight was that $\hat q$ can be defined (with caveats) even when the medium is strongly coupled, using long, narrow Wilson loops whose two long edges are light-like Wilson lines. The medium effects for the original calculations of in-medium splitting rates can be formulated in terms of 3-body imaginary-valued `potentials' that are defined with three long, light-like Wilson lines. Corrections due to overlap of two consecutive splittings can be calculated using similarly defined 4-body potentials. I give a simple argument for how such $N$-body potentials can be determined in the appropriate limit just from knowledge of the values of $\hat q$ for different color representations. For $N > 3$, the $N$-body potentials have non-trivial color structure, which will complicate calculations of overlap corrections outside of the large-$N_c$ or soft bremsstrahlung limits.