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

physical review. d/physical review. d.

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