  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.
