Abstract
A powerful historical insight about the theory of in-medium showering in QCD backgrounds was that splitting rates can be related to a parameter $\stackrel{^}{q}$ that characterizes the rate of transverse-momentum kicks to a high-energy particle from the medium. Another powerful insight was that $\stackrel{^}{q}$ can be defined (with caveats) even when the medium is strongly coupled, using long, narrow Wilson loops whose two long edges are lightlike Wilson lines. The medium effects for the original calculations of in-medium splitting rates can be formulated in terms of three-body imaginary-valued ``potentials'' that are defined with three long, lightlike Wilson lines. Corrections due to the overlap of two consecutive splittings can be calculated using similarly defined four-body potentials. I give a simple argument for how such $N$-body potentials can be determined in the appropriate limit just from the knowledge of the values of $\stackrel{^}{q}$ for different color representations. For $N>3$, the $N$-body potentials have a nontrivial color structure, which will complicate calculations of overlap corrections outside of the large-${N}_{\mathrm{c}}$ or soft bremsstrahlung limits.
Highlights
In theoretical studies of p⊥ broadening and jet quenching of very high-energy partons that travel through a quarkgluon plasma, a very important parameter describing scattering of the parton from the medium is known as q
It is the proportionality constant in the relation hQ2⊥i 1⁄4 q Δz, where hQ2⊥i is the typical squared transverse momentum that the parton picks up after traveling a distance Δz through the medium, in the limit that Δz is large compared to characteristic scales of the medium such as mean-free paths for collisions
N-body potentials for overall color singlets are a tool for consolidating all interactions with the medium that occur over timescales small compared to the timescale of the splitting processes shown in Figs. 2 and 4, i.e., on timescales small compared to formation times
Summary
In theoretical studies of p⊥ broadening and jet quenching of very high-energy partons that travel through a quarkgluon plasma, a very important parameter describing scattering of the parton from the medium is known as q. The potential term in that Schrödinger equation represents medium-averaged effects of interactions with the medium over timescales short compared to the long formation time Over those timescales, the transverse positions of the particles can be treated as constant. N-body potentials for overall color singlets are a tool for consolidating all interactions with the medium that occur over timescales small compared to the timescale of the splitting processes shown in Figs. 2 and 4, i.e., on timescales small compared to formation times
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