Consider a high-energy parton showering as it traverses a QCD medium such as a quark-gluon plasma. Interference effects between successive splittings in the shower are potentially very important but have so far been calculated (even in idealized theoretical situations) only in soft emission or large-$N_{\mathrm{c}}$ limits, where $N_{\mathrm{c}}$ is the number of quark colors. In this paper, we show how one may remove the assumption of large $N_{\mathrm{c}}$ and so begin investigation of $N_{\mathrm{c}}=3$ without soft-emission approximations. Treating finite $N_{\mathrm{c}}$ requires (i) classifying different ways that four gluons can form a color singlet and (ii) calculating medium-induced transitions between those singlets, for which we find application of results for the generalization of Wigner $6\text{−}j$ symbols from angular momentum to $\mathrm{SU}(N_{\mathrm{c}})$. Throughout, we make use of the multiple scattering ($\widehat{q}$) approximation for high-energy partons crossing quark-gluon plasmas, and we find that this approximation is self-consistent only if the transverse-momentum diffusion parameter $\widehat{q}$ for different color representations satisfies Casimir scaling (even for strongly coupled, and not just weakly coupled, quark-gluon plasmas). We also find that results for $N_{\mathrm{c}}=3$ depend, mathematically, on being able to calculate the propagator for a coupled nonrelativistic quantum harmonic oscillator problem in which the spring constants are operators acting on a five-dimensional Hilbert space of internal color states. Those spring constants are represented by constant $5\times 5$ matrices, which we explicitly construct. We are unaware of any closed form solution for this type of harmonic oscillator problem, and we discuss prospects for using numerical evaluation.

• Received 22 May 2019

DOI:https://doi.org/10.1103/PhysRevD.100.034030

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society