# The LPM effect in sequential bremsstrahlung: from large-N QCD to N=3 via the SU(N) analog of Wigner 6-j symbols

## TL;DR

This paper advances the understanding of parton shower interference effects in QCD media by extending calculations from large-Nc to N=3, employing SU(Nc) Wigner 6-j symbols, and analyzing the implications for quark-gluon plasma physics.

## Contribution

It introduces a method to analyze finite N_c effects in sequential bremsstrahlung using SU(N_c) Wigner 6-j symbols, moving beyond large-N_c approximations.

## Key findings

- Casimir scaling of the transverse momentum diffusion parameter q̂
- Explicit 5x5 matrices for color state harmonic oscillator problem
- Dependence of N_c=3 results on numerical evaluation of coupled oscillators

## Abstract

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-$N_{\rm c}$ limits, where $N_{\rm c}$ is the number of quark colors. In this paper, we show how one may remove the assumption of large $N_{\rm c}$ and so begin investigation of $N_{\rm c}{=}3$ without soft-emission approximations. Treating finite $N_{\rm c}$ 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-$j$ symbols from angular momentum to SU($N_{\rm c}$). Throughout, we make use of the multiple scattering ($\hat q$) 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 $\hat q$ 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 $N_{\rm c}{=}3$ depend, mathematically, on being able to calculate the propagator for a coupled non-relativistic quantum harmonic oscillator problem in which the spring constants are operators acting on a 5-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.