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.
Highlights
Very high-energy particles traveling through a medium lose energy primarily through splitting via bremsstrahlung and pair production
In the case of QCD, the basic methods for incorporating the LPM effect into calculations of splitting rates were developed in the 1990s by Baier et al [3,4] and Zakharov [5] (BDMPS-Z)
Readers need not appreciate the full history and formalism of the subject to follow most of this paper: the problem we most need to address will be an application of SUðNÞ group theory to (i) the different ways to make color singlets from four partons and (ii) transitions between those singlets due to interactions with the medium
Summary
Very high-energy particles traveling through a medium lose energy primarily through splitting via bremsstrahlung and pair production. Readers need not appreciate the full history and formalism of the subject to follow most of this paper: the problem we most need to address will be an application of SUðNÞ group theory to (i) the different ways to make color singlets from four partons and (ii) transitions between those singlets due to interactions with the medium. The latter will involve the SUðNÞ generalization of Wigner 6-j coefficients. The matrix structure of the spring constants makes computing and using the propagator for the system (1.2) more difficult for Nc 1⁄4 3 than for the large-Nc limit
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