High-energy gluon bremsstrahlung in a finite medium: harmonic oscillator versus single scattering approximation

TL;DR

This paper compares two approximations—harmonic oscillator and single scattering—for calculating high-energy gluon bremsstrahlung in a finite QCD medium, clarifying their validity ranges.

Contribution

It provides a detailed analysis of the parametric validity of the harmonic oscillator and single scattering approximations at leading logarithmic order.

Findings

Both approximations yield similar results in a broad intermediate regime.

The validity ranges of the two approximations are clarified.

Results are derived for small coupling and large logarithms, with alpha_s log << 1.

Abstract

A particle produced in a hard collision can lose energy through bremsstrahlung. It has long been of interest to calculate the effect on bremsstrahlung if the particle is produced inside a finite-size QCD medium such as a quark-gluon plasma. For the case of very high-energy particles traveling through the background of a weakly-coupled quark-gluon plasma, it is known how to reduce this problem to an equivalent problem in non-relativistic two-dimensional quantum mechanics. Analytic solutions, however, have always resorted to further approximations. One is a harmonic oscillator approximation to the corresponding quantum mechanics problem, which is appropriate for sufficiently thick media. Another is to formally treat the particle as having only a single significant scattering from the plasma (known as the N=1 term of the opacity expansion), which is appropriate for sufficiently thin media. In a broad range of intermediate cases, these two very different approximations give surprisingly similar but slightly differing results if one works to leading logarithmic order in the particle energy, and there has been confusion about the range of validity of each approximation. In this paper, I sort out in detail the parametric range of validity of these two approximations at leading logarithmic order. For simplicity, I study the problem for small alpha_s and large logarithms but alpha_s log << 1.