The LPM effect in sequential bremsstrahlung

TL;DR

This paper investigates the Landau-Pomeranchuk-Migdal (LPM) effect in sequential bremsstrahlung within QCD, focusing on overlapping coherence lengths of hard and soft gluon splittings, and develops methods for numerical evaluation in a simplified model.

Contribution

It generalizes the analysis of overlapping bremsstrahlung to include both hard and soft gluon splittings using large-Nc and multiple scattering approximations.

Findings

Derived formulas for overlapping double bremsstrahlung rates.

Reduced the problem to a numerically solvable 1D integral.

Provided insights into interference effects in high-energy QCD media.

Abstract

The splitting processes of bremsstrahlung and pair production in a medium are coherent over large distances in the very high energy limit, which leads to a suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. We analyze the case when the coherence lengths of two consecutive splitting processes overlap, which is important for understanding corrections to standard treatments of the LPM effect in QCD. Previous authors have analyzed this problem in the case of overlapping double bremsstrahlung where at least one of the bremsstrahlung gluons is soft. Here we show how to generalize to include the case where both splittings are hard. A number of techniques must be developed, and so in this paper we simplify by (i) restricting attention to a subset of the interference effects, which we call the "crossed" diagrams, and (ii) working in the large-$N_c$ limit. We first develop some general formulas that could in principle be implemented numerically (with substantial difficulty). To make more analytic progress, we then focus on the case of a thick, homogeneous medium and make the multiple scattering approximation (also known as the $\hat q$ or harmonic approximation) appropriate at high energy. We show that the differential rate $d\Gamma/dx\,dy$ for overlapping double bremsstrahlung of gluons with momentum fractions $x$ and $y$ can then be reduced to the calculation of a 1-dimensional integral, which we perform numerically. [Though this paper is unfortunately long, our introduction is enough for getting the gist of the method.]