The LPM effect in sequential bremsstrahlung: gluon shower development

We give details of our study of whether high-energy gluon showers inside a QCD medium can be treated as a sequence of individual splitting processes $g \to gg$, or whether there is significant quantum overlap between where one splitting ends and the next begins (neglecting effects that can be absorbed into an effective value of the jet quenching parameter $\hat q$ that characterizes the medium). The study is carried out by imagining in-medium gluon shower development in the simplest theoretical situation, which includes imagining a very large, static, homogeneous medium and taking the large $N_{\rm c}$ limit. Along the way, we also show how in-medium shower evolution can be written in terms of a "net" splitting rate $[d\Gamma/dx]_{\rm net}$, and we provide a moderately simple analytic fit to our numerical results for the overlap effects included in that rate, which we hope may be of use to others wishing to study possible consequences of overlapping splittings.