The LPM effect in sequential bremsstrahlung 2: factorization

TL;DR

This work extends the Landau-Pomeranchuk-Migdal (LPM) framework to overlapping formation times in sequential bremsstrahlung within a QCD medium, focusing on real double gluon emission from an initial gluon. By defining and computing Δ dΓ/dx dy as the difference between the full double-splitting rate and an idealized Monte Carlo (IMC) approximation, the authors isolate genuine overlap effects, regularize divergences, and apply the harmonic-oscillator (hat q) approximation in the large-Nc limit. They derive and numerically evaluate the sequential diagram contributions, including color-routing and pole terms, and show that the overlap correction scales parametrically as Δ dΓ/dx dy ∼ (αs^2)/(x y^{3/2}) √(ħq/E) in the regime y ≪ x ≪ 1, with the total result exhibiting cancellations of logarithmic enhancements akin to Gunion-Bertsch behavior. The findings illuminate how to incorporate overlap-time corrections into Monte Carlo shower models and kinetic-theory frameworks, highlighting regions where corrections are positive or negative and indicating the need for including 4-gluon vertices and virtual corrections for a complete treatment. These insights advance quantitative jet-energy-loss predictions in dense QCD media and guide future refinements of medium-modified parton showers.

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. In this paper, we continue analysis of 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), avoiding soft-gluon approximations. In particular, this paper analyzes the subtle problem of how to precisely separate overlapping double splitting (e.g.\ overlapping double bremsstrahlung) from the case of consecutive, independent bremsstrahlung (which is the case that would be implemented in a Monte Carlo simulation based solely on single splitting rates). As an example of the method, we consider the rate of real double gluon bremsstrahlung from an initial gluon with various simplifying assumptions (thick media; $\hat q$ approximation; large $N_c$; and neglect for the moment of processes involving 4-gluon vertices) and explicitly compute the correction $Δ\,dΓ/dx\,dy$ due to overlapping formation times.

Figures (37)

• Figure 1: (a) A depiction of a high-energy particle showering in a medium as it moves from left to right. The magenta ovals crudely depict the formation lengths (transversely as well as longitudinally) associated with each splitting. We show here a case where consecutive splittings are well separated compared to formation times [see text]. (b) A corresponding depiction of the approximation made in a simple idealized Monte Carlo (IMC), where the formation times are treated as effectively zero. (c) shows the case where two consecutive splittings (colored magenta and green) happen to overlap. In all of these figures, we have exaggerated the transverse direction: for high-energy particles (and high-energy daughters), splittings will be very nearly collinear.
• Figure 2: A pictorial version of the definition of $\Delta\,d\Gamma/dx\,dy$ as the difference between (i) the double-splitting rate (represented by the gray box) and (ii) the comparable rate given by idealized Monte Carlo (IMC) restricted to two splittings. Above, $z \equiv 1{-}x{-}y$.
• Figure 3: A pictorial summary of the cancellation of the contributions to fig. \ref{['fig:correct']} for splittings that are well-separated in time.
• Figure 4: An example of a process in a corrected Monte Carlo that could easily be implemented if $\Delta\,d\Gamma/dx\,dy$ is positive. Here, the $1\to3$ splitting, representing the inclusion of an additional possible process in the Monte Carlo with probability distribution $\Delta\,d\Gamma/dx\,dy$, would account for the correction due to the possibility of fig. \ref{['fig:MC']}c.
• Figure 5: A depiction of decay processes $X \to bb$ and $X \to \bar{b}\bar{b}$ for our first kinetic theory analogy, based on ref. KolbWolfram.