[Paper Review] On the Angular Dependence of the Radiative Gluon Spectrum

R. Baier, Yuri L. Dokshitzer+2 more|CERN Bulletin|May 7, 2001

High-Energy Particle Collisions Research2 references4 citations

TL;DR

This paper corrects a critical error in the angular dependence calculation of radiated gluon spectra from high-energy quarks traversing a QCD medium, as formulated in the BDMPS approach. It derives the fractional energy loss outside a cone, R(θ_cone), as a universal function of θ²_cone L³ q̂, showing strong peaking at small angles (θ²_cone q̂ L³ ≈ 1/10), and establishes equivalence with the Zakharov-Wiedemann formalism via explicit derivation of the momentum spectrum.

ABSTRACT

The induced momentum spectrum of soft gluons radiated from a high energy quark produced in and propagating through a QCD medium is reexamined in the BDMPS formalism. A mistake in our published work (Physical Review C60 (1999) 064902) is corrected. The correct dependence of the fractional induced loss $R(θ_{ m cone})$ as a universal function of the variable $θ^2_{ m cone} L^3 \hat q$ where $L$ is the size of the medium and $\hat q$ the transport coefficient is presented. We add the proof that the radiated gluon momentum spectrum derived in our formalism is equivalent with the one derived in the Zakharov-Wiedemann approach.

Motivation & Objective

• To correct a previously published error in the calculation of the angular distribution of soft gluons radiated from a high-energy quark in a QCD medium.
• To re-derive the fractional energy loss R(θ_cone) outside a cone of opening angle θ_cone, defined as ΔE(θ_cone)/ΔE.
• To demonstrate that R(θ_cone) is a universal function of the variable θ²_cone L³ q̂, where L is the medium size and q̂ the transport coefficient.
• To prove equivalence between the gluon momentum spectrum derived in the BDMPS formalism and the one in the Zakharov-Wiedemann approach.

Proposed method

• Re-express the induced gluon spectrum in impact parameter space using rescaled time variables and Fourier transforms.
• Correct the initial condition for the hard vertex contribution f_h, fixing a prior error that incorrectly replaced 1/B² with 1/B₁² in the integrand.
• Apply the harmonic oscillator Green function G to evolve the initial conditions, ensuring proper time evolution of the amplitude.
• Use partial integration and Green’s function identities to simplify the spectrum expression, eliminating divergent terms via κ=0 subtraction.
• Transform the final spectrum into a form that matches Eqs. (A.5) and (A.6) from Wiedemann’s work, confirming equivalence with the Zakharov-Wiedemann approach.
• Verify that the resulting R(θ_cone) depends universally on θ²_cone L³ q̂ through explicit evaluation of the momentum spectrum integral.

Experimental results

Research questions

• RQ1 What is the correct angular dependence of the induced radiated gluon spectrum in the BDMPS formalism, after correcting a prior error?
• RQ2 How does the fractional energy loss R(θ_cone) outside a cone of angle θ_cone depend on the medium parameters L and q̂?
• RQ3 Is the gluon momentum spectrum derived in the BDMPS approach equivalent to the one derived in the Zakharov-Wiedemann framework?
• RQ4 What is the physical origin of the strong peak in R(θ_cone) at small θ_cone, and how is it quantitatively described?

Key findings

• The corrected expression for R(θ_cone) is a universal function of the variable θ²_cone L³ q̂, confirming the scaling behavior previously observed in Wiedemann’s work.
• R(θ_cone) exhibits a strong peak at small angles, with maximum fractional energy loss occurring when θ²_cone q̂ L³ ≈ 1/10.
• The momentum spectrum derived in the BDMPS formalism is explicitly shown to be equivalent to the one derived in the Zakharov-Wiedemann approach, matching Eqs. (A.5) and (A.6) of Wiedemann’s paper.
• The correction resolves a prior error in the treatment of the hard vertex contribution f_h, where the initial condition was incorrectly factorized, leading to incorrect angular dependence.
• The subtraction of the κ=0 term successfully removes medium-independent contributions, ensuring a finite and physical spectrum.
• The final spectrum expression (29)–(30) is consistent with the known form of gluon radiation from a quark traversing a QCD medium, validating the formalism.