Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gluon Radiation off Hard Quarks in a Nuclear Environment: Opacity Expansion

Urs Achim Wiedemann

TL;DR

This paper unifies the BDMPS and Zakharov approaches to medium-induced gluon radiation by deriving a colour-trivial, $N$-th order opacity expansion using a non-abelian Furry framework. It demonstrates that resumming the opacity series reproduces Zakharov’s path-integral and thereby proves the formalisms’ equivalence for the $k_ot$-differential cross section, extending to arbitrary orders and to the case of nascent quarks produced inside the medium. Explicit results are provided for the first few opacity orders, clarifying coherence and interference effects and enabling practical comparison to finite-thickness media. The work also establishes a clear probabilistic interpretation, connections to the dipole cross section, and a tractable low-opacity expansion suitable for phenomenology in RHIC and LHC environments.

Abstract

We study the relation between the Baier-Dokshitzer-Mueller-Peigne-Schiff (BDMPS) and Zakharov formalisms for medium-induced gluon radiation off hard quarks, and the radiation off very few scattering centers. Based on the non-abelian Furry approximation for the motion of hard partons in a spatially extended colour field, we derive a compact diagrammatic and explicitly colour trivial expression for the N-th order term of the kt-differential gluon radiation cross section in an expansion in the opacity of the medium. Resumming this quantity to all orders in opacity, we obtain Zakharov's path-integral expression (supplemented with a regularization prescription). This provides a new proof of the equivalence of the BDMPS and Zakharov formalisms which extends previous arguments to the kt-differential cross section. We give explicit analytical results up to third order in opacity for both the gluon radiation cross section of free incoming and of in-medium produced quarks. The N-th order term in the opacity expansion of the radiation cross section is found to be a convolution of the radiation associated to N-fold rescattering and a readjustment of the probabilities that rescattering occurs with less than N scattering centers. Both informations can be disentangled by factorizing out of the radiation cross section a term which depends only on the mean free path of the projectile. This allows to infer analytical expressions for the totally coherent and totally incoherent limits of the radiation cross section to arbitrary orders in opacity.

Gluon Radiation off Hard Quarks in a Nuclear Environment: Opacity Expansion

TL;DR

This paper unifies the BDMPS and Zakharov approaches to medium-induced gluon radiation by deriving a colour-trivial, -th order opacity expansion using a non-abelian Furry framework. It demonstrates that resumming the opacity series reproduces Zakharov’s path-integral and thereby proves the formalisms’ equivalence for the -differential cross section, extending to arbitrary orders and to the case of nascent quarks produced inside the medium. Explicit results are provided for the first few opacity orders, clarifying coherence and interference effects and enabling practical comparison to finite-thickness media. The work also establishes a clear probabilistic interpretation, connections to the dipole cross section, and a tractable low-opacity expansion suitable for phenomenology in RHIC and LHC environments.

Abstract

We study the relation between the Baier-Dokshitzer-Mueller-Peigne-Schiff (BDMPS) and Zakharov formalisms for medium-induced gluon radiation off hard quarks, and the radiation off very few scattering centers. Based on the non-abelian Furry approximation for the motion of hard partons in a spatially extended colour field, we derive a compact diagrammatic and explicitly colour trivial expression for the N-th order term of the kt-differential gluon radiation cross section in an expansion in the opacity of the medium. Resumming this quantity to all orders in opacity, we obtain Zakharov's path-integral expression (supplemented with a regularization prescription). This provides a new proof of the equivalence of the BDMPS and Zakharov formalisms which extends previous arguments to the kt-differential cross section. We give explicit analytical results up to third order in opacity for both the gluon radiation cross section of free incoming and of in-medium produced quarks. The N-th order term in the opacity expansion of the radiation cross section is found to be a convolution of the radiation associated to N-fold rescattering and a readjustment of the probabilities that rescattering occurs with less than N scattering centers. Both informations can be disentangled by factorizing out of the radiation cross section a term which depends only on the mean free path of the projectile. This allows to infer analytical expressions for the totally coherent and totally incoherent limits of the radiation cross section to arbitrary orders in opacity.

Paper Structure

This paper contains 28 sections, 135 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The formalism
  3. Medium average
  4. Leading $O(x)$ approximation
  5. Opacity expansion
  6. Notation and identities
  7. Colour triviality of the gluon radiation cross section
  8. Classification of diagrams
  9. Iteration
  10. $N=1$ Gunion-Bertsch radiation spectrum
  11. $N=2$ Opacity expansion
  12. A path-integral for arbitrary orders in opacity
  13. Gluon rescattering for $\xi > \bar{y}_l$
  14. Rescattering for $y_l< \xi < \bar{y}_l$
  15. Radiation cross section in terms of the colour dipole
  16. ...and 13 more sections

Figures (3)

  • Figure 1: Contribution to the gluon radiation amplitude (\ref{['2.4']}), involving $M$-, $L$-, and $N$-fold rescattering of the in- and outgoing partons respectively.
  • Figure 2: Diagrammatic representation of the radiation probability (\ref{['2.17']}) in configuration space. For details, see text.
  • Figure 3: All non-vanishing contributions to the $N=2$ radiation cross section as given by (\ref{['3.13']}). Diagrams are labelled with a double index $(ix)$, see text for details.