Gluon radiation off hard quarks in a nuclear environment: opacity expansion

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 Nth order term of the k(perpendicular to)-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 ki-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 Nth 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. (C) 2000 Elsevier Science B.V. All rights reserved.