The medium-modified $g\to c\overline{c}$ splitting function in the BDMPS-Z formalism

Abstract

The formalism of Baier-Dokshitzer-Mueller-Peigné-Schiff and Zakharov determines the modifications of parton splittings in the QCD plasma that arise from medium-induced gluon radiation. Here, we study medium-modifications of the gluon splitting into a quark-anti-quark pair in this BDMPS-Z formalism. We derive a compact path-integral formulation that resums effects from an arbitrary number of interactions with the medium to leading order in the $1/{N}_c^2$ expansion. Analyses in the N = 1 opacity and the saddle point approximations reveal two phenomena: a medium-induced momentum broadening of the relative quark-anti-quark pair momentum that increases the invariant mass of quark-anti-quark pairs, and a medium-enhanced production of such pairs. We note that both effects are numerically sizeable if the average momentum transfer from the medium is comparable to the quark mass. In ultra-relativistic heavy-ion collisions, this condition is satisfied for charm quarks. We therefore focus our numerical analysis on the medium modification of $g\to c\overline{c}$, although our derivation applies equally well to $g\to b\overline{b}$ and to gluons splitting into light-flavoured quark-anti-quark pairs.