Medium-induced gluon radiation in hard forward parton scattering in the saturation formalism

We derive the medium-induced, fully coherent soft gluon radiation spectrum associated with the hard forward scattering of an energetic parton off a nucleus, in the saturation formalism within the Gaussian approximation for the relevant correlators of Wilson lines and for finite number of colors. The validity range of the result is rigorously specified by keeping track of the order of magnitude of subleading contributions to the spectrum. The connection between the saturation formalism and the opacity expansion used in previous studies of the same observable is made apparent. Our calculation sets the basis for further studies of the interplay between saturation and fully coherent energy loss in hard forward parton scattering.

Figure 1

Generic diagrams contributing to the cross section for $a+\mathrm{A}\to a+g+\mathrm{X}$, where the parton $a=q$, $g$ is represented by the dashed line. The scattering off the nucleus consists in two-gluon exchanges between the $(a,g)$ system and any number of nucleons in the nucleus, as encoded in the correlators of Wilson lines $S^{(i)}$ defined in Eqs. (7)–(10). We represented a particular event where only two nucleons interact: for instance a first nucleon can interact elastically in the amplitude and not interact in the conjugate amplitude, and a second nucleon can interact inelastically. The parton $a$ can radiate the gluon from the initial state [generic diagram (a)], the final state [generic diagram (b)], or coherently from the interference between initial-state and final-state emissions [generic diagrams (c) and (d)].

Figure 2

Amplitude squared for the elastic scattering of the energetic parton $a=q$, $g$ (represented by the dashed line) off a nucleus.