Forward trijet production in proton-nucleus collisions: gluon initiated channel
In this paper, we present results for the forward three-parton production differential cross section in the gluon-initiated channel in proton-nucleus collisions. This result is the leading-order contribution to forward trijet production and provides the real-emission building block required for the NLO dijet/dihadron cross section. The calculations are carried out within the Color Glass Condensate (CGC) effective theory, and in the dilute-dense approximation, using effective vertices for the quark and gluon propagators interacting with the small-$x$ background gluon field. We employ the covariant perturbation theory approach and disentangle the amplitudes into regular and instantaneous contributions. Our results are expressed as convolutions of multiparton color correlators of light-like Wilson lines and perturbative impact factors, organized in compact expressions in terms of the ``bare" topologies of the contributing diagrams. The gluon-initiated channel receives contributions from a $q\bar{q}g$ and a $ggg$ final state. Interestingly, when considering the $ggg$ final state, we observe, for the first time, that the four-gluon vertex topology follows a structure similar to the instantaneous contributions. This observation suggests a simplification in the organization of multigluon CGC amplitudes and may prove useful for future one-loop calculations. Furthermore, when integrating (one of) the real gluon(s) in the final state, we identify that the rapidity divergence is absorbed into the real part of the JIMWLK evolution of the leading-order Wilson-line correlator. In addition, we isolate the divergences arising when the unobserved parton becomes collinear to one of the observed hadrons in the final state. These divergences are absorbed by renormalizing the initial-state parton distribution functions and final-state fragmentation functions, which are shown to obey DGLAP evolution.