Collinear Singularities and Running Coupling Corrections to Gluon Production in CGC

TL;DR

This work investigates how running coupling corrections enter gluon production in the CGC framework and reveals that final-state collinear splittings introduce infrared-sensitive contributions, while total cross sections exhibit cancellations via the optical theorem. The authors develop a formalism with quark-bubble dressing to track running coupling effects, finding that $dN^G/d^2q\,dy$ acquires a factor $\\alpha_s(\\Lambda_{coll}^2)$, tying the coupling scale to an IR resolution parameter. They show that, despite these divergences in spectra, the energy density of produced matter, $\\epsilon$, remains infrared-safe due to cancellations that survive the $p_T$-weighting integral, a result supported by a toy-model demonstration. The study emphasizes the need for factorization (e.g., into fragmentation functions) and motivates further complete RC calculations to improve CGC-based predictions for high-energy hadronic and nuclear collisions.

Abstract

We analyze the structure of running coupling corrections to the gluon production cross section in the projectile-nucleus collisions calculated in the Color Glass Condensate (CGC) framework. We argue that for the gluon production cross section (and for gluon transverse momentum spectra and multiplicity) the inclusion of running coupling corrections brings in collinear singularities due to final state splittings completely unaffected by CGC resummations. Hence, despite the saturation/CGC dynamics, the gluon production cross section is not infrared-safe. As usual, regularizing the singularities requires an infrared cutoff Lambda_coll that defines a resolution scale for gluons. We specifically show that the cutoff enters the gluon production cross section in the argument of the strong coupling constant alpha_s(Lambda_coll^2). We argue that for hadron production calculations one should be able to absorb the collinear divergence into a fragmentation function. The singular collinear terms in the gluon production cross section are shown not to contribute to the energy density of the produced matter, which is indeed an infrared-finite quantity.

Figures (16)

• Figure 1: The time-ordered picture of projectile-target scattering amplitude introducing notations to be used below. All multiple rescatterings are denoted by a broad (blue) vertical line. The solid narrow vertical line denotes the final state cut. (Color on-line.)
• Figure 2: On the left: the scattering amplitude squared for the diagram from Fig. \ref{['fig:notations']}. On the right: an abbreviated notation for the same amplitude squared.
• Figure 3: One-loop quark bubble final-state corrections to the contribution of the diagram in Fig. \ref{['fig:abbrev']} to the total cross section. Here the quark bubble does not interact with the target. The dotted vertical line denotes an intermediate state labeled by $\alpha$.
• Figure 4: One-loop quark bubble final-state corrections to the contribution of the diagram in Fig. \ref{['fig:abbrev']} to the total cross section: here the quark loop interacts with the target.
• Figure 5: Two-loop quark bubble final-state corrections to the contribution of the diagram in Fig. \ref{['fig:abbrev']} to the total cross section. Here the quark bubbles do not interact with the target.