Forward Hadron Productions in Proton-Proton Collisions in Small-$x$ Formalism

TL;DR

This paper calculates inclusive hadron production cross sections in proton-proton collisions at forward rapidity within small-$x$ formalism, incorporating next-to-leading order corrections and renormalization of divergences, providing a baseline for studying non-linear QCD effects.

Contribution

It presents a one-loop order calculation of hadron production in small-$x$ formalism using the hybrid approach, including divergence renormalization and comparison with non-linear models.

Findings

Divergences are absorbed into parton distributions and wave functions.

Results agree with previous non-linear proton-nucleus calculations in dilute limit.

Provides a baseline for identifying non-linear QCD effects in experiments.

Abstract

Employing the so-called hybrid formalism, we calculate the cross section of inclusive hadron production in proton-proton collisions at forward rapidity in small-$x$ formalism at one-loop order. For the case of hadron production at forward rapidity, we can uses collinear parton distributions for projectile proton and $k_\perp$ dependent gluon distribution for target proton. We show that collinear divergences associated with initial and final state parton radiations are renormalized into parton distributions and fragmentation functions in terms of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equation, respectively. Furthermore, rapidity divergence can be absorbed into the wave function of target proton which gives rise to the well-known Balitsky-Fadin-Kuraev-Lipatov equation. These divergences are completely separated from the short distance partonic hard parts, which is now finite at the next-to-leading order accuracy. The result presented in this paper can be reckoned as a baseline calculation without any non-linear QCD effects in small-$x$ formalism. As a consistency check, we compare our results with the previous calculation for non-linear proton-nucleus collisions in the small-$x$ formalism and find complete agreement in the dilute and large $N_c$ limit. In phenomenology, the direct comparison of the above two separate calculations can reveal the role and strength of the non-linear dynamics in high energy QCD, and thus help us reliably study the onset of gluon saturation when genuine non-linear interactions become important.