Forward hadron productions in proton-proton collisions in small-$x$ formalism

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 use collinear parton distributions for the projectile proton and $k_{\perp }$-dependent gluon distribution for the target proton. We show that collinear divergences associated with initial- and final-state parton radiations are renormalized into parton distributions and fragmentation functions, respectively, in terms of the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equation. Furthermore, rapidity divergence can be absorbed into the wave function of the 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 next-to-leading-order accuracy. The result presented in this paper can be reckoned as a baseline calculation without any nonlinear QCD effects in small-$x$ formalism. As a consistency check, we compare our results with the previous calculation for nonlinear proton-nucleus collisions in 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 nonlinear dynamics in high-energy QCD, and thus help us reliably study the onset of gluon saturation when genuine nonlinear interactions become important.

Figure 1

Lowest-order diagrams for $q+\mathrm{p}\to q+X$ (left) and $g+\mathrm{p}\to g+X$ (right) with one-gluon exchange in the $t$ channel.

Figure 2

Real diagrams at NLO for the $q\to q$ channel. Notations are the same as in Fig. 1.

Figure 3

Virtual diagrams at NLO for the $q\to q$ channel. Mirror contributions are omitted.

Figure 4

Real diagrams at NLO for the $g\to g$ channel.

Figure 5

Virtual gluon loop corrections at NLO for the $g\to g$ channel. Mirror contributions are omitted.

Figure 6

Virtual quark loop corrections at NLO for the $g\to g$ channel. Mirror contributions are omitted.

Figure 7

Real diagrams at NLO for the $g\to q$ channel.