Next-to-Leading Order Hard Scattering Using Fully Unintegrated Parton Distribution Functions

TL;DR

This work develops and applies a fully unintegrated factorization framework based on parton correlation functions to preserve exact four-momentum conservation in QCD calculations. By systematically classifying leading regions and using region-specific approximators plus Ward identities, the authors derive both LO and NLO fully unintegrated hard scattering coefficients for gluon-induced deep inelastic scattering, with explicit double-counting subtractions that yield ordinary, well-behaved functions. A key result is the explicit NLO coefficient tilde W^{μν}_{γ^* g → q qbar}, expressed as a subtraction from the naive amplitude by terms involving Xi_a and ar{Xi}_d, ensuring cancellation of divergences point-by-point in momentum space. The formalism enables direct coupling with fully unintegrated gluon PDFs and jet factors, offering a path to more precise, momentum-conserving descriptions of final-state spectra in Monte Carlo event generation and small-x QCD phenomenology.

Abstract

We calculate the next-to-leading order fully unintegrated hard scattering coefficient for unpolarized gluon-induced deep inelastic scattering using the logical framework of parton correlation functions developed in previous work. In our approach, exact four-momentum conservation is maintained throughout the calculation. Hence, all non-perturbative functions, like parton distribution functions, depend on all components of parton four-momentum. In contrast to the usual collinear factorization approach where the hard scattering coefficient involves generalized functions (such as Dirac $δ$-functions), the fully unintegrated hard scattering coefficient is an ordinary function. Gluon-induced deep inelastic scattering provides a simple illustration of the application of the fully unintegrated factorization formalism with a non-trivial hard scattering coefficient, applied to a phenomenologically interesting case. Furthermore, the gluon-induced process allows for a parameterization of the fully unintegrated gluon distribution function.

Figures (7)

• Figure 1: Graphs contributing to gluon induced production of two jets at order-$g_s^2 \,$.
• Figure 2: Figure \ref{['fig:boxdiagrams']}(a) with the application of the target collinear approximation for region $R_1$. The graph on the right-hand side of the arrow shows the separation into the factors of Eq. (\ref{['eq:grapha_app']}).
• Figure 3: Figure \ref{['fig:boxdiagrams']}(b) with the red circles showing the application of the target collinear approximation. The graph on the right hand side of the arrow shows the separation into the factors of Eq. (\ref{['eq:pmhardfactb']}). The gluon is shown attaching to an eikonal line represented by double lines.
• Figure 4: Dashed blue boxes symbolizing the application of the approximation appropriate for region $R_2$ to Figs. \ref{['fig:boxdiagrams']}(a) and (b). Analogous graphs are needed for Figs. \ref{['fig:boxdiagrams']}(c) and (d), though we do not show them here explicitly.
• Figure 5: Basic amplitude separated into an upper part $\mathcal{U}$ and a lower part $\mathcal{L}$, see Eq. (\ref{['eq:amp']})