NLO prescription for unintegrated parton distributions

TL;DR

This work presents a formalism to generate unintegrated parton distributions f_a(x,k_t^2,μ^2) at NLO accuracy from known integrated PDFs, extending the LO last-step kt-factorisation by incorporating NLO splitting functions and precise kinematics. It accounts for angular ordering via coherence-induced cutoffs and uses Sudakov form factors to resum virtual contributions, ensuring correct normalization and scale dependence. Numerical results using MSTW2008 PDFs show small NLO corrections to the splitting kernels and a notable enhancement of the gluon density at small x, with the scale choice k^2 = k_t^2/(1−z) being important for NLO reliability. The findings have practical implications for predictions of exclusive processes like Higgs production at the LHC and pave the way for extending the method to generalized (skewed) unintegrated PDFs.

Abstract

We show how parton distributions unintegrated over the parton transverse momentum, k_t, may be generated, at NLO accuracy, from the known integrated (DGLAP-evolved) parton densities determined from global data analyses. A few numerical examples are given, which demonstrate that sufficient accuracy is obtained by keeping only the LO splitting functions together with the NLO integrated parton densities. However, it is important to keep the precise kinematics of the process, by taking the scale to be the virtuality rather than the transverse momentum, in order to be consistent with the calculation of the NLO splitting functions.

Figures (8)

• Figure 1: A schematic diagram of inclusive jet production in DIS at LO which shows the approximate equality between, on the left-hand side (a), the formalism based on the doubly-unintegrated quark distribution, $f_q(x,z,k^2_t,\mu^2)$, where the off-shell quark has virtuality $-k^2_t/(1-z)$, and on the right-hand side (b), the conventional QCD approach using integrated parton densities, $a(x,\mu^2)$, where the incoming partons are on-shell.
• Figure 2: The ladder diagram describing DGLAP evolution at NLO. $C^{(1)}$ is the appropriate NLO coefficient function, and $P^{(0)}$ and $P^{(1)}$ are the appropriate LO and NLO splitting functions.
• Figure 3: DGLAP splitting functions, $z\tilde{P}_{ab}(z)$, given by Eq. \ref{['eq:Ptilde']}, at LO and NLO, after the subtraction for $z>\mu/(\mu+k_t)$ due to angular ordering, where we take $\mu^2 = 10^4$ GeV$^2$ and $k_t^2=100$ GeV$^2$.
• Figure 4: Unintegrated parton distributions, $f_a(x,k_t^2,\mu^2)$, given by Eq. \ref{['eq:fNLO']}, at $\mu^2 = 10^4$ GeV$^2$, for different orders of integrated PDF and 'last-step' splitting function. We use MSTW 2008 MSTW2008 integrated PDFs to generate the numerical predictions throughout this paper, except for Fig. \ref{['fig:f7']} which shows the relative insensitivity to the choice of input PDF set.
• Figure 5: Unintegrated parton distributions, $f_a(x,k_t^2,\mu^2)$, given by Eq. \ref{['eq:fNLO']}, at $\mu^2 = 100$ GeV$^2$, for different orders of integrated PDF and 'last-step' splitting function.