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Unintegrated parton distributions

M. A. Kimber, A. D. Martin, M. G. Ryskin

TL;DR

Kimber, Martin and Ryskin develop a framework to obtain unintegrated parton distributions $f_a(x,k_t^2,\mu^2)$ from single-scale auxiliary functions $h_a(x,k_t^2)$ that solve unified DGLAP- and BFKL-type evolutions. A final last-step evolution at scale $\mu$ imposes angular ordering, producing true two-scale PDFs that extend into the $k_t>\mu$ domain and incorporate major LO virtual and subleading $\ln(1/x)$ effects. They validate the formalism by computing the deep inelastic structure function $F_2(x,Q^2)$ and show good agreement with data without direct fits, highlighting the practical utility for exclusive processes and global analyses. The work clarifies the relationship to integrated PDFs, demonstrates the dominance of angular ordering over BFKL effects in the HERA region, and provides a path toward global fits in terms of unintegrated distributions.

Abstract

We describe how to calculate the parton distributions $f_a(x, k_t^2, μ^2)$, unintegrated over the parton transverse momentum $k_t$, from auxiliary functions $h_a(x, k_t^2)$, which satisfy single-scale evolution equations. The formalism embodies both DGLAP and BFKL contributions, and accounts for the angular ordering which comes from coherence effects in gluon emission. We check that the unintegrated distributions give the measured values of the deep inelastic structure function $F_2(x, Q^2)$.

Unintegrated parton distributions

TL;DR

Kimber, Martin and Ryskin develop a framework to obtain unintegrated parton distributions from single-scale auxiliary functions that solve unified DGLAP- and BFKL-type evolutions. A final last-step evolution at scale imposes angular ordering, producing true two-scale PDFs that extend into the domain and incorporate major LO virtual and subleading effects. They validate the formalism by computing the deep inelastic structure function and show good agreement with data without direct fits, highlighting the practical utility for exclusive processes and global analyses. The work clarifies the relationship to integrated PDFs, demonstrates the dominance of angular ordering over BFKL effects in the HERA region, and provides a path toward global fits in terms of unintegrated distributions.

Abstract

We describe how to calculate the parton distributions , unintegrated over the parton transverse momentum , from auxiliary functions , which satisfy single-scale evolution equations. The formalism embodies both DGLAP and BFKL contributions, and accounts for the angular ordering which comes from coherence effects in gluon emission. We check that the unintegrated distributions give the measured values of the deep inelastic structure function .

Paper Structure

This paper contains 7 sections, 27 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Unintegrated DGLAP partons
  3. Inclusion of $\ln (1/x)$ effects
  4. The unintegrated gluon distribution
  5. $F_2$ calculated from the unintegrated partons
  6. Relation of $f$ to integrated partons
  7. Conclusions

Figures (5)

  • Figure 1: An illustration of our procedure, in which the evolution of a single-scale unintegrated parton is followed by a final step of the ladder which introduces dependence on the second hard scale, $\mu$.
  • Figure 2: part of the evolution chain. We commonly write $k_t$ for $k_{tn}$ and then the parent's transverse momentum as $k'_t$. The radiated transverse momentum is $q_t$. Unified evolution is naturally performed KKMS in terms of the rescaled transverse momentum $q_n=q_{tn}/(1-z_n)$.
  • Figure 3: plots of the $k_t$-dependence of the unintegrated gluon $f_g(x,k_t^2,\mu^2)$ for various values of $x$, at $\mu=10$ GeV. The solid curves are our version (a) of $f_g$ from (\ref{['eq:a14']}); for comparison we show with dashed lines the unintegrated gluon from KKMS (as in KKMS, the dashed lines have been smoothed in the transition region $k_t\sim \mu$). Also we plot our "DGLAP" unintegrated gluon (b) from (\ref{['eq:a4']}) in dotted lines, which with the correct angular ordering cut-off is very close to the new $f_g$, especially at high $x$.
  • Figure 4: The quark box, and crossed-box, diagrams which mediate the contribution of the unintegrated gluon distribution $f_g(x/z,k_t^2,\mu^2)$ to $F_2$.
  • Figure 5: This is not a fit but the results of using our "DGLAP" unintegrated partons (b) to calculate $F_2$; the gluon-originated contributions are shown as dashed lines and the quark-originated parts are shown as dotted lines. Recent data are plotted DATA, and compare well with the sum of the gluon and quark contributions (solid curves), especially at high $Q^2$.