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A Global Fit to Scattering Data with NLL BFKL Resummations

C. D. White, R. S. Thorne

TL;DR

This paper develops a global QCD fit incorporating next-to-leading-log BFKL resummations with running coupling, combined with a fixed-order NLO expansion, to stabilize small-$x$ behavior in DIS data. By constructing resummed splitting and coefficient functions, including approximate NLL heavy-flavour inputs and a DIS($ ext{χ}$) scheme, they obtain a positive-definite gluon and improved predictions for the longitudinal structure function $F_L$. The NLL+NLO fit provides a markedly better description of HERA, fixed-target, and collider data than fixed-order fits, while maintaining momentum conservation and reasonable $ ext{α}_s(M_Z^2)$ values. The results support the significance of small-$x$ resummations for reliable high-energy parton distributions, with remaining work focused on refining heavy-flavour inputs and scheme transformations.

Abstract

We calculate DIS-scheme splitting and coefficient functions for electromagnetic deep inelastic scattering with small x resummations, as given by the NLL BFKL equation with running coupling and approximate NLL resummed impact factors. The resummations are combined with a NLO fixed order expansion, and the improved quantities thus obtained are stable at small x and significantly suppressed with respect to LL results. These results are implemented in a global fit to DIS and related data, and the results compared to a NLO fixed order DIS-scheme fit and with a previous LL resummed fit. The NLL resummed fit quality is excellent, and constitutes a marked improvement over the purely fixed order approach. The input gluon, at Q_0^2=1GeV^2, obtained from the resummed fit is positive and slightly increasing as x-->0, in contrast to the result obtained at fixed order. A resummed prediction for the longitudinal structure function F_L is presented. It is positive definite and growing with energy at low x and Q^2, where the fixed order results show a significant perturbative instability.

A Global Fit to Scattering Data with NLL BFKL Resummations

TL;DR

This paper develops a global QCD fit incorporating next-to-leading-log BFKL resummations with running coupling, combined with a fixed-order NLO expansion, to stabilize small- behavior in DIS data. By constructing resummed splitting and coefficient functions, including approximate NLL heavy-flavour inputs and a DIS() scheme, they obtain a positive-definite gluon and improved predictions for the longitudinal structure function . The NLL+NLO fit provides a markedly better description of HERA, fixed-target, and collider data than fixed-order fits, while maintaining momentum conservation and reasonable values. The results support the significance of small- resummations for reliable high-energy parton distributions, with remaining work focused on refining heavy-flavour inputs and scheme transformations.

Abstract

We calculate DIS-scheme splitting and coefficient functions for electromagnetic deep inelastic scattering with small x resummations, as given by the NLL BFKL equation with running coupling and approximate NLL resummed impact factors. The resummations are combined with a NLO fixed order expansion, and the improved quantities thus obtained are stable at small x and significantly suppressed with respect to LL results. These results are implemented in a global fit to DIS and related data, and the results compared to a NLO fixed order DIS-scheme fit and with a previous LL resummed fit. The NLL resummed fit quality is excellent, and constitutes a marked improvement over the purely fixed order approach. The input gluon, at Q_0^2=1GeV^2, obtained from the resummed fit is positive and slightly increasing as x-->0, in contrast to the result obtained at fixed order. A resummed prediction for the longitudinal structure function F_L is presented. It is positive definite and growing with energy at low x and Q^2, where the fixed order results show a significant perturbative instability.

Paper Structure

This paper contains 18 sections, 86 equations, 27 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The BFKL equation at NLL order
  3. Fixed coupling solution
  4. Running Coupling Solution
  5. Solution for the BFKL Anomalous Dimension
  6. Inclusion of Impact Factors
  7. Transformation from the $Q_0$ Scheme
  8. Matching to the NLO expansion
  9. Treatment of Heavy Flavours
  10. Approximate NLL Coefficients
  11. NLL Heavy Matrix Element $A_{Hg}$
  12. A Global Fit at NLL Order
  13. Implementation
  14. Results
  15. Global Fit with Positive Large $x$ Gluon
  16. ...and 3 more sections

Figures (27)

  • Figure 1: Power-series result for the BFKL anomalous dimension $P_{+,NLL}$ for $t=10$, $n_f=4$ obtained from using the full gluon solution of equation (\ref{['g1sol']}), together the result obtained after truncating to formal NLL order and the asymptotic NLO (fixed order in $\alpha_S$) result.
  • Figure 2: The total NLL resummed splitting functions $P_{qg}$ and $P_+$ for $n_f=4$ and $t=6$, compared with the NLO fixed order results.
  • Figure 3: The NLL resummed splitting functions $P_{qg}$ and $P_+$ for $n_f=4$ and $t=6$ (solid), together with the LL result with (dashed) and without (dot-dashed) running coupling corrections. Also shown is the NLO fixed order result (dotted).
  • Figure 4: The NLL resummed splitting functions $P_{qg}$ and $P_+$ for $n_f=4$ and $t=8$, compared with the NLO fixed order results.
  • Figure 5: Comparison of the splitting function in figure (\ref{['splitsplot']}) with the CCSS and ABF results for $\alpha_S=0.2$ ($t\simeq7.5$ for the LO coupling). The CCSS and ABF results have $n_f=0$. Results taken from ABF.
  • ...and 22 more figures