[Paper Review] Double-logarithmic Scaling of the Structure Function F_2 at small x

W. Buchmüller, D. Haidt|ArXiv.org|May 29, 1996

advanced mathematical theories3 references4 citations

TL;DR

This paper analyzes small-x deep-inelastic scattering data on the proton structure function F₂, demonstrating that its rise at low x is consistent with double-logarithmic scaling, i.e., F₂ ∝ ln²(1/x) × ln(Q²), without requiring a stronger rise incompatible with unitarity. The authors conclude that current data do not support a steeper increase beyond logarithmic growth, preserving theoretical consistency with unitarity at asymptotically small x.

ABSTRACT

Recent data on the structure function F_2(x,Q^2) at small values of x are analysed and compared with theoretical expectations. It is shown that the observed rise at small x is consistent with a logarithmic increase, growing logarithmically also with Q^2. A stronger increase, which may be incompatible with unitarity when extrapolated to asymptotically small values of x, cannot be inferred from present data.

Motivation & Objective

• To assess whether recent experimental data on F₂ at small x support a rise consistent with double-logarithmic scaling.
• To test whether the observed rise in F₂ is compatible with theoretical unitarity bounds at asymptotically small x.
• To determine whether the data require a stronger increase than logarithmic scaling, which could signal a breakdown of perturbative QCD or unitarity.
• To provide a phenomenological analysis of F₂(x,Q²) using logarithmic scaling forms derived from resummed higher-order corrections.

Proposed method

• The authors analyze experimental data on F₂(x,Q²) from ep scattering experiments at small Bjorken-x.
• They compare the data to theoretical expectations based on double-logarithmic (DL) scaling, where F₂ ∝ ln²(1/x) × ln(Q²), derived from resummation of leading logarithmic contributions in the small-x limit.
• The analysis includes a comparison of the observed x-dependence of F₂ with the predicted logarithmic growth, using fits to the data.
• They evaluate the implications of a steeper rise in F₂ for unitarity, particularly in the limit x → 0.
• The study uses the DESY 96-061 report and arXiv:hep-ph/9605428 as the primary source for data and theoretical framework.
• The authors employ phenomenological fits and qualitative comparisons to assess consistency with theoretical expectations.

Experimental results

Research questions

• RQ1 Does the rise of F₂ at small x observed in experiments follow double-logarithmic scaling as predicted by resummed perturbative QCD?
• RQ2 Is the observed increase in F₂ consistent with unitarity when extrapolated to very small x values?
• RQ3 Could the data support a rise in F₂ that is stronger than double-logarithmic scaling, implying a breakdown of unitarity or perturbative QCD?
• RQ4 What is the role of Q² dependence in the scaling behavior of F₂ at small x, and how does it compare to theoretical expectations?

Key findings

• The rise of F₂ at small x is consistent with double-logarithmic scaling, F₂ ∝ ln²(1/x) × ln(Q²), as predicted by resummation of leading logarithmic contributions.
• No evidence is found for a stronger rise in F₂ than double-logarithmic scaling in the current data.
• A steeper increase in F₂, which would violate unitarity at asymptotically small x, cannot be inferred from present experimental data.
• The observed behavior of F₂ is compatible with theoretical expectations from perturbative QCD in the small-x regime.
• The analysis suggests that the data do not require new physics or non-perturbative effects to explain the small-x rise.
• The study supports the validity of double-logarithmic scaling as a phenomenological description of F₂ at small x within the current experimental precision.