[Paper Review] Structure Function Resummation in small-x QCD

Guido Altarelli, Richard D. Ball+1 more|ArXiv.org|Feb 7, 2008

High-Energy Particle Collisions Research6 references4 citations

TL;DR

This paper presents a fully resummed perturbative QCD framework for small-x evolution in deep-inelastic scattering, resolving instabilities in NNLO calculations by incorporating physical constraints like momentum conservation and renormalization group invariance. The key result is that resummation suppresses small-x enhancements seen in fixed-order NNLO, reducing errors in extracted parton distributions by up to 20% at x ~ 10⁻⁶ when compared to unresummed NNLO evolution.

ABSTRACT

We summarize our recent results on small x resummation in full QCD with n_f quark flavours and discuss their phenomenological impact in the extraction of parton distributions from present day structure function data and their extrapolation to the kinematics relevant for future colliders such as the LHC.

Motivation & Objective

• To resolve the instability of NNLO perturbative QCD evolution in the small-x (high-energy) limit, where corrections diverge as x → 0.
• To construct a consistent, physically motivated resummation of small-x logarithms that satisfies momentum conservation, renormalization group invariance, and gluon exchange symmetry.
• To enable accurate extraction of parton distribution functions (PDFs) from HERA data for use in LHC phenomenology by correcting for unphysical enhancements in fixed-order NNLO evolution.
• To compare and reconcile two distinct resummation approaches—ABF and CCSS—showing their agreement within theoretical uncertainties.
• To provide fully resummed, scheme-consistent splitting functions and coefficient functions for physical observables like F₂ and Fₗ in the MS̄ scheme.

Proposed method

• Construct resummed singlet splitting functions by identifying and resumming the single eigenvalue of the anomalous dimension matrix with leading N=0 singularities at small x.
• Implement physical constraints—momentum conservation, renormalization group invariance, and gluon exchange symmetry—by including formally subleading terms that stabilize the small-x expansion.
• Use the BFKL framework in the CCSS approach and GLAP anomalous dimension improvement in the ABF approach, demonstrating their duality in describing the same physics at leading twist.
• Derive resummed coefficient functions h₂ and hₗ for F₂ and Fₗ in deep-inelastic scattering, ensuring consistency with the MS̄ or DIS factorization schemes.
• Construct the full n_f ≠ 0 two-by-two matrix of resummed splitting functions and combine them with resummed coefficient functions in a consistent factorization scheme, as implemented in ref. [22].
• Perform numerical comparisons of K-factors (resummed/NLO) across different schemes and x values to quantify the phenomenological impact on structure functions and PDFs.

Experimental results

Research questions

• RQ1 How do unresummed NNLO corrections lead to unphysical divergences in the small-x limit, and why is resummation necessary to stabilize perturbative evolution?
• RQ2 To what extent do physical constraints such as momentum conservation and renormalization group invariance affect the structure of small-x resummation in full QCD with n_f quark flavors?
• RQ3 What is the quantitative impact of small-x resummation on the extraction of parton distribution functions from HERA data, particularly at x ~ 10⁻⁶ relevant for LHC physics?
• RQ4 How do the ABF and CCSS resummation approaches compare in their predictions for structure functions, and what explains their agreement despite different formalisms?
• RQ5 What is the magnitude of the error in PDF extraction if small-x resummation is neglected, and how does it vary with x and scale?

Key findings

• Resummation stabilizes the small-x behavior of perturbative evolution, suppressing the strong enhancement or suppression seen in leading and subleading log approximations.
• The resummed K-factor for F₂ is less than one at small x, indicating that resummed evolution yields smaller structure functions than fixed-order NLO evolution, in contrast to the NNLO case.
• At x ~ 10⁻³, the error in PDF extraction from HERA data due to neglecting resummation is approximately 5%, increasing to 20% at x ~ 10⁻⁶.
• The suppression of quark and gluon distributions at small x under resummation is more significant than the enhancement seen in NNLO, and this effect is reduced but still non-negligible when evolved to higher scales.
• The ABF and CCSS resummation approaches yield consistent results within theoretical uncertainties, validating the physical consistency of the resummed framework.
• The full n_f ≠ 0 resummed evolution matrix and coefficient functions have been constructed in the MS̄ scheme for the first time, enabling consistent phenomenological applications.