[Paper Review] An Improved Splitting Function for Small x Evolution

Guido Altarelli, Richard D. Ball+1 more|ArXiv.org|Oct 1, 2003

Astronomy and Astrophysical Research3 citations

TL;DR

This paper proposes an improved splitting function for small-x evolution by resumming running coupling effects in the BFKL kernel, combining double-leading logarithmic accuracy with momentum conservation. The resulting anomalous dimension softens the unphysical rise of structure functions at small x, achieving excellent agreement with HERA data without free parameters, matching the phenomenologically tuned exponent from prior fits through an analytic Airy function-based resummation.

ABSTRACT

We summarize our recent result for a splitting function for small x evolution which includes resummed small x logarithms deduced from the leading order BFKL equation with the inclusion of running coupling effects. We compare this improved splitting function with alternative approaches.

Motivation & Objective

• To resolve the unphysical rapid rise of structure functions at small x predicted by leading-order BFKL evolution.
• To incorporate running coupling corrections into the small-x splitting function while preserving factorization and smooth behavior.
• To eliminate the need for fine-tuning a phenomenological exponent λ by deriving it analytically from resummed dynamics.
• To provide an analytic, factorization-compatible expression for the improved splitting function that matches NLO GLAP evolution in the HERA data region.

Proposed method

• Resum the running coupling corrections to the BFKL kernel by adding effective subleading Δχi contributions to χ1(M), which are then resummed to all orders in the splitting function.
• Construct a double-leading expansion that includes both (αs log 1/x)^n terms from χ0(M) and αs(αs log 1/x)^n terms from χ1(M), ensuring consistency with momentum conservation.
• Introduce an 'Airy' anomalous dimension γA(c0, αs, N) to replace the singular behavior of γs at N = αsc0, using the square root of the Airy function's zero to define the softening scale.
• Implement a subtraction scheme to avoid double-counting: remove the αs term from γs(αs/N) to prevent overcounting the leading-order contribution.
• Use the duality relation χ(γ(N, αs), αs) = N to map the BFKL kernel to the anomalous dimension, ensuring consistency between BFKL and GLAP evolution at leading twist.
• Derive an analytic expression for the improved splitting function γI^NL(αs, N) that includes NLO perturbative terms, resummed running coupling effects, and the Airy function contribution, with momentum subtraction for consistency.

Experimental results

Research questions

• RQ1 Can running coupling corrections in the BFKL kernel be resummed in a way that softens the small-x rise of structure functions without introducing unphysical singularities?
• RQ2 Does the inclusion of both χ1(M) and running coupling effects lead to a splitting function that matches NLO GLAP evolution in the HERA data region without free parameters?
• RQ3 Can the phenomenologically tuned exponent λ ≈ 0.21 from prior fits be reproduced analytically through the zeros of the Airy function in the resummed splitting function?
• RQ4 How do the resummed splitting functions compare to alternative approaches that include χ1(M) but not γ1(N), or vice versa, in terms of preasymptotic behavior and resummation uncertainty?

Key findings

• The improved splitting function γI^NL(αs, N) achieves a soft small-x rise that closely follows NLO GLAP evolution for αs = 0.2, with no free parameters.
• The effective exponent λ ≈ 0.2112 derived from the Airy function's rightmost zero matches the phenomenologically tuned value λ ≈ 0.21, confirming the analytic origin of the softening.
• The Airy term γA(c0, αs, N) cancels the branch cut of γs(αs/N) at N = αsc0, replacing the singular behavior with a smooth, physically acceptable rise.
• The inclusion of γ1(N) and the double-counting subtraction significantly improves agreement with NLO GLAP, even though γ1 is not fully included in the final expression.
• The resummed splitting function is analytic and factorization-compatible, enabling straightforward fitting to data and matching to standard evolution equations.
• The preasymptotic behavior of the resummed splitting function differs from alternative approaches, with differences serving as a measure of residual resummation uncertainty.