[Paper Review] A Detailed Comparison of NLO QCD Evolution Codes

Blümlein, J., M. Botje+5 more|ArXiv.org|Sep 18, 1996

Algorithms and Data Compression21 citations

TL;DR

This paper presents a high-precision comparison of seven next-to-leading order (NLO) QCD evolution codes, resolving discrepancies caused by different truncation prescriptions in the perturbative series. After correcting for theoretical differences, the five most consistent codes achieve numerical agreement better than 0.1%, establishing reference results for future global QCD fits and high-precision structure function analyses at HERA and beyond.

ABSTRACT

Seven next-to-leading order QCD evolution programs are compared. The deviations of the results due to different theoretical prescriptions for truncating the perturbative series are clarified, and a numerical agreement between five codes of better than 0.1% is achieved. Reference results for further comparison are provided.

Motivation & Objective

• To resolve discrepancies among NLO QCD evolution codes arising from different theoretical prescriptions in truncating the perturbative series.
• To achieve numerical agreement at the 0.1% level across multiple codes, enabling reliable global QCD fits.
• To provide reference results for future comparisons, especially for high-precision structure function measurements at HERA.
• To clarify the origin of numerical differences between codes, particularly those related to moment-space versus x-space implementations and iterative evolution schemes.
• To ensure that theoretical uncertainties in αs(MZ²) extracted from HERA and e⁺e⁻ data do not stem from numerical or implementation artifacts in evolution programs.

Proposed method

• The study compares seven NLO QCD evolution programs, including both moment-space and x-space implementations, under identical theoretical and numerical conditions.
• The perturbative series is truncated at NLO, and differences due to distinct prescriptions for handling higher-order terms (NNLO and beyond) are systematically analyzed.
• The comparison uses the same initial parton distributions, evolution scales, and values of ΛQCD and αs(MZ²), ensuring consistency across codes.
• A local evolution code based on the solution of the evolution equation in x-space is used as a reference to validate global codes.
• The results are normalized to a common reference code (e.g., [SR]) to isolate numerical differences from theoretical assumptions.
• Reference results for F₂ and parton distributions at Q² = 100 GeV² and 10⁴ GeV² are provided in Table 1 for benchmarking.

Experimental results

Research questions

• RQ1 What causes numerical differences between NLO QCD evolution codes when using the same input parameters and perturbative order?
• RQ2 To what extent can numerical agreement between NLO evolution codes be improved by correcting for differing theoretical truncation prescriptions?
• RQ3 Can a local x-space evolution code serve as a reliable reference for validating global moment-space evolution programs?
• RQ4 How small are the residual numerical differences after aligning theoretical assumptions across codes, and do they meet the 0.1% accuracy goal for high-precision QCD fits?
• RQ5 What reference values for F₂ and parton distributions at Q² = 100 GeV² and 10⁴ GeV² can be used for future code validation?

Key findings

• After correcting for differences in theoretical prescriptions for truncating the perturbative series, five of the seven NLO evolution codes achieve numerical agreement better than 0.1% at Q² = 100 GeV².
• The residual numerical differences between codes are found to be at most 0.05% at Q² = 100 GeV², except for very large x where parton densities are small.
• The local x-space evolution code [BvN] agrees perfectly with the x-space codes [MB, PZ], confirming the reliability of these implementations.
• The 1% difference observed between [AV] and [SR] at small x is fully explained by different truncation schemes in their iterative evolution procedures.
• The study provides two sets of reference results in Table 1 for F₂ and parton distributions at Q² = 100 GeV² and 10⁴ GeV², normalized to a common reference code, enabling high-accuracy benchmarking.
• The results confirm that numerical artifacts from evolution code implementations do not contribute significantly to the theoretical uncertainty in αs(MZ²), supporting the reliability of global QCD fits.