[Paper Review] The polarized Bjorken sum rule analysis: revised

V. L. Khandramai, O. P. Solovtsova+1 more|arXiv (Cornell University)|Feb 16, 2013

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper revises the QCD analysis of the polarized Bjorken sum rule at low momentum transfers using four-loop perturbative QCD corrections and higher-twist contributions. It demonstrates that the perturbative series exhibits asymptotic behavior at $ Q^2 \lesssim 1 \, \text{GeV}^2 $, where the four-loop term dominates, and shows that analytic perturbation theory (APT) and modified perturbation theory (MPT) improve infrared behavior and stability in extracting the $ \mu_4 $ higher-twist coefficient, yielding $ \mu_4 = -0.044 \pm 0.004 \, \text{GeV}^2 $.

ABSTRACT

We present progress in the QCD analysis of the Bjorken sum rule at low momentum transfers. We study asymptotic structure of the perturbative QCD expansion at low $Q^2$ scales based on analysis of recent accurate data on the Bjorken sum rule and available now four-loop expression for the coefficient-function $C_{Bj}(Q^2)$. We demonstrate that the standard perturbative series for $C_{Bj}(Q^2)$ gives a hint to its asymptotic nature manifesting itself in the region $Q^2 \lesssim 1$ GeV$^2$. It is confirmed by the considered integral model for the perturbative QCD correction. We extract a value of higher-twist $μ_4$ coefficient and study the interplay between higher orders and higher-twist contributions. Results of other approaches to the description of Bjorken sum rule data are discussed.

Motivation & Objective

• To analyze the perturbative QCD series of the Bjorken sum rule at low $ Q^2 $ using four-loop corrections.
• To assess the convergence and asymptotic nature of the perturbative expansion in the infrared regime.
• To extract the higher-twist $ \mu_4 $ coefficient reliably by accounting for interplay between higher-order perturbative and non-perturbative contributions.
• To compare the performance of standard perturbation theory (PT), analytic perturbation theory (APT), and modified perturbation theory (MPT) in describing low-$ Q^2 $ data.

Proposed method

• Utilizes the four-loop coefficient function $ C_{\text{Bj}}(Q^2) $, known up to $ \alpha_s^4 $, to compute the perturbative correction $ \Delta_{\text{Bj}}^{\text{PT}}(Q^2) $.
• Applies the running coupling $ \alpha_s(Q^2) $ derived from the four-loop $ \beta $-function to model the $ Q^2 $-dependence of the perturbative series.
• Uses an integral model for the perturbative correction to study asymptotic behavior and validate the series' convergence properties.
• Employs analytic perturbation theory (APT) to replace the standard $ \alpha_s^n $ terms with spectral integrals, ensuring causality and eliminating unphysical singularities.
• Applies modified perturbation theory (MPT) by introducing an effective gluonic mass $ m_{\text{gl}} $, modifying the logarithmic running to $ \ln(\xi + Q^2/\Lambda^2) $.
• Performs $ \mu_4 $-fits to Jefferson Lab and COMPASS data using PT, APT, and MPT frameworks to compare infrared behavior and stability.

Experimental results

Research questions

• RQ1 Does the four-loop perturbative series for the Bjorken sum rule exhibit asymptotic behavior at low $ Q^2 $?
• RQ2 How do higher-order perturbative corrections affect the extraction of the higher-twist $ \mu_4 $ coefficient?
• RQ3 Can analytic perturbation theory (APT) improve the stability and reliability of $ \mu_4 $ extraction in the infrared region?
• RQ4 Does modified perturbation theory (MPT) allow for a consistent description of the Bjorken sum rule data down to $ Q^2 \to 0 $?
• RQ5 How do the results from APT and MPT compare with standard perturbation theory in fitting low-$ Q^2 $ data?

Key findings

• The perturbative series for the Bjorken sum rule shows asymptotic behavior at $ Q^2 \lesssim 1 \, \text{GeV}^2 $, where the $ \alpha_s^4 $ term dominates, indicating a breakdown of convergence.
• The four-loop correction does not improve theoretical precision at low $ Q^2 $, confirming the series' asymptotic nature.
• The integral model for the perturbative correction supports the asymptotic behavior observed in the series expansion.
• Using APT, the higher-twist coefficient $ \mu_4 $ is extracted as $ -0.044 \pm 0.004 \, \text{GeV}^2 $, with consistent values across all loop orders, indicating stability.
• MPT allows a reliable description of the Bjorken sum rule data down to $ Q^2 \sim 0.1 \, \text{GeV}^2 $, with a preliminary fit showing good agreement with Jefferson Lab data.
• APT and MPT both improve the infrared behavior compared to standard PT, with APT providing better stability in $ \mu_4 $ extraction across loop orders.