[Paper Review] Analyticity of QCD observables beyond leading-order perturbation theory

N. G. Stefanis, A. I. Karanikas|ArXiv.org|Feb 21, 2008

Particle physics theoretical and experimental studies3 references3 citations

TL;DR

This paper extends Analytic Perturbation Theory (APT) to Fractional Analytic Perturbation Theory (FAPT) to treat QCD observables beyond leading order, incorporating non-integer powers of the running coupling via dispersion relations that include all spectral density contributions. The method eliminates unphysical singularities, reduces renormalization scale dependence, and resums $π^2$ terms, yielding stable, scheme-independent predictions for processes like Higgs decay and the pion form factor at next-to-leading order and beyond.

ABSTRACT

A theoretical framework is presented to treat hadronic observables within analytic perturbative QCD beyond the leading order of the coupling and for more than one single large momentum scale. The approach generalizes and extends the pioneering work of Shirkov and Solovtsov on an analytic strong running coupling. Some applications to hadronic observables at the partonic level are also discussed.

Motivation & Objective

• To generalize the analyticity principle of QCD coupling beyond one-loop order and for multiple large momentum scales.
• To address the limitations of standard perturbative QCD, particularly the Landau pole and renormalization scale sensitivity.
• To develop a framework that incorporates non-integer powers of the coupling through fractional analytic continuation.
• To achieve scheme- and scale-independent predictions for hadronic observables at next-to-leading order.
• To enable accurate calculations of processes involving both factorization and evolution scales, such as the pion form factor and Higgs decay.

Proposed method

• Adopt the Karanikas-Stefanis (KS) analytization principle, requiring all terms affecting the spectral density to be included in the dispersion relation.
• Use dispersion relations to define analytic images of real powers of the strong coupling, extending the Shirkov-Solovtsov approach to non-integer powers.
• Incorporate higher-loop renormalization group effects through coefficients $\Delta_m^{(l)}$ and $\mathfrak{a}_{m+\nu_0}^{(l)}$, which include resummed $\pi^2$ terms.
• Apply the formalism to both spacelike (Euclidean) and timelike (Minkowski) regions, ensuring singularity-free expressions.
• Utilize the $\overline{\text{MS}}$ scheme with $N_f = 5$ active flavors and match results to standard perturbative QCD at two-loop order.
• Implement numerical techniques to compute analytic coupling images and expansion coefficients at one- and two-loop levels.

Experimental results

Research questions

• RQ1 How can the analyticity of QCD coupling be extended beyond one-loop order to include non-integer powers of the coupling?
• RQ2 What is the impact of including all spectral density contributions on the renormalization scale dependence of hadronic observables?
• RQ3 How does Fractional Analytic Perturbation Theory (FAPT) improve the convergence and stability of perturbative expansions in QCD?
• RQ4 To what extent does FAPT resum $\pi^2$-terms induced by analytic continuation in the timelike region?
• RQ5 Can FAPT yield scheme- and scale-independent predictions for processes like the Higgs decay into $b\bar{b}$ pairs at next-to-leading order?

Key findings

• FAPT reduces sensitivity to the factorization scale in observables like the pion’s electromagnetic form factor, verified at next-to-leading order.
• The method achieves quasi renormalization-scale independence for the same observable at the next-to-leading order level.
• The inclusion of evolution effects beyond one-loop order is naturally incorporated through the analytic coupling images.
• The perturbative expansion in FAPT converges faster due to the resummation of $\pi^2$ terms via analytic continuation.
• For the Higgs decay into $b\bar{b}$, FAPT predictions agree with standard perturbative QCD at two-loop order but show slightly larger values due to resummed $\pi^2$ contributions in the coefficients $\mathfrak{a}_{\nu}$.
• The framework enables the computation of leading-order power corrections for reactions such as the pion form factor and Drell-Yan process, embedding the scheme in a broader theoretical context.