[Paper Review] QCD coupling up to third order in standard and analytic perturbation theories

B. A. Magradze|ArXiv.org|Oct 8, 2000

Particle physics theoretical and experimental studies12 citations

TL;DR

This paper presents exact analytical solutions for the QCD running coupling up to third order in both standard perturbation theory (PT) and analytic perturbation theory (APT), using the Lambert W function to solve the renormalization group equations. It derives universal analytic coupling functions $\mathcal{A}_n(Q^2,f)$ and $\mathfrak{A}_n(s,f)$ in spacelike and timelike regions, respectively, and provides numerical tables for these functions at second and third orders, significantly improving infrared behavior and reducing scheme dependence.

ABSTRACT

We analyze two sets of specific functions, that/which form the basis of the nonpower asymptotic expansions both in the timelike and spacelike regions for single scale dependent QCD observables in the Shirkov--Solovtsov's Analytic Perturbation Theory (APT) free of unphysical singularities. These functions are explicitly derived up to the third order in the closed form in terms of the Lambert-W function. As an input we used the exact two loop and the three loop (corresponding to Padé transformed beta-function) RG solutions for common invariant coupling α_s. The elegant recurrence formulas, helpful for numerical analysis, are obtained for the both sets of the APT functions. Then we construct the global versions of APT functions using the continuity conditions (at the quark thresholds) on the α_s in the \bar{MS} scheme and give numerical results. For first three of these functions \mathfrak{A}_n(s) and {\cal A}_n(Q^2); n=1,2,3 in the large interval of the momentum transfer and energy (1 GeV

Motivation & Objective

• To derive exact analytical solutions for the QCD running coupling up to third order in the $\overline{MS}$ scheme using the Lambert W function.
• To construct universal analytic coupling functions $\mathcal{A}_n(Q^2,f)$ and $\mathfrak{A}_n(s,f)$ valid across the full momentum range, independent of quark flavor thresholds.
• To reduce scheme dependence in QCD observables by expanding the coupling in terms of a scheme-independent two-loop solution.
• To provide numerical evaluations of the analytic couplings $\mathcal{A}_1(Q^2,f)$, $\mathfrak{A}_1(s,f)$, and their powers up to third order, including comparisons with standard PT and Pade approximants.

Proposed method

• Solving the two-loop and three-loop renormalization group equations for the QCD coupling $\alpha_s(Q^2,f)$ using the Lambert W function, yielding exact closed-form expressions.
• Using the solution to define analytic couplings $\mathcal{A}_n(Q^2,f)$ and $\mathfrak{A}_n(s,f)$ via dispersion relations, ensuring correct analyticity and infrared behavior.
• Deriving the spectral functions $\tilde{\rho}_n^{(3)}(t,f)$ from the imaginary parts of the Pade-approximated three-loop coupling in the complex $Q^2$-plane.
• Constructing global universal functions $\mathcal{A}_n(Q^2)$ and $\mathfrak{A}_n(s)$ by matching across quark flavor thresholds using analytical solutions.
• Applying iterative and Pade approximants to the three-loop coupling and comparing their numerical behavior and range of validity.
• Numerically evaluating the analytic couplings $\mathcal{A}_n(Q^2,f)$ and $\mathfrak{A}_n(s,f)$ for $n=1,2,3$ at various $Q^2$ and $s$ values, with $\Lambda_{\overline{MS}}=450$ MeV.

Experimental results

Research questions

• RQ1 How can the QCD running coupling be solved exactly at two- and three-loop orders using special functions like the Lambert W function?
• RQ2 What are the analytical structures and spectral properties of the analytic couplings $\mathcal{A}_n(Q^2,f)$ and $\mathfrak{A}_n(s,f)$ in the spacelike and timelike regions?
• RQ3 How do the analytic couplings $\mathcal{A}_n(Q^2,f)$ and $\mathfrak{A}_n(s,f)$ differ from the standard perturbative coupling $\alpha_s(Q^2,f)$, especially in the infrared?
• RQ4 What is the numerical behavior and range of validity of iterative versus Pade approximants for the three-loop coupling?
• RQ5 How can universal, flavor-independent analytic coupling functions $\mathcal{A}_n(Q^2)$ and $\mathfrak{A}_n(s)$ be constructed across all energy scales?

Key findings

• The two-loop QCD coupling is solved exactly using the Lambert W function, with the solution expressed as $\alpha_s^{(2)}(Q^2,f) = -\frac{\beta_0}{\beta_1} \frac{1}{1 + W_{-1}(\zeta)}$, where $\zeta$ depends on $Q^2/\Lambda^2$ and the beta-function coefficients.
• The three-loop coupling is expressed in Pade-approximated form as $\alpha_{\text{Pade}}^{(3)}(Q^2,f) = -\frac{\beta_0}{\beta_1} \frac{1}{1 - \beta_0\beta_2/\beta_1^2 + W_{-1}(\xi)}$, enabling analytic continuation and spectral function construction.
• The analytic couplings $\mathcal{A}_n(Q^2,f)$ and $\mathfrak{A}_n(s,f)$ are derived via dispersion relations, ensuring correct analyticity and oscillatory behavior in the infrared.
• Numerical tables (Tables 7–12) provide values for $\mathcal{A}_1(Q^2,f)$, $\mathcal{A}_2(Q^2,f)$, $\mathcal{A}_3(Q^2,f)$, $\mathfrak{A}_1(s,f)$, $\mathfrak{A}_2(s,f)$, and $\mathfrak{A}_3(s,f)$ at $\Lambda = 450$ MeV, covering $Q^2$ and $s$ from 1 to 200 GeV$^2$.
• The iterative approximant for the three-loop coupling is valid up to $Q^2 \sim 100$ GeV$^2$, beyond which Pade approximants show better convergence.
• The analytic couplings $\mathcal{A}_1(Q^2)$ and $\mathfrak{A}_1(s)$ remain finite and well-behaved in the infrared, avoiding the Landau pole, with $\mathfrak{A}_1(s)$ decreasing from 0.385 at $\sqrt{s}=1$ GeV to 0.121 at $\sqrt{s}=200$ GeV.