Analytical Inverse QCD Coupling Constant approach and its result for $α_s$

TL;DR

The paper tackles constructing an analytic, infrared-confining model for the QCD running coupling by focusing on the inverse coupling $\varepsilon_s(q^2)=1/\alpha_s(q^2)$ within Analytic Perturbation Theory (APT). It extends APT by introducing three parametric regularizing functions in the spectral density, enforcing $\varepsilon_s(Q^2)\to0$ as $Q^2\to0$ to mimic confinement while preserving the correct ultraviolet behavior and deriving two-loop analytic expressions. The authors fit the regularization parameter $p$ (with $\Lambda$ around 300 MeV) to experimental $\alpha_s$ data, obtaining values that yield $\alpha_s(M_Z^2)$ in line with the PDG world average $0.1180\pm0.0009$, thereby validating the IR-confining, analytically well-behaved construction. This work provides a framework for incorporating confinement in perturbative QCD predictions via an ICC-centered dispersion approach and suggests further exploration of multi-parameter fits and broader observables.

Abstract

We propose a model for the QCD running coupling constant based on the Analytical Inverse QCD Coupling Constant concept with an additional regularization in the low momentum region. Analyticity in the $q^2$-complex plane, where $q$ is the 4-momentum transfer, is imposed by methods of the Analytic Perturbation Theory. The model incorporates a peculiar low-momentum behavior for $α_s(q^2)$ as a divergence at $q^2=0$ to retrieve color confinement, without spoiling its correct high-momentum behavior. This was achieved by means of a two-parameter regularization function, for which we considered three possible analytic expressions. In fact, in the framework of the Analytic Perturbation Theory, $α_s(q^2)$ assumes a finite value for $q^2=0$, at all perturbative orders (\emph{infrared stability}), hence the infrared divergence can not be implemented. For this reason, we found it more straightforward to work with its reciprocal, namely $\varepsilon_s(q^2) = 1/α_s(q^2)$, imposing its vanishing at the origin of the $q^2$-complex plane via the multiplication of the aforementioned regularizing functions to the spectral density. Once the two free parameters of the regularization functions are settled by fitting to the experimental values of $α_s(q^2)$ at the momenta where these data are available and reliable, the model can reproduce the QCD running coupling constant at any other momentum transferred.} \\ \noindent \textbf{Keywords: }APT, Analytical Inverse QCD coupling constant ICC, regularization functions, $α_s(M_Z^2)$.

Figures (4)

• Figure 1: Momentum spectra for the three regularizing functions, $p=0.8$ and $\Lambda=300 \,\, \textrm{MeV}$.
• Figure 2: 4-momentum squared distribution for $[\overline{\varepsilon}_s(t)]_{\rm an}$, $p=0.8$, $\Lambda=300 \,\, \textrm{MeV}$ and $n_f=5$.
• Figure 3: 4-momentum squared distribution for $[\overline{\alpha}_s(t)]_{\rm an}$, for $p=0.8$, $\Lambda=300 \,\, \textrm{MeV}$ and $n_f=5$.
• Figure 4: Optimal curves for momentum evolution of the running coupling constant ${\overline{\alpha}}_s(t)$, $\Lambda=300$ MeV.