Analytical Inverse QCD Coupling Constant approach and its result for $\alpha_s$

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 $\alpha_s(q^2)$ 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, $\alpha_s(q^2)$ 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 $\varepsilon_s(q^2) = 1/\alpha_s(q^2)$, 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 $\alpha_s(q^2)$ 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, $\alpha_s(M_Z^2)$.