Tachyonic AdS/QCD, Determining the Strong Running Coupling and \beta-function in both UV and IR Regions of AdS Space

In this paper we investigate the QCD running coupling $\alpha_s^{AdS}(Q^{2})$ and the associated $\beta$-function $\beta(Q^{2})$ in the spirit of tachyonic AdS/QCD. We distort the $\text{AdS}_5$ conformal symmetry using \textit{color dielectric function} $G(\phi(z))$ associated with free tachyons. Results obtained in this region are expected to present characteristics similar to the perturbative QCD. We further study the distortion of the conformal symmetry with \textit{condensed tachyon color dielectric function} $G(\eta(z))$ which leads to \textit{color confinement} in the IR region similar to low energy nonperturbative QCD. Results obtained in this region are compared with AdS/QCD with positive dilaton profile, $e^{\kappa^2 z^2}$. The \textit{color dielectric function} is carefully defined to generate the needed characteristics in both regions with dependence only on fifth dimensional coordinate $z$.