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

In this paper, we investigate the Quantum Chromodynamics (QCD)-like running coupling, ${\alpha }_s^{AdS}(Q^2)$, and its associated $\beta$-function within a tachyonic Anti-de Sitter (AdS)/QCD framework. The ${\text{AdS}}_5$ bulk geometry is deformed through the introduction of a color dielectric function $G(\phi (z))$, associated with a tachyon field $\phi (z)$. This function governs the behavior of ${\alpha }_s^{AdS}(Q^2)$ across all momentum scales by modifying the AdS background at both small and large values of the holographic coordinate $z$.In the ultraviolet (UV) regime (small $z$), the deformation is driven by free tachyons and reproduces features consistent with perturbative QCD. In contrast, in the infrared (IR) regime (large $z$), tachyon condensation dominates, yielding behavior characteristic of nonperturbative QCD. This construction enables a unified description of the running coupling and its $\beta$-function over the full range of momentum transfer $Q^2$, where $Q^2$ denotes the space-like momentum scale.