$S$ Parameter in the Holographic Walking/Conformal Technicolor

Abstract

We explicitly calculate the $S$ parameter in entire parameter space of the holographic walking/conformal technicolor (W/C TC), based on the deformation of the holographic QCD by varying the anomalous dimension from $γ_m \simeq 0$ through $γ_m \simeq 1$ continuously. The $S$ parameter is given as a positive monotonic function of $ξ$ which is fairly insensitive to $γ_m$ and continuously vanishes as $S \sim ξ^2 \to 0$ when $ξ\to 0$, where $ξ$ is the vacuum expectation value of the bulk scalar field at the infrared boundary of the 5th dimension $z=z_m$ and is related to the mass of (techni-) $ρ$ meson ($M_ρ$) and the decay constant ($f_π$) as $ξ\sim f_πz_m \sim f_π/M_ρ$ for $ξ\ll 1$. However, although $ξ$ is related to the techni-fermion condensate $\condense$, we find no particular suppression of $ξ$ and hence of $S$ due to large $γ_m$, based on the correct identification of the renormalization-point dependence of $\condense$ in contrast to the literature. Then we argue possible behaviors of $f_π/M_ρ$ as $\condense \to 0$ near the conformal window characterized by the Banks-Zaks infrared fixed point in more explicit dynamics with $γ_m \simeq 1$. It is a curious coincidence that the result from ladder Schwinger-Dyson and Bethe-Salpeter equations well fits in the parameter space obtained in this paper. When $f_π/M_ρ\to 0$ is realized, the holography suggests a novel possibility that $f_π$ vanishes much faster than the dynamical mass $m$ does.