Walking on the Ladder: 125 GeV Technidilaton, or Conformal Higgs -Dedicated to the late Professor Yoichiro Nambu-

The walking technicolor based on the ladder Schwinger-Dyson gap equation is studied, with the scale-invariant coupling being an idealization of the Caswell-Banks-Zaks infrared fixed point in the "anti-Veneziano limit", such that $N_C \rightarrow \infty$ with $N_C \cdot \alpha(\mu^2)=$ fixed and $N_F/N_C=$ fixed ($\gg 1$), of the $SU(N_C)$ gauge theory with massless $N_F$ flavors near criticality. We show that the 125 GeV Higgs can be naturally identified with the technidilaton (TD) predicted in the walking technicolor, a pseudo Nambu-Goldstone (NG) boson of the spontaneous symmetry breaking of the approximate scale symmetry. Ladder calculations yield the TD mass $M_\phi$ from the trace anomaly as $M_\phi^2 F_\phi^2= -4 \langle \theta_\mu^\mu \rangle = - \frac{\beta(\alpha (\mu^2))}{\alpha(\mu^2)}\, \langle G_{\lambda \nu}^2(\mu^2)\rangle \simeq N_C N_F\frac{16}{\pi^4} m_F^4$, independently of the renormalization point $\mu$, where $m_F$ is the dynamical mass of the technifermion, and $F_\phi={\cal O} (\sqrt{N_F N_C}\, m_F)$ the TD decay constant. It reads $M_\phi^2\simeq (\frac{v_{\rm EW}}{2} \cdot \frac{5 v_{\rm EW}}{F_\phi})^2 \cdot [\frac{8}{N_F}\frac{4}{N_C}]$, ($v_{\rm EW}=246$ GeV), which implies $F_\phi\simeq 5 \,v_{\rm EW}$ for $M_\phi \simeq 125\, {\rm GeV}\simeq \frac{1}{2} v_{\rm EW}$ in the one-family model ($N_C=4, N_F=8$), in good agreement with the current LHC Higgs data. The result reflects a generic scaling $M_\phi^2/v_{\rm EW}^2\sim M_\phi^2/F_\phi^2 \sim m_F^2 /F_\phi^2 \sim 1/(N_F N_C) \rightarrow 0$ as a vanishing trace anomaly, namely the TD has a mass vanishing in the anti-Veneziano limit, similarly to $\eta^\prime$ meson as a pseudo-NG boson of the ordinary QCD with vanishing $U(1)_A$ anomaly in the Veneziano limit ($N_F/N_C \ll 1$).Comment: revtex4, 36 pages, 7 eps figures, some corrections made, references added; a version to appear in JHEP; typo correcte

SS Parameter in the Holographic Walking/Conformal Technicolor

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 $\gamma_m \simeq 0$ through $\gamma_m \simeq 1$ continuously. The $S$ parameter is given as a positive monotonic function of $\xi$ which is fairly insensitive to $\gamma_m$ and continuously vanishes as $S \sim \xi^2 \to 0$ when $\xi \to 0$, where $\xi$ 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-) $\rho$ meson ($M_\rho$) and the decay constant ($f_\pi$) as $\xi \sim f_\pi z_m \sim f_\pi/M_\rho$ for $\xi \ll 1$. However, although $\xi$ is related to the techni-fermion condensate \condense, we find no particular suppression of $\xi$ and hence of $S$ due to large $\gamma_m$, based on the correct identification of the renormalization-point dependence of \condense in contrast to the literature. Then we argue possible behaviors of $f_\pi/M_\rho$ as \condense \to 0 near the conformal window characterized by the Banks-Zaks infrared fixed point in more explicit dynamics with $\gamma_m \simeq 1$. 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 $f_\pi/M_\rho \to 0$ is realized, the holography suggests a novel possibility that $f_\pi$ vanishes much faster than the dynamical mass $m$ does.Comment: typo, a version to be published in Progress of Theoretical Physic