pi^+ - pi^0 mass difference and S parameter in the large N_f QCD

Abstract

In the framework of the Schwinger-Dyson equation and the Bethe-Salpeter equation in the improved ladder approximation, we calculate the S parameter and an analogue of the \pi^+ - \pi^0 mass difference \Delta m_\pi^2 \equiv m_{\pi^+}^2 -m_{\pi^0}^2 as well as the NG boson decay constant f_\pi on the same footing in the large N_f QCD, through the difference between the vector current correlator \Pi_{VV} and the axial-vector current correlator \Pi_{AA}. Approaching the chiral phase transition point \alpha_*\to \alpha_{\rm cr} (=\pi/4) from the broken phase, where \alpha_* is the the gauge coupling on the infrared fixed point, \Delta m_\pi^2 as well as f_\pi^2 goes to zero with the essential-singularity scaling (Miransky scaling), while the ratio indicates a blowing up enhancement reflecting the characteristic behavior of the large N_f QCD as a walking theory which is expected to scale as \Delta m_\pi^2/f_\pi^2 \sim (\alpha_*/\alpha_{\rm cr}-1)^{-1/2}. On the other hand, the S parameter takes values somewhat smaller than that of the real-life QCD and indicates slightly decreasing tendency as we approach the phase transition point.