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 π^+ - π^0 mass difference Δm_π^2 \equiv m_{π^+}^2 -m_{π^0}^2 as well as the NG boson decay constant f_πon the same footing in the large N_f QCD, through the difference between the vector current correlator Π_{VV} and the axial-vector current correlator Π_{AA}. Approaching the chiral phase transition point α_*\to α_{\rm cr} (=π/4) from the broken phase, where α_* is the the gauge coupling on the infrared fixed point, Δm_π^2 as well as f_π^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 Δm_π^2/f_π^2 \sim (α_*/α_{\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.