Anomalous Dimension in QCD

Abstract

The anomalous dimension $γ_m =1$ in the infrared region near conformal edge in the broken phase of the large $N_f$ QCD has been shown by the ladder Schwinger-Dyson equation and also by the lattice simulation for $N_f=8$ for $ N_c=3$. Recently Zwicky claimed another independent argument (without referring to explicit dynamics) for the same result, $γ_m =1$. We show that this is not justified by explicit evaluation of each matrix element based on the ``dilaton chiral perturbation theory (dChPT)'' : $<π(p_2)| 2\cdot \sum^{N_f}_{i=1} m_f \bar ψ_i ψ_i |π(p_1)>= 2M_π^2 + [(1-γ_m) M_π^2\cdot 2/(1+γ_m)]= 2 M_π^2 \cdot 2/(1+γ_m) \ne 2 M_π^2$ in contradiction with his estimate, which is compared with $<π(p_2)| (1+γ_m) \cdot \sum^{N_f}_{i=1} m_f \bar ψ_i ψ_i |π(p_1)> =(1+γ_m) M_π^2+ [(1-γ_m) M_π^2]=2 M_π^2$ (both up to trace anomaly), where the terms in $[ \,\,]$ are from the $σ$ (pseudo-dilaton) pole contribution. Thus there is no constraint on $γ_m$ when the $σ$ pole contribution is treated consistently for both.