Anomalous Dimension in QCD

Abstract

The anomalous dimension $\gamma_m =1$ 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, $\gamma_m =1$. We show that this is not justified by explicit evaluation of each matrix element based on the ``dilaton chiral perturbation theory (dChPT)'' : $<\pi(p_2)| 2\cdot \sum^{N_f}_{i=1} m_f \bar \psi_i \psi_i |\pi(p_1)>= 2M_\pi^2 + [(1-\gamma_m) M_\pi^2\cdot 2/(1+\gamma_m)]= 2 M_\pi^2 \cdot 2/(1+\gamma_m) \ne 2 M_\pi^2$ in contradiction with his estimate, which is compared with $<\pi(p_2)| (1+\gamma_m) \cdot \sum^{N_f}_{i=1} m_f \bar \psi_i \psi_i |\pi(p_1)> =(1+\gamma_m) M_\pi^2+ [(1-\gamma_m) M_\pi^2]=2 M_\pi^2$ (both up to trace anomaly), where the terms in $[ \,\,]$ are from the $\sigma$ (pseudo-dilaton) pole contribution. Thus there is no constraint on $\gamma_m$ when the $\sigma$ pole contribution is treated consistently for both.