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)'' : $<\pi (p_2)|2\cdot {\sum }_{i=1}^{N_f}{m}_f{\bar{\psi }}_i{\psi }_i|\pi (p_1)>=2M_{\pi }^2+[(1-{\gamma }_m)M_{\pi }^2\cdot 2/(1+{\gamma }_m)]=2M_{\pi }^2\cdot 2/(1+{\gamma }_m)\ne 2M_{\pi }^2$ in contradiction with his estimate, which is compared with $<\pi (p_2)|(1+{\gamma }_m)\cdot {\sum }_{i=1}^{N_f}{m}_f{\bar{\psi }}_i{\psi }_i|\pi (p_1)>=(1+{\gamma }_m)M_{\pi }^2+[(1-{\gamma }_m)M_{\pi }^2]=2M_{\pi }^2$ (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.

Comments: Submitted to Special Issue `Emergence of Symmetries in Strong Nuclear Correlations' of Symmetry