On the Anomalous Dimension in QCD.

The anomalous dimension γ m = 1 in the infrared region near the 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 and for N c = 3 . Recently, Zwicky made another independent argument (without referring to explicit dynamics) for the same result, γ m = 1 , by comparing the pion matrix element of the trace of the energy-momentum tensor π (p 2) | (1 + γ m) · ∑ i = 1 N f m f ψ ¯ i ψ i | π (p 1) = π (p 2) | θ μ μ | π (p 1) = 2 M π 2 (up to trace anomaly) with the estimate of π (p 2) | 2 · ∑ i = 1 N f m f ψ ¯ i ψ i | π (p 1) = 2 M π 2 through the Feynman–Hellmann theorem combined with an assumption M π 2 ∼ m f characteristic of the broken phase. We show that this is not justified by the explicit evaluation of each matrix element based on the dilaton chiral perturbation theory (dChPT): π (p 2) | 2 · ∑ i = 1 N f m f ψ ¯ i ψ i | π (p 1) = 2 M π 2 + [ (1 − γ m) M π 2 · 2 / (1 + γ m) ] = 2 M π 2 · 2 / (1 + γ m) ≠ 2 M π 2 in contradiction with his estimate, which is compared with π (p 2) | (1 + γ m) · ∑ i = 1 N f m f 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. We further show that the Feynman–Hellmann theorem is applied to the inside of the conformal window where dChPT is invalid and the σ pole contribution is absent, and with M π 2 ∼ m f 2 / (1 + γ m) instead of M π 2 ∼ m f , we have the same result as ours in the broken phase. A further comment related to dChPT is made on the decay width of f 0 (500) to π π for N f = 2 . It is shown to be consistent with the reality, when f 0 (500) is regarded as a pseudo-NG boson with the non-perturbative trace anomaly dominance. [ABSTRACT FROM AUTHOR]