{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T08:05:58Z","timestamp":1761897958832,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,12,19]],"date-time":"2023-12-19T00:00:00Z","timestamp":1702944000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The anomalous dimension \u03b3m=1 in the infrared region near the conformal edge in the broken phase of the large Nf QCD has been shown by the ladder Schwinger\u2013Dyson equation and also by the lattice simulation for Nf=8 and for Nc=3. Recently, Zwicky made another independent argument (without referring to explicit dynamics) for the same result, \u03b3m=1, by comparing the pion matrix element of the trace of the energy-momentum tensor \u03c0(p2)|(1+\u03b3m)\u00b7\u2211i=1Nfmf\u03c8\u00afi\u03c8i|\u03c0(p1)=\u03c0(p2)|\u03b8\u03bc\u03bc|\u03c0(p1)=2M\u03c02 (up to trace anomaly) with the estimate of \u03c0(p2)|2\u00b7\u2211i=1Nfmf\u03c8\u00afi\u03c8i|\u03c0(p1)=2M\u03c02 through the Feynman\u2013Hellmann theorem combined with an assumption M\u03c02\u223cmf 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): \u03c0(p2)|2\u00b7\u2211i=1Nfmf\u03c8\u00afi\u03c8i|\u03c0(p1)=2M\u03c02+[(1\u2212\u03b3m)M\u03c02\u00b72\/(1+\u03b3m)]=2M\u03c02\u00b72\/(1+\u03b3m)\u22602M\u03c02 in contradiction with his estimate, which is compared with \u03c0(p2)|(1+\u03b3m)\u00b7\u2211i=1Nfmf\u03c8\u00afi\u03c8i|\u03c0(p1)=(1+\u03b3m)M\u03c02+[(1\u2212\u03b3m)M\u03c02]=2M\u03c02 (both up to trace anomaly), where the terms in [] are from the \u03c3 (pseudo-dilaton) pole contribution. Thus, there is no constraint on \u03b3m when the \u03c3 pole contribution is treated consistently for both. We further show that the Feynman\u2013Hellmann theorem is applied to the inside of the conformal window where dChPT is invalid and the \u03c3 pole contribution is absent, and with M\u03c02\u223cmf2\/(1+\u03b3m) instead of M\u03c02\u223cmf, we have the same result as ours in the broken phase. A further comment related to dChPT is made on the decay width of f0(500) to \u03c0\u03c0 for Nf=2. It is shown to be consistent with the reality, when f0(500) is regarded as a pseudo-NG boson with the non-perturbative trace anomaly dominance.<\/jats:p>","DOI":"10.3390\/sym16010002","type":"journal-article","created":{"date-parts":[[2023,12,19]],"date-time":"2023-12-19T06:03:11Z","timestamp":1702965791000},"page":"2","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the Anomalous Dimension in QCD"],"prefix":"10.3390","volume":"16","author":[{"given":"Koichi","family":"Yamawaki","sequence":"first","affiliation":[{"name":"Kobayashi-Maskawa Institute for the Origin of Particles and the Universe (KMI), Nagoya University, Nagoya 464-8602, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1335","DOI":"10.1103\/PhysRevLett.56.1335","article-title":"Scale Invariant Technicolor Model and a Technidilaton","volume":"56","author":"Yamawaki","year":"1986","journal-title":"Phys. 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