| Summary: | The anomalous dimension <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mi>m</mi></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula> in the infrared region near the conformal edge in the broken phase of the large <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>N</mi><mi>f</mi></msub></semantics></math></inline-formula> QCD has been shown by the ladder Schwinger–Dyson equation and also by the lattice simulation for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>8</mn></mrow></semantics></math></inline-formula> and for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>N</mi><mi>c</mi></msub><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula>. Recently, Zwicky made another independent argument (without referring to explicit dynamics) for the same result, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>γ</mi><mi>m</mi></msub><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>, by comparing the pion matrix element of the trace of the energy-momentum tensor <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="⟨" close="⟩"><mi>π</mi><mrow><mo>(</mo><msub><mi>p</mi><mn>2</mn></msub><mo>)</mo></mrow><mrow><mo stretchy="false">|</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msub><mi>γ</mi><mi>m</mi></msub><mo>)</mo></mrow><mo>·</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>N</mi><mi>f</mi></msub></msubsup><msub><mi>m</mi><mi>f</mi></msub><msub><mover accent="true"><mi>ψ</mi><mo stretchy="false">¯</mo></mover><mi>i</mi></msub><msub><mi>ψ</mi><mi>i</mi></msub><mo stretchy="false">|</mo></mrow><mi>π</mi><mrow><mo>(</mo><msub><mi>p</mi><mn>1</mn></msub><mo>)</mo></mrow></mfenced><mo>=</mo><mfenced separators="" open="⟨" close="⟩"><mi>π</mi><mrow><mo>(</mo><msub><mi>p</mi><mn>2</mn></msub><mo>)</mo></mrow><mrow><mo stretchy="false">|</mo><msubsup><mi>θ</mi><mi>μ</mi><mi>μ</mi></msubsup><mo stretchy="false">|</mo></mrow><mi>π</mi><mrow><mo>(</mo><msub><mi>p</mi><mn>1</mn></msub><mo>)</mo></mrow></mfenced><mo>=</mo><mn>2</mn><msubsup><mi>M</mi><mi>π</mi><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> (up to trace anomaly) with the estimate of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="⟨" close="⟩"><mi>π</mi><mrow><mo>(</mo><msub><mi>p</mi><mn>2</mn></msub><mo>)</mo></mrow><mrow><mo stretchy="false">|</mo><mn>2</mn><mo>·</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>N</mi><mi>f</mi></msub></msubsup><msub><mi>m</mi><mi>f</mi></msub><msub><mover accent="true"><mi>ψ</mi><mo stretchy="false">¯</mo></mover><mi>i</mi></msub><msub><mi>ψ</mi><mi>i</mi></msub><mo stretchy="false">|</mo></mrow><mi>π</mi><mrow><mo>(</mo><msub><mi>p</mi><mn>1</mn></msub><mo>)</mo></mrow></mfenced><mo>=</mo><mn>2</mn><msubsup><mi>M</mi><mi>π</mi><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> through the Feynman–Hellmann theorem combined with an assumption <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>M</mi><mi>π</mi><mn>2</mn></msubsup><mo>∼</mo><msub><mi>m</mi><mi>f</mi></msub></mrow></semantics></math></inline-formula> 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): <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="⟨" close="⟩"><mi>π</mi><mrow><mo>(</mo><msub><mi>p</mi><mn>2</mn></msub><mo>)</mo></mrow><mrow><mo stretchy="false">|</mo><mn>2</mn><mo>·</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>N</mi><mi>f</mi></msub></msubsup><msub><mi>m</mi><mi>f</mi></msub><msub><mover accent="true"><mi>ψ</mi><mo stretchy="false">¯</mo></mover><mi>i</mi></msub><msub><mi>ψ</mi><mi>i</mi></msub><mo stretchy="false">|</mo></mrow><mi>π</mi><mrow><mo>(</mo><msub><mi>p</mi><mn>1</mn></msub><mo>)</mo></mrow></mfenced><mo>=</mo><mn>2</mn><msubsup><mi>M</mi><mi>π</mi><mn>2</mn></msubsup><mo>+</mo><mrow><mo>[</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msub><mi>γ</mi><mi>m</mi></msub><mo>)</mo></mrow><msubsup><mi>M</mi><mi>π</mi><mn>2</mn></msubsup><mo>·</mo><mn>2</mn><mo>/</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msub><mi>γ</mi><mi>m</mi></msub><mo>)</mo></mrow><mo>]</mo></mrow><mo>=</mo><mn>2</mn><msubsup><mi>M</mi><mi>π</mi><mn>2</mn></msubsup><mo>·</mo><mn>2</mn><mo>/</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msub><mi>γ</mi><mi>m</mi></msub><mo>)</mo></mrow><mo>≠</mo><mn>2</mn><msubsup><mi>M</mi><mi>π</mi><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> in contradiction with his estimate, which is compared with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="" open="⟨" close="⟩"><mi>π</mi><mrow><mo>(</mo><msub><mi>p</mi><mn>2</mn></msub><mo>)</mo></mrow><mrow><mo stretchy="false">|</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msub><mi>γ</mi><mi>m</mi></msub><mo>)</mo></mrow><mo>·</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><msub><mi>N</mi><mi>f</mi></msub></msubsup><msub><mi>m</mi><mi>f</mi></msub><msub><mover accent="true"><mi>ψ</mi><mo stretchy="false">¯</mo></mover><mi>i</mi></msub><msub><mi>ψ</mi><mi>i</mi></msub><mo stretchy="false">|</mo></mrow><mi>π</mi><mrow><mo>(</mo><msub><mi>p</mi><mn>1</mn></msub><mo>)</mo></mrow></mfenced><mo>=</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msub><mi>γ</mi><mi>m</mi></msub><mo>)</mo></mrow><msubsup><mi>M</mi><mi>π</mi><mn>2</mn></msubsup><mo>+</mo><mrow><mo>[</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><msub><mi>γ</mi><mi>m</mi></msub><mo>)</mo></mrow><msubsup><mi>M</mi><mi>π</mi><mn>2</mn></msubsup><mo>]</mo></mrow><mo>=</mo><mn>2</mn><msubsup><mi>M</mi><mi>π</mi><mn>2</mn></msubsup></mrow></semantics></math></inline-formula> (both up to trace anomaly), where the terms in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mspace width="0.166667em"></mspace><mspace width="0.166667em"></mspace><mo>]</mo></mrow></semantics></math></inline-formula> are from the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> (pseudo-dilaton) pole contribution. Thus, there is no constraint on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>γ</mi><mi>m</mi></msub></semantics></math></inline-formula> when the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula> pole contribution is absent, and with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>M</mi><mi>π</mi><mn>2</mn></msubsup><mo>∼</mo><msubsup><mi>m</mi><mi>f</mi><mrow><mn>2</mn><mo>/</mo><mo>(</mo><mn>1</mn><mo>+</mo><msub><mi>γ</mi><mi>m</mi></msub><mo>)</mo></mrow></msubsup></mrow></semantics></math></inline-formula> instead of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>M</mi><mi>π</mi><mn>2</mn></msubsup><mo>∼</mo><msub><mi>m</mi><mi>f</mi></msub></mrow></semantics></math></inline-formula>, we have the same result as ours in the broken phase. A further comment related to dChPT is made on the decay width of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mn>0</mn></msub><mrow><mo>(</mo><mn>500</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>π</mi><mi>π</mi></mrow></semantics></math></inline-formula> for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>N</mi><mi>f</mi></msub><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula>. It is shown to be consistent with the reality, when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mn>0</mn></msub><mrow><mo>(</mo><mn>500</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is regarded as a pseudo-NG boson with the non-perturbative trace anomaly dominance.
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