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13 pages, 295 KiB

Open AccessArticle

by Koichi Yamawaki

Symmetry 2024, 16(1), 2; https://doi.org/10.3390/sym16010002 - 19 Dec 2023

Cited by 2 | Viewed by 1466

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 [...] Read more.

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}) \cdot \sum_{i = 1}^{N_{f}} m_{f} {\bar{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 \cdot \sum_{i = 1}^{N_{f}} m_{f} {\bar{ψ}}_{i} ψ_{i} | π (p_{1})⟩ = 2 M_{π}^{2}

through the Feynman–Hellmann theorem combined with an assumption

M_{π}^{2} \sim 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 \cdot \sum_{i = 1}^{N_{f}} m_{f} {\bar{ψ}}_{i} ψ_{i} | π (p_{1})⟩ = 2 M_{π}^{2} + [(1 - γ_{m}) M_{π}^{2} \cdot 2 / (1 + γ_{m})] = 2 M_{π}^{2} \cdot 2 / (1 + γ_{m}) \neq 2 M_{π}^{2}

in contradiction with his estimate, which is compared with

⟨π (p_{2}) | (1 + γ_{m}) \cdot \sum_{i = 1}^{N_{f}} m_{f} {\bar{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} \sim m_{f}^{2 / (1 + γ_{m})}

instead of

M_{π}^{2} \sim 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)

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. Full article

(This article belongs to the Special Issue Symmetries and Ultra Dense Matter of Compact Stars II: Emergence of Symmetries in Strong Nuclear Correlations)

19 pages, 909 KiB

Open AccessArticle

Genuine Dilatons in Gauge Theories

by R. J. Crewther

Universe 2020, 6(7), 96; https://doi.org/10.3390/universe6070096 - 10 Jul 2020

Cited by 33 | Viewed by 2987

Abstract

A genuine dilaton

σ

allows scales to exist even in the limit of exact conformal invariance. In gauge theories, these may occur at an infrared fixed point (IRFP)

α_{IR}

through dimensional transmutation. These large scales at

α_{IR}

can be separated from [...] Read more.

A genuine dilaton

σ

allows scales to exist even in the limit of exact conformal invariance. In gauge theories, these may occur at an infrared fixed point (IRFP)

α_{IR}

through dimensional transmutation. These large scales at

α_{IR}

can be separated from small scales produced by

θ_{μ}^{μ}

, the trace of the energy-momentum tensor. For quantum chromodynamics (QCD), the conformal limit can be combined with chiral

S U (3) \times S U (3)

symmetry to produce chiral-scale perturbation theory

χ

_{σ}

, with

f_{0} (500)

as the dilaton. The technicolor (TC) analogue of this is crawling TC: at low energies, the gauge coupling

α

goes directly to (but does not walk past)

α_{IR}

, and the massless dilaton at

α_{IR}

corresponds to a light Higgs boson at

α ≲ α_{IR}

. It is suggested that the

W^{\pm}

and

Z^{0}

bosons set the scale of the Higgs boson mass. Unlike crawling TC, in walking TC,

θ_{μ}^{μ}

produces all scales, large and small, so it is hard to argue that its “dilatonic” candidate for the Higgs boson is not heavy. Full article

(This article belongs to the Special Issue Spontaneous Breaking of Conformal Symmetry)

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