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Estimation of the Lin-Yang bound of the least static energy of the Faddeev model

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Minoru Hirayama, Hitoshi Yamakoshi, Jun Yamashita

High Energy Physics - Theory · Submitted 20 Feb 2006 · 11 references

Pith's reading: A numerical trial configuration gives the upper bound E_Q ≤ 6233 sqrt(c2 c4) Q^{3/4} for the least static energy of the Faddeev model with Hopf index Q.

T0 · AI review

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arXiv IDhep-th/0602192 (version 3)
Pith IDYSRIN2QI, Pith's permanent record of this paper and its review
CitationHitoshi Yamakoshi, Jun Yamashita, Minoru Hirayama. "Estimation of the Lin-Yang bound of the least static energy of the Faddeev model." arXiv:hep-th/0602192 (2006). https://pith.science/paper/YSRIN2QI
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Claim paper
The paper

The paper claims

A numerical trial configuration gives the upper bound E_Q ≤ 6233 sqrt(c2 c4) Q^{3/4} for the least static energy of the Faddeev model with Hopf index Q.

The reading

The Faddeev model describes a three-component unit vector field that forms knotted solitons. Each knot carries an integer called the Hopf index Q. A known theorem by Lin and Yang guarantees that the minimum energy E_Q in each Hopf sector is bounded above by a constant times Q^{3/4}, but it does not give the constant. This paper tries to estimate that constant by building an explicit trial field configuration.

The construction combines three maps: a stereographic projection from a ball, the Hopf map from the three-sphere to the two-sphere, and an additional map of chosen degree. This produces one large 'knot' with Hopf index related to the square of an integer, plus several simple unit Hopf balls placed far away. The total Hopf index is Q = n^2 + m, where n is the degree of the extra map and m is the number of added unit knots. The energy of the configuration is computed as a sum of contributions from each ball, and after optimizing the scale of each ball, the total energy takes the form of a constant times Q^{3/4}.

Several integrals must be evaluated numerically to fix the constant. The authors report the values f = 14.9, g(k) < 19.3 and h(k) < 268.4, which lead to the final bound E_Q ≤ 6233 sqrt(c2 c4) Q^{3/4}. This is a smaller coefficient than the analytic bound from a recent paper if the quoted comparison value of 7.667 x 10^5 is correct, but that expression appears to be misprinted. No code or error bars are provided for the numerical integrals.

Extended reading notes

Core claim

The paper states in Section 4: 'we have obtained the bound EQ ≦ 6233 sqrt(c2 c4) Q^{3/4}' (Eq. (3.48)). If this is correct, the Lin-Yang constant C satisfies C ≤ 6233, improving on the analytic bound quoted from Adam et al.

Load-bearing premise

The final bound depends on the numerically evaluated integrals f = 14.9, g(k) < 19.3 and h(k) < 268.4 (Eqs. (3.38)-(3.40)), which are stated without derivation, error bars, or reproducibility artifacts. A change in these values would directly change the constant 6233. The paper also invokes the inequality ak^3+bm ≤ (a+4b)(k^4+m)^{3/4} (Eq. (3.47)) without proof.

Original abstract

Lin and Yang's upper bound E_Q <= cQ^(3/4) of the least static energy E_Q of the Faddeev model in a sector with a fixed Hopf index Q is investigated. By constructing an explicit trial configuration for the Faddeev field n, a possible value of the coefficient c is obtained numerically, which is much smaller than the value obtained quite recently by analytic discussions.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,342 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.57 · compares with prior results 0.96 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

Lin and Yang's upper bound E_Q <= cQ^(3/4) of the least static energy E_Q of the Faddeev model in a sector with a fixed Hopf index Q is investigated. By constructing an explicit trial configuration for the Faddeev field n, a possible value of the coefficient c is obtained numerically, which is much smaller than the value obtained quite recently by analytic discussions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Lin and Yang's upper bound E_Q <= cQ^(3/4) of the least static energy E_Q of the Faddeev model in a sector with a fixed Hopf index Q is investigated. background 0.78 0.24 0.12 asserted 0.84
s1 By constructing an explicit trial configuration for the Faddeev field n, a possible value of the coefficient c is obtained numerically, which is much smaller than the value obtained quite recently by analytic discussions. result 0.73 0.92 0.71 hedged possibility 0.82

What kind of paper the classifier reports this is: simulation study 0.65 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read
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  • references and citing sentences
  • relations to other papers
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  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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