{"title":"非酉CFT的稳定性分析","authors":"Masataka Watanabe","doi":"10.1007/jhep11(2023)042","DOIUrl":null,"url":null,"abstract":"A bstract We study instability of lowest dimension operator (i.e., the imaginary part of its operator dimension) in the rank- Q traceless symmetric representation of the O ( N ) Wilson-Fisher fixed point in D = 4 + ϵ . We find a new semi-classical bounce solution, which gives an imaginary part to the operator dimension of order $$ O\\left({\\epsilon}^{-1/2}\\exp \\left[-\\frac{N+8}{3\\epsilon }F\\left(\\epsilon Q\\right)\\right]\\right) $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mrow> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>exp</mml:mo> <mml:mfenced> <mml:mrow> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:mfrac> <mml:mi>F</mml:mi> <mml:mfenced> <mml:mi>ϵQ</mml:mi> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:math> in the double-scaling limit where $$ \\epsilon Q\\le \\frac{N+8}{6\\sqrt{3}} $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ϵQ</mml:mi> <mml:mo>≤</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>6</mml:mn> <mml:msqrt> <mml:mn>3</mml:mn> </mml:msqrt> </mml:mrow> </mml:mfrac> </mml:math> is fixed. The form of F ( ϵQ ), normalised as F (0) = 1, is also computed. This non-perturbative correction continues to give the leading effect even when Q is finite, indicating the instability of operators for any values of Q . We also observe a phase transition at $$ \\epsilon Q=\\frac{N+8}{6\\sqrt{3}} $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>ϵQ</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>6</mml:mn> <mml:msqrt> <mml:mn>3</mml:mn> </mml:msqrt> </mml:mrow> </mml:mfrac> </mml:math> associated with the condensation of bounces, similar to the Gross-Witten-Wadia transition.","PeriodicalId":48906,"journal":{"name":"Journal of High Energy Physics","volume":"72 1","pages":"0"},"PeriodicalIF":5.0000,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":"{\"title\":\"Stability analysis of a non-unitary CFT\",\"authors\":\"Masataka Watanabe\",\"doi\":\"10.1007/jhep11(2023)042\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A bstract We study instability of lowest dimension operator (i.e., the imaginary part of its operator dimension) in the rank- Q traceless symmetric representation of the O ( N ) Wilson-Fisher fixed point in D = 4 + ϵ . We find a new semi-classical bounce solution, which gives an imaginary part to the operator dimension of order $$ O\\\\left({\\\\epsilon}^{-1/2}\\\\exp \\\\left[-\\\\frac{N+8}{3\\\\epsilon }F\\\\left(\\\\epsilon Q\\\\right)\\\\right]\\\\right) $$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>O</mml:mi> <mml:mfenced> <mml:mrow> <mml:msup> <mml:mi>ϵ</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>exp</mml:mo> <mml:mfenced> <mml:mrow> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>3</mml:mn> <mml:mi>ϵ</mml:mi> </mml:mrow> </mml:mfrac> <mml:mi>F</mml:mi> <mml:mfenced> <mml:mi>ϵQ</mml:mi> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:mrow> </mml:mfenced> </mml:math> in the double-scaling limit where $$ \\\\epsilon Q\\\\le \\\\frac{N+8}{6\\\\sqrt{3}} $$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>ϵQ</mml:mi> <mml:mo>≤</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>6</mml:mn> <mml:msqrt> <mml:mn>3</mml:mn> </mml:msqrt> </mml:mrow> </mml:mfrac> </mml:math> is fixed. The form of F ( ϵQ ), normalised as F (0) = 1, is also computed. This non-perturbative correction continues to give the leading effect even when Q is finite, indicating the instability of operators for any values of Q . We also observe a phase transition at $$ \\\\epsilon Q=\\\\frac{N+8}{6\\\\sqrt{3}} $$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>ϵQ</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> <mml:mn>8</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>6</mml:mn> <mml:msqrt> <mml:mn>3</mml:mn> </mml:msqrt> </mml:mrow> </mml:mfrac> </mml:math> associated with the condensation of bounces, similar to the Gross-Witten-Wadia transition.\",\"PeriodicalId\":48906,\"journal\":{\"name\":\"Journal of High Energy Physics\",\"volume\":\"72 1\",\"pages\":\"0\"},\"PeriodicalIF\":5.0000,\"publicationDate\":\"2023-11-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of High Energy Physics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/jhep11(2023)042\",\"RegionNum\":1,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"PHYSICS, PARTICLES & FIELDS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of High Energy Physics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/jhep11(2023)042","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, PARTICLES & FIELDS","Score":null,"Total":0}
A bstract We study instability of lowest dimension operator (i.e., the imaginary part of its operator dimension) in the rank- Q traceless symmetric representation of the O ( N ) Wilson-Fisher fixed point in D = 4 + ϵ . We find a new semi-classical bounce solution, which gives an imaginary part to the operator dimension of order $$ O\left({\epsilon}^{-1/2}\exp \left[-\frac{N+8}{3\epsilon }F\left(\epsilon Q\right)\right]\right) $$ Oϵ−1/2exp−N+83ϵFϵQ in the double-scaling limit where $$ \epsilon Q\le \frac{N+8}{6\sqrt{3}} $$ ϵQ≤N+863 is fixed. The form of F ( ϵQ ), normalised as F (0) = 1, is also computed. This non-perturbative correction continues to give the leading effect even when Q is finite, indicating the instability of operators for any values of Q . We also observe a phase transition at $$ \epsilon Q=\frac{N+8}{6\sqrt{3}} $$ ϵQ=N+863 associated with the condensation of bounces, similar to the Gross-Witten-Wadia transition.
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