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On the large charge sector in the critical O(N) model at large N

Author(s): Giombi, Simone; Hyman, Jonah

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Abstract: <jats:title>A<jats:sc>bstract</jats:sc> </jats:title><jats:p>We study operators in the rank-<jats:italic>j</jats:italic> totally symmetric representation of <jats:italic>O</jats:italic>(<jats:italic>N</jats:italic>) in the critical <jats:italic>O</jats:italic>(<jats:italic>N</jats:italic>) model in arbitrary dimension <jats:italic>d</jats:italic>, in the limit of large <jats:italic>N</jats:italic> and large charge <jats:italic>j</jats:italic> with <jats:italic>j/N</jats:italic> ≡ <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \hat{j} $$</jats:tex-math><mml:math xmlns:mml=""> <mml:mover> <mml:mi>j</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math></jats:alternatives></jats:inline-formula> fixed. The scaling dimensions of the operators in this limit may be obtained by a semiclassical saddle point calculation. Using the standard Hubbard-Stratonovich description of the critical <jats:italic>O</jats:italic>(<jats:italic>N</jats:italic>) model at large <jats:italic>N</jats:italic>, we solve the relevant saddle point equation and determine the scaling dimensions as a function of <jats:italic>d</jats:italic> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \hat{j} $$</jats:tex-math><mml:math xmlns:mml=""> <mml:mover> <mml:mi>j</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math></jats:alternatives></jats:inline-formula>, finding agreement with all existing results in various limits. In 4 <jats:italic>< d <</jats:italic> 6, we observe that the scaling dimension of the large charge operators becomes complex above a critical value of the ratio <jats:italic>j/N</jats:italic>, signaling an instability of the theory in that range of <jats:italic>d</jats:italic>. Finally, we also derive results for the correlation functions involving two “heavy” and one or two “light” operators. In particular, we determine the form of the “heavy-heavy-light” OPE coefficients as a function of the charges and <jats:italic>d</jats:italic>.</jats:p>
Publication Date: 28-Sep-2021
Electronic Publication Date: 28-Sep-2021
Citation: Giombi, Simone, Hyman, Jonah. (2021). On the large charge sector in the critical O(N) model at large N. Journal of High Energy Physics, 2021 (9), 10.1007/jhep09(2021)184
DOI: doi:10.1007/jhep09(2021)184
EISSN: 1029-8479
Keywords: 1/N Expansion, Conformal Field Theory, Field Theories in Higher Dimen- sions, Global Symmetries
Language: en
Type of Material: Journal Article
Journal/Proceeding Title: Journal of High Energy Physics
Version: Final published version. This is an open access article.

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