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

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="http://www.w3.org/1998/Math/MathML"> <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="http://www.w3.org/1998/Math/MathML"> <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>