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More on analytic bootstrap for O(N) models

Parijat Dey, Apratim Kaviraj, Kallol Sen

TL;DR

This work extends analytic bootstrap methods to $O(N)$-symmetric CFTs by incorporating trace, antisymmetric-traceless, and symmetric-traceless exchanges in the OPE, and analyzes the resulting large-spin double-twist operators.It derives detailed expressions for the anomalous dimensions $ abla(n,bell)$ in terms of minimal-twist exchanges from stress-tensor, current, singlet, and symmetric-traceless scalars, including the $n$-dependence for all representations ($I,A,S$) and the special case $n=0$, where an exact match with the $oldsymbol{ε}$-expansion is found.The leading large-spin, large-twist behavior is shown to be governed by the universal scaling $ abla(n,bell) icommod{n^{2y}}{ ield}$ with $y$ determined by the exchanged operator’s twist, and the coefficients are expressed in closed form in terms of $N$, $Δ_phi$, and the minimal-twist data. A concrete holographic example for $N=2$ demonstrates consistency between bulk gravitational/gauge interactions in AdS$_5$ and the boundary CFT anomalous dimensions, supporting the validity of the analytic bootstrap results and providing a bridge to holography for $O(N)$ vector theories.

Abstract

This note is an extension of a recent work on the analytical bootstrapping of $O(N)$ models. An additonal feature of the $O(N)$ model is that the OPE contains trace and antisymmetric operators apart from the symmetric-traceless objects appearing in the OPE of the singlet sector. This in addition to the stress tensor $(T_{μν})$ and the $φ_iφ^i$ scalar, we also have other minimal twist operators as the spin-1 current $J_μ$ and the symmetric-traceless scalar in the case of $O(N)$. We determine the effect of these additional objects on the anomalous dimensions of the corresponding trace, symmetric-traceless and antisymmetric operators in the large spin sector of the $O(N)$ model, in the limit when the spin is much larger than the twist. As an observation, we also verified that the leading order results for the large spin sector from the $ε-$expansion are an exact match with our $n=0$ case. A plausible holographic setup for the special case when $N=2$ is also mentioned which mimics the calculation in the CFT.

More on analytic bootstrap for O(N) models

TL;DR

This work extends analytic bootstrap methods to $O(N)$-symmetric CFTs by incorporating trace, antisymmetric-traceless, and symmetric-traceless exchanges in the OPE, and analyzes the resulting large-spin double-twist operators.It derives detailed expressions for the anomalous dimensions $ abla(n,bell)$ in terms of minimal-twist exchanges from stress-tensor, current, singlet, and symmetric-traceless scalars, including the $n$-dependence for all representations ($I,A,S$) and the special case $n=0$, where an exact match with the $oldsymbol{ε}$-expansion is found.The leading large-spin, large-twist behavior is shown to be governed by the universal scaling $ abla(n,bell) icommod{n^{2y}}{ ield}$ with $y$ determined by the exchanged operator’s twist, and the coefficients are expressed in closed form in terms of $N$, $Δ_phi$, and the minimal-twist data. A concrete holographic example for $N=2$ demonstrates consistency between bulk gravitational/gauge interactions in AdS$_5$ and the boundary CFT anomalous dimensions, supporting the validity of the analytic bootstrap results and providing a bridge to holography for $O(N)$ vector theories.

Abstract

This note is an extension of a recent work on the analytical bootstrapping of models. An additonal feature of the model is that the OPE contains trace and antisymmetric operators apart from the symmetric-traceless objects appearing in the OPE of the singlet sector. This in addition to the stress tensor and the scalar, we also have other minimal twist operators as the spin-1 current and the symmetric-traceless scalar in the case of . We determine the effect of these additional objects on the anomalous dimensions of the corresponding trace, symmetric-traceless and antisymmetric operators in the large spin sector of the model, in the limit when the spin is much larger than the twist. As an observation, we also verified that the leading order results for the large spin sector from the expansion are an exact match with our case. A plausible holographic setup for the special case when is also mentioned which mimics the calculation in the CFT.

Paper Structure

This paper contains 14 sections, 89 equations, 1 figure.

Table of Contents

  1. Introduction and summary of results
  2. $O(N)$ Fundamentals
  3. Stress-tensor exchange
  4. Current exchange
  5. Singlet scalar exchange
  6. Symmetric tensor scalar exchange
  7. Pattern for Anomalous dimensions for $n \neq 0$
  8. Leading $n$ dependence of anomalous dimensions
  9. Integer y
  10. Half integer y
  11. $\epsilon$-expansion from Bootstrap
  12. Holographic calculation: Example O(2) model
  13. Discussion
  14. Acknowledgements

Figures (1)

  • Figure 1: The figure shows the variation of the $\log (-\gamma_n)$ with $\log n$ for the current and the stress tensor exchange for different $\Delta_\phi$. The normalizations $P_T$ and $P_J$ are for each of the current and stress tensor exchange.