Comment on CFT in AdS and boundary RG flows: O(1/N) Result

TL;DR

This work resolves the discrepancy between Giombi and Khanchandani's $O(1/N)$ series for the boundary anomalous dimension and the authors' simple analytic result in the ordinary transition by proving analytic equivalence and correcting the two ${}_3F_2$ representations. It extends the analysis to all surface transitions, deriving explicit $O(1/N)$ corrections to boundary exponents, including $oxed{ \\hat{\\gamma}^O = rac{(4-d)\\Gamma(2d-3)}{d\\Gamma(d-2)\\Gamma(d-1)} }$, $\\hat{\\gamma}^S$ for the special transition, and showing $\\hat{\\gamma}^T = 0$ with $\\\\Delta_{\\phi}^T = d-1$, while confirming consistency with known epsilon expansions. The authors develop a robust large-$N$ framework based on the inverse of a single bubble in mixed space and solve the resulting differential equations to obtain the anomalous dimensions for all transitions, including the extraordinary one. This provides a cohesive, cross-validated picture of boundary critical behavior in AdS/CFT-like settings and lays groundwork for higher-order $1/N$ corrections and related boundary phenomena.

Abstract

In a recent paper [JHEP 11 (2020) 118], S. Giombi and H. Khanchandani studied the 1/N expansion of the O(N) model in semi-infinite space within the framework of conformal field theory in anti-de Sitter space. They presented a series expansion for the O(1/N) correction to the boundary anomalous dimension in the case of the ordinary transition. Although they were unable to sum the series or simplify its form analytically, they demonstrated numerically that their result matches our earlier, simple analytic expression given in Prog. Theor. Phys. 70 (1983) 1226. In this paper, we show that their series expansion is in fact exactly equivalent to our original expression. However, since the final formula in eq. (4.57) of their paper, which is expressed in terms of two different 3F2 functions, cannot produce the correct values, we derive the correct formulae involving two 3F2 functions in the Appendices. We comment on the similarity and difference between their analysis and ours in the cases of the ordinary and special transitions, and present full details of deriving the anomalous dimension for all the cases including the extraordinary transition, which were not written in our earlier paper.

Figures (6)

• Figure 1: Phase diagram of the $O(N)$ model in semi-infinite space. Depending on the interaction strength at the surface, the phase boundary at the bulk critical temperature $T_c$ is classified into the ordinary, special, and extraordinary transitions, each belonging to distinct universality class.
• Figure 2: Plots of $\hat{\gamma}^O$ versus the space dimension $d$ in the case of the ordinary transition. Black solid curve is eq. (\ref{['Eq_O']}) of the $1/N$ expansion PLA1PTP1, red dashed curve near $d=4$ is eq. (\ref{['4-d']}) of the $\epsilon=4-d$ expansion ReeveGuttmannDiehlDietrich, and blue dashed curve near $d=2$ is eq. (\ref{['d-2']}) of the $\epsilon=d-2$ expansion DiehlNusser.
• Figure 3: A single bubble (a) and the relation to its inverse (i.e., the bubble summation represented by a double wavy line) (b), which are relevant in the large $N$ expansion. Here, a double solid line represents the correlation function in the large $N$ limit.
• Figure 4: Two self-energy diagrams contributing at order $1/N$.
• Figure 5: Diagrammatic representation of the correlation function contributing at order $1/N$.