Bootstrapping $O(N)$ Vector Models in $4<d<6$

Abstract

We use the conformal bootstrap to study conformal field theories with $O(N)$ global symmetry in $d=5$ and $d=5.95$ spacetime dimensions that have a scalar operator $φ_i$ transforming as an $O(N)$ vector. The crossing symmetry of the four-point function of this $O(N)$ vector operator, along with unitarity assumptions, determine constraints on the scaling dimensions of conformal primary operators in the $φ_i \times φ_j$ OPE. Imposing a lower bound on the second smallest scaling dimension of such an $O(N)$-singlet conformal primary, and varying the scaling dimension of the lowest one, we obtain an allowed region that exhibits a kink located very close to the interacting $O(N)$-symmetric CFT conjectured to exist recently by Fei, Giombi, and Klebanov. Under reasonable assumptions on the dimension of the second lowest $O(N)$ singlet in the $φ_i \times φ_j$ OPE, we observe that this kink disappears in $d =5$ for small enough $N$, suggesting that in this case an interacting $O(N)$ CFT may cease to exist for $N$ below a certain critical value.

Figures (5)

• Figure 1: Bounds on $\Delta_\sigma$ in terms of $\Delta_\phi$ in $d=5.95$ for $N=600,1000,1400$ under the assumption that $\sigma$ is the only scalar operator with dimension less than $\Delta_{\sigma^2}\geq3.986, 3.993, 3.996$ respectively. These bounds were computed with $J_\text{max}=20$ and $\Lambda=17$. The red dot denotes the large $N$ expansion $(\Delta_\phi,\Delta_\sigma)$ values for the critical $O(N)$ vector model for $N=1400$. The crosses denote the $\epsilon$ expansion $(\Delta_\phi,\Delta_\sigma)$ values for the CFT with three relevant operators that exists for all $N>0$.
• Figure 2: Bounds on $\Delta_\sigma$ in terms of $\Delta_\phi$ in $d=5.95$ for $N=1000$ under the assumption that $\sigma$ is the only scalar operator with dimension less than $\Delta_{\sigma^2}\geq3.993$. The black line was computed with $J_\text{max}=30$ and $\Lambda=21$, the brown line was computed with $J_\text{max}=25$ and $\Lambda=19$, and the orange line was computed with $J_\text{max}=20$ and $\Lambda=17$. Note that the lower kink corresponding to the interacting $O(N)$ CFT is well converged, but the second higher kink diminishes significantly as $\Lambda$ is increased.
• Figure 3: Bounds on $\Delta_\sigma$ in terms of $\Delta_\phi$ in $d=5$ for $N=500$ under the assumption that $\sigma$ is the only scalar operator with dimension less than $\Delta_{\sigma^2}\geq3.965$. The black line was computed with $J_\text{max}=25$ and $\Lambda=19$, while the orange line was computed with $J_\text{max}=30$ and $\Lambda=21$. The red dot denotes the large $N$ expansion $(\Delta_\phi,\Delta_\sigma)=(1.500414,2.022)$ for the critical $O(N)$ vector model. Note the extremely zoomed in scale of this plot.
• Figure 4: Bounds on $\Delta_\sigma$ in terms of $\Delta_\phi$ in $d=5$ for a range of $N$ under the assumption that $\sigma$ is the only scalar operator with dimension less than $\Delta_{\sigma^2}\geq3.8$. These bounds are computed with $J_\text{max}=25$ and $\Lambda=19$.
• Figure 5: Bounds on $\Delta_\sigma$ in terms of $\Delta_\phi$ in $d=5$ for $N=6$ (left) and $N=40$ (right) under the assumption that $\sigma$ is the only scalar operator with dimension less than $\Delta_{\sigma^2}$. The solid lines were computed with $J_\text{max}=25$ and $\Lambda=19$ for a variety of assumed lower bounds for $\Delta_{\sigma^2}$.