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

TL;DR

This paper employs the conformal bootstrap to explore $O(N)$ symmetric conformal field theories in dimensions between 4 and 6, identifying a critical point where the interacting theory may vanish for small N.

Contribution

It extends the conformal bootstrap approach to higher dimensions ($d=5$ and $d=5.95$) to constrain and analyze the existence of $O(N)$ CFTs and their phase structure.

Findings

Identifies a kink near the conjectured $O(N)$ CFT in $d=5$ and $d=5.95$.

Suggests the $O(N)$ CFT may not exist below a critical N in $d=5$.

Provides bounds on operator dimensions in these theories.

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 $\phi_i$ 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 $\phi_i \times \phi_j$ 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 $\phi_i \times \phi_j$ 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.