Bootstrapping Mixed Correlators in the Five Dimensional Critical O(N) Models

We use the conformal bootstrap approach to explore $5D$ CFTs with $O(N)$ global symmetry, which contain $N$ scalars ${\varphi }_i$ transforming as $O(N)$ vector. Specifically, we study multiple four-point correlators of the leading $O(N)$ vector ${\varphi }_i$ and the $O(N)$ singlet $\sigma$. The crossing symmetry of the four-point functions and the unitarity condition provide nontrivial constraints on the scaling dimensions (${\mathrm{\Delta }}_{\varphi }$, ${\mathrm{\Delta }}_{\sigma }$) of ${\varphi }_i$ and $\sigma$. With reasonable assumptions on the gaps between scaling dimensions of ${\varphi }_i$ ($\sigma$) and the next $O(N)$ vector (singlet) scalar, we are able to isolate the scaling dimensions $({\mathrm{\Delta }}_{\varphi }$, ${\mathrm{\Delta }}_{\sigma })$ in small islands. In particular, for large $N=500$, the isolated region is highly consistent with the result obtained from large $N$ expansion. We also study the interacting $O(N)$ CFTs for $1⩽N⩽100$. Isolated regions on $({\mathrm{\Delta }}_{\varphi },{\mathrm{\Delta }}_{\sigma })$ plane are obtained using conformal bootstrap program with lower order of derivatives $\mathrm{\Lambda }$; however, they disappear after increasing $\mathrm{\Lambda }$. We think these islands are corresponding to interacting but nonunitary $O(N)$ CFTs. Our results provide a lower bound on the critical value $N_c>100$, below which the interacting $O(N)$ CFTs turn into nonunitary. The critical value is unexpectedly large comparing with previous estimations.