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

TL;DR

This work uses a mixed-correlator conformal bootstrap approach to study five-dimensional $O(N)$ vector CFTs, focusing on the leading vector $\phi_i$ and singlet $\sigma$. By combining crossing symmetry with unitarity and imposing dimension gaps, the authors carve out islands in the $(Δ_φ,Δ_σ)$ plane, finding a small island for $N=500$ that matches large-$N$ predictions and shrinks with increasing derivation order $Λ$. For $N\le100$, islands appear only at low $Λ$ and vanish at higher $Λ$, implying a critical value $N_c>100$ and suggesting the interacting fixed point becomes nonunitary below $N_c$. Overall, the results provide nonperturbative, high-precision constraints on 5D $O(N)$ CFT data and support a large-$N$ unitary fixed point while highlighting potential nonunitarity at smaller $N$.

Abstract

We use the conformal bootstrap approach to explore $5D$ CFTs with $O(N)$ global symmetry, which contain $N$ scalars $φ_i$ transforming as $O(N)$ vector. Specifically, we study multiple four-point correlators of the leading $O(N)$ vector $φ_i$ and the $O(N)$ singlet $σ$. The crossing symmetry of the four-point functions and the unitarity condition provide nontrivial constraints on the scaling dimensions ($Δ_φ$, $Δ_σ$) of $φ_i$ and $σ$. With reasonable assumptions on the gaps between scaling dimensions of $φ_i$ ($σ$) and the next $O(N)$ vector (singlet) scalar, we are able to isolate the scaling dimensions $(Δ_φ$, $Δ_σ)$ 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 $1\leqslant N\leqslant100$. Isolated regions on $(Δ_φ,Δ_σ)$ plane are obtained using conformal bootstrap program with lower order of derivatives $Λ$; however, they disappear after increasing $Λ$. 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.

Figures (4)

• Figure 1: Bounds on the conformal dimensions $(\Delta_\phi,\Delta_\sigma)$ in the interacting $5D$$O(500)$ CFT. The colored regions represent the conformal dimensions allowed by conformal bootstrap. Specifically the light blue region is obtained from single correlator bootstrap, while the dark blue island is isolated through bootstrapping the multiple correlators. We used the derivative at order $\Lambda=19$ and spins $S_{\Lambda=19}$ in the numerical calculations. Besides, we assumed a gap $\Delta_{S,0}^*=3.965$ in the S-channel. An extra gap $\Delta_{V,0}^*=5$ has been used in the V-channel for bootstrapping multiple correlators. The black dot and cross relate to the predictions from $\epsilon$ expansion and large $N$ expansion, respectively.
• Figure 2: Isolated regions for the conformal dimensions $(\Delta_\phi,\Delta_\sigma)$ in $5D$$O(500)$ vector model. The light, medium and dark blue regions are corresponding to the results from multiple correlator conformal bootstrap with $\Lambda=21, ~23, ~25$, respectively. In the graph we have used the dimension gaps $\Delta_{S,0}^*=3.965$ and $\Delta_{V,0}^*=5$. The black cross denotes the prediction from large $N$ expansion.
• Figure 3: From top to bottom, the islands represent the allowed regions of $(\Delta_\phi,\Delta_\sigma)$ in the $5D$$O(N)$$N=40,60,70$ vector models. The results are obtained from conformal bootstrap with $\Lambda=19$ and spins $S_{\Lambda=19}$. The black dots and crosses denote predictions from $\epsilon$ expansion and large $N$ expansions, respectively. The dimension gaps used in conformal bootstrap program are: $(\Delta_{S,0}^*,\Delta_{V,0}^*)=(3.4, 4.1)$ for $N=40$, $(\Delta_{S,0}^*,\Delta_{V,0}^*)=(3.5, 4.3)$ for $N=60, 70$. The perturbative methods, especially the large $N$ expansion get abnormal and stay away from the region allowed by conformal bootstrap at $N=40$.
• Figure 4: Bounds on the conformal dimensions $(\Delta_\phi,\Delta_\sigma)$ in $5D$$O(100)$ vector model. The light blue region is obtained from single correlator bootstrap. The multiple correlators bootstrap leads to a small island colored in dark blue. In the bootstrap program we adopt the setup with $\Lambda=19$ and the correspond spins provided in (\ref{['spins']}). We apply a dimension gap $\Delta_{S,0}^*=3.6$ in the S-channel. Besides, an extra dimension gap $\Delta_{V,0}^*=5$ has been used in the V-channel for bootstrapping multiple correlators. The black dot and cross relate to the predictions from $\epsilon$ expansion and large $N$ expansion, respectively.