Exploring the Minimal 4D $\mathcal{N}=1$ SCFT

TL;DR

This work applies the conformal bootstrap to 4D $\mathcal{N}=1$ SCFTs with a chiral operator $\phi$ obeying $\phi^2=0$, aiming to uncover a minimal interacting fixed point. By analyzing the four-point function $\langle\bar{\phi} \phi \bar{\phi} \phi\rangle$, the authors demonstrate the necessity of a $\text{U}(1)_R$ current multiplet when the chiral ring constraint is saturated and extract central-charge bounds that converge to $c/c_{\text{free}} \approx 8/3$, with $\Delta_{\phi}$ approaching $1.415$ as the numerical cutoff $\Lambda$ grows. They identify a kink at $\Delta_{\phi} \approx 1.407$ when $\phi^2$ is excluded from the $\phi\times\phi$ OPE, provide preliminary estimates for higher-dimension operators (e.g., $\Delta_V \lesssim 3$, $\Delta_{V'} \lesssim 4.25$, $\Delta_{R'} \lesssim 7.2$), and discuss the potential realization as a non-Lagrangian 4D $\mathcal{N}=1$ SCFT with a remarkably small central charge. The study lays groundwork for future mixed-correlator analyses that could isolate an island and illuminate the theory’s UV completion, possibly connecting to Argyres–Douglas-type fixed points or other non-Lagrangian constructions.

Abstract

We study the conformal bootstrap constraints for 4D $\mathcal{N}=1$ superconformal field theories containing a chiral operator $φ$ and the chiral ring relation $φ^2=0$. Hints for a minimal interacting SCFT in this class have appeared in previous numerical bootstrap studies. We perform a detailed study of the properties of this conjectured theory, establishing that the corresponding solution to the bootstrap constraints contains a $\text{U}(1)_R$ current multiplet and estimating the central charge and low-lying operator spectrum of this theory.

Figures (11)

• Figure 1: Upper bound on the allowed dimension of the operator $\bar{\phi}\phi$ (the leading relevant nonchiral scalar singlet) as a function of the dimension of $\phi$. The generalized free theory dashed line $\Delta_{\bar{\phi}\phi}=2\space\Delta_{\phi}$ is also shown. The shaded area is excluded. If we assume that $\phi^2$ is not in the spectrum then everything to the left of the dotted line at $\Delta_\phi=1.407$, which is the position of the kink, is excluded. Here we use $\Lambda=21$.
• Figure 2: Lower and upper bounds on the OPE coefficient of the operator $\phi^2$ in the $\phi\times\phi$ OPE. The vertical dotted line is at $\Delta_\phi=1.407$ and the horizontal dashed line is at the free theory value $\lambda_{\phi^2}=\sqrt{2}$. The shaded area is excluded. Here we use $\Lambda=21$.
• Figure 3: Upper bound on the OPE coefficient of an operator $\bar{\phi}\phi$ with dimension $\Delta_{\bar{\phi}\phi}^{(\text{bound})}$ as a function of the dimension of $\phi$. Here we do not assume that $\bar{\phi}\phi$ is the scalar with the lowest dimension in the OPE $\bar{\phi}\times\phi$. The shaded area is excluded. In this plot we use $\Lambda=21$.
• Figure 4: Upper bound on the dimension of the leading superconformal primary vector operator in the OPE $\bar{\phi}\times\phi$ as a function of the dimension of $\phi$. The shaded area is excluded. Everything to the left of the vertical dotted line at $\Delta_{\phi}=1.407$ is excluded due to the assumption that there is no $\phi^2$ operator. The generalized free theory dashed line $\Delta_V=2\space\Delta_{\phi}+1$ as well as its intersection with the bound are also shown. In this plot we use $\Lambda=21$.
• Figure 5: Upper bound on the dimension of the second superconformal primary vector operator in the OPE $\bar{\phi}\times\phi$ as a function of the dimension of $\phi$, assuming that the first vector has dimension 3. the shaded area is excluded. everything to the left of the vertical dotted line at $\Delta_{\phi}=1.407$ is excluded due to the assumption that there is no $\phi^2$ operator. In this plot we use $\Lambda=21$.