Bootstrapping Mixed Correlators in 4D $\mathcal{N}=1$ SCFTs

TL;DR

This work executes the first numerical bootstrap study of mixed correlators in four-dimensional $\mathcal{N}=1$ SCFTs, focusing on systems involving a chiral scalar $\phi$ and a real scalar $R$ (including the linear-multiplet case $J$). By constructing novel superconformal blocks for mixed channels and assembling seven crossing relations into a semidefinite-programming problem, the authors derive tighter bounds on operator dimensions and OPE coefficients, and sharpen constraints on the central charge $c$. Their analysis reveals features consistent with a special minimal SCFT near $\Delta_{\phi}\approx 1.4$ (and $c_{\text{minimal}}=\tfrac{1}{9}$ in some extrapolations) while failing to isolate an island at current precision, highlighting both a tantalizing signal and the need for greater computational power and methodological advances. Overall, the results substantiate a structured minimal-like region in the 4D $\mathcal{N}=1$ bootstrap landscape and chart concrete steps toward a more decisive identification of the putative theory.

Abstract

The numerical conformal bootstrap is used to study mixed correlators in $\mathcal{N}=1$ superconformal field theories (SCFTs) in $d=4$ spacetime dimensions. Systems of four-point functions involving scalar chiral and real operators are analyzed, including the case where the scalar real operator is the zero component of a global conserved current multiplet. New results on superconformal blocks as well as universal constraints on the space of 4D $\mathcal{N}=1$ SCFTs with chiral operators are presented. At the level of precision used, the conditions under which the putative "minimal" 4D $\mathcal{N}=1$ SCFT may be isolated into a disconnected allowed region remain elusive. Nevertheless, new features of the bounds are found that provide further evidence for the presence of a special solution to crossing symmetry corresponding to the "minimal" 4D $\mathcal{N}=1$ SCFT.

Figures (12)

• Figure 1: Upper bound on the dimension of the operator $R$ as a function of $\Delta_\phi$ using only \ref{['crossRelphi']}. The generalized free theory dashed line $\Delta_R=2\space\Delta_{\phi}$ is also shown. The shaded area is excluded. In this plot we use $\Lambda=21$ for the thin and $\Lambda=29$ for the thick line.
• Figure 2: Lower bound on the central charge as a function of $\Delta_\phi$. The shaded area is excluded. In this plot we use $\Lambda=25$.
• Figure 3: The thick line is the lower bound on the central charge as a function of $\Delta_\phi$, assuming that $\Delta_R$ lies on the bound of Fig. \ref{['fig:dim_phibphi']}. The thin line is the bound of Fig. \ref{['fig:cc']}. The shaded area is excluded. In this plot we use $\Lambda=25$.
• Figure 4: Upper and lower bounds on the OPE coefficient of the operator $\bar{\phi}$ in the $\bar{\phi}\times R$ OPE as a function of $\Delta_\phi$, assuming $\Delta_R$ lies on the bound of Fig. \ref{['fig:dim_phibphi']} and demanding $c_{\bar{\phi}R\phi}=c_{\bar{\phi}\phi R}$. We also impose a gap equal to one between $\Delta_R$ and $\Delta_{R^\prime}$. The shaded area is excluded. In this plot we use $\Lambda=17$.
• Figure 5: Lower bound on the central charge as a function of $\Delta_\phi$, assuming that $\Delta_R$ lies on the bound of Fig. \ref{['fig:dim_phibphi']} and demanding $c_{{\bar{\phi}} R\phi}=c_{{\bar{\phi}}\phi R}$. We also impose a gap equal to one between $\Delta_R$ and $\Delta_{R^\prime}$. The shaded area is excluded. In this plot we use $\Lambda=17$.