Bounds on 4D Conformal and Superconformal Field Theories

TL;DR

<p>We develop a nonperturbative conformal bootstrap framework in four dimensions to bound operator dimensions, central charges, and OPE coefficients in general 4D CFTs and in 4D ${\cal N}=1$ superconformal theories. By combining crossing symmetry with a linear-programming approach and, for SUSY, explicit ${\cal N}=1$ superconformal blocks, we derive general bounds on the dimension of $\Phi^{\dagger}\Phi$, flavor-current data $\tau^{IJ}$, and the central charge $c$, with stronger constraints in the supersymmetric case. We explicitly construct the ${\cal N}=1$ superconformal blocks and show how SUSY ties OPE data to anomaly-like quantities via Ward identities, enabling checks against known theories such as SQCD in the conformal window. The results provide universal, nonperturbative constraints on four-dimensional conformal dynamics and have potential implications for AdS/CFT and model-building in beyond-Standard-Model physics.>

Abstract

We derive general bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and N=1 superconformal field theories. In any CFT containing a scalar primary phi of dimension d we show that crossing symmetry of <phi phi phi phi> implies a completely general lower bound on the central charge c >= f_c(d). Similarly, in CFTs containing a complex scalar charged under global symmetries, we bound a combination of symmetry current two-point function coefficients tau^{IJ} and flavor charges. We extend these bounds to N=1 superconformal theories by deriving the superconformal block expansions for four-point functions of a chiral superfield Phi and its conjugate. In this case we derive bounds on the OPE coefficients of scalar operators appearing in the Phi x Phi* OPE, and show that there is an upper bound on the dimension of Phi* Phi when dim(Phi) is close to 1. We also present even more stringent bounds on c and tau^{IJ}. In supersymmetric gauge theories believed to flow to superconformal fixed points one can use anomaly matching to explicitly check whether these bounds are satisfied.

Figures (9)

• Figure 1: The "search space" $\mathcal{S}\subset \mathcal{V}^*$ is the intersection of the hyperplane $\alpha(F_{\Delta_0,l_0})=1$ with the convex cone of linear functionals $\alpha$ satisfying $\alpha(F_{\Delta,l})\geq 0$ for all $(\Delta,l)$ in the spectrum.
• Figure 2: Picking a metric on $\mathcal{V}^*$, we arrive at the following picture. The linear functional that minimizes $\alpha(1)$ is the unique point $\alpha_*$ on the boundary of $\mathcal{S}$ where $-1$ is in the positive span of the "normal vectors" $-F_{\Delta_i,l_i}$ to $\mathcal{S}$ at $\alpha_*$. Here, the three parallel arrows illustrate the direction of steepest descent of $\alpha(1)$. We have suppressed an infinite number of dimensions (including the $F_{\Delta_0,l_0}$ direction) in order to draw this figure in the plane.
• Figure 3: A plot of $\alpha_*(F_{\Delta,l})$ for various $\Delta,l$, where $\alpha_*\in\mathcal{S}$ gives the strongest bound on the OPE coefficient of the lowest-dimension scalar $\mathcal{O}_0\in\phi\times\phi$. Here, we have taken $\dim\phi=1.1$, $\Delta_0=\dim\mathcal{O}_0=1.15$, and $k=4$. We show only pairs $\Delta,l$ satisfying unitarity. Note that $\alpha_*(F_{\Delta,l})$ is never negative in this range, consistent with the constraints of our linear program, although it has zeros $(\Delta,l)\in\{(2.2,0),(8.2,0),(4,2),(7.7,2),(6,4)\}$. Note also that $\alpha_*(F_{\Delta_0,0})=1$, as required.
• Figure 4: An upper bound on the dimension of $\Phi^\dag\Phi$ (the lowest-dimension scalar appearing in the $\Phi\times\Phi^\dag$ OPE), as a function of $d=\dim \Phi$. Here, we have taken $k=6$. The bound appears to approach $2$ as $d\rightarrow 1$, as expected. On the other hand, we do not find a dimension bound for $d\gtrsim 1.16$. It is possible that pushing the numerics beyond $k=6$ could lead to bounds in this region.
• Figure 5: An upper bound on the OPE coefficient $|\lambda_{\mathcal{O}_0}|$ of a scalar primary $\mathcal{O}_0$ appearing in $\Phi\times\Phi^\dag$, as a function of $\Delta_0=\dim\mathcal{O}_0$, for $d=\dim\Phi=1.05, 1.1, 1.25, 1.5$ (with $k=5$). Here we no longer assume $\mathcal{O}_0$ is the lowest-dimension scalar in $\Phi\times\Phi^\dag$.