Super Sum rules for Long-Range Models

Abstract

We study sum rules that control the Regge limit of one-dimensional conformal field theory (CFT) correlators and relate them to dual bulk scattering processes at high energies in $\mathrm{AdS}_2$. By imposing the condition that no scattering takes place in the bulk, these sum rules single out special solutions to crossing symmetry that describe long-range models, which can be understood as free fields in AdS with boundary interactions tuned to criticality. We test these sum rules perturbatively in several distinct theories, namely the 1d long-range versions of the Ising, $O(N)$ and Lee--Yang models, and find that they correctly predict the CFT data characterising these theories. Along the way we compute for the first time the leading contributions of quadruple-twist operators to the long range Ising correlator and analyse their role in the new sum rules. Finally, we explore the consequences of imposing these sum rules in a numerical bootstrap framework and find that they lead to substantial reductions in the allowed parameter space.

Figures (8)

• Figure 1: The functional $\tilde{\beta}_0(\Delta)$ is shown for $\Delta_{\phi}=1/5$ (top left), $\Delta_{\phi}=1/4$ (top right), $\Delta_{\phi}=3/10$ (bottom left) and $\Delta_{\phi}=3/4$ (bottom right). For clarity, the functionals have been rescaled. Note the presence of a negative region just above $\Delta=1$.
• Figure 2: Bound on the gap above the identity in the $\phi \phi$ OPE, when imposing the $\tilde{\beta}_0$ sum rule. For ${\Delta_\phi}\geq 1/2$, the bound is the generalized free boson $\Delta_0=2{\Delta_\phi}$. The bound becomes non trivial for $0\leq {\Delta_\phi}<1/2$. The red dashed curve is the maximum gap without imposing the $\beta_0$ sum rule, corresponding to the generalized free fermion at $\Delta_0=2{\Delta_\phi}+1$.
• Figure 3: Bounds on near marginal operators and comparison with Long range Ising perturbative data. The gray region is a bound on the dimension of an operator above 1, allowing operators with $\frac{1}{2}<\Delta<1$. When the bound is saturated one finds an operator in the extremal solution whose dimension is shown as the black dot dashed curve. For comparison, we show the dimensions of the operators $\phi^2$ and $\phi^4$ for the long range Ising model obtained from Padé resummations of available perturbative data, plotted as the bottom and top red dashed curves respectively. The black vertical line divides the plot into two regions according to which the resummed $\Delta_{\phi^2}$ is larger or smaller than $1/2$; in the former our bound does not apply to LRI anyway.
• Figure 4: Tree-level and one-loop diagrams for the two-point correlator.
• Figure 5: One-loop diagrams for the three-point vertex function, whose divergent part will be canceled by the counterterm.