The Effective Bootstrap

TL;DR

The paper introduces an effective multipoint conformal bootstrap by sampling at multiple points in the cross-ratio plane ($z$,$\bar{z}$) and truncating the OPE at $\Delta_*$, with a rigorous remainder bound $\mathcal{R}(z,\bar{z},\Delta_*,\Delta_\phi,\gamma)$ (using $\gamma=1$ in $d=3,4$). It recasts the truncated crossing equations as a linear program over discretized operator spectra and evaluated points, enabling fast numerical bounds on operator dimensions and central charges, tested on the 3D Ising and $O(n)$ vector models as well as generic 4D CFTs. The method yields new bootstrap-derived spectra, including $S'$ and $T'$ in $O(n)$ with $n=2,3,4$, and reproduces known kink structures, while highlighting different UV sensitivities between scalar dimensions and the central charge. Overall, the work demonstrates that the multipoint, $\Delta_*$-bounded bootstrap is a practical, rapidly converging alternative to derivative-based approaches, with clear paths toward higher precision and extensions to mixed correlators.

Abstract

We study the numerical bounds obtained using a conformal-bootstrap method - advocated in ref. [1] but never implemented so far - where different points in the plane of conformal cross ratios $z$ and $\bar z$ are sampled. In contrast to the most used method based on derivatives evaluated at the symmetric point $z=\bar z =1/2$, we can consistently "integrate out" higher-dimensional operators and get a reduced simpler, and faster to solve, set of bootstrap equations. We test this "effective" bootstrap by studying the 3D Ising and $O(n)$ vector models and bounds on generic 4D CFTs, for which extensive results are already available in the literature. We also determine the scaling dimensions of certain scalar operators in the $O(n)$ vector models, with $n=2,3,4$, which have not yet been computed using bootstrap techniques.

Figures (9)

• Figure 1: $\eta$ defined in eq. (\ref{['etas']}) as a function of $\Delta_*$ in a generalized free theory in $d=4$ dimensions evaluated at the symmetric point $z=\bar{z} = 1/2$. We have taken $\Delta_\phi=1.5$ and $\gamma=1$.
• Figure 2: The region in the $z$-plane with $\lambda(z)\leq 0.6$ and a sample of 100 points in a fundamental domain of that region. The crossed lines are the two branch-cuts where the bootstrap equations do not converge.
• Figure 3: Bounds on $\Delta_\epsilon$ as a function of $\Delta_\sigma$ for $N=100$ points and different values of $\Delta_*$. The regions above the lines are excluded. The black cross marks the precise values of $\Delta_\sigma$ and $\Delta_\epsilon$ for the 3D Ising model as determined in ref.El-Showk:2014dwa. The curves and the labels in the legend have the same order from top to bottom.
• Figure 5: Bounds on $\Delta_\epsilon$ as a function of $\Delta_\sigma$ for fixed $\Delta_*=16$ and different values of $N$. The regions above the lines are excluded. The black cross marks the precise values of $\Delta_\sigma$ and $\Delta_\epsilon$ for the 3D Ising model as determined in ref.El-Showk:2014dwa. The curves and the labels in the legend have the same order from top to bottom.
• Figure 7: Bounds on $\Delta_S$ as a function of $\Delta_\phi$ for 3D CFTs with different $O(n)$ symmetries, with $\phi$ in the fundamental representation of $O(n)$. The regions above the lines are excluded. All the bounds have been determined using $N=80$ points and $\Delta_*=16$. The curves and the labels in the legend have the same order from top to bottom.